Pipeflow analysis

PIPEFLOWANALYSIS

DEVELOPMENTS I N WATER SCIENCE, 19

OTHER TITLES I N TH/SSERfES

1

G. BUGLIARELLO AND F. GUNTER

COMPUTER SYSTEMS A N D WATER RESOURCES

2

H.L. GOLTERMAN

PHYSIOLOGICAL LIMNOLOGY

3

Y.Y. HAIMES, W.A.HALL AND H.T. FREEDMAN

MULTIOBJECTIVE OPTIMIZATION I N WATER RESOURCES SYSTEMS: THE SURROGATE WORTH TRADE-OFF-METHOD

4

J.J. FRIED

GROUNDWATER POLLUTION

5

N. RAJARATNAM

TURBULENT JETS

6

D. STEPHENSON

PIPELINE DESIGN FOR WATER ENGINEERS

7

v. HALEK AND J. YSVEC

GROUNDWATER HYDRAULICS

8

J.BALEK

HYDROLOGY A N D WATER RESOURCES I N TROPICAL AFRICA

9

T.A. McMAHON AND R.G. MElN

RESERVOIR CAPACITY A N D Y I E L D

10 G. KOVACS SEEPAGE HYDRAULICS

1 1 W.H. GRAF AND C.H. MORTIMER (EDITORS) HYDRODYNAMICS OF 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

1 3 M.A. MARIGO AND J.N. LUTHIN SEEPAGE A N D GROUNDWATER

14 D. STEPHENSON STORMWATER HYDROLOGY A N D DRAINAGE

15 D. STEPHENSON PlPLELlNE DESIGN FOR WATER ENGINEERS (completely revised edition of Vol. 6 in the series)

16

w. BACK AND R.

LETOLLE (EDITORS)

SYMPOSIUM ON GEOCHEMISTRY OF GROUNDWATER

17 A.H. EL-SHAARAWI (EDITOR) IN COLLABORATION WITH S.R. ESTERBY TIME SERIES METHODS I N HYDROSCIENCES

18 J. BALEK HYDROLOGY AND WATER RESOURCES I N TROPICAL REGIONS

Civil Engineering Department, University of the Witwatersrand,

1 Jan Smuts Avenue, 2001 Johannesburg, South Africa

ELSEVIER Amsterdam - Oxford

- New York - Tokyo

1984

ELSEVIER SCIENCE PUBLISHERS B.V. Molenwerf 1 P.O. Box 21 1,1000 A E Amsterdam, The Netherlands

Distributors for the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, N Y 10017

ISBN 0444422838 (Vol. 19) ISBN 0444-41669-2 (Series) 0 Elsevier Science Publishers B.V., 1984 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., P.O. Box 330, 1000 A H Amsterdam, The Netherlands

Printed i n The Netherlands

V PREFACE

The

advent

excuse

of

for

economical

engineers

H y d r a u l i c e n g i n e e r s now flows

in

complex

simulations

can

water

engineer

trunk

mains,

who

has

engineer

often

tends

occasion

arises

he

h e a d loss, al

of

use

steady

hammer

to

has to

left

no

problems.

Tedious

with

graphical

relief.

T h i s book

and

analogue

i s aimed a t

the

to

design

water

reticulation

pipe

networks,

lines

and

storage

reservoirs.

The

practising

to

negelect

requires

available

simple

to w a t e r

computers solutions

a

the

rapid,

theoretical simple

side,

answer

but to

when

the

p r o b l e m s of

d i s c h a r g e c a p a c i t y a n d p r e s s u r e s . The v a r i o u s c o m p u t a t i o n -

methods

with

micro

computer

networks.

discarded

pumping

simple

avoid

h a v e t o o l s f o r model I i n g s t e a d y a n d u n s t e a d y

pipe

be

and

to

to

him

flow

of

summarized

in

problems a n d a d v a n c i n g

i n complex

students

are

networks.

hydraulic

this

book,

starting

t h r o u g h slow

motion

The s u b j e c t m a t t e r w i l l a l s o be

engineering

and

those

contemplating

research in t h i s f i e l d . The i t e r a t i v e t e c h n i q u e s f o r f l o w a n a l y s i s of p i p e n e t w o r k s such a s H a r d y Cross method a r e k n o w n

the been

applied

extensively

known techniques linear

method.

computer and

are

analysis

can

changes

are

When

without

in f a c t

it

to most the

simpler

comes

to

computers.

Some

to a p p l y o n computers, flow,

e.g.

for

many

more

factors

such

as

column

compares

pressure v a r i a t i o n s reservoir

are given

various

in pipelines, gravity

in many c h a p t e r s a n d

comprehensive

methods whether

systems.

the

hammer, method

separation,

N u m e r i c a l methods f o r computers

and accurate provided simple r u l e s a r e followed. and

lesser

e.g.

water

i s much more r a p i d t h a n t h e o l d e r g r a p h i c a l

account

condenses

mu1 t i p l e

of

unsteady

i n section a n d b r a n c h pipes.

easy

water engineers and have

aid

for they

analysing

T h i s book flows

and

b e p u m p i n g systems o r

Simple BASIC computer

programs

these w i l l s e r v e a s a b a s i s f o r more

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

r e a d i n g t h i s book. In design using

addition

to

of

systems

computer

graphics pipes.

pipe

is

That

those

on

simulation

given

but

i s covered

flow

using

the

programs. book

in a n o t h e r

Design f o r Water E n g i n e e r s '

analysis,

optimization

does

An not

sections programs,

introduction cover

are

given

and

operation

to

computer

structural

d e s i g n of

book b y the same a u t h o r ,

(Elsevier,

1981 1.

on

'Pipeline

This Page Intentionally Left Blank

VII

PAGE

CONTENTS

PREFACE CHAPTER 1

HYDRAULICS AND HEAD LOSS EQUATIONS

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

. . . .

. . . . .

. . . . .

. . . . .

Basic Equations Flow-Head L o s s Re l a t i o n s h i p s M i n o r Losses Use o f H e a d L o s s C h a r t s f o r S o l u t i o n of Simple P i p e Systems References

. . . . .

. . . . .

. . . . .

. . . . .

. . . . . . . . . .

CHAPTER 2 ALTERNATIVE METHODS OF P I P E NETWORK FLOW ANALYSIS

. . . T y p e s of P i p e f l o w P r o b l e m s . . M e t h o d s of S o l u t i o n . . . . . . Simple P i p e Problems . . . . . Inter-Connected Reservoirs . . . Pseudo S t e a d y F l o w . . . . . . Compound P i p e s . . . . . . . Node Head C o r r e c t i o n M e t h o d . . Computer P r o g r a m B a s e d o n Node Head C o r r e c t i o n M e t h o d . . . . Newton-Raphson M e t h o d . . . . References . . . . . . . . . .

. . . . . . . . . . .

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

. . . . . . . . . . .

CHAPTER 3 LOOP FLOW CORRECTION METHOD OF NETWORK ANALYSIS

. . . I nt roduct ion . . . . . . Method o f F l o w C o r r e c t i o n Loop S e l e c t i o n . . . . . Mu1 t i p l e R e s e r v o i r s . . . Branch Pipes . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

Pumps a n d P r e s s u r e R e d u c i n g V a l v e s P r a c t i c a l D e sign f o r Loop Flow Correct i o n Method Computer P r o g r a m References

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

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. i . 1 . . 16 . 18 .20

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. 21 . 21 . 21 .2 3 .2 4 .25 .2 6

. . . .

CHAPTER 5 OPTIMUM DESIGN OF BRANCHED P I P E NETWORKS BY L I N E A R PROGRAMMING

. . . . . . . I nt r o d u c t io n . . . . . . . . . . . . . . . . . Simplex Method f o r T r u n k M a i n Diameters . . . . Network D e s i gn . . . . . . . . . . . . . . . . Looped N e t w o r k s .L P O p t i m i z a t i o n . . . . . . . Linear Programming Program . . . . . . . . . . References . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

.2 8 .3 0 . 31 .3 3

. . . . . .35 . . . . . -35 . . . . . .35

. . . . . .3 7 . . . . . .3 8

. . . . . .3 8 . . . . . .3 9

. . . . . . . . Computer P r o g r a m f o r L i n e a r M e t h o d References . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 4 LINEAR METHOD . . . . Introduction . . . . . . . . . . . L i n e a r Method A p p l i e d to Loop Flows L i n e a r M e t h o d f o r Node H e a d s . . .

. . . .

. . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

.4 0 .41 .4 4 .4 5 .4 5 .4 7 .4 8 .49 .54

. . . . . .

. . . . . .

. . . . . .

. . . . . .

.55 .5 5 .5 5 -60

.63 .6 3 . . . . . .6 9

VIII

.

CONTENTS ( c o n t )

PAGE

CHAPTER 6 DYNAMIC AND NON-LINEAR FOR LOOPED NETWORKS

PROGRAMMING

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

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 Pipes 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 f o r Least-Cost A l l o c a t i o n o f Resources Steepest P a t h Ascent T e c h n i q u e f o r Extending Networks Design of Looped Networks References

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

CHAPTER 7

. . . . .

. . . . .

. . . . .

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

CONTINUOUS SIMULATION

. . . . . . . .

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

. . . . .

. . . . .

. . . . .

Systems A n a l y s i s T e c h n i q u e s and t h e Use of S i m u l a t i o n Models Mathematical M o d e l l i n g of Water Q u a l i t y Numerical Methods f o r the Solution of Single Differential Equations R e a l - T i m e O p e r a t i o n o f W a t e r S u p p l y Systems Computer P r o g r a m to S i m u l a t e R e s e r v o i r Level V a r i a t i o n s in a Pipe Network References

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

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

. 7 0 .71 . 7 4

.77 .81 . 8 6 . 8 7 . 8 8 . 9 0 .96

. . . . . . . .102 . . . . . . . . . . . .102

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

CHAPTER 8 UNSTEADY FLOW ANALYSIS BY R I G I D COLUMN METHOD

. . . . . . R i g i d Water Column Surge Theory . Derivation of Basic Equation . . . Solution o f Equation of Motion . . SurgeTanks . . . . . . . . . . References . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 9 WATER HAMMER THEORY . . . . . . . B a s i c W a t e r Hammer E q u a t i o n s . . . . . . . . . E f f e c t of A i r . . . . . . . . . . . . . . . . . Methods o f A n a l y s i s . . . . . . . . . . . . . . Valves . . . . . . . . . . . . . . . . . . . . A c c u r a c y and S t a b i l i t y o f F i n i t e D i f f e r e n c e Schemes . . . . . . . . . . . . . . B a s i c Computer P r o g r a m f o r A n a l y s i n g G r a v i t y L i n e s References . . . . . . . . . . . . . . . . . . CHAPTER 10 HAMMER

. . . .

. . . . . . . .

.105 .106

. . . .

. . . . . . . .

.106 .108 .108 -112 -114

. . . .

-115

. . . .

-115 -118 .120 -122

. . . .

. . . .

. . . .

. . . . . . . .

-124 131 -135

BOUNDARY CONDITIONS I N WATER

. . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . W a t e r 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 1 1 WATER COLUMN SEPARATION . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction

Non-Return Valves A i r Vessels References

. . . .

. . . . *

-136 -136 -138 .141 .143 .145 .146 -146

IX

.

CONTENTS ( c o nt )

PAGE

CHAPTER 1 1 ( c o n t . ) C o m p u t a t i o n a l Technique f o r Column Separation S i m p l i f i e d R i g i d Column A n a l y s i s P r o g r a m f o r S i m u l a t i o n o f W a t e r Hammer i n Pump1 i n e s f o l l o w i n g P u m p t r i p References

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

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

147 148 153 159

CHAPTER 12 WATER HAMMER AND FLOW ANALYSIS I N COMPLEX P I P E SYSTEMS

160

The P r o b l e m o f F l o w A n a l y s i s i n C o m p l e x Pipe Networks C o n v e n t i o n a l Methods o f Network A n a l y s i s Method o f S o l u t i o n o f t h e E q u a t i o n s Boundary Conditions Valve Stroking References

160 162 164 167 168 172

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

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

CHAPTER 13

GRAPHICAL WATER HAMMER ANALYSIS

.

*

. .

*

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 14 P I P E GRAPHICS . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . Interactive Drawing . . . . . . . . . . . . . . . . . . Computer P r o g r a m f o r P i p e G r a p h i c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References CHAPTER 15 COMPUTER PROGRAMMING I N BASIC . . . . . . ...................... Descri p t i o n Arithmetic . . . . . . . . . . . . . . . . . . . . . . . Logi ca I Operat o r s . . . . . . . . . . . . . . . . . . Reasons f o r G r a p h i c a l A p p r o a c h B a s i c s of M e t h o d M i d - P o i n t s and C h a n g e i n D i a m e t e r Pumping Lines References

*

. .

* *

175 175 175 178 179 182

. 183 . .

183 185 186 190

.

192

.

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

Variables P r e c is i o n Functions Spec i a I C h a r a c t e r s B a s i c P r o g r a m m i n g Statements Graphics Statements INDEX

*

192 193 193 194 194 194 195 195 198 202

This Page Intentionally Left Blank

CHAPTER 1

HYDRAULICS AND HEAD LOSS EQUATIONS

BAS I C EQUAT I ONS

Most with

hydraulic

one

or

adaptation be

they

problems

more

of

of

them.

the It

analytical,

in

pipe

basic

is

systems can be solved s t a r t i n g

equations

the methods of

graphical

or

described

below,

solution of

numerical,

or

an

the equations,

which

this

book

in

various

is

about. The

three

throughout

equations

the

equation

book

and

the

one-dimensional equating section there

the

along

is

no

is

flow

the

energy

flow

flow

which

are

the

the

any

stream

concerned

tube.

By

to

flow

flow'

any p o i n t .

equation

has

forms

momentum

incompressible

simply

is

the

'steady

the

steady,

equation

at

continuity

equation,

For

section

i n velocity

the

appear

equation.

continuity

r a t e at

variation

will

continuity

obtained

r a t e at

it

by

another

i s implied that

A s f a r as unsteady an

additional

term,

namely the change i n storage between the sections. The and

momentum equatior!

states

equals

the

steady,

one-

that sum

stems

the change

of

the

from

Newton's

i n momentum f l u x

forces

on

basic

law of

between two sections

the f l u i d causing

the change.

F

is

volumetric

For

dimensional flow t h i s i s

AFx = P Q A V x where

motion

(1.1 the

flow

force,

rate,

V

is

P

the

fluid

mass

density,

Q

is

1 the

i s velocity and subscript x r e f e r s to the ' x '

direct ion. The b a s i c energy equation on

an

element

change

in

excluded energy

from

the

+

by

gravitational Mechanical

equation.

to account f o r

incompressible

conveniently v,2

fluid

In

and pressure forces to the

and heat energy

most

systems

there

transfer are is

a

loss

of

due to f r i c t i o n a n d turbulence and a term i s included i n the

equation of

of

k i n e t i c energy.

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

p1

29 Y

this.

fluids

is

The r e s u l t i n g equation f o r termed

the

Bernoulli

steady flow

equation

and

w r i t t e n as:

+ z,

=

v22 29

+ P1

+

Y

z2

+

he

(1.2)

is

where V = mean velocity at a section V2/2g = velocity head ( u n i t s of length) 9 = g r a v i t a t i o n a l acceleration = pressure P p/Y = pressure head ( u n i t s of l e n g t h ) Y = u n i t weight of f l u i d Z = elevation above an a r b i t r a r y datum = head loss due to f r i c t i o n or turbulence between he sections 1 a n d 2. The s u m of the velocity termed by

a

the total head. coefficient

section

of

the

or

account

conduit.

for

The

head should be m u l t i p l i e d

in v e l o c i t y across the

the v a r i a t i o n

average

value

of

the

and f o r l a m i n a r flow i t i s 2.0.

i s 1.06

t u r b u l e n t flow conduit

to

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

S t r i c t l y the velocity

coefficient

for

Flow through a

i s termed e i t h e r uniform o r non-uniform depending on whether

not

there

is

a

variation

in

the

cross-sectional

velocity

d i s t r i b u t i o n a long the conduit. For

the

i.e.

there

The

flow

fluid

Bernoulli

equation

to

apply

the

should be no change i n velocity is

assumed

should be

to

be

flow at

one-dimensional

incompressible,

should be steady,

any and

point with

time.

irrotational.

The

or else a term f o r s t r a i n energy has

to be introduced. The

respective

practical

cases

heads,

and

minor

head

it

the may

losses

heads

are

velocity be due

illustrated

head

is

small

neglected.

In

fact

to

bends,

in

Figure

1.1.

compared w i t h it

i s often

expansions,

For

most

the other

the case that

etc.

can

also

be

neglected a n d f r i c t i o n need be the o n l y method whereby head i s lost.

ENTRANCE L O S S

FRICTON LOSS CONTRACTION L O S S

F R I C T I O N LOSS

VELOCITY

"hP H AD

PRESSURE H E A D

PI 6

E L E VAT!ON

Fig.

1.1

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

3

LOSS RELAT I ONSH I PS

FLOW-HEAD

Empirical Flow Formulae

The t h r o u g h p u t

o r c a p a c i t y of a p i p e of f i x e d dimensions depends

o n t h e t o t a l h e a d d i f f e r e n c e between t h e e n d s .

T h i s h e a d i s consumed

b y f r i c t i o n a n d o t h e r losses. The

first

friction

field observations. waterworks developed.

head

practice The

conventional

loss/flow

These e m p i r i c a l although

head

loss/flow

relationships

more

rational

formulae

thus

formulae and are usually given

were

derived from

a r e s t i l l popular in

relationships

formulae

have

been

e s t a b l i s h e d a r e termed in a n exponential

form

of the type

or

v

=

KD~SY

(1.3)

s

=

K'Q"/D~

(1.4)

V

where

is

mean

flow

velocity,

m,

n,

x,

y,

K

and

K'

are

D i s the inside diameter of the c i r c u l a r p i p e a n d S i s the

constants, head

the

loss g r a d i e n t

( i n m head

loss p e r

m l e n g t h of

pipe).

Some o f

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: Basic Equations HazenWilliams

S.I.

S=Kl(V/Cw) 1.85 /D 1.167 1.33 S = K 2 ( n V l 2/D

units

K =3.03 1 K =2.86

=6.84

Chezy

S=K3(V/cz)* /D

1 K =6.32 2 K =13.13

Darcy

S=XV2/2gD

Dimensionless

Manning

-

Except universal

for and

the the

of

in m i n d t h a t

f o r standard water

variations

coefficients

in gravity,

vary

with

the

(1.6)

(1.7)

K =4.00 3

above

(1.8)

equations

are

not

in the equation depends on the u n i t s .

'constant'

I t should be borne purely

formula

(1.5)

2

3

Darcy

ft-sec

some o f

t h e f o r m u l a e were i n t e n d e d

e n g i n e e r i n g p r a c t i c e a n d take no account temperature

pipe

diameter,

or

t y p e of

liquid.

The f r i c t i o n

surface configuration

and age

of pipe. The c o n v e n t i o n a l f o r m u l a e a r e c o m p a r a t i v e l y s i m p l e t o u s e a s t h e y do not

involve f l u i d viscosity.

They

may

be solved directly

a s they

4 do

not

the

require

friction

an

initial

factor

estimate

(see next

of

Reynolds number

s e c t i o n ) . On the other

to determine

hand,

the more

modern equations cannot be solved d i r e c t l y f o r r a t e of flow. of

the

formulae

simple w i t h

for

velocity,

for

friction

or

head g r a d i e n t

the 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,

o r graphs plotted on use

diameter

analysing

log-log

flows

in

paper. pipe

Solution is

nomograph

The equations a r e of p a r t i c u l a r

networks

where

the flow/head

loss

equations have to be i t e r a t i v e l y solved many times.

A

popular

liams

tabulated

in

solution

against

water,

the

of

i n waterworks

coefficients

1.1.

If

the

a i d of

a

chart

organizations

flow

value

formula

Friction

Table

with

waterworks

the

flow

formula.

for

various

C

decreases

use

for

formula is

is

the

graphs

of

age,

in

this

be

to

equation

used

most e f f i c i e n t head

p i p e diameters, with

p r a c t i c e i s the Hazen-Wiluse

loss

are

frequently, way.

gradient

Many plotted

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

type of

As

p i p e and properties of

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 1.1

Type o f

f r i c t i o n coefficients C

Hazen-Williams

Pipe New

PVC : Smooth c o n c r e t e , A C : S t e e l , b i tumen I ined, galvanized: Cast i r o n : Riveted steel, v i t r i f i e d , woodstave

25 y e a r s old

Condi t i on 50 y e a r s old

Bad I y C orr o de d

150

140 130

130 120

130 100

150 130

130 110

100

60

90

50

120

100

80

45

150

Rational flow formulae

Although use f o r

acceptance scientific universally used

the

many

and

conventional

years,

amongst basis

flow

formulae

more r a t i o n a l engineers.

backed

applicable. l i q u i d s of

the proposed formulae.

by

Any

are

formulae

The

new

numerous consistent

l i k e l y to remain

are

gradually

formulae

measurements u n i t s of

v a r i o u s viscosities

have and

in

gaining a

sound

they

are

measurements may be

a n d temperatures conform

to

3

The r a t i o n a l for

flow

flow

past

formulae f o r flow

bodies

or

over

flat

o r i g i n a l research was on smal I-bore Lack of

i n pipes a r e sim I a r to those

plates

1960).

(Schlichting,

The

roughness.

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

roughness f o r l a r g e pipes has been one deterrent t o

d a t a on

the use of the r e l a t i o n s h i p s i n waterworks practice. The velocity maximum and

a

in

i n a f u l l p i p e v a r i e s from zero on the boundary to a

the

centre.

boundary

imparting annulus.

a

kinematic

layer

shear

The

resistance

mixing

onto to

and

with

forces

on

established

is

force

viscosity,

turbulent

Shear

an

with

of

oppose

of

the

flow

particles

of

is

flow fluid

concentric

fluid

it of

the

annulus

neighbouring

motion

turbulent

transfer

walls each

inner

relative in

the

termed

is

imparted

different

by

momentum

between one l a y e r a n d the next.

A boundary

layer

i s established at the entrance to a conduit and

t h i s l a y e r g r a d u a l l y expands u n t i l i t reaches the centre. point

the flow becomes uniform.

established flow

5=0.7 D

The length of p i p e r e q u i r e d f o r f u l l y

i s g i v e n by

(1.9)

f o r t u r b u l e n t flow

The

Reynolds

number

incorporating

the

conventional

flow

Re

fluid

VDh

=

viscosity

formulae.

is

Flow

in practice). boundary

The basic

shear

force

head over

a

length

is

absent

number in

i n a pipe i s laminar for higher Re

loss equation

a

dimensionless

which

u

( < 2 0 0 0 ) and becomes t u r b u l e n t f o r

Re

Beyond t h i s

of

the low

( n o r m a l l y the case

i s d e r i v e d b y setting the

p i p e equal

to

the

loss

in

pressure mu1 tip1 ied b y the area:

(1.10)

rnDL = yhf rrD2/4

(1.11) (1.12) where

X

stress, a

( 4 . r / v ) / ( V 2 / 2 g ) i s the Darcy f r i c t i o n factor,

=

D i s the p i p e diameter and h

l e n g t h L.

For

laminar

X

i s a function of flow,

Poiseuille

T

i s the shear

i s the f r i c t i o n head loss over

f Re and the r e l a t i v e roughness k / D .

found

that

=

64/Re

i.e.

is

independent of the r e l a t i v e roughness. Laminar flow w i l l not occur i n normal engineering practice. turbulent

flow

i n practice.

i s complex

The t r a n s i t i o n zone between l a m i n a r and

and undefined b u t i s also of l i t t l e interest

6 Turbulent rough

flow

conditions

boundary.

conditions

are

distribution

The

derived

in

a

may

occur

equations

for

from

general

the

turbulent

layer,

with

the

either

friction

is

or

smooth

factor

equation

which

a

for

for

the

derived

a

both

velocity

from

mixing

Ieng t h theory :

(1.13) w i t h t h e c o n s t a n t k f o u n d to b e 0.4 a n d c o n v e r t i n g to

Integrating

=

Y log Y '

5.75

G77 where

v

is

the

(1.14)

velocity

a

at

distance y

hydrodynamical l y

smooth b o u n d a r y

Nikuradse

found

that

thickness,

so

y'av

For a

from the b o u n d a r y .

there i s a l a m i n a r sub-layer,

/m

where

y'

is

the

boundary

and layer

(1.15) The c o n s t a n t 5.5 Where and

the

was f o u n d e x p e r i m e n t a l l y .

boundary

Nikuradse

is

found

rough

that

y'

=

the

laminar

k/30

sub-layer

where

k

is

is

the

affected boundary

roughness. Thus

V

-

rn

5.75

=

Rearranging

log

+ 8.5

(1.16)

1.15 a n d 1.16 a n d e x p r e s s i n g v

equations

the average v e l o c i t y

_ _-

$ V

=I

b y means o f t h e e q u a t i o n Q

i n terms of

vdA r e s u l t s i n

2 l o g Re / T - 0.8

(1.17)

n(turbulent boundary

- _-

D 2 log k

Jx'

layer,

+ 1.14

(turbulent boundary Notice relative

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

that

( 1 .la) layer,

for

a

smooth

k/D

roughness

rough boundary).

and

boundary, for

a

very

is

independent

rough

boundary

of it

the is

i n d e p e n d e n t of 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 an

a n d White combined

equation covering

both

smooth

Equations

1.17 a n d 1.18 to p r o d u c e

a n d rough boundaries as well

as

the t r a n s i t i o n zone:

f 1i

=

k

1.14 - 2 l o g ( -

D

9.35

+ ___ 1 Rev"

(1.19)

7

Their

equation

reduces

to

1.17 f o r smooth pipes,

Equ.

and

Equ.

to

1 .I8 f o r r o u g h pipes. T h i s semi-empirical equation y i e l d s satisfactory results

for

original Natural

various

commercially

experiments roughness

used

is

sand

available

as

evaluated

pipes.

artificial

according

Nikuradse's

boundary the

to

roughness.

equivalent

sand

roughness. Table 1.2 g i v e s values of k f o r v a r i o u s surfaces. TABLE 1.2

Roughness of p i p e m a t e r i a l s ( H y d r a u l i c s Research Station,

1969). Value of k i n mm Finish

Smooth

G l a s s , drawn m e t a l s S t e e l , PVC o r AC Coated s t e e l Galvanized, v i t r i f i e d clay C a s t i r o n o r cemen t I i n e d Spun c o n c r e t e o r wood s t a v e R i v e t e d s t eel Fou I s e w e r s , t u b e r c u I a t ed water mains Unl ined r o c k , e a r t h

Fortunately increases

i s not v e r y

A

linearly

formulae

for

with

were

A

age

Rough

Average

____

0 0.015 0.03 0.06 0.15 0.3 1.5

0.003 0.03 0.06 0.15 0.3 0.6 3

0.006 0.06 0.15 0.3 0.6 1.5

6 60

15 150

30 300

6

sensitive to the v a l u e of k assumed. for

water

p l o t t e d on

a

pipes.

single

The

various

k

rational

g r a p h b y Moody and t h i s

g r a p h i s presented as F i g u r e 1.2.

A close approximation to A A = 0.0055

This

{

i s often g i v e n b y the f o l l o w i n g equation:

1 + ( 2 0 0 0 0 k / D + 106/Re)1/31

equation

approximation

avoids

an

implicit

(1.20)

situation

which should be substituted

but

i s only

i n the r.h.s.

of

a

first

(1.19)

to

o b t a i n a better value. Unfortunately solution

for

variables, the

any

and

velocity

required.

produce

a

for

The

re-arranged

the

diagram

variable trial the

for

Reynolds

explicit

in

not

values

number

the

flow/head

very

amenable of

the

to d i r e c t dependent

a n a l y s i s may be necessary

Research

variables

is

given

and e r r o r

Hydraulics

the

simple

Moody

if

Station

reasonable at

Colebrook-

loss

graphs.

accuracy

Wal l i n g f o r d White Thus

to get is

(1969)

equation equation

to

1 .I9

may b e a r r a n g e d i n the form

v

=

-2 J2gDS' log (-

k

3.70

+

2.51~ D

1 m

(1.21)

0.10 0 09 0.08

0.07

0.06 1.05

f-

0

; .0. .-

0.02

C

U

z

0.015

0.01 0.00:

0.001

Fig.

1.2

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

i n pipes

9

Thus

for

k,

roughness Figures

any a

1.3

to

fluid

graph 1.7

at

a

may

are

certain

be

such

temperature

plotted graphs

and

V,

in

terms

of

for

water

at

defined

D

5.

and

15"C

and

for

v a r i o u s roughnesses K . I n s u m m a r y t h e D a r c y - W e i s b a c h e q u a t i o n f o r h e a d loss i n c i r c u l a r p i pes

(1.22) w h e r e h i s t h e h e a d loss ( o r t h e e n e r g y the

Darcy-Weisbach

pipe

diameter,

g

friction is

the

average fluid velocity. replaced

by

f,

loss p e r u n i t w e i g h t ) ,

is

gravitational

I t may

whereas

L

factor,

British

the

pipe

length,

acceleration

V

and

is

the

in North America X

be noted t h a t practice

is

X

D i s the

i s to use another

f'in

is the

equation

hf = f 'LV2/2gR where R

(1.23)

i s the

hydraulic

r a d i u s D/4

for

a circular

pipe.

Thus

f'=

64. Since V = 4Q/rD2

(1.24)

we c a n a l s o r e w r i t e e q u a t i o n 1.22 a s --h

L

f - 8XQ2 V'gD

(1.25)

Solution of the basic equations

The

Colebrook-White

formula

(1.19)

Moody d i a g r a m m a y a l s o b e w r i t t e n a s

-0.8686 _ _ _ Pn

1 = -

(-

k

I

+ ___ 2.51

3.7D

vf-r

A l s o Re = VD/v

= 4Q/nDv

The

must

equation

conveniently

done b y

be

R

e

which

forms

the b a s i s of

,

the

(1.26)

r (1.27)

solved

b y ,iteration

letting x = X

-

-' a n d

for

h.

This

can

be

u s i n g t h e Newton Raphson

i t e r a t i v e scheme, (1.28) w h e r e in g(x) = x

t h i s c a s e we h a v e f r o m e q u a t i o n 1.26

+

k 3.7D

0.8686 en (-

+ 2.51~ ) Re

and g ' ( x ) i s t h e d e r i v a t i v e o f g ( x ) . We t h e n o b t a i n

(1.29)

10

k + 2.51~ 1 3.7D Re 2.51 Re k + /2.5lX

+ 0.8686 en(-----

x x+ = x - I

o'8686

+

where

x+

is

a

(m Re1

successive a p p r o x i m a t i o n to

u s i n g the p r i o r s e v e r a l times,

approximation

and

can

be e a s i l y

calculator. a n d (1.27)

x.

(1.30)

Setting x

the s o l u t i o n

of

g(x)

r a p i d l y for

solved

on

a

almost

computer

any or

starting value of

on

a

programmable

in c o m b i n a t i o n w i t h e q u a t i o n s

Using t h i s equation

(1.22)

(1.31)

hf2gD

that

+

one c a n s o l v e f o r h e a d loss.

Note t h a t s i n c e A =

so

= 0

x+ a n d s o l v i n g f o r x

=

t h e s o l u t i o n c a n be o b t a i n e d to a n y d e s i r e d a c c u r a c y .

T h i s e q u a t i o n converges x

1

1

(1.26)

Lvz

can

be

solved

directly

V

for

given

hf,

L,

9,

D,v

and k:

V

= J2gDhf/L'

en (-

-0.8686

+

k

3.7D If

V

however,

is

given

then

2.51~

either

or

A

(1.33)

)

1 2gD3h f / L ' hf

must

be

obtained

e m p l o y i n g a n i t e r a t i v e procedure.

Comparison of F r i c t i o n Forrnul ae

The D a r c y e q u a t i o n may be w r i t t e n as

v or

(1.35)

v

=

cZ

fi

(1.36)

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

=m

cZ

The

Hazen-Wi I I iams

(1.37) equation

may

be

rewritten

for

al I

practical

p u r p o s e s i n t h e f o l l o w i n g dimensionless f o r m : S

= 515(V/CW)'

(Cw/Re)'*15/gD

By c o m p a r i n g t h i s w i t h t h e Darcy-Weisbach

(1.38) equation

(1.25)

i t may

be deduced t h a t cW

= 42.4

(A0'54Re0*08)

(1.39)

FIG.

1.3

FRICTION LOSS CHART FOR PIPES FLOWING FULL

1.4

FRICTION LOSS CHART FOR PIPES FLOWING FULL

12

FIG.

WOE

OOOP

ow 1

008

009

8

01

9

P

c

FIG. 1M)

1.5

FRICTION LOSS CHART FOR PIPES FLOWING FULL ffor water at 15OC)

1

k = 0 . 0 6 mm

I

80 60

40

Typical example< Galvanised iron Coated cast iron Glazed vitrified clay

30 20

-e

10

g 8 x L

0

a

4

Y)

b ,

-2 I

-

7

P I 1

0

01

0:

0.: WhCrU

0.:

0

w

14

FRICTION LOSS CHART FOR PIPES FLOWING FULL

n

FIG. 1.6

FIG. 1 . 7

FRICTION LOSS CHART FOR PIPES FLOWING FULL

15

16

The H a z e n - W i l l i a m s c o e f f i c i e n t C Re a n d v a l u e s may

It

will

be

in

the

from

Figure

1.2

that

lines

for

c o e f f i c i e n t c o i n c i d e w i t h the Colebrook-White

transition

non-smooth Reynolds

pipes

zone.

In

the

the coefficient

wi II

i .e.

number,

and

be p l o t t e d on a Moody d i a g r a m (see F i g u r e 1 . 2 ) .

observed

Hazen-Williams

i s t h e r e f o r e a f u n c t i o n of X

W

one c a n n o t

completely actually

constant

I ines o n l y

turbulent

zone

for

r e d u c e the g r e a t e r t h e Hazen-Wi I I i a m s

associate a c e r t a i n

coefficient w i t h a p a r t i c u l a r p i p e as i t v a r i e s w i t h the flow r a t e . The

Manning

equation

p a r t f u l l pipes.

V

=

i s widely

used f o r open c h a n n e l

flow

The e q u a t i o n i s

KnR 2 k sf

where K

and

(1.40)

i s 1.000

in SI

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

units and

1.486

in f t

I b u n i t s , a n d R i s the

where A i s t h e c r o s s s e c t i o n a l a r e a of f l o w a n d

P t h e wetted p e r i m e t e r . R i s D/4 f o r a c i r c u l a r p i p e , a n d i n g e n e r a l for nov-circular

TABLE 1 . 3

sections,

Values of

4 R may b e s u b 2 t i t u t e d f o r D .

Manning's

In' __._____

Smooth g l a s s , p l a s t i c C o n c r e t e , s t e e l ( b i tumen I i n e d ) , g a l v a n i z e d Cast I r o n S I imy o r g r e a s y s e w e r s R i v e t t e d s t e e l , v i t r i f i e d , wood-stave Rough c o n c r e t e

0.010 0.01 1 0.012 0.01 3 0.01 5 0.017

MINOR LOSSES

One method of e x p r e s s i n g h e a d section

is

the

conventional express

equivalent

friction

losses

opening.

formulae

through f i t t i n g s

h = KV2//2g where c' loss coefficients supplementary

loss

K

loss t h r o u g h f i t t i n g s

length

i s the although

method, are

used.

often

a n d changes used

Modern

practice

i n terms of t h e v e l o c i t y

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

T a b l e 1.4

m a n u f a c t u r e r s may

when

head,

in

the i s to

i.e.

gives typical also

provide

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

The v e l o c i t y V to use i s n o r m a l l y t h e mean t h r o u g h t h e f u l l

bore of the p i p e or f i t t i n g .

17

TABLE 1.4

Bends

Loss c o e f f i c i e n t s f o r p i p e f i t t i n g s .

hB

KBV2/2g

=

Bend a n g l e

Sharp

r/D=l

0.16 0.32 0.68 1 .27 2.2

0.07 0.13 0.18 0.22 0.40

30° 45O 60° goo 1 80°

Valves

hv

=

KVVf/2g Open I n g :

Sluice Butterfly G I obe Need I e Ref l u x b

Contract ions and expansions

7.5O 15' 30° 180'

0

1/2

3/4

Ful I

24 120 160 4

5.6 7.5 40 1

1 .o 1.2 20 0.6

0.2 0.3 10 0.5 1-2.5

Expansions:

hc = KcV22/2g

hc = KcVl2/2g

.5

0.4

.37

En t r a n c e a n d ex i t

.25

A 1 /*2

0.6

.15

I osses:

0.8

1.0

.07

0

0 .13 .32 .78 1.0

0.2 .08 .24 .45 .64

he= K e V 2 / 2 g

En t r a n c e Protruding Sharp Bevel I ed Rounded

0.06 0.08 0.08 0.08 0.15

in cross section

Contract ions:

0.2

6

0.07 0.10 0.12 0.13 0.25

1/4

A2/A 1 Wal I angle

2

0.8 0.5 0.25 0.05

Exit 1 .o 1 .o 0.5 0.2

0.4 .05 .15 .27 .36

0.6 .02 .08 .13 .17

0.8

1.0

0 .02 0 .03 0 .04 0 .O

18

It

i s frequently

fittings

if

in

unnecessary

fact

they

velocity

head

V2/2g

velocity

head

is

losses. loss

The

for

due very

calculate

to

be

typically

small

losses

except

is

to

need

less

in

comparison

to

fittings

high

the

losses i n a l l

considered than with

at

line

pressure

full

the

friction

full head

5% of the f r i c t i o n

r a r e l y exceeds

design

pipe

The

so even

0.2m many

all.

heads

when

it

is

economical to reduce t h e diameter of f i t t i n g s such as valves. On

the

other

hand,

the f r i c t i o n loss. pocket

in

frequently of

free

the presence

comparison air

air

with

the

cross

i n free b u b b l e form

is

of a i r

i n p i p e l i n e s can a d d to

Although a i r w i l l seldom b u i l d up to create a l a r g e

not

impossible

sectional

i n transport.

and

since

the

there

is

The presence of

area

1%

head

loss

is

nearly

to V 2 the corresponding head loss increases 2%.

proportional

F i t t i n g s can be accounted f o r b y a d d i n g the losses to the f r i c t i o n loss: Thus

h

+

hf

=

h

(1.41)

r

(1.42) (1.43) One method i s to add on an equiva ent same head loss as the f i t t i n g .

length of p i p e to g i v e the

The equ valent

length i s from (1.43) (1.44)

L’ = (D/~)zK Obviously t h i s way

of 1 w i t h Re i s not s t r i c t l y accounted f o r

the v a r i a t i o n

and n e i t h e r i s the v a r i a t i o n of K

or

discharge coefficient

of valves w i t h opening.

USE O F HEAD LOSS CHARTS FOR SOLUTION OF SIMPLE PIPE SYSTEMS

Many

of

head losses of

unknowns.

diameter as

the

the

branches,

following

chapters

describe

i n complex p i p e networks. For

along total the

Some a d d i t i o n a l

instance

i t s length,

flow

rate

flows

as

facts

c o n t i n u i t y equation

if

may well

a

pipe

the head be as

with

a

known

head changes

head

Similarly losses

a r e needed to solve f o r

is invariably

calculating

loss f o r each section,

unknown. the

methods of

Those problems have a number

if

as well the

pipe

become unknowns.

a l l the unknowns. The

u t i l i z e d i n such cases.

I n general,

19

however,

it

somewhat

where simultaneous solutions of

involved.

i s necessary to s i m p l i f y

Thus

friction

factor

if

the

X

could

t h e flow-head

loss relationships

a number of equations are

Darcy

head

loss

be

assumed

a

equation

is

constant.

utilized

the

Alternatively

an

equation of the f o l l o w i n g form i s used h

(1.45)

KLQn/Dm

=

where K and

i s assumed constant f o r a l l pipes.

iterative

essential

methods e x p l a i n e d

and

friction

simplifications

f r i c t i o n factor

I n many of t h e numerical

these

varying

simp1 i f i c a t i o n s

with

pipe

and

flow

are

not

can

be

I f e q u i v a l e n t p i p e o r a n a l y t i c a l methods are employed

accounted f o r . the

factors

later

are

necessary,

or

at

least

the

assumption

that

i s independent of flow r a t e .

One method of accounting f o r simple pipes i n series o r in p a r a l l e l i s the e q u i v a l e n t p i p e method.

The diameter of an equivalent

or

more

simply,

if

computations

are

to

be

pipe to

(see Chapter 2 )

replace a compound p i p e may be d e r i v e d a n a l y t i c a l l y

performed

by

hand,

obtained from a h e a d loss c h a r t such as F i g u r e 1 . 5 Consider

as

an

example

a

compound

1000 m long p i p e i n series w i t h

bore

pipe 2000 m

long.

rate

possible

is

not

uniform

bore

diameter

of

m/km

and

the

300

mrn

( l e a d i n g i n t o ) a 400 mm bore

unless

a

the

3000

say pipe

pipe

is

replaced

A

long.

m

by

method

an

of

i s to assume a reasonable flow

so that

equivalent

the mean head loss g r a d i e n t pipe

diameter

to discharge

equivalent

obtaining rate

100 e / s ) and read o f f the total head loss from F i g u r e 1 . 5 : 1 . 3 X 2 km = 8 rn,

a

A simple r e l a t i o n s h i p between head loss and flow

pipe,

such

p i p e comprising

the (e.g.

h = 5.4 +

i s 8 / 3 = 2.67

100 e / s

i s 345

mm. As

another

example consider

l a i d p a r a l l e l to a connected

at

say

10 m.

and

the

the

both ends.

The

flow

mm

bore

I n t h i s case a total

head g r a d i e n t

along

r a t e from F i g u r e 1 . 5

700 mm bore p i p e

equivalent

500

a

pipe,

3000

m

long

700 mm bore p i p e 4000 m long a n d both pipes a r e

pipe with

a

i s 2.5

a t a g r a d i e n t of 10 m/4 = 2.5

i s 300.

m/km

length of,

and

say,

head

loss

i s assumed,

500 mm p i p e i s 3.33

the

m/km

The head g r a d i e n t along

the

flow

rate

620 P / s .

An

4000 m to discharge 920 e / s

m/km would r e q u i r e a bore of 820 mm.

I t should be noted that F i g u r e 1 . 5

i s f o r a p a r t i c u l a r roughness and

20

slightly charts

different were

assumptions

results would be obtained

employed were

made

or to

even start

if with.

v a r y i n g Reynolds number was ignored. have White seldom

been

obtained

equation justified

and in

by

using

the

iterating. real

systems

That

as

a l t e r n a t i v e head flow

or

head

loss

loss

i s b e c a u s e t h e e f f e c t of

A more a c c u r a t e s o l u t i o n c o u l d

Moody

The

if

different

diagram

additional the

losses

or

effort

the Colebrookis,

however,

due to many factors

( i n c l u d i n g roughness) a r e o n l y estimates anyway.

REFERENCES

and Simon, D.B., 1960. F l u i d MechA l b e r t s o n , M.L., B a r t o n , J.R. anics for Engineers, Prentice H a l l , N.J. Diskin, M.H., Nov. 1960. The l i m i t s of applicability of the H a z e n - W i l l i a m s f o r m u l a e , La H o u i l l e B l a n c h e , 6. H y d r a u l i c s R e s e a r c h 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 D e s i g n of C h a n n e l s a n d P i p e s , 3 r d Edn., H.M.S.O., London. 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 H i l l , N.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 C o l e b r o o k - W h i t e method. C i v i l E n g . i n S.A., 2 1 ( 7 ) , pp. 169-171.

21 CHAPTER 2

ALTERNATIVE METHODS O F PIPE NETWORK FLOW ANALYSIS

TYPES O F P IPEFLOW PROBLEMS

The

hydraulic

planning, problems

engineer

design can

is

confronted w i t h

a n d o p e r a t i o n of

be

divided

into

many problems

piped water

analysis

and

supply

design

i n the

systems.

types,

The

both

for

steady flow a n d unsteady flow. The a n a l y s i s of s t e a d y f l o w s in s i m p l e p i p e s may b e f o r f l o w r a t e (given same

the

head

loss)

calculations

although

solution

it

to

comes

or

apply of

for

head

compound

to

( g i v e n the flow

loss

pipes

(in parallel

more t h a n one e q u a t i o n

branched or

i s then

rate). or

series)

involved.

looped n e t w o r k s more s o p h i s t i c a t e d

The

When

methods

become n e c e s s a r y . The d e s i g n

pcoblem i s u s u a l l y t r e a t e d a s a s t e a d y s t a t e problem.

That

i s f o r k n o w n heads a n d d r a w o f f s ,

pipe

l a y o u t a n d diameter

aspect,

namely

reservoir

a n d r e s e r v o i r l o c a t i o n a n d size. sizing

w h i c h may o f t e n be s o l v e d over

t h e e n g i n e e r h a s to select t h e

is

r e a l l y an

using steady-state

The l a t t e r

unsteady flow equations.

problem

Net o u t f l o w

t h e p e a k d r a w o f f p e r i o d may b e assessed b y m u l t i p l y i n g d r a w o f f

r a t e b y time. For more r a p i d v a r i a t i o n s even

elastic

water

and transient suitable

The

are

available

problem

of

in flow

theory

' r i g i d column'

is

necessary

to

s u r g e theory o r determine

heads

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

steady-state

design

determination valves,

flows.

programs

to d e t e r m i n e

hammer

pipe

they could,

however,

Once

e v e n b e used

f l o w s o r heads. associated

wal I

with

thicknesses,

unsteady

and

flows

is

the

the o p e r a t i n g r u l e s f o r

pumps e t c .

METHODS OF SOLUTION

Where complex

pipe networks a r e u t i l i z e d f o r water d i s t r i b u t i o n i t

i s n o t easy

to c a l c u l a t e

point.

if

each

Even given

the

pipe

the flow

flow-head length,

i n each p i p e o r

loss

diameter

equation and

t h e h e a d a t each

assumed

roughness,

is explicit the

for

non-linear

22 relationship In

between

unlooped

head loss a n d flow

tree-like

drawoffs b u t

if

networks

the

the p i p e network

makes c a l c u l a t i o n

flows

will

be

difficult.

defined

by

the

incorporates closed loops flows a r e

unknown as well as heads a t the v a r i o u s nodes. The

complexity

available for to

be

the

computation,

utilized.

manually

of

Many

whereas

of

pipe

network,

will

dictate

the

following

computers

are

as

well

as

the

which method of methods

required

methods, p a r t i c u l a r l y where unsteady flow

for

analysis

can

be

the

more

Equivalent pipes f o r compound pipes i n series. Equivalent pipes f o r complex pipes i n p a r a l l e l .

3.

T r i a l a n d e r r o r methods f o r mu1 t i p l e r e s e r v o i r problems.

4.

Analytical

flow-head

complex

i s involved:

1.

of

is

performed

2.

solution

facilities

loss

equations

for

compound

pipes.

5.

Analytical

solution of flow-head

loss equations f o r pseudo-steady

flow.

6.

Iterative

node

head

correction

for

predominantly

branched

networks ( b y h a n d o r computer).

7.

Iterative

a.

Simultaneous

loop

flow

correction

for

looped

networks

(by

h a n d or

computer).

matrix or

9.

solution of

the head loss equations f o r a l l pipes b y

i t e r a t i v e methods

(easiest f o r

loss i s l i n e a r l y proportional

to f l o w ) .

Linearization

of

l a m i n a r flow

when

head

head

loss

equation

and

iterative

solution

for

head

loss

equation

and

iterative

solution

for

heads a t nodes.

10. L i n e a r i z a t i o n

of

flows i n piEes.

1 1 . A n a l y t i c a l solkition of r i g i d column unsteady flow equation. 12. Numerical acceleration

solution

of

equation,

finite head

difference loss

equation

form and

of

rigid

continuity

column equa-

tion.

13. Graphical a n a l y s i s f o r unsteady r i g i d column flow. 14. Graphical a n a l y s i s for- unsteady e l a s t i c water hammer.

15.

F i n i t e difference and c h a r a c t e r i s t i c

solution of

hammer

Valves,

equations u s i n g computers.

release systems and branches may be considered. Genera I I y the equations used a r e :

d i f f e r e n t i a l water

pumps,

vaporization,

23 Continuity

of

flow

at

junctions

(net

inflow

less d r a w o f f must

be z e r o ) . between nodes e q u a l f r i c t i o n h e a d loss in t h e

Head d i f f e r e n c e s pipes

linking

generally

them.

neglected

Minor

losses

included

or

and

velocity

the

in

friction

head

term,

are

or

an

e q u i v a l e n t l e n g t h o f p i p e i s a d d e d to t h e p i p e l i n e to a l l o w f o r m i n o r losses. Dynamic

equations

of

-

motion

only

where

acceleration

or

d e c e l e r a t i o n of w a t e r i s s i g n i f i c a n t .

SIMPLE P I PE PROBLEMS

Calculation OF Head Difference, Given F l o w Rate the

For

diameter direct,

case

and

a

known

roughness,

flow

rate

in a

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

of

u s i n g f o r e x a m p l e t h e Hazen-Williams

hf = 6.84L(V/C) C

where

of

is

head

known

length,

loss i s s i m p l e a n d

equation,

/D 1 * 1 6 7 (m-s u n i t s )

less

the

p i p e of

Hazen-Williams

r e d u c i n g f o r rough pipes,

(2.1)

coefficient

e.g.

140 f o r

o r t h e Darcy-Weisbach

smooth p i p e s

equation,

vz =3XL

hf

where

is

A

Reynolds

a

function

of

Re =

VD/v,

v

is

velocity,

D

the p i p e bore,

number,

V

fluid,

(2.2)

is

the

mean

the

pipe

relative

the

roughness

kinematic

and

viscosity

of

the the

L the p i p e l e n g t h

and g i s g r a v i t a t i o n a l acceleration.

Calculation of F l o w Rate, Given Head Difference

The solved one

exponential directly

for

type flow

of

equation,

velocity

reason

why

this

type

empiricism.

The

more

scientific

iteratively flow

for

velocity

calculate

the

this

which

velocity

as

equation

Re,

t h e Colebrook-

from

the

head

remains

formula

the

Hazen-Williams,

a n d hence d i s c h a r g e r a t e .

of

Reynolds

i s a n unknown.

corresponding

c a l c u l a t e i t from flow

case

of

e.g.

popular

Darcy number

The p r o c e d u r e

read

1

from

has is

to a

can

be

This

is

despite be

its

solved

function

of

i s to guess a V ,

a Moody d i a g r a m ( o r

White e q u a t i o n ) a n d s o l v e f o r a new loss

equation.

It

is

generally

not

24 necessary

to

repeat

twice

the

flow

as

utilized

manually

(see

converges

with

the

rapidly.

a

1).

Chapter

If

head

loss c h a r t

If

the

reservoirs.

to

more

the

procedure

is

than to

be

based on

are

performed

the equations c a n be used to

RESERVO I RS

more c o m p l i c a t e d

p i p e network is

velocity

computations

problem

involves

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

r a t e a n d h e a d d r o p a l o n g compound p i p e s of

new

the calculations.

I NTER-CONNECTED

A

process

i n a c o m p u t e r a Newton R a p h s o n c o n v e r g e n c e m e t h o d c o u l d b e

programmed

simplify

the

For

example

drawoff

inter-connecting

may

occur

from

a

a number point

in a

i s fed from a number of reservoirs - the problem

which

estimate

a

both flow

the

flow

rate

each

in

pipe

and

the

head

at

the

drawoff point. It all

i s assumed

pipes

are

that

the f r i c t i o n

known. the

Also,

pressure

factors,

the head

level is

diameters

in

each

nowhere

and

lengths of

reservoir

negative

in

remains the

constant

and

network.

A s i m p l e s u c h s u p p l y system i s d e p i c t e d i n F i g u r e 2.1

pipe

Water l e v e l 30

I

I

Fig.

2.1

The head

at

a n d 2-3

Inter-connected

problem n o d e 3.

1

/

can

be

Then

I

Reservoirs

reduced the

can be calculated.

to

the

corresponding

estimation flow

of

the

in each of

residual pipes

1-3

I f t h e n e t f l o w to node 3 does n o t e q u a l

25 the

drawoff,

should

be

lOO@/s,

the

A

revised.

assumed

trial

and

head

obtain a more accurate head at 3.

node 3

at

error

method

is

could

incorrect

and

be employed to

Thus i f the net flow towards 3 i n

both pipes exceeded lOOP/s

the head should be increased, and i f the

flow

should be decreased.

was too low the head

should

be above

ground

unobtainable as

air

level

would

be

or

else

drawn

the into

The f i n a l head at 3

flow

of

lOO@/s

would be

the p i p e l i n e s through a i r

valves a n d f i t t i n g s .

k

If

O.Olmm,

=

0.014.

X

T r y H3

=

3 5 4 / s , Q23 = + 8 O @ / s , so H3 i s too h i g h . w i l l be found Q

A of

23

methodical

the

head

= 11OP/s,

25m and solve f o r Q

Repeating w i t h H

3

Q31 = 10 @ / s .

a n d converging method f o r

3

at

head correction

at

each

method).

improving

the estimation

i t e r a t i o n i s demonstrated l a t e r

The

methods

( t r i a l and e r r o r ,

reservoirs,

more

than

one

drawoff

and

( t h e node

or

r e l a x a t i o n ) could be employed f o r more complex situations, more

=

31 = 21m, i t

compound

iterative e.g.

3 or

pipes

with

changes i n diameter. A l t e r n a t i v e l y an a n a l y t i c a l solution could be o b t a i n e d . Thus i f H1 - H3 = klQIZ

(2.3)

H2 - H3 = k2Q22

and

Then u s i n g Q1

Q1 = -2k,Qg

+

(2.4)

Q2 = Q to e l i m i n a t e Q2 and H 3 3

(2.5)

- 4 ( k l - k 2 ) ( H 2 -H 1-k 2 Q3' f

/(2k2Q3)'

(2.6)

2(kl-k2) For

X

m'/s.

=

0.014

The

then kl

latter

so Q2 = 0.1

=

solution

+ 0.014

=

0.114

3622 and k 2 = 712 so Q1 = -0.014 i s found on

inspection to be

o r -0.035

incorrect and

rn'/s.

PSEUDO STEADY FLOW

I f the water time, flow

levels

i t may s t i l l

i n two

inter-connected

r e s e r v o i r s change w i t h

be possible to o b t a i n an a n a l y t i c a l

r a t e a n d water

levels at

any

time.

solution to the

Acceleration i s neglected i n

the c a l c u l a t i o n f o r the example i n F i g u r e 2.2.

Assuming a head loss equation of the form h

=

(e.g.

H, K =

- H2

=

K Qn

X L/2gD5 ( ~ / 4 ) ' and n = 2 f o r the Darcy e q u a t i o n ) .

(2.7)

(2.8)

26

Fig.

2.2

Inter-connected reservoirs w i t h heads converging

then Q

(h/K)l/n

=

(2.9)

For c o n t i n u i t y

...

(2.10)

= dH2A2 = Qdt

dHIAl

b

-dh

= dH,+dH2

(2.11)

= Qdt(l/A1+l/AZ)

(2.12)

I n t egra t i ng i.e.

starting

between h The

t

Kl/n

,

with

a

1- 1/ n )

( hlt1 /n -

( 1 - l / n ) ( l / A 1 +1 /A2) known

0

head

difference

ho,

=At the

(2.13)

relationship

and time t may be determined from t h i s equation.

problem

could

also

be solved

n u m e r i c a l l y a n d such methods

a r e discussed i n chapters 7 a n d 8.

COMPOUND

P I PES

Equivalent Pipes for Pipes

It

i s often useful

in Series

to know

the equivalent

p i p e which would g i v e

the same head loss a n d flow as a number of inter-connected pipes i n series or

parallel.

The equivalent

p i p e may be used

i n place of

the

compound pipes to perform f u r t h e r flow calculations. The e q u i v a l e n t diameter of

a compound

p i p e composed of

sections

27

of

different

equating

diameters

the

total

and

lengths

head loss f o r

in

series

any flow

may

be c a l c u l a t e d b y

to the head loss through

the e q u i v a l e n t p i p e of length equal to the length of compound pipe:-

(2.14) (2.15) (m

is

5

in

the

Darcy

formula

and

4.85

in

the

Hazen-Williams

formula).

Fig. 2.3

Pipes i n Series

Complex Pipes in P a r a l l e l

Fig.

2.4

Pipes i n P a r a l l e l

Similarly,

the

equivalent

diameter

of

a

system

of

pipes

parallel

i s d e r i v e d b y e q u a t i n g the total flow through the equivalent

pipe

to the sum of

'el

the flows through the i n d i v i d u a l pipes ' i ' 'i n

para1 l e l : Now h

=

h.

(2.16)

28

KPeQnD!

i.e.

so ai

(2.17)

(Pe/ei)'/n(~i/~e)m/n~

=

and Q

= KeiQin/Dm

ZQi

=

C[(Pe/Pi)'/n(Di/De)m/nQ]

=

and b r i n g i n g D

Cancelling out Q,

(Dem/Pe)'/"

= C

and i f each

P

De = ( C D i

(2.18) (2.19)

ee

and

to the left h a n d side,

( D i m / P i ) 1/ n

(2.20)

i s the same,

m/n n/m

(2.21

)

The e q u i v a l e n t diameter could also be d e r i v e d u s i n g a flow/head

For pipes i n p a r a l l e l , assume a reasonable head loss and

loss c h a r t .

read

off

the

flow

through

each

pipe

from

the

chart.

Read off

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. the

For

pipes

total

head

equivalent

in

series,

loss

with

pipe

assume the

diameter

a

reasonable flow

assistance of

which

wduld

the

and

chart.

discharge

the

calculate

Read o f f

the

assumed

flow

w i t h the total head loss across i t s length. It

often

much

as

series

speeds

network

possible

or

using

parallel.

described

below

compound

pipes

Of

could and

analyses

course

this

is

diameters

the

always

be

in

simplify

to

equivalent

methods used

fact

the

to

pipe for

of

networks

minor network

analyse

as

pipes

in

analysis

flows

through

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

more

complex systems than those discussed above.

NODE HEAD CORRECT ION METHOD

A

converging

nodes

and

complex

iterative

branched

termed

the

node

initial

heads

at

networks

multiple

reservoirs.

method.

It

each

Draw

(2)

Assume

initial

(except

if

the

node.

drawoffs,

the

Heads

at

heads at

is

nodes

T h i s method i s

necessary

are

for

then

to assume

corrected

by

The steps i n an a n a l y s i s a r e as follows:-

the p i p e network

initial

the correct

i s often used, especially

with

(1)

inputs,

obtaining

p i p e flows

head correction

successive approximation.

all

method of

the corresponding

schematically

to

a c l e a r scale.

Indicate

f i x e d heads and booster pumps.

arbitrary head a t

assignments,

but

that

reasonable

heads

node i s f i x e d ) .

at

each

node

The more accurate

the speedier w i l l be the convergence of

29 the solution.

(3)

Calculate

the

head u s i n g

a formula of

Calculate the net zero,

in each

pipe

to

any

the form Q =

node

with

(hDm/Kt)'/"

a variable or

using a

loss c h a r t .

f low/head

(4)

flow

inflow

the specific

to

node and i f t h i s i s not

correct the head b y a d d i n g the amount

(2.22)

AH = C Q

m

This equation i s d e r i v e d as follows:-

Q =

Since

(2.23)

(hDm/K@)'/"

(2.24)

Qdh/nh

dQ =

We r e q u i r e Z ( Q + dQ) = 0 Qdh i.e. C Q + C = o nh

(2.25)

dH = -dh where h i s head loss and

But

(2.26)

H i s head a t node

so

AH =

\

ZQ

(2.27)

zo

Flow

Q

and

head

loss

are

considered

node.

H i s the head a t the node.

positive

if

towards

the

I n p u t s ( p o s i t i v e ) and drawoffs

( n e g a t i v e ) a t the node should b e included i n Q .

(5)

Correct

(6)

Repeat

the head a t each variable-head

node i n s i m i l a r manner,

repeat steps 3 and 4 f o r each node.

i.e.

the procedcre (steps 3 to 5 ) u n t i l a l l flows balance to a

sufficient

degree

of

accuracy.

If

the

head

difference

between

the ends of a p i p e i s zero at any stage, omit the p i p e from the p a r t i c u l a r b a l a n c i n g operation. I t should be noted that cases converges s l o w l y the

case

head loss

if

the

head correction flow,

i f at

system

i s adjacent at

the

the node head correction method i n some

is

to

all

i t can be unstable. i.e.

one

pipe

common

node could cause a

even a flow r e v e r s a l i n the low head loss pipe.

the

node

in

question.

This can be with

another p i p e w i t h a low head loss.

correction should be less than to

or

unbalanced,

It

the head loss may

work a correction out of the system.

therefore

a

high

A small

l a r g e change i n I n fact a head

i n any p i p e connecting take

many

iterations

to

A l t e r n a t i v e l y the flow r a t e could

o s c i l l a t e w i l d l y a n d no longer comply w i t h the l i n e a r i z e d r e l a t i o n s h i p between a s m a l l change i n head loss and change i n flow.

30 O n the other h a n d , correction

method.

No

need n o t

be estimated.

node

start

to

estimated. rather results. head

pipes

to

sometimes to

they

heads

the

have

so

are

the

analysis

with

loops

loops

losses a s f o r

preliminary

is very

easier

the

Final

input

be

simple for assumed

the node head

and

initial

flows

Heads c a n b e t a k e n a s g r o u n d l e v e l s f o r e a c h

with is

It

than

data

no

identify

pipes b y

separate.

given

for

done

it

node method.

It

heads

the

node

have

nodes

i s easier

each

loop method.

is

initial

they

to

be

to

connect

interpret

instead

of

the

just

line

I f c h a n g e s a r e r e q u i r e d once a

is

easy

Many of

add,

to

subtract

these a d v a n t a g e s

or

alter

are retained

in the l i n e a r method described in Chapter 4 .

COMPUTER PROGRAM BASED ON NODE HEAD CORRECTION METHOD

The

method

computer simple,

program and

very

easy

appended

easy

to

change

in

cases

to

head

loss

combinations.

together

with

to to

converge

slow

listing,

is

use this

after of

An

a'

chapter).

is

input

method

layouts

depicted

l i s t i n g a n d output.

analysed in chapter 3 u s i n g a different

program

to

Data

The

run.

unbalanced

example

input

simple

and

or

is

is,

also

however,

low

after

(see

and

the

high

program

T h e same e x a m p l e i s

method.

Program Input

The

program

following Line 1: L i n e 2:

m

No.

of

(the

pumps;

3

'error');

Length

and

Darcy

and

will

prompt

for

the

(the metres;

Pipe data can have

nodes ( t o t a l ) ; No.

its

o f f i x e d h e a d nodes

maximum change in head p e r i t e r a t i o n

factor

(assumed constant reservoir)

for

i n metres.

all

pipes);

Note t h a t i n

the f i x e d heads should be numbered f i r s t . lines

correct

in

is

for not

metres;

each

pipe):

important First

Top at

estimate

node

this

of

No.;

stage); head

at

D r a w o f f a t b o t t o m n o d e in m 3 / s .

be p u t head

(one

order

Diameter

b o t t o m node in m e t r e s ;

should

of

( w h i c h must be a

subsequent

node in

No.

permissible f i n a l

n u m b e r i n g nodes, Line

interactively

information:

Head a t n o d e 1

Bottom

run

N e t w o r k name

(reservoirs);

in

is

in a n y

o r d e r b u t e v e r y n o d e i n t h e system

and drawoff

defined

in t h e

data

input,

i.e.

31 each

node should

(head a n d the

last

be

drawoff)

a

bottom

for

any

node

node

i s recorded.

d a t a which

in

at

least

i s supplied

one

twice

If

pipe. thus

it

data

i s only

Drawoffs a r e ignored at f i x e d head

nodes. When the answers are p r i n t e d , will

be

in

the correct

order

the top head node and bottom node

thus d e f i n i n g flow

direction.

The head

at the bottom node i s also given.

i2=998.5m

500m

25

300m

25

F i g . 2.5

1

l\

i

'

f

Network analysed b y program

NEWTON-RAPHSON METHOD

Most

methods of

linearization methods to

a

are

linear

require

such

of

network

the

head

based on a increment a

in

a n a l y s i s described here a r e based on a equations.

loss

linear

increment

head loss.

linearization

by

Even

i n flow

the

Hardy

Cross

b e i n g proportional

The method hereunder does not

the

analyst

but

a

mathematical

32 the n o n - l i n e a r

a p p r o x i m a t i o n to

I inear

using

increments.

The

head

loss term

Newton-

Raphson

solve the non-l i n e a r head-flow in the network.

convergence, instability

The method

but is

be

equations simultaneously

i s s a i d b y Jeppson

requires

to

i s improved i n steps technique

a

close

avoided.

It

initial also

i s u s e d to

f o r each p i p e

(1976) to h a v e r a p i d

estimate

requires

of

flows

relatively

if

little

computer s t o r a g e . The

method

equation f o r

F

where the

h

used

,

Chapter

of

iteration.

1

to

solve

the

Colebrook-White

i s b a s e d on t h e a p p r o x i m a t i o n

( x ) i s the f u n c t i o n of

differential

m-th

in

F

This

with

x

respect

equation

t o b e s o l v e d ( F ( x ) = 0 ) and F ’ to

x.

comes

Subscript

directly

from

m

refers

the

first

to

is the

order

a p p r o x i m a t ion F ’ ( x m ) = F ( x m + , ) - F ( x m ) where F ( X ~ + =~ 0) X

The iteration This

is

- x

m+l

m

convergence

is

i s proportional termed

simultaneous

(2.29)

rapid to

quadratic

s o l u t i o n of

a

because

the e r r o r

error

i n t h e m-th

convergence. number of

the

The

the

m+l-th

i t e r a t i o n squared.

method

non-linear

in

applied

equations f o r

to

heads

a t nodes i s d e m o n s t r a t e d w i t h a n e x a m p l e below

H=80 m

1OOOm x200m m I

t

30 t / s Head

losses H =

0.02LQZf/s

D Sm l 0 0 0 2x

. ’ .Q 812

(71

/4 )

=

0.0325~10-~ L Q 2 / D 5 ( D i n m,

=

5547h’D5I2 / L 4 1

= 3.14

H12I 1

H232

823

=

2.16

834

=

4.96 H 3 4 I

1

the

Q i n f / s , L in m )

33 Set

F2

-30

0 f o r f l o w b a l a n c e a t node 2

=

'212-823

=

3.14(8O-H2)'-2.16(H2-H3)'

=

Q23-834

=

1

1

F3

- 30

0 f o r f I ow b a l a n c e a t n o d e 3

=

1

1

=

2.16 (H2-H3) '-4.96

(H3-100)'

dF 2 -

=

1 .57/(80-H2)?

dF3 -

=

- 1 .08/(H2-H3)i-2.48/(H3-100)i

=

55m,

1

dH 2 dH 3

T r y H2

1

- 1 .08/(H2-H3)e

H 3 = 90m 1

Then

=

H22

1

5 5 - 3.14(80-55)'-2.16(55-90)'-30 I

=

56.06

I

1.57/(80-55)'-1.08/(55-90): 1 =

H32

9 0 - -2.16(55-90)'-4.96(90-100)p

= 87.0

1

-1.08/(55-90)~-2.48/(90-100)' 1

Note (55-90)' direction.

1

i s assumed

to be -(90-55)?

etc.

T h e new h e a d s c a n b e r e - s u b s t i t u t e d

to account f o r flow

into the equations for

t h e n e x t i t e r a t i o n a n d so o n . It will same

as

equations

be observed that those were

for

the

written

the r e s u l t i n g equations a r e i n effect the

node

head

i n terms of

correction unknown

method.

flows

If

the

instead of

basic heads,

t h e r e s u l t i n g e q u a t i o n s w o u l d b e t h e same a s t h e loop f l o w c o r r e c t i o n equation

(see

simultaneously

Ch. for

3). all

The new

equations

flows

instead

may, of

for

however, one

be

loop a t

solved a

time

(Ch. 3 ) .

REFERENCES

Davis,C.V., 1952. H a n d b o o k o f A p p l i e d H y d r a u l i c s , McGraw Hi1 1,2nd E d n . N.Y. 1976. A n a l y s i s o f F l o w i n P i p e N e t w o r k s . A n n A r b o r Jeppson, R.W., Science. 161pp. Stephenson, D., 1981. P i p e l i n e D e s i g n f o r W a t e r E n g i n e e r s , 2 n d E d n . E I sev i e r , A m s t e r d a m . 234pp. Webber, N.B., 1971. F l u i d M e c h a n i c s f o r C i v i l E n g i n e e r s , C h a p m a n and H a l I , L o n d o n .

34

Network a n a l y s i s by Node Head C o r r e c t ion-Program Output and Input

F'I'

NUDE HERO t;OFF'

H2M

397 p 9'7.4 994.8 994 .e

5 e Gz9.8 ! 32 I F F T - S U N I T S 68. D I S P "HPlPES..NHODES,NRESS,E~ Rm.. DARCY f

I

TOPHm"

990 3 998.3 993.8

i

78 I NPUT N 1 N2 > t.13, H 3 , F 2 H < 1 3 80 FOR J = l TO N 1 I F I X E D HEADS NUMBEF'ED F I R S T 90 D I SP " NODE 1+ . NODE2- I Lnr I Dm I I N ITLH2,Q2m?./s"; 180 I NPClT K i; J L 1:. J 3 X < J > ,D < J ? ,H 2

,42

118

120 138 148 158

168 170 172

174 188 190

208

.

H < L C . J >j = H 2 p ( L < J j ? = ~2 F < J > =F NEXT 3 FOR Il=l TI2 50 ! M A X I T H S U=B FOR J = l TO N1 I F H < K < J ) ~ : ~ < > H ( L ( J > THEN > 188 H=H+ 01 I F H I K C . J j l > = H < L ( J l j THEN 228 lJ=L < -J1 L < J > =K ( J >

218 228

238

S=N

F O R J = l TO N l I F K ( J > < > I THEN 338 R=R-Qi;J, S=S +Q ( J 1 i < H ( K < I. ? - H < L ( J ? ! ? GOTO 368 I F L < . . J > O I THEN 368 )-H
410 428 438 44u

THEN 468

. .. I F U < H 3 THEN 438 NEXT I 1 Lb .P .R. -I N ..T. . P R I N T " H+ NXIM)

@ < M ? QM

3/5 H2M" 450 FOR J = l TO Nl 468 P R I N T U S I N G 478

;

K!:J?,L(J!..

X C J ) , D < . J j .,Q<.J> .H:> , . DUD, I? 470 1tlAGE DDD DDD, ~ @ @ C I @D@

D.DDD>DDDDD.D 488 NEXT J 498 END

991.1 NETWORK NHME? TEST I?CIN NPIPES., NNClDES, NRESS ERRm, D A R C Y f ., TOF'Hm? e..7.3.. 1.. .815.. 1880 NODEl+.. NODE2-, La, DNI: INITLH2..C!2m3 I

35 CHAPTER 3

LOOP FLOW CORRECTION METHOD OF NETWORK ANALYSIS

I NTRODUCT I ON

An

efficient

method of

developed

by

performed

manually

methods with

of

alternative

routes

may

(1936).

by

By

the

The

computer

closed

loops

and

i n p i p e networks was

numerical i s one

method

of

the

can

most

be

rapid

flows and head losses in a network

estimating

supplying

Provided

iterations

or

loops.

a n a l y s i n g the flows

Cross

manually

closed

system.

Hardy

loops

any

of

are

be considerably

it the

selected less

than

is

implied

(known)

that

drawoffs

judiciously the

the

there

are

from

the

number

of

number of

individual

the network

has to b e

pipes i n the network and convergence i s r a p i d . The method suffers drawn

the disadvantages that

i n t h e ’ f o r m of

a number of closed loops o r routes and i n i t i a l

\

flows have to be estimated such intersection.

There

are

that

further

flows balance a t each node o r

complications

when

more

than

one

r e s e r v o i r o r f i x e d head i s included o r when b r a n c h pipes feature. Pipes j o i n the nodes.

nodes o r junctions

The number of

number of pipes of off

drawoffs, a

central

number of hand,

i n the network.

e.9.

the

and drawoffs should be taken from

nodes should be minimized to minimize the T h i s may be done b y lumping a lot

from a block of houses, point.

nodes may longer

Where

local

lengths of

losses

are

individual

pipes

them

important

to b e reduced much.

not be able

the

head

together and t a k i n g

the

On the other

the

more r a p i d l y

d a t a assembly a n d a n a l y s i s can be done.

METHOD OF FLOW CORRECTION

The

method

requires

an

a r e successively corrected. in The

closed basis

loops, of

using

the

initial

r e a l i s t i c estimate of flows which

Corrections a r e made i n steps to the flows

a r e l a x a t i o n o r converging

correction

is

derived

as

i t e r a t i v e method.

follows.

A

first

order

approximation to the d i f f e r e n t i a l of the head loss equation i s made:

36 Assuming

h

Then

dh

The and

it

KLnQ"-'dQ/Dm

=

total

if

KLQn/Dm

=

net is

head

not,

loss

the

= (nh/Q)dQ

around

head

any

losses

closed

are

loop

should be zero

incorrect

and

should

be

adjusted by dh: Then

Z(h+dh)

0

=

(3.3

Ch

+

Zh

+ Z(nh/Q)dQ

Cdh = 0

(3.4 = 0

(3.6)

-Lh

AQ =

nco

I f the Darcy-Weisbach m=5,

whereas

m=4.85. also

the

Some

be

(3.5

degree

noted

respect i v e l y

equation

Hazen-Williams

that

of

freedom

equation

head losses,

would

give

be i n any

units,

e.g.

n=2 a n d

n=1.85

i s a v a i l a b l e i n selecting

and h may

Q

i s used f o r

and

n.

I t may

P/s

and m

.

Barlow a n d M a r k l a n d

(1969) showed how a second order a p p r o x i -

mation to the d i f f e r e n t i a l produced a more r a p i d convergence. Steps

in

Engineers,

1.

analysis

may

be

set

out

as

follows

(lnstn.

of

Water

1969);

Draw

the network p l a n to a c l e a r scale a n d set a l l d a t a such as

pipe

lengths a n d diameters,

plan.

In

fact

r e s e r v o i r heads and drawoffs on the

i s often convenient

it

to set

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

consecutively

and

mark p o s i t i v e direc-

in

assign

on such a p l a n .

2.

Number

3.

Starting

with

any

pipe

sensible

flow

and

flow

the

tions, e.g.

adjacent

closed

clockwise.

pipe

taking

intermediate node, must b e zero. to

each

the

at

The

of

care

i.e.

each

loop, to

that

this the

an

pipe.

flow

arbitrary Repeat

balances

but

for

an

at

the

net inflow to less drawoff from each node

number

inner

any

direction

Proceed thus

pipe.

number

assumed

loops

through of

loops,

new

the network

assumptions i.e.

loop.

a

The

new more

a l l o c a t i n g flows

necessary flow

will

accurate

will have the

equal to

be

initial

estimates of flows the speedier w i l l be the solution.

4.

Calculate t h e head loss i n each p i p e u s i n g a formula o f the form

h

=

KLQn/Dm.

Also

calculate

the

term

h/Q

for

r e t a i n i n g the p o s i t i v e v a l u e of h and Q i n t h i s case.

each

pipe

37

5.

Select

a

loop

analysis

1

loop the

such

as the most prominent one

manually,

or

proceed

i f u s i n g a computer.

loop,

adding

head

in

numerical

Calculate the net

drops,

or

if

performing

order

subtracting

the

starting

at

head l o s s around head

increases i f

the flow i s i n the adverse direction.

6.

If

the

net

head

around

loss

the

loop

is

not

zero,

correct

the

flows i n each p i p e i n the loop u s i n g the formula

(3.6)

A Q = -Zh/nZ(h/Q).

7.

Repeat steps 5 a n d 6 f o r each loop.

8.

Repeat

4

steps

7

to

until

the

head

losses

around

each

loop

balance to a s a t i s f a c t o r y degree o r u n t i l the flow corrections a r e negligible.

9.

Calculate point

the

and

head

going

at

from

each pipe

node

by

starting

to pipe.

at

a known head

Compare the heads at each

node w i t h ground levels to determine the r e s i d u a l pressure head.

It to

i s easier

to estimate head losses from a

use an equatizn

if

head

loss c h a r t than

the procedure i s done m a n u a l l y .

The value of

n can also be set a t 2 f o r ease of manual computations. If h the

( a n d consequently

computations,

h/Q

Q)

should

Zh/Q f o r the corresponding

works out a t zero f o r any pipe d u r i n g be

assumed

be zero

to

in calculating

loop o r loops.

LOOP SELECT I ON

It

is

most

convenient

adjacent

to each other

however,

not

numerical

necessarily

convergence

to

when

select

and

number

proceeding across

the method which w i l l

when c o r r e c t i n g flows.

ing

loops the convergence can be speeded

can

only

proceed

at

the

rate

of

one

consecutively

the network. result

i n most r a p i d

By c a r e f u l l y

up.

loop

loops

This i s ,

designat-

Remember a correction at

a

time

through

the

system. It

i s often

l a r g e r pipes, outer

convenient

i.e.

the

whether

or

identified by outer loop.

imagined as a

network not

have

some of

the

loops embracing the

c u t t i n g across small minor loops. A l t e r n a t i v e l y

pipes can be

outside

to

thus

such

two

loop

has

a

additional numbers

loop turned inside out.

loop number. loops then

are it

is

The method w i l l

included.

If

the

The space

a

work

pipe

is

necessary to number the

38 The

procedure

of

identifying

pipes

by

loops

the

may also be used to i n d i c a t e the d i r e c t i o n of flow. a sign convention such as clockwise positive, the flow

is

they

If all

separate

loops obey

then the loop i n which

i n the p o s i t i v e d i r e c t i o n can a l w a y s be g i v e n f i r s t when

i d e n t i f y i n g a p i p e a n d i t s flow direction. When computing the flow loss c a l c u l a t i o n

or

and flow corrections may

else

all

the

correcting

all

the

loop

easiest

computerize

time,

easiest be

to

to

correction f o r successive loops,

when

doing

paid

to

flow

be made f o r one loop at a

corrections

flows

may .be

together.

whereas

the

The

latter

and

directions,

computed

former

is

Careful

however,

or

before

procedure

procedure

the calculations manually.

signs

the head

often

is the

a t t e n t i o n has else

the

flow

balance can be lost f o r some of the nodes.

MULT I PLE RESERVO I RS

The head

loop network method becomes t r i c k y when more than one f i x e d

reservoir

is

incorporated

in

the

pipe

network.

method of coping w i t h m u l t i p l e r e s e r v o i r s i s to connecting

the

above one).

reservoirs

(one dummy

One a d d i t i o n a l

The

length a n d diameter of

and

the estimated

flow

in

loop

pipe

insert

per

The

simplest

a dummy p i p e

additional

reservoir

i s thereby created per dummy pipe.

dummy pipes may be selected a r b i t r a r i l y should correspond to the

a dummy p i p e

head difference between the r e s e r v o i r s i t connects. The flow

and head loss i n the dummy p i p e a r e taken i n t o account

i n c a l c u l a t i n g z h and c h / Q is

not

corrected

in dummy

around the respective loops, b u t the flow pipes

when

the

flow

correction

i s made

around the r e l e v a n t loop.

BRANCH PIPES

Although

predominantly

tree-like

networks

can often be handled most e f f i c i e n t l y method,

it

The flow the loop

is

possible

i n an

with

include

closed

b r a n c h pipes i n the

a

reservoir

a dummy p i p e ) .

r e s o r t i n g to

loops

loop method.

isolated b r a n c h p i p e should be p r e - defined

b r a n c h connects

without

to

without

u s i n g the node head correction

when

it

will

(unless

be incorporated i n a

Therefore the head loss can be estimated

i t e r a t i o n s except

if

it

i s a compound p i p e made

39 up

of

a

number of

diameters

i n series.

Even then,

equivalent pipe

methods a r e possible. In

general

defined.

either

I n either

the

case

head

at

a

the other

node,

variable

or

the

drawoff,

can be

has to be c a l c u l a t e d i n

the network a n a l y s i s . The

method

network the

of

including

programme

number

of

real

flows i n those

branch

pipes

with

defined

flows

in

a

i s to a s c r i b e a r b i t r a r y loop numbers g r e a t e r than loops

the

to

branch

pipe,

and

not

correct

the

loops.

PUMPS AND PRESSURE REDUCING VALVES

Pumps may be assumed to increase or boost the head i n a pipe i n one

direction

the

other

by

hand,

a

specified

may

reduce

amount.

Pressure

total

fixed-head

h e a d at

a

valves,

on

the head b y a specified head and can

be treated as a negative pumping head. fixed

reducing

point

that

( I f e i t h e r operates to g i v e a

point

may

be

treated

like

a

reservoir).

A pumping head would be included i n an a n a l y s i s b y s u b t r a c t i n g head generated from loop flow

is

pumping when

the f r i c t i o n head drop

i n the d i r e c t i o n of against

the

p o s i t i v e flow pumping

i f proceeding around the

i n the p i p e w i t h pump.

head

add

the

friction

I f the

head

and

T h i s pumping head i s included i n 1 h b u t not i n Zh/Q

head.

proceeding around the

loop.

The computation f o r

Q i s then as

before. Where a

the

preset v a l v e or f i t t i n g

d i r e c t i o n of

equivalent the

head

loss

in

flow

the

fitting

i n a pipe causes a head loss i n

can

generally

b e converted

length o f pipe and added to the r e a l p i p e length.

loss the

through pipe

is

the

fitting

( A L/D)V2/2g,

is

KVZ/2g and then

the

the

to

friction

equivalent

an

Thus i f head

additional

l e n g t h of p i p e is

(3.7)

AL = KD/A

I f the pumping head i s a function of the flow r a t e , e.g.

2

h

= a + a Q + a 2 Q P O 1 = the where a

0

(3.8) pump

shutoff

head

and

cons tan ts, then equation (3.6)

w i l l be replaced as follows:

al

and

a

2 are

pump

40 The r e v i s e d Q1

(Q +

=

(3.9) (3.10)

dQ)

For any pipe head drop h = KLQn/Dm

-

alQ-a Q 2

a.

2

or i f h i s the uncorrected head drop,

..)/Dm-aO-a,

h+dh = KL(Qn+n Qn-'dQ+.

(Q+dQ -a2(Q2+2QdQ+...)

(3.11)

The total head drop around a loop must be zero: = 0

C(h+dh)

dQ =

Hence

(3.3)

- c ( KLQn/Dm-a0-alQ-a2QZ n

or

A Q =

(3.12)

c K L Q n - l /Dm-a -2Qa2

-C(h -h ) f P n C ( h /Q)-C(al+2Qa2) f

(3.13)

PRACT I CAL DES I GN

The water engineer may be r e q u i r e d to analyse a p i p e network to flows o r heads. He may have to check pressure heads f o r aged

check

networks to see i f they a r e s u f f i c i e n t , a

revised

designed

demand to

conditions

pattern.

cope

have

with

to

a

be

o r to re-analyse a network f o r

Alternativety certain

checked.

the

system

demand

For

may

pattern

example,

may have been based on peak p e r i o d drawoffs,

have been

and

design

abnormal p i p e sizes

of

b u t an abnormal

load

may come on the system when a f i r e h y d r a n t i s operated.

Then lower

pressures

demand

met.

generally

may

be

tolerable

For example spread over

provided

a township,

the

peak

fire

house demands

is

may

@ / s whereas i n d i v i d u a l f i r e h y d r a n t s may have

be assumed to be 0.1

to d e l i v e r more than 10 e / s . Pipe will

sizes

be

have

analysed

diameter

until

tree-I i k e

network

smaller

a

to

for

of

flows and

selected

each

with

a

by

assumed

satisfactory

trial

and e r r o r .

pipe

arrangement

radiating

network i s at

distribution

The network

layout

hand. of

and

pipe

Generally

pipes

a

becoming

i n diameter towards the t i p s i s the most economic, b u t closed

loops a r e r e q u i r e d f o r out

be

commission. can

security

The

be optimized

error

design

is

i n case pipes a r e damaged or

tree - l i k e using

necessary

network

linear for

with

taken

consequent

known

programming methods b u t

looped

systems.

As

a

guide

trial the

head loss g r a d i e n t s should be lesser the l a r g e r the p i p e diameter o r flow,

b u t care

valleys(e.g.

i s necessary to ensure pressures a r e not

lOOm

upper

limit

for

household

fittings)

or

too h i g h i n too

low a t

41

high

points

(e.g.

u n d u l a t i n g areas, these

pressure

15m

lower

limit

for

residential

areas).

In

pressure r e d u c i n g v a l v e s may be r e q u i r e d to meet

limits,

or

many

r e s e r v o i r s could

be

used

supply

to

separate zones.

COMPUTER PROGRAM FOR LOOP FLOW CORRECTION METHOD

on

The

accompanying

the

loop

computer

flow

and

correction

as

the

pumps are permitted. a d j u s t i n g G from 9.8 m

is

indicated

The

method.

program It

Darcy

head

is

adapted

it

can

to

handle

suit

a HP-85

50 pipes.

No

u n i t s (metres a n d seconds) b u t b y

to 32 i t may b e r u n - i n ft-sec ft

loss equation

(equivalent

heads

It

stands

is in SI

i n BASIC language i s based

u n i t s . Also where

should be assumed i f calculations

units.

program represents pumping

program

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

are to be i n ft-sec

factor

computer

X

could

to

f

and in

in

is

used

with

North America).

a

The

constant symbol

friction

F i n the

i s assumed the same f o r each pipe. X fact

and

be r e a d f o r each p i p e b u t t h i s would

increase d a t a i n p u t . I n p u t requested i n t e r a c t i v e l y is: Line 1;

Name of system

Line 2 ;

No.

of

pipes;

No.

of

loops;

No.

of

reservoirs;

Error

permitted i n m3/s; Darcy f r i c t i o n f a c t o r . Line 3 etc. -

(One loop;

for

each

Negative

pipe

starting

loop;

Length

with

(m);

dummy

pipes);

Diameter

estimate of flow (m’/s). Running time i s r o u g h l y 1s per 3 pipes per i t e r a t i o n .

(m);

Positive Initial

42

n

E c .r

.r

c

\

-

b

n o N m w N - - I '-E

o

N

m

W

V

r

E

m X

0

N

0 0 0

t

E &

X

0

E

E

In 7

X

0

E

0 0 0

-

88t

BLP 89P 8SP BPC Bet? BZP 91t BBP 86E

88E

BLE 09.2

BEE

BPE

BEE 0ZE BIZ B0E

862 08Z BLZ B9Z BSZ

8tZ

96 I 08 I

BL I 89 I 0sI

8t I Br" I 8.2 I 81 1

na I B6

68 8L 89 8s OP

rl iJI

43

44

pseudo-pipe /

%

w i t h head loss=Hl-H2

\

\ n

I

'

fi

1

\

H1=1000m

I a 1 a s s u m e d f 1o w s oop method (ch 3 ) 08m3 / s

"

11

400i Q.3

ed l o o p s

F i g . 3.2

Network analysed by program u s i n g loop method

REFERENCES

a n d M a r k l a n d , E., June 1969. Computer a n a l y s i s of B a r l o w , J.F. p i p e n e t w o r k s , Proc. I n s t n . C i v i l E n g r s . , 48, p a p e r 7187, p249. 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 o f c o n d u i t s o r conduct o r s . U n i v e r s i t y o f I l l i n o i s , B u l l e t i n 286. I n s t i t u t i o n o f Water E n g i n e e r s , 1969. M a n u a l o f Water E n g i n e e r i n g P r a c t i c e , 4 t h E d n . , London.

45 CHAPTER 4

LINEAR METHOD

I NTRODUCT I ON

The

trial

analysis proceed

and

error

and

relaxation

methods

were o r i g i n a l l y developed f o r manual from

matrices or methods

one

pipe

solution of

suffer

to

the

next

in

a

of

solution.

routine

linear

With

inversion

equations,

the

of

e.g.

such

as

poor

development

matrices Gauss'

gence of

non-l i n e a r equations,

that

Newton-Raphson

of

and

large

simultaneous equations are not required.

disadvantages

1969).

methods of

network

Calculations

manner

convergence

systems unless improved solution methods a r e used Markland,

pipe

of

of

Martin

some

computers

numerical

and

for

Barlow and

solution of

elimination.

employing

(e.g.

digital

and methods of

method

(e.g.

The

came

sets of

Rapid convermethod

Peters

such as

1963)

is

also

for

pipe

possible. The

possibility

of

solution

networks r e l a t i n g flows each

node

thus

encountered for

pipes

I inear

Except

waterworks

i s non-linear.

fact

makes

an

approximation into

a

of

equations

for

practice,

laminar the

flow

which loss

flow-head

is

rarely

relationship

A

difficult.

method of

converting

the

to a l i n e a r form was proposed b y Wood and Charles (1972).

linearization

in

matrix

T h i s makes the establishment and solution of

matrix

equations r e l a t i v e l y easy. is

a

to head losses and i n c l u d i n g flow balance at

arises.

simultaneous equations

equations The

in

of

is

improved in

of

a

set

of

simultaneous

The l i n e a r form of head loss-flow

approximation

'constant'

solution

to

iteratively

the

head

true

the

by

loss

equation

equation

substituting

equation.

and

revised

The

the flows

linearization

techniques can b e a p p l i e d i n s o l v i n g f o r flows around loops o r heads at

nodes

methods method

(Isaacs was

made

reduces

solutions

and

and

by

Mills, Wood

and

computational averaging

convergence

(Wood a n d

convergence

for

the

1980).

Rayes

effort,

methods

Charles,

node

A

method

comparison

of

the

(1981). Although can

it

may

1972). where

be

There some

result

the

in

also

pipes

linear

oscillating

required are

various

to

speed

problems of

have

low

losses, as f o r the node head correction method of Hardy Cross.

head

46

I n general a set of equations i s established as follows:

A t each node, f o r c o n t i n u i t y

XIin There only

drawoff

=

are

equations

j

nodes

junctions

(4.1

which

where

for

can

'closed'

j

vary

the head

in

nodes

head

is fixed

or

are

junctions.

considered

That

thus,

is

and

have to have a v a r i a b l e volume.

For such nodes the c o n t i n u i t y equation becomes = increase i n volume stored

AtCQin

If

the r e l a t i o n s h i p between volume and head i s known t h i s could

be u t i l i z e d b u t

it

fixed,

for

that

constant. fixed the

(4.2)

is

i s common p r a c t i c e to assume r e s e r v o i r levels a r e

Therefore

head

nodes

networks

is

Alternatively

steady the

are

state

mass

not

reduced

conditions

balance

required by

the

the above equation

equations

as

the

number

at

are

reservoirs

or

variables

in

number of

of

fixed

assumed

head reservoirs.

i s replaced b y one of the form f o r \

each known head j u n c t i o n

H

volumes

= constant

There

are

also

i

equations

(one

for

each

closed

loop)

of

the

f o l l o w i n g form I h = 0 where

(4.3)

h

forming

is the

the

head

loop,

i.e.

must be zero.

loss the

in net

Now each head

a

selected

head

loss h

loss

direction around

i s r e l a t e d to

in

one

pipe

each closed loop the flow r a t e i n

the pipe, Q , b y an equation of the form

(4.4)

h = KQn so

that

pipe.

heads

can

be

I t can be shown

m ioops,

that

by

the

term

involving

i n a network w i t h i pipes,

Q

f o r each

j nodes a n d

( o n i y counting i n t e r n a l loops)

i = j + m This

replaced

(4.5)

1

holds

for

all

networks

tree-like

branches.

equations

( t h e a d d i t i o n a l one

thus g i v i n g

a

total

as the unknowns.

There

are

with j

closed

-

1

loops as well

continuity

i s r e d u n d a n t ) and m

or

as open

fixed

head

loop equations,

of i equations f o r i unknowns i f flows a r e used

The j f equations f o r

known heads can be omitted

where j f i s the number of f i x e d head nodes.

47

oniitted where j

f Alternatively, are

there

i s the number of f i x e d head nodes. if

heads

equations f o r 1/n

j

There

where each flow Q i n m pipes i s

nodes,

.

replaced b y ( h / K )

so

a t nodes a r e regarded as the unknowns

are m equations r e l a t i n g head loss to flow

i n the pipes,

once the Q ' s i n the head loss equations a r e replaced b y H

that

1

- H2 the number of equations i s s t i l l j f o r s o l v i n g f o r j unknown heads. Unfortunately,

the

There

in

laminar

h a n d Q a r e non-linear

relating both

except

node equations

is

no

easy

and

method

situations

so d i r e c t

loop

the

of

flow

equations

simultaneous solution of

flow

solution

the

of

equations non-linear

is difficult. simultaneous

equations a n d t r i a l

and e r r o r o r numerical methods u s u a l l y have to

be

method

employed.

approximate they

be

can

computer

The

the

head

solved

storac&

method,

and

a v a i lable,

is

if

loss

used

here,

equations

simul taneously no problem,

storage

is

the

by by

Gauss

limited

successive approximation,

linear

method,

is

to

l i n e a r equations and then various

methods.

elimination

but e.g.

i s an

computing

time

is

Where

efficient readily

successive over-relaxation

a n d Newton Raphson methods can be employed.

LINEAR METHOD APPLIED TO LOOP FLOWS

If

head loss

the

i n any p i p e can be expressed b y an equation

such as

(4.6)

h. = K.Q." I I i t can be r e w r i t t e n approximately

h.

I

n-1

K.Q. I

where

K.

is

roughness, by

Qi

10

the

iteration

Q.

KfQ.

=

a

(4.7)

f u n c t i o n of is

previous and

as follows

a

the

flow

iteration. set

of

the

length

of

pipe i,

i t s diameter and

r a t e and Q.

i s the flow

Both

h.

10

€Ia in d

equations

r a t e indicated

are unknowns f o r each

r e l a t i n g flow

and

head

loss

established: Around each loop,

Che = 0

(4.8)

is

48

i.e.

aim = o

CK.:

(4.9)

a n d f o r e a c h node j :

= q.

ZQ..

Q.. i s the f l o w

where

(4.10)

J

IJ

i n p i p e i to node j , Qim

in pipe i

i s the flow

IJ

i n loop m a n d q . i s t h e d r a w o f f a t node j . J

I f there a r e i pipes,

j nodes a n d m loops,

t h e n i t was s t a t e d t h a t

i = j + m - 1 There

are

i+l

equations

for

i flows.

a

(4.11)

number

revised

node

j

equations

in total

and

w h i c h one

of

The r e s u l t i n g set o f

of

times.

Each

(improved)

m loop e q u a t i o n s

time,

before

i s redundant

so one c a n

l i n e a r equations new

there a r e

so

solve

h a s to b e s o l v e d

Q.'s emerge a n d the K."s

re-solving

r a t e s h a v e to be e s t i m a t e d p r i o r to

the

equations.

the f i r s t

Initial

are flow

s o l u t i o n of t h e l i n e a r

equations. Although

the

above

method

converges

we1 I

fairly

the

following

method i s e a s i e r to v i s u a l i z e a n d i s e x p l a i n e d i n more d e t a i l . \

L I N E A R METHOD FOR NODE HEADS

In

this

simplify then

case

the

written

to node j ,

the

Darcy

calculations.

Q..lQ../I J

as

h..

= IJ

head

The

loss

friction

h../K..

where

IJ

IJ

equation

will

be

loss e q u a t i o n f o r

Q.. i s

used

a

pipe

t h e f l o w from node i

i and j

and K . .

from

a

Q.. i s a n a p p r o x i m a t e s o l u t i o n to Q.. ( o b t a i n e d

If

IJ

IJO

IJ

previous

iteration

substitutes C . . = l / ( K . . l Q . . IJ

IJ

ZQ.. 'J

IJO

or

I),

from

an

then a

initial

estimate)

and

i f one

' l i n e a r ' equation results; (4.12)

= C..h.. IJ

=

IJ

8 X..C../gd?.n'. IJ

is

IJ

i s t h e h e a d d i f f e r e n c e between

IJ

IJ

to

IJ

S u b s t i t u t i n g i n t o t h e c o n t i n u i t y e q u a t i o n a t e a c h node j, (4.13)

= q.

CQ..

J

IJ

ZC..h.. IJ

IJ

(4.14)

= q.

J

R e p l a c i n g h . . b y H.-H. IJ

I

J

C(C..H.-C..H.) = q . IJ I IJ J J Hence H . = CC..H.-q. J

I J I

(4.15) (4.16)

J

CC.. IJ

One

has

J

linear

equations

(equal

to

the

number

of

variable

49

head

nodes)

equations tion

for

can

or

over-relaxation as

unknowns

(the

heads

H

at

be solved b y v a r i o u s techniques,

method,

program

J

the

Gauss

method

is

iterative

employed

requires

it

Siedel

little

the

in

memory

each

e.g.

node).

Gauss e l i m i n a -

method.

A

successive

accompanying

whereas

The

computer

matrix

a

would

r e q u i r e a l a r g e computer storage c a p a c i t y . To

avoid

overshoot

averaging

an

procedure

can

be

introduced

a f t e r each step, = wH. + ( l - w ) H .

H.' J

J1

(4.17)

JO

w h e r e 0 'w

' 1

and s u b s c r i p t

t h e r e c e n t H.. J After solution

of

the

o r e f e r s t o t h e p r e v i o u s H. and 1 to J

equations

for

each

p i p e a r e c a l c u l a t e d and t h e n e a c h C . . .

The

IJ

H . , f l o w s 8.. i n e a c h J IJ linearization procedure

i s t h e n r e p e a t e d and a new set o f e q u a t i o n s s o l v e d f o r t h e h e a d s a t each

node

H.. J

The

procedure

is

repeated

until

convergence

is

s a t i sf a c t o r y . \

Pumps

If

a

pump

a

in

h e a d loss i s H.-H.+h I J P = q. J

h then the P' Equation (4.10) i s therefore replaced b y

line

.

generates

a

specific

head,

X..(Hi-H.+h ) IJ J P

CC..(Hi+h :.H.=

P CC..

IJ

J

If

(4.18)

)-q. J

(4.19)

IJ

the pumping head i s a function of the flow r a t e ,

the convergence

c a n be slow.

COMPUTER PROGRAM FOR LINEAR METHOD

A

BASIC

minimize are is

program

data

input.

possible as assumed

and

modifications,

Thus

presented. no

Reynolds but

the

I n fact number former

The

pumps o r

the program stands.

constant.

with

is

Also

program pressure

is

written

reducing

to

valves

the Darcy f r i c t i o n factor

i t could be v a r i e d from p i p e to p i p e and

would

pipe

roughness

increase

input

with

and

the

small latter

50

w o u l d increase computational time. A

great

assumed. and

are

Flows

the

This

advantage

is

no

initially

corresponding

procedure

that

can

set

heads

lead

initial

calculated

slow

or

correspond

to

are

to

flows

at

heads

need b e

unit

velocity

to

s u c c e s s i v e nodes.

convergence f o r

some

cases

but

i s i n f a c t one o f t h e m a i n a d v a n t a g e s o f t h i s m e t h o d . Input The

and o u t p u t

input

HP85

is

i s 5s

prompted

per

in metres

are

pipe

at

each

and

the

line.

and c u b i c

metres

Typical

running

number of

pipes

could

per

second.

time o n be

an

increased

a b o v e 30 b y a l t e r i n g t h e d i m e n s i o n s t a t e m e n t . The appended p r o g r a m follows ly.

The

above will

variable

section

also

program).

It

be

improved

360

and

This

in

work

t h e p r o c e d u r e Slescribed p r e v i o u s -

used follow

the

general. Although

in

ft-

should for

410

would

names

s

were

if

be noted t h a t

large

require

units

networks

limited a

to

the

nomenclature

is

data

G

is

in S.I.

units

a l t e r e d t o 32.2

t h e speed o f if

the

in

used

here

the it

( l i n e 69 o f

the program could

iterations

p i p e a connecting

between to

that

lines node.

new d i m e n s i o n e d v a r i a b l e f o r e a c h n o d e and

a connectivity search.

D e s c r i p t i o n of V a r i a b l e s in P r o g r a m

K L M NO N1 N2 N3

SD/FX Sums H ( I ) f o r e a c h S.O.R. iteration Sums AF Pipe diameter ( i n m) H o l d s t h e o l d v a l u e o f H ( I ) t o comp u t e AH Darcy f r i c t i o n factor F of p i p e K Darcy f r i c t i o n factor f o r a l l pipks Head a t j u n c t i o n Node c o u n t e r Number of nodes in system Joint b e g i n n i n g number Joint end number Number o f f i x e d h e a d nodes (Numbered first) Iteration counter Node c o u n t e r Pipe counter Maximum number of m a i n i t e r a t i o n s e.g. J+5 M a x i m u m n u m b e r of S.O.R. iterations w i t h i n e a c h m a i n i t e r a t i o n e.g. J+10 Counts the number of m a i n i t e r a t i o n s Counts the total n u m b e r o f S.O.R. iterations

51

Number o f p i p e s i n s y s t e m F l o w i n p i p e m’/s D r a w o f f i n m’/s Drawoff from junction (+ve out of junction) gn2/8 CK.. ‘J Counts the number of H( 1 ) ’ s CK. .H. IJ J H o l d s o l d Q ( K ) v a l u e f o r c a l c u l a t i o n of average of o l d value a n d continuity value Tolerance on head c a l c u l a t i o n in F e.g. 0.0001 loop i n metres o n T o l e r a n c e o n S.O.R. h e a d s e . g . 0.01 S.O.R. factor w e.g. 1.3m. Must be between 1 a n d 2. Pipe length (in m)

TO T1

Data Required Heading. This can be any alphanumeric e x p r e s s i o n up t o 18 c h a r a c t e r s l o n g . W i l l be p r i n t e d out a t head of results. P : Number o f p i p e s J : T o t a l n u m b e r o f j u n c t i o n s o r nodes J3 : Number o f j u n c t i o n s w i t h f i x e d h e a d s F l : Darcy friction factor, assumed t h e same f o r a l l p i p e s e . g . 0.015 J ( I ) : Head a t s u c c e s s i v e f i x e d - h e a d nodes ( w h i c h must b e n u m b e r e d f i r s t ) ( P lines, i n o r d e r s u c h t h a t a n y node except f i x e d head i s r e f e r r e d to f i r s t a s a J2 t h e n a s a J1) J l ( K ) : Joint b e g i n no. J 2 ( K ) : J o i n t e n d no. M a k e s u r e e a c h node is an end number at least once to allocate a drawoff. X ( K ) : Pipe length (in m ) D(K): P i p e diameter ( i n m) Q 2 ( J 2 ( K ) ) : Drawoff from end node (in m’/s); i f t h i s information i s r e a d twice the l a s t v a l u e i s r e t a i n e d .

Line 1:

L i n e 2:

L i n e 3 to J3+3: (J3 l i n e s ) L i n e 4+J3 t o 4+J3+P:

G e n e r a l Comments

Generally

t h e l i n e a r method c o n v e r g e s in f a r

t h e H a r d y Cross method. to to

the flow 100

is all

pipes.

that

The

less i t e r a t i o n s than

Between 4 a n d 10 s u c c e s s i v e a p p r o x i m a t i o n s i s r e q u i r e d even f o r

snag

is

the

solution

networks of

large

i n v o l v i n g up numbers

of

52

s i m u l taneous

equations.

If

successive

least t h a t number of substitutions The

initial

program

flow

assumption

assumes a v e l o c i t y

of

over-relaxation

is

used

at

may be required.

may

be critical

lm/s

and calculates

i n some cases.

The

the correspond-

ing f l o w r a t e i n m 3 / s f o r e a c h p i p e .

The i n i t i a l h e a d s a t e a c h node

are

and

also

based on

SOR

start

the

each

pipe

read

node

into

is

be

such

calculated,

this can

that

the

at

assumed

each

on

the this

values node.

head

r a t e in each p i p e f o r

it

could

be

are

When at

basis

used data

the

to for

assumed

provided

the

in w h i c h d a t a a r e s u p p l i e d

upstream

large errors especially

Alternatively

these

node

i n t h e case o f r e s e r v o i r s .

o r supplied

lead to

head

computer

Hence t h e o r d e r

(1972) u s e a u n i t f l o w

widely.

of the

calculated

u p s t r e a m head i s k n o w n . should

assumption

improvement

are

downstream

this

head

has

been

Wood a n d C h a r l e s

the f i r s t estimate b u t

i f the p i p e diameters v a r y

assumed

to

start

that

flows are

laminar. Although

the

that

successive

this

the

next

convergence trials

oscillate

approximation

p r e v i o u s two v a l u e s

is fairly about can

be

rap\id,

i t h a s been o b s e r v e d

the f i n a l taken

value.

as

Linear

I NF'IUT ,, NO01 SP N P I PE'?, NODES F I i:H DHF'C

-

DISP L : It4PCIT H i Li NEXT L

of

the

Method Network Analysis with Output

and Input (same example as in Chapters 2 and 3 ) .

"

mean

(whether flows o r heads a r e the unknowns).

Computer Program for

Y 4

the

To overcome

.

53

NEXT K F O F C = 1 TO P -,.-'i - .-I 1 i K ?=S2i(.-II ..-.

..

,

F'I F'ENET TOPN BgTI.4 X u 1 3 588 1 4 788 3 4 688 2 4 588

3 5 4

E.

5 6 t; 7

Dni 258 28R 158 458 256

308 588

288

488 28R

256

3R8

t+APIE TESTL I H tJF'IPES> NODES, F I X H * OHRCY f

54

REFERENCES

Barlow,J.F. and M a r k l a n d , E., June, 1969. C o m p u t e r A n a l y s i s of p i p e n e t w o r k s . P r o c . ICE, 43 p 249-259. Campbell, G.V., 1981. Pipe network analysis program (linear m e t h o d ) . P r o j e c t , U n i v e r s i t y of t h e W i t w a t e r s r a n d . l s a a c s , L.J. and M i l l s , K.G., J u l y , 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 a n a l y s i s . P r o c . ASCE 106 ( H Y 7 ) . p1191-1201 and P e t e r s , G., 1963. The a p p l i c a t i o n of N e w t o n ' s M a r t i n , D.W., m e t h o d t o n e t w o r k a n a l y s i s b y d i g i t a l c o m p u t e r . J. I n s t n . W a t e r E n g r s . 17, p115-129 Wood, D.J. a n d C h a r l e s , O.A., J u l y , 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 . P r o c . ASCE, 98 (HY7) p1157-1170. Wood, D.J. and R a y e s , A.G., Oct. 1981. R e l i a b i l i t y o f a l g o r i t h m s for p i p e n e t w o r k a n a l y s i s . P r o c . ASCE, 107 ( H Y l O ) p 1145-1161.

55

CHAPTER 5

OPT IMUM DESIGN O F BRANCHED P I PE NETWORKS BY L I NEAR PROGRAMM I NG

I NTRODUCT I ON

In the previous chapters the p i p e diameters, were

assumed

computed. trial.

known

Design

and

of

the

pipe

corresponding

networks

could

lengths and layouts

flows only

and

heads

were

undertaken

by

The design problem i s not a s easy as the a n a l y s i s problem.

In

the n e x t

t w o c h a p t e r s a p p r o x i m a t e methods f o r

networks

are

given.

be

direct

d e s i g n of p i p e

Economics d i c t a t e s t h e most p r a c t i c a l

design

in

e a c h case. Linear may

programming

only

be

used

if

is

a

the

L i n e a r programminb cannot networks

with

tions.

It

can

where

the

between

flow

the

treated

in the

certain

range.

The

to

in

the

and

Any

it

linear.

the

the

are

system

pre-selected

approxima-

tree-like

Since cost

different

other

or

networks

relationships

nonlinear,

linear:

diameters

diameter

the

For each i s allowed

i s treated a s the

linearly proportional

type of

linear

to t h e

constraint can be

I t may be r e q u i r e d to m a i n t a i n the pressure

the

network

above

a

fixed

minimum

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

total

but

is

successive

to

known.

losses a n d c o s t s a r e

analysis.

technique,

variables

mains

render

number of

lengths.

points

inequal i t y of

is

each p i p e of

The h e a d

respective pipe

a

resort

diameter

used

main pipe, length of

variable.

at

is

between

trunk

branch

loss,

technique

branch or

without design

to

each

head

optimization

b e u s e d f o r o p t i m i z i n g t h e d e s i g n of p i p e

loops

used in

flow,

following

and

closed be

powerful

relationship

length

of

pipe

of

a

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

or

certain

( a linear

within a certain diameter

pipe available,

may

be

etc.

SIMPLEX METHOD FOR TRUNK M A I N DIAMETERS

The points. mm,

following The

and

variables, different

of

example

permissible the'

X1,

second

X2,

diameters.

manual comparison of

Xj

concerns

diameters leg, and

This

of

200

X4

a

trunk the

and

simple

first

150 mm.

which

are

example

t h e cost o f a l l

main

the

leg

with

two

250 a n d 200

are

There

are

lengths

could

be

drawoff

of

thus four pipe

optimized

of

by

a l t e r n a t i v e s g i v i n g the correct

56

loss,

head

linear

but

programming

is

used here to demonstrate

the

technique.

rl H:lOrn

LENGTH m 11s

DIAMETER m m

Fig.

5.1

500 LO

The

head

X,

tr, 2 00

150

x3 0 0

4 00

X2 L 50 3.2

50

a12

losses

LOO

2 00

250

Least-cost

1w5m

f

FLOW

UNKNOWN LENGTH m OPTIMUM L E N G T H m HEAD LOSS m

14 11s

2 6 11s

RESERVOIR

XL 1.68 TOTAL 5.0

trunk main b y linear programming. per

100

mm

of

pipe

and

costs

per

m for

the

\

v a r i o u s pipes a r e i n d i c a t e d below:TABLE 5.1

P i p e D i a m e t e r s a n d Costs

250 200 150 linear

below

T a b l e 5.3

5

TABLE 5.2

the

4 3

0.1 0.42

constraints

and

cost $1 O O / ~OOm

m/IOO m

0.25 0.71

The form

@ 14 P / s

Head loss @ 40 P / s m/100 m

Diameter mm

on

the

coefficients

system of

the

are

in equation

expressed

equations

are

tabulated

( I ) . L e n g t h s a r e e x p r e s s e d i n h u n d r e d metres.

Constraint Equations

Lengths

x1

+

= 5

x2 x3

Head Loss Objective Function:

0.25X,

+ 0.71X2 +

5X1 +

4X2

+

0.1X3 4x 3

+

x4

+ 0.42X4

= 4 =

5

3X4 = m i n .

in

57

The c o m p u t a t i o n s p r o c e e d b y s e t t i n g a l l r e a l v a r i a b l e s t o zero, it

is

necessary

to

equation to s a t i s f y a,

b

and

very

c

high

and

the

the

amount

by

m.

are

To

initiate

assigned

of

the

the

the

program

was

whether

by

+

(0.25

values

the column

entries

i s worthwhile

other

which value

that

in

any

number

is

for

each

column b y

column

is calculated

as

in

the

cost

reduction

the

coefficient

of

the

per u n i t (or negative opportunity

lowest

opportunity

maximum

introduced,

value

amount

calculate

of

The

v a l u e ) . The

The k e y c o l u m n i s t h a t

O n l y one v a r i a b l e m a y b e i n t r o d u c e d a t a time. the

variable.

s i n c e i t shows t h e

shows

be

i.e.

coefficients of

column

case).

determine

value,

i n t h e second c o l u m n a n d s u b t r a c t i n g t h e t o t a l

the

i s now d e s i g n a t e d t h e k e y c o l u m n .

may

variable

by multiplying

t h e c o r r e s p o n d i n g cost

which

which

be

T h u s one

known

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

p r o f i t a b l e v a r i a b l e to i n t r o d u c e w o u l d b e X2,

To

5

I f one u n i t o f X 1

each column.

most

2

from

cost

would

variable.

formed

column

and

t h e n t h e cost w o u l d i n c r e a s e b y ( 5 - ( 1 x m ) - ( 0 +

the p r o g r a m v a r i a b l e

greatest

4

5,

u n i t s of c.

thus

X

the slack

which

replacing a

variable,

i s calculated for

,,m ))

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

it

any

number

introduced,

-

solution,

variable

program

i n t r o d u c i n g o n e u n i t of

determine

opportunity

m)

so

each

variables are designated

u n i t of X 1 w o u l d d i s p l a c e 1 u n i t o f a a n d 0.25 To

into

in a n y p a r t i c u l a r l i n e of the m a i n body of the t a b l e

The n u m b e r s

displaced

The s l a c k

variables

(see t h e t h i r d c o l u m n of T a b l e 5 . 3 ( 1 ) .

respectively

indicate

c

slack

and t h e i r cost c o e f f i c i e n t s a r e s e t a t

5.3(1),

designated

b

a,

artificial

the e q u a l i t y .

Table

in

values

variables

introduce

( i n t h e cost

the

key

minimization

column

the replacement

variable

r a t i o s f o r each

row as follows:Divide

the

corresponding replacement

amount

of

ratio

is

the

in

number

program variable

the

selected

key

as

for

column.

that

each row b y

The

lowest

i s t h e m a x i m u m amount

c o u l d b e i n t r o d u c e d w i t h o u t v i o l a t i n g a n y of the c o n s t r a i n t s . with

the

lowest

positive

a n d the number a t

the

replacement

the

positive which

The r o w

r a t i o i s designated the key row

intersection of

the key column a n d key row,

the k e y number. After (Table

introducing 5.3(1l))

so

a

that

new the

variable, replacement

the

matrix

ratios

is

remain

rearranged correct.

The

p r o g r a m v a r i a b l e a n d i t s cost c o e f f i c i e n t in t h e k e y r o w a r e r e p l a c e d

58 Limear p r o g r a m m i n g s o l u t i o n of p i p e problem

TABLE 5.3

I r

Variable Cost coef. 1

Prcg. Cost Amt. X, Var. Coef. 5

x2

x3

4

4

a

1

b C

r n m m

5 4 5

WPORTUN 1lY VALUE :

1

5-1.2%

b

c

rn

m

m

0.71

0.10

4-1.71rn

4-l.lm

3-1.42m 0

RepI.

ratio 5 / 1 ’*

1 1 0.42

1 0.25

x4 3

a

1

0

m

1

5/.71

0

*key

KEY C O L M

3 C

4 rn rn

5

x1

x2

5

4

1

1

4 1.45

row

x3 4

a

b

c

m

m r n

1 1

-0.46

x4 3

0.1

1

0.42

m

1

-0.71

1

4 3.458

\

1+0.46m

0

4-l.lm

3-1.4h

1.71rn-4

0

x2

x3

b

c

4

x4 3

a

4

rn

m

m

Ill

2 x4

4 rn

3

5 0.55 3.45

1 1.1 -1 , l

1 1

0

3.28-0.76m 0

x1

x2 4

x3 4

1

-0.69 0.69

IV

x2 x1

x4

4

4.5

5

0.5

3

4

1 0

0

5

1

0.76 0.24

1.1-l.lm

5

0

x4 3

-2.38 0.5*& 2.38 -

1.69 -1 -69 t.1-0.6%

3.38-8.2

a

b

rn

m r n

2.16 -2.16

1 1.52

1

1

0.31

0

(NO FURTHER IWROVEhENT POSS l WE )

rn-

c

m

m

59

by

the new v a r i a b l e a n d

i t s cost coefficient.

The amount column a s

well as the body of the 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

in a non-key

number

row,

s u b t r a c t the corresponding

number

i n the key row m u l t i p l i e d b y the r a t i o of the o l d row number

in

key

the

column

divided

by

the key number.

The new tableau

is

g i v e n as Table 5.3(1 I ) . The ratios

procedure and

of

studying

revising

the

table

negative o p p o r t u n i t y value. positive

opportunity

(indicated by

opportunity is

repeated u n t i l

I n the example,

values

so

values

the

and

there

i s no f u r t h e r

( I V ) shows a l l

Table 5.3

least-cost

replacement

solution

is

at

hand

the c u r r e n t program v a r i a b l e s a n d t h e i r corresponding

values). The

reader

programming technique.

should

(e.g.

There

refer

Dantzig,

are

to

a

standard

1963)

for

a

textbook

full

on

linear

description

many other cases a few of

of

the

which o n l y can be

men t ioned be low :4 If

the

and

constraints

not

just

are

of

the

equations,

< = (less-than-or-equal-

slack

variables

coefficients a r e introduced

i n t o the 1.h.s.

make

artificial

them

equations.

The

with

to)

type

zero

cost

of each constraint to

slack

variables with

high

cost coefficients a r e then omitted. I f the c o n s t r a i n t s a r e of the > = (greater-than-orintroduce into

the

with

zero

artificial i.h.s.

of

cost

slack

variables

the c o n s t r a i n t

coefficients from

with

equal-to)

h i g h cost

a n d s u b t r a c t slack each

inequality

to

type,

coefficients variables make

them

equations. If

the

value

objective with

function

function

the

is

to

highest be

is

to

be minimized,

negative

maximized,

value

the

is

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

selected,

opportunity

but

value

if

the

with

the

of

the

highest p o s i t i v e v a l u e i s selected. Note a l l v a r i a b l e s are assumed to be positive. The

opportunity

corresponding

values

variables

represent

i .e.

they

shadow

values

indicate

the

value

of

i n t r o d u c i n g one u n i t of that v a r i a b l e i n t o the program. I f two replacement r a t i o s a r e equal, the when

amount the

of

program

matrix

is

variable

rearranged.

whichever row i s selected,

i n the other Merely

row w i l l

assume

it

to

be zero have a

60 v e r y small

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

NETWORK DESIGN

Most n e t w o r k s c a n b e s i m p l i f i e d t o a t r e e - l i k e design

flows.

network pipe

and

legs

The

most

economic

loops a r e p u r e l y

can

be

made

up

for

backup.

of

lengths

diameters w i t h costs a n d head as

a

function

of

flows

network

is

network with known in

fact

In tree-like of

a

tree-like

networks the

commercially

available

losses p e r m e t r e o r k i l o m e t r e i n s e r t e d

beforehand.

The

r a n g e of

diameters

can

be

l i m i t e d b y experience. Example

-

Determine the

least-cost

p i p e diameters f o r the network

i I l u s t r a t e d below

x1 P i p e b o r e , mm e 3 5 0 H e a d l o s s 2.5m/km cost R1 O O / m

x2 300 5.5m/km R85/r @

x3 250 18m/km R70h

e

H=50m 0

Total length 2000m 100 e / s (Sol-992m)

( 100811

800m 2 0 t / s

-

I

x4 @ 300 3.8m/km R85/m x5 250 9m/ km R70/m

e

F i g . 5.2

I

(0) X6 @ 200 2m/km R50/m

( 8001111 x7 Q 150 8m/km R40/m

1O O O m 1

P l a n o f t r a n c h network f o r example

Denote t h e u n k n o w n l e n g t h s o f i n d i v i d u a l s e c t i o n s a s X.

H=20m

61

I n algebraic

form the constraints a r e :

Head losses a l o n g each route:

0.0025X1+0.0055X2+0.018X3+0.0038X4+0.009X5(50 0.0025X1+0.0055X2+0.018X3+0.002X6+0.008X7~30 Lengths:

Xl+X2+X3

=

x4+x5

=

2000 1000

X6+X7

=

800

Objective f u n c t i o n : Minimize 1OOX1+85X2+7OX3+85X4+7OX5+5OX6+4OX7

62

lee

85

88 8 5 88 48 88 8 88 8 Be

ee

7 8 80 8 ee 8 80

OPTIMAL SiiLIUTION ~ f i ~ 1 5 AFTEP ,

78 88 5 0 80 8 88 8 08

SURPLIJS VARIABLES ARE SUBTRACTED FROM THE LEFT S I D E OF > = INEQUALITIES SLACK VARIABLES ARE ADDED TO THE LEFT S I D E O F <= I N E Q U A L I T I E S

PIPE

5 ITERATIONS

'*!APIHBLE SLACK 1

';$LyiM

TABLEAU AFTEF' 9 98 - 81 8 Bb'r 8 88 17.48

8 . 88 .81 -1.8B .81

- 24

8.88

8.88

- 44 1688.88

8 88

8.88 8 . 88

,'

64

8.88 8 8 8 . 88

11 - 4 8 15.88

8.80 -91

.ha

-256888.88

L3

1 . 88 8.80 1.68

8.88 2.80 1200. 8Ei -49.68

17.48 1311 11 B0 .98

38 88 2008 88 1088 8 0 880 08

42 41 4290 9 2933 3 2375 81

O B J FLINC COEFF RANGItiG B A S I S VAPIHRLES

UHR x3 :1. 2 X5 :.:7

LObJEP LIMIT 6 4 17 7 8 68 88 UNBND

O B J FNC 'sJFiLUE 7 @ 08

IJPPEI LIMI 85 0 ,

85 88

96 8 -

70 08 48 88

85 6( 42 8i

OBJ FllNC COEFF RAHGING NON-BASIS VARIABLES

6.88 8 .68

8.80 8.88 1 -88

L2

1 . BE1

8.88 .4 :3 -88.88

8 . 88 18 8 8 . 88

H2 L1

- 4.a 5 8 ,8B - .E.4

1.24 8.08 8.88 1.44 992.88

1 .8rl 8.88

A R T I F I C I A L 'JAPIABLES ARE ADDED TO THE LEFT S I D E OF E Q U A L I T I E S 8 > = I N E Q U A L I T I E S TO GENERATE AN I N I T I A L BASIC FEASIBLE SOLUTION

VAR X1 x4 X6

LONER LIMIT 88.68

70.80 47.28

O B J FNC VALUE 100.98 85.00 58.08

UPPEF LIMIl UNBNC UNBNC IJNBNC

W I T H I N THE L I M I T S Y O l l M A Y CHANGE THE VALUE OF ANY ONE CONSTRAINT RHS OR O B J FUNC COEFF WITHOUT CAIJSING 'JRRIABLES TO ENTER OR LEAVE THE SOLCfTION BUT, VALUES FOP THE OBJ FUNC: AND SOLUTION VARIABLES M A Y CHANGE LdHEN T H I S I S DONE

63 LOOPED NETWORKS - LP OPTIMIZATION

When p i p e loops a r e created b y connecting pipes a t more than one place, and

cannot

reduced

simplified.

The

-

one

a

If,

network

network being

is

to

supplying

any

to

to the best

in

however,

the

fact

identify

i n pipe transport,

close approximation

linear

l i n e a r programming methods.

tree-like

problem

scale

pipe

to

least-cost

the

economy of only

by

because the flows a r e unknown.

be

network

the problem description becomes non-

be solved d i r e c t l y

is largely can

i n t o the system

That

the network

problem

is

again

i n v a r i a b l y a tree-like

the

tree.

Because of

the

the most economic layout i s w i t h

point.

If

t h i s i s accepted,

then a

( l e a s t cost) t r e e - l i k e network can be

obtained b y l i n e a r programming as follows. Starting

i,

pipe the

with

define

following

a

looped

arbitrarily (linear)

each p i p e i :

network the

number

each

p o s i t i v e flow

constraints

in

node j ,

directions,

terms

of

the

and

each

then set

up

Q,

in

unknown,

\

For flow balance at each node j , ZQ.

into node j

q . ( d r a w o f f from node j )

=

J Objective f u n c t i o n : Minimize

Z QiLi

where L . i s the known length o f p i p e i. This

will

times flow

bulk this

transport,

will

not

i s more

likely

to

be

a more accurate b u t non-linear Separable

employed

to

engineering sented

a

i.e.

be

the

litres

per

the optimum f o r

second

non-linear

r e l a t i o n s h i p s since economy of scale i s not introduced.

p i p e cost

Z LiQim.

that

the

Strictly

rate-cost

Each that

minimize

metres.

programming

optimize

such

judgement

manual

proposed

should

method of

here,

a

and

proportional

(m
so

objective f u n c t i o n would b e M i n

methods

problem

and

(Hadley,

but

generally

obtaining Powell

Qm

to

a

the

suffice.

1964)

approach Bhave

could

be

here p l u s

(1978),

pre-

s i m i l a r optimum network Barnes

(1982)

proposed

to an

a l ternat i ve h i e r a r c h ica I method.

L I NEAR PROGRAMM ING PROGRAM

The

appended

network,

the

program network

i s suitable 'layout'

for

optimizing

both

a n d the p i p e diameters.

minimizes the objective function,

stages of

a

The program

a n d supplies the coefficients f o r a l l

64

dummy

variables

and

artificial

variables

required

for

the

simplex

method. The p r o g r a m a

branched

input

for

i s f o l l o w e d b y an e x a m p l e .

system

each

and

section

then

the

follows

optimum program v a r i a b l e s ,

pipe

and

A n e t w o r k i s r e d u c e d to

diameters

the

optimal

are

selected.

solution,

The

namely

t h e i r magnitudes a n d costs.

Symbols i n linear p r o g r a m m i n g o p t i m i z a t i o n p r o g r a m b y m i n i m i z a t i o n

12 J J2

Objective coefficient o f v a r i a b l e in p r o g r a m Objective coefficient o f v a r i a b l e in p r o g r a m Net cost B - ZX/A Min E Row no. Key I Column no. Key J

N N1 M M1 M2 M3 R(I) R2

T o t a l no. o f v e W i a b l e s p l u s d u m m y s No. v a r i a b l e s Total columns No. o f < = c o n s t r a i n t s No. o f = constraints No. o f > = c o n s t r a i n t s Rep I acemen t r a t i o Min replacement r a t i o

V(I) X(J, X1 ( J Z(I)

V a r i a b l e no. i n p r o g r a m M a t r i x coefficient M a t r i x coefficient M a g n i t u d e of v a r i a b l e i n p r o g r a m M a g n i t u d e of v a r i a b l e i n p r o g r a m

A(I) B(J) 82

E(J) E2

I

Zl(l

Note i n p u t n u m b e r s s h o u l d b e b e t w e e n 0.001

F i r s t Problem:

S U P PlY Reserv o i r

Node

F i g . 5.3

and 1000

Network Layout

P i p e ( 1 1 400111A n s w e r Q=(280)e/s

(2001

65 Constraints:

node 2:

8 , - Q2 - Q 3 = 0

node 4 :

Q4 - Q

node 3: node 5:

M in i m i z e

5

2'

+

Q3

+ Q5 -

'6

= o =

Q6 =

200 80

E QL=4OOQ +300Q +500Q3+500Q4+450Q5+700Q6

1

2

2

.7 R H S 4 '7

66 Second P r o b l e m :

P i D e Sizina

280 e / s ,

Q,L =

200 P / s , 3 0 0 m

400m

A

/

V a r i a b l e X1

*

x2

D i a , mm 400 Grad m/km 9 C o s t $ / m 100

350

350

x 5' 300

9

20

/x 4

?3 300

17 80

(01

(400 1

(300 1

80 e / s 500 m

Fig.

5.4

Constraints;

Head l o s s t o 3:

5': Length Xl-X3:

.009X1+.017X2+.04X3+.009X4+.02X5~50

. 0 0 9 X 1 + . 01 7 X 2 + . 0 4 X 3 + . 009X 6 + . 0 3 X 7 ~ 7 0

Xl+X2+X3 = 400

x4-x5:

x4+x5

= 300

X6-X7:

X6+X7

= 500

67

I N P U T COEFS I N CONSTFAINTS I N O R DEH I,=..=,>= COEFS IN CONSTRAINT 1 X l ?

.889 X 2 ? .017 X 3 ? .a4 x 4 ?

:.;7? . 8: COEFS 114 CONSTRAINT 3 X

i

?

1

x 1 x 1 x

2 ? 3 ? 4 '7

8 X S ?

L P OPTIP1Zt.I P I P E D I H S 'JAR I ABLE .- MHGb4 I TLICIE s. COST COEF

68 L i n e a r programming program

10

zij

550 EZ=@ =.-J 8 M FUR J = l T O N 578 I F EcJ)>=E2 THEN 658 538 E 2 = E I J >

538 J2=J

600 618 620 624

626 630

G 48 656 668

NEXT J I F E 2 > = 8 THEN 938 F O R 1 = 1 TO M I F Z i I > > @ THEN 638 Z r 3 TFEN 658 X<J2,1)=.@@0OBBI R=Z NEXT I

676 R2=999999939999 G8El FOR 1=1 T @ M 698 I F R < I > < = 8 THEN 738 7 8 B I F R ( I > > = R 2 THEN 730 718 R 2 = R < I ) 726 I 2 = I 738 NEXT I 748 FOR 1=1 TO M 758 Z l I I > = Z < I > 768 FOR J=l TO N 779 X l I J , I ) = X ( J , Ii 7B8 NEXT J 798 NEXT I 888 FOR 1=1 TO M 818 Z < I > = Z l ( I ~ ~ - Z l < I 2 > * X 1 ~ . - II2> . ', X

--

-r

.:. 3 b-j 348 3 5 t3 368 328

378 388 338 488

418 428

4.38 448

458 FOR .J=l TO N l 468 D I S P " O B J . C n E F 478 IHPIJT BilJS 486 NEXT J 498 FOR .J=1 T @ N 588 E i J .:-=B r J j 510 F O R 1=1 T O M 528 E < J > =E C I. i -'A ( J , 538 NEXT I 548 NEXT J

1 ,:Jz. I.?) 8 2 0 FCIR J = l TO t.1 S30 Xi.!, I > = X l ' J . . I i - X l ~ ~ ~ J , I ~ i ~ : ~ i r . t 2 .- Ij / X 1 :t J Z .. 12 ) 8 4 0 NEXT J 858 NEXT I 868 Z I I ~ ) = Z ~ I I ~ ? I X ~ 1~ 2. ) J ~ , 878 FOR J=l TO N 8 8 8 X r J ? I ; ' j = X l C J , I 2 . > / X l * : J 2 . , 12) 8 9 8 NEXT J 98 8 H 1: I 2 Z =B r; .J2 > 518 U C I Z > = J 2 928 GOTO 4 9 8

938 BZ=8

3 4 8 FOR 1=1 TO M

558 BE=B2+H I I> 1:zc I

>

968 NEXT. I 378 P R I N T " L P O P T I M Z N " i N f 988 P R I N T " V A R I A B L E , MAGt.(ITUDE, C@ST COEF" 998 FOR 1 = 1 TO M 1988 P R I N T U S I N G 1818 i V C I ? , Z < I j , A< I :I 1818 IMHGE DODDnD,@D@@@DD.DDD,DD DDDD . DUD I 0 2 0 NEXT I 1836 P R I N T U S I N G 1835 i BZ 1035 I M A G E " C @ S T = " , @@@DDDDR.@DCI 1 0 4 0 STOP 1858 END

69 REFERENCES Bhave, P.R., A u g . 1978. O p t i m i z a t i o n of s i n g l e s o u r c e n e t w o r k s , P r o c . ASCE, 104 ( E E 4 ) p799-814. 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 ton U n i v . Press, Princeton. H a d l e y , G., 1964. N o n l i n e a r and D y n a m i c P r o g r a m m i n g , A d d i s o n Wesley, R e a d i n g . P o w e l l , W.F. and B a r n e s , J.W., Jan., 1982. O b t a i n i n g l a y o u t of w a t e r d i s t r i b u t i o n systems. P r o c . ASCE., 108 ( H Y 1 ) .

70

CHAPTER 6

DYNAMIC AND NON-LINEAR

PROGRAMMING FOR LOOPED NETWORKS

2 to 4 described

Chapters

methods

p i p e networks w i t h o r without network fixed

layout

and

drawoffs

design

a

or

new

necessary

to

diameters,

inputs at

network

compare

demands

acceptable. diameters

and If

for

number

pipe

be

repeated

and

i s at

for

could

corresponding be

A

the

necessary of

calculated. it

to

be

layout

possible

layout would be try

alternative

i s repeated u n t i l

a

and e r r o r process would

layout.

Each

of

the

so d e r i v e d would then have to be costed a n d that

networks

in

To

would

proposed

to

flows

This t r i a l

another

flows

flows were j u s t s u f f i c i e n t to

satisfactory,

analysis

the

particular pipe

drawoffs,

possibilities.

be

hand.

pattern

nodes

of

would

sizes

For any

flow

i f corresponding

it

calculating

certain

pressures were

not,

s a t i s f a c t o r y solution then

the

meet

for

loops.

various

to

a

would be analysed a n d meet

closed

final

network

w i t h least cost selected.

A

technique

without and

positive

networks between and

of

recourse

trial

technique

with pipe

determining

to

closed

and is

least-cost would

possible

for

loops.

diameters,

the

error,

The

flows,

optimization cases

are

confined each

There

are

techniques discussed

a can

and

of

number be

used

described

is

losses

most r o u t i n e mathematical optimization

relationships.

directly, No d i r e c t

general

problem

head

network

be desirable.

that

and

to

optimize

below.

The

layouts cases

To

optimize

a

known.

relationship linear

techniques r e q u i r e l i n e a r

networks f o r

is

of

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

tree-like

branch

the

costs i s not

to s i n g l e mains o r

random search techniques o r

optimization

network

which with

successive approximation

and

are

these

normally

the flow

closed

in

loops,

techniques a r e

needed. Mathematical

optimization

a n a l y s i s techniques design

techniques

techniques

(again

mathematical tecniques a

( w h i c h i s an not a

analysis name

optimization

technique

such

are

also

known

as

systems

incorrect nomenclature as they a r e techniques),

not

really

techniques

include simulation

selection

searching.

techniques

will

or

operations

descriptive). be

retained

research

The

name

here.

Such

( o r mathematical model l i n g ) coupled w i t h as

steepest

path

ascent

or

random

71 The

direct

which

is

optimization

useful

for

methods

optimizing

t r a n s p o r t a t i o n progamming, demands,

I inear

and

which

include

a

dynamic

series

of

i s useful f o r

programming,

for

programming,

events

or

things,

a l l o c a t i n g sources to

i n e q u a l i t i e s , ( v a n der

Veen,

1967 and Dantzig,

1963.) L i n e a r programming u s u a l l y r e q u i r e s the use

of

but

a

computer,

there

are

standard

optimization

programs

available.

DYNAM I C PROGRAMM I NG FOR OPT I M I Z I NG COMPOUND P I PES

One of can

the simplest optimization techniques,

normally

be

programming. selecting

an

The

along

instance,

its

consider

from a r e s e r v o i r . drawoff

length a

in

fact

a n d indeed one which computers,

only

technique a

a

may

compound

depending

trunk

to

is

dynamic

systematic

way

of

from a series of events a n d does not

The

of

diameters

recourse

is

program

mathematics.

economic

diameter

without

technique

optimum

i n v o l v e any most

used

main

be

pipe

on

used

which

pressures

supplying

a

to

select

may

and

number

the

in

vary

flows.

of

For

consumers

The diameters of the t r u n k main may be reduced as The problem i s to select the most

takes place a l o n g the line.

economic diameter f o r each section of pipe.

A simple example demonstrates the

pipeline

neither

should

profile at of

Figure

any

independent of

the

the use of the technique.

Two

hydraulic

point.

each section of

mm diameter

account

6.1.

consumers

grade

water

Consider from

the

line

drop

below

the

pipe

The elevations of each p o i n t a n d the lengths

pipe are

indicated.

pipe i s $0.1

The cost of

per

( I n t h i s case the cost i s assumed to be

per m of pipe. of

the

pressure

such

a

variation).

head, The

although analysis

downstream end of the p i p e ( p o i n t A ) . will

draw

a n d the head a t each drawoff p o i n t i s not to drop below 5

pipeline,

m,

in

be w i t h minimum r e s i d u a l

it

is

will

simple

to

be s t a r t e d

take

at

the

The most economic arrangement

head i.e.

5 m, a t p o i n t A.

The head,

H, at p o i n t B may be a n y t h i n g between 13 m a n d 31 m above the datum,

but

to

simplify

the

analysis,

we

will

only

consider

three

possible heads w i t h 5 m increments between them a t p o i n t s B and C. The each of

diameter

D of

the

pipe

between

A

and

B,

corresponding

to

the three allowed heads may be determined from a head loss

chart and

i s i n d i c a t e d in Table 6.1

( 1 ) a l o n g w i t h the corresponding

72

Fig.

=

OIA

ANSWER

6.1

260nm

3 LO mm

310m

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

.

cost

We

will

also consider only

three possible heads a t

point

C.

The

n u m b e r o f p o s s i b l e h y d r a u l i c g r a d e l i n e s between B a n d C i s 3 x 3 = 9,

but

one

of

disregarded.

these

is

grade

an

adverse

gradient

diameter

required for

of

The cost

of

case

there

is

one

an

only

which

pipe.

000

(from

minimum

asterisk.

there

110 e / s i s 310 mm ( f r o m F i g u r e 1 . 3 ) . x 310 x 1 000 = $31 000.

Table 6.1(1)).

total

It

is

cost

this

need b e r e c a l l e d

I n t h i s example,

and

a n d the

Now

m u s t b e a d d e d t h e cost o f t h e p i p e between A a n d B,

$60

with

be

i f HB = 13 a n d

Thus

f r o m C t o B i s 0.006

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

to t h i s cost this

flow

may

so

i s presented f o r each

l i n e between B a n d C.

HCE = 19 t h e n t h e h y d r a u l i c a

gradient

( I t ) a set of f i g u r e s

I n T a b l e 6.1

possible hydraulic

at

is

only

cost when

the next

one

of

For

pipe and

each

between the

possible

A

and

corresponding

proceeding to the

in

head

C,

H

C

marked

diameters

next section of

s e c t i o n between C a n d D i s t h e l a s t

possible

head

at

D,

namely

the

reservoir

level. In

Table

6.1

(I1I )

the

hydraulic

gradients

and

corresponding

diameters a n d costs f o r Section C - D a r e indicated. pipe

for

this

arrangement the

least

possible

260,

310

section

up t o C.

total total and

are This

added

the

costs

340

is mm

To t h e costs o f

the

optimum

i s d o n e f o r e a c h p o s s i b l e h e a d a t C,

c o s t s e l e c t e d f r o m T a b l e 6.1 cost

of

Sections

A

-

and

( I I I ) . Thus the minimum

$151 000 a n d t h e most economic for

pipe

8, B

-

C

diameters a r e and

C

-

D

73 TABLE 6 . 1

I

Dynamic programming o p t i m i z a t i o n

HEAD HYDR. /AT B /GRAD.

I_

DIA.

24

60000

9 1000":

87000

.001

$

I

62000

91000

153ooo

.0035 340

68000

83000

l?A-tiEi* 29

.oo

1

430

I

430

86000

1 60000 85000

I

43000 50000

COST

24

I

I

I 1

19

29

60000

I

Ill

a compound p i p e

1

19

I

of

I

.006

310

i l 31000

50000

74 respectively.

in

diameters selected

It

may

which

be

desirable

case

the

to

nearest

keep

pipes

standard

to

standard

diameter

could

f o r each section as the c a l c u l a t i o n s proceed o r each

be

length

could be made up of two sections; one w i t h the next l a r g e r s t a n d a r d diameter a n d one w i t h the next smaller s t a n d a r d diameter, the same total head loss as the theoretical

but with

result.

Of course many more sections of p i p e could be considered a n d the accuracy

would

each section.

A

booster

case

be

The cost of

cost

and

the pipes could be v a r i e d w i t h pressures.

could b e considered a t

station

pump

its

increased b y considering more possible heads a t

capitalized

power

A computer may prove useful

tables.

considered,

and

there

are

cost

in which

any point,

should

be added

in

the

i f many p o s s i b i l i t i e s a r e to be

standard

dynamic

programming

programs

available. It

will

be

seen

that

reduces the number of least-cost

the

technique

of

dynamic

programming

p o s s i b i l i t i e s to be considered b y selecting the

arrangement

at

each

step.

(1969)

Kally,

and

Buras

and

(1969) describe a p p l i c a t i o n s of the technique to s i m i l a r a n d

Schweig,

other oroblems.

TRANSPORTATION

PROGRAMMING

FOR

LEAST-COST

OF

ALLOCATION

RE SOURCES

Transportation does

not

programming

require

primarily

for

the

use

allocating

of

a

the

of consumers such that

cost

delivering

of

the

to

technique

sizes. pattern friction

It

is

a

of

is

probably use, an

technique

number

least-cost

which

system

of

normally i s of

sources

use to

i s achieved.

a

The

resource along each route should be l i n e a r l y

of

is

The

a

the throughput a l o n g that

through head

the

technique

computer.

yield

number

proportional

another

is

of

no

use

in

however,

existing small

in

pipe

in

route a n d f o r selecting

selecting

a

distribution

comparison

with

t h i s reason

the optimum p i p e least-cost

system, static

pumping

provided head,

or

the for

o b t a i n i n g a p l a n n i n g guide before demands a r e accurately known.

An

example

serves

to

there a r e two sources of N.

A a n d B could

require

10

and

15

illustrate water,

deliver O/s

the

technique.

I n t h i s example,

A a n d B, and two consumers, M a n d

12 a n d 20 O/s

respectively.

respectively

Thus

there

is

and M and N a

surplus

of

75 water.

The cost of p u m p i n g a l o n g r o u t e s A -

M,

A

-

N,

B

-

M and 6

- N a r e 5,7, 6 a n d 9 c/l 000 l i t r e s r e s p e c t i v e l y .

comuwu

N

M

REQUIREMENT

10 LIS

SOURCE YIEU)

12 LfS

15 L I S

A

F i g . 6.2 Least-cost 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 p r o g r a m m i n g example. TABLE 6.2 T r a n s p o r t a t i o n programming - o p t i m i z a t i o n of a n a l l o c a t i o n system

(1) SOURCE YIELD

CONSUMER:

M

REQUIREMENTS:

10

N

SURPLUS

EVALUATION

7

NUMBER: 0

A

-2

12

0 B

p

7

5

15

(11) A

B

12

20

2

/7

20

EVALUATION NUMBER:

0

-2

7 7

0

9

2

12 3

10

4

7

-2

76 The

data

are

set

in

out

tabular

form

for

solution

in

Table

6.2( I ) . Each row represents a source a n d each column a demand. u n i t cost of corner of

delivery

the corresponding

make an a r b i t r a r y that

each

block

yield

of

the

satisfies bottom

first

a n d demand

column i s 2,

row

table

making

block AM.

which

Starting with

possible

maximum

an

initial

methodically would

Proceeding to the next column,

namely column N.

since

(even i f

So the next block

Proceed through tke

possible assignment

until

be

the

allocation

a

is

least-cost

most

at

to the slack

Assign

the

0 to

value

each

stage

column).

until

Thus the

then the 7 in the t h i r d

the sum of

evaluation

i s equal

number

a n d so on.

numbers this

sum

pay

to

is

the

bigger

introduce

easy

to

see

each

evaluation

cost

than

a

sum

from

The biggest possible r a t e of difference biggest

between

and

introduce put

occupied

an

in

in

amount

block

BM

i n our

stems

the cost

of

each

into

from

the

the

any occupied f o r row

B

is

sum

the block, block.

method of of

If

i t would

This

i s not

determining

occupied

and

blocks.

the cost coefficient.

The

improvement would be to

The maximum amount by

block.

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

case the o n l y ,

i s determined

for

unoccupied

coefficients

improvement

i n t o Block BM.

blocks as corners

relative

the row a n d column e v a l u a t i o n

cost coefficient of

the e v a l u a t i o n

fact

a

f o r column N i s 7,

coefficient

but

To decide which

assign

the cost coefficient

resource a l l o c a t i o n

immediately,

rearranged

1 a n d work out the other e v a l u a t i o n

to

the

emerges.

are

the row e v a l u a t i o n number a n d column

Now w r i t e the sum of

beneath

figures

arrangement,

The v a l u e f o r column M i s 5,

block.

the

a n d column as follows:-

row

numbers such t h a t

made

distribution

profitable

e v a l u a t i o n number to each row

be

i n the

.

Once

2,

left This

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

B,

a r e allocated

i s to

10.

is

is written

s a t i s f i e s the y i e l d of row A.

i s i n row

step

the top

allocation

next a l l o c a t i o n i s the 13 in the second row, co I umn

The f i r s t

resources i n such a manner

i s satisfied.

maximum

i s completed,

the

resources

i n the table.

column M and the amount

demand of

to be considered

all

the

l e f t corner of

the f i r s t

block

i n i t i a l assignment of

table,

the

The

along each r o u t e i s i n d i c a t e d i n the top r i g h t

drawing

a closed

(see the dotted c i r c u i t

which can loop u s i n g

in Table 6 . 2 ( 1 ) ) .

77 Now f o r each u n i t which to

be

subtracted

from

block

this

case

AM

from to

the

keep

the

amount 10 to

the

maximum

evacuate block AM.

by

i s added to

block

in

BN,

block BM, one u n i t would have

added

yields

to block AN a n d subtracted

and

allocation

to

requirements consistent.

BM

i.e.

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

the block a t each corner of

satisfy

10,

is

y i e l d s and requirements.

since

this

In

would

10 i s made,

and

the closed loop adjusted Only one re-

distribution

of resources should be done a t a time. After

making

the best new a l l o c a t i o n ,

re-calculate

i n Table 6.2

number and e v a l u a t i o n sums as

the evaluation

( I I ) . Allocate resource

to the most p r o f i t a b l e block and repeat the r e - d i s t r i b u t i o n procedure until fact

there that

i s no f u r t h e r

there

possible cost improvement,

indicated b y the

i s no e v a l u a t i o n sum greater than the cost coefficient

i n any

block.

I n our example we a r r i v e d a t the optimum d i s t r i b u t i o n

in

steps,

but

two

more compl icated

p a t t e r n s i n v o l v i n g more sources

a n d consumers may need many more attempts. The

example

transportation are

dealt

can

only

programming.

with

in

serve There

textbooks

to are

on

introduce many

the

subject

optimization techniques such as v a n d e r Veen and

t h i s example o n l y

serves

as

an

other

the

subject

conditions of

mathematical

(1967)and Dantzig ( 1963)

introduction.

For

instance,

two blocks i n the t a b l e happened to be evacuated simultaneously, of

the

'e'

blocks could

say.

be

allocated

Computations then

a

proceed

very as

of

which

small

quantity

before and

if one

denoted b y

the q u a n t i t y

'el

disregarded at the end.

STEEPEST PATH ASCENT TECHNIQUE FOR EXTENDING NETWORKS

A

'steepest

path

ascent

technique'

p i p e l i n e networks a t minimum cost

can

be

(Stephenson,

used

for

extending

1970). The technique

i s p r i m a r i l y f o r a d d i n g new pipes to e x i s t i n g networks when demands exceed the capacity of the e x i s t i n g p i p e networks. It

i s u s u a l l y possible to s u p p l y p o i n t s i n a system along v a r i o u s

routes o r from a l t e r n a t i v e sources.

I t may not be necessary to l a y a

new

the

pipeline

all

the

way

from

source

to

the

demand,

or

a l t e r n a t i v e l y the diameter of the new p i p e may v a r y from one section to another.

The fact that elevation,

a n d hence pressure head,

a l o n g a p i p e l i n e route causes d i f f e r e n t p i p e diameters a t d i f f e r e n t sections.

varies

to be optimal

78 O n account o f t h e c o m p l e x i t y of t h e o p t i m i z a t i o n t e c h n i q u e , a

computer

be

supplied

computer

i s essential. to

Alternative

routes a l o n g which

t h e node in q u e s t i o n a r e pre-selected

program

is

used

to

determine

the

optimum

use of

water

could

manually.

The

pipelines

and

c o r r e s p o n d i n g d i a m e t e r s t o meet t h e s p e c i f i c demands.

An

informal

demonstration

that

the

method

d e s i g n i s g i v e n w i t h t h e a i d of d i a g r a m s .

R e l a t i o n s h i p between d i s c h a r g e a n d F i g . 6.3 cost of two p i p e s .

Fig.

6.4

Steepest p a t h p r o j e c t e d o n t o

C,-C2 p l a n e

yields

an

optimum

79

Pipel ine thickness.

costs Wall

thickness

increasing

usually

increases

per metre w i l l

so the cost

diameter,

with

increase

diameter in

proportion

be a f u n c t i o n of

wal I

and to

the

the square of

the diameter. Now the

for

pipe

pipe

is

not

demand, be

a g i v e n head g r a d i e n t ,

diameter to a power of laid

the

entire

route

i s p r o p o r t i o n a l to

2.5.

from

However,

the

b u t merely reinforces p a r t of the network,

limited b y

may

along

the discharge

approximately

be

source

that

discharge

varies

section to a power g r e a t e r than u n i t y ,

with

the

to

the

the capacity w i l l

the c a p a c i t y of the remainder of the network.

deduced

if

cost

of

a

Hence i t new

pipe

b u t i s l i m i t e d b y the capacity

of the remainder of the p i p e network. If

more than one proposed new p i p e i s i n v o l v e d the r e l a t i o n s h i p

between

discharge

i I l u s t r a t e s the

and

pipe

relationship

costs

between

i s mu1ti-dimensional. discharge

at

a

F i g u r e 6.3

particular

node

a n d cost of two possible pipes in the network. The

curves

on

C1

the

-

C2

plane

in

F i g u r e 6.3

are

lines of

constant

discharge Q.

The shortest p a t h between two Q l i n e s spaced

a small

distance a p a r t

is a

l i n e p e r p e n d i c u l a r to

i s the p a t h w i t h the steepest discharge/cost sought. in

the Q lines.

gradient,

This

a n d i s the one

The procedure i s therefore to s t a r t at the o r i g i n and proceed

increments

perpendicular

on to

the

-

C1

the next

Q

C2

line,

plane, until

each

increment

being

the desired discharge Q i s

attained.

To

determine

increments step

on

in

the

t r i a n g l e XYZ,

the

cost,

C1

-

increments

in

=

aQ ac2

to

increment i n cost of each p i p e f o r a

C2

has

plane

to

e n l a r g e d in F i g u r e 6 . 4 .

-/](*Q/

corresponding

the a c t u a l

increment YW i n €I a r e to be determined.

Now cos 0

diameter

-$;'+

( AQ/

aQ -I2 acl

be

calculated.

C1 and

C

2

Consider

the

corresponding to

80

AC2

Similarly

cos 0

YW

AC, =

ac1

+

YW c o s

=

aQ cos20

AQf

=

L

for n possible pipes,

In a s i m i l a r manner,

The any

rate

a

increase of

a Q/aC

pipe,

without

of

small

analyzing

d i a m e t e r D..The

increase

in

cost

by

in diameter,

per

by

respect

in

increase

discharge

determined

with

increment

divided

in

the

is

discharge

i t may be p r o v e d t h a t

gives

step,

the

pipe

the network increase

i

pre-selected

so t h a t

with

and

in discharge,

associated

required relationship.

is

AQ,

of

C. of

t o t h e cost

The the

with

the

increment

increase

cost of e a c h p i p e i s y i e l d e d b y t h e a b o v e e q u a t i o n a t e a c h step. corresponding known pipes

increases

diameter/ are

cost

increased

diameter

in

relationships. steps

in

until

are

then

calculated

in

The

from

the

The

diameters

of

the

proposed

the

discharge

at

the

specified

node i s s u f f i c i e n t .

A network a n a l y s i s should be performed a f t e r each

s t e p to r e - b a l a n c e

t h e system.

It will local

be observed

maxima

charge/cost

will

be

curves

are

from

F i g u r e 6.3

reached generally

with

that the

concave

it

i s unlikely that any

technique upwards,

as and

the

dis-

have

few

points of inflection. So f a r only

the

t e c h n i q u e h a s been u s e d to s u p p l e m e n t

one n o d e a t

a

time.

I t h a s been f o u n d t h a t

the supply

t h e p i p e l i n e system

should be well conditioned f o r s a t i s f a c t o r y convergence. necessary

to

initialize

the

diameters

of

proposed

to

pipes

It is usually at

a

value

81

greater

than

diameter).

zero

( s a y of

Otherwise

equations

is

the

unrealistic

1/4 of

the order of linear

for

the

the a n t i c i p a t e d f i n a l

approximation initial

steps

to

and

the

differential

false

r e s u l t s are

yielded.

DESIGN O F LOOPED NETWORKS

It

was

explained

previously

pipe

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

and

error

or

sizes

and

be

it

i s not possible to design a

loops without

approximations.

fact

that

that

pipe

flow

The

should

the solution f o r

the

pose problems.

least-cost

flow/head

and directions

conform

l a r g e r than specified minimum sizes, all

recourse to t r i a l

non-ljnear

magnitudes

diameters

pressures a r e r e q u i r e d ,

approaches to which,

the

unknown,

minimum

w i t h closed

successive

loss r e l a t i o n s h i p , initially

that

to

are

standard

a n d that c e r t a i n There are many

looped network,

none of

i t should be noted, overcome a l l the problems a n d ensure that

a t r u e least-cost solution,

and not a local peak

i s at

a r e nevertheless

hand.

The solutions

than a network

which

i n the cost function,

i n v a r i a b l y more economic

i s designed b y s t a n d a r d methods,

and offer a

s t a r t i n g system f o r m a n i p u l a t i o n b y the design engineer. Some

techniques

proposed

for

achieving

least-cost

solutions,

together w i t h t h e i r l i m i t a t i o n s , a r e o u t l i n e d below.

( i ) Loop/node correct ion method

A

method of

least-cost

design,

which

does

not

from the f a m i l i a r

methods of Hardy Cross a n a l y s i s ,

the

optimization

author.

non-l i n e a r engineers. node

and

The

programming Instead f o r each

successive loop

procedure

techniques cost

is

which

not

radically

was developed b y

based on

are

unfamiliar

revisions are

i n the network

depart

linear to

or

most

performed f o r each

u s i n g a correction based on

the d i f f e r e n t i a l of the cost function determined as follows:Assume any p i p e cost C = a

D bt

(6.2)

where a a n d b a r e constants. Now

the

diameter,

D,

can

be expressed

head loss h f o r a n y p i p e : D=(KtQn/h)'/m where K , m a n d n a r e constants So C = a(KtQn/h) b/mt

in

terms of flow

Q and

82 Different i a t i n g , dC= ( nb/m) ( C/Q) dQ-( b/m ) ( C / h ) d h the cost of any p i p e can be v a r i e d in two ways:

i.e. flow,

(6.3) by varying

and b y v ary i ng h while

Q , keeping the head loss, h , constant,

m a i n t a i n i n g Q constant.

Actually,

both factors must be considered in

designing

network.

The

a

least-cost

fact

that

the

and

diameters

corresponding costs are functions of the two independent v a r i a b l e s i s often overlooked i n mathematical optimization models. The complete optimization

procedure f o r a network i s therefore as

follows: (1)

Assume

a

pipe

layout

and

assume

any

reasonable

initial

diameter f o r each pipe.

(2)

Analyse

the

determine

network

in

flows

using, each

say,

pipe

the

and

Hardy

heads

Cross method,

at

each

node.

to Any

number of constant head r e s e r v o i r s a n d drawoffs i s permitted.

(3)

c a l c u l a t e the sum of dC/dQ = (nb/m) C/Q

For each loop in t u r n , for

each

pipe

direction

around

otherwise

the

increases

flow

in

ZdC/dQ loop.

loop.

the

loop

negative

cost

the

the

in

if

the

positive if

it

cdC/dQ would

around

the

assumed value

positive

of

dC/dQ,

i s positive,

pay loop.

to

i.e.

reduce

the

Conversely

if

i t would p a y to increase the flow around the

Subtract

add

or

an

increment ZdC/dQ,

losses constant.

The

maximum size of

would reduce any flow specified

minimum

half

this

to zero, size.

say

loop,

repeating t h i s analysis. decreasing

should

be

value,

An

this,

order of

in

flow

or

reduce a n y

increment

is

loop

that

which

p i p e diameter to

slightly

i s preferable.

in

the

to keep the head

increment

smaller

Proceed from

than

loop to

I t i s p r e f e r a b l e to proceed i n the

ZdC/dQ,

absolute v a l u e of ranked

around

and decrease o r increase the

each p i p e in the loop respectively

diameter of

loops

in

i s negative,

depending on the s i g n of

a

the

Now

increases, direction

is

Q

take

value.

flow

positive

If

order

before

which means the making

the

flow

correct ions.

(4)

For

each

node

in

turn

other

than

c a l c u l a t e the sum of dC/h = ( b / m ) ( C / h ) the node.

fixed-head

reservoirs,

f o r each p i p e connecting

Take the p o s i t i v e v a l u e i f the head drops towards the

node i n question a n d the negative v a l u e i f the head drops away

83 from

the

node.

H,

head,

increase

at the

If

CdC/dh

the

node,

head.

By

is

positive

and

if

it

increasing

it

is

pays

negative,

the head a t

reduce

to

it

pays

the node,

have to be increased in diameter

the

reduced

pipes

pipe

leading

away

Conversely a decrease leading

leading sible

to

away.

to

the

directions

or

a

the

head

limit

the

head

cost

maintain

diameters

without

diameters

the

node

above the specified minimum. preferably

increase the

in

pipe

of

to

the

of

pipes

in head permis-

maximum change

decrease

reducing

The

diameter

and

i n head w i l l decrease diameters of

and

Determine

produce

minimums.

node

in

to

pipes

l e a d i n g to the node w i l l

flows.

the

a l t e r i n g any flow

to

less

than

should

also

be

specified

maintained

Vary the head correspondingly,

change

to,

say,

half

the

or

maximum

permissible a n d c a l c u l a t e the new p i p e diameters connecting the node.

(5)

Repeat

steps

No f u r t h e r

discernable. the

initial

flow

unbalanced. been

3 and 4 u n t i l

established

important

that

in cost

step 2 has been achieved i t

however they

the

improvement

is

network analyses a r e necessary as once

balance of

Notice

no f u r t h e r

that

once

cannot

initially

be

the

flow

i s not

directions

altered.

It

is

have

therefore

assumed diameters a r e r e a l i s t i c and

that the corresponding flow p a t t e r n i s g e n e r a l l y correct. The will or

technique

will

yield

non-standard

pipe

diameters and these

have to be corrected b y assuming the nearest s t a n d a r d p i p e size

by

size

l e t t i n g each p i p e comprise two sections, one the next s t a n d a r d

greater

diameter section equal

and

yielded is

by

indicated by

other

the

next

the a n a l y s i s .

calculated

that

performed

the

computer

from by as

the the

standard

less

T h e corresponding

fact

that

analysis.

they

size

are

the

The

lengthy

total

the

length of each head

calculations and

than

loss must should

definitely

not

be as

simple as those f o r a Hardy Cross network a n a l y s i s . (ii)

Flow correction b y l i n e a r programming I n the previous section i t was demonstrated that f o r each p i p e dC

is

linearly

proportional

to

dQ

increments in Q a n d H a r e s m a l l ) .

and

dH.

(This

is

provided

the

I f the objective f u n c t i o n i s taken

as the minimization of CdC f o r each p i p e , the problem may be set up

84 as a

The I inear c o n s t r a i n t s

l i n e a r programming optimization problem.

would be: ZdQ i n = 0 where dQ could be p o s i t i v e or negative,

For each node, and H 'a

specified minimum.

The objective

function

C dC

is

minimum,

=

where dC

is a

linear

function of increments i n flows i n each p i p e and heads a t each node.

A

standard

changes

in

initial

flow

network

meters head

linear

could are

programming program could be used to select along

is

each

then

confined

very

to

the I inear

analysis

will

required.

network

analysis

reduction

should

i n the total

the l i n e a r

and

head

analysed.

be calculated.

unbalanced a f t e r be

pipe and

assumed

Unless the

small

values,

The be

linear

network cost.

core

storage

required

whereas

it

node once

corresponding

increments i n flow

the

head

losses

until

and

will

be

and a network

there

is

no

and

further

The subprogramme f o r s e t t i n g up

i s complicated and a

is

an

dia-

programming optimization

iterated

core storage i s r e q u i r e d f o r reasonably The

each

programming optimization

programming tableau

number of pipes,

at The

the

l a r g e computer

l a r g e networks.

proportional

i s proportional

to

to

the

the

square

linear

of

the

number of

pipes f o r the Loop/Node Correction Method. Non-standard

diameters

are

yielded,

and the flow

directions

are

not a l t e r e d once an i n i t i a l assumption i s made.

( ii i )

As

Non-l i n e a r programming

the

problem

of

s t a n d a r d or "canned" be

used.

ascent

Many

of

technique.

constraints

and

expressed as

design

of

pipe

network

programs

A sub-program objective

are

based

on

non-linear

the

steepest

a

path

would be r e q u i r e d to formulate function.

The

constraints

could

the be

l i n e a r functions b u t the objective f u n c t i o n i s n o n l i n e a r .

T h e c o n s t r a i n t s are:For each node C Q For fixed-head

is

non-l inear programming computer program could

these

the

a

in = 0

nodes H = a specified value.

For v a r i a b l e head nodes,

HL

a specified minimum.

For each loop C h = 0. The objective f u n c t i o n i s CC = Z a =

(Ke

Qn/h) ( 1 /m)

minimum

85 Non-standard assumed fairly

diameters

beforehand.

complicated,

As it

are

yielded

and

the technique of may

be

flow

d i r e c t i o n s must be

n o n l i n e a r programming

difficult

to

debug

the

similar

to

program

is if

e r r o r s occur.

(iv)

Optimum length method

(1971)

Kally

proposed

a

method

programming method of optimization of

very

the

linear

tree-I i k e networks w i t h known

flows. Since the flows

are

looped network b y balance

flows

likely

to r e - d i s t r i b u t e a f t e r optimization of a

t h i s method,

at

each

node,

Hardy Cross a n a l y s i s i s necessary to after

which

a

further

optimization

is

performed, a n d so on. An

initial

calculates diameter

estimation of

flows

by

along

changes

at

a

and

linear,

the

and

calculated

by

Hardy

Cross

portion

various

head change

diameter

nodes

length

are of

optimum

linear

of

i s fed

analysis.

any

enlarged of

programming.

For

pipe,

calculated.

lengths

i n t o the program, a

the

small

which

change

corresponding

in

head

The r e l a t i o n s h i p s between

( o r reduced) pipes i s assumed each

section

Diameters

of

may

new

be

diameter

confined

to

s t a n d a r d sizes.

(v)

Equivalent p i p e method

(1973)

Deb,

predefined

replaces a l l length

and

pipes

in a

equivalent

layout b y p i p e s w i t h a common

( i .e.

diameters

such

that

head

losses remain unaffected). An

initial

adjusting loop

is

a

flow

pattern

is

assumed

p i p e sizes f o r successive loops. minimum at

and

corrected

in

steps

by

The total p i p e cost f o r any

some flow extreme i.e.

w i t h the flow

i n some

pipes i n the loop equal to the specified minimum.

A constraint may

l i m i t i n g the minimum r a t e of

flow through each p i p e

be imposed ( f o r r e l i a b i l i t y a n d c o n t i n u i t y of

b u r s t s or

supply

i n case of

blockage closed loops a n d specified minimum flows i n pipes

are u s u a l l y r e q u i r e d ) .

A which

minimum will

cost

if

b r i n g the

assumed loop cost

for

each

loop

to the assumed

a n d a flow figure

correction

i s calculated.

86

If

the

minimum

figure flows

is are

assumed within

flow

and

constraint then

the

permissible

is

a

limits,

and f l o w s c o r r e c t e d again.

assumed,

violated,

a

new

minimum

flows corrected accordingly. slightly

This

lower

loop

i s repeated u n t i l

cost

If the cost

is

the cost

cannot b e reduced a n y more w i t h o u t v i o l a t i n g the c o n s t r a i n t s . The

technique

dependent

on

the

yields

non-standard

initial

flow

diameters,

assumptions.

The

and

is

equations

highly involved

tend to obscure the i n i t i a l assumptions.

REFERENCES

Appleyard, J.R., Aug. 1975. D i s c u s s i o n o n l e a s t cost d e s i g n o f b r a n c h e d p i p e n e t w o r k systems. P r o c . Am. SOC. C i v i l E n g s . , 101, (EE4). B u r a s , N. and S c h w e i g , Z . , Sept. 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 , P r o c . Am. SOC. C i v i l Engs., 95 (HY5). 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. Deb, A.K., J a n . 1973. L e a s t - c o s t p i p e n e t w o r k d e r i v a t i o n , W a t e r and W a t e r E n g g . , 77 ( 9 2 8 ) . Kally, E., April 1971. A u t o m a t i c p l a n n i n g of t h e l e a s t c o s t w a t e r d i s t r i b u t i o n n e t w o r k , W a t e r and W a t e r E n g g . , 75 (8902). Kally, E., March 1969, P i p e l i n e p l a n n i n g b y dynamic computer p r o g r a m m i n g , J. Am. W a t e r W o r k s Assn., (3). Lam, C.F., June, 1973. Discrete gradient optimization of water s y s t e m s , Proc., Am., SOC. C i v i l E n g r s . , 99 ( H Y 6 ) . Stephenson, D., 1970. O p e r a t i o n r e s e a r c h t e c h n i q u e s f o r p l a n n i n g and operation of pipeline systems, Proc. P i p e l i n e Engg. Convn., B r i n t e x E x h i b s . , London. Van 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 o f O p e r a t i o n a l Research, C l e a v e r Hume, L o n d o n .

87 CHAPTER 7

CONT I NUOUS S I MULAT I ON

The

previous

chapters

have

p i p e networks w i t h constant unsteady column tions

flow

considered.

decelerations the

As

There

are

first

estimated on

c o m p r e s s i b i l i t y of

steady

approximation

that

the water

are the

many

unsteady

compressibi l i t y

flow

nor

basis.

flow

in

the

water

I n subsequent

and the p i p e i s consider-

ber

of

tem.

hours,

The

the

is

tions

can

tion

an

is

the

order

to

changes

It this

in

situation

stored

which An

i n a r e s e r v o i r over a num-

in a r e t i c u l a t i o n sys-

flows

be

refilled by

to

term

but

variations

volume

which

pumping

the steady flow

appreciable error.

loss

head the

in

significant.

when

example d u r i n g the n i g h t s and weekends.

flow

consequently

frequently in

rates for a and

the

the

That

continuity

is,

is

water

level the

equa-

pipe f ri c -

also

i n storage levels.

heads on

time

order

water

interval

applied

Corresponding

will

i n the reservoirs

reticulation

coninui ty general

terms

can

reservoir

makes

be

hours

considerably.

be

equation

analyse

determine

to

based on

would

to

r e t i c u l a t i o n system.

time

program it

necessary

elasticity

computational puter

for

however, are

system

will

These i n t u r n w i l l affect t h e discharges. is

ployed

subsequently

a p p l i e d without major

and

basis

tion

lower,

observe

to

change.

will

unsteady

be

in

change

is

storage

instance d u r i n g peak

reservoir

demand

This

for

conditions,

accelerations

example i s the slow depletion of

a

a

to

I n subsequent chapters

The l a t t e r i s termed water hammer.

neither

the

confined

i s assumed to b e incompressible and the accelera-

i n a pipe

and

chapters ed.

is

been

r e s e r v o i r heads.

the

purpose

pipe

simulation

instead

and

of

hammer

for

pumping

easier

which

equations

could

used

studying

and

reduces

global

com-

be em-

I n t h i s chapter are

on

the accelera-

so-called

equations

program

systems

and

much

seconds

the

unjustified.

friction

simulation

capacities

The omission of

Although

water

inefficient and

the

reticulation

the

to develop fluctuations

i n storage in a complex p i p e water supply system. Most water components, be

a

supply

one

a

systems can be assumed to comprise two major

link

or

pipe

and

pipe junction or a reservoir.

the

other

a

node which could

The h a n d l i n g of

the node would

88 depend voir at

on

the

would

the

description

be

other

. a g e volume

regarded

extremity which

in

of

as a node of the

turn

A

i t s operation.

outflow

constant

head reser-

i n f i n i t e surface area whereas

could be a f u n c t i o n of

controls

the

head on

the

the stor-

discharge o r i -

fice. The

simulation

design o f niques is

would

that

tions

approach

a water

it

be

can

can

frequently

r e t i c u l a t i o n system complicated.

too

incorporate

and does not

be

used

to optimize the

where d i r e c t optimization

Another

non-linear

advantage

of

tech-

simulation

equations and specified func-

have to be based on average o r

assumed steady

state conditions.

SYSTEMS ANALYSIS TECHNIQUES AND THE USE O F SIMULATION MODELS

Developments i n operation research have led to numerous extremely

powerful

systems

analysis

techniques.

These

techniques

can

be

b r o a d l y c l a s s i f i e d i n t o two main categories: 1)

Direct optimization

2)

Simulation techniques. Direct

solution cation

optimization certain

to

of

techniques

problems.

transportation

I inear

gramming, are

techniques

estimated

raw

be

programming,

programming

for

can

Grosman

and

water,

used

the

describes

extended

optimum

the

appli-

transportation

separable

conveyancing

find

to

(1981)

programming.

and

proCosts

desalination.

The

techniques a r e used to c a l c u l a t e the average flows from each source to

each

and

demand

quantity

point.

The

flows

satisfy

constraints and result

water

minimum

i n the minimum

total

quality

cost solu-

t ion.

The tions.

techniques Average

used

flows

i n many

and

studies assume steady

constant

all

the sources a n d demand points.

ed

which

result

i n a steady between demand ferent of

sources

pollutants

depending

on

several

i t may not in

in

are

Real

water

assumed

at

are calculat-

systems are never

Water r e q u i r e d at demand p o i n t s g e n e r a l l y v a r i e s

and

periods

quality

The average flows

i n the minimum cost.

state.

zero

water

state condi-

the some

what

hundred

optimum sources

source

litres

a

second.

be possible to draw ratio varies

was

used

determined. throughout and

the

During

peak

water

from the d i f -

The

concentration

the

day

proportion

and

week

of

clean

89 water.

It

can

deterministic quality tee

may

that

not

the

solution

be

concluded

models

is

which

that

be r e a l i s t i c .

constraints

practical

optimum

assume

solutions

flows

and

The

'optimum

be

satisfied

at

that

solution

will

or,

steady

indeed,

solution'

the

derived

from

constant

water

cannot guaran-

all

times, is

that

an

the

optimal

one a t a l l . Simulation components the

in

provides

of

a

and

find

to

way. it

is

ing

of ways

thus

is

observing

varying

sought.

relationships

among

to

make

them

work

not

yield

an

necessary

simulate

to

the

behaviour

No

conditions.

The

does

objective

is

components

together optimal

in the

the

an

system

directly

order

in

gain

best possible

solution

iteratively

the

'solution' to

of

of

and

achieve

to

E v e n w h e n c o m b i n e d w i t h e f f i c i e n t t e c h n i q u e s f o r select-

the values of

effort

under

sense

the

Simulation

an optimum.

means o f

system

mathematical

understanding

a

may

lead

each decision a

to

solution

variable,

which

a n enormous c o m p u t a t i o n a l

is still

f a r from

t h e b e s t pos-

sible. To ly

i t s credit,

non-linear

techniques

simulation

are

seldom

non-linearities

which

el.

can

Simulation

solutions'

and to

or

no cost,

time o r

be

can

be

planning.

identified b y tain

A

tions. the

the

risk

is

and

are

direct

and and

their

for

generally

used

real

as

an

and can

be

decision

the

determined by

than

Little

quantity

most

and

making

parameters

visualised

may

The t i m e s c a l e

effects of

studying

convincing

it

optimum.

to

mod-

'optimum

techniques

global

aid

and

simulation

alternative

and

be

optimization

complexities

simulation.

term

variables

can

the

into a

with

with

values

more

a

short

relationships model

all

direct

optimization

involved

important,

changing

with

The

incorporated

search

long

determined

simulation

results

deal

experiment

to

with

and

More

parameters

constraints.

to

used

narrow

controlled

quality and

able

be

possible

and

are easily

together

be

can

c a n b e u s e d to s o l v e m o d e l s w i t h h i g h -

relationships

can

effects. from

be

Cer-

simula-

people

and

those o b t a i n e d f r o m

d e t e r m in i st i c a p p r o a c h e s .

to

The

r a t e of

time

can

differential at The

each

change of

always

be

equations.

iteration

solution

to

in a

the

water

described

quantity by

a

and quality,

set

of

first

with

order

respect

ordinary

These e q u a t i o n s c a n b e s o l v e d s i m u l t a n e o u s l y simulation set

of

using powerful

equations

yields

numerical

the

volume

methods.

of

water

90 in each storage component of tion

time-step.

These

operating

rules

settings,

make-up

the model depends

and

values various

flows,

the

can

be

the end of

used

relationships

in to

validity

of

each simula-

conjunction

determine

demands, overflows etc.

represents the r e a l on

the system at

with

pump

the

on/off

The degree to which

system a n d the accuracy of the r e s u l t s the

model

and

the

accuracy

of

the

solution of the set of equations.

A general simulation program has been w r i t t e n which can be used to of

simulate water models. first-order

ordinary equations

consisting of j

The model must be described b y a system differential and

equations,

involving q variables,

Such

a

model,

can be w r i t t e n

i n the general form: dx.

-- 1 dt

dxl,dx2, -

Q(t,x1,x2,---x q ' dt

where i = 1 , 2

---

dx. J ) dt ---d t

j

According to James (1978), the o r d e r l y

procedure f o r constructing

simulation models i s : Systems Analysis:

the s a l i e n t components,

inter-actions,

relation-

ships a n d dynamic mechanism of a system a r e i d e n t i f i e d . Systems Synthesis:

the model i s constructed and coded i n accord-

ance w i t h Step 1 ) . Verification which

:

would

the

model's

be expected

responses

if

are

compared

the model's s t r u c t u r e

with

those

was prepared

as intended. V a l i dation: -

the responses from the v e r i f i e d model a r e compared to

corresponding

observations o f ,

a n d measurements from the a c t u a l

system. Inference:

experiments w i t h ,

a n d comparisons of

responses from,

the v e r i f i e d a n d v a l i d a t e d model - t h i s i s the design stage.

MATHEMAT I C A L MODELL I NG O F WATER QUALI T Y

Model I ing Concepts

A f i e l d to which many of

the present concepts can be a p p l i e d i s

that of water qua1 i t y d e t e r i o r a t i o n i n i n d u s t r i a l systems. washing

Cooling and

systems a r e examples where q u a l i t y w i l l deteriorate i n time.

91 I t i s not easy

to p r e d i c t

equi I i b r i u m

the

the r a t e of

concentrations

in

build-up

water

of

dissolved s a l t s o r

reticulation

systems,

even

w i t h an u n d e r s t a n d i n g of the o r i g i n s a n d methods of concentration of salts.

This

is

because

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

of

the

complex

One way of

nature

of

industrial

water

accounting f o r a l l these effects i n

a r e a l system appears to be b y modelling the system on a computer. Once improve

a

model

is

produced

the operation

of

and

existing

validated,

tives of

be

used

to

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

I t i s one of the objec-

and f o r o p t i m i z i n g the design of new systems.

which

i t may

service water

a research programme to produce such a mathematical model

will

be

formulated

in

general

terms

for

adaptation

to

any

p a r t i c u l a r system. The b u i l d - u p cally in

i m p u r i t i e s in water can b e simulated mathemati-

together w i t h the water

conduits

can

of

be

or

in vapour

calculated.

The flows of water

r e c i r c u l a t i o n cycle.

form

in a n d out of the system

i n the a i r

The processes of evaporation,

l u t i o n a n d make-up

condensation,

pol-

can a l l be modelled.

Mass Ba I ances For the purposes of mathematical simulation of water systems, system

must

be

described

can be described

express

the

numerically.

terms

i n terms of

solved a n a l y t i c a l l y . to

in

of

equations.

One-stage

the

systems

a mass balance equation which can be

I n other more complex s i t u a t i o n s i t i s necessary

equations

Different

in

types

finite of

difference

models a n d

form

the

and

solve

assumptions

them

therein

are described below. Parameters whereby pol l u t i o n i s measured may e i t h e r be conservat i v e o r non-conservative.

I n a conservative system i n p u t to any p a r t

of the system equals outflow. flow

then

Similarly

evaporation if

the

there i s no reaction, The model

will

parameter

Thus, be is

i f the parameter studied i s water

neglected a

chemical

in

a

conservative

compound

it

model.

i s assumed

deposition o r solution i n a conservative model.

may be steady-state o r time-varying.

D u r i n g the s t a r t -

up p e r i o d of a mine as concentrations b u i l d up the system i s s a i d to be is,

unsteady. in

the

After case

dissolved solids

a w h i l e the system may reach e q u i l i b r i u m . of

salts

in

solution,

the

increases

in

That

mass

of

i n the system due to leaching o r evaporation equals

92

t h e loss b y p u m p i n g o r d e p o s i t i o n .

Mixed a n d Plug Flow Systems

In

a

plug-flow

system,

the

water

is

the pipes a n d d r a i n s a t a c e r t a i n r a t e , rate.

The s a l t s c o n t e n t

series of point. be

steps

In a

the

same

at

can

m i x e d system, point. the

system

salt

input

by

the

the

system.

mass o f

Real

completely

mixed,

turbulence

and

therefore

which

input

systems as

exhibit

will

there

cross

will

is

so

connections.

through

be affected

assumed

that

divided

by

gradual

probably be

travel

impurities a t that

in a

concentrations arrives a t that

T h i s s i m p l i f i e d mechanism

systems

concentrations.

An

to

salts w i l l

the concentration of

every

increases

describe

point

through

water to

any

as water with different

completely

instantaneously

in

at

assumed conveying

to

the the

total

i s often rates

spread

concentration volume

of

satisfactory

of

in

change

b e between p l u g f l o w a n d

diffusion

I n general

and

mixing

due

to

s a l t s a r e conveyed

by

advection (lateral transport) and dispersion.

Examples

i l l u s t r a t i o n o f t h e u s e of t h e mass b a l a n c e e q u a t i o n s

The s i m p l e s t

is f o r a steady-state concentration

in

system.

mg/P.

Q i s f l o w r a t e i n P / s o r MP/d,

Inflow

of

water

and

of

salts

per

C i s the unit

time

equals outflow rate:

Fig.

7.1

Point

Q2 Q I C l + Q2CZ Q1

..

Node

=

Q3

(7.2)

=

Q3C3

(7.3)

+

c3

= QlC1+Q2C2

+Q2

(7.4)

93

e.g.

= 5 MP/d,

i f Q,

c, c3

then

400 mg/e,

=

200 mg/@

and t h e t o t a -

mass o f

Q3C3

A completely tions:

Q2

=

=

-

C2 =

(water flow r a t e )

00 mg/e

(salt

concentration)

d scharged p e r day

salt

15 x

0 Me/d

200 = 3 0 0 0 k g / d .

(7.5)

m i x e d system c a n b e d e s c r i b e d b y d i f f e r e n t i a l e q u a -

Subscript

i

refers

to

inflow,

e

to

exit,

s

to

initial

condi-

tions.

Volume S Conc. C F i g . 7.2 QiCi

M i x e d f l o w node = QeC

= Q

.’.

dt

Fig.

=

7.3

C

+

S

D i f f u s e node

S en

-

dt

SdC QiCi-QeC

fact that C = C

=

dt

+ S-dC f o r c o n s t a n t

Integrating

t

d(SC) -

and

evaluating

the

constant

of

integration

from

the

a t t = 0:

QiCi-Q C (QiCi-QeC s ,

(7.9)

‘e Q i C i - QiCi/Qe- Cs or C = Qet/S ‘e e

(7.10)

94

at t = 0, C = C

e.g.

c

=

a n d at t = - ,

5’

S = 0,

o r Qe = - o r

(ai/ae)ci

Observe

that

if

Qi

g a i n s o r losses, e.g.

does

not

equal

Be,

there

must

be

internal

due to evaporation.

The previous example could be studied n u m e r i c a l l y . Although t h i s requires

specific

numbers,

it

is

often

the

only

practical

way

of

s o l v i n g more complex problems. Assume S = 1000 m 3 , Qi = 1 m 3 / s = Qe, Choose solution,

At the

=

100 s.

accuracy

computations.

It

The

of

must

choice of

C s = 0, C i = 500 m g / t . At

can

affect

r e s u l t s and the numerical

be

determined

by

trial,

the speed of

s t a b i l i t y of

from

the

experience o r

from theoret ica I cons i d e r a t ions. NOWQ . C . I

..

Q C = S

-

c 2 -c 1

-

At + S

C2 = C1

(7.11)

At

I

Q.(C.-C l

l

l

)

= C

The c o m p u t a t i o n s can b e s e t

t

500-C,

cl

0 100 200 300

1000

0 50

Equation

out

0.1(500-C1) i n tabular

~ 0 . 1

c,

95 135

50 45 40 37

50 95 135 172

326

174

17

343 m g / t

with

would the

i n d i c a t e C = 316 m g / t result

indicated by

(7.12)

form as follows:

500 450 405 365

(7.10)

comparable

+

1

a t t = lOOOs, which i s

the

numerical

solution of

343 m g / t .

Systems A n a l y s i s

A

more

sophisticated

approach

than

c r i b e d above

i s the use of

niques,

the assistance of computers

with

a l l o w an optimum design

the

simulation

method des-

systems a n a l y s i s a n d optimization

to be selected

if

necessary.

tech-

The methods

from numerous a l t e r n a t i v e s .

95

t

Plug flow

a.

C

t

b.

Completely mixed system

I

CI

C

ti

C t

D i f f u s e system

C.

Fig.

7.4

Comparison o f

p l u g f l o w and m i x e d s y s t e m s

96

The a l t e r n a t i v e s t a n d a r d engineering option

from

a few

i s to select the best

approach

selected designs.

The

i s tedious

l a t t e r approach

where there a r e many a I terna t i ves. The

design

general

optimization

configuration

variables defined

has

and

not

the

in

been

system

approach

which

the

fixed.

involves

numerical

An

overall

i s described

the

creation

v a l u e of economic

a

of

independent objective

is

i n terms of equations or con-

s t r a i n t s.

NUMER I CAL

METHODS

FOR

SOLUT I ON

THE

OF

D I FFERENT IAL

S I NGLE

EQUATI ON5

Numerical values

of

variable have the

solutions

the and

the

appear

in

the

functions

at

not

functional

ability

disadvantage

as

a

solve

to

that

various

entire

of

a

any

table

tabulation

the

relationship.

practically

the

form of

values

Numerical

equation

must

be

of

the

independent time

but

methods

they

recomputed

have if

the

i n i t i a l conditions a r e changed. If

a

certain

function interval

f(t)

can

then

it

be can

expanded about a p o i n t t = t

represented be

0'

by

a

represented b y

i.e.

power the

series

Taylor

in a series

about the i n i t i a l value: (7.13)

L e t t i n g n represent

represent

time t o a n d n + l

the previous step a t

the next step a t t +h, the series can be w r i t t e n as:

0

(7.14)

'n+1 Consider the example problem

)/'

=

dy dt

=

(7.15)

y+t

w i t h i n i t i a l conditions Y(0) = 1 This

(7.16) is

a

The a n a l y t i c a l compare

the

linear

time

solution numerical

to

variant

1st

the problem,

results

i l l u s t r a t e the e r r o r at a n y step.

of

order

differential t

y = 2e -t-1

some

of

the

equation.

w i l l be used to methods

and

to

97

The Euler Method

The

Euler

methods

method

discussed.

example problem

the

is

To

simplest

obtain

(7.15),

initial

after

a

then

be

next

time

cannot

values

time

y

increment

calculated

at

increment

easily

be

y'l,

Y l l I , etc.

this

is

Taylor

of

not

generally

problem

Neglecting a

of

could

Knowing

be e v a l u a t e d

the d e r i v a t i v e s

Derivatives

in computer

case.

The

of

arbitrary

programs.

e r r o r of

I t c a n b e shown

,

that

i .e.

Substituting

the

could

Euler

the

functions

The d e r i v a t i v e s

example

the

method

(7.15)

but

truncates

the

t h e terms a f t e r t h e f i r s t d e r i v a t i v e a n d having

to

evaluate

the

second

and

(7.17) the

order

subsequent

h2 w h i c h

terms

(7.14)

in

results

i s denoted O ( h 2 ) .

e r r o r and r e s u l t s f r o m one step o n l y ,

becomes O ( h )

the

Then

h2yni1/2 and

truncation

local

yn+l

I + 0 ( h 2) e r r o r

Yn+,=Yn+hYn

in

...,

to e v a l u a t e f o r

the

to

a n d yn+2 c o u l d b e e v a l u a t e d a f t e r

so on.

formulated

the

... must

,

i n t o the T a y l o r series (7.14).

'1

all

numerical solution I1 , y I l l yiV

'n7 'n The v a l u e s of a l l

n+l,

and

subsequent d e r i v a t i v e s .

the

n' h.

a r e easy

the

exact

I

series b y excluding

eliminates

least a c c u r a t e of

a l l the d e r i v a t i v e s y

be e v a l u a t e d a n d s u b s t i t u t e d the

an

but

i.e.

This

is

f r o m n to n + l .

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

a n e r r o r o f o r d e r h. example

(7.15)

into

the E u l e r a l g o r i t h m

(7.17)

gives: Y n+ 1= Y n + h . ( y n + t n )

The time

initial

increment

values

for

y

(7.18)

condition h=0.02 can

be

y ( O ) = l means

and

t h a t y=O a t t=O.

Choosing t h e

l e t t i n g t h e s t e p n u m b e r n = O a t t=O,

evaluated

at

successive

time

the

increments a s

f o l lows: Y1=Yo

+h(y +t

0

0

Y2=Yl+h(Yl+tl)

) = 1+0.02(1+0) =

1.0200

(7.19)

1.0200+0.02(1.0200+0.02) = 1 . 0 4 0 8

(7.20)

1 .0624

(7.21)

=

y

=y + h ( y + t ) = 1 . 0 4 0 8 + 0 . 0 2 ( 1 .040+ 0 . 0 4 ) 3 2 2 2 Y 4=

= =

1.0848

(7.22)

Y 5=

=

1.1081

(7.23)

etc.

98 The numerical solution a f t e r 5 steps i s y(0.10)=1 .lo81 whereas t y=2e -t-1 gives the exact a n a l y t i c a l solution as y(O.10)=1.1103. Hence

the

accuracy. to h, to

i.e.

gain

absolute

global

error

i s 0.0022,

i.e.

two-decimal-place

Since the g l o b a l e r r o r of the Euler method i s p r o p o r t i o n a l O(h),

the step size h must be reduced a t

four-decimal

the computational the b e g i n n i n g of

accuracy,

effort the

22-fold.

interval

y

h 0.004.

i.e.

I

Fig. . IS

7.4

This

shows

least 22would

how

the

fold

increase slope at

used to determine the function

value a t the end of the i t e r a t i o n i n the Euler method.

I

Analytical

t

F i g . 7.4 The E u l e r method

The

slope

at

the

unless the solution suffers

from

the

beginning

of

the

i s a str a i g h t line.

disadvantage

of

interval

is

always

wrong

Thus the simple Euler method

lack

of

accuracy,

requiring

an

extremely small step size.

The Modified Euler Method

Fig.

7.4

and

the

subsequent

discussion

suggest

how

the

Euler

99

method

can

b e improved w i t h

l i t t l e a d d i t i o n a l computational effort.

The a r i t h m e t i c a v e r a g e of the slopes a t of the i n t e r v a l

the b e g i n n i n g a n d the end

i s used ( o n l y the slope a t the b e g i n n i n g i s used i n

the E u l e r method). (7.24)

The

Euler

a l g o r i t h m must f i r s t b e used to predict Yn+1 n+l I c a n b e estimated. A p p l y i n g the same example (7.15) that Y 1 before a n d s u b s t i t u t i n g y = x+t i n t o (7.24) g i v e s

as

(7.25)

S u b s t i t u t i n g the E u l e r equation (7.18) = y

'n+1

+h(yn+tn)

+ (yn +h(Yn+tn

1

+

f o r yn+l

gives (7.26)

t n+ )l

2 U s i n g h=0.02 and t h e i n i t i a l

condi t i o n s : y o = l , t 0 =O (7.27)

=1+0.02

( 1 + 0 ) +(1+0.02(1+0)+0.02) 2

(7.28)

=1 . 0 2 0 4

(7.29)

y =1.0204+0.02(1.0204+0.02)+(1.0204+0.02(1.0204+0.02~+0~0~~ 2 2 (7.30) =l .0416

y5=1.1104 The Nearly

(7.31)

cf analytical

answer twice

certainly

agrees

solution

to

a s much work

not

the

22

within was

1

1.1103 in "the

done

as

in

fourth the

the

local

and

place.

E u l e r method b u t

times more t h a t would h a v e been needed w i t h

t h a t method to a t t a i n f o u r decimal p l a c e a c c u r a c y . that

decimal

global

I t c a n b e shown

e r r o r s of the M o d i f i e d E u l e r method a r e

100

O(h3)

and

respectively.

O(hZ)

The

E u l e r methods a r e o f t e n r e f e r r e d

Modified

Euler

and

the

simple

t o a s second a n d f i r s t o r d e r meth-

ods r e s p e c t i v e l y .

Runge-Ku t t a Met hods

The

Fourth-Order

Runge-Kutta

p r o v i d e the greatest accuracy development

of

t h e Second-Order behind

the

those w h i c h effort.

The

E u l e r method a s

a n d H a i n e r (1978) w h i l e G e r a l d (1980) d e r i v e s

Runge-Kutta

methods.

h.

amongst

p e r u n i t of computational

a

All

first

algorithm and explains

the

Runge-Kutta

estimate.

A

weighted

methods

the p r i n c i p l e s use

the

simple

I m p r o v e d e s t i m a t e s a r e t h e n made

u s i n g previous estimates a n d different interval

are

the method i s a l g e b r a i c a l l y complicated a n d i s g i v e n

i n Stummel

completely

methods

average

of

timeall

values within

the

estimates

is

the time used

to

calculate widely

y The F o u r t h - O r d e r R u n g e - K u t t a methods a r e t h e most n+l * used because of t h e i r power a n d s i m p l i c i t y . The f o l l o w i n g i s

a particular

Fourth-Order

method w h i c h

i s commonly u s e d a n d w h i c h

i s included in the simulation program: yn+,=y

+ l (k1+2k2+2k3+k4)

(7.32)

"6 (7.33)

kl

= hf(tn,yn)

k2

= h f ( tn+ih,yn+ikl

k3

=

k4

= hf ( t n + l

)

(7.34)

hf ( t n + $ h , y n + i k 2 )

(7.35)

Again ample:

the

,yn+k3) problem

(7.36) given

in

dy/dt=f(t,y)=t+y,y(O)=l.

one step

(h=0.1)

m e n t s (h=0.02)

whereas

(7.15) This

y(O.1)

above time

is

y(O.1)

was calculated

solved is

as

an ex-

calculated

in f i v e time incre-

u s i n g t h e s i m p l e a n d m o d i f i e d E u l e r methods a b o v e .

k = h ( t n + ~ n)

=o.

in

1 (0+1)

= 0.10000

(7.37)

k2=0.1 (0.05+1 .05)

= 0.11000

(7.38)

k3=0.1(0.05+1.055)

= 0.11050

(7.39)

101

k4=0.1 (0.10+1 .1105) y (0.1 ) =1 .OOOO+-,(

1

= 0.12105

(7.40)

0.10000+2x0.11000+2x0.11050+0.12105)

(7.41)

=1 .11034 This

(7.42)

agrees

to

five

decimals

with

the

analytical

result

and

i l l u s t r a t e s a f u r t h e r g a i n i n accuracy w i t h less e f f o r t than r e q u i r e d by

the previous

than

the

Euler methods.

modified

Euler

It

method

i s computationally

because,

the f u n c t i o n a r e r e q u i r e d f o r

each step

can

the

be

many-fold

method

would

five-decimal evaluation

have

of

the

methods

In

this

approximately

50

(220/4).

local

The

algorithm

in

(7.36)

the

can

order

but

The

be

particular

O(h )

is

for

and

220

the

the

the

to

involves the

by

for

the steps

steps

of

compared

than

two,

The simple Euler

step

required

example

term

5

of

each

times more e f f i c i e n t error

than

efficiency

roughly

evaluations

rather

same accuracy.

y(O.1)

function.

function

accuracy.

for

r e q u i r e d of

accuracy of

Runge-Kutta number

larger

more e f f i c i e n t

w h i l e f o u r evaluations of

achieve only

one

Euler

and

calculating

the

same

Runge-Kutta

the

order

of

method

is

the simple Euler method

Fourth-Order

global

error

Runge-

would

Kutta

be

about

0(h4).

M u l t i s t e p Methods

The called

simple single

from the form

step

last

the

Euler,

Modified

methods

step

with

b e g i n n i n g the solution ble.

of

y

and/or

derivative

interval. use

a

a n d Runge-Kutta they

use o n l y

I n t h i s they different

methods

the

information

have the a b i l i t y

step

size

and

are

are

to per-

ideal

for

where o n l y the i n i t i a l conditions are a v a i l a -

The p r i n c i p l e b e h i n d a m u l t i s t e p method i s to u t i l i z e the past

values the

because

step computed.

next

Euler

a

Most

yl

to construct

function

and

mu1 tistep

constant

step

to

extrapolate

methods

size

h

a polynomial

have

to

this

the

make

that

approximates

i n t o the next time

disadvantage the

that

construction

of

they the

polynomial easier. Another disadvantage of multistep methods i s that several are

past p o i n t s a r e r e q u i r e d whereas o n l y the i n i t i a l conditions

available

at

the

start.

The

starting

values

are

generally

102

calculated

from

the

initial

conditions

using

a

single-step

method

such as a Runge-Kutta method.

REAL-TIME OPERAT I ON OF WATER SUPPLY SYSTEMS

Although

numerical

e v e n t u a l l y design of

models

are

can e q u a l l y well be a p p l i e d on-line

between

data

operation

of

minimization

loggers systems

of

(example

and

of

these

and

for

planning

supply

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

reservoirs

and

system they

and

can

for

water

the connection optimizing

supply

the

pipes.

Cost

be performed on a continuous

1975).

Coulbeck,

On

the

other

hand,

the methods can be used to i d e n t i f y s h o r t f a l l s

1977).

i n the system (Rao a n d Bree, The

use

f o r the operation of the system.

mini

operations

Sterling

when a p p l i e d o f f - l i n e

flow

direct

(1981) described a number of

Shamir

basis

of

the components in a water

telecommunication

of

reservoir

levels,

pipe

pressures

and

r a t e s i s r e l a t i v e l y simple whether i n analogue o r d i g i t a l form.

Although

there

rate

accuracy

the

are

problems of

adequate to cope w i t h fact

with

simple

the

methods

measurement is s t i l l

of

discharge

probably

the predicted f u t u r e demands.

more

Forecasting in

i s the most d i f f i c u l t aspect of the r e a l time simulation.

be

p a t t e r n s may possible

to

approximated

prepare

a

daily

deterministic

basis

but

the

calculations

more

cumbersome.

by

F o u r i e r methods.

and

weekly

introduction

demand

of

than

It

Demand

i s generally

pattern

probability can

on

a

makes

the

also

be

Growth

in

demand

program can

be

used

to

investigate alternative

Constraints

on

resource

included i n the simulation. The

simulation

operating water, be

methods.

minimized

sales.

availability

power o r manpower can be b u i l t into the program.

Quality

by

including

constraints

energy can

tariffs

also

be

and

income

such

as

Costs can from

water

included.

COMPUTER PROGRAM TO SIMULATE RESERVOIR LEVEL VARIATIONS IN A PIPE NETWORK

The accompanying computer

program w i l l

simulate the v a r i a t i o n s

103

in

water

level

in

reservoirs

in

addition

to

performing a

network

flow balance. The

program

2 with

Chapter 'fixed

head'

simulation

each

or,

iterations

minimal since the and o n l y to

in

this

11

in

and 1,

reservoir

network

the

additional

duration

f o r example 24 in

i s based on an

node

case, hours

be corrected a t

area of

'reservoir and

time

program i n

r e s e r v o i r f o r each

type'

increment

node.

If

the

T2 are input,

then the heads at each node and water level

will each

be

p r i n t e d out

time

for

interval

every

after

network flows a r e balanced

unbalance due to r e s e r v o i r

time-fixed

head correction

variable,

subsequent

hour.

the

first

The a c t u a l should

be

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

level changes which w i l l have

time

i n t e r v a l s . Although drawoffs a r e

i n the present program,

they could b e a l t e r e d a t pauses

i n the r u n n i n g o r inserted i n equation form. The

output,

required

namely

level

variations,

could be used to estimate

r e s e r v o i r depths ( u s i n g t r i a l r e s e r v o i r surface a r e a s ) and

i n fact to see at which r e s e r v o i r locations the storage i s most

100

e/s Res.

A

\

F i g . 7.5

System f o r continuous simulation example

H,

= 100m2 = lOOm

104

required.

Data

requirements

are

similar

to

the

analysis

program

w i t h the following additions. In

and pipe with

the

first

increment data, the

the

the reservoir supply

represent

at

after

is

pipes

areas

the

added

of

should the

name,

at

the

be

the simulation d u r a t i o n end of

from

the

up-stream

the

line.

various

reservoirs

t h e e n d of t h e p i p e d a t a l i n e s .

In the

reservoirs in

square

In order to d i s p l a y

l e v e l s i n t h e b i g g e s t r e s e r v o i r i t i s n e c e s s a r y to h a v e

pipe a

line

hours

first

surface

metres g i v e n

a

data in

from a pseudo f i x e d

pumped

supply

feeding

head, into

very the

large, actual

reservoir biggest

to

level

r e s e r v o i r in t h e d i s t r i b u t i o n system.

REFERENCES

G e r a l d , C.F., 1980. A p p l i e d N u m e r i c a l A n a l y s i s . 2 n d Ed., A d d i s o n Wesley. Grosman, D.D., 1981. O p t i m u m A l l o c a t i o n of M i n e S e r v i c e W a t e r Subject to Q u a l i t y Constraints. MSc(Eng) Dissertation, U n i v e r s i t y of the Witwatersrand. H o l t o n , M.C., 1982. A Computer f r o g r a m m e for t h e S i m u l a t i o n o f W a t e r R e t i c u l a t i o n Systems i n G o l d M i n e s . MSc ( E n g ) D i s s e r t a t i o n U n i v e r s i t y of t h e W i t w a t e r s r a n d . 1978. D e v e l o p i n g and U s i n g Computer S i m u l a t i o n M o d e l s James, W., o f H y d r o l o g i c a l Systems. C o m p u t a t i o n a l H y d r a u l i c s I n c . H a m i l t o n . and Bree, D.W., 1977. E x t e n d e d p e r i o d s i m u l a t i o n of Rao, H.S. w a t e r systems, P r o c . ASCE, 103 (HY2) p97-108. S h a m i r , U., June 1981. R e a l - t i m e c o n t r o l o f w a t e r s u p p l y systems. Proc. I n t l . Symp. o n R e a l - t i m e O p e r a t i o n o f H y d r o - s y s t e m s . U n i v . of W a t e r I oo, p555-562. Sterling, M.J.H. and C o u l b e c k , B., 1975. O p t i m i z a t i o n o f w a t e r p u m p i n g c o s t s b y h i e r a r c h i c a l methods. Proc. ICE, 59, p789-797. Stumme1,F. and H a i n e r , K., 1980. I n t r o d u c t i o n to N u m e r i c a l A n a l y s i s . S c o t t i s h Academic P r e s s .

105 Computer program for umtinwus simu lation of p i p e network w i t h raservoirr

l@

3 4 6 p=R+O < I.

28

358 S=S+Q<J>#LH}-H(LCJ,S? 360 NEXT J 378 HLI>=H+E*R/S

38 48

>

388 I F A B S < 2 t R 6 ) < = V THEN 40@ 390 V-ABSCE?R,S) ! 32 IF FT-S UNITS 480 NEXT I D I S P 'NPIPES,NNODES*NRES,ERc 4 1 8 YF V QM 4 4 8 PRINT ' N+ NT2 3 / S H2M"iTS FOR J=l TO N l ! FIXED HEADS 458 FOR J-1 TO N l YUHBERED F I R S T 468 PRINT USING 478 i V < - l > a L < J : ' , DISP mNODE1+,NODE2-.L~~D~,IN X<J>,D<J),Q<J),H,H2 488 NEXT J > 62 4 8 5 NEXT T 3 GOTO i r e 4 9 8 END INPUT K < J ) , L ( J > r X < J ) , D < J ) , H 2 ,QE,A) NETUORK NAME? H2>3088#, 15890, .8S#180 NOOE1+rNODE2-rLn,Dn,INTLH2,Q2~3~ ~ADEFORFHUS? 2 , 3 , 1 8 8 8 , .18,78~ .1~208 NODEl+,NODE2-,L~,D~~XNTLHZ,Q2r3s ,ADEFGRFHUS? 1r3,2080, . 2 * 7 8 , . 1

_...-. --*

TUPIIT I

58 G = 9 . 8 68

78

80 90 95

180 183

188 118

12e

130 148 142 145

158

21 22 23

G / F / X < J) > 2 312 NEXT J 13 IF. 11>i THEN 258 22 4 FOR 101 TO M J 231 2 315 H/A%T2%3608 2 4i3 GOTO 2 4 9 2416 H(I)=H 27 0 s=0 0 FOR J=1 TO N l 2Et 2510 I F K < J ) < > I THEN 330 8 RoR-QC J) 381 31.0 SrS+Q<J)/-H(L<J,)) 32!8 GOTO 368 0 I F L<J)C>I THEN 360 3 31

P I P E MET ANRL BY NODE HERD CORR TESTRES N+ NX<M> D 1 2 3009 ,158 2 3 ism .i8e 1 3 2888 ,286 N+ NX < H > O<W> 1 2 3808 .158 2 3 1800 ,188 1 3 2800 .208 N+ NXCH) O < M > 1 2 3808 150 2 3 1800 ,180 i 3 2009 .200 N+ HX < M > D 1 2 3800 .158 2 3 1880 .188 1 3 2800 .288

QH3/S H2H 8 013 89.3 ,841 63.2 ,859 63.2 BM3/S H2H 1 ,013 88.6 .841 63.2 .859 63.2 Q H 3 4 H2M 2 ,813 88.8 .a41 62.1 .cis9 62.1 QM3/S H2M 3 813 87.3 841 62.1 .859 62.1

106

CHAPTER 8

UNSTEADY FLOW ANALYSIS BY R I G I D COLUMN METHOD

RIGID WATER COLUMN SURGE THEORY

Transients i n closed conduits a r e n o r m a l l y classed i n t o two categories: to

slow

as

motion

surge,

mass o s c i l l a t i o n of

and

rapid

change

s t r a i n of

the f l u i d a n d conduit

mer.

slow

For

or

small

in

flow

which

changes

the f l u i d

in

which

accompanied

i s referred by

elastic

i s r e f e r r e d to as water ham-

flow

rate or

pressure the two

theories y i e l d simi t a r r e s u l t s . It

is

theory

normally

than

by

elastic

s i t u a t i o n s where simplified rigid

easier

is

it

theory,

column

bends

applied

and

across

acceleration

water

ends

throughout

the or

in

conduit

the

to

the

the

of

rigid

column

water

is

column

etc.

to a p p l y

this

must be a p p l i e d .

conduit

column

i t s length.

by

other h a n d there a r e many

expansions

of

rate

system

even dangerous

the

A

treated

is

free

pressure

produces

an

With

as to

an

move

difference

instantaneous

The basic equation r e l a t i n g the

head difference between the ends of bore

a

hammer theory

although

through

the

On

water

the

incompressible -mass, around

theory.

inaccurate

and

theory

analyse

to

the water column

change

in

velocity

is

i n a uniform derived

from

Newton's basic law of motion, a n d i s h = --L -

dv dt

g

h

where conduit

(8.1 )

is

the difference

length,

v

in head between the two ends,

i s the flow

velocity,

L i s the

g i s g r a v i t a t i o n a l accelera-

tion and t i s time. The with

equation

slow

calculating power changes

is

useful

deceleration the

trip

or

water

for calculating

of level

starting

up

a

water

variations in

a

in

a

pumping

i n a hydroelectric installation

The equation

the head r i s e associated

column.

It

may

be

surge

shaft

line,

or

for

following

power

load

fed b y a pressure p i p e l i n e .

may be solved i n steps of A t b y computer,

form or g r a p h i c a l l y .

used

i n tabular

107

Example 1

Numerical A n a l y s i s of Surge Shaft

100 m long penstock

A

w i t h a cross-sectional

area,

A t , of 1 m2

i s protected a g a i n s t water hammer b y a s u r g e s h a f t a t the t u r b i n e , with

a

cross-sectional

orifice.

The

initial

sudden

complete

area,

velocity load

of

A2,

2

m2

in the c o n d u i t

rejection

at

and

an

unrestricted

i s 1 m/s a n d there i s a

the

turbine.

Calculate

the

maximum r i s e i n w a t e r level i n the s u r g e s h a f t n e g l e c t i n g f r i c t i o n . Take

At = 1 sec.

=-0.098h.

t

Ah=0.5v

0- 1

0.5 0.476 0.428 0.359 0.272 0.172 0.064

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

The

maximum

analytical could be mean

v

F i g . 8.1

rise

A h = A,vAt/A2

=

h

A v = -ghA t/L

Av= - 0 . 0 9 8 h

V

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

0.951 0.855 0.717 0.544 0.345 0.129 -0.094

0.5 0.976 1 .404 1 .763 2.035 2.207 2.271*

i s 2.27

m,

= -9.8h/100

lv/2 = 0 . 5 ~ .

w h i c h may be compared w i t h the

s o l u t i o n of 2.26 m. The a c c u r a c y of the n u m e r i c a l method improved b y

and

respectively. losses,

Then from Equ. 8.1,

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

h

The

over

taking the

time

method c a n

smaller

time

intervals readily

to

i n t e r v a l s o r t a k i n g the calculate

be extended

A h

and

A V

to i n c l u d e head

a n d i s c a l c u l a t o r - o r i e n t a ted.

D e r i v a t i o n of r i g i d column e q u a t i o n of motion

108

DERIVATION OF BASIC EQUATION

Net b o d y f o r c e a l o n g p i p e l i n e = wAL s i n 0 = wA(h

2

-h ) 1

(8.2)

F o r c e = mass x a c c e l e r a t i o n w ( h -h -h ) A = - ( w / g ) A L d v / d t 2 1 f .*. ( h2-h 1 ) = hf - ( L / g ) d v / d t

(8.3)

(8.4)

SOLUTION OF EQUATION OF MOTION

The

equation

graphical

can

means

be

solved

(Jaeger,

1956)

analytically or

some

in

numerically

cases,

or by

(manually

or

by

computer). Only

the

and

section

simplest no

relationships derived

friction)

between

in F i g .

l e g to s t a r t .

8.2

surge can

be

velocity

algebraic

in

U-tube

of

form.

which

systems studied

(constant

analytically.

amplitude

and

Consider

as

conduit

and

That time

example

an

crossi s the can

the

be

simple

i s d i s t u r b e d b y f o r c i n g t h e l i q u i d up one

The e q u a t i o n o f motion from (8.1)

dy and s i n c e v = - d v - d 2 y dt 'dt I n t e g r a t i n g t w i c e w i t h r e s p e c t to t g i v e s y =

is

dtz

'ma,

cos( t

m

,

(8.6) where dy/dt

the =

constants

of

integration

are

0 w h e n t=O.

mean l e v e l s Y

F i g . 8.2

U-tube

from

'='ma,

at

t=O,

and

109 The

oscillations

obviously

repeat

every

2

n

m which

period. The v e l o c i t y i s d y / d t

= -y

and

-- _

"max For

with

the

case

constant

of

a

level

to

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

sin (

max

'max

t

the

(8.7) (8.8)

a

J2g/L'

c o n d u i t of

is

(8.9)

area A

a surge shaft

t

l e a d i n g from a

w i t h cross-sectional

reservoir area A

5'

is slightly different.

h

AS

Surge shaft

0

sudden valve closure 8.3

Fig.

It shaft.

is

Simple s u r g e s h a f t

customary

neglect

to

Then f o r t h e f l o w

L

-

Dynamics:

9 C o n t i n u i t y : vA

:.

t

the

i n e r t i a of the water

*

d t + Y = O

(8.10)

= Asdy/dt

(8.11)

3

(8.12)

dv = d2y dt At A d2Y g A t dt2'

dt2

... C s

=

The g e n e r a l s o l u t i o n to t h i s i s y=a cos 2nt/T+b where T = 2

I f a t t=O,

n

/

v

Then y = v

(8.13) s i n 2nt/T(8.14)

(8.15)

t

y=O t h e n a=O cos

I f a t t=O,

in the surge

in the tunnel,

v=v

0

then

fi

0

9 As

sin

2 nt

T

(8.16) (8.17)

110

A More Precise Method for Manual Numerical A n a l y s i s

The

simple

applied may

with

very

therefore

digital

time

the

the

time

and

A

method

manual

should

is

more

one

may

written

average h over

a

area

It

work

terms

in

and

is

implicit o r

in

time

interval

is

of

flow

rate

Q

Alternative-

instead

of

v.

The

a r e s o l v e d f o r h2 and Q2. A h e a d loss term,

introduced for

i s A,,

to

it

h )/2 t o c a l c u l a t e Av, a n d t h e a v e r a g e v e l o c i t y 1 A h over each t u n n e l ( v 2 + v )/2 t o c a l c u l a t e 1 i s t h e n n e c e s s a r y to s o l v e t h e two e q u a t i o n s

following equations i s also

be

appropriate

methods a r e employed

( d y n a m i c a n d c o n t i n u i t y ) s i m u l t a n e o u s l y f o r h2 a n d v2. ly

only

l a r g e number of steps

equations be

the

really

+

or

interval.

the

When

Thus

illustrated

increments.

difference

form.

conduit

time

involved

(h2

namely

method

solution.

finite

averaged

used,

small

be

computer

suggested

in

numerical

the

throttling a t head

loss

the surge shaft inlet.

term

i s expressed

he

The i n l e t

i n terms o f

the

p r e v i o u s f l o w r a t e a s a n i m p l i c i t s o l u t i o n w o u l d b e more c o m p l i c a t e d for

t h e q u a d r a t i c term.

Thus head

losses must b e s m a l l r e l a t i v e to

surge rises. (8.18)

+ hf}

D y n a m i c : Q2 = Q 1 - ( g A t A t / L ) { ( h 2 + h , ) / 2 where he = K Q t l Q t l / ( 2 g A z i ) Continuity:

(8.19)

h -h =(Q2+Ql)At/2As 2 1

(8.20)

S o l v i n g f o r Q2 a n d h

2:

Q, - ( gAt A t / L ) { hl + ( Q1 A t/4AS)+KQ, Q2 -

I Q, 1 / ( 2 g A i 2

1

(8.21)

l+gAtA t2/4LAS

a n d h2 = hl+(Q2+Q1 ) A t / ( 2 A s )

Another useful

(8.22)

application of

w i t h water column separation.

the upstream end o f a pumping sufficiently cases

the

slowly sis.

line,

to c a u s e v a p o r i z a t i o n a t water

column

beyond

and r i g i d c o l u m n

Equ.

the r i g i d water column equation i s

Following the stopping of

8.1

may

be

theory

the

the pressure frequently drops peaks along the vapour

pocket

i s sufficiently

integrated

a pump a t

twice

with

line.

will

decelerate

accurate for respect

to

In such

analy-

time

t

to

determine the distance the water column wi II t r a v e l before stopping.

111

Simple open surge tank

Variable area surge tank

Throttled inlet/outlet

Closed surge tank

Differential area surge tank Fig.

8.4

Types of

surge

tanks

112

If

the pumps stop

b e h i n d the w a t e r

instantaneously column o f

t h e volume of

e

length

will

the v a p o u r

pocket

b e Q=Aev 2 / 2 g h where

v

i s the i n i t i a l flow v e l o c i t y .

SURGE TANKS

By

breaking

face,

the

duced

greater

and

The

use

surge

installations

with

surge tanks

with

a free

amplitude for

surge

tank

this

tunnels

p i p e systems where heights

water can

closed

level

surge

variations

numerically

or

dynamic equation

tanks, and

the

employing

b u t n o t encount-

b e excessive.

hydraulic

two

can

A p a r t from

calculations, be

equations.

performed

One

is

the

(8.1) a n d the o t h e r i s the c o n t i n u i t y equation,

v3A3=G2A2 = A 2 d y / d t

=

is

the pressures a r e

would

pressures,

sur-

b e re-

purpose

s u r g e t a n k s a r e i n d i c a t e d i n F i g u r e 8.4.

and

water

analytically

of

i n pumped o r

consequently

throttled

namely

v A 1 1

pressure and

i n hydroelectric

regularly

Some shapes of the

l e n g t h of closed c o n d u i t

hammer

considerably.

common ered

a

water

where v

i s velocity,

i s time,

subscript

(8.23)

A i s cross-sectional

area,

y i s water depth,

t

1 r e f e r s to t h e c o n d u i t a n d 2 to t h e open s u r g e

s h a f t a n d 3 to t h e i n l e t . In

the

higher

case

than

through

of

the

the

throttles,

water

throttle.

The

r e p r e s e n t e d b y he = K . I

on

whether

throttle shaft

the

serves

and

variations

flow to

since

head

the

shaft the

where K . I

out

the water

head 8.5)

loss

of

out not

/o

the

level

is

it will

the

in

surge

through

loss

into or

reduce

(see F i g u r e

in

v31v31/2g

/o

is

the

the

level

conduit by

rise loss

restriction

could

be

could v a r y depending

opening.

Generally

fluctuations of

could

the head

phase

the

in the s u r g e

with

the

level

i n c r e a s e t h e maximum h e a d

i n t h e c o n d u i t i f n o t excessive. Approximations

to

t h e damped s u r g e s c a n

t i c a l methods ( P i c k f o r d , Alternative

methods

be o b t a i n e d b y a n a l y -

1969). of

a n a l y s i s of

surge shafts are g r a p h i c a l l y

( J a e g e r , 1956 a n d 1977) a n d w i t h t h e a i d o f c h a r t s ( R i c h ,

1963).

113

Load rejection

head loss into surge shaft

8.5

Fig.

Surge in t h r o t t l e d surge shaft

Example 2

N u m e r i c a l A n a l y s i s of Penstock P r o t e c t e d w i t h an A i r C h a m b e r

From Boyles law H S

0 0

assuming

=

(8.24)

HS

isothermal expansion

w h e r e h e a d H i s a b s o l u t e i.e. Continuity:

AS = vAAt

Dynamics:

Av =

3 e

+ atmospheric head (8.25)

L = 1000m, Ho = 30m = 40m a b s o l u t e , = lm3,At

v

=

1.5m/s,

A= 0 . 2 m Z ,

.5s

=

40 H = -C

t

(8.26)

HAt ( r i g i d c o l u m n e q u . )

A V=

- .005H

A S=-O.1 v

V

S ~

0-.5 .5-1 -1.5 -2 -2.5 -3 -3.5

30+10=40 46.0 52.6 58.8 63.5 6 5 . 6::

-.2 23 26 29 - . 31 33 32

-. -. -. -. -.

64.5

-.13 11 08 05 - .0 2 .01 .04

1.3 1 .07 .81 .52 .21 -.12 - .44

-. -. -.

.87 .76 .68 .63 .61 .62 .68

-d

Max H = 6 5 . 6

- 10

This analysis

i s not p a r t i c u l a r l y accurate as accelerations

great

to

permit

=

55.6m.

accuracy

w i t h r i g i d column

d e m o n s t r a t e s u s e of t h e t e c h n i q u e .

theory.

were too

I t nevertheless

114

Fig.

8.6

Numerical

analysis

of

penstock

protected

with

an

air

chamber

REFERENCES

1956. E n g i n e e r i n g F l u i d Mechanics. B l a c k i e C? Son L t d . Jaeger, C., L o n d o n . 529 p p . C., 1977. F l u i d T r a n s i e n t s in H y d r o - E l e c t r i c Engineering Jaeger, P r a c t i c e . B l a c k i e 0 Son L t d . L o n d o n . 413 pp. P i c k f o r d , J., 1969. A n a l y s i s of Surge, M a c m i l l a n . 203 pp. 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 , Dover, N.Y. 409 pp.

115 CHAPTER 9

WATER HAMMER THEORY

BAS I C WATER HAMMER EQUATIONS

The

fundamental

wave

equations

water hammer may be a r r i v e d a t of

momentum

and

general

case of

friction

head

of

mass.

The

following

a pipe inclined at

loss

varying

describing

the

phenomenon

of

from consideration of conservation

with

derivation

is

for

the

any a n g l e to the horizontal a n d

the

square

of

the

velocity.

The

notation used i s g i v e n i n F i g u r e 9.1.

pd dx 2

Fig.

9.1

L o n g i t u d i n a l and cross sections through p i p e

Conservation of Momentum

The

following

balance, energy element

but

equation

it

is

equation. of

fluid

possible

is to

derived derive

Newton's Second Adx

The r e s u l t a n t force

moving

it

Law

of

i n the x-

i n the x-direction

from

a

starting motion

direction

is

force-momentum from

Bernoulli's

applied

to an

(see F i g u r e 9.1).

equals the r a t e of change of

momentum i n that d i r e c t i o n : pA - (p+*dx)A ax dv and since -.clt

- W A A c x - W A d x s i n e=-AdxW dv 2gd -

av at

av ax

- + -v.

dt

(9.1 )

(9.2)

116 t h e e q u a t i o n becomes

(9.3) (9.4)

t h e n W-

ah ax

az ax

=% +

W-

ax

w v av -

+

g

(9.5)

ax

az a-x =sin 0 .

where

(9.6)

Hence t h e momentum e q u a t i o n becomes (9.7)

To account f o r t h e d i r e c t i o n a l c h a n g e i n h e a d loss w i t h v e l o c i t y one can w r i t e v l v l

In

i n s t e a d of v 2 :

t h i s equation

total

head h .

the v e l o c i t y

h e a d v 2 / 2 g h a s been

One c a n t a k e h a s

included

t h e p i e z o m e t r i c h e a d p/W

in the

+ z and

w r i t e t h e momentum e q u a t i o n t h u s :

ah ax

+

1g (a” + at

Generally

the

be neglected,

v-)av ax

term but

n e c e s s a r y , e.g.

xv2 = + -

0

2gd

vav

/ax

(9.9)

i s small

compared

i t can be accounted f o r

with

av/at

i n numerical

a n d can

solutions

if

i n flexible plastic piping.

Conservation of Mass

The

second

differential

difference

between

elemental

length

storage

caused

the of

by

equation

r a t e of

pipe elastic

is

mass

arises inflow

equated expansion

to

of

from

continuity.

The

to a n d o u t f l o w from the

rate

the

of

pipe

increase and

i s ( d / 2 b ) d p/a t

The c o r r e s p o n d i n g r a t e o f s t r a i n i s (d/2bE)a p/a t a n d t h e r a t e of Now A =

-

..

aA at

nd2/4

nd 2

ad at

of

elastic

compression o f t h e w a t e r .

The r a t e of s t r e s s i n c r e a s e i n t h e p i p e w a l l

an

increase in diameter i s (d2/2bE)ap/at (9.10) (9.11)

i17

(9.12)

_- - nd

d2 ap 2 2bE at

The r a t e of i n c r e a s e i n v o l u m e o v e r a l e n g t h d x i s g i v e n b y (9. 3 )

- n d 3 d x ap 4bE Longitudinal assumed

that

expansion

(9. 4 )

at strains

h a v e been n e g l e c t e d h e r e .

longitudinal

stress

which

would

I t h a s a l s o been

affect

the

lateral

due to the Poisson's r a t i o effect i s i n s i g n i f i c a n t a l t h o u g h

they can be included if desired. The r a t e o f

i n c r e a s e i n s t o r a g e c a u s e d b y e l a s t i c compression of

water i s (9.15) E q u a t i n g i n f l o w m i n u s o u t f l o w to r a t e o f c h a n g e of s t o r a g e , VA -

av -dx)A ax

+

(V

=

(-

nd2dx 4K

+

nd3dx 4bE)

(9.16)

ap

a V + ( L + - )d * = o K bE a t

'

(9.17)

ax

1.e.

Again,

i f we p u t h = p / W

a n d r e c a l l az/at

= 0,

we g e t (9.18)

C-T

where c = 1/

+

p(-

-)

The b a s i c d i f f e r e n t i a l term,

water

hammer e q u a t i o n s ,

including

a friction

t h u s become

-+ah

1

av+xvlvl=o

ax

g

at

2gd

(9.19)

and

-ah + - - c 2 av at

g

= o

ax

(9.20)

O m i t t i n g t h e f r i c t i o n t e r m t h e e q u a t i o n s become

ah + l a-v ax gat

=

O

(9.21) (9.22)

The g e n e r a l s o l u t i o n to these e q u a t i o n s i s

118

(9.23) v

+ 2f

= v

c 1

0

which at

indicates

speed +c

(t

X

that

along

+ 2f (t + c 2

-) c

-

X

-)

(9.24)

pressure and velocity

the p i p e .

c

changes a r e propagated

i s r e f e r r e d to a s

t h e w a t e r hammer

wave c e l e r i t y .

Where

longitudinal

expansion

is

allowed

for

a

more

accurate

expression f o r c is

c =I/$-

(9.25)

5

where k = - - p f o r p i p e s u p p o r t e d a t

- p 2 f o r b o t h ends f i x e d

= 1

1

=

f o r a p i p e w i t h expansion j o i n t s

p i s the Poisson r a t i o which f o r

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

P

i s t h e f l u i d mass d e n s i t y ,

d

and

is

total

head,

(1000 kg/m3 f o r w a t e r ) ,

is

distance

f r i c t i o n coefficient

practice).

indicated b y

X

is

actually

a

t h e Colebrook-White

also affected b y

unsteady

wall

along

(9.81 m / s 2 ) , t i s time,

Darcy-Weisbach

is

x

i s 0.3

(2100 N/mm2 f o r w a t e r ) ,

a r e the p i p e diameter a n d

acceleration

U.S.

steel

(210 000 N/mmZ f o r s t e e l ) ,

K i s the b u l k modulus of f l u i d , b

one end o n l y

v

(

thickness

pipe,

g

respectively, is

i s flow velocity,

X

function

motion

gravitational and A

i s the

in British practice and f

of

the

flow

e q u a t i o n o r a Moody but

h

velocity diagram.

in

as It

no q u a n t i t a t i v e assessment

of t h i s i s a v a i l a b l e

EFFECT O F A I R

The presence of water

hammer

free a i r

considerably.

in p i p e l i n e s c a n r e d u c e t h e s e v e r i t y of

Fox

(1977)

indicates

that

the

celerity

(speed) of a n e l a s t i c wave w i t h f r e e a i r i s L

m p(-

+

-

+

- )

(9.26)

119

For l a r g e a i r contents t h i s reduces to c = where

p i s the absolute pressure and f

(9.27) i s the free gas f r a c t i o n b y

volume. c

is

Thus

reduced

2% of

remarkably

air

at

a

for

pressure

even

relatively

head of

low

50 m of

gas contents.

water 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 steel p i p e l i n e to 160 m/s. The Joukowsky water hammer head i s Ah

-C

=-

where

Av

(9.28)

Lv

9

in velocity

i s the change

reduction

i n water

of

flow.

There

i s thus a l a r g e

hammer head h f o r a r e l a t i v e l y small f r a c t i o n of

I f the a i r collects at the top of the p i p e there i s no reason to

air. see

why

other

the same equation cannot a p p l y .

hand

partly

derived

full

pipe.

an

equation

for

the

Stephenson

(1967) on the

celerity

a bore

of

in

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

f o r small a i r proportions c

= JgAh/f

where

Ah

is

There

the

head

is

a

valves

school

in

overpressures.

The

columns. large

that

air

reduction

in

air

in

a

air

head.

The

hammer is

according

at

to

the equation

The size of low

pUk

a i r valves

favours of

the

a

i n s t a l l a t i o n of

reducing

water

to cushion

the

hammer

impact of

however,

indicate

required

to

that an

produce

idea stems from the use of

i n pipelines. high

smal I

It will

pressure volume.

any air

be realized

initially Upon

and

pressure

the a i r from an a i r vessel expands

= constant

(vacuum) pressures w i l l

indicates

is

to draw

s i v e f o r l a r g e diameter pipelines.

This

will,

under

relatively

reduction f o l l o w i n g a pump t r i p ,

air.

bore.

primarily

is

of

vessels

occupies

means

Calculations

volume

the

h = 50 m.

which

a

as

to a l l e v i a t e water

therefore

behind

thought

intention

approaching

vessels

of

pipelines

excessively significant

rise

158 m/s f o r f = 0.02 and

c e l e r i t y of

air

(9.29)

where U i s the volume of

i n the necessary volume of a i r

be found on a n a l y s i s to b e exces-

120

METHODS O F ANALYSIS

A common pressures

method

used

assumed

points along

h plotted

line,

pipe

graphically

concentrated

the

simultaneously

analysis of

be

to

be

to

of

at

end

water

against

most

v,

for

usually

by

which by

p a r t i c u l a r systems

differs

little

of

in

differences

hammer e q u a t i o n s for

intervals

and

solution

known

pipe

Friction

was

or

few

at

a

time

of

intervals.

Chaudhry,

solution

is by

principle

from

(Stephenson,

of

digital

characteristics

T h i s method

the

water

computer.

hammer

Solution i s

(Streeter a n d Wylie,

the

old

1966).

is

1979.

graphical

The

1967)

method o r

differential

water

a r e e x p r e s s e d in f i n i t e d i f f e r e n c e f o r m a n d s o l v e d

successive time

which

method

the method of

finite

the

hammer e q u a t i o n s were s o l v e d

successive

economical

equations for

of

hammer

w i t h the v a l v e o r pump c h a r a c t e r i s t i c s on a g r a p h of

now l a r g e l y r e p l a c e d b y c o m p u t e r s e . 9 . The

water

(Lupton,1953).

one

a n d the

systems f o r

intervals. is

At

takes

conditions

set

equal

place

along

The c o n d u i t

is

the

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

A x/c.

to

depicted pipeline

Fig.

in

at

The

time

x

9.2. t,

t

-

grid on

S t a r t i n g from

one

proceeds

to

c a l c u l a t e the head a n d v e l o c i t y a t each p o i n t a l o n g the l i n e a t time t

+

At.

By a d d i n g e q u a t i o n s (9.19) differential

terms

can

be

a n d (9.20)

replaced

by

divided b y c,

total

the p a r t i a l

differentials

and

one

obtains (9.30)

for

dx dt

-

- c :

Equs.(9.30) at and

time

t

r

at

dh=h'-h

9

and

(9.31)

+ A t

v

time

t.

Thus

D

and

+v

may

in terms of

dv=v'-v ~r ~r characteristic equations, h +hr + c v - v r hl =L --9 P 2 9 2 v l

cX v I v l d t 29d

dh - Z d v -

g h -h

for

.

be solved for

c

2

h'

known h and v

are cdth

+

v

r

(v

r

1-v

~

29d

- hdt 2d

(9.31) P at

and v '

at point p P two other points q

dh=h'-h dv=v'-v P q P q' resulting equations,

qp,

The

= 9 + - qr 2

= o

qlvq

I

and

for

termed

rp, the

(9.32)

2 v , I v r ~ + v q l v q I~ 2

(9.33)

121

TIME

0

0

F i g . 9.2 x - t G r i d f o r w a t e r 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 s method. At

the

imposed; pump

terminal

v

h is fixed,

either

speed.

The

simultaneously at

and

time are

with

+

t

points,

At.

an or v

correct

additional

condition

is

usually

i s a f u n c t i o n o f a g a t e o p e n i n g or

Equ.

(9.30)

or

(9.31)

is

solved

t h e k n o w n c o n d i t i o n to e v a l u a t e t h e new h a n d The c o m p u t a t i o n s commence

terminated

when

the

at

known conditions

pressure fluctuations

are sufficiently

damped b y f r i c t i o n . a b r a n c h p i p e s o c c u r s ( f l o w o u t o f p to s ) o r t h e r e i s a

Where change

in

diameter,

then

r e p l a c e d b y Equs. (9.34)

h'

(9.32)

to

(9.33)

should

-Q,-Qs)-(chdt/2g).

9

(QqIQqI/dqAq-QrIQrI/drAr-QsIQsI/dSAs)

]/(A

q

+Ar+As)

(9.34)

t h e n 8' = Q + ( A g / c ) ( h - h ' )-ha Q d t 2d A 4 P d q 1 / q q qP q 4

(9.35)

QIpr

= Qr+(Arg/c)

lar[ dt/2dr

Ar

(9.36)

Q'

= Qs+(ASg/c) ( h ' p - h s ) - h Q S I Q S l d t / 2 d S

As

(9.37)

Where

be

(9.37):

[ hqAq+hrAr+hSAS+(c/g) ( Q

=

P

to

Equs.

PS

Q

Pr

( hSp-hr)-h Qr

i s f l o w o u t of p to r e t c .

and

Q

q

i s the flow out of q to

p etc. It

should

equations

be

above

noted

that

the

i s termed e x p l i c i t

finite

difference

form

of

the

s i n c e h e a d losses a r e e x p r e s s e d

122

i n terms of losses

the velocities a t the p r e v i o u s time i n t e r v a l .

are

significant

compared

with

i m p l i c i t solution may be necessary The l a t t e r

is,

however, and

the

water

Where head

hammer

heads,

1981 ) .

(Chaudhry and Yevjevich,

more complicated

simultaneous

as

solutions

the of

equations

equations

an

involve

more

unknowns

point

i n space a r e necessary. The equations also become non-linear.

for

every

A method of overcoming these problems i s e x p l a i n e d i n Chapter 12.

VALVES

At a v a l v e o r other c o n s t r i c t i o n i t the

characteristic

equations

A

simultaneously.

valve

increases head losses.

and

acts

in

i s necessary to solve one of

the

valve

effect

like

f r i c t i o n head loss was assumed

which

I t w i l l be r e c a l l e d that

to be a function

previous point a n d previous time i n t e r v a l . becomes i n c r e a s i n g l y

equation

constriction

One may therefore e n q u i r e why the head 1055

be treated as f o r f r i c t i o n head loss.

cannot

discharge a

inaccurate

of

velocity

at

the

Unfortunately t h i s method

( a n d unstable) for

increasing

head

tosses a n d an e x p l i c i t (new time) f u n c t i o n i s g e n e r a l l y r e q u i r e d f o r the

loss

head

at

a

constriction.

(It

is

however

exceedingly

laborious to account f o r a new v e l o c i t y head loss at a p i p e branch, so

in

the

case

of

network

programs

a

weighted

pseudo-implicit

method has been employed). The discharge c h a r a c t e r i s t i c of

a v a l v e can be expressed i n the

f o l l o w i n g way:

Q = CdFAm where

Q

is

the

valve a n d C d of

the

flow

in

the

valve,

H

i s a discharge coefficient.

valve

coefficient

(9.38)

(0 i s closed

i s often

a

and

f u n c t i o n of

1

is

i s the head

loss across

the

F i s the f r a c t i o n a l opening fully

open).

The

discharge

F unfortunately but t hi s w i l l

be

accounted f o r here b y assuming the o n l y v a r i a b l e i s F.

The degree of v a l v e opening as a function of time i n the case of uniform

stem

travel

generally

non-linear.

near

end

the

illustrates

the

of

the

or

proportional

For most valves travel

proportional

of

area

the open

turns

of

a

handwheel

is

the hole closes more r a p i d l y gate. for

The

following

different

valves

figure as

a

function of time assuming steady t u r n i n g of the handwheel o r actua-

123 tor,

( t h e a c t u a t o r can be controlled to close n o n - l i n e a r l y

i f desired

a n d t h i s w i l l be discussed under chapter 12.

1

.o

A

-(l-x)sin arcos ( 1 -

Azi 0.8

s i n ( a r c o s x!l /

-. \

'

0.6

I

Gate valve

0.4

/ - \

0.2

Spherical

0

0.2

0.4

0.6

.o

0.8

Butterfly

Proportional rotation o f handwheel from open to closed

F i g . 9.3

Proportional areas f o r some valves

For gate v a l v e s the F - T

r e l a t i o n s h i p can be approximated b y

a parabol ic function : F = 1 -(t/T)'

(9.39)

I n general the head loss f u n c t i o n can be w r i t t e n

Q

= { l - ( t / T ) N I A p m

where

Ho =

i n i t i a l H/Vo'

(9.40) i.e.

initial

head loss through

f u l l y open

A i s the p i p e area. P Solving t h i s a n d the c h a r a c t e r i s t i c equation at the downstream end v a l v e d i v i d e d b y i n i t i a l velocity

squared.

of a p i p e dh + s v + 9

X v l v l d x= 29d

0

(9.30)

124

J

(9.41)

.

c

.AD

QZ

and h =

(A where Z i s

the

at

the

P

is

head.

previous is

t

proportional

(9.42)

H i s t h e head above Z and h

t h e downstream head,

total

valve.

+ z

[1-(t/T)N])2/Ho

time

interval,

T

time,

1

Subscript

is

operation of

the

r e f e r s t o the upstream p o i n t

and

p

closure

to the p i p e upstream of time

of

a

the

v a l v e , t/T i s

the

the handwheel assuming a constant r a t e of

operation and A i s the open area.

ACCURACY AND STABILITY O F FINITE DIFFERENCE SCHEMES

The

water

hammer

equations.

As

they

numerical

techniques

equations

cannot are

be

are

non-linear

partial

differential

solved a n a l y t i c a l l y f o r most

employed.

Most

existing

cases,

methods can

be

c l a s s i f i e d as follows: a)

e x p l i c i t f i n i t e difference methods

b)

i m p l i c i t f i n i t e difference methods

c)

f i n i t e element methods. The

method of

implicit

taneous solution o f a l l matrix;

its

interval, method

main

&/At,

solution

advantage i s not

is

a

method

whereby

is

that

governed b y

the

ratio

of

space

to

i s considered to be s t a b l e f o r any choice of Ax a n d At. i n v e s t i g a t o r s considered t h i s to be an advantage.

however,

found

this

simul-

that

"advantage"

r e s u l t s was h i g h .

time

any s t a b i l i t y c r i t e r i a and the

previous

of

a

the flow properties i s obtained b y s o l v i n g a

Most

Others,

i t i s not a l w a y s possible to make p r a c t i c a l use as f o r h i g h r a t i o s of

Ax/At

inaccuracy

i n the

I m p l i c i t methods a r e also not convenient f o r

use

as one cannot keep t r a c k of r e s u l t s a t d i f f e r e n t time periods. F i n i t e element

methods

are

usually

avoided as they

a r e expen-

s i v e to r u n and accuracy a n d s t a b i l i t y c r i t e r i a can be tedious.

125

Explicit finite past

for

the

equations. variable certain time, as

They

time

thus

they

of

as

but

a

a

fixed

the flow

economical

when

and

an

however, scheme

use

those of and

non-l inear

the

flow

grid

and

they

flow

They

at

the

a

a

previous

a r e simple to use

easier

is

define

properties a t

properties

it

i n the

differential

to

follow

the

properties a l o n g the catchment as the solution They

of

it

have

used.

an

accuracy

using

of

the

used

partial

i n the way

express

e x p l i c i t solution.

properly

the

all

regular

performed e x p l i c i t l y .

choice

they

function

permitting

use

different

d i f f e r from each other

gradients,

v a r i a t i o n of is

difference schemes have been widely

solution

been found to be accurate and

The

explicit

problems

finite

and s t a b i l i t y .

accordingly

accompanying

difference

scheme

the are,

Choosing the most proper

is,

therefore,

important

in

o b t a i n i n g s t a b l e a n d accurate results.

Basic Terms Related to Accuracy and S t a b i l i t y f o r Difference Schemes

Many

natural

described cannot

by

be

solved

techniques b y ational two

basic

differential

the

equations

to

the

and

usually

the

"How

represented b y more

it

approximate equations

numerical

This

procedure raises

system modelled by the

is

solution

to the

computa t iona I a l g o r i thm?" is

paid

to

the

second

be answered b y s t u d y i n g

be

the

to

natural

the

will

be

equations to a comput-

system and comparing

Therefore

differential

resorts

can

equations

well

attention

question can o n l y

functions

differential

schemes.

is

well

follow

equations

the

"How

the n a t u r a l it.

differential between

to

The f i r s t of

continuous If

one

difference

equations?",

analysis

question.

applied

using

questions.

behaviour

are

mathematically

algorithm

differentia 1

which

equations.

a p p r o x i m a t i n g the d i f f e r e n t i a l

the

In

systems

differential

assumed

system

and

the

the

i t to t h e equations here

we1 I .

that

The

the

difference

difference

scheme

a p p r o x i m a t i n g them i s c a l l e d a Truncation e r r o r ( T r ) , i.e.

D i f f e r e n t i a l equations = Difference scheme The

truncation

expansion.

There

error

can

i s also

a

+ Tr

(9.43)

r e a d i l y be established u s i n g T a y l o r ' s difference

schemes which one c a l l s the E r r o r ( E ) ,

in

the

solutions

of

the two

126

i .e.

Solution

equations The

of

Differential

equations

Solutions

=

of

Difference

(9.44)

E

f

exact

v a l u e of

the E r r o r cannot r e a l l y be obtained

i n this

case as the d i f f e r e n t i a l equations cannot be solved a n a l y t i c a l l y . say

that

We

a difference scheme i s consistent w i t h a set of D i f f e r e n t i a l

equations

if

the

Truncation

error

tends

to

zero

as

the space a n d

time increments tend to zero, i.e.

i f limit Tr = 0

Consistent

as Ax, At We with

say

that

+

0

(9.45)

the solution of

the solution of

the difference scheme i s convergent

the d i f f e r e n t i a l

equations

i f the E r r o r tends to

zero as the space a n d time increments tend to zero, i.e.

if limit E = 0

Convergent

as Numerical formed.

It

is

the

(9.46)

Ax,At+O

Diffusion

is

the

process

development

of

the numerical

the

error

( E ) through

fests

in the form of a n attenuation

in

which

the

truncation

Error

(Tr)

error

technique used.

(E)

is

to the

I t g e n e r a l l y mani-

a n d spreading of

wave fronts.

I f computations a t p o i n t s distance Ax a p a r t a r e a t time i n t e r v a l s A t then numerical

diffusion w i l l

proceed through

the system a t

a rate

Ax/ At.

Stability and Accuracy Criteria for an Explicit Finite Difference Scheme

Since tions

one

is

(p.d.e's)

dealing there

with is

non-linear

no

rigorous

partial proof

differential

specifying

equa-

stability

c r i ter i a. In

the

solution

Hammer equations influenced b y the

time

i.e. under

a

non-l near

( Ax/ AtIcr.,

set

of

p.d.e's

like

the

Water

i t was found that both s t a b i l i t y a n d accuracy a r e

the values chose

increment

stable

of

( At).

exists

conditions

In for or

f o r the space

particular, determining not.

The

a

increment

critical

whether effects

( A x ) and

r a t i o of

Ax/A

a scheme w i l l of

Ax

s t a b i l i t y a n d accuracy a r e summarized i n F i g u r e ( 9 . 4 ) .

and

At

t

run on

127

+

--

A

solution is unstabl e

solution is stable

Accuracy of Solution decreases because of numerical diffusion

If the difference scheme i s

convergent for a fixed ( A X / A t ) the smaller Ax and A t the more accurate the solution. t

0

Fig. 9.4 Effect of values on Ax a n d At on s t a b i l i t y a n d accuracy f o r a n e x p l i c i t f i n i t e difference scheme From F i g u r e (9.4) selection

of

Ax

one can deduce that

and

At

values

for

an

the main c r i t e r i a in the explicit

finite

difference

scheme are: a)

that the scheme s h a l l proceed u n d e r s t a b l e conditions i.e. (9.47)

b)

Ax At

shall

be

close

to

AX

(-)

At c r

to minimize d i f f u s i o n e r r o r s a n d

o b t a i n optimal accuracy. c)

The difference scheme s h a l l be convergent.

tained

by

running

the

comparing w i t h a n a l y t i c a l

scheme w i t h

different

T h i s could be ascerAx's

and

At's

wave

as

and

r e s u l t s in a simple case.

Determining (Ax/AtIcr

(Ax/At)cr propagated.

has

been

shown

to

be

the

speed of

T h i s can be demonstrated b y considering

characteristics

it

is

the method of

128

The

method

of

characteristics

waves

travelling

dx/dt.

The f a m i l y of

are

called

total

the

depth,

along or

against

the

at

case of

The

the

flow

water

in

the

a specific

form

of

velocity,

i n the x - t p l a n e

properties,

hammer

velocity

equations

are

and des-

r e l a t i o n s h i p s obtained from the wave equations ( 1 and 2 )

cribed by

u s i n g the r e l a t i o n s h i p s of derived

the flow

flow

curves described b y dx/dt

characteristics.

in

describes

describe

t r a v e l l i n g along

the

dx/dt.

flow

the flow

at

I n other

words,

properties

as

seen

a velocity

defined b y

the r e l a t i o n s h i p s by

an

observer

the characteris-

tics. I n the case of the water hammer equations, a velocity e.g.

gradient,

shutting

against dx/dt

the

=

As

2

of

flow

i.e.

valves at

a

This of

wave

2

c

is

as

i n the flow,

propagated

given

by

the

to

with

or

equation

(9.48)

different

information

propagation

i I lustrated

by

equation

etc.

velocity

a wave i s caused by

anywhere

c

the wave t r a v e l s

gradient

change i n velocity

i t propagates information about the velocity

points

in

the

conduit.

by

the

wave

considering

the

characteristic

in

time

The and

concept

space

curves

of

can

be

defined

by

( 9 . 4 8 ) . This i s i l l u s t r a t e d i n Figure ( 9 . 5 )

t

X

Fig. 9.5 Propagation of information along c h a r a c t e r i s t i c s o f the water hammer eauations

129

C 1 and C2 a r e a set of c h a r a c t e r i s t i c s described

I n F i g u r e 9.5, by

equation

and

(9.48).

velocity

are

known

from the o r i g i n at at

Suppose

that

the

points A

at

time tl.

flow

properties,

0,

and

xA

total

head

and xB distances

One can then o b t a i n the flow properties

a p o i n t R w h i c h l i e s on the same c h a r a c t e r i s t i c s as points A and

.

B, xR distance from the o r i g i n a t time t The shown

critical to

scheme

be

for

space

the

to

wave

solving

time

interval

speed

the

by

water

ratio,

considering

hammer

t)cr,

(Ax/A

a

central

equations.

The

can

be

difference scheme

is

i l l u s t r a t e d i n F i g u r e 9.6.

i represent a space i n t e r v a l ,

Let val

in F i g u r e 9.6.

as shown

flow

properties

Information time

are

about

interval.

to

the

In

a n d k represent a time inter-

The point

be calculated, flow

properties

Figure

9.6(a)

the

i n question,

i.e.

where the

( i , k).

has the co-ordinates is

sought

true

from

the

propagation

previous speed

is

smaller than the numerical p r o p a g a t i o n speed w h i l e in F i g u r e 9.6(b) the

converse

have

a

is

slope

true.

Ax/At

have a slope d x / d t

Numerical i n the x-t

in the x-t

i s obtained w i t h i n

the i - I ,

In

information

Figure

lines range

9.6(b)

outside is

not

the

i-I,

plane.

lines are

lines

t r u e propagation

I n F i g u r e 9.6(a)

is

sought

by

range.

Since

the

numerical

by

the

information

true

information scheme,

that lines

i + l r a n g e b y the t r u e propagation

i+l

propagated

propagation p l a n e while

lines.

propagation outside it

this

cannot

be

found a n d thus i n s t a b i l i t y w i l l r e s u l t . For s t a b i l i t y of an e x p l i c i t f i n i t e difference scheme the f o l l o w i n g must,

therefore,

hold:

>dx

A5 At - dt

(9.49)

T h i s i s r e f e r r e d to as the "CFL condition" richs

and

Lewy

(1956), o r simply

a f t e r Courant,

Fried-

the Courant c r i t e r i o n f o r s t a b i l -

ity. To minimize d i f f u s i o n e r r o r s as mentioned e a r l i e r Ax/A t must be as close as possible to dx/dt. &/At

Using the r e s u l t

i s chosen to be equal to the wave c e l e r i t y ,

in equation (9.48),

i.e. (9.50)

130

t

t

A

\

\y

k

k-1

.

A,

I k

-

k- 1

.

i-I

W

i ,i+l

-

X

i-I

i

* X

i+l

-

Numerical propagation 1 ines; slope(Ax/

-----------

True propagation 1ines; slope (dx/dt)

F i g . 9.6 Comparison o f n u m e r i c a l a n d t h e o r e t i c a l i n f o r m a t i o n in a c e n t r a l d i f f e r e n c e scheme

Equation The

(9.50)

relationship

stable

conditions

i s used to define

ensures with

that

no

the

p r o p a g a t i o n of

t h e space a n d t i m e i n t e r v a l .

difference

diffusion

At)

errors.

scheme The

is

run u n d e r

solution

of

the

e q u a t i o n i s f u r t h e r made more a c c u r a t e b y t h e c h o i c e of s m a l l e r

Ax

a n d a t A t i n t e r v a l s as c a n be seen f r o m F i g u r e (9.4).

131

BAS IC COMPUTER

PROGRAM FOR ANALYS 1 NG GRAV ITY

DRAWOFF

THE

ALONG

LINE,

VARIABLE

L I NE W I TH ONE

P I P E DIAMETER,

NUMBER OF

INTERVALS AND CLOSURE T I M E OF DOWNSTREAM GATE VALVE. I n p u t i s as follows: Line 1;

Title

L i n e 2:

Pipelength,

m;

C e l e r i t y m/s;

Number of

pipe divisions;

iterations

(in

division

so

1

E n d f l o w m'/s;

iteration

At= A L / C ) ;

V a l v e c l o s u r e time,

wave

Point

at

N u m b e r of

travels which

s;

1

pipe

draw-off

occurs. L i n e 3; L i n e 4 etc.;

D r a w o f f m'/s Elevations

( p u t 0 if none).

i n m of

each point

fixed reservoir level), L a t e r lines;

( f i r s t one b e i n g t h e

(one p e r l i n e ) .

D i a m e t e r s o f e a c h p i p e d i v i s i o n in m ( o n e p e r l i n e ) .

80 N 2 = N l t l 90 D I S P DRFtUOFF mA3 108 FOR J = l TO N2

110 P(J?=8 120 NEXT J 138 INPUT P ( M 1 ) 1 4 0 D I S P "ELEVHTIONSm 1 5 8 F O R J = l T O N2 1 6 B INPUT Z C J 1 7 0 NEXT J

132

768 NEXT I 7 7 8 STOP 7 8 0 END P I P E NRME? SAMPLE RClN

133

.Be1 -25.8

-.

B 8 8 8 -29.3 FLcl . M3"S HER@

L

4

1

132 883

.3-.

t4 0 D E 1

89.9 -23.7

.j

4

i

t.4 U0E 1

7 &*

132

9

.882

i

3

4

HOUE 1 2

13' 080

. a 8 1 297.2 8 . 8 8 9 286.4 NCfUE FLO. M3^S H E A D 1 .12Q 1 8 B . 8 L . B e 2 288.9

3 4

-

'7

.@@I1 4 E1.889 NODE FLO. M 3 1 - 824 2 888 7 .091 4 8 . B88 NODE FLU. M 3 1 - .824 975 - .891 4 9.L398 2 1 1 . 6 H0DE FLO. M3*S HERD 3

NUDE FLO . M3^S HERO 1 -.838 188.8 '2 .El88 327 4 L

t4 U 0E F L U . N3"5 H E R D - . @ 3 S lee.@ 1

-.BY9 181.8

4 tJODE 1 3

L

7

L .

4

NODE 1

- . 8 8 1 229.5 8.808 234.1 F L O . M3"S HEQ@ -.a48 1B0.9 -.a89 181.9 - . 1389 1 8 5 . 2 8.088 228.1 FLO.M3^S HEAD -.049 188.8 - . El913 1883.7 -.El87 1 8 2 . 2 8.688 -20.3 FLO . M 3 ^ S H E R D -.a48 188.8 -.@I88 9 7 . 7 ~

4 t4ODE 1 2

~~

'-I

- ,

1 2

-.825 -.0?5

1sa.e

199.7

3

-.@?5 193.5

4

0 . 8 0 8 286.7

134

O.lm 3 / s

t

c = 1000m/s L i s t of symbols i n g r a v i t y p i p e w a t e r hammer a n a l y s i s p r o g r a m

Symbols

A(J) C D(J) Dt D2 F G H(J) H2 H3 H1 I1 I J J2 L M3 NS N1 N2 M1 P(J 81 Q(J 43

R(J S(J

Area o f p i p e Celerity Diameter o f n e x t section Increment i n X Increment in T D a r c y f r i c t i o n f a c t o r i n H = FXV2/2GD 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 , 9.81 Head Dummy v a r i a b l e Dummy v a r i a b l e Head loss t h r o u g h valve/V’ Number o f i t e r a t i o n s I t e r a t i o n number Node n u m b e r Counter Length M2-1 Name Number o f sections i n m a i n p i p e N1 + 1 J a t w h i c h d r a w - o f f o c c u r s ( a n y node 2 to N1) Draw -off Draw-off f r o m m a i n p i p e e n d Flow Draw-off b r a n c h P a t p r e v i o u s time i n t e r v a l H a t p r e v i o u s time i n t e r v a l

s1

P2/D

T

Time i n seconds t o c l o s e ‘ m p i n p i p e l i n e downstream v a l v e Dummy v a r i a b l e Power i n v a l v e a r e a - c l o s u r e e q u a t i o n Length of pipe Elevation

T1

v1 X Z(J)

135

REFERENCES

C h a u d h r y , M.H. a n d Yevjevich, V. (Eds.), 1981. Closed Conduit Flow. Water Resources Pub1 ics. Colorado. C h a u d h r y , M.H., 1979. A p p l i e d H y d r a u l i c T r a n s i e n t s , Van N o s t r a n d R e i n h o l d Co. 503 p p . F r i e d r i c h s , K . a n d Lewy, H., 1956. O n the P a r t i a l C o u r a n t , R., D i f f e r e n t i a l E q u a t i o n s of Mathematical Physics. N.Y. U n i v . I n s t . Maths. Fox, J.A., 1977. H y d r a u l i c A n a l y s i s of Unsteady Flow in P i p e Networks. MacmiI I a n , London. Lupton, H.R., 1953. G r a p h i c a l A n a l y s i s of Pressure Surges i n Pumping Systems. J. I n s t . Water Engs. Stephenson, D . , Oct. 1967. P r e v e n t i o n o f v a p o u r pocket c o l l a p s e i n I n s t n . C i v i l Engs. 9(10) p 255-261. a p u m p i n g l i n e . t r a n s . , S.A. Stephenson, D . , Sept., 1966. Water hammer 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 . Proc. ASCE., 92 (HY5), p71-94. Streeter,V.L. a n d Wylie, E.B., 1967. H y d r a u l i c T r a n s i e n t s , McGraw H i l l , N.Y. Wood, D.J. a n d Jones, S.E., J a n u a r y 1973. Water hammer c h a r t s f o r v a r i o u s types of valves. Proc. ASCE 99, HY p a p e r 9500, p 167-1 78.

136

CHAPTER 10

BOUNDARY COND I T IONS I N WATER HAMMER

DESCRIPTION

The starts The

finite from

conditions

by

the

flow

may

analyst

or

relationship

end of to

difference

the

the

by

the the

steady-state

be

user,

it

and

it

These

is

these

equations

specified

heads

as

time.

defined

head loss-

heads

at

either

i s unnecessary to r e f e r

intermediate

known.

but

a

reservoir

iteration at

at

a given f ri c t i on

given

the f i r s t

characteristic

pipe

flows

from

with

conditions still

the

of

along

determined

After

flow

should

specified

be as

together

line.

initial

conditions

solution

known conditions

points

conditions

but may

conditions

the also

which

end be

dictate

i n what way the flows are to v a r y . Thus the

if

valve

between

valve

closure

is

opening

has

to

discharge

and

head

to

be

the

cause

be specified as well

of

water

hammer,

as the r e l a t i o n s h i p

The

loss across the v a l v e .

l a t t e r can

be g i v e n in equation form a n d the v a l v e opening can also be specified

as

(termed

a

function

valve

(Wylie and Streeter, Where is

a

pipe

generally

level

does

of

time,

stroking

even

can

controlled

be

used

to

by

the

limit

program

head

rises)

1978).

leads

to

specified.

not

or

and

vary

o r from a r e s e r v o i r ,

This much,

may or

be at a f i x e d

the

water

level

the r e s e r v o i r head level may

i f the water vary

as

flow

occurs i n t o o r out of the r e s e r v o i r . Other pipe

boundary

diameter,

conditions

closed

end

encountered

pipes,

or

included

branches

for

changes

both of

in

which

the r e l a t i o n s h i p between head and flow may be determined b y equations

such

as

( 9 . 3 4 ) to

(9.37)

( t h e cross sectional

area of

a pipe

beyond the closed end i s taken as zero).

Conditional B o u n d a r y Conditions

During

the

assumption equation

may

(9.34)

course be was

of

computations

encountered. to

be

below

Thus

the if

vapour

physical

the

limits

head computed

head of

the

to

an

using

liquid,

the

137 liquid the

would

vapour

and the (the

vaporize.

The

so

pressure

would

could

be computed

stream

and

downstream

(9.36).

The

increase

flow

rates

of

in

either

be

u n a b l e to drop

should be set equal

l i q u i d assumed to vaporize.

volume)

two

it

head

the

side

of

pressure

The extent of the v a p o r i z a t i o n

by

calculating

point

vapour

to vapour

below

using

volume

the

is

point,

the

flow

equations the

r a t e up(9.35)

multiplied

by

or

in

difference

the

the time

increment. Chapter 1 1 elaborates on t h i s type of problem. It to

may

limit

limit

be

that

flow

rates

maximum

and

other

the or

flow

forms

engineer

imposes

controls

pressures.

Control

valves

rates,

of

or

water

pressures.

hammer

on can

the

system

be

used

The use of control

protection,

e.g.

to

valves

surge

tanks,

n o n - r e t u r n v a l v e s a n d a i r vessels i s described l a t e r . Spring

loaded

specified figure. head

rate

the

it

has

opened,

above

ate,

and the

there

a

can

pressure

be

set

against

to

them

discharge exceeds

a

at

a

preset

can be programmed as w e l l as the discharge-

unless

open

the

(Fig.10.1 there

A

valve.

further

b ) i s that

the

is

control

also

a

disadvantage

l i n e could d r a i n a f t e r valve

on

the

Tee.

methods r e q u i r e the head to increase before they oper-

by

pipe

this

on

the

valves

through

r u p t u r e disc

Both

in

when

T h i s condition

relationship

of

on

release

that

time

system.

direct

are

There

principle

pilot

sensing

the

is

be

are

too

many

e.g.

operated

first

sub-program

may

it

valves

required

to

to protect other points

control

Figure

downsurge,

late

which

e.g.

valves

1 0 . 1 ~ . On could

Figure

check

the

which operate

the

other

hand,

commence opening

10.le.

I n such cases

operating

condition

of

the v a l v e . It any

should

be

constriction

cavitation

noted is

number

that

pressure

liable

to

too

high.

is

reducing

valves,

in fact

or

cause

cavitation

downstream

The

cavitation

index

if

the

i s generally

of the form

P -P K = - d v P -P u

where vapour as

(10.1

1

d

P

is

and

pressure

and

u to upstream.

low as 0.3

subscript

d

refers

to

downstream,

Depending on the v a l v e design,

a r e possible before the onset of c a v i t a t i o n .

tation

occurs,

vapour

stream

which

actually

bubbles

and

gas

reduce

the

water

release

may

hammer

to

I f cavi-

occur

wave

v

values

down-

celerity.

138 T h i s can closing,

i n t u r n reduce the downsurge downstream i f

in t u r n provides a s t a b i l i z i n g c h a r a c t e r i s t i c

which

On

valve.

the v a l v e

the

other

hand,

cavitation

erosion

of

is

to the

pipework

is

a

troublesome phenomenon to be avoided.

WATER HAMMER PROTECTION OF PUMPING LINES

The pressure t r a n s i e n t s f o l l o w i n g power f a i l u r e to e l e c t r i c motor d r i v e n pumps

are

usually

the most extreme that

w i I I experience.

Nevertheless,

pumps

also

should

c h a r a c t e r i s t i c s often

be

a pumping system

the pver-pressures caused b y s t a r t i n g

checked:

Pumps

with

steep

high over-pressures

induce

when

head/flow

the power

is

switched on so a wave w i t h a head equal to the closed v a l v e head

is

generated.

starting, If

By

partly

closing

the

pump

the

suddenly,

pumps the

supplying

will

flow

an

unprotected

a l s o stop.

water

column

may

cause

than atmospheric pressure. drop

valves

during

is

vapour

separation pressure

may

is

are

stopped

the sudden deceleration of

pressure to drop

to a v a l u e

less

The lowest v a l u e to which pressure could

pressure.

thus

wave

the

pipeline

I f the p i p e l i n e p r o f i l e i s r e l a -

t i v e l y close to the h y d r a u l i c g r a d e I ine, the

delivery

the over-pressures can be reduced.

Vaporization

occur

at

returned

peaks

as

a

or

along

positive

even

the

water

pipeline.

wave

the

column

When the

water

columns

w i l l r e j o i n g i v i n g r i s e to water hammer over-pressures. Unless

some method of

water hammer protection

i s installed,

or

f r i c t i o n p l a y s a s i g n i f i c a n t r o l e in r e d u c i n g water hammer pressures before p o s i t i v e r e t u r n surge occurs, have

to

cv / g

be

designed f o r maximum water hammer overhead equal

(termed the Joukowsky

0

high-pressure

lines

comparison w i t h most

economic

installed

it

a p u m p i n g p i p e l i n e system may

where

head). water

and

may be prudent

I n f a c t t h i s i s often done w i t h

hammer

the pumping head. solution,

even

to

heads

may

be

small

in

For short l i n e s t h i s may be the if

to check

water that

hammer

protection

is

the u l t i m a t e strength of

the 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 against reduce

philosophy water the

behind

hammer

downsurge

is in

the design of most methods of similar. the

The

object

pipeline

in

caused

most by

protection

cases stopping

is

to the

139

Screw - down t o preset s p r i n g load Isolating valve

dh a.

e

c.

b.

Spring l oa de d r e l e a s e valve

A

c

c

u

m

u

Rupture disc

l ator

Pneumatically loaded re1 ease valve

Valve e.g. sluice valve Electric motor actuated contact

d.

Electrically operated valve (opening and closing)

Needle control

Non return valve

e.

Piston seal

, valve

Self actuated pil ot controlled valve could be opened during the downsurge

F i g . 10.1

R e l e a s e valves

140 pumps. even

The

be

upsurge

entirely

will

then be correspondingly

eliminated.

the downsurge i s to feed

The

water

most

common

reduced,

o r may

method of

limiting

i n t o the p i p e as soon as the pres-

sure tends to drop.

S UR

HYDRAULIC GRADE LINE

DELIVERY RE SER H)IR

JAN

3

I

F i g . 10.2 P i p e l i n e p r o f i l e i l l u s t r a t i n g s u i t a b l e locations for various devices f o r water hammer protection

Suitable in

10.2.

Figure

the of

locations

pipe. the

water

Most

Observe

water

umn

i s converted water

ate

the

the

in

of

the

cases

the

is

gradually

the

feeding

prevented to

a

so

slow

k i n e t i c energy of energy

instead of

decelerates

would gather

i I lustrated water

into

sudden momentum change

is

converted

i n heads between the ends.

water column

devices a r e

involve

tank

original

into potential

column

protective

systems

all

beyond

phenomenon

Part

The

difference

various

of

that

column

hammer

phenomenon.

for

under

the

elastic

motion

surge

the water colelastic

the

effect

energy. of

the

I f i t was allowed to deceler-

momentum i n the reverse direc-

tion a n d impact a g a i n s t the pump to cause water hammer over-pressures.

If,

maximum

however,

potential

mum k i n e t i c

the

water

energy,

energy,

there

column

i s arrested

which coincides w i t h will

be

no

at

its

point

of

the p o i n t of m i n i -

sudden change in momentum

141 a n d consequently may at

be the

no

stopped

by

entrance

to

A

pipeline.

small

water

hammer over-pressure.

installing the

a

reflux

discharge

orifice

bypass

valve

tank to

or

the

The reverse flow

or

air

throttling vessel,

reflux

or

valve

device in

the

would

then

allow the pressures on e i t h e r side to g r a d u a l l y equal ize. Fortunately charts

are

and for

i n v e s t i g a t i o n of

hammer

analysis

Rigid

water

is

column

surge tank action, If

the

of

solutions

could

be

system

similar

developed.

characteristics have of

be

to

in-line

mined b y

of

employed

the

protective by

that

so

a water

1981 ) .

for

the

a n a l y s i s of

of discharge tanks.

in-line

reflux

valves

or

hammer a n a l y s i s i s u s u a l l y

size

such

as

and error.

if

a number

a computer

location,

device

trial

a i r vessels

(Stephenson,

systems a r e envisaged,

determined

of

be done g r a p h i c a l l y o r ,

Normally

a

and

a

The

program discharge

discharge

location

tank

and

size

o r bypass r e f l u x v a l v e s may s i m i l a r l y have to be deter\

trial.

In

these

the most economical be

be

incorporates

may

design

necessary

an e l a s t i c water

analysis

of

may

the

i n e r t i a effects,

normally

a n d in some cases,

a pump bypass v a l v e , The

the pump

not theory

pipeline

necessary.

available for

developed,

instances a computer

method of

and

solution,

by varying

program i s u s u a l l y

as a general program could

the design parameters methodically,

an optimum solution a r r i v e d at.

NON-RETURN V A L V E S

or

I n some s i t u a t i o n s the s t r a t e g i c

location of

check

is

least

valve

reduce

or

water

separation

occurs

stream

the

of

reflux

valve,

hammer

the

over-pressures.

installation

pocket

could

of

prevent

a non-return

sufficient

Where

a

non-return

flow

reversal

valve,

prevent

to

water

or

col umn

valve and

at

down-

the

subse-

quent over-pressures. In the

another

type

of

application,

pipeline

from

the

in the main

pipe

drops

suction

water

reservoir

below

or

could a

be

tank

the head outside.

drawn

when

the

into head

T h i s water would

f i l l the c a v i t y a n d likewise reduce the r e t u r n surge. Non-return

in computer line

valve

valves programs.

remains

are When

open

included the

with

flow

minimal

with

conditional

statements

r a t e i s p o s i t i v e then an head

loss.

When

the

inflow

142 reverses,

o r attempts to reverse the v a l v e closes.

head a r e

initially

calculated

Then

the

flow

check

equal

to

zero

on

and

and

if

made.

is

either

downstream

respectively.

If

side of

re-compiled

There

u s i n g equations

may

be

negative, the

valve

using a

(9.32)

the

and

flow

(9.33).

r a t e i s reset

a n d the head upstream

equations

further

The flow r a t e a n d

(9.30)

conditional

and

check

the upstream head i s less than vapour head,

(9.31)

for

head,

i t i s reset a n d

the upstream flow recalculated. The

use of

off-line

non-return

valves

with

discharge

tanks

is

i l l u s t r a t e d i n F i g u r e 10.3.

Fig.

10.3

Discharge t a n k

I n p r i n c i p l e many forms of lines

operate

pipeline This

when

the

water hammer protection f o r pumping

That

is

in

pressure

they

the

the

end

of

The

types

pump

the

line of

will

it

with

hammer

be reduced which

different

effect

over-pressures.

protecting The

analyis

described b y Rich (1963).

the of

then

a

into

pump

the trip.

r e t u r n s from the f a r

thus

valves,

principle,

after

in amplitude

operate

non-return

liquid

least reduces the downsurge.

wave

tanks a n d in-I ine non-return

slightly

insulation

water

protection

bypasses

discharge a

(positive)

discharge

l i n e drops

f i l l s the p o t e n t i a l vacuum o r a t

When

on

similarly.

vessels,

on

pumps,

Surge tanks operate

they

pipeline various

air

flywheels

valves.

i.e.

correspondingly.

include

offer

beyond types

a

continuous

them

of surge

against tank

is

143 AIR VESSELS

Air

vessels

sures o r ping

pumps.

pump

a r e often

to feed water They

of

are

Air

systems.

pressure

used

to

i n t o the also

vessels

the adjacent

cushion water

hammer over-pres-

low pressure zone created b y stop-

used

to

balance flow on r e c i p r o c a t i n g

generally

pipeline.

contain

They

air

at

the

operating

a r e connected to the pipe-

l i n e v i a a p i p e which may have a constriction.

This outlet constri-

tion w i l l

reduce the volume of water forced from the a i r vessel

the

when

pipe

constriction

a

has

low-pressure the

zone

disadvantage

lower a f t e r the pump t r i p than without The a i r vessel outlet porated

in

the outlet

occurs.

This

in

i s created

that

pressures

the

prevents

reduce the

This

pipe are

the constriction.

could also have a non-return

to

pipe.

the

in

into

backflow

returning

when

the

water

v a l v e incor-

p o s i t i v e surge

column

gathering

momentum a n d reduces the volume of a i r needed to cushion the f i n a l maximum r e t u r n

flow.

The

most

efficient

r e t u r n flow

is

usually

a

ff

r e s t r i c t e d bypass p i p e around lets back some flow optimum outlet

the discharge non-return

b u t also acts as a c o n s t r i c t i o n or t h r o t t l e .

combination

of

air

vessel

capacity,

initial

size a n d i n l e t size must be found b y t r i a l .

available for

valve. ,This

preliminary

1981; Thorley and Enever,

selection of

air

vessel

air

The

volume,

There qre c h a r t s size

(Stephenson,

1979).

A i r Vessel Equations

The with

air

the

in

isothermal volume)

the

laws

of

(PS =

and

air

vessel,

physics.

constant,

adiabatic

on

The where

release, expansion P

expands is

in

accordance

usually

in-between

i s absolute pressure and S i s

( P S l e 4 constant).

The r e l a t i o n s h i p generally

adopted i s P S l m 2= constant. The

increment

(10.2)

i n volume of

air,

dS

i s obtained from the contin-

ui t y equation

d S = (Q2-Ql)dt

(10.3)

where

Q2

vessel

connection

is

time period,

the

dt.

discharge and

rate

in

the

pipeline

8, before the a i r vessel,

the

air

averaged over

beyond

the

144

The change i n pressure head

i n the a i r

can be c a l c u l a t e d from the a i r expansion. adjacent

to

the

air

vessel

is

i s out

or

in).

The

new

The head i n the p i p e l i n e

calculated by

loss i n the connecting p i p e o r bypass, discharge

vessel over the t i m e dt

subtracting

(depending on

rates

in

the

the

whether

pipeline

are

head flow then

ca I c u lated u s i n g the respective character ist i c equations (10.4) Qi=Qr+(Ag/c) ( h ’ -hr ) - XQr where prime’

q

is

an

1 Qr I dt/2dA

upstream

point,

(10.5) r

is

a

downstream

point

and

the

refers to the new values.

I f dt i s l a r g e i t may be necessary to use the mean values of Q and Q2 over

the time i n t e r v a l ,

a n d not the

initial

values.

1

This i s

a n i m p l i c i t solution a n d would mean the above equations would have to be i t e r a t e d a few

times

to o b t a i n the discharge rates at the end

of the time i n t e r v a l .

COMPRESSOR

WATER

THROTTLED INLET

F i g u r e 10.4

A i r Vessel

145

REFERENCES

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 Pub1 i c s . N . Y . R i c h , G.R., Sharp, B.B., 1981. Waterhammer P r o b l e m s and s o l u t i o n s . E d w a r d Arnold, London. Stephenson, D., 1981. P i p e l i n e D e s i g n f o r W a t e r E n g i n e e r s . E l s e v i e r , Amsterdam. T h o r l e y , A.R.D. and E n e v e r , K.J., 1979. C o n t r o l and S u p p r e s s i o n of P r e s s u r e S u r g e s i n P i p e l i n e s and T u n n e l s . C o n s t r u c t i o n I n d u s t r y R e s e a r c h and I n f o r m a t i o n A s s o c i a t i o n , R e p o r t 84. L o n d o n . Wylie, E.B. and S t r e e t e r , V.L., 1978. F l u i d T r a n s i e n t s . McGraw Hill.

146

CHAPTER 1 1

WATER COLUMN SEPARATION I NTRODUCT I ON

Water

cannot

is

which

a

f u n c t i o n of

absolute head. instance

tolerate a pressure

by

then

i t s vapour pressure,

i s only

temperature b u t

a

few

metres

When the pressure i n a water column i s reduced, f o r

stopping

atmospheric and energy

the

less than

to

a

pump,

the

pressure

may

reduce

to

below

T h e inherent

the vapour pressure of the water.

no longer converts to s t r a i n energy b u t k i n e t i c energy.

The head may not drop low enough to stop the water column so that downstream of the

the c a v i t y i t proceeds,

same d i r e c t i o n

as

initially,

a l b e i t a t a lower velocity,

creating

a

in

cavity

in

the conduit

between the pump a n d the water column.

The

cavity

will

which

vacuous

come out of

i n the water. air

is

b u t contains water

solution

elastic

subsequent found,

a n d some a i r

due to the lowering of the pressure

i s also possible that a i r w i l l be drawn i n through

It

valves on the l i n e or even

limited

vapour

effect

rise

on

however,

return

that

noticeable effect

in

the

through

the pumps.

cushioning of

the

mass

the

water

of

column.

air

This a i r has a

pressure

is

too

i n r e d u c i n g the subsequent

It

drop is

small

and

generally to

have

a

water hammer pressure

rise. The form of dependent

although

u s i n g mass cases. pies

It the

the c a v i t y created b y the drop i n pressure i s time-

balance

the

total

and

even

i s often s u f f i c i e n t l y full

cross

length

can

thence

known.

Initially

sectional be

volume the

area

determined

the c a v i t y occurs

and

create a is

sharp,

may

pocket. and

also I n fact

in

travel

of at

be

computed

column

rigid

accurate

across the section of the conduit. section

can

accurately

equations

in

some

to assume the c a v i t y occuthe

pipe

and

any

time

if

the

the

effective

volume

is

i n the form of bubbles dispersed

The bubbles r i s e to the top of the

longitudinally

before

coalescing

to

i t i s o n l y i n steep pipes t h a t the interface

longitudinal

pipes

the

cavity

or

vapour

pocket

147

spreads cavity water

longitudinally. laterally.

A bore may t r a v e l u p the l i n e to spread the

I f the c a v i t y

spreads over more than one 'node',

hammer c a l c u l a t i o n s a r e complicated b u t often

be imagined as occuring

i n separate

the c a v i t y can

pockets at nodes (see M a r t i n ,

1981 1.

Spreading o f Cavity

ine

Figure 1 1 . 1

The tion

Shape of c a v i t y

mechanics of

can

(see F i g .

most

water

readily

hammer

be

accompanied b y column separa-

visualized

using

the

graphical

11.2).

Whether employed,

the a

graphical

check

method

or

a

computational

vapour pressure head. Once i t does,

entered s e t t i n g

i t equal

vapour

pocket

method

is

i s made a t each stage to ensure the head does

not drop below

of

method,

to vapour

by adding

a subroutine i s

head a n d accumulating a volume

A S = A V A dt to S each time step, where

AV i s the difference in downstream a n d upstream velocities.

COMPUTATIONAL TECHNIQUE FOR COLUMN SEPARATION

The method of numerically

handling

water column separation o r vaporization

i s to an extent a process of t r i a l .

head

a n d flow

point,

that

may f i r s t

At any time i n t e r v a l

be computed assuming c o n t i n u i t y at each

i s i n f l o w equals outflow.

The head i s computed without

148

heed

separation.

to

pressure head,

it

If

can

the be

resulting

set

equal

head

to

is

vapour

less

than

head,

e.g.

vapour

+

1

m

absolute or -9 m gauge head.

I f , however, there i s an a i r v a l v e on

the

ine

even

the

low,

near

mass drawn

with

the

reveal

the

this).

or

i s generally

size

.IC, .1.e.

point,

'separation'

in

when

ef fec

that

the

water of

At

an

head

air be

subsequently

receiver

the upper

in fact

transported

higher.

The

in air

n e g l i g i b l e a n d w i l l have no cushioning

columns air

considerable

may

to

rejoin

prevent

( a comparison

water

will

hammer

the head w i l l approach atmospher-

limit

i t w i l l be the e l e v a t i o n of the p i p e above datum.

The

flow

r a t e s upstream

and

downstream

of

the

pocket

a r e now

recomputed u s i n g the respective c h a r a c t e r i s t i c equations a n d substituting

vapour

vapour

pocket volume over the next time i n t e r v a l

-Qi)

head

where Q.

A point this

tally

for

i s inflow a n d

of

the

is

at

Q

the

point.

Then

the

i s computed,

vapour

pocket

volume

at

d u r i n g the successive computations.

greater

increase

in

At(Qo

outflow at the p o i n t .

cumulative

i s maintained volume

head

than

zero

the

head

must

be

each As

such

long as

equated

to

vapour head a n d the flows each side of the p o i n t computed thus. When the vapour

pocket

the head i s recalculated

r e v e r t s to zero ( o r t u r n s n e g a t i v e ) then

f o r a continuous water column.

The vapour

pocket may collapse between computational

time

intervals,

in which

case

than

would

predicted

the

head

assuming with

M1

rises

i t rises a t

a

lesser

extent

discharge tank

computational

procedure

i s used to reduce water hammer the i s employed.

The

connected to the p i p e l i n e through a non-return only

be

(compare p o i n t M2

i n F i g u r e 11.2).

I f a one-way same

to

the end of the time i n t e r v a l

when

the

r e t u r n flow volume of

head

in

the p i p e drops below

discharge

is

v a l v e a n d discharges the tank

and vapour pocket collapse a r e eliminated.

separation

tank

head.

Thus

The t a l l y on

i s thus m a i n t a i n e d as f o r a vapour pocket b u t

as soon as i t attempts to reduce,

the non-return

v a l v e prevents t h i s

a n d the head r i s e s as f o r a water column without separation.

SIMPLIFIED R I G I D COLUMN ANALYSIS

If

the

water

column

can

be assumed

to

separate

at

a

known

149

M1

L H

M2

0

\

/ /

\

/

\

/

\

/

\

/ /

\

/

\ \

/

\ \

/

/ /.

\ \

F i g u r e 11.2

)c

I

Drop i n head due to pump t r i p and subsequent r i s e i n r e j o i n i n g of water columns

150

point

and

the

following

subsequent

analysis

reversal.

The

will

downstream

indicate

downstream

head changes a r e small,

the extent

column

may

of

be

column

travel

the

before

to

decelerate

the

difference

assumed

according to the f o l l o w i n g equation:

h

where

is

between

the decelerating

the

elevation

vapour

head of

head,

pressure

head

if

direction

the water column

while

the

water

solution

it

is significant,

downstream

column

r e t u r n velocity

in

could

the

and

it will

reverses.

friction w i l l

without

here

-

v

velocity.

act

i s decelerating,

and subsequently

This

will

cause

a

reduction

in

the

i n d i c a t e a r e t u r n velocity equal to the

i n magnitude: we get (11.2)

the

constant

Now when

of

integration

which

is

the

the column reaches i t s extremity

initial

-\ flow

before revers-

v = 0 hence t = voL/gh

ing,

the

i n an upstream

ght/L is

and

(towards discharge end) d i r e c t i o n when

I n t e g r a t i n g the previous equation w i t h time, 0

pocket

F r i c t i o n head can also

of the water column whereas the f o l l o w i n g a n a l y t i c a l

o r i g i n a l velocity

v = v

be

vapour

the downstream open end.

be added to h

i n the opposite,

which

(11.3)

I n t e g r a t i n g a g a i n gives the distance the water column t r a v e l s (11.4)

x = v t-ghtz/2L S u b s t i t u t i n g f o r t a t v=O gives X

max

(11.5)

= v 'L/2gh

Hence the maximum volume of c a v i t y i s Ax

max

= ALvo2/2gh

(11.6)

Rising Mains

Along occur vapour

gently

along

a

pocket

rising

pumping

considerable front

will

pipelines

length occur

of

pipe.

initially

the

vaporization

The

spreading

as

fast

as

an

of

may the

elastic

151 wave,

but

additional In

the in

following the

analysis

rising

small

proved

travelling

will

refilling

be

slower

owing

to

the

i n the vapour pocket. the

length of

The cross sectional

be v e r y is

subsequent

'elasticity'

formed form.

the

magnitude

pipeline

area occupied b y

i n r e l a t i o n to the total

that

the

vapour

pocket

along

the

p i p e l i n e from

of

the

vapour

i s evaluated the vapour

in

pocket

equation

i s shown to

p i p e cross sectional area. is

rapidly

both

ends,

filled and

by

that

It

surges

no

water

hammer pressure r i s e i s associated w i t h the f i l l i n g of the pocket. To point C

simplify

the

analysis

i s i n c l i n e d at

a r e absolute values,

it

is

assumed

that

a constant a n g l e e to

that

velocity

at

(see F i g u r e 11.3) v

X

= v -9[Ah o c

The

second

caused

by

-(x-x

the

on

X

)sine]-g[t--]

initial

right

negative

deceleration due to g r a v i t y .

a

-(av) ax

aA

+ at

=

0

x

point

the

Heads

between

c

and

study

the

delivery

a f t e r the i n i t i a l negative wave has passed,

0

term

any

p i p e beyond

i s gauge p l u s atmospheric.

F i g u r e 1 1 .3 S i m p l i f i e d p i p e p r o f i l e f o r theoretical

The

the

the horizontal.

is

(11.7)

sine hand

end,

side

surge,

is

while

the v e l o c i t y reduction the

last

term

is

the

The c o n t i n u i t y equation a p p l i c a b l e i s (11.8)

152

1

F i g u r e 11.4

Detail at surge front

where

the

A

is

cross sectional

cross sectional

a r e a of

in c o m p a r i s o n w i t h

area

the v a p o u r

the t o t a l

of

'a'

water may

the

in

I f the

pipe.

be assumed to b e s m a l l

c r o s s s e c t i o n A,

t h e l a s t e q u a t i o n may

b e simp1 i f i e d to

(11.9) (11.7) a v / a x

from

= 29 s i n 8 / c ,

(11.9) may b e i n t e g r a t e d to

so t h a t

yield

(11 T h i s equation At

ha. into

i s s m a l l compared w i t h A.

'a'

t h e open d e l i v e r y e n d t h e h e a d r e m a i n s a t

atmospheric

When t h e n e g a t i v e w a v e r e a c h e s t h i s p o i n t w a t e r the

vapour with

illustrates that

A

pipe.

the

surge

In order

pocket. surge,

at

travels

back

to s t u d y

celerity

c

.

.lo)

along

t h i s surge Apply

the let

head

i s forced back

pipe,

filling

the

the a n a l y s t t r a v e l

momentum

principles

to

the

f r e e b o d y o f w a t e r shown in F i g u r e 11.4.

P ( v +c

1 s

)2

Here centroid expressed

( A - a ) + p g y l ( A - a ) = ( v +c )2AP+PgAhAtPgy2A 2 s P

i s t h e mass d e n s i t y o f

of

the

in

cross section

terms

cont inui t y e q u a t i o n :

of

of

v l , cs,

water and water.

The

a

A

and

7

(11.11)

i s t h e d e p t h to t h e velocity

with

the

v2

may

be

aid

of

the

153

=

( v +c ) A 2 s

(A-a)

i s very

(vl+cs)(A-a) Since to y A. 2

c

(11.12) n e a r l y equal

to A,

y (A-a) 1

i s n e a r l y equal

S o l v i n g t h e l a s t two equations f o r c (11 .13)

am

e

=r

(11 .14)

2 sine(t-)

For m n y practical

s i t u a t i o n s t h i s equation w i l l

between 500 and 1000 m/s,

i I l u s t r a t ing t h a t

it

y i e l d v a l u e s of

i s of

c

t h e same o r d e r o f

m g n i t u d e a s t h e Mave c e l e r i t y o f a f u l I p i p e . The v e l o c i t y change a t t h e surge f r o n t m y b e expressed i n terms o f A h b y r e w r i t i n g (11.12) and (11.13) a s f o l lows:-

I

A[ ( V , + C ~ ) - ( V ~ + C= ~gAhA/cS )

:.

( v -v

1

2

= 9Ah cs

)

PROGRAM FOR SIMULATION OF WATER HAMMER I N PUMPLINES FOLLOW I NG PUMPTR I P

The

accompanying

calculates pumptrip. diameter

water The is

basic

hammer

pump

is

constant,

computer

pressures assumed

but

program

in

at

friction

a

for

pipeline

an

HP-85

following

the

upstream

end,

is

accounted

for.

the

a

pipe

Data

is

pipe

is

r e q u e s t e d i n t e r a c t i v e l y as f o l l o w s ; Line 1:

T h e name of t h e s y s t e m

L i n e 2:

The number of sections

L i n e 3:

The

number

divided travel

of

into 4 time

iterations intervals,

for

pipe) L i n e 4:

Length (m)

L i n e 5:

Diameter (m)

L i n e 6:

Wavespeed ( m / s )

L i n e 7:

F l o w r a t e (m’/s)

L i n e 8:

Friction head (m)

the

4

water

(e.g.

if

the

iterations represent the hamrnmer

wave

up

the

154

Line 9 on wards :

Elevation units)

The

first

and

last

of each successive p o i n t

(all

in m a n d s

.

elevations

should

represent

the

water

level

in

t h e suction sump a n d d e l i v e r y r e s e r v o i r ( b o t h assumed constant).

The

program

starts,

and

assumes

the

p r i n t s out

pump

trips

immediately

the

simulation

heads a n d flows a t each p o i n t every

inter-

val. Water

column

separation

is

accounted f o r

When the head drops to the e l e v a t i o n of t r i e s to drop below

the elevation,

it

the flow

r a t e b o t h before a n d a f t e r

the

head.

new

difference negative

Vapour

between value)

the

the

pocket two

head

difference equations f o r

reverts

a full

i s made.

pocket

i s set equal

can

can

way.

any p o i n t , o r

to e l e v a t i o n a n d

i s computed

and

when

to

pipe.

that

by

i t turns

summing to zero

indicated by

The spreading o r

only

occur

at

the

overestimate

water

hammer

for the

(or a

the

finite

longitudinal

a n d no reduction i n wave

D u e to t h i s s i m p l i f i c a t i o n and the f a c t

closure

program

simplistic

the p i p e a t

movement of vapour pockets a r e neglected, speed

a

the point a r e re-calculated

volume

flows

in

that vapour

computational

time,

pressure

to

pocket

are

plotted

due

the

collapse. The

maximum

heads

at

each

head

along

the

line

above the p i p e p r o f i l e a t the end of the r u n .

L i s t of symbols in p r o g r a m for p u m p l i n e w a t e r hammer a n a l y s i s

A

B D D1 E F F1 F2 F3 G

H H2 I 19

J

cross sectional area c e l e r i t y of wave diameter 5 sum of l / D mass density of l i q u i d unit friction total f r i c t i o n head f r i c t i o n term f r i c t i o n term g r a v i t a t i o n a l acceleration head head term i t e r a t i o n number p r i n t o u t i n t e r v a l (0 = o n l y summary r e q u i r e d , 1 = f u l l l i s t i n g of heads) p i p e number

155

K L M M1 P P1 Q 82 R S

T T2

U X x2 Y

2

number o f iterations name number of p i p e i n t e r v a l s

M + l maximum head h e a d term in p l o t flow beyond point flow term flow in f r o n t of p o i n t a t p r e v i o u s time i n t e r v a l flow beyond p o i n t a t p r e v i o u s time i n t e r v a l flow in f r o n t of p o i n t time increment head a t p r e v i o u s time i n t e r v a l length length interval v a p o u r pocket volume e Ie v a t i o n

Computer Program f o r A n a l y s i s of Water Hammer a f t e r Pumptrip WATER HEMWER I N PIF'ELINES33E' D ~ : J > = I S i l i AFTER PUMP T R I P - WAP 3 4 9 a<.))=. 7 s 5 m 1 , 1 ) ^ 2 28 l31F A<21),B(21j1F<2]).H(21), 358 nl=nl+l/D<J>*5 P<21>~Q<21>rRC21~~U~Z1~~ NEXT 2 ~ 2J1 3 6 9 378 H i 1 > = Z C fi 1> +F 1 ) 38 @ I N D C 2 8 j . Y C E l > , T ( 2 1 ~ , 5 ( ~ 1 ) 3 8 8 FOR J = l TO M l ! P I F E SECNS 4 8 DISP "PUl'lPST@F P I F E NAME'; 398 C!iJ>=C!*:l) 5 395 Tc.J)=Q(.-I> -8 - TNPIIT - .. - . I- S 52 D15.P "HCI. SEWS': 68 IHPIJT M 7ci . - H. I_= M + l 8 8 D I C P "NO ITERHTIl3NS ' I . 428 P ( J ) = H < J ) 99 IHF'UT Y 438 NEXT J 1 8 9 @ I E P " P I P E LENGTY.rn". 4 4 8 P P I N T 'WATEPHAtlMER " J L ) 1 1 1 7 TUF'IIT S 45fr FOP I = 1 TO Y 469 I F I S < l THEN 518 478 P R I N T a PNT HEfiDmi FLOWIS"i I

16 !

'

156

778 PRINT WEADS UE PlUHP

e ieew

786 P l = P < 1 > + 2 6 8 799 21=2<1~-168

868 GCLERR 919 SCRLE l . M l , Z l , P l 826 X A X I S Z ( l > , l 839 Y A X I S 131ee

848 MOVE 1,2(1) 359 FOP J=2 TO Ill 868 @RAW J , Z ( J ) 878 NEXT J €!Re MOVE 1 , P C l ) Y98 FOP J=2 TO IT1 988 DRAW J , P C J ) 918 NEXT J 928 COPY 938 STOP 948 END

N ICF

FIE

157

output

PHT HEflDa FLOms 10 I 250 @.Be@ 1 197 - . 0 3 4 3 195 - . % 3 2 4 193 -.El34 = 268 - . R 3 8 FGT HEflDn FLOms 11 256 9.8@@ I 24s .mi 2 ‘ ;95 - . e x 3

4

282

5

206

PN r W R O n 245 1 248 2 255 3 Z@2 4 288 5 PHT HEFlOm 245 1 252 2

3 1

255 252

5

-.a36 -.I338 FLOms 12 0.@B@ .gel

-.00J -.@34 -.035 FLOrns 13 0.0B0

-.@84 -.Re3

-.eel

208 -.635 PNT HEFlOmr FLOas 14 1 259 8 . 0 0 0

2 3 4

5 4

. lee

288

F!4T YEADm FLOms i 38 6.8@@ 97 . P26 ,834 i em 3 4 97 I032 e33 5 286 PHT HEAQm FLOas 138 e . eee @(?2 141 . e34 I80 282 - . # 3 1 4 . e33 288 5 PHT HERDm FLOms 143 8.088 1 2 141 - . B e 2 3 195 - . e x 4 292 - . a 3 1 5 288 -.e2Es PNT HERDm FtOns 143 e.eee 1 197 - . 9 3 4 2 3 195 - . 8 X 193 - . 0 3 4 4 5 2013 -.ECG

-

-

252 208 ~~

~

7

-.

-

-.a04 -.@a3

-.Be1 .a31 PNT HEAOm FLOns 15 1 259 @ . e m 2 z57 .@el 3 250 - . 8 0 3 4 la8 .@38 5 200 .031 PNT HEROm FLOns 16 1 254 @ . m e 2 257 .eel 3 285 .ew 4 198 ,038 5 200 .828 PNT YEROm FLOlrs 17 1 254 e . 0 m

5

6

252 250

8

3

2 3 4 5

243

3 4

205

.@31

207

.a32

.832

205 .834 287 .032 298 .828 PNT HERDs FLOms 18

150

HE

I nput

159

REFERENCES F o x , J.A., 1977. H y d r a u l i c A n a l y s i s of U n s t e a d y F l o w in P i p e Networks. Macmillan, London. Martin, C.S., 1981. Two-phase f l o w s . I n 'Closed Conduit F l o w ' , and Y e v j e v i c h , V., W a t e r Resources Edd b y Chaudhry, M.H. Pub1 i c s . , C o l o r a d o , p225-251. Stephenson, D., O c t o b e r 1957. P r e v e n t i o n o f v a p o u r p o c k e t c o l l a p s e in a p u m p i n g l i n e . Trans. SAICE, 9 ( 1 0 ) , p255-261.

160

CHAPTER 12

WATER HAMMER AND FLOW ANALYSIS I N COMPLEX P I P E SYSTEMS

THE PROBLEM O F FLOW ANALYSIS

In

many

municipal

the

pipework

as

the

be

predesigned, and

even

columns ping

more

of

a

grow.

fact

are

pump

can

often

often

of at

hammer

be

the

and

main

ends

often

most

severe

pipe

may

diameter of

these

in

arise

pressures

systems

i s developed

supply

variable the

problems

water

distribution

layout

the

system

Similar

the

water

geometric

Although

distribution

complex. in

industrial

regular

extensions

the

and

and

no

requirements

lengths is

follows

IN COMPLEX P I P E NETWORKS

due

and pipes

pumping

to

the

pressure

trip-

condition

i n the p i p e l i n e . Where h y d r a u l i c of

a

e.g. as

mine,

the

above

5

it

the

of

high and

are

quently

per

pipe

an

else

the

diameter

mains

water

directly accurate

a

the

such

are

The

velocity

high

thus

saving

water

in

is

to

flow

be

with

velocity

protecting

cost.

In

very

be v e r y h i g h .

pressure analysis of

high

accepted

can

also

associated

the

very

capital

heads

hammer method

often

gradients

pipe velocities w i l l

proportional

as down t h e s h a f t

velocities

hydraulic

hammer

sophisticated

a g a i n s t w a t e r hammer

be h i g h ,

pipe

second.

the corresponding

flow

or

the

gravity

Unfortunately of

metres

reduces

case

pressures can

corresponding

the

changes

and is

conse-

necessary

pipe

system

i s required.

T h e r e a r e t h r e e m a i n s t a g e s i n t h e a n a l y s i s of a w a t e r r e t i c u l a tion to

be

system. sized

During and

instal lation. in

order

to

off-takes

Pipe

the

available

head

below

supply

the

losses between

along

should

wall

planning

will

wal I

depend

of

on

sections of

the

and

the

of

the

the

the

Where

be r e d u c e d .

elevation a

pipes

of

pumping pipe

the

can

pressure there The

supply

head.

p i p e s d e p e n d i n g on f l o w

have

selected b e f o r e

greater

costs.

t h e p i p e may

r e s e r v o i r o r else on

lengths

stages

thicknesses

reduce

thicknesses

diameter

individual

various

initial

bores

minimize

then

the

the corresponding

be

The

are total

point head-

distributed

rates.

I n fact

161

simple can

linear

be

used

network. select

select

to

successively

Pipe

cost

methods

(Stephenson,

p i p e diameters

i n such

1981 )

a

branch

d y n a m i c p r o g r a m m i n g methods c a n b e used to

decreasing

diameters

down

deep

mine

shafts,

1982).

sizes

later

optimization

least

Alternatively

(Bernstein,

in

programming

are

years

based

the

on

flows

an

initial

may

be

estimate of

increased

or

design flow

altered

and

but it

is

n e c e s s a r y to a n a l y s e t h e system to d e t e r m i n e t h e f l o w s w h i c h c a n b e obtained from pipe

for

be

head

each

from

there

a

are

successive

by

or

Weisbach

i s not

unknowns

it

the

friction

will

bores

change.

The

with

affected

iterative

resistance equation

i s often not

determine

pipe.

As be

matter as

the

to

flows

problem

time.

the

i t must

i s often necessary

lengths.

be

that

i s not a n easy

factors

Although

important

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

it

to

pipe

is

i s taken from a

known e x p l i c i t l y

This

and

methods

individual

scale

flow

tication

of

changing

losses

then

headloss equations.

the

where water

a t each b r a n c h a l o n g

approximation

aggravated

coils

branch point

number

in

pressures

friction

For example

cooling

known

at

calculated

corrode

system.

supplying

pressure the

the

As

and

may the

use

be

pipes

a n d the corresponding solution

of

the

Darcy-

i s possible t h i s degree of sophis-

w a r r a n t e d as the exact roughness effect on the

h y d r a u l i c c a p a c i t y can o n l y be estimated r o u g h l y anyway. The

third

problem

determination valves

or

hundred

of

tripping

metres

if

heads

so

water

pumps. the

d u e to the c l o s u r e of static

in a n a l y s i n g

transient

that

The

a it

water

in

flow

a

valve. may

p i p e r e t i c u l a t i o n systems hammer

pipe

pressures

due

hammer h e a d c a n b e s e v e r a l i s reduced

rapidly

such as

T h i s head should be added onto the reduce

the

factor

of

safety

system a n d i n f a c t h a s been k n o w n to b u r s t p i p e s o r f i t t i n g s . closure

times

pressures taken

to

Water

hammer

branch main

must

being minimize

pipes pipe

systems c a n

therefore

excessive water

i s the closing

to

be

and

hammer

selected simi l a r l y pressures

to

prevent

water

precautions due

to

have

of

the

Valve hammer to

be

pumps t r i p p i n g .

pressure waves c a n a l s o be reflected from the ends of and

where

be the

more

severe

flow

b e f a i r l y complex

accurate analysis.

is

in

the b r a n c h p i p e

altered.

Since

water

than

in the

reticulation

the use of computers i s necessary f o r

162

The the

two

problems

calculation

of

of

analysis

water

hammer

of

flow

in

pressures

have

been

combined

which

i s described

here.

in

and

a

program

i n feeding

data

the

The t w o pro-

approach.

developing

The e f f o r t

networks

due to c h a n g i n g of

flows can be done u s i n g one computational blems

pipe

the

b a s i s of

i n t o a comput-

er f o r the v a r i o u s analyses i s thus minimized.

CONVENT I ONAL METHODS OF NETWORK ANALYSIS

The

standard

complex

methods

piping

systems

of

analysing

flows

developed

were

by

and

headlosses

Hardy

Cross

in

1936

in

and are s t i l l

used b y many engineers on account of t h e i r s i m p l i c i -

ty,

of

of

the ease doing

ers.

In

the

visualization

calculations

earlier

of

the

manually

chapters

these

procedure and the p o s s i b i l i t y

instead of

methods

r e s o r t i n g to comput-

and

others

more computer

orientated a r e described. Very

mathematical

simultaneous ships

in

proach mer

a

methods

equations pipe

such

describing

network

have

as

the

been

the

flow

solution of

proposed.

An

Starting

with

any

assumed

flows

flows w i l l e v e n t u a l l y s t a b i l i z e at t h e i r steady-state Although

slightly

determination in

of

fact

hanced b y

flow

more

the ease of

r e l i e s on

the

operated,

then a f t e r a

waves and

that

travelling

flows

and

extended

predict

in

the system.

ly

occur

analysis

will

pressures to

at

computationally

s u p p l y of if

a

d a t a to

valve

length of

backwards

a steady-state

if

the

apham-

heads

the

values. i s only

the

i s considerably

the computer.

en-

The method

i n a p i p e network

is

forwards

in

the

pipe

network

One therefore has the steady-state

throughout pressure

-

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

and

pump

of

time f r i c t i o n w i l l damp the e l a s t i c

and

emerge.

or

set

water

and

problem

r a t e s i n closed networks,

i s efficient

fact

complicated

a

relation

alternative

i s to simulate the system u s i n g the d i f f e r e n t i a l

equations.

dure

of

headloss

the and

network. flow

The

histories

method at

can

every

be

point

These a r e the pressures and flows which w i l l a c t u a l -

any

methods

interval, described

which

is

not

previously.

the

case

for

Consequently

the if

network transient

conditions a r e to be studied the method w i l l y i e l d them as well.

163

There

have

analysing

water

networks. for

also

of

the

as

applied

varying

to

pressures

pumping

finite

and

are

boundary

in

programs

more

All

is

more

and more

motion,

easily

the methods s t a r t

for pipe

rapid

the meth-

i s generally

straightforward

conditions

t r e d f i n i t e difference methods.

1967)

Wylie,

developed

mains

differences

equations of f l u i d

(Streeter

computations

the

computer

method of

the d i f f e r e n t i a l

characteristics

ferred

many

hammer

Although

solution of

od

been

and

than

pre-

can

be

the cen-

w i t h the basic

d i f f e r e n t i a l equations of c o n t i n u i t y a n d momentum,

(12.1)

(12.2) The

city

h

symbol

a n d potential averaged

diameter, is

Reynolds vlvl

is

the

of

across

is

t

the

denotes

friction

is

its bulk its

ing

characteristic velocity

an be

space.

at

point

explicit the

v

at

when

d

is

For

form, the

that

the

along

of

pipe,

the

is

d

head

i s the p i p e

direction

which

can

acceleration

flow

which the

of

flow,

vary

with

the

term

and

changes may

mass

steel

give

direction.

be

shown

density

water

the

or

t

of

its wall

v time

in

friction

the

friction

interval

and

c

is

equal

K

thickness

is

generally

pipes.

Adopt-

then the ensubetween

conditions

the

to

the f l u i d ,

pipes c

relationship

boundary

Invariably

previous

pressure

i s the water velo-

increases f o r t h i c k - w a l l

initial

time.

v

i n d i c a t e d i n F i g u r e 12.1

equations

in

of

the p i p e diameter,

but

grid

known

the

is

P

modulus.

1100 m/s

and

in

sum

i t accounts f o r c h a n g i n g d i r e c t i o n

celerity

where

the time-distance

to

(the

coefficient

v 2 as

wave

modulus, elastic

the order of

new

is

headloss

+ d/Et)]

a

distance

gravitational

g

i n place of

P(l/K

at

is

number, used

1/J[

ing

cross section

friction

hammer

of

the

x

time,

water

E

head

Darcy-Weisbach

the

and

water

head above a specified d a t u m ) ,

and

equation term

head those

i s used

i s assumed

previous

point

in

164

TIME

12.1

Fig.

x-t

for

Grid

water

hammer

analysis

by

characteristic

method.

METHOD OF SOLUT ON O F THE EQUATIONS

the f i n i t e d i f f e r e n c e f o r m of t h e d i f f e r e n t i a l dx and substituting - = + c yields dt

equations

Starting with

hb' - h where after

+

t p

p + c(v ' g b is

t

pumping

iteration,

are kv2/2g. to

and f

top

+ fvlvl

head,

and

=

(AL/D + k)/2g,

(upstream)

(12.3)

= 0

'

the p r i m e

b r e f e r s to

Subscript

the

v ) t

-

end

values

where losses d u e to f i t t i n g s

t h e bottom of

r e f e r s to new

a

(downstream) end a n d

pipe

although

these

are

a r b i t r a r i l y d e f i n e d as the flow d i r e c t i o n can v a r y . Generally i.e.

it

the v e l o c i t y

difference friction

loss

is

this

shaft can

result

several

valve.

the

small

equations.

compared

not

the

is,

That

with

case,

for

is

water

satisfactory hammer

head.

provided I n some

e x a m p l e a p i p e b u r s t down a

i n a s t e a d y - s t a t e f r i c t i o n h e a d loss down t h e s h a f t

hundred

It

XLv2/2gd,

t h e p r e v i o u s t i m e i n t e r v a l i s used i n t h e f i n i t e

of

is

to use t h e e x p l i c i t f o r m of

metres.

in many s i t u a t i o n s ,

significant a

at

forms

situations

of

is sufficient

nevertheless,

The e.g.

head

loss

when

term

the head

kv2/2g

is

also

loss i s t h r o u g h

much e a s i e r from. t h e p o i n t of v i e w o f

c o m p u t a t i o n s to lump t h e two h e a d loss terms t o g e t h e r .

165

is,

It

on

equations

in

term

that

at

sion

results

the

implicit the

head

loss t e r m known

downstream

hammer

form,

difficult

i.e.

with

was solved

in

are

the

equation. and

head

A

of

nodes.

The

head

at

effects

from

each

branch

a

such

form f o r valve.

a

by

In

head

loss

at

general

node

arises

the

is

when

the

head

the

water

there

or

by

i.e.

the

b y simultan-

and

upstream

obtained

employing

the

a s i m p l e case,

In

equation

complication

9

chapter

a n d must be obtained

diameter

in

characteristic

in the

cumbersome.

loss/discharge

further

changes

the

the velocity

implicit

downstream

i s also a n unknown of

solve

to

I f t h i s i s attempted a q u a d r a t i c expres-

computations

head

solution

branches

hand,

new time.

and

with

eous

other

are

downstream

summating

continuity

the

equation

at

t h e node.

I t should a l s o be noted t h a t h e a d loss

the the

line.

i s assumed

An

averaging

c a t i o n s and,

in a n y

the answer w i l l depend on whether

to b e a t

the f a r

procedure

case,

would

concentrated

e n d or

t h e n e a r e n d of

introduce

further

compli-

h e a d losses s u c h a s v a l v e s

a r e u s u a l l y a t one e n d o r the other. to a v o i d m a n y o f these d i f f i c u l t i e s f o r a b r a n c h e d p i p e

I n order network

some s i m p l i f i c a t i o n

obtained,

but

solution.

This

complete

i s necessary.

a degree of implicitness method

imp1 i c i t method

is

sometimes

i s complex.

a f u l l y e x p l i c i t f o r m a n d a semi fvlvl

=

F fv

b

Iv

b

I

Substituting i s the drawoff solving for h

b

solution i s

i s i n t r o d u c e d to s t a b i l i z e t h e unstable,

but,

as

stated,

a

A w e i g h t e d compromise between

i m p l i c i t form i s

+ (I-F) f v ' ( v b l . vt+?(ht-hlb-p-F

T h e n s o l v i n g (12.3)

A semi-explicit

vb'

=

f v b I vb 1

)

(12.4)

I

1+( 1 - F ) f g v b I/c

i n t o the c o n t i n u i t y a t node b c o n n e c t i n g

' A . v . = qb where q I bi b i pipes each w i t h area A , , a n d

equation

I

(12.5) This

i s then

substituted

into

(12.4)

to o b t a i n

velocity

vb'

i n each

p i p e leading head

equations.

Explicit

computations i f I n fact

I t w i l l be observed that the e x p l i c i t form of the

to b.

equation

loss

when

solutions

is

not

solution

the time

the

pipe

i s based on

head

take the average flow interval.

iterative

This

and

it

the

the

above

instability

i n the

i s to be considered the numerical

through

I n fact

the

valve

or

errors.

through

the

velocity

would

each

the most accurate method would be at

the b e g i n n i n g a n d end of

render

the

equations

node.

The procedure

previously

described

equations

have

I n order

microcomputer solution

to

therefore made. of

serious

to

in

selected f o r a n a l y s i s i s excessive.

loss

quadratic

basis fo r

and adapt

lead

used

the v e l o c i t y at the previous time i n t e r v a l and not

a t the new time i n t e r v a l .

time

can

interval

form

been known to become unstable and magnify

have

because

is

cumbersome

only

r a p i d v a l v e closure

This

to

the

to

be

simplify

to

the

exceedingly

solved

on

an

the procedure

the above s i m p l i f i c a t i o n was

i n effect adopted a weighted average

pseudo-

explicit-implicit

method and a

p u r e l y e x p l i c i t method. It

also

is

determination stream end of flow

assumed

i s that

at

the pipe.

variations

the

that

the

relevant

velocity

the end specified as the

for

loss

head

'bottom' o r down-

Although flow d i r e c t i o n s can change d u r i n g

'bottom

end'

is

in

fact

specified

as

a

fixed

position b y the a n a l y s t . Where is

used

water purely

hammer a n a l y s i s

i s not

for

and

the

analysis

important

and the program

determination

of

steady-state

flows i n a p i p e network then damping can be increased b y assuming a very each being

low o r a r t i f i c i a l water hammer wave c e l e r i t y .

p i p e i n the system to

be analysed

i t s pipe length divided by

successive

analyses.

i s defined b y

the user as

the selected time i n t e r v a l between

Thus b y selecting a long time i n t e r v a l between

successive computations the water

hammer

friction

The

controls

The c e l e r i t y of

the

equations.

converge to steady-state

flows.

waves a r e e l iminated a n d

flows

Obviously

will

therefore

rapidly

t h i s technique should

not

be used where t r u e water hammer heads a r e r e q u i r e d a n d i t may be necessary one second,

to

select

a

smaller

time

interval,

f o r water hammer analyses.

f o r example

less than

167

BOUNDARY COND I T IONS

The

head

fixed by

at

the head

head

changed

very

low

the

a pump

the

water.

the

head

travel

p o i n t s and discharge

points w i l l

computations.

There

The

is

also

stopped

the

problem of

in a

low-head

drop

will

and

below

therefore

down

the

the

specified

grow

pipes.

When f o r

pumping main

drops and t h i s may vaporize

program accommodates t h i s effect not

pocket up

i s suddenly

hammer head i n i t i a l l y

will

be

i n a r e s e r v o i r and such nodes w i l l not have the

in

the water

vapour

input

heads which may cause water column separation.

instance, then

certain

and The

automatical l y and

head a t

diminish

in

any size

corresponding

A

node. as

water

waves hammer

head when the water columns r e j o i n w i l l be computed automatically. D i s t r i b u t i o n pipes often have pressure reducing relief

valves

pressure

installed

reducing

to

ensure

valve

can,

no

in

excessive

fact,

be

valves or surge

pressure

treated

as

rises. a

A

reverse

PUMP. The p o s i t i o n i n g of control

i n the p i p e network i s a l s o of

valves

importance from the point of view of r a p i d a c c e s s i b i l i t y and closure in

times of emergency or for to

timed

close

over

hammer pressures. be

installed

limits.

valves at of

a

to

The

operate

the

when

can

point

in

specified

purposes.

period

in

Such valves can be

order

to

I n the case of automatic control

program

any

valve

a

control

pressures

accommodate

or

the

control

water

valves they can

flows

exceed

opening

or

certain

closing

of

i n time a n d the combined opening and closing same simulation

can be made by

i m a g i n i n g two

v a l v e s i n the same position. An

application

analysis

of

Although

pipes

with

selected

abnormal system

of

to

a

program

are

normally flows

be

to

used

specified

that

will

meet

have

to

be

loads w i l l

flows

various

duties,

the

is

in

the

conditions.

there are often

Where the water

fire-fighting at

for

a reasonable load factor

certain

be considered. for

steady-state

system

designed on

to

points and

other

to

reticulation

design

pressures

be assumed

the

in

situations

is

required at mum

flow

then same

achieved.

In

high

reticulation

peaks

may

be

time specified m i n i such

cases

it

can

be reduced o r eliminated i n order

to achieve the necessary peak flows.

168

300 mm d i a . 6 0 0 m 1 5 0 mm d i a .

mm dia.

250

X (4) H = 0

(2)

12.2

Fig.

Simple p i p e network analysed f o r flows and heads

The a n a l y s i s of er

pressures at

The a

water

pump

valves,

to

be

down

in

is

The

and

the

the

at

may

was

shut

the

system

is

be

caused

by

more p r o b a b l y

the

An

a

have

example

to

t a b u l a t e d at can

closure

time

For

rapidly of

in

adjusted

can

be

closing of which

such

then

likely

a h i g h velocity the

an

the

The

intervals.

supply

to

a

analysis

are

valve

by

trial

in

pipe

intervals).

heads

at

each

The control valves

operate over

selected

a valve, is

control led as e l e c t r i c a l

seconds.

to

of

The system was i n i t i a l l y

heads

5

interest.

tripping

i t e r a t i o n 20 u s i n g 0.55

specified time be

the

case

instance

results

steady-state (at

also of

supplies water at

be

the

of

i n F i g u r e 12.2.

uniformly

turbines

u s i n g an a d d i t i o n a l

in may

pipe

10 sec

=

t

determining maximum water ham-

h i g h pressure.

determine

to

valve

such

a

node number 2

was shut

node were to

where

at

I n f i g u r e 12.5

load i s shed.

analysed

or

system.

turbine

plotted for

2-4

the

shaft

hydraul ic

points

pressures

the opening

severe a

various

hammer

or

or

the system f o r

specified

and error

times or

by

a l g o r i t h m on the program described here w h i c h

enables t h e program to recommend the v a l v e closure t i m e .

VALVE

The

STROKING reader

has

been

presented

with

methods of

analysing pipe

169

systems f o r flows and

for

transient

indirectly hammer

the

conditions.

diameters wi II

analysis

both under steady

The network

required

then

for

reveal

flow conditions

analysis for

the

various

maximum

flows gives

pipes.

pressures

Water

in

which,

i n d i c a t e p i p e wal I thicknesses.

turn,

Both and

and pressures,

types

error

pipes

can

comes

with

is

directly

sized

a

pipe

of

a

to

it

answers,

the

case

of

optimization the

direct

comes

to

Thus

if

view.

system and

resort

In

using

analysis

when

point

indirect

required.

hammer

difficult

hammer

h e may

yield

be

more

water

analysis

approach

water

to

often

of

a

some

When

approach

operation

engineer

i s r e q u i r e d to specify

trial

networks,

methods.

design

optimum the

i.e.

from

it is the

i s confronted

v a l v e closure

times

and e r r o r approach - r e q u i r i n g an a n a l y -

trial

s i s f o r each assumed v a l v e closure r a t e . In

some

closure

cases

a

direct

relationship

specified

limits.

complicated in

turn,

of

a

by

solution

where

The

pressures

control

the fact

may

of

that

are

flow

flow

be obtained to

in

a

gives

rise

to

a

valve

maintained

pre-determined

within way

is

r a t e depends on pressure whi'ch,

i s affected b y changes i n flow.

valve

be

for

pressure

For instance,

increases

the closure

upstream,

which

in

t u r n can affect the flow r a t e through the valve. procedure of

The pressure

rises

(Streeter closure In

and time

fact

the

closure set

to

Wylie, is

of

values

A

1967).

possible f o r

minimum

valve

a

valve

i n a p a t t e r n which

is

referred

mathematical

single

pipes

closure

time

as

to

valve

solution

w i t h or

stroking the

to

without

i s yielded

limits

valve

friction.

as well

as the

r a t e of closure i n v a r i o u s steps. For (while is

a

single

frictionless

maintaining

obtained

mencement uniformly,

with

of

the

the

that

the

at

some

head

following

operations

such

pipe

at

the

the

head

minimum point

operating discharge at

point

B

valve less

procedure. end 'B'

closure than At

valve

along

HBmax )

the

A

the

time

com-

i s closed l i ne rises

i n the time 2L/c (see F i g u r e 12.3). T h i s i s assuming HBmax i s less than Ho+cv/g. The actual flow r a t e through the HBmax v a l v e a f t e r 2L/c seconds may be solved from the water hammer to

equation

applied

charge equation V

to

A

=

the

valve,

K

5.

AHA

=

cAv/g

and the v a l v e

dis-

170

Now

the

closing

of

the

valve

m a i n t a i n i n g the head at B equal

i s continued

at

a

uniform

rate

to HBmax.

Ho -

B 1'2.3

Fig.

A

Maximum head along pipe

Complex pipe system

If a valve a point ship

where

between

branch

the

pipes,

relationship

some

changes

re1at ionsh i p

'A'

and

in

even

to o b t a i n

point

diameter

flat

too

can and

become

cumbersome.

friction

more complicated.

head

AhB

computed a t each successive or

effect

are

For

involved

such

cases

it

If the is

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

and

between

p i p e lengths away from

i s to be l i m i t e d , the mathematical relation-

cause

becomes

often simplest at

i s to be closed a number of the head

rise

and

at

a

partial

time step.

the v a l v e closure over

'B'

point valve

by

closure

I f the g r a d i e n t

the next

trial.

time

could

The be

i s too steep

interval

could be

adjusted geometrically.

function

a t a specified node B. After one i t e r a t i o n w i t h an assumed

trial

valve

than

expected

ratio S(H

12.4.

i n F i g u r e 1 2 . 4 i s a target head r i s e

l i n e HBOD

Thus assume the

closure

time

the

so

the

valve

closure

C)

where

S

BO correction

to

the

HB

head

D)/S(H

60

The

T,

is

could

the

closing

rises

slope

be

C.

to

speeded

of

the

speed

of

the

if

time

T h i s i s less up

by

the

l i ne i n Figure valve

could

be

intervals

are

too

continued each step.

It

will,

however,

be found

that

the

l a r g e o r the p o i n t B i s too close to the v a l v e A unexpected o r even misleading

answers can occur.

that overshoot

or

A target closure

171

Target HB

t

/

t

T 1st t r i a l closure time

Feedback method of c o n t r o l l i n g head a t B

Fig.

12.4

time

and

gate

and

this

can

closure c h a r a c t e r i s t i c only

occur

somewhere

drop,

obviously

be

in

must

refined using

the

system

be estimated beforehand

this

technique.

causing

the

the p r o p o r t i o n a l correction

If

heads

reflections

to

reverse

or

i s unappl i c a b l e a n d the

v a l v e i s being closed too q u i c k l y or the time step i s too great. The

accompanying

p r i n t o u t s and p l o t s o f

computer

heads versus

time a r e from a program based on the above p r i n c i p l e . The programmer may e n q u i r e from the program over what t i m e he should close the v a l v e i n o r d e r to not exceed a specified head a t a specified

point

feedback

in

the

pipe

network.

The

method

is

to

employ

a

p r i n c i p l e a n d g r a d i e n t method f o r p r o j e c t i n g water hammer

heads a t the specified node.

The program w i l I therefore successively

correct

the

the

closure

increase a t node

will

whereas

v a l v e keeping an eye on

the designated node. be

subsequently maximum

time of

plotted be

head

on

the

transferred

specified

for

I n fact the head a t the designated

screen

to

the head

a

as

graph

node 2

was

the maximum which a c t u a l l y

computations such

as

1100 m

proceed

and

F i g u r e 12.6.

The

above

the datum,

occurred was 1121 m c.g.

1167

m f o r the v a l v e closure in 5 seconds i n F i g u r e 12.5

The most severe such conditions a r e is

a

burst

closure

in

a

upstream

pipe of

the

which burst.

is

to

At

likely

to

be followed steady-state

a r i s e where there by

a

rapid

runaway

valve

conditions

172

the flow the

heads

most

of

fittings the

velocities can at

i n the

damping

hammer under

the

the head

reach very of

end will

the

pressure

of

rising

normal o p e r a t i n g

the other

system a r e c o n s e q u e n t l y

very

hand

low

as

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

p i p e system.

effect

On

h i g h values.

the

The f r i c t i o n i s t h e r e f o r e v e r y h i g h a n d friction

may

often

n o t much more t h a n conditions.

result

that

for

in

the

water

valve closure

N e v e r t h e l e s s e a c h system s h o u l d

be a n a l y s e d i n d i v i d u a l l y b e f o r e s u c h c o n c l u s i o n s a r e made.

REFERENCES

Fox,J.A., 1977. Hydraulic Analysis Networks. M a c m i l l a n , London. a n d W y l i e , E.B., 1967. Streeter, V . L . Hill.

of

Unsteady

Flow

Hydraulic Transients.

in

Pipe

McGraw

173

PIPE

PIPE 0 1 1 2 2 4 1 3 PIPE e1 21 2 4 1 3

... . ... . .

i

,

..., ,.

- - _H E R D ,

,

AT2 .IVAL~~!EUPO-I

FIFE 0 1

1 2

2

4

1

x

E 1

J 3 FIFE 0 1 1 2

FIFE 0

1

1

2

2

Tfi JT I 14 E - I 1.0 T E F: A T I0 H

4

1 3 FIFE E l ! 1 z

2

4

1 3 FIFE a 1 1 2 2 4 1 3 FIFE 8 1 1 2

2

4

1 3 PIPE 0 1

1

2

2

4

1

3

Fig. 12.5 Head v a r i a t i o n s a t node 2 due to v a l v e c l o s u r e i n 5 s a f t e r 10 s convergence

'1.0

174

T

F I F E t l E T WATEP H A M M E F HtlkiL FFCll;

2 4 1 3 PIPE 8

1

1

2

2

4

1

3

8 185 FLOW L.5 PIPE 1 SE' a 1

1

2

2

4

@

B 1

1 2

136

2

4

- 3

1 3 195 F I F E FLOW L'S

1

3

PIPE B 1

1 2

2

4

1

3

1 8 192 F'FLCI 196

F i g . 12.6 Head v a r i a t i o n s a t node 2 a n d v a l v e c l o s u r e p a t t e r n i f maximum h e a d a t 2 i s l i m i t e d t o 1lOOm a b o v e d a t u m

T 11 H X 17.00 16.58 8 .0 0 8 . 88

175

CHAPTER 13

GRAPH I CAL WATER HAMMER ANALYSIS

REASONS FOR GRAPHICAL APPROACH

Although water

not

advocated

hammer

analysis.

for

general

calculation

The

still

graphical

use,

retain

method

the

graphical

in

place

a

proposed

by

methods of

water

Allievi

hammer

(1925)

and

developed b y Bergeron (1935) a n d Schnyder (1937) i s well suited f o r illustrating

the

mechanics

of

water

clearer

understanding

of

students

than

use of

with

speed,

changes

column

separation

The

technique

and

the

in

does

Computer

especially

the

diameter, and

manpower

excessive. cases,

the

look

solution

wave

wave

reflections

the

effects

can

all

laborious at

is

of be

if

more

possible

valve

closure,

steps

in

water

graphically. are

designs

economic

by

V a r i a b l e wave

portrayed

many

alternative

A

reflection.

is

a computer program.

friction

become to

hammer

required,

often the

becomes

majority

of

i f m u l t i p l e solutions a r e required.

BASICS O F METHOD

The

graphical

method

If

the

linear relationship

one

+

the known crosses

plots

head

o r -c/g.

another

way

H against

velocity

V

have

a

The procedure i s therefore to p l o t l i n e s through

known

of

line w i l l

the

( s t a r t i n g ) p o i n t s on an H versus V

v a l v e o r another c/g a

AH =

(13.1)

slope of

fact

i s based on

M

(c/g)

relationship

between

plot.

H

and

Where the l i n e

V

(e.g.

l i n e ) the new conditions a r e obtained.

solving

two

equations

graphically

for

at

a

It is in

2 unknowns,

namely H and V . The method,

graphical

for

dx = dt

+c ;

for

dx =

-c

dt

method i s i n fact

identical

to

the c h a r a c t e r i s t i c s

since one s t a r t s w i t h the same equations (see chapter 9 ) :

;

AH = - $lV AH =

9

- xVIVIL/Zgd

A V + xVIVIL/Pgd

(13.2) (13.3)

176 where L

i s the

length of

p i p e over which the flow velocity changes

V a n d head changes b y AH.

by

The procedure employing the equations i s to c a l c u l a t e g r a p h i c a l ly

at

intervals

in

spaced L a p a r t i s necessary

from

either

selected pipes

the

the

each L

that end at

time

time

along

w i t h changes

and

i s such

between

velocity

On the H-V head,

significant

the

R

is

plotted

Note that

on

Thus each L/c

computed

I n the l a t t e r case,

XVIVIL/2gd The

should equal

conditions.

In

way

a n d i f b r a n c h pipes e x i s t p l o t (see l a t e r ) .

g r a p h one marks the known conditions, the i n i t i a l

relationship a

this

the

the procedure follows w i t h reference to F i g u r e

a n d Vo

as

points

p i p e length i t

i n wave c e l e r i t y and even changes i n diameter

for.

An e x p l a n a t i o n of

13.1.

selected

that waves a r r i v e a t the j u n c t i o n

the diagram should be replaced by a H-Q

the s t a t i c

at

I f L i s not the total

the same time.

interval

can be accounted

head

line.

parabola

between below

l i n e velocity

H

the

namely H

If line friction is

line velocity.

and

l i n e head at

(the curve H -Hf).

line

the negative V side the p a r a b o l a curves upwards since

i s i n the opposite d i r e c t i o n .

valve

discharge

to

be of

directly the

the form H

characteristic

K

=

L p r o p o r t i o n a l to flow

valve

i s closed

discharge factor

hammer

a n d head drop

K

and

it

characteristic.

If

i s known at

through

through

on

equation.

a valve

The

and

the

since

water

discharge

plotted

the downstream v a l v e i s r e q u i r e d to be solved simultaneously

between flow

between

also

across

the

relationship

are

graph

with

a

characteristics

relationship

i s generally

where V i s the p i p e velocity rate

i n m’/s.

it

is

assumed which i s

The f a c t o r K reduces as

i s a f u n c t i o n of open reduced

each step of L/c

head

in

seconds.

a

area as well defined

way

as the

I t s i n i t i a l value i s

obtained from

KO = Hvo/

Jvd

(13.4)

i s the i n i t i a l head loss through the v a l v e ( H - H f ) a t a vo A p a r a b o l a can be drawn through the p o i n t s ( H V ) l i n e velocity V

where

H

0

with

its

.

apex

0’

at

(0,O). Other

parabolae

can

be

drawn

d i f f e r e n t v a l v e openings p r o v i d e d K i s known. F i g u r e 13.1

i l l u s t r a t e s the g r a p h i c a l procedure f o r the case

0

at

177

Lc C

H (m 20

S

-201

Sp-VALVE

R

F i g u r e 13.1 Graphical line with friction

X

-t

CHART

S

a n a l y s i s f o r slow v a l v e closure i n g r a v i t y

SHUT

I78

of

slow

the downstream end ( p o i n t S ) of a g r a v i t y

v a l v e closure at

l i n e w i t h i n i t i a l f r i c t i o n head loss H f‘ The waves emanating a t the v a l v e pipeline The

in

time

as

characteristic

the waves.

at

defined

corresponding

H-V

the /2gd

the

g r a p h the

and then

the

(13.2)

full

and

up

and

l i n e s on

(13.3)

are

down

the

the x-t

chart.

applied

across

i n order to compute conditions a t p o i n t S

For instance,

commence teristic

indicated by

equations

proceed

boundary

to

dx/dt

R

l i n e s t a r t i n g at

proceeds along a

and

to o b t a i n V

+c

=

Ro

point

(Ho,Vo)

use

1’

the charac-

a n d H a t S1.

On

drops b y Hf=XVIVIL The p o i n t S,

l i n e w i t h slope -c/g.

i s also on the v a l v e c h a r a c t e r i s t i c f o r time L/c. It

may

at

the

Ro

which

be

noted

that

upstream end was

used

the

line friction

i n t h i s case since in

establishing

i s the velocity

the f r i c t i o n head.

at

point

For charac-

i n the opposite d i r e c t i o n to compute conditions a t point R ,

teristics

the velocity

at

the downstream end

i s used to assess f r i c t i o n loss.

p o i n t S 1 on the H-V

Thus,

s t a r t i n g at

by Hf

( t h e difference between the H

0

draws a

l i n e w i t h slope +c/g.

R2.

i s point

R6

S5,

it

i s assumed concentrated

d i a g r a m one draws a l i n e up and Hf

lines at V S 1 ) and then

l i n e intersects the H

Where t h a t

line

One proceeds i n t h i s manner to e s t a b l i s h p o i n t s S

etc.

Similarly,

starting

at

R3, Sq etc.

successively p o i n t s R 1 , S2,

So

point

one

3’ R4’ establish

can

(not done i n F i g u r e 13.1)

MID-POINTS AND CHANGE I N DIAMETER

It points

is

frequently

along

the

necessary

line,

for

thickness can be v a r i e d the

pipe

end

points are

known

is

divided

point

into

using

determine

instance

to s u i t

determined

and

to

in

as

order

that

at

intervals.

before commencing relevant

intermediate

the

the maximum heads.

a number of

the

heads

pipe

The heads at from

characteristic

an

between

interval.

Each

vals

there

interval

are.

computations

AL/c

is

AL/c

where

AL

is

the

adjacent

equation

e s t a b l i s h the boundary condition at the next time i n t e r v a l . interval

wall

I n such cases

to

The time

the

length

should be the same no matter how many i n t e r -

There

to the other,

from

one The

is

the

possibility

of

c

which can be accommodated i n t h i s way.

varying

179

f r i c t i o n head per i n t e r v a l The

head

and

flow

i s XVIVlAL/2gd. at

intermediate

established

by

(slope - c/g from the upstream p o i n t

Where the two lines meet

+ c/g from the downstream p o i n t )

and

is

I ines from n e i g h b o u r i n g p o i n t s on each

p r o j e c t i n g two c h a r a c t e r i s t i c side.

points

i s the head a n d l i n e Jeloc i t y

at the mid-point. The

same procedure In

meter.

such

cases,

flow r a t e Q not V .

A

where on

the

slopes

i s applicable i f however,

it

graph

could

be

2

are

13.5)

- - 9-

i s the p i p e c r o s s sectional H-Q

to p l o t H against

i s convenient

A Q + XQ1Q1AL/2gdAZ]. +[-c A

AH =

Then

i s a change i n d i a -

there

c/gA,

area so the slope of

A

where

(and c)

d i f f e r e n t f o r each section.

the lines

and

hence

the

The procedure i s i l l u s -

t r a t e d i n F i g u r e 13.2.

PUMP I NG L I NES

When head

pumps

trip

downstream

in

(on

a

pumping

the

line

delivery

there

side)

of

is first

the

a

pump.

drop

The

in

same

procedures can be followed to determine i n i t i a l head drop

graphical

a n d subsequent head rises. It

often

water

occurs

that

vaporization

separation described pocket

and in

at

the

sary

to

time

Chapter

the

divide

drop

points

subsequent 10.

volumes as they

tify

the

in

along

head the

rejoining

is

sufficient

line.

of

The

the

A separate t a l l y must

pipe

into

a

number

of

columns

be kept of

head r i s e occurs on r e j o i n i n g .

the

mechanics

water

expand and l a t e r contract

cause

to

of was

vapour

i n order to idenI t may be neces-

increments

i n order

to

p i c k up the locations o f v a p c u r pockets.

A

similar

procedure can

tanks

are

used

cases

the

vapour

be

eliminate

to

pocket

adopted

water

does not

if

hammer

water

hammer discharge

overpressures.

In

such

need to collapse before the head

rises. Many systems of overpressures valves. valve

A

are

valve

protecting possible.

immediately

i s opened a f t e r

pumping One

lines against

method

downstream

the pump

is of

the the

water

use pump

of

hammer release

non-return

i s t r i p p e d a n d subsequently slowly

180

R

T

S

H

Ho

0 F i g . 13.2 G r a p h i c a l c a l c u l a t i o n of w a t e r h a m m e r h e a d s a t m i d - l i n e w i t h c h a n g e in p i p e d i a m e t e r

181

R

P

;"-I-i neglect f r i c t i o n

Head H (m)

300 RO 9

R1

R2

R3

'

R4

R5

R6

c/g=lOOs Velocity

7

P1

HO

F i g . 13.3 G r a p h i c a l a n a l y s i s of pump stop w i t h water hammer release v a l v e

182

closed.

The a n a l y s i s o f

such a v a l v e i s done g r a p h i c a l l y

in F i g u r e

13.3. The

rotational

water

hammer.

thods

of

with

the

Parmakian

graphical pump

analyses of

inertia

of

the pump can also assist

(1955)

analysis

rotational

of

and

the

Pickford transients

characteristics

(1969)

d e s c r i b e me-

following

included.

s u r g e s h a f t s and r i g i d c o l u m n t r a n s i e n t s ,

a 1 so r e f e r r e d to J a e g e r ( 1956)

in r e d u c i n g

For

pumptrip graphical

the reader i s

.

REFERENCES

A l l i e v i , L., 1925. T h e o r y of W a t e r h a m m e r , t r a n s l a t e d b y E.E. Halmos, R i c c a r d o G a r o n i , Rome. B e r g e r o n , L., 1935. E t u d e des v a r i a t i o n s d e r e g i m e d a n s l e s conduites d ' e a u : Solution g r a p h i q u e generale, Revue Generale de I ' h y d r a u l i q u e , P a r i s . V o l . 1 p12-69. C., 1956. E n g i n e e r i n g F l u i d M e c h a n i c s , B l a c k i e & Son, Jaeger, L o n d o n , 529pp. P a r m a k i a n , J, 1955. W a t e r h a m m e r A n a l y s i s , D o v e r Pub1 i c s . , N.Y., 161pp. P i c k f o r d , J. 1969. A n a l y s i s of S u r g e , M a c m i l l a n , 203pp. 1937. C o m p a r i s o n s b e t w e e n c a l c u l a t e d and t e s t S c h n y d e r , 0. Nov., r e s u l t s on waterhammer in p u m p i n g p l a n t . Trans. ASME, 59, P a p e r Hyd-59-13, p695-700.

183

CHAPTER 14

P I PE GRAPH I CS

INTRODUCTION

It to

i s frequently

scale.

Sketch

work

analysis

water

hammer

in

this

heads

be

would of

very

manner,

illustrative

cause

bursts,

will

pressures,

v i s u a l ize

water

also

tory

complexes

o r d of

are

on

a

can

of

of

The computer

performed

A

screen

before

considerable

pipe

cost

relation-

head

at

and

all

grad-

stand out.

resulting

This

type

particularly display

fighting

as

High

in

of

insuf-

depiction

use

in

i s available.

examples.

grows,

and

additional

to

Wat-

supplying

facThese

a n d a rec-

l a t e r extensions.

Suitable

storage

can

drawing. ideal

place to depict

and

amendments

to

runs,

drawings

and

savings

complicated

systems

the factory

duplications

i s the

construction

heads

levels are t y p i c a l

be v i s u a l i z e d on a 3-dimensional

Interactive

be

fire

extended

connections,

low

graphical

and

number

frequently

viewing

The

limits

for

graphics.

systems

networks

drawn or

pressure

pressure

identified.

p i p e positions would be of

positions

useful.

the

when

and

be

unless some form of

distribution

systems

depiction,

more

and

were

Bottlenecks can f r e q u e n t l y be pinpointed

i s possible i n 3-dimensional

er

profiles

as steep h y d r a u l i c g r a d i e n t s w i l l

to

Industrial

pipe

a p i p e network

the chapters on p i p e net-

pipe

be even

the

be studied.

likely

ficient

Three

angles

elevation

to

in

dimensional

between would

utilized

analysis.

alternative

have

were

and

ship

ients

plans

to draw

a n d optimization

from

points

informative and useful

is

possible

a

if

pipe

orders

such systems. system can are

mistakes

be

finalized. are

ironed

out on the screen a n d not on site. Computer

graphics

enables the d r a w i n g scale,

a n g l e to be a l t e r e d at is

obtained.

viewed part

of

from the

the computer.

Various another system

will, pipes,

angle can

be

until

connections and

size and v i e w i n g

the best p i c t u r e f o r any purpose

detail

called

on

or

valves

p i c t u r e s of while

the

may any

be

best

particular

engineer

sits

at

184

Graphics are p a r t i c u l a r l y useful pipework

subject

close

to

in

the f a b r i c a t i o n of i n d u s t r i a l

tolerances.

In

fact

for

problems

before

problem areas a r e pipes which get

construction

the

identified. other, orifices is

High

stage.

of

stacks of the

erection

head

loss

i n accessible,

is

Tight

likely

areas,

e.g.

to

model

sort or

sharp

bends

and

difficult

will

be

bends one

after

the

Locations f o r measuring

long, s t r a i g h t sections w i l l be f a c i l i t a t e d .

It

out

a

such

graphical

Erection

corners,

be

to

problems

time

at

portrayal

will

be

design

rather

stage

than

at

with

erection

of incorrect assemblies w i I I

Erection costs and r e - f a b r i c a t i o n

reduced.

For example

pipes to minimize space and

system.

and tees can be ironed out.

preferable

physical

be

where

valves

model I i n g

the

i d e n t i f i e d a n d remedial steps taken.

strength

intersections

During

i n the way of each other can be relocated. Support

systems can be designed f o r maximize

commences.

plants,

i s made to i r o n

factories or r e f i n e r i e s a p h y s i c a l model of the p l a n t out

many

more r a p i d a n d f u r t h e r

savings

are possible. It

will

be found

that

the cost of

less than that of a p h y s i c a l model. can match many of for

pipes,

ease of of

viewing

angle,

it

is

screen

zooming

it

for

For

is

graphics

suffer

closeups,

e.g.

view

and

valves,

and

rapid variation

linking

to

is

colour coding

a

to

the set-up.

to

instance,

the

hydraulic

I n some other plant

from

a

angle may r e q u i r e a coded i n s t r u c t i o n , a c l e a r i n g

and

re-drawing

the

less p o r t a b l e than

computer

i s superior,

and r a p i d alterations inferior.

s l i g h t l y different the

I t can be assembled f a s t e r ,

the features of a r e a l model, e.g.

I n some ways

a n a l y s i s model,

of

g r a p h i c s model

s t a n d a r d codes f o r v a r i o u s devices such as

viewing.

features

a computer

poor

system

a physical

resolution,

although

which model

takes

time.

and colour

The

screen

t h i s can be overcome b y

p l o t t i n g the p i c t u r e on paper ( h a r d c o p y ) .

A computer program f o r d r a f t i n g a n d design of p i p e layouts can easily

be

linked

to

other

as

programs

pressures,

as

movements,

superimposed

stresses, cause

while

well

for

for

analysing

stress

analysis.

flows

and

Temperature

loads and supports a l I a d d t o l o n g i t u d i n a l

internal

circumferential

programs

pressures,

stressing.

supports

Supports

and

can

be

external

loads

repositioned

to

185

reduce

stress

concentration

by

going

back

and

forth

between

the

l a y o u t a n d the a n a l y s i s programs.

I NTERACT I V E DRAW I NG

On (CAD)

some o f

the

commercially

systems

for

piping,

screen.

drawing

or

drawing

with

point

is

arrow

on a

trolled Then

by

the

simple

changed

a

pointer

not

the

computer

while

it

the

points

correct

program

used, is

done

be

input

That

on

a

is,

aided d r a f t i n g directly

off the

drawing

a

on

sketch

location a

or

the

or

of

cursor

a

or

to

the

key,

a

correct

spot

on

the

symbol

such

as

a

screen.

valve,

a

i s r e p r o d u c e d on t h e d r a w i n g . accompanying d a t a can only

the p i c t u r e

that

computer be

generally digitizer.

i n a l p h a n u m e r i c mode,

and

drawing,

a

can

The c u r s o r c a n b e moved a r o u n d b y h a n d con-

until

pushing

of

with

screen.

buttons

In

aid

identified

f l a n g e o r reducer

or

Data can

the

available

editing

but

i s on large

adding

to

that

t h e screen. systems

and

be r e a d i n

i s b y means o f c o - o r d i n a t e s T h i s i s a l i m i t a t i o n of

can

editing

use

while

more the

interactive

picture

is

in

f r o n t o f one.

I n m a n y modern C A D systems of

him

and

identifies

pushing a button of

the

device

des i g n a t ed

Fig.

14.1

can

the

t h e d r a f t s m a n h a s a menu i n f r o n t

correct

symbol

to r e p r o d u c e t h e symbol. also

be c o n t r o l l e d

by

with

arrow

before

The s i z e a n d o r i e n t a t i o n keys

or

.

3-dimensional

an

d e p i c t i o n o f a catchment

buttons suitably

186

Many

of

the

symbols

characteristics assists

in

of

the

are

device

distinguishing

globe valves,

diagrammatic, rather

between,

i.e.

than

for-

distinguishing appearance.

its

example,

butterfly

the This

valves,

gate v a l v e s a n d control valves.

F i g u r e 14.2

was abstracted

b y Lamit (1981) from ANSI standards

a n d covers a wide r a n g e of symbols. Annotation and

l e t t e r i n g can also be done b y l o c a t i n g the label

s i z i n g the letters a n d then t y p i n g the labels on a keyboard. the screen (cathode r a y tube o r C.R.T.)

Once the d r a w i n g on to

the

proper and,

satisfaction

of

sheet.

can

This

the

draftsman

be

a

to

codes

to

depict

d i s t i n g u i s h between gas, dotted o r f u l l

be

scale

as colour

different water,

reproduced

than

on

the

is

on

a

C.R.T.

the q u a l i t y of l i n e s a n d reso-

f o r b l a c k a n d whife on a C.R.T.

different

may

than on the C.R.T.

be much better

so i f colour g r a p h i c s a r e used, that

larger

i f done on a d r a f t i n g machine,

lution w i l l

it

This i s p a r t i c u l a r l y i s o n l y 1/3 of

resolution

I t i s s t i l l often useful to use types

of

line,

products, etc.

for

example

to

b y means of dashed,

lines.

COMPUTER PROGRAM FOR PIPE GRAPH I CS

The

appended computer program was w r i t t e n f o r a micro computer

depiction of p i p i n g systems.

Program Description

The devices

program

as

written

accommodate

pipes w i l l

Alternatively

drawings

were

if

a

reproduced

more than c l u t t e r

colour

screen

graphical l y

could b e used to depict d i f f e r e n t pipes, sewers, e.g.

ventilation

the

dotted,

100

pipes

( f l a n g e s , v a l v e s o r t a n k s ) . Unless d e t a i l close-up

used t h i s number of HP-85.

can

HP87,

ducts o r

present

high

different

on

were paper,

e.g.

pressure. types

of

and

100

views a r e

the screen of the available different

or

if

colours

cold water,

hot water,

Alternative

computers,

lines,

e.g.

dashed o r

which a r e also useful f o r d i s t i n g u i s h i n g lines.

Input

is

typed

below

the

program s t a r t i n g

from

l i n e 700,

and

187

TYPE

,

32

CmIFtCE

33

REDUCtWG

34

SOCKET

35

WELD NECK

WELO

43

EXPANSION

46

LATERAL

87 LOCISHELD VALVE YOTOP CONTROL VALVE PLUGS a BULL

48

td;3

+& +c,

XAR

F i g . 14.2

Symbols for pipe f i t t i n g s

188

r e q u i r e s l i n e n u m b e r s a s w e l l a s a DATA statement Line 1:

Left

hand

limit

of

limit,

Right

screen

hand

display

to y

x

i n g m e a s u r e d f r o m x-z

plane.

should

upper

and

practical

4/3

be

lower

angle

times

limit

or

axis,

limit,

ft.),

the

the

for

left

Angle

screen i s

between

undistorted

for

of

and right hand

difference

an

combination

Upper

A n g l e up o f v i e w -

Since t h e HP-85

3 , t h e d i f f e r e n c e between

x

limits

Lower

metres

(in

v i e w i n g measured from

4

limit,

followed b y :

isometric

the

scale.

A

viewing

is

30°, 15". Line 2 onwards:

Pipe

data

i.e.,

Xl,Yl,Zl,X2,Y2,Z2

e n d p o i n t s on the pipe, After

the

last

information

along

resented

X,

3

and

pipe

line,

insert

a

data

line with

on

devices:

Pipe

no.

(in

the

order

i n ) , D i s t a n c e f r o m s t a r t p o i n t t o e n d p o i n t meas-

typed ured

start

to i d e n t i f y the last pipe.

O,O,O,O,O,O,O

D e v i c e 1ines:The

real

of

Cost p e r m e t r e o f p i p e .

pipe,

Type of

device

a

vertical

line,

2 = valve

by tank

=

(1

the

represented

by

U),

Size

= flange

rep-

represented b y of

device

(m o r

ft.). After

the

last

O,O,O,O to program, or

it

until

is

input), ment.

the

that

the

cases.

whole

in

Note

picture

that

data

has

to

screen of

with the

and

amend

limits or

display

original

C

and

for

pipe

amend-

pushing

i s altered

this

screen

a

order

(in

the p i c t u r e

altering

line

running

PAUSE i s p r e s s e d ,

and

this

data

l a t t e r case the program

pipe

number

a

After

until

I n the

any

typing

for

insert data.

X1 ,Y1 ,Z1 ,X2,Y2,Z2

p i p e on

corrected.

available as

pipe new

Upon

LINE, also

amend

the

of

activated

i s pressed.

to

requests

line,

end

remains

K1

reset

device

identify

END

a n d t h e cost

facility

is

viewing

angles

be drawn again

not

i n these

189

Fig.

14.3

F i r e D i s t r i b u t i o n P i p i n g Depiction

Example

The appended program i s used to depict distribution

network

from a roof t a n k ing

valve

nection

is

through

a double

the

from

the

system

lower

follows

the f i r e f i g h t i n g water

bui Iding.

Water

i s supplied

looped p i p e network.

i n d i c a t e d downstream of

extends

describing

a double-storey

to

the tank

floor the

to

a

future

computer

An isolat-

a n d a f l a n g e d conextension.

program.

The

Data

cost

of

each p i p e i s estimated to be $10 per metre length. Upon r u n n i n g the program the general view depicted and copied from the x - a x i s supply

tank

and

15" up.

from

an

? e screen.

A more d e t a i l e d view

(0,O

angle

i n F i g u r e 14.3a was

The v i e w i n g a n g l e i s 30° from

was

( F i g u r e 14.3b) of the

also obtained.

Note that

in

each case the total cost of the system i s also indicated.

Symbols in pipe g r a p h i c s p r o g r a m

A1 A2

angle of d i r e c t ion a n g l e of

viewing viewing

from from

x-axis z-axis

measured measured

in

y

towards

190

x-y plane. cost of p i p e per u n i t length total cost of pipes dimension of device 0 = o r i g i n a l d a t a , 1 = amended p i p e data length distance a l o n g p i p e from s t a r t to device p i p e counter device counter device type. 1 = flange, 2 = v a l v e , 3 = tank x - co-ordinate l e f t h a n d l i m i t on screen s t a r t p o i n t i n screen x-co-ordinates end point i n screen x-co-ordinates x-co-ordinate of device i n p l a n e of screen r i g h t h a n d l i m i t on screen y-co-ord i na te z-co-ord i na te z-co-ordinate of device in p l a n e of screen lower l i m i t on screen s t a r t point i n screen z-co-ordinate end p o i n t i n screen z-co-ordinates z-co-ordinate of device i n p l a n e of screen upper l i m i t on screen s t a r t x of l i n e end x of l i n e

x1 x2 x-co-ordinate of device s t a r t y of l i n e end y of l i n e Yl Y2 y-co-ord i na te of dev ice s t a r t z of l i n e end z of l i n e 21 22 z-co-ordinate of device.

191

,

, 1.0

DATUM

I

I COVER

Fig.

I

I

LEVEL

14.4

Computer f l a t b e d p l o t of graded sewer

COMPUTER GRAD I NG

Pipelines a r e well drawing largely

of on

sections empirical

s u i t e d f o r automatic g r a d i n g by computer and by

a

linked

bases

which

plotter.

A

can

readily

be

network

is

designed

programmed

as

constraints. Although

it

is

possible

to

start

from d a t a from a contour p l a n

a n d develop the p l a n layout w i t h i n the computer, lay feed

the network the

data

out on a p l a n , peg and survey from

s u i t a b l e program w i l l

pegging

sheets

directly

i t i s preferable to i t i n the f i e l d and

to

the

computer.

A

then select the most economic depths and p i p e

diameters.

The

data

inspection

and

adjustment

may

be d i s p l a y e d if

desired,

i n summary form f o r taking

off

visual

quantities.

The

192

results drawing

are

then

submitted

longitudinal

sections.

to

a

separate

plotting

routine

for

Such a plot i s i l l us t rat ed i n Figure

14.14.

REFERENCES

ANSI 232.23. Symbols f o r p i p e f i t t i n g s Lamit, L . G . , 1981. P i p i n g Systems, D r a f t i n g H a l l , Eaglewood C l i f f s . 612 p p .

and

Design.

Prentice

193

498 588 518 528 5.38

548 558 568 578 588 598 688 618 628 638 648 658 666

676 688 708 718 728 738 748

758 768 770 788 798 808 818 828

238 848 858 868 878 889 898

"fP

194

CHAPTER 15

COMPUTER PROGRAMM I NG IN BAS IC

DESCRIPTION

BASIC i s an e l e m e n t a r y c o m p u t e r l a n g u a g e o r i e n t a t e d to t e c h n i c a l problem solving, i s often

and somewhat s i m i l a r

to

FORTRAN l a n g u a g e .

used on microcomputers w h i c h a r e designed

the human

input effort.

BASIC

streamline

to

the statements in the l a n g u a g e a r e

Some o f

a c t u a l l y commands to t h e c o m p u t e r a s w e l l a s p r o g r a m s t e p s . The

programming

Statements, program

Functions,

lines

which

may

be

classified

Operators have

to

be

into

in

numbered

although not necessarily sequentially,

various

Commands.

and

types,

e.g.

Statements the

correct

are order

e.g.

10 20 21 50 200

Computer Commands

Some s t a t e m e n t s a c t i v a t e t h e c o m p u t e r ,

AUTO [ b e g i n s t a t e m e n t n o . ,

e.g.

[increment]]e.g.

AUTO 100,5 m a k e s

the computer s t a r t a t statement

100 and a u t o -

m a t i c a l l y increment each number b y 5 during input. CAT

Numbers in

CONT [ s t a t e m e n t n o . ] ,

optional.

continues execution of a program a t the

s p e c i f i e d s t a t e m e n t no. COPY

[I a r e

produces a l i s t o f e v e r y t h i n g o n the tape.

a f t e r a PAUSE.

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

DELETE f i r s t s t a t e m e n t n o .

[ l a s t statement no.]

r e m o v e s those

statements from the program. ERASETAPE

initializes a tape

INlT

resets the program

to t h e f i r s t

v a r i a b l e s to u n d e f i n e d v a l u e s .

l i n e and s e t s a l l

195

L I S T [ b e g i n s t a t e m e n t no.

[ e n d statement no.]]

l i s t s the program

o n screen. LOAD " p r o g r a m

name" c o p i e s t h e p r o g r a m f r o m a t a p e o r d i s c to c o m p u t e r memory.

P L I S T [ b e g i n s t a t e m e n t no.

[ e n d statement n o . ] ]

l i s t s the program

on paper. REN [ f i r s t s t a t e m e n t n o .

[ increment]]

a s specified.

renumbers the program I ines

Default values a r e 1 0 , l O .

s t a r t s e x e c u t i o n of t h e p r o g r a m f r o m t h e s p e c i -

RUN [ s t a t e m e n t n o . ]

f i e d statement. d e l e t e s t h e p r o g r a m f r o m memory.

SCRATCH STORE " p r o g r a m

name".

SECURE " f i lename",

Stores t h e p r o g r a m onto t a p e o r disc.

" s e c u r i t y code",

type.

Type 0 secures a g a i n s t

LIST,

P L I S T and E D I T .

T y p e 1 a g a i n s t STORE,

LIST,

P L I S T and E D I T .

T y p e 2 a g a i n s t STORE,

PRINT#,STORE e.g. UNSECURE " f i lename",

B I N and T y p e 3 a g a i n s t CAT.

SECURE " P R O G l " , " X Y ~ ' 1 .

" s e c u r i t y code",

type.

AR I THMET I C

T h e f o l l o w i n g s y m b o l s a r e u s e d in a r i t h m e t i c s t a t e m e n t s

+

add

-

subtract

'>

multiply divide

A

r a i s e to the power. The

dence i s

arithmetic A+,*

and

LOG I CAL OPERATORS

=

equal to

>

greater than

f

not equal to

less t h a n

functions

, +

and

p r i o r i t i e s and t h e p r e c e D t h u s A+B*C"D i s A + { B * ( C ) } .

have certain

-,

196

VARIABLES

Any letter

letter may followed

confined e.g.

to

D(4,5). Arrays

by

letters,

be used to represent a v a r i a b l e , a

number,

e.g.

C(3).

e.g. Up

A,B3. to

as well as any

Subscript

variables

are

two s u b s c r i p t s a r e possible,

Nested parentheses a r e e v a l u a t e d from i n w a r d s out. should

be

preceded

s u b s c r i p t s e x c e e d 10, e . g .

by

a

DIM A(20),

DIM

(dimension)

statement

if

B(30,lOO).

PRECISION

REAL

v a r i a b l e s a r e accurate to

variables

are truncated

to 5 d i g i t s .

v a r i a b l e s a r e d e c l a r e d SHORT o r

12 d i g i t s ,

w h i l e SHORT

REAL p r e c i s i o n

INTEGER

is used unless

INTEGER.

FUNCT IONS

ABS(x) ACS(x) ASN(x) ATN(x) CEIL(x) COS(X) COT(x) CSC(X) DTR(x) EXP ( x ) FLOOR(x) FP(x) INT(x) IP(x) LGT(x) LOG(x) MAX(x, y ) MIN(x,y) MOD PI RMD(x, y ) RND RTD(x) SEC(x) SGN(x) SIN(x) SQR(x)

absolute, positive, value of x arcos ( x ) arcsin x arctan x smal lest integer<=x cosin x cotangent x cosecant x degrees to r a d i u s X

same a s I N T ( x ) fractional p a r t of x largest integer <=x i n t e g e r p a r t e.g. l P ( - 3 . 2 7 6 ) = -3 loglox n a t u r a l log x x o r y whizhever i s largest x o r y whichever i s smallest i n t e g e r e.g. MOD B = p o s i t i v e i n t e g e r v a l u e o f B 3.14159265359 remainder of x/y-yqlP(x/y) n e x t number x in a r a n d o m sequence o<=x
197

TAB(n) TAN(x) VAL (S$) VAL$(x)

s k i p s to column n tangent of x n u m e r i c a l e q u i v a l e n t o f s t r i n g S$ s t r i n g e q u i v a l e n t of x

SPECIAL CHARACTERS

permits

multi-statement

e.g.

100

A=B!A

IS

line,

A=B

@

G O T 0 200 delimits

a

remark

100

e.g.

A

NEW

VAR I ABLE prompts f o r I,

I,

input

delimits a s t r i n g of text which i s displayed

BAS I C PROGRAMM I NG STATEMENTS

The f o l l o w i n g statements c a n be i n c o r p o r a t e d in programs. ASSIGN n u m b e r TO "name", CHAIN " f i l e n a m e " .

ASSIGN 2 TO "DATA"

e.g.

L o a d s and r u n s t h e p r o g r a m . T h e is

deleted

"assigned"

except

for

current program

"common"

variable

and

buffers.

CLEAR

Clears the screen

COM

Common v a r i a b l e l i s t .

CREATE " f i l e n a m e " ,

number of records [number of bytes p e r record]

DATA l i s t .

Provides constants and/or

Used w i t h CHAIN

Establishes a d a t a f i l e of the specified size. s t r i n g s from which

READ s t a t e m e n t s a b s t r a c t d a t a . DEFAULT ON/OFF

p r e v e n t s o r c a n c e l s some m a t h e r r o r s f r o m h a l t ing e x e c u t i o n

DEF FN n a m e [ ( p a r a m e t e r ) ] [ =

expression].

w i t h i n a program,

e.9.

Defines a special f u n c t i o n DEF FN A(X)=SQR ( X + 2 + 2 * X )

DEG

Sets d e g r e e mode f o r t r i g f u n c t i o n s

DIM list.

Declares the maximum subscripts f o r a r r a y s . e.g.

DIM A(20),

B(50,lOO)

198

DlSP [ l i s t ]

d i s p l a y s t h e l i s t o n the screen.

DlSP USING f o r m a t

line

EQUALS",

[;

format that

e.g.

DlSP "A

B

list).

D i s p l a y s o n screen a c c o r d i n g to the

in format

line.

I f a l i n e number i s used

l i n e h e a d e d IMAGE a c t s a s t h e f o r m a t ,

e.g.

DISP USING 50;A,B END

Terminates program execution

FOR c o u n t e r = f i r s t v a l u e TO l a s t v a l u e [ s t e p

increment].

The

s t a t e m e n t s up t o a l i n e NEXT c o u n t e r a r e repeated f o r the counter proceeding from the f i r s t v a l u e to the last value.

The counter

step i s one unless otherwise specified. GOSUB s t a t e m e n t n o .

The n e x t statement executed i s the specified statement in a subprogram.

Control

is returned

to the m a i n p r o g r a m w i t h a RETURN s t a t e m e n t . GOT0 s t a t e m e n t n o .

Control

i s t r a n s f e r r e d to the designated

st a t emen t

GRAD

s e t s t r i g o p e r a t i o n s to grads.

( 400 grads in

360") I F e x p r e s s i o n THEN s t a t e m e n t n u m b e r [ELSE s t a t e m e n t n u m b e r 1. 1 2 I f the expression is true control i s transferred to s t a t e m e n t ,

IMAGE f o r m a t .

and i f n o t ,

to the next

statement [ o r statement ] 2 Used w i t h PRINT USING o r DlSP USING s t a t e m e n t s t o specify the format o f the output. T h e f o l l o w i n g can b e s p e c i f i e d : n ( .....) r e p e a t s t h e o p e r a t i o n i n p a r e n t h e s i s n times A string character Z L.H. d i g i t p o s i t i o n o r l e a d i n g z e r o D L.H. d i g i t p o s i t i o n o r l e a d i n g b l a n k decimal p o i n t position S s i g n (+ o r - ) M m i n u s o r blank E e x p o n e n t i a l f o r m ESDDD X blank / carriage return Is I' literal e.g. IMAGE 2D.DDD,PXZDD

.

INPUT v a r i a b l e name

[,

v a r i a b l e name

...I V a r i a b l e s

a r e assigned

199

v a l u e s o n the k e y b o a r d when a prompt

'I?"

appears on the screen INTEGER v a r i a b l e [ s u b s c r i p t s ]

. . ..

Specifies v a r i a b l e s a s integers

and d i m e n s i o n s them

KEY L A B E L

D i s p l a y s l a b e l s a s s o c i a t e d w i t h ON KEY s t a t e ments

LOAD B I N " f i l e n a m e " .

Loads a b i n a r y coded f i l e from tape i n t o

memory NEXT c o u n t e r .

Returns control

ON ERROR GOTO s t a t e m e n t n u m b e r .

t o a FOR s t a t e m e n t T r a n s f e r s to d e s i g n a t e d statement

if a recoverable execution e r r o r ON e x p r e s s i o n GOTO l i s t .

i s encountered

T r a n s f e r s c o n t r o l t o t h e s t a t e m e n t no.

in t h e I i s t c o r r e s p o n d i n g t o e x p r e s s i o n

being 1, PAUSE

2 , 3 etc.

H a l t s e x e c u t i o n u n t i l CONT i s e n c o u n t e r e d

P R I N T [USING s t a t e m e n t n u m b e r ] [ l i s t ] . u s i n g the format

Prints the l i s t on paper

in t h e s p e c i f i e d s t a t e m e n t

number o r in free format.

Items in the

l i s t m u s t b e s e p a r a t e d b y commas o r semicolons, e.g. PRINTER

I S code number.

PRINT USING 1 0 0 ; A , B ( 5 ) , Redefines the p r i n t e r .

" I S THE ANSWER'' l=screen,

2=paper

PURGE f i l e name.

Eliminates the designated f i l e from tape

RAD

Sets r a d i a n mode f o r t r i g f u n c t i o n s

RANDOMIZE [ e x p r e s s i o n ] .

G e n e r a t e s a r a n d o m n u m b e r seed.

By

s p e c i f y i n g an e x p r e s s i o n t h e r a n d o m n u m b e r sequence c a n b e r e p e a t e d READ [ b u f f e r n o . ]

name,[,name2

...].

Reads a s t r i n g o f v a r i a b l e s

f r o m DATA s t a t e m e n t s o r f r o m a b u f f e r REAL v a r i a b l e [ ( s u b s c r i p t s ) ]

. ..

D e c l a r e s v a r i a b l e s r e a l and

d i m e n s i o n s them REM [ a n y t h i n g

remark statement, not f o r execution,

RENAME o l d n a m e TO newname. RESTORE [ s t a t e m e n t n o , ] .

same a s !

F i l e i s renamed

Resets d a t a p r i n t e r t o t h e s p e c i f i e d

s t a t e m e n t o r to t h e b e g i n n i n g o f t h e d a t a f i l e RETURN

Transfers control from the l a s t l i n e of a s u b r o u t i n e back t o t h e o r i g i n a l p r o g r a m l i n e

200

f o l l o w i n g a GOSUB s t a t e m e n t

... D e c l a r e s

SHORT v a r i a b l e [ ( s u b s c r i p t ) ]

v a r i a b l e s short a n d

d i m e n s i o n s them STOP

T e r m i n a t e s e x e c u t i o n and r e t u r n s p o i n t e r to f i r s t statement

STORE B I N "name".

Stores the f i l e named on tape

TRACE

Used t o f o l l o w s t a t e m e n t s e x e c u t e d .

NORMAL w i l l

c a n c e l t h e TRACE o p e r a t i o n

GRAPH I CS STATEMENTS

The HP s e r i e s 80 m i c r o c o m p u t e r e n a b l e s

g r a p h s to b e p l o t t e d o n t h e

s c r e e n and o n p a p e r u s i n g t h e f o l l o w i n g s i m p l e s t a t e m e n t s . ALPHA BPLOT s t r i n g ,

Sets d i s p l a y to a l p h a n u m e r i c mode number of characters p e r line.

P l o t s a g r o u p of d o t s

as specified b y the s t r i n g DRAW x - c o - o r d i n a t e ,

y-co-ordinate.

D r a w s a l i n e f r o m c u r r e n t pen

p o s i t i o n to s p e c i f i e d ( x , y ) GCLEAR [ y ]

c l e a r s screen below specified y v a l u e

GRAPH

Sets d i s p l a y t o g r a p h i c mode

IDRAW x - i n c r e m e n t ,

y-increment.

Draws a l i n e from c u r r e n t pen

p o s i t i o n to p o s i t i o n d e t e r m i n e d b y t h e x and y increments IMOVE x - i n c r e m e n t ,

y-increment.

Moves t h e p e n b y t h e s p e c i f i e d

increments, without d r a w i n g a l i n e LABEL s t r i n g .

Writes the s t r i n g s t a r t i n g a t the c u r r e n t pen posit ion

LDlR angle.

Specifies d i r e c t i o n of y-co-ord i na te.

MOVE x - c o - o r d i na te,

to ( x , y ) PEN n u m b e r .

abel

( a n g l e = O o r 90°)

Moves p e n f r o m c u r r e n t p o s i i o n

w i t h o u t d r a w ng a l i n e

Specifies whether dots a r e b l a c k ( n e g a t i v e number) o r white (positive number)

PENUP

L i f t s p e n up

PLOT x - c o - o r d i n a t e , SCALE x - m i n ,

y-co-ordinate.

x-max,

y-min,

y-max.

Makes a dot a t

(x,y)

Sets t h e x and y s c a l e o n t h e

screen between the l i m i t s sDecified

20 1

X A X l S y i n t e r c e p t [ , t i c s p a c i n g [xrnin,

xrnax]].

Draws a h o r i z o n t a l

a x i s with t i c marks and within Negative t i c s specify

YAXIS x intercept [ , t i c

spacing [yrnin,

yrnax]].

Draws a v e r t i c a l

axis (By c o u r t e s y

of

the Hewlett

l i m i t s specified.

the r i g h t s i d e as reference

P a c k a r d company.)

202

INDEX

Acceleration, 1 13 A c c u r a c y , 97,124 Advec t i o n , 92 A i r , 18,118,148 A i r v a l v e s , 119, 141 A i r vessels, 143 A l b e r t s o n , M.L. 20 A l l i e v i , L , 175, 182 A l l o c a t i o n , 72 A n a l y t i c a l s o l u t i o n , 22 , 96 A r i t h m e t i c , 193 Assumpt ions, 36 A v e r a g i n g , 45 B a r l o w , J . F . , 44,45,54 BASIC, 192 Basic equations, 1 Bend, 17 B e r g e r o n , L, 175, 182 Bernoul t i equation, 1 , 2 B h a v e , P.R., 63, 69 B o u n d a r y c o n d i t i o n s , 136, 167 Boundary layer, 5 B o y l e s l a w , 113 B r a n c h p i p e , 38, 55, 161, 165 Cathode r a y t u b e , 186 C a v i t a t i o n , 137, 138 C e l e r i t y , 118, 129 C h a r a c t e r s , 187, 195 C h a r a c t e r i s t i c s , 128, 144, 175 C h a u d h r y , M.H., 120, 135 Chezy, 3, 10 Closed loops, 70, 85 Colebrook a n d W h i t e , 6 Col ebrook-W h i t e e q u a t i o n , 6,7,16,20, 23, 32, 118 Command, 193 Compound p i p e s , 26 Computer p r o g r a m s , 30,41 ,49,63 68,102,131 ,153,186 Computer s t o r a g e , 149 C o n s t r a i n t s , 63, 86 C o n t i n u i t y e q u a t i o n , 1, 109, 112 Convergence, 26,29,45,51 Cost, 188 Courant, F r i e d r i c h s & Lewy, C o n d i t i o n , 129 Cross, H , 31,35,45,51, 82,84,162

D a n t z i g , G.B.,71, 86 D a r c y f r i c t i o n f a c t o r , 3,5,10,23 27,30,49 D a r c y h e a d loss e q u a t i o n , 41,48 D a r c y - W e i s b a c h , 9,10,23,25,36, 118,161,163 Deb, A.K. 85 Demand, 40 D e s i g n , 21 D i a m e t e r s , 56,66,71 ,178 D i f f e r e n t i a l e q u a t i o n s , 125 D i f f u s i o n , 92, 126 D i s c h a r g e c o e f f i c i e n t , 122, 176 D i s c h a r g e . t a n k s , 142 D i s k i n , M.H,,20 D i s t o r t i o n , 188 D i s t r i b u t i o n system, 160 D r a w i n g s , 28,183 D r a w o f f , 25,55 Dummy p i p e s , 38 D y n a m i c e q u a t i o n , 23, 110 D y n a m i c p r o g r a m m i n g , 71,161 E l a s t i c i t y , 116 Elevation, 2 Empirical, 3 E n e r g y , 1,2, 140 E n t r a n c e loss, 17 E q u i v a l e n t l e n g t h , 18 E q u i v a I en t p i p e , 19,22,38,85 E r r o r , 125 E u l e r , 97 E v a p o r a t i o n , 94 E x i t loss, 17 E x p l i c i t , 128,165 F e e d b a c k , 171 F i n i t e d i f f e r e n c e , 124 F i r e h y d r a n t , 40,183 F l o w c o r r e c t i o n , 83 FORTRAN, 113 F o u r i e r methods, 102 F o u r t h o r d e r , 100 F r i c t i o n , 1 17 Friction factors, 4 F r i c t i o n loss c h a r t s , 1 1 F u n c t i o n , 193

203

Gauss m e t h o d o f e l i m i n a t i o n , 47, 49 G e r a l d , C.F., 100, 105 G l o b a l E r r o r , 97 Grading, 192 G r a p h i c a l m e t h o d , 175 G r a p h i c s , 183, 195 G r a v i t y l i n e s , 131 Grosman, D.D., 88, 105

45

Hadlev - ,, G.. , 63., 69 H a z e n - W i l l i a m s , 3,4,10,16,23 27.36 Head loss, 1 ,3,16,35,40,71 ,122 Houses, 35 H y d r a u l i c g r a d e l i n e , 2, 72 H y d r a u l i c r a d i u s , 16 H y d r a u l i c s r e s e a r c h s t a t i o n , 7,20 I m p l i c i t , 165 I n d u s t r i a l s y s t e m s , 183 I n e r t i a , 709 I n i t i a l f l o w s , 51 I n i t i a l v a l u e s , 57 I n t e g r a t i o n , 110 I n t e r a c t i v e compu t i n g , 102 I n t e r a c t i v e d r a w i n g , 185 lrrotat ional, 2 I s a a c s , L.J., 45, 54 I t e r a t i v e methods, 10, 153 J a e y e r , C . , 112, 114, James, W . , 90, 105 J o u k o w s k y h e a d , 138 K a l l y , E., 85 K e y c o l u m n , 57 Kinematic viscosity,

5,

tech -

N e t w o r k s , 21,35,36,45,63,70,160 New ton-Raphson, 9,24,32 New t o n ' s I a w , 45,47,105,115 Nikuradse, 7 Node h e a d c o r r e c t i o n , 28 Nodes, 28,35,87 Non c i r c u l a r s e c t i o n s , 16 Non l i n e a r equations, 21, 124 Non l i n e a r p r o g r a m m i n g , 84 Non r e t u r n v a l v e , 131, 141 N u m e r i c a l d i f f u s i o n , 126

182

O b j e c t i v e f u n c t i o n , 63 One d i m e n s i o n a l , 2 O r i f i c e , 88 O p e r a t i o n s r e s e a r c h , 70, 88 O p p o r t u n i t y v a l u e , 59 O p t i m i z a t i o n , 55, 70 O p t i m u m l e n g t h , 85 O p t i m u m s o l u t i o n , 89 O v e r p r e s s u r e s , 138,140,142,143

23

L a m i n a r f l o w , 5, 22 185 L a m i t , L.G., L a y o u t , 63 L i n e a r method, 45 L i n e a r p r o g r a m m i n g , 63, 83 L i n e a r i z a t i o n , 22, 29 L i n k , 87 L o a d r e j e c t i o n , 103 L o c a l e r r o r , 97 L o g i c , 194 L o o p e d n e t w o r k s , 63, 81 L o o p f l o w c o r r e c t i o n method, L o o p / n o d e c o r r e c t i o n , 81 L u p t o n , H.R., 120, 135

M a n n i n g , 3, 16 M a n u a l a n a l y s i s , 35, 110 M a r k l a n d , E., 4 5 3 4 M a r t i n , C.S., 147, 159 Mass, c o n s e r v a t i o n , 115 Mathemat i c a l opt imiza t ion n i q u e s , 70, 82 M a x i m u m h e a d , 172 Menu, 185 M i d p o i n t , 178 M i n e , 161 M i n i m u m f l o w , 85 M i n i m u m p r e s s u r e , 4 0 , 167 M i n o r h e a d losses, 16 M i x e d f l o w , 96 M o d e l s , 90, 184 Momentum, 1 , 115, 152 Moody d i a g r a m , 7,8,16,20,23 118 M u l t i s t e p , 101

35

P a r a l l e l , 26 P a r r n a k i a n , J . , 182 P a r t f u l l p i p e s , 16 P e n s t o c k , 106 P e r i o d , 109 P i c k f o r d , J., 112,114,182 P l a n , 36 P l u g f l o w , 96 P o l y n o m i a l , 101 P o i s s e u i I le, 5 P o i s s o n ' s r a t i o , 117, 118

89,

204

P o w e l l , W.F., 63,69 P r e c i s i o n , 96, 195 P r e s s u r e , 40 P r e s s u r e r e d u c i n g v a l v e s , 39,41 P r e s s u r e t r a n s i e n t s , 138 Pseudo e x p l i c i t & i m p l i c i t , 166 Pseudo r e s e r v o i r , 105 Pseudo s t e a d y f l o w , 22,25,98 Pumps, 39,49,137,160,182 P u m p t r i p , 153, 167 Q u a d r a t i c convergence, Q u a l i t y , 90

32

Rao, H. S . , 102, 105 Real time o p e r a t i o n s , 102 R e l a x a t i o n , 35, 45, 47 Rep I acemen t r a t i o , 57 R e s e r v o i r s , 24,28,38,88,102,103 Resources, 102 Reynolds n u m b e r , 4,5,6,7,16,20

23,49,163 Rich, G.R., 112,114,142,145 R i g i d col umn, 21 ,22,23,106,110,113

T a b u l a t o r s o l u t i o n , 106 T a y l o r s e r i e s , 97, 125 T a y l o r ' s e x p a n s i o n , 125 Time s t e p , 103 T r a n s i e n t p r e s s u r e s , 162 Transportation programming, T r e e - l i k e n e t w o r k s , 66, 85 T r i a l a n d e r r o r , 22,45,70 T r u n c a t i o n e r r o r , 125, 126 T r u n k m a i n s , 45, 55 T u n n e l , 109 T u r b i n e , 106 Turbulent flow, 5 U n k n o w n s , 46 Unsteady flow, U-tube, 106

74

21, 106

V a l v e c h a r a c t e r i s t i c s , 176 V a l v e s , 17,122,137,141 ,169 V a l v e s t r o k i n g , 172 Van d e r Veen, 71, 86 Vapour p o c k e t , 1 12,137,148,150,

1 51 , 1 54,167,179

141 ,146,148,182 R i s i n g m a i n , 150

V a p o u r p r e s s u r e , 138,146,147,

Rough b o u n d a r y , 6 Runge-Kutta, 100, 101 R u n n i n g time, 41

V a r i a b l e s , 195 V e l o c i t y , 116, 172 V e l o c i t y head, 2, 16

S c h l i c h t i n g , H., 5, 20 S c h n y d e r , O., 175, 182 Screen, 183 Series, 26 S h a m i r , U., 102, 105 Simplex method, 55 Simultaneous s o l u t i o n , 22, 33, 176 S t a b i l i t y , 124 S t a n d a r d d i a m e t e r , 74 Statements, 196 Steady f l o w , 1 , 21 Steepest p a t h method, 77 Stephenson, D . , 33,70,86,120

W a l l t h i c k n e s s , 21,79,115 Water column s e p a r a t i o n , 110,

150,

135 ,143,145 S t e r l i n g , M.J.H., 102, 105 Stream t u b e s , 1 Streeter, V.L., 120,135,136,145,

138, 141,146,147,148,154,167,175,179 Water hammer ,21,87,106,112,1 15, 118,119,120,136,137,138,140,141, 142,143,147,148,151,153,160,175, 182,183 Water hammer equations,22,117,

124,126,128,129 Water hammer p r o t e c t i o n , 140 Water r e t i c u l a t i o n systems, 91 Water s u p p l y , 102 Watson, M.D., 20 Wave, 118,126,175 Wetted p e r i m e t e r , 16 Wood, D . , 45,54

169,172 Stummel, F., 100, 105 Successive o v e r r e l a x a t i o n , Surge, 106 Surge t a n k s , 112, 137 Symbol, 185 Systems a n a l y s i s , 70, 88

Yields,

49

75