Hydraulic Analysis of Unsteady Flow in Pipe Networks

Hydraulic Analysis of Unsteady Flow in Pipe Networks

Hydraulic Analysis of Unsteady Flow in Pipe Networks J. A. FOX Reader in Qvil Engineering University of Leeds

M

© J. A. Fox 1977

Softcover reprint of the hardcover 1st edition 1977 978-0-333-19142-2 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission First published 1977 by THE MACMILLAN PRESS LTD London and Basingstoke Associated companies in New York Dublin Melbourne Johannesburg and Madras

ISBN 978-1-349-02792-7 ISBN 978-1-349-02790-3 (eBook) DOI 10.1007/978-1-349-02790-3 This book is sold subject to the standard conditions of the Net Book Agreement

Text set in 10/11 pt IBM Press Roman

Contents

Preface

ix

Notation 1 Simple water hammer theory

xi

1.1 1.2 1.3

2

3

1 2

Introduction Rigid pipe-incompressible fluid theory Sudden valve opening at the downstream end of a pipeline 1.4 Slow valve closure 1.5 Distensible pipe-elastic fluid theory 1.6 Instantaneous valve closure 1.7 Separation 1.8 The calculation of the magnitude of the transient caused by complete instantaneous valve closure at the end of a simple pipeline 1.9 Pressure rise caused by instantaneous valve closure 1.10 Sudden valve closure

17 21 21

Analytic and graphical methods

23

2.1 2.2 2.3 2.4 2.5

23 23 23 25 36

Introduction Analytic methods of solution Stepwise valve closures at pipe period intervals The Allievi interlocking equations The Schnyder-Bergeron graphical method

Boundary conditions for use with graphical methods 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

Introduction Pumps Four quadrant pump operation Surge tanks Types of surge tanks Transient analysis of surge tanks Mass oscillation of surge tanks Pressurised surge tanks or air vessels Methods of integrating the surge tank equations

v

4 6 9 10 16

55 55 55 60 62 63 65 66 68 70

vi

Contents

4

The method of characteristics 4.1 4.2

4.3 4.4 4.5 4.6 4.7 4.8 4.9

5

Introduction Method of deriving the characteristic forms of the waterhammer equations The characteristic forms of the waterhammer equations The zone of influence and the domain of dependency The zone of quiet The integration of the characteristic equations Boundary conditions The method of the regular rectangular grid Other finite difference methods

Variable parameters in unsteady flow 5.1 5.2 5.3

Variation of wavespeed Gas evolution The magnitude of variable wave speed and the inclusion of gas release 5.4 The use of the variable wavespeed equation 5.5 Vaporous cavitation 5.6 Calculation of friction 5.7 The use of variable f values 5.8 Interpolation 5.9 The calculation of the free bubble content 5.10 Evaluation of velocities and potential heads at internal points in a pipe length

6

Boundary conditions: pumps 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18

Introduction Pumps equipped with a nonreturn valve The derivation of the pump's characteristic equation Dynamometer/turbine operation of a pump with forward flow Pump efficiency Pump power Pump start up Pump run down The in-line pump boundary condition Suction well pumps Four quadrant pump operation The use of the Suter curves Pump run up to steady speed of pumping Pumps with by-pass valves Pump stations Surge suppression of transients generated by pump trip Line pack and attenuation Lock in

72 72

74 77

78 79 79 81 82 85 87 87 88 90 94 94 95 96 97 98 99 100 100 100 101 102 105 105 106 107 108 110 Ill

118 120 120 121 124 127 128

Contents

7

Other boundary conditions

130

Junctions Joints Air vessels The motorised valve Servocontrolled valves Reservoirs Bends

130 132 134 136 142 144 146

7.1 7.2 7.3 7.4 7.5 7.6 7.7

8

Unsteady flow in gas networks 8.1 8.2 8.3 8.4 8.5

9

Introduction Basic equations Characteristic equations The value of 1 Boundary conditions

Impedance methods of pipeline analysis 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 9.14

10

vii

In traduction The analogy between electrical and hydraulic impedance The linearisation of the waterhammer equations The solution of the linearised waterhammer equations The evaluation of -y The impedance concept Receiving and sending ends The equation of impedance Boundary conditions The impedance of a network Harmonic analysis The forcing oscillation The oscillating valve A network in which resonance can be excited by forcing oscillations located at different points in the network

Unsteady flow in open channels 10.1 10.2 10.3 10.4 10.5 10.6 10.7

Introduction The equations of unsteady flow in open channels The characteristic forms of the open channel equations The travelling surge The profile of a free surface flow when a travelling surge is present The method of analysis of an unsteady free surface flow in which travelling surges are present Other methods of analysis

147 147 147 149 150 154 155 155 156 157 160 162 164 165 165 167 170 172 173 174 176

177 177 178 181 192 195 195 199

Contents

viii

11

Global programming 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10

Introduction The route or link method of global programming Pipe description Longitudinal profiles Upstream reservoirs Downstream reservoirs Pump description Pipe longitudinal profiles at & intervals Calls of procedures Time level scanning

201 201 201 203 203 204 204 204 205 205 206

References

207

Bibliography

209

Index

211

Preface

The reader may be interested to know how this book came to be written. The author has always found the subject of unsteady flow of great interest and throughout his career has studied it with special application. As a consequence most of his research effort has been in this area and he has guided many of his Ph.D. students into this topic also. In 1969 an engineer from a local Consulting Engineer's office approached him requesting information concerning surge analysis methods which could take into account variations in wavespeeds caused by free bubbles in the fluid. At that moment in time the author had already developed a computer program which could analyse surge in simple rising mains but had not included this wavespeed effect. The effect was soon incorporated into the program and it was used to analyse a main which had a history of bursting to decide what was the main cause of the bursts. At the same time, unknown to the author, one of his own ex-Ph.D. students had been employed to take measurements of the pressure history of the main. When the analytic results were compared with the measured results it was found that agreement was extremely good, the only error of significance being in the timing of the pressure peaks. The actual magnitudes of the maximum and minimum pressures were excellently predicted. Upon seeing these results the author and the engineer from the Consulting Engineer's office, Bryan Smith, decided to open an office, Hydraulic Analysis Ltd, Leeds, which would routinely undertake the analysis of proposed or existing systems. This venture turned out quite successfully and with the passage of time the firm has been called upon to analyse more and more complex systems, ranging from simple rising mains delivering sewage to a sewage works, to undersea oil pipelines such as that of the Forties field in the North Sea. The firm has been called in to analyse water supply networks for various authorities throughout the world, oil pipelines in the Middle East, most of the pipelines built or proposed for the North Sea, water injection schemes to improve oil delivery from underground strata, pipe networks in Condeeps and other complex networks such as those found in oil refineries and gas liquefaction plants. The firm has had to grow to handle this work, taking in a computer specialist, Andrew Keech,

ix

X

Preface

as a partner and employing more staff. Throughout this period it has been necessary to develop the original program and now it has reached a considerable level of sophistication. As an academic, the author feels that the essential material in this program should be published and so he decided to write this book. It is not possible to include within the confines of one book all of the material that has gone into the program; there are many facilities which have not been included but the main material on which the program is based has been described. The author would like to warn the reader that he has not tried to write the definitive book of waterhammer. It probably could not be written at present as the subject is still undergoing rapid development. Even so, this book is an idiosyncratic view of waterhammer and many people who have contributed greatly to the subject may feel slighted by the omission of their material or by the failure even to mention its existence. The author would like to apologise to such people and would plead, in advance, the limitations of space. The book is idiosyncratic in other ways, techniques of finite difference integration such as those due to Lax, Wendroff and coauthors have only been given passing mention and no mention at all has been made of what the author believes is a potential technique for the future - the finite element method. However, he has demonstrated, to his own satisfaction at least, the complete adequacy of the method of characteristics and offers this as partial justification for his limited presentation of a very large, very complex subject.

Leeds, 1976

J.A. F.

Notation

Throughout this book, symbols are defined wherever they are used and these are listed below. However, variables of local interests only are not included in this list but are defined in the text.

A A0 At

ae

area of flow area of valve opening at time zero (chapter 1) area of valve opening at time t (chapter 1) area of pipe (chapter 1) area of valve opening pump constant in equation H = AN 2 + BNQ- CQ 2 (chapter 6) exit area of pump impeller (chapter 6) plan area of a suction well (chapter 6) effective valve area (chapter 9)

B B b

pump constant in equation H = AN 2 + BNQ- CQ 2 (chapter 6) channel surface breadth (chapter I O) mean channel breadth (chapter 10)

Cct

coefficient of discharge of a valve celerity of a small pressure wave constant in friction formula used in the surge tank analysis (chapter 3) constant in pump equation H =AN 2 + BNQ- CQ 2 (chapter 6) coefficient in the stroke equation of a servocontrolled valve (chapter 7) specific heat of gas at constant volume (chapter 8) specific heat of a gas at constant pressure (chapter 8) electrical capacitance/unit length of a transmission line (chapter 9) celerity of a small surface wave (chapter 10) Chezy C (chapter I 0) coefficient of discharge of a sluice gate (chapter I 0) celerity of a surge wave in an open channel (chapter I 0)

ap

av A Ae Asw

c C

C c5

Cv Cp C

c

C Cct Cw

xi

xii dt dx dp dp dv d D

De d sw d1

E

time increment (infinitesimal) distance increment (infinitesimal) pressure increment (infinitesimal) density increment (infinitesimal) velocity increment (infinitesimal) pipe diameter pump impeller diameter constant in the pump efficiency equation depth in a suction well internal diameter of an air vessel Young modulus of elasticity pump efficiency (chapter 3) constant in the pump efficiency equation (chapter 3) internal energy of gas/unit mass (chapter 8)

f:

Ec e E1

Notation

~:: J

constants in the characteristic forms of the unsteady gas equations as defined in text (chapter 8)

E

g(j-i) (chapter 10)

f

Darcy fin the Darcy formula

\

2 hf=~y _gm fh

f

F

Fe F

f

Fn

g

Gr

h

hr hs hn hi

(as defined in text)

hoop stress in pipe wall 'function or and wave height when wave is travelling downstream (chapter 2) 'functiOit of' and wave height when wave is travelling upstream (chapter 2) constant in the pump efficiency equation in chapter 8, the force acting upon the fluid/unit length of pipe frequency of applied head oscillation (chapter 9) Froude number (based on absolute velocity) (chapter 10) intensity of the local gravitational field throughout the text gradient of the pump's speed ~ time rundown curve (chapter 6) potential head - the sum of local pressure head and elevation of the point above an arbitrary datum. head lost due to friction sometimes static head. sometimes head at pointS, according to context head immediately upstream of a valve or nozzle. potential head change caused by momentum changes (note Pi= whi)

Notation hair h3 hp hw htr hcnt

h

h' H hsp hw

xiii

pressure head of air in an air vessel expressed as height of the equivalent liquid column (chapter 3) atmospheric pressure head height of the base of an air vessel above the pipe centre line as hair above (chapter 7) head sensed by a pressure transducer controlling a servo-operated valve critical head at which a servo-operated valve will start to move. steady state head (chapter 9) unsteady head component (chapter 9) amplitude of pressure head wave (chapter 9) height of the reservoir surface above the spillway crest (chapter 10) the height of a wave crest above channel bed level (chapter I 0)

I

inertia of the rotating parts of a pump and motor set (chapter 6) electrical current (chapter 9) v'=J(chapter 9) (see context) channel bed slope - taken positively downwards (chapter 10)

j

frictional head loss/unit weight of fluid/unit length of channel (chapter I 0)

K

constant in valve loss formula hr = K 2g

v2

K K k

4[L =d + k (chapter

1 and chapter 5)

bulk modulus of liquid constant, sometimes describing local losses, 2

i.e. ht = k

~g

(due to bends. junctions etc)

In chapter I also used: k

k k

kr kv

k L L /1

L

= voLh gT s

constant in head ~ q equation h = kq 2 , i.e. the friction formula used in the Schnyder-Bergeron method (chapter 5) mean height of pipe roughnesses in the Colebrook-White formula pump impeller head loss coefficient (chapter 6) pump volute head loss coefficient (chapter 6) spillway constant (chapter I 0) length, usually pipe length L constant in the development of the characteristic form of the differential equations of waterhammer (chapter 7) internal height of an air vessel (chapter 7) electrical inductance/unit length of a transmission line (chapter 9)

xiv

Notation

=~(defined in text)

m

hydraulic mean radius

n

area

N

running speed of a pump in rev rnin- 1 (chapter 3 and 6) polytropic index (chapter 8)

n

ratio~: (chapter I)

Q

pressure generated by momentum change wetted perimeter pump power in chapter 3 and chapter 6 pressure of air in an air vessel (chapter 3) atmospheric pressure pump power in chapter 6 flow flow at time t heat flow/unit area (chapter 8) steady flow rate (chapter 9) unsteady flow component (chapter 9) amplitude of flow oscillation (chapter 9)

Re

pvd vd Reynolds number = - or-

Pi p p Pair Pa Pwr

q

qt

q

q

q

R R Rei Si

Sf Sreq

s T

T T T

IJ.

ZJ

universal gas constant (chapter 8) hydraulic resistance/unit length (chapter 9) electrical resistance/unit length of a transmission line (chapter 9) valve stroke at beginning of a At period (chapter 7) valve stroke at end of a At period (chapter 7) valve stroke required as defined by a pressure transducer (chapter 7) an integer taking the value of+ I or -1 (chapters 7 and 10) pipe period 2L (chapters 2 and 3) c pipe wall thickness (defined in text) torque applied in the pump equation (chapter 3) absolute temperature (chapter 8) time

u

velocity of impeller blade tips (chapter 6)

v

mean flow velocity (chapter I) velocity in pipe when t---* oo (chapter I) velocity at time zero (chapter I) velocity at time t (chapter I) air volume in an air vessel (chapter 3) volume of dissolved gas

V=

v0 Vr

Vair

Vg

Notation

XV

velocity of whirl at exit from a pump impeller (chapter 6) relative velocity at exit from a pump impeller (chapter 6) absolute velocity at exit from a pump impeller (chapter 6) velocity of flow at exit from a pump impeller (Chapter 6) electrical voltage (chapter 9) velocity of a surge wave in an open channel (Chapter I 0) weight density of fluid dimensionless heat parameter in the head Suter curve (chapter 6) dimensionless torque parameter in the torque Suter curve (chapter 6)

X

X

z

z

z Zc

z

Q Q

distance along pipeline distance along pipeline elevation of the pipe centre line above datum elevation of surface in a surge tank above reservoir static surface level (chapter 3) elevation above a datum of a suction well base (Chapter 6) the elevation above datum of the centre line of the pipeline at the point of its junction with an air vessel hydraulic impedance (chapter 9) characteristic impedance (chapter 9) depth of the centroid of a channel cross section (chapter 10) real component of 'Y (propagation constant) (chapter 9) constant defining nature of a channel cross section (chapter 10) CctA the product ~y2g (chapter 2) p

imaginary component of 'Y (propagation constant) (chapter 9) pump blade angle (chapter 6) propagation constant in chapter 9 ratio of specific heat of gas (chapter 8) ratio of channel cross sectional area/surface breadth (chapter 10) distance increment (finite) time increment (finite) volume increment (with subscripts to define which volume change is intended) (chapter I) pressure increment due to momentum change (finite) (chapter I) potential head change: related to l:lpi by l:lpi = wl:lhi (chapter I) fractional volume of free gas in liquid (chapter 5) ratio of aefae, (chapter 9) steady component of e (chapter 9)

xvi €

,

Notation unsteady component of € (chapter 9)

I h)O·S

the square root of the head ratio: \ho

(chapter 2)

the slope of the characteristic line

av

av.

1/

fractional valve opening:

e

angle in Suter presentation of four quadrant pump characteristic B=tan- 1

(~

gs)(chapter6)

constant in the characteristic formulation of the waterhammer equa· tions (chapter 4) d aB ax (chapter 10)

;>..

cxs

J1

dynamic viscosity of fluid

v

kinematic viscosity of fluid

7T

3.14159 mass density . . ha . . CVo All leVI c ractenstlc: 2 gho

p

p

T T

X

l/1 l/1

D. D.

(chapter 2)

=!:!. p

(chapter 2)

surface tension coefficient in chapter 5 viscous sheer stress in chapter 2 and chapter 10 the compound line produced by the summing of two eagres in the Schnyder- Bergeron method (chapter 2) the valve characteristic in the Schnyder - Bergeron method (chapter 2) phase angle (chapter 9) angular velocity of pump impeller (chapter 3 and chapter 6) angular velocity of the applied head oscillation (chapter 9)

1 Simple waterhammer theory

1.1 Introduction The hydraulic analysis of flow in networks is usually based upon the consideration of steady state conditions. This is due to historic reasons; the analysis of unsteady state is an order of magnitude more difficult than that of steady state and was only possible at all if grossly simplifying assumptions were made. Until the relatively recent development of computers the only methods available were graphical in type and these could only be applied to simple networks in which the hydraulic controls were of an elementary nature and in which the number of pipes was small. Now that computers are available, a very great improvement has been made to the quality of analytic techniques that can be used and it is no longer necessary to confine the mathematical modelling of a network to that of steady state. The analysis of unsteady state can include steady state as a special case but it yields much more information than this. The behaviour of the system during starting, its run up to steady state and the transient phase that occurs after shut down can all be described with considerable accuracy. It is usually found that the conditions occurring during steady state operation are of only passing interest, what happens during the starting and shut down phases being of much greater importance. The operation of complex hydraulic controls can be simulated and the only limitation upon the size and complexity of the network is the size of the computer store. Waterhammer is the name commonly used for pressure transients. The reason for the name is that when a steep pressure wave front passes through a pipe it generates a sound that resembles the noise that occurs when a pipe is struck by a hammer. In fact, all transients do not generate sound but the name has gained such a wide acceptance that there is no point in trying to change it. Wherever the word waterhammer is used in this text it should be understood to include all pressure transients even if they are not sufficiently steep fronted to cause noise. In the usual Newtonian approach to the analysis of the motion of a body it is usually assumed that a force causes an acceleration which is

2

2

Hydraulic analysis of unsteady flow in pipe networks

simultaneously applied to all particles within the body. In fact, when a force is applied to a body those particles at the point of application of the force are immediately accelerated. The movement of these particles relative to adjacent particles causes forces to be applied to them and they in turn are accelerated. The process then operates upon the next layer of adjacent particles and these are accelerated. Eventually all the particles in the body will be accelerated. In effect, a wave of compressive stress has passed through the body and this wave will have propagated at a speed that is usually large but not infinite. Most bodies are not sufficiently long in the direction of application of the force for the wave passage time to be in any way significant but the effect is always present. In the case of a long pipeline containing a fluid, the passage of a compressional wave through the fluid can take a significant time and the pressures caused by the compressional wave may be sufficiently large to burst the pipe. In such a case an analysis which did not include the effect of such transient behaviour would be of little value. However, if the pipeline were short, and a pressure fluctuation were applied at one end of it over a time which was much greater than the time taken for the compressional wave to traverse the pipe, then an approach which assumed that all fluid particles were being simultaneously accelerated would represent a reasonably accurate model of the fluid's behaviour. Two ways of predicting the behaviour of a fluid column when under the action of a force are thus available:

(I) Rigid column theory which considers the entire fluid column to be accelerating at the same value throughout its length, the wavespeed being infinitely large. (2) Elastic theory which considers any pressure change to be transmitted through the fluid column at a large, but finite wavespeed. Rigid column theory can only be used if the time of operation of the hydraulic control is considerably greater than the time taken for a wave of pressure to pass through the fluid column. Elastic theory can always be applied and gives more accurate results but it is usually more complex in nature. 1.2 Rigid pipe-incompressible fluid theory Historically, the development of waterhammer theory has followed a pattern of increasing complexity. It started by using the solid body type analysis which is now called 'rigid pipe-incompressible fluid' theory. Later, 'distensible pipe-compressible fluid' theories were developed. The second category of theories has been the basis of most of the work performed recently and can now be considered to be in a high state of development. Rigid column theory can be of considerable value, as situations arise in which pressure transients are not of great interest but in which fluid movements are important. Rigid column theory is capable of describing such motions moderately accurately.

Simple waterhammer theory

3

A simplified form of the dynamic equation First, one of the fundamental equations of waterhammer must be developed. Consider flow through a pipe of length L experiencing a pressure gradient

~~which is decelerating the fluid.

Note Pressures are assumed to increase in the direction of x increasing. The velocity vat time tis assumed to be the same at all points in the pipeline. The fluid mass contained in the elemental length Ax is pA ~x. The force decelerating the fluid is A

;~ Ax neglecting friction.

instantaneous pressure grade line

p

X

8x Figure 1.1

By Newton's second Law of Motion

ap

dv

A ax Ax + pA ~x dt = 0

so

ap dv -+p-=0 ax dt

(1.1)

This is an extremely simplified version of the Euler equation If dv is constant throughout the pipe length, and it is if the pipe is dt rigid and the fluid incompressible, the equation can be integrated to give

dv dt

~p=-pL

where

~p

(1.2)

is the pressure difference over the pipe length L which it is neces-

sary to produce to generate the

acceleration:~

Note that if the downstream pressure exceeds the upstream pressure by the amount

~p then~~ will be negative, i.e. a deceleration.

4

Hydraulic analysis of unsteady flow in pipe networks As pressures and heads are related by the expression p

= wh = pgh

this result can be cast into the form L dv t:.h=--g dt

(1.2a)

This solution is valid for frictionless flow but requires modification if frictional effects are to be included. 1.3 Sudden valve opening at the downstream end of a pipeline As an illustration of how the above simple theory can be used and how frictional effects can be taken into account the case of a sudden downstre'lm valve opening will next be examined. It suffers from the defects of all rigid pipe-incompressible fluid theories.

reservoir

lenglh L diameter d

Figure 1.2

!::.hi in equation I .2a is the excess of head at the downstream end of the pipe over that at the upstream end. At the downstream end the head is atmospheric as the valve is full open (treated as zero head here) while upstream it is hs so !::.hi = 0- hs if friction were not present, but as friction is present hs must be reduced by the amount of hr so

( 1.3) where hr is the friction head. Now

hr

=

4fLv 2 2gd

(I .4)

This is the Darcy-Weisbach equation (Fanning equation in USA); the fused is one quarter that used in the USA version of the Darcy equation but is the same as that used in the Fanning equation. Therefore if local losses are also included !::.hi=_ (h _ 4fLv 2 s 2gd

_

kv 2 ) 2g

= _f:dv gdt

5

Simple waterhammer theory where!';; is the head losses caused by local losses such as those occurring at bends, junctions etc,

h _ 4fL ~ _ kv2_!:_ dv d 2g 2g- g dt s

so

v2 2g

Ldv g dt

h -K-=--

or

s

(1.5)

where

dv dt

2gh 5 - Kv 2

- = -- ---·-

2L

so

dt

t =

-J 2Lglz/(

loge

d v___ _ ) = '"lL(' _ _ 2 2gh 5 - Kv

-

-.JKV)

( V2ifi s + v'2ifis _ .JKij

(1.6)

where vis the velocity at time t. V~ When t becomes int1nite V2ilzs where V~ denotes the asymptotic velocity as t tends to infinity

=YR.

fiihs

V = ~~K

so

a standard result from normal steady state theory as would be expected. Rearranging equation 1.6

so

t

= y2g~sK

loge (

:t~:)

. V ~ ~_v_ = e(2ghsK)o.s t/L

..

v~

-v

6


Rearranging

v

(1.7)

~ = e(2ghsK)o.st/L + 1

This can be plotted as shown in figure 1.3. The dotted line shows how the velocity curve is affected by the elasticity of fluid and the elasticity of the pipeline wall material and demonstrates how the rigid pipe-incompressible fluid theory predicts the mean velocity changes. It is a commonplace experience that when a tap is suddenly opened the flow is very fast for an instant then drops and rises to a steady state which is less than that of the first spurt. The high velocity v

~

f\--- actual velocity history

t-~-----1

theoretical velocity history

(2qh, K )0 "5 /L

Figure 1.3

from the first spurt is approximately equal to the spouting velocity and its rapid fall off is due to the inability of the strain energy of the fluid and pipe wall material to maintain this value for very long. A more accurate solution of this problem is described in chapter 2.

Vfihs

1.4 Slow valve closure A further example of the use of rigid pipe-incompressible fluid theory is that of slow valve closure. Assume that the general equation for the effective valve area is:

ae

=aof(t)

(1.8)

where C4J is the full open valve area and f(t) is some function of time. Effective valve area means the actual area multiplied by the coefficient of discharge. Also assume that the Bernoulli equation can be applied to the flow

7


through the valve even though it is in an unsteady state. This assumption is always used and it has been experimentally justified on many occasions. Then

Qt =

ae~

{1.9)

where Qt is the flow through the valve at time t and hn is the head immediately upstream of it at this time. Therefore, where A is the area of the pipe, so by differentiation with respect to time

dvt=dae dt dt

~+.!v'li(hro.saedhn A

ho _h

now

n

A dt

2

_ 4 fLvl = L dvt 2gd g dt

where h 0 is the head in the supply reservoir. This equation is derived from equation 1.2a. Substituting for ~Vt and rearranging gives t

(ho _

dhn = nVfihn h _ 4[Lvt 2 ) _ 2h d (f(t)) (1.10) dt L[(t) n 2gd f(t) dt A where n =ao This equation can be integrated by finite difference methods. An estimate of the maximum head can be made however. Multiply equation 1.10 by f(t). This gives 2 (fi( t)) dhn = n~ (h 0 _ h _ 4[Lvt ) dt L n 2gd

_

2 z.

"n

d (f(t)) dt

If the maximum head occurs at the time when the valve closes then Vt = 0 and the frictional term will then have no effect. If the valve closure is of a type which generates maximum head prior to the moment of incomplete valve closure then Vt will not be zero at the instant when dd~n = 0. If friction is important, a finite difference integration of equation 1.10 will have to be undertaken to obtain the maximum head but if friction can be neglected the following analysis can be used to obtain it. If a maximum head, in the mathematical sense, occurs anywhere within the closure period [(t)

~~will equal zero. If the head rises throughout the

valve's closure and reaches its largest value at the instant of closure without producing a turning point then the expression f(t) d:r" will be zero because the function[(t) must be zero when the valve is closed. So, irrespective of whether a mathematically maximum value is produced at some time within

8


the valve closure, or a largest value at the end of the closure is produced, setting the expression f(t)

~~n to zero will define the largest value.

0 = nVfihmax (ho- hmax ) - 2hmax -d (/(t )) L dt

(d

n2

2 2ghmax (ho- hmax) 2 = 4hinax - (f(t))

L

dt

Rearranging gives

hmax ( -~

)2

)2 - (2 + 2 L :t (f(t))\l) ~ h max + I = 0

•

~2~

~

Let

Then

(1.11)

As an example, consider a valve closure in which the valve flow area reduces linearly with time, i.e. ae

=a0 (1 - ~)

where Tis the time of valve closure. t

f(t)=1-T

Then

d 1 - (f(t)) = - dt T

and

Then

but

2g~o n

=

v~ where v0 is the steady state velocity in the pipeline before

the commencement of valve closure. Then

voL

k=ghoT

so it is simple to evaluate hmax/h0 from equation 1.11. Simple finite difference techniques can be used to integrate equation 1.10 to obtain the curve of hn ~ t if this should be required.


9

Consider the linear valve closure mentioned above in which the valve area ae =a0 (I - t/T) and ignore friction: when t

=0, hn =hs ( dhn) dt t

= 2hs

o T

=

If the time Tis split up into m increments T

t:J.t=-

m

so

The next step of integration can now be performed

h

_h

nt=2 .:l.t-

nt=.:l.t

(dhn)

+ dt

t=.:l.t x

T

m

This process can be repeated until a sufficient time period has been explored. As the integration scheme is of an initial value type, m must be large and so the process is best performed on a computer. The program required can be written in a very short time and the run time will be very small even when m is made large. As stated already this analytic method is subject to very grave defects if the focus of interest is the magnitude of pressure transients, as it omits all elasticity effects and so only begins to approximate to reality when valve closure times are large, in which case the problem becomes trivial. The method of analysis can be useful when pressure transients are not of importance, however.

1.5 Distensible pipe-elastic fluid theory The remainder of this book will be concerned with elastic theory. The most accurate of the methods, described later, requires the use of computers. The various theories of waterhammer described will be presented in a sequence of increasing complexity and the first three chapters will be devoted to describing what are considered by the author to be obsolescent theories. The reader would be wise to master these obsolescent theories in the sequence in which they are presented as it is the sequence in which they were developed. He will then be able to assess the improvement in accuracy attainable from the use of progressively more rigorous methods.

10


1.6 Instantaneous valve closure This section is important, as it provides the reader with an insight into the mechanism of wave propagation and reflection in pipelines. Pressure rises generated by very rapid velocity fluctuations are known as transients and have a period equal to four times the pipe length/wave celerity, i.e. 4Ljc. Instantaneous valve closure is a theoretical concept, as no valve can be closed in zero time but the study of instantaneous closure leads to an understanding which can be used to solve real problems. When a valve at the end of a pipeline is closed in zero time the layer of fluid immediately upstream of it is instantly brought to rest, and its impact upon the valve will cause its pressure to rise. This increase in pressure will cause the section of pipe containing the fluid layer to distend and the fluid in the layer to compress. The fluid layer immediately upstream of the now stationary first layer will next be arrested a very short time later. The time delay is caused by the second layer continuing in motion for a small time while it moves forward to occupy the volume made available by the pipe distension and fluid compression of the first layer. The third layer is brought to rest in the same way as the first and second, its loss of momentum due to its impact upon the second layer causing within it a pressure rise identical to that experienced by the first and second layers. As the first and second layer cannot rebound from the closed valve their pressure cannot fall and is maintained at its initial impact value. Progressively, layer after layer of fluid is brought to rest. The situation is depicted in figure 1.4a. L

I

distended pipe

pressure wave magnitude ~ f>;

static

I<

f-

head~

wavespeed 'c

t>, pressure head plat

(a)

Figure 1.4

Eventually, the entire pipe length is full of fluid which is at rest but at a pressure head of hi + h 5 where hi is the head rise caused by the impact, i.e. an inertia head, and h 5 is the static head of the fluid in the upstream


11

reservoir (neglecting local losses). The situation is then as depicted in figure 1.4b. The process of impaction of successive layers with the small time delays involved in each layer's impact mentioned earlier is, in effect, the propagation of a wave of pressure hi at a velocity c. The time taken for this wave to traverse the length of the pipe L is L/c. When the wave has traversed the pipe the entire mass of fluid in the pipe is at rest but it is also at a pressure hi + h 5 • This situation is unstable as the reservoir is at a lower pressure h 5 • The fluid therefore starts to flow out of the pipe in a direction towards the reservoir. Successive layers of fluid move towards the reservoir at the original velocity v, each layer of fluid expanding and its associated pipe L

~

-

••0

. 7.

d1stended p1pe

"'reservoir

I=£.

c

pressure head plot (b)

Figure 1.4 (continued)

section contracting back to its original diameter. Figure 1.4c depicts an intermediate stage in this process. Eventually the reflected wave arrives at the valve. Figure 1.4d depicts the situation that then prevails. The flow circumstance is now exactly the same as that which existed at t = 0 except that the flow is now away from the valve instead of towards it. Again this condition is unstable. As soon as it occurs, the fluid at the valve end attempts to leave the closed valve and to move in an upstream direction. It cannot do this, so its momentum is converted into a pressure decrease. The fluid layer next to the valve is brought to rest and its pressure is reduced by an amount equal to the original pressure rise, i.e. by hi. Successive layers are brought to rest as before but this involves a pressure drop as opposed to the original pressure rise. Figure 1.4e depicts an intermediate stage in this process. Eventually the entire pipe length is filled with fluid at rest but with a pressure head of hs- hi as shown in figure 1.4f. Again this situation is unstable because fluid will flow in from the reservoir at the original velocity v. This will cause the pressure to rise back up

12

;

Hydraulic analysis of unsteady flow in pipe networks L

--- v

.....,_v~o

h,~

I

lh, pressure head plat

l' .,

z I

0c_ >I> _1,_

c

c

(c)

L

~ .....,;,-'

X pressure head plot (d)

L

h,

(e)



J

13

L

:4

""reservo1r

X

~reduced pipe diameter

pressure head plot

(f)


to its original pressure head h 5 and the velocity will revert to its original value of v directed towards the valve. An intermediate stage of this process is depicted in figure 1.4g. The final consequence of this reversal of flow is shown in figure 1.4h. This situation is exactly the same as the initial circumstance so the process repeats endlessly. In fact, friction rapidly attenuates the transients that have been described so the reflected waves are sequentially reduced in magnitude. In fact, the above description has ignored the effect of friction but this is described later in this section and in section 6.17. Typically, five or six reflected waves of significant magnitude will be seen. In the description of the mechanism of wave formation given above it has been said that a wave reflects completely and negatively at a reser· voir and completely and positively at a closed end (see figures 1.4a, band c, and figures 1.4e and f.). This means that a wave travelling over a fluid at pressure h 5 with a magnitude /::;.h is reflected at a constant head point (a reservoir) with a magnitude h 5 - /::;.hand at a zero velocity point (a closed end or closed valve) at a head of h 5 + Ah. This is an automatic consequence of the laws of conservation of energy. In a circumstance in which fluid has

v2

p·2

velocity energy 2g Nm/N but no relative strain energy w~K Nm/N, i.e. at L

~~=~~v==:;~r.....3E:·o=T;~X ""original pipe diameter

~>!>~ c c

pressure head plot (g)


reduced pipe d1ameter

14


~

"reservoir

L

-·

pressure head plot

I= 4L

c

(h)


a closed end, there is a direct conversion of kinetic energy to strain energy, i.e. a positive wave reflection and vice versa at a reservoir. (Note K is the bulk modulus of the fluid.) A major principle has been enunciated: 'a complete positive reflection occurs at a closed end of a pipeline and a complete negative reflection occurs at an open end'. This suggests that partial (positive or negative reflections) occur at ends which are not completely open (constant head) or completely closed (zero velocity), i.e. at junctions. This will be more definitively discussed in later chapters. By careful consideration of figures 1.4a-1.4f, pressure plots at various points can be produced. At the downstream end of the pipeline a pressure-time plot can be evolved, for example see figure l.Sa. 2L

2L

h

I'

c

I

c

2L

I

c

I_ hi I

h;

valve closure occurs

....___; (a)

Figure 1.5

At a point!' upstream from the valve the pressure-time diagram is as shown in figure l.Sb, and at the reservoir end of the pipeline the pressuretime diagram is as shown in figure l.Sc

15


h

valve closure

occurs

J/ L

h,

_{ c

(b)


Note that although the wave shape changes greatly as the point of observation of the wave moves upstream there is no attenuation of the wave magnitude. The effect of friction on the wave is, in some ways, surprising. The wave shape at the valve is as illustrated in figure 1.6. This diagram requires explanation. At point A the valve has just closed, the head hi has been generated because the velocity v has been destroyed. At point B the velocity v has also been destroyed and an inertia head hi has been consequently generated but the wave arriving at the valve at time B was generated by stopping fluid moving at a point!' up the pipeline. At the instant when this fluid was stopped the pressure head was greater than the original pressure at the downstream end by an amount 4fl'v 2 /(2gd) and the fluid was stopped at a time l'/c after the valve closure. The abrupt stopping of the flow at a point l' upstream of the valve thus causes a total pressure rise of hi+ 4fl'v 2 /(2gd) but this rise arrives at the valve at a time l'/c later. So, the 2L c

c

h

2L c

I

I_ h,

/

t

valve closure occurs

(c)


16

Hydraulic analysis of unsteady flow in pipe networks 2L h

T

h = 4fLv2 f

2qd

4flv 2 +h.

2qd

'

Figure 1.6

pressure rise caused by stopping the flow /'upstream takes a further 1'/c to be transmitted to the valve at wavespeed arriving there at a time 2l'lc after the valve closure. Considering the circumstance at the reservoir end, the liquid stops at a time L/c after the valve closure and the pressure rise is communicated to the valve after a further L/c time interval. The pressure rise is hi+ 4fLv 2 /(2gd). Immediately following this pressure wave there will be a large negative pressure wave. This negative pressure wave will be running through stationary liquid so the pressure will drop from hi+ 4fLv 2 /(2gd) to -hi. Due to energy losses in friction the value ofv will be less than the original v value so hi will be smaller than the original hi and the friction head 4fLv 2/(2gd) will also be smaller than the original value of the friction head. Thus the waves attenuate. (See section 6.17 for further explanatory comments.) 1.7 Separation If a negative wave created by a reflection at the reservoir end of the pipe should attempt to reduce the pressure to a value less than vapour pressure the liquid will boil at the ambient temperature and a hole will appear within the liquid. The pressure will not be able to fall below the vapour pressure. The pressure trace will then appear as illustrated in figure I. 7. Because an equivalent negative transient to the initial positive transient cannot occur (as the pressure is unable to fall below the vapour pressure) the fluid moving away from the valve at a time a little greater than 2L/c cannot be brought to rest quickly. Consequently a long delay occurs whilst the inadequate pressure difference operates to reverse the flow. The situation may then repeat until the initial transient has been so attenuated as to be unable to reduce the pressure during its negative phase to vapour pressure. Once this occurs the transient behaves in an exactly similar manner to that of any other transient. A similar phenomeon known as gas release


absolute zero head

17

Figure l. 7

can simulate a very similar condition to boiling. In water, if the pressure falls below 2.4 m (8ft) absolute (approximately), air bubbles will evolve from the air dissolved in all natural water. These bubbles reduce the rate of reduction of water pressure whereas boiling prevents its reduction below vapour pressure absolutely. When the fluid column is at vapour pressure and a hole appears within it the phenomenon is called column separation. 1.8 The calculation of the magnitude of the transient caused by complete instantaneous valve closure at the end of a simple pipeline Allievi expression

Consider a length of pipe ~x long through which a transient pressure of magnitude ~Pi passes in time ~t, reducing the velocity from v to zero (valve closure case). ~p· = _w~x dv g dt I from equation 1.2. ~t is the time taken for the transient to traverse the ~x length reducing the velocity by ~v and is equal to ~xjc where cis the celerity of the transient w~x

~pi=---

g

-~v

x--

&jc

~Pi= ~h· = C~V W

I

g

(1.12)

This is sometimes known as the Allievi expression but is also variously attributed to Moens, Korteweg and Joukowsky. As the ~v in the above expression is the velocity decrement occurring in time ~t it can be replaced by v if the valve closure is total (but still instant) and then ~h·

I

=cgv

The remaining problem is to calculate the value of c.

(1.13)

18


Wavespeed The magnitude of the wavespeed in part depends on the bulk modulus of the fluid and in part on the distensibility of the pipe. It can be calculated quite simply if the pipe distensibility is definable. A simple case will now be demonstrated and results for other cases will be quoted.

Pipe fitted with expansion joints so that it can extend longitudinally without generation of longitudinal stress and free to distend diametrally When a pressure 11pi is applied to a pipe it will distend diametrally. The fluid within it will compress. The pipe distension plus fluid compression effects enable it to contain more fluid than it would do in its normal unpressurised condition. This increase in volume can be calculated as follows. Volume increase due to fluid compression= 11 Vp

=~Pix~ d 2 L

where K is the fluid bulk modulus and dis the pipe diameter. The hoop stress fh in the pipe wall = 11,{~d where Tis the wall thickness of the pipe. The circumferential strain in the pipe wall equals the diametral strain ah =fh

E being Young's Modulus. The increment in pipe radius = ah x

E

(1.14)

~

The increment in pipe volume 11Vpipe = circumference x length x radial increment d 1 2 11 Vpipe = rrdL x ah 2 =2 ahrrd L - 1 l1pjd d 11 Vpipe- E 2 T x 2 rrdL

so

The total volume now made available by pipe distension and fluid com· pression is therefore 11 Vtotal and

(1.15) Remembering that if the wave has not reached a section the fluid will


19

continue travelling at its original velocity v, then the time taken for the continuing undisturbed flow to occupy this additional volume will be

so

This time is the same time as that required for the wave to traverse the pipe compressing the fluid and distending the pipe. From the Allievi equation (1.13) D.pi

wcv =pcv =---g

and equation 1.2

D.v = -v

and as

PLv

pcv=D.t

so

1

.!!_) ('_!+ TE

c = D.pi K v

but from equation 1.12

·· c=wc(1 - + d) g

K

TE

(1.16) If the pipe had been infinitely rigid this result would have reduced to

c=-1

Jrl

or

c=Jf

20


so the effect of distensibility can be imagined as reducing the effective bulk modulus of the fluid from K to K' where

For the case of a lined tunnel in rock: steel pipe through rock tunnel with concrete infilling between pipe and tunnel

c= where

where

ds de

external diameter of steel liner

= external diameter of concrete pipe

Es = Young's modulus of steel Ec = Young's modulus of concrete

ER = Young's modulus of rock 1

Poisson's ratio for rock

m T

=

wall thickness of steel liner.

For a plain tunnel in rock

cJf!i=/w -+I

K

2 ER

For a thick walled pipe

where

d1 = external diameter of pipe d 2 = internal diameter of pipe K = 2.03067 x 10 9 N/m2 =42.336 x 10 6 lbf/fe for water E = 2.10915 x IOn N/m 2 = 44.064 x 10 8 lbf/fe for steel w "' 9810 N/m 3 = 62.4 lbf/ft 3 for water g = 9.8lm/s2 = 32.2ft/s2


21

1.9 Pressure rise caused by instantaneous valve closure The pressure rise can now be calculated from Allievi's formula wcv !::.pj=-g wv

-

g

-j;(k+ !e) (1.17)

When using this formula it is vital to remember that units must be completely consistent, e.g. in SI units, g must be in m/s, w in N/m 3 , K and E in N/m 2 and d and Tin m. The results obtained predict very large pressure rises if a valve in a pipeline is closed instantaneously, e.g. for a typical steel pipeline of normal dimensions an instantaneous valve closure will generate a pressure head rise of 125 metres of working fluid for every metre per second of velocity destroyed. As some pipelines are now working at high velocities (10 metres per second is not an extreme value) great care must be taken to make either the pipeline extremely strong or ensure that no instantaneous valve closures can occur.

1.10 Sudden valve closure The very large pressure rise created by an instantaneous valve closure can, unfortunately, be generated by valve closures which are far from instantaneous. (This may not be the largest rise as line pack can produce even higher pressures especially in long pipelines- see section 6.17.) If a slow valve closure is thought of as occurring in a sequence of small steps of closure, each step occurring instantaneously but separated in time by a small time interval, then each step will generate a small velocity decrement t::.v associated with a small pressure rise t:.p. t:.p will be given by the Allievi equation t::.p = wct::.v g

and its wave form will resemble that described in figure !.Sa. Each of the steps will produce such a wave but each will start a small time interval after its predecessor. The waves so generated will superimpose upon one another and the pressure at the valve will rise. If the last closing

22


motion of the valve is completed before the first wave's negative reflection returns to it the sum total of the !lp values of all the waves will exactly equal that produced by an instantaneous valve closure in which the same initial velocity was destroyed. Such a valve closure is called 'sudden'. The wave's shape will be different from an instantaneous closure but its peak magnitude will be the same. It is produced if the closure occurs in a time less than the pipe period 2L/c. If the valve closure is slower than this, reflected expansion waves will be returning while the later steps of valve closure are still occurring. These pressure decrements superimposing upon pressure increments still being generated by the continuing valve closure will cause a reduced rate of increase of pressure or even a decrease to occur so ensuring that the pressure rise generated by a valve closure which takes longer than 2L/c will produce a peak pressure which is less than that generated by a sudden closure. Pipelines are now being built which may be as long as I 00 kilometres without any intermediate booster stations. The pipe period for such a line could be as great as 200 seconds. The closure of a valve at the end of such a pipeline in a time of 3 minutes 20 seconds might seem slow but in fact it would be fast being a sudden closure and giving rise to transients of maximum magnitude. It will be realised that it is not possible to discuss valve closure rates in terms of being fast or slow without reference to the period of the pipe (2L/c) to which the valve is fitted.

Note The pipe period 2L/c must not be confused with the period of oscillation of the water hammer wave 4L/c.

2 Analytic and graphical methods

2.1 Introduction

It is necessary to discuss the work that has been done in the past on the analysis of transients generated by slow movements of hydraulic controls. As this chapter is still concerned with providing a background to the more modem techniques of analysis that will be presented in later chapters only an outline will be provided.

2.2 Analytic methods of solution Two analytical techniques for solving slow valve closures exist. Both of them require the assumption that friction in the pipeline can be ignored. The two methods are completely equivalent although this may not be obvious upon a superficial examination. The assumption that friction can be ignored is extremely dangerous in the hands of an inexperienced analyst. It can lead not only to grossly wrong solutions but to unsafe solutions. Therefore, before either of the following techniques is used, the circumstance to which they are to be applied must be very carefully examined bearing in mind the above comments.

2.3 Stepwise valve closure at pipe period intervals The method is based upon the idea of considering the pressure and velocity conditions in the pipe at every pipe period (2L/c) interval throughout the valve closure. It is necessary, of course, to know the position of the valve at the end of each of these intervals. The first step of valve closure will not have generated a negative reflection from the reservoir end of the pipe at time zero so it will be dealt with separately. A small time after the first closing movement of the valve has occurred the situation in the pipeline will be as shown in figure 2.1.

23

24


Figure 2.1 There is a wave of magnitude ilhi, travelling up the pipeline at velocity c. This wave is classifed as an F wave. Waves travelling down the pipe are

classified as/waves. The reason for this terminology will be explained in the next section. Initially, before any valve closure movement has occurred, the velocity v0 exists throughout the pipe and the prevailing head is h 5 •

(2.1)

Therefore

Where av o is the full open valve area and Cd 0 is the coefficient of discharge. Immediately after the first step of valve closure

- cd, avl

l'j; (h

v•---11/""g ap

s

cilvl) 0·5 +--

(2.2)

g

where c is the wavespeed. Denoting

Cdavj2i 2gby

Then

Vo

ap

= 13oh~ 5

cilv Vt =13. ( hs+T

and

13

rs

(2.3) (2.4)

ilv1 = Vo -v1

but

Vf = l3~ (hs +CVo; CV1)

v~ + l3~cv1 -13~ ( h5 + C:)

=0

Solving this quadratic

vl

=J3~c +_! 2g

2

13jc2 +413lhs + 413g~cvo IS

(2.5)

Analytic and graphical methods 13ic vi=--+131 2g

2 v0 c (l31c) +hs+2g

g

25 (2.6)

In the foregoing, the assumption has been made that Bernoulli's equation can be applied at the partly open valve. As stated in section I .4 this assumption has been shown to work well even in unsteady flow conditions. Having calculated v1 the magnitude of the F wave: f1h 1 is readily obtained from f1h 1 =c(v 0 -vi)/g. After a period 2L/c a further closing valve movement occurs, the 13 value of the valve becoming:

~=

cd 2 av v'fi/ap l

However, an f wave will have been reflected from the reservoir at a time Ljc after the initial step of closure and will be arriving back at the valve at the time 2L/c just as the valve makes its next closing movement. The magnitude of this F wave will equal -F because it was generated by a negative reflection of the f wave at the reservoir. The head at the valve at time 2L/c will thus be hs+ f. The velocity in the pipe will be v1 = v0 - f1v1 • The head at the valve will rise from hs +/to hs + f + c(v1 - v2 )/g V2

Now as

= 132 (hs + f + c(vi- V2 )/g) 0 "5

f1v1 =!..Fandf= -F c f1v 1 = -[gjc v1

so

-

v2 = v0 - f1v 1 - v2= v0 +!..t- v2 c

v~ = 13~(hs + f + c~o + f- c;2) Solving this quadratic gives V

2

=-

+ 13 rrl32c)2 + CVo + h + 2/ 13~c s g 2,.f\2i 2g

(2.7)

As f equals -f1h 1 , v2 can be calculated and f1h 2 can then be evaluated from f1h 2 =c(v 1 - v 2 )/g The head h 2 = hs +f + f1h 2 • The entire process can be repeated until the valve closure has been explored. 2.4 The Allievi interlocking equation Allievi1 developed the following analytic technique in 1903. It is more complex than the method given in section 2.3 and it is more elegant to the mathematically minded reader but it is no more accurate and in fact the technique described above can be manipulated so as to produce the Allievi interlocking equations quite easily.

26


Before the Allievi interlocking equations can be presented it is necessary to develop the differential equations of waterhammer. These two equations are differential forms of the continuity and dynamic equations and as they form the basis of all accurate analytic methods they will be developed here.

The continuity equation In figure 2.2 a section of pipe is shown in which a wave is travelling in the upstream direction. The mass of fluid entering the elemental length ox in time otis pAv&t and the mass leaving it in time otis

(p + ~~

ox) (A + ~:ox) (v + ~~ ox) o

t

The additional mass that can be accommodated due to fluid compression and pipe distension within the ox length during the ot interval is due to the increase in mean pressure that occurs over the 0t interval and is:

pAox

~:&t (k + ~)

(see equation 1.15)

The net mass inflow to the ox length must equal the amount of mass that the fluid compression and pipe distension can accommodate, so

av aA ap pAv&t- ax oxAv&t- pv ax ox&t- pA ax ox&t- pAvot ap

= pAox at ot

d) ( K1 + TE

wave moving upstream at velocity c

z datum

Figure 2.2

(2.8)

27

Analytic and graphical methods neglecting second order small quantities,

d) = 0

1+ap ( av pAaA ap +pv-+pA -+ Avat ,K TE ax ax ax

so

g wc 2

but

d

1

=K+ TE

(2.9)

(see equation 1.16)

(2.10)

so but

ap

ap Po ax

ax

K

-=--

~ aA=~ aA ap

and

A ax

A ap ax

A pipe of circular cross section was assumed when developing equation rr 2

1.16 on page 197 so A

=4 d

~ aA A ax ad ap

but

= _v_ ap • !1: • 2d ad = 2v ip_ ad '!!: dz ax 4 4

=~ ap

ax

(ih d) _a ( E

pd - ap 2TE x

d ax ap

d)

d2 2TE

~ aA A ax

so As w(h - z)

=p

and w

d 2 = v ap !i ax TE d ax 2TE

= 2 ~ ap

~~ = ~equation 2.1 0 becomes (2.11)

where p0 is the density at the origin pressure above which pressure is measured.

£. = 1 + PK and asK is extremely large in comparison with any pracPo tical p value pjp0 can be accurately approximated to unity. But

:. .£.. ah + vw c2 at

av =0 _£_) + ax (ahax - ax~) (K_!_ + TE

28


A.)= cK

w(.!+ K TE

but

2

az ah c2 av ah -+v-+-- -v-=0 ax ax g ax at

so

(2.12)

This is the continuity equation of waterhammer in differential form. The dynamic equation A force balance equation can be written for the section of pipe shown in figure 2.2. Force acting L -+ R =

pA (I)

(P +~~ox) (A +~:ox) +(P +~ ~~) (~:ox) - rPox- wAox ~~ (2)

(4)

(3)

(5)

Term (I) is the normal pressure force acting on the left end of the pipe segment. Term (2) is the normal pressure force acting on the right end of the pipe segment. Term (3) is the longitudinal component of the reaction of the mean pressure force from the pipe wall upon the fluid. Term (4) is the frictional force opposing flow. Term (5) is the weight component acting along the pipe centreline opposing the flow. Note Pin term (4) is the mean wetted perimeter of the pipe segment and T is the frictional shear stress between the fluid and the pipe wall. Ignoring second order small quantities, force acting in the L -+ R direction is

dz ap -A -ox-rPox-wAoxdx ax This force causes an acceleration of the fluid in the segment according to Newton's Second Law.

dv dz ap -A -ox -rPox -wAox- = pAoxdt dx ax

(2.13)

:. dividing through by Aox and rearranging T _ dV dz ap ax+ w dx + p dt +A/P-O

(2.14)

Now A/P = m, the hydraulic radius of the pipe, where A = cross sectional area and P =the wetted perimeter of the flow, and dv can be expanded by

dt

the use of the definition of a total differential derivative, i.e.

29


dv av av dt = v ax+ at

dz

but

dx

az

ax

au

a

au

T

ax (p + wz) + pv ax + p at + m =0 Dividing through by w (=pg)

]__ (!!._ + z) + !:'_ av + .!_ av + __!___ = 0

ax w

g ax gat

pgm

From the Darcy-Weisbach formula (Fanning in USA) _!__

pgm

=fvlvl 2gm

(This implies the use of a steady-state friction formula.) Now.e_ + z w

=h:

the potential head

ah +_!:'_ av +_!_ av +fvlvl ax g ax gat 2gm

so

=0

(2.15a)

If the pipe is of circular cross section this becomes (2.15b) as m = d/4. Note These two equations, continuity and dynamic, are a pair of quasi-linear hyperbolic partial differential equations and as such cannot be solved analytically. Together they represent the problem of transient propagation in distensible pipes but, as they cannot be solved analytically, various simplifications have been made to them in an attempt to obtain an analytic solution. The best known of these simplified theories is that due to Allievi and this will be presented here. Allievi decided to ignore the nonlinear terms

and friction. This means that he ignored the v ~~ term in the continuity

-aX

ah an d so, m · some p1pe · li nes, 1s · ---a X

. · o f t h e ord er o f v equa t ton as v ah 1s

small. He ignored

v+ C

the~g aavX term and the 2 fvdlvl g,

The~g aavx term is of the order of(__!!___) aav v+c~ t

term in the dynamic equation. so it is usually small but to

2fvlvl term 1s · fnction · · 1tse · If an d t h.IS can on Iy be d one · to 1gnore · 1gnore t h e --g;J" if frictional head losses are trivial fractions of static heads. This assumption of no friction is critical; without it there is no chance of obtaining an ana-


30

lytic solution but with it the value of the analytic solution becomes severely limited. The simplified equations that Allievi used are:

av g ah ax=- c 2 at ah av at= -g ax

continuity equation

(2.16)

dynamic equation

(2.17)

By differentiating the first with respect to t and the second with respect to x the wave equation in classic form can be obtained, i.e.

av ax at av at- -ax=

and

a 2h at2

=

g - c2

a2 h

3("2

a2 h

-g ax 2

a2 h c2ax 2

(2.18)

Once the form has been recognised it will be realised that an analytic solution must be possible. The solution is usually ascribed to Riemann, i.e.

h= h0 + F(t +~)+ t(t -~)

(2.19)

v=v0 -~ [F(t+~)-t~-~)]

(2.20)

(It can be found in any mathematical textbook covering the solution of partial differential equations.) The reader may have noticed that in other texts the solution quoted differs from that given above. It will be found that this is due to the author adopting the convention in this book that xis measured from an upstream origin (the reservoir) and vis assumed positive if the flow is in the direction of x increasing, whereas other authors have chosen their x origin at the downstream end of the pipe (i.e. at the valve) and taken their velocity as positive if it is in a direction of x decreasing. This is mathematically inconsistent so the author has not accepted it.

The symbols Fand

f denote 'function of. Clearly F(t + ~) and f(t -~)

must have the dimensions of head and in fact must represent contributions of head from waves; this will be more fully demonstrated in the next paragraph. Consider the Reimann head equation. It could be plotted on an x base for an instant of time t =t 0 . For an observer travelling upstream at wavespeed c and located at x =X at time t = t0 , the head seen will be hx (see figure 2.3) and hx

= ho + F

(to +~),ignoring the f (t- :) term for the


X

31

..

X

Figure 2.3

moment, but the observer is travelling, so at time t 1 he will be located at position X 1 obeying the equation X 1 ==X- (t1 - t 0 )c. Now, according to the Reimann equation hx, == ho +F(t, + ~1 ) but

F(t 1 +~1 )==F(t 1 +X-(~-to)c) ==F(t0 +~)

i.e. the original unchanged value ofF at time t == t 6 at position x ==X. If, to an observer travelling upstream at wavespeed, the F term does not change then the F term can only be a wave travelling upstream at velocity c. Similarly it is argued that thefterm represents a wave travelling downstream at speed c. Thus, like all equations, the Reimann head equation is a statement of the obvious. The head at any point on a pipeline x at time t is made up of the static head h 0 plus the head contributed at that point and time by a wave travelling up the pipe (F wave) plus the head contributed by any wave travelling down the pipe (!wave). From these equations it is simple to deduce the reflection circumstances at reservoirs and closed ends by mathematical means rather than by the use of engineering insight as in chapter 1. Consider a wave generated at the downstream end of a pipe travelling upstream towards a reservoir. Consider a point on it generated at time t

== 0 at the downstream end (x ==L ). The wave magnitude will be F(o + ~).

As it travels upstream its shape will be unchanged (see before) so at time t and position x

When this wave reaches the reservoir at time L/c it will still have the same magnitude and the function will be F

({+~),i.e. F (~)·At the reservoir the

head must remain constant at h 0 , so

h== ho == ho + F(~) + t(~ + o)

32 hence


F(~) which is still equal to F(o +~)equals-[~+ o).

After a further period of!:. the f wave which has been generated will

c

travel back to the valve arriving at x = L at time 2L/c, i.e. j 2L J

\'c

-!::.) c ,

i.e.t(~), indicating that thefwave has travelled back to the valve with unchanged magnitude. Thus the initial wave

F(o +f) will travel upstream

to the reservoir with unchanged magnitude and shape at which it will be reflected completely and negatively as an/wave, which will then travel downstream to the valve with unchanged magnitude and shape arriving there 2Ljc after the first F wave started out. So, in a simple pipeline,[ waves arriving at the valve were generated one pipe period earlier as F waves which were totally and negatively reflected at the reservoir. Similarly at a closed end F waves are totally and positively reflected. This can be shown from the Reimann velocity equation. At a closed end the velocity must always be zero, so at such an end v and v0 must always be zero. Thus i.e.

o= o-~ [1~+ ~)-t(~- o)J F(~) =t(~)

Summarising: if time periods ofT(

(2.21)

T= :L the pipe period) are considered

and any particular time is denoted by iT where i is any appropriate integer number then: for the case of a reflection at an open end (a reservoir) /i =-Fi-1

(2.22)

and for the case of a reflection at a closed end

/i=Fi-1 Denoting the F(t

(2.23)

+~)wave at a time iT by Fi and the t(t -~)wave at

time iT by fi the Reimann equations can be written as (2.24) (2.25)

33

Analytic and graphical methods Considering the sequence of events at a downstream valve caused by its slow closure at pipe period intervals at t = 0

= ho + Fo + fo

(2.26)

= Vo -~c (Fo- fo)

(2.27)

=ho + Fi +ft

(2.28)

ho Vo

at t= T h1

=vo-~(F, -[t)

(2.29)

= ho + F2 +!2

(2.30)

=Vo -~c (F2- f2)

(2.31)

V1

at t

c

= 2T h2 V2

and so on, but / 0 = 0 as there cannot be an f wave until the initial F wave has reflected at the reservoir and returned to the valve. Now

ft = -Fo

(2.32)

!2=-F,

(2.33)

= -F2

(2.34)

[3

and so on, so h 0 =h 0 +F0

(2.35)

Vo=vo-~Fo

(2.36)

h 1 = ho + F, - Fo

(2.37)

c

v1

=v 0 -~(F1 +F0 ) c

(2.38)

= h 0 + F2 - F 1

(2.39)

=Vo - ~ (F2 + Ft)

(2.40)

h2 V2

c

and so on. Adding successive pairs of head equations and subtracting successive pairs of velocity equations h1 + h 0

etc

= 2h 0 + F 1

(2.41)

h2 + h, = 2ho + F2- Fo

(2.42)

h 3 + h 2 = 2h 0 + F3

(2.43)

-

F1

34


and

(2.44) {2.45) (2.46) etc, generally then

c fl-fl-2 =- (Vj-1 so

g

c

VJ

hi+ hi-t - 2h 0 =- (vi-t -Vi) g

(2.47) (2.48)

and the F and f functions have been eliminated from the problem. Nothing more can be done with these equations unless further information is supplied. If Vi and Vi-t can be specified it is possible to solve these equations sequentially by setting ito 1, 2, 3, etc, in turn. The boundary conditions at the valve can next be introduced and this leads to a solution. Using the usual theory for defining conditions at a valve (see equation 1.9 et seq.)

(2.49) or

(2.50) When i= 0 so

{2.51)

Denote and TJ is greek letter eta

r is greek letter zeta vi= voT/iri

(2.52)

hi=horf

(2.53)

35

Analytic and graphical methods Substituting back into equation 2.48

VoC b 2 gho y p

Denote then

(2.54) The symbols used are those suggested by Allievi in his original paper. The symbol p must not be confused with the symbol used elsewhere to denote the specific mass of a fluid. Here it is called the Allievi pipe CVo l d. . . ch aractenshc an 1s equa to 2gho. Equation 2.54 represents a family of equations obtained by making

i = I, 2, 3, etc, successively. They are known as the Allie vi interlocking equations. When i =I

(2.55) will be I since, up to the instant of valve closure commencement, heads will be steady.71 0 will be I if the valve is fully open, so when i = I

~0

~~ -1 = 2p(l -r~~~.)

(2.56)

If the fractional valve opening 71 1 is known, ~ 1 can be calculated by the usual quadratic solution. Wheni = 2

(2.57) ~ 1 has just been calculated 711 and 71 2 must be specified, i.e. the valve closure pattern must be known, ~ 2 can therefore be calculated. The entire valve closure period can be explored. As the solution from one step is used in the next step, the solutions interlock, hence the name of this technique. Having obtained ~ 1 t 2 t 3 , etc, heads can be readily calculated

h1

=ho~~. h2 =hon,h3 =hon. etc

and velocities can be similarly acquired: V1

=Vo111tt. V2 =Vo712~2• V3 =Vo113t3, etc

A complete solution has thus been obtained.

36


This method has been well worked out and techniques of applying it to pipe networks as well as to simple pipes have been evolved. These methods are well presented in books such as Hydraulic Transients by Rich and Engineering Fluid Mechanics by Jaeger. However, the author feels that as it is not possible accurately to include the effects of friction in this method and also believes that it has probably reached the end of its useful life there is little point in using space to describe such developments. 2.5 The Schnyder-Bergeron graphical method The Allievi interlocking equations lead to an elegant solution of frictionless waterhammer but become difficult and cumbersome to use if the pipe network is not simple and if the hydraulic controls are in any way complex. Essentially the Schnyder2 -Bergeron 3 graphical method solves the same fundamental equations as the Allievi method. The graphical method is much easier to generalise than the Allievi method. It does not involve the calculation of reflection and transmission coefficients if there is more than one pipe in the network and is capable of making a moderately accurate allowance for frictional effects in the system. It is able to deal with quite difficult boundary conditions and has been used, by highly skilled practitioners, to great effect. It suffers from a number of defects and one of these is the level of skill needed to deal with anything other than a simple situation. When considering most schemes it is necessary to investigate a large number of modes of operation of the scheme. As graphical techniques are time consuming this can rarely be done. Any graphical technique consists of the following:

(I) Graphically representing the boundary conditions present in the network. (2) Graphically representing the equations of waterhammer which describe the conditions within the pipe as the waves traverse it. (3) Linking the events across the ends of two pipes (or more) at which a boundary condition (i.e. a hydraulic control) is present. In any waterhammer problem four variables are involved: head, velocity (or flow), position in the pipe ~x, and time ~t. To represent these four variables on a graph an ingenious device is used which employs the fact that a transient is propagated at a speed c. In the graphical method this speed is treated as being constant and this is not true in many circumstances. This is another defect of the graphical method. In the early parts of this presentation friction will be ignored, but methods of introducing friction will be described later. The waterhammer or eagre lines

An eagre is a small wave that travels at constant velocity. In this context


37

the word is used to describe the small incremental waves that continually traverse the pipe during and after the operation of a hydraulic control. Rewriting the Reimann equations:

h=h 0 +F(t+~)+t(t-~)

[2.18]* [2.19]

Rearranging the velocity equation gives

i(v0- v) = F(t +n- t(t -n

(2.58)

Writing them both for two points at x and x' and at times t and t'

hx,t -ho =F(t +n+ tt -~) hx•,t•-ho =F(t'+~')+ t(t'-~')

(2.59) (2.60)

and

~(vo- Vx,t) = F(t +~)-f~ -~)

(2.61)

i(vo -vx•,t')=F(t'+{)-t~'-~')

(2.62)

Subtracting the head equations gives

hx,t -hx',t' =F(t +~) +t(t -~) -F(t' +%)- t(t' -~')

(2.63)

Subtracting the velocity equations gives

i(vx• Vx,t) = F(t + ~)- t(t- ~) ,t' -

-F(t'+~') +t(t'-~')

(2.64)

Up to this point these equations are applicable to any value of x, t, x' and t'but if it is assumed that x, x', t and t' are related as follows x = x' + c(t- t ')then a result of great importance emerges. This relationship is that which would describe the motion of an observer

* Square brackets indicate that the equation with this number was introduced earlier.

38


travelling downstream in the direction of x increasing. This observer will be called a waverider from now on. For this waverider

x x'+ct-ct' , x1 t +- = t + = 2t - t +c c c

(2.65)

x x' + ct- ct' , x' t--=t=t - c c c

(2.66)

and

Thus for such a waverider travelling downstream F(t + xja) changes but f(t -xja) does not- an idea that has been implied previously. The head relationship for such a waverider becomes

hx,t-hx',t' = F(2t-t'+x'/c)+f(t'-x'fc) - F(t'+x'fc)-f(t'-x'fc)

= F(2t- t'+ x'fc)- F(t'+ x'fc)

(2.67)

The velocity expression becomes

~ (vx• ' t'- Vx , t) = F(2t- t' + x'fc)- F(t' + x'fc)

g

(2.68)

Therefore

c h X, t-h X,'t'=--(v g X, t-v X,'t')

(2.69)

It must be emphasised that this result is only valid for a waverider moving downstream. For a waverider moving upstream obeying the equation x =x'- c(t- t') it can be shown by an identical method that

c h X, t- h X,, t' =-g (v X, t-v X,, t')

(2.70)

As v = q/A where q is the flow and A the cross sectional area of the pipe

h x,t - h x,t "= + .£. (qx,t - qx,t' ·) - Ag

(2.71)

This equation describes two lines of equal but opposite slope in an

h ,._ q space. It must always be remembered that the line of positive slope

implies a waverider moving up the pipe whilst the line of negative slope implies a waverider moving down the pipe. The line of positive slope is called an eagre I and the line of negative slope is called an eagre II. Movement along either of these lines implies movement along the pipe at velocity c and hence movement in time also. The direction of increasing x in any pipe must be chosen so as to be in the same direction as the initial steady flow.


39

Boundary conditions The events that occur at the end of a pipeline are defined by the interaction of a wave with whatever hydraulic control is present. Many devices can be present at the end of a pipe, for example, a reservoir, a valve, a pump, a turbine, a surge tank, an air vessel, a dump tank, a junction. At this stage only two simple cases will be considered, i.e. a reservoir and a valve. (I) The reservoir is a particularly simple device as all it does it to pro· vide flow into the pipe at an (assumed) constant head if located upstream or accept flow from it at a constant head if located downstream. Its graphi· cal representation is therefore a horizontal straight line on an h - q plot. (2) Valve. At any instant in the valve closure

(2.72) where ~ and av 0 are the fractional valve opening and the full-open valve flow area respectively. Note~ does not have the same meaning as it did when used earlier. h= ( -qcd~avo

)2/

2g

(2.74)

or where

(2.73)

2g(Cdav0 ) 2 ~ 2

(2.75)

Equation 2.74 is the equation of a family of parabolae, each member of which is defined by the value of t/1 which in turn depends upon the cur· rent value of~- As a valve closes, ~ decreases, so t/1 increases. ~ ranges from 1.0 for a fully open valve to zero for a fully closed valve and takes a posi· tive value less than unity for a partial closure. At times T, 2T, 3T, etc (Tis the pipe period 2L/c) the value of~ must be known and hence values of t/1 for each step of closure can be calculated. Thus for every time there will be a parabola which describes all possible h values corresponding to all possible q values for the fractional valve opening then current.

The graphical solution For a simple pipeline connecting a reservoir to a valve there will be three elements necessary to define the problem: the reservoir characteristic line, the eagre lines of waterhammer and the valve's characteristic para· bolae. (Note The word characteristic is not used here to mean the charac· teristic p mentioned in connection with the Allievi interlocking equation.) t/10 is the valve characteristic line for the fully open valve. The fact that this line is not a horizontal straight line through the origin shows that the full-open area of the valve is less than that of the pipe cross sectional area. In figure 2.4 the valve is shown as closing fully, in a period of four pipe periods (the subscripts to the t/1 symbol denote how many pipe periods have elapsed since the commencement of closure). The t/14 parabola has

40

Hydraulic analysis of unsteady flow in pipe networks h

hs is the height of the reservoir static surface level above the valve

Figure 2.4

degenerated into a vertical straight line because l/1 4 = oo as~= 0 when the valve is closed. At point A, see figures 2.5, the reservoir head equals that behind the valve, so point A defines the steady state flow before valve closure commences. To solve the problem, the eagre lines must next be plotted. Tis the pipeline period (2L/c ). At time 0 the flow through the valve is defined as at A 0 . The flow at B (and throughout the pipe length) is the same as that at A at time zero and the head at B is also the same as the head at A as there is no friction in the pipeline. The head and flow at B cannot alter until a wave arrives there so the point for Bo.ST is the same as that for B0 and this in turn is coincident with A 0 as shown in figure 2.5. If a wave rider is started from Bat time 0.5T he will travel downstream along an eagre II. Plotting this line onto the diagram produces a line Bo5T to Al.OT· i.e. as the waverider leaves Bat time 0.5T he will arrive at time I.OT at the valve. At this instant the valve characteristic l/11.0T comes into existence and the intersection of the valve characteristic and the eagre h

t/lo

q

)

r------8

-----::~ plan of pipeline

Figure 2.5

41


II line defines the h ~ q conditions at the point A at time l.OT. Now imagine a waverider travelling back up the pipe at wavespeed c travelling from A to B. As he is travelling upstream he will move along an eagre I (positive slope), i.e. line A LOT- B1.5T, and he will arrive at Bat time l.STwhen the eagre I line will intersect the reservoir characteristic so defining the h - q conditions at Bat time l.ST. Reversing the waverider will give an eagre II joining B1.5T to A2.0T; the valve characteristic l/12.0T comes into existence at the instant that the waverider arrives so defining the h - q conditions at A at time 2.0T. The diagram can now be completed by exactly similar methods. The diamond shape B3.5T "'* A4.0T "'* B4.5T "'* As.OT can only be produced in the absence of friction and it represents the pendulation of flow that occurs after valve closure is complete. Plotting the head at A against time produces a curve of the type illustrated in figure 2.6.

h-ht I

s

/

_.--......,

-- ----

..........

\

5T

/

/ I

..

Figure 2.6

The prediction of heads at time intervals other tlwn pipe periods To calculate the pressure heads at time periods which are fractions of a pipe period it is necessary to plot additional l/1 curves. As an example, consider the case of the prediction of heads at times of half a pipe period (see figure 2. 7). The symbol Twill be omitted from the subscripts in the following sections, i.e. the subscript 1.5 should be read as l.ST. At time 0, heads and flows at A will be defined by the point A 0 , the conditions at C will be the same as those at A 0 until a wave reaches Cat time 0.25Tand at B until a time O.STas before, so, A 0 , C0 .25 , B0 , B0 •25 and Bo.s will all be coincident. Starting a waverider from Bat time O.STwill give an eagre II line Bo.s -A l.O so conditions at the valve at time l.OT will be defined by the intersection of this eagre and the valve characteristic for the time l.OT. A waverider can be started from Cat time 0.25T and this will produce an eagre II which will intersect the l/lo.s characteristic, so defining the heads and flows at the valve at time O.ST, i.e. at A 0 •5 • A waverider starting off from A at time l.OT will reach Bat time l.ST, i.e. at B~, 5 and similarly one starting from A at time O.STwill arrive at Bat time l.OT, i.e. at B~, 0 . Additionally a waverider starting from A at time l.OTwill arrive at the pipe's midpoint Cat time 1.25T and one starting from Bat time l.OTwill arrive at Cat l.25T also. The intersection of the

42


/5 1/1.

I

)•

Q

c

•

-

~

Note C is the midpoint of the pipe

Figure 2.7

two eagres will define conditions at Cat time 1.25T. The rest of the diagram shown in figure 2.7 can then be completed. Thus by inserting the 1/Jo.s, 1/1 1•5 , 1{12.5, family of valve characteristics the conditions at the valve at mid-period values have been obtained, and the conditions at the pipe midpoint have also been established. Slow llOive opening In this case the 1/10 line coincides with the zero flow ordinate because the valve is initially closed. The valve is assumed to open over a 3Tperiod in figure 2.8. Once it is full open the 1{13 characteristic is the characteristic for all subsequent times because the valve does not move after it has reached its full open state. At time zero the flow is zero and the head is h5• The conditions at B do not change until a wave caused by the first step of valve opening arrives, at time O.ST. Starting a waverider from B at time O.ST moving towards A gives an eagre II. This eagre intersects the 1/1 1. 0 curve at A 1.0 . Reversing the waverider's direction produces an eagre I which intersects the reservoir characteristic at B1.5. The process can be repeated to complete the diagram. Depending upon the slope of the eagre and the l/1 3 .oetc line two geometries can be produced as shown in figures 2.9a and 2.9b. Thus if .£.. is relatively small (i.e. in the case of a highly distensible pipe

gA

with a low wavespeed and/or a pipe of large cross sectional area) it is possible to produce transients which exceed the reservoir static head.


43

lj;20 lj;3 0 4.0 ere

A

Figure 2.8

-

..__

Slow partial valve closure

The closure is assumed to occur over two pipe periods and the !J; 2 line is the lJ; line for the {3 2 .o value of the partial opening when the valve has completed its movement. The !J;2 line is thus the !J;3 , !J;4 , !J;5 line also. The analysis is performed as usual but the eagre lines eventually spiral in upon the intersection of the !J;2 line and the reservoir characteristic. Thus, even in the case of a frictionless flow, pressure transients attenuate for the case of a partial valve closure; they also do so for the case of an opening valve. See figure 2.11. If the slope of the eagre line is relatively large, the spiral shape of the eagres may be modified as shown in figure 2.1 0. Plotting of eagre lines

The slope of the eagres is given by tan a=±!!...... Ag

(2.76)

The values so calculated may be very large, i.e. if c A= 0.5m 2 .

~:::>.200 Ag

(b)

(a)

Figure 2.9

= 1000 metre/sec and

44

Hydraulic analysis of unsteady flow in pipe networks V;z.o

h

q

Figure 2.10

h

Figure 2.11

h

100

Figure 2.12

c

B pipe 1

Figure 2.13

Tz pipe 2

A

valve


45

As the h- q plot is not to natural scales the value of a calculated is not of great use. The suggested method of obtaining the eagre slopes is illustrated in figure 2.12. If cfAg = 200, say, draw a line from the q = 0.3 point on the abscissa, i.e. point D, to the h = 60 point on the ordinate B, i.e. BD. Complete the rectangle ABCD. Draw in the other diagonal A C. By the use of a parallel rule eagre II lines can be drawn parallel to BD and eagre I lines can be drawn parallel to AC. The method of dealing with joints Consider a pipeline in which a pipe is joined to a smaller diameter pipe as in figure 2.13. The lengths of BC and CA must be adjusted so as to be in a ratio such that the pipe periods T1 and T2 for the two pipes are in a relatively simple ratio to one another,

e.g.~~= 1 or 2 or 3 or~ or

t·

If this ratio is not simple, say 1.42, the graphical analysis will be complicated and take a very long time to perform. To illustrate the method, T 1 has been assumed to be equal to 2T2 and arbitrary eagre slopes have been chosen. Note, the eagre slope of the upstream pipe of larger cross section will be less than that of the downstream pipe. The closure of the valve at the downstream end of pipe II is to take 512 pipe periods so five 1/1 characteristic parabolae must be calculated and plotted. (More than this number may be necessary if the TtfT2 ratio is not a simple integer ratio.) The following discussion applies to figure 2.14. The horizontal abscissa of the diagram has been moved up from the true zero head to the h = h 1 head value to save space and this is a usual practice when performing graphical analyses. Only the portions of 1/1 lines which lie above the reservoir line at h = hs need to be plotted. At A 0 steady state is defined. A 0 also defines conditions at Cup until the time at which a wave will arrive there, i.e. Co.s. It also defines conditions at B up until time 1.5 T2 , i.e. Bu. Note, the subscripts express times in multiples of T2 (also remember that T1 = T2 ). Thus a waverider starting from Cat time 0.5T2 will arrive at A at time T2 - on the diagram this gives an eagre II, i.e. from CO.s to At.o· Starting from Bat 0.5 T2 will give an eagre II arriving at Cat time 1.5 T2 and starting a waverider from A at time 1.0T2 will give an eagre I arriving at Cat time 1.5 T2 • Thus the intersection of these two eagres defines conditions at Cat time 1.5 T2 • Having obtained C1.5 two waveriders can be started, one travelling upstream and one downstream. The downstream eagre will be an eagre II and the upstream eagre will be an eagre I. The downstream waverider will arrive at A at time 2.0T2 and the associated eagre will intersect the 1/1 2 .0 line at A 2 .0 • The upstream eagre I will intersect the reservoir characteristic at

t

B2.S.

46

Hydraulic analysis of unsteady flow in pipe networks h-h,

~

The subscripts indicate multiples of

72 Figure 2.14

A waverider starting from B at time 1.5 T2 will have an associated eagre II and he will arrive at Cat time 2.5T2 • A waverider starting from A at time 2.0T2 will move along an eagre I and will arrive at Cat time 2.5T2 • The intersection of these two eagres will define C2.s· As B0 , Bo.s, B1 .0 and B1.s are coincident, eagres B05 -+ C1•5 and Bl.S -+ c2.5 are colinear over part of their length. After the time of 1.51; the origin of any waverider starting from B will no longer be at the steady-state point but it will always be found that new starting points forB will have been established from earlier stages in the analysis, e.g. Bo.s, B~. 0 , and Bu are coincident at the steady state point but Bz.s will have been established before it becomes necessary to use it. From C2.s, A3.o and B3.5 can be located. From A 3 .0 and /h.s the point C3.5 can be obtained and the rest of the diagram can be completed similarly. The method of dealing

~th

junctions

At junction C in figure 2.15 the equation of continuity must always be true, i.e. ql = q3 -q2 C 1 denotes conditions at the junction end of pipe 1. C 2 denotes conditions at the junction end of pipe 2. C 3 denotes conditions at the junction end of pipe 3. At the junction, neglecting local losses and kinetic energy changes

(2.77) i.e. the heads at the junction ends of pipes joining at the junction at any given time t must be equal.

47

Analytic and graphical methods A

8

Figure 2.15

(Note that the indices 1, 2 and 3 define locations, not the first, second and third power; the subscript t denotes time.) The method of analysing junctions given below is based on the method of analysing surge tanks given by Hawkins and Zienkiewicz 4 • Consider figure 2.15 and let the pipe periods of pipes 1, 2, and 3 be T\ T 2 and T 3 . On an h ~ q graph (figure 2.16) assume that conditions at A at time t - 0.5T 1, at Bat time t - 0.5T2 and at D at time t - 0.5T 3 are known and can be located at At-O.ST', Bt-O.ST, and Dt-0.5T3. Through these points draw the eagre lines appropriate to waveriders travelling towards the junction C. Now, at C 2 the head must equal that at C 3 . eagre 12

h

eagre II 3

q

Figure 2.16

The flow into pipe 1, i.e. ql must equal q3 -q2 and the head at C1 ' h 1 ' must equal h 2 and also equal h 3 • It is therefore possible to draw a line upon the h ~ q graph which is a combination of the two eagres, eagre 1h and eagre 12 . Point X on eagre 11 3 and Point Yon eagre 12 have the same head so the requirement that he• = hc3 is fulfilled at all points on line PQ. The abscissa of the required line must be given by qx- qy so if PSis set off along line PQ and PS = XY, Swill define one point on the required line obtained from eagre 11 3 and eagre 12 . This process can be repeated for another pair of points, e.g. E and G and so point T can be obtained (RT= EG). The required line is thus ST; it is called ax line (Greek letter

48


chi). This Xt line intersects the remaining eagre 11 line at Ct, and so conditions at the junction end of pipe l have been obtained for time t. By drawing a horizontal line through ct, points Cf and Cf are obtained. This process is valid because of the requirement that hcf =hcf =hcl Points cL Ci and c~ have now been obtained and new eagres can be started off from these points to establish a new generation of points At+O.ST' , Bt+0.5T' and Dt+0.5T3 ~nd the]e can be u~ed in turn to establish a further generation of points, C t+T' C t+T' and C t+T'• etc. These new points are not shown on figure 2.16. The construction of the x line can be simplified as shown in figure 2.17. eagre 11

h

R

----

Xt

eagre II3

line

-----

D, -0.5T'

q

Figure 2.17

By drawing the horizontal lineRS through S, point R on the Xt line is established. At S, q 2 = q 3 so q 1 = q 3 - q 2 = 0 and this must define a zero flow point on the required Xt line, i.e. at R. By drawing the horizontal line PQ, Q is established. At P, q 2 is zero so at Q the flow q 1 in pipt 1 at C 1 is equal to q 3 , i.e. q 1 equals the flow in pipe 3 at C 3 . Therefore Q is another point on the Xt line. The Xt line must be straight so by joining RQ the Xt line is obtained. In some junctions q 1 =q 2 + q 3 . The combination process performed above for the q 1 =q 3 - q 2 equation can be applied with a very simple modification.

An illustration of the analysis of a three way junction This is shown in figure 2.18. Ann way junction can be analysed by the methods given in the previous section. The combination of the eagres of two pipes produces a x line, by combining this x line with the eagre of another pipe a further line (say a Aline) can be obtained. This A line can be combined with the eagre line of the next pipe to produce a further line (say a ¢J line) and, finally, after all the eagre lines of all the n pipes except one have been combined into one line, the operating conditions of the excepted pipe at its junction end can be determined by the intersection point of the compounded line with

Analytic and graphical methods B

pipe I

D 2

c

49

pipe2

h h-h,

D'

pipe 3

D 03

A

T, o Tz o r3 a, o 25° a 2 o 30° a 3 o 45•

q

Note

X lines are

parallel

Figure 2.18

the eagre of the excepted pipe. The heads of all pipes (at their junction ends) being equal, a horizontal line drawn through the intersection point defines the flows of all the other pipes by its intersection with the other pipes' eagres. An analysis such as this is difficult to perform as it requires considerable concentration and is tedious to carry out. Networks can be analysed in which there are a number of junctions. The technique outlined above can be used over and over again if the analyst can maintain his concentration for a long enough time. The many construction lines involved in this method should be drawn very lightly and erased when the required points have been established.

Inclusion of friction Friction is a distributed phenomenon but it cannot be included as such in a graphical analysis. Schnyder and Bergeron separately advanced the suggestion that the frictional head losses could be concentrated at one point in the pipe, for example as if the friction loss were caused by an orifice. Such an imaginary orifice is called a choke or throttle. Schnyder suggested placing the orifice at the reservoir end whilst Bergeron suggested placing it at the downstream end just upstream of the valve. Both the Schnyder and the Bergeron techniques produce relatively accurate answers for one point on the pipeline - Schnyder's method gives correct answers at the valve and Bergeron's at the reservoir. Modern techniques permit the use of any number of throttles located at uniform spacings along the pipe. They give more accurate results but require a very considerably greater effort to perform. Systems in which most of the applied head is used to overcome friction require the use of a relatively large num-

50


her of throttles if friction is to be well described but such analyses are rarely performed adequately because of the heavy commitment of time and effort involved. Knowledge of the pressure head at points along a pipeline's length may be very important as the largest and smallest pressures may occur at points other than at the ends. The lowest pressures can, in some circumstances, fall to vapour pressure {but not below it) and then local boiling of the fluid occurs causing two phase flow. This phenomenon is usually called column separation and this title suggests that at the point at which vapour pressure occurs two vertical faces of fluid form which move away from one another leaving a vapour filled space in between. This picture of the events implies that the two vertical faces after a period of separation will later accelerate towards one another, finally impinging upon one another and so causing a very high pressure rise. The author does not believe that this sequence occurs at highpoints in a pipeline constructed to an engineering scale although it may do so at closed valves and other closed ends. Instead, he believes that a foaming mass of liquid is generated when local pressures fall to vapour pressure and two-phase flow then develops with the formation of a free surface. The cavity so created does not usually occupy the diameter of the pipe and opens and closes by surface wave action. No significant pressures are created by the closure of the cavity as the interaction of free surface waves cannot generate pressures of magnitudes of engineering importance. The author has generated the 'column separation' phenomenon in his laboratory but has been unable to detect any pressure transient that could be ascribed to the collapse of the cavity. Graphical methods can predict the likelihood of column separation but are not likely to produce accurate predictions of the pressures that it causes. The prediction of extremely low pressures in pipes may be as impor· tant as that of high pressures since a running buckle failure or simple collapse of the pipe may occur in near-vacuum conditions The method of modifying the analytical technique to include friction is best demonstrated by describing Schnyder's solution (see figure 2.19).

\,......_:s·_ _ _AC><] .}

Figure 2.19

The throttle is located at the reservoir end of the pipe just downstream of the reservoir and B 1 is located just downstream of it. The orifice is assumed 2 to cause a head loss of 4{~~ , i.e. fLq 2 /(3d 5 ) where L and dare in metres and q is in cubic metres/sec. This expression can be written as

h =kq 2 where k

=fL/(3d 5 )

Thus at B 1 the potential head h = H5 - kq 2 • H.~

{2.78) {2.79)

is the height of the reservoir water surface level above the valve. By

51

Analytic and graphical methods h

characteristic of point B'

q

Figure 2.20

plotting kq 2 onto the h - q diagram as shown in figure 2.20 and subtracting the ordinates of this graph from the reservoir characteristic the frictional reservoir characteristic for point B' is obtained. The valve characteristic 1/J lines can now be plotted onto this graph and the analysis performed. See figure 2.21. h

q

Figure 2.21

Note This method of including friction is the analysis obtains the accurate steady state flow and head (at points A and B) and also produces frictional attenuation as shown by the spiral around the intersection of the reservoir characteristic and the ordinate through the origin. The Bergeron method will next be demonstrated. The throttle is now located at the valve. At A the head "' quantity relationship is as defined earlier and is described by the 1/J line. At A' the head is increased by the friction loss of the pipeline which is assumed to be concentrated at the downstream throttle. Therefore the A' characteristic must lie above the 1/J characteristic by the amount kq 2 • So plotting the kq 2 graph onto the h "' q graph, as for the Schnyder analysis, and then adding the kq 2 ordin)

_ B_ _ _ _ . _ , A ' A r---

:

[><]

downstream throttle

Figure 2.22

52


Figure 2.23

ates to the t/1 characteristic ordinates gives the A' characteristic. This method describes the steady state conditions accurately, i.e. at A and B but does not produce frictional attenuation after valve closure is complete. The use of multiple throttles

This technique is due to Angus 5 •6 . It will be illustrated by the use of two throttles. Assume two throttles located at B, the reservoir end, and at C, the midpoint of the pipe. Half the pipe's friction loss is assumed to occur at Band half at C. Throughout the section AC" and C'B' waveriders can travel without experiencing friction and eagre lines can be drawn which define conditions in these pipe segments. By plotting two friction curves,

c'

c"

Figure 2.24

one for! kq 2 and the other for kq 2 the friction from one throttle is given by the he= 1 kq 2 line and for both by the he= kq 2 line. The ordinates of these two graphs can next be deducted from the Hs ordinate to give the graphs of Hs- t kq 2 and Hs- kq 2 • The valve characteristic lines must next be drawn, i.e. t/lo, t/lo.s, t/lt.o, t/lt.s, t/12.0, t/12.s and t/13.0 · The intersection of the full open valve characteristic t/1 0 with the h =Hskq 2 line defmes steady flow conditions at the valve, i.e. A 0 and also defines conditions just downstream of the midpoint throttle, i.e. C0 " and C~~s. Just upstream of this throttle conditions are defined by the intersection of a vertical line through A 0 with the h = Hs- 0.5 kq 2 line, i.e. at C~. The conditions at Cwill not change until 0.25Tafter valve closure starts soC~ and C~.25 are coincident. Conditions at B' will be the same as those at C~

53

Figure 2.25

q

until a time of O.SThas elapsed soC~, C~_25 , B~ , B~.s are coincident. Continuation of the vertical line to intersect with the reservoir characteristic will give conditions upstream of the upstream throttle so defining B 0 --+ Bo.s. By starting a waverider off from C" at time 0.25T travelling downstream, an eagre II will be obtained. The waverider will arrive at the valve at time 0.5T so the intersection of the eagre line with the 1/10 •5 line defines A0 .5 • Waveriders starting from A at time O.STand from B' at time O.STwill meet at Cat time 0.75T. The head at c" must be less than that at C'by an amount of! kq 2 so point c~: 75 on the eagre I coming from A0 .5 must be so chosen that point Cd. 75 on the eagre II from B~. 5 is vertically above it and the distance C~. 75 - cd:1 s isi kq 2 where q is the current q. This means that the process is one of trial and error but in actuality the process of selection of the two points is quite rapid. Starting two eagres from C, i.e. an eagre I from Cd. 75 and an eagre II from c~:75 gives B 1 by the intersection of the eagre I and B' characteristic, and A1.o by the intersections of the eagre II and the 1/11. 0 valve characteristic. Starting an eagre I from At.o and an eagre II from B{.0 produces results for c,:25 and 25 . As before, c;_ 25 must be vertically above Ct2sby the amount! kq 2 , q being the relevant q value and the estimation process used above must be used again to define c,:2s and c,:~s. The process can be continued to fill in the remainder of the diagram. The process described above can be used to analyse as many throttles as required but the higher the number the more the work involved. If

:o

c;:

54


throttles are to be inserted at quarter points along the pipeline length there must be a corresponding increase in the number of 1/1 curves and so the work involved increases approximately with the square of the number of throttles. In a very long pipeline four throttles will not be sufficient to give an accurate description of frictional effects.

3 Boundary conditions for use with graphical methods

3.1 Introduction

In chapter 2 the fundamental techniques of graphical analysis have been demonstrated but, throughout, the downstream boundary condition was always assumed to be that of a closing valve. In this chapter, methods of dealing with other boundary conditions are described.

3.2 Pumps When pumps start up, transients are generated, the pump operating point moves along its characteristic curve attenuating these transients until eventually the steady state operating point is achieved. After a period of steady state operation the pump will be switched off (pump trip) and this will generate a negative transient which travels to the downstream end where it will be negatively reflected, if the downstream boundary condition is a reservoir, and return to the pump as a positive transient. If the pump is fitted with a nonreturn (or reflux) valve the positive wave will try to generate a reverse flow which will close the reflux valve producing a closed end. This will create a positive reflection and a high positive transient will be transmitted back downstream. As the starting transient cannot be larger than the no flow head (or closed valve head) of the pump it is usual in graphical analysis to ignore this transient and only consider the transient caused by pump trip. The power supplied to a pump is used to accelerate the pump, to supply the water power produced by the pump and the energy lost within it. Eventually the water power generated by the pump plus the energy lost within it equals the power supplied to it and then the pump ceases to accelerate and steady state is reached. At pump trip the kinetic energy of the rotating impeller and motor is all that is available to continue pumping. This energy is thus rapidly reduced by the pumping of water and so the pump decelerates until finally it comes to rest. The rate at which energy is absorbed

55

56


from the rotational energy of the pump impeller, the motor armature and other rotating masses is P and is given by P=- wqH E

(3.1)

where His the head across the pump, q is the flow, E is the efficiency of the pump. The negative sign indicates that power is being absorbed from the pump. Note wqH is the water power delivered by the pump but the power P absorbed from the pump and motor must be greater than this to supply losses occurring within the pump, so it is necessary to divide by the efficiency E, not multiply by it. Now

P=Tr?.

(3.2)

where in this context T denotes torque and n denotes angular velocity in radians per second, but

T=Ir?.

(3.3)

I is the moment of inertia of the pump and motor's rotating parts and n is the angular acceleration in_radians per second squared. (Note As Pis negative, Twill be negative son will actually be a deceleration.) wher~

Thus

P=Inn

n=In .!!._

so Now

(3.4)

,. . = 2rrN,

~~

60 an

d A H

= 2rr (N2 -

At

60

Nd

where N 1 is the rotational speed in revolutions per minute of the pump at the start of the At time interval and N 2 is the rotational speed in rev min- 1 at the end of the At time interval.

(3.5)

So

N2

PAt

=N, + ( 4rr2

)

(3.6)

I 3600 N,

N =N _ 3600 wqHM 2 1 4rr 2 IN1E

(3.7)

where E is the pump efficiency at the flow q and speed N 1 • Thus equation 3. 7 gives the pump running speed N 2 after the lapse of a time at the beginning of which the running speed was N1 rev min- 1. Consider the characteristic curve of a pump. At steady state the charac-

Boundary conditions for use with graphical methods

57

teristic curve is that given in figure 3.1, i.e. curve N0 where N 0 is the steady state pump running speed. After pump trip the pump slows down and finally stops. Consider conditions at, say, pipe period intervals T, i.e. tJ. twill equal T. The running speed at the end of any tJ.t interval will be given by equation 3. 7 where

M= T.

The dimensional analysis of rotodynamic machines gives the following result: Nd 2 H k ) _ gH .( q \Nd 3 ' N 2 d 2 ' p---;;- 'd'd' E - O

(3.8)

J

where dis the diameter of the impeller, Jl is the dynamic viscosity of the fluid, k is the roughness of the impeller surfaces and the other variables are as already defined. This result is a standard result and is given in most

Figure 3.1

Q

textbooks on engineering fluid mechanics. In the author's book An Introduction to Engineering Fluid Mechanics 14 , a complete derivation is given. The k/d term has no effect because as the pump speed changes k and d remain constant. The term H/d is found to have a trivial effect in practice and provided that the flow in the impeller remains turbulent, i.e. pNd 2 !JJ. remains large whilst flow continues during the pump run down, the pNd 2 !JJ. term has little effect also. As flow only continues during the period that the running speed of the pump remains above that necessary to cause pumping against the head applied to the pump and pNd 2/JJ. remains large during this period, it is considered reasonable to ignore variations in the value of pNd 2 /JJ. in this context. The result therefore reduces to )gH q [ (ii(j3• N2d2' E - 0

(3.9)

For a particular pump dis constant and g will not change so the following result applies:

(3.10)

58


-- --

h

---......

system characteristiC

q

Figure 3.2

At full pump speed N 0 , the usual technique of steady state analysis can be used- see figure 3.2. From this graph hso and qso can be obtained. It is necessary to know the efficiency -q curve of the pump (see figure 3.3). This curve can then be used to obtain theE- q/Nd curve which is dimensionless presentation of theE- q curve but, as d is constant for a given pump, theE- q/N plot can be used instead (figure 3.4). E

Figure 3.3

After a time of one pipe period T the pump running speed will be given by equation 3.7, i.e.,

=N.

N '

qso No

Now and

_ 3600wqsohso T

o

47T2 INoEso

=qTo

(3.12)

M

-hso = hTo - and E-r:0 =E Ni

Nt

(3.11)

so

(3.13)

so it is very easy to calculate qT0 and hTo and this establishes one point on the pump characteristic for speed N 1 at time T. A second point can then

59

Boundary conditions for use with graphical methods E

Figure 3.4

q

7V

be chosen on the steady state pump characteristic lzs1 , qs 1 and the corresponding point on the N1 curve can be calculated, i.e. qsl = qTl

N0

{3.14)

N1

and hsl - hTl

N~- Nt

{3.15)

and the process repeated for still further points to establish the pump characteristic for speed N 1 at time T. Mter the lapse of a further T period

{3.16) where qT, hT and ET are the instantaneous flow head and efficiency values at time T. The hT and qT values are obtained from the first step of the graphical analysis, and the ET value from the efficiency -q/N curve. The process can be used to develop the head - flow curve for the second time step, i.e. qs1 = qT1 _ q2T 1

No

and

N1

-

N2

hsl - hTl - h2Tl

NJ- N{- N?

(3.17) {3.18)

q2T1 and h2T1 can be obtained. Similarly other points on the pump characfor speedN2 , e.g. q2T2 , h2T2 ;q2T 3 , h2T 3 ;q2T 4 , h2T 4 , etc, can be teristic •

obtatned. A complete set of pump characteristic curves for the necessary number of speeds can thus be derived. There are a number of assumptions made in this method.

(I) The time interval Tis assumed to be sufficiently small for an initial value finite difference calculation to be adequately accurate. If Tis large then O.STor 0.25Tvalues should be used instead.

60


(2) The efficiency ET taken from the efficiency -q/N curve is not completely accurate. The efficiency curve was obtained for speed N 0 and the assumption is made that this curve remains unchanged as the speed changes. This would be correct if the pNd 2 /JJ. group had no effect upon the efficiency but in fact, of course, it has. (3)

The~ and~ groups which have been used to establish points on

the characteristic curve which is being developed can strictly only be used if E is the same at both speeds but, as in assumption (2), this is not perfectly true so the characteristic curves produced are also not perfectly accurate. Figures 3.5a, 3.5b, 3.5c demonstrate how the graphical analysis is performed. Knowing hT 1 and qT 1 the N2T curve can be obtained as already discussed. The completed diagram is given in figure 3.5c. The case illustrated could only arise in the circumstance of a pump set having a relatively large inertia delivering to a short delivery pipe. In such a case the pipe period Tis very small and the run down period of the pump is relatively long. If the pipe period Tis large in terms of the pump run down it will be necessary to plot pump characteristics for fractions of the time period. It will then be possible to obtain pressures and flows at intermediate pipe points in the way which was illustrated in chapter 2. In certain texts the series of pump h - q characteristics for various pump speeds are derived using a technique which does not in any way depend upon the transient heads in the pipeline. It seems to the author that at the instant when pump trip occurs, the initial parameters of steady state- h 5, q 5 and £ 5 are available so it is possible to predict the rate of pump speed decrease and so obtain the pump speed at a time !:1t seconds later. After !:1t seconds the pump speed is therefore known but the head, flow and hence the efficiency are determined by the interaction of the eagre I line with the pump characteristic, i.e. the precise flows, heads and efficiencies at the instant when the next prediction of the pump speed (for At later) is to be made, are dependent upon the waterhammer circumstances. Therefore it would appear that a technique which does not take this waterhammer circumstance into account cannot be correct. The method advanced in this book is free from the defects listed above and it is simple and straightforward to use.

3.3 Four quadrant pump operation If no reflux valve is fitted upstream of the pump it may happen that pressures downstream of the pump may rise during transient operation to such values that flow may be forced backwards through the pump even though it is still running forwards. Again, the pressure may fall so low that flow is passed forwards through the pump tending to speed it up or

Boundary conditions for use with graphical methods steady state pump characteristic

h

throttle

(a)

h

coordinates q2T h 2 T

q From q2r and h2 r the N 3 r curve can be established (b)

h

(c)

Figure 3.5

61

62


reduce its rate of run down. In other words, the pump may operate in (a) pumping mode, (b) dynamometer mode or (c) turbine mode. All three modes may occur both when the flow is forward or reversed and when the pump rotation is forward or reversed. The number of pumps that have been tested in all modes of operation during transient conditions is extremely limited and the analyst will not be able to obtain the complete characteristic curves of the pump he is wishing to analyse. It is possible to estimate suitable curves from the few complete characteristic curves that are available but this is best done using techniques that are not well suited to graphical analysis. For this reason the graphical techniques are not presented here although, of course, they exist; a technique of dealing with this problem is described in chapter 5.

3.4 Surge tanks Surge tanks are devices connected to pipelines to turn the high frequency, high pressure transients into low frequency, low pressure, mass oscillations. The simplest examples is that of a relatively large diameter vertical pipe connected to the pipeline, situated near to the hydraulic control which is generating the transients which it is wished to suppress. When a hydraulic reservoir static water level

reservoir

Schematic layout of a typical hydroelectric power scheme

Figure 3.6

control (such as a valve, or turbine gate or pump) attempts to impose a rapid velocity change upon the fluid, the fluid can enter the tank experiencing almost no retardation. As more and more fluid enters the tank the level within it rises, slowly applying a decelerating head to the fluid in the pipeline. As relatively large volumes of fluid must enter the surge tank to increase its level, the period of the oscillatory surge pressures set up in the pipeline is long. The decelerations are therefore low so the pressure magnitudes are greatly reduced. However, to prevent completely pressure tran-


63

sients from passing up into the pipeline a surge tank of infinite cross section is needed. Even so, surge tanks of normal dimensions are capable of reducing pressure transients to very small values. A technique of estimating the magnitudes of such transients is outlined later in this chapter. When the hydraulic control in question is a turbine, a very important function of a surge tank is to supply fluid during the early phases of a gate opening operation. During the early part of an opening operation of the turbine gate the fluid in the pipeline will be travelling too slowly to supply the demand made by the turbine but fluid drawn from the surge tank can meet this demand with a consequent decrease in the level within it. This decrease will cause the fluid in the pipeline to accelerate and eventually the flow in the pipeline will equal the turbine's demand. The fluid level in the surge tank will be set into oscillation and it is possible that its frequency or a sub or superharmonic of it may match the natural frequency of the governor mechanism (or one of its harmonics). If this should happen the governor may go into resonance and the system will then be inoperable. For this reason variant forms of surge tanks have been developed which are capable of damping any applied oscillation.

3.5 Types of surge tanks Figures 3.7a, 3.7b, 3.7c, 3.7d, illustrate the four major types of surge tanks. The simple surge tank has already been described. It has the following advantages:

(I) It transmits very small transients. {2) It behaves extremely well during opening phases of the downstream control. {3) It has a long period. There is minimal attenuation of the mass oscillation in this type of tank. The choke ring surge tank is equipped with a short riser entering the main tank and the upper end of the riser is sealed. Fitted into its vertical periphery are valves which are forced open during the rising surge providing large orifices through which flow can enter the tank; during the falling surge the valves close but in the valves small orifices are fitted which provide a frictional resistance to outflow from the tank. This type of surge tank has the quality of attenuating any surge rapidly because of the turbulent dissipation that occurs as flow passes through the orifices and generates turbulence in the surrounding fluid masses. This attenuation has to be paid for as the orifices cause transmission of transients up the pipeline. The Johnson differential surge tank has a central riser of small cross section. This means that flow up the riser causes a rapid but limited increase of pressure in the pipeline when the turbine gates close. The top of the riser is usually set at reservoir static level so the pressure rise is small. The rapid rise causes faster retardation of flow in the pipeline than would occur if a simple surge tank had been installed. Once the riser is overtopped a

64


(a)

(b)

Simple surqe tank

Choke ring surge tank

(c)

(d)

'----

0 (

r-=-

.k>hnson differential surge tank

Pressurised surge tank or air vessel

Figure 3.7

weir flow occurs over the lip of the riser filling the annular portion of the tank. Some flow also enters the annular portion through the orifices at the base of the riser. The pressure in the pipeline remains approximately constant during this phase. Once the level in the annular portion reaches the lip of the central riser, the surface rises uniformly and slowly across the entire cross section of the tank. During a falling surge the level drops until the level reaches that of the top of the central riser. The level in the central riser then drops rapidly and flow occurs through the orifices in the base of the riser. This causes the level in the annular space to drop relatively slowly. The graph of pressure in the pipeline against time is thus a complex shape made up of discontinuous portions. This tank attenuates surges well, has a complex wave form which is less likely to resonate with the turbine governor and has a shorter period than the simple surge tank. It will transmit higher pressure transients to the pipeline than will a simple surge tank and will not attenuate surges as well as the choke ring surge tank. Like the choke ring surge tank it will not behave as well as the simple surge tank in supplying flow during the starting phase of a turbine's operation. The pressurised surge tank is nothing more than an air vessel. It is used


65

when any other type of surge tank would have to be excessively high, or when, for strategic reasons, the surge tank must be buried inside a mountain. Water dissolves air; the higher the air pressure the more it can dissolve, i.e. up to approximately 2% by volume per atmosphere. Consequently, pressurised surge tanks need to be fitted with air compressors which can maintain the necessary volume of air in the surge tank. Automatic equipment is necessary to control the compressors and such systems need to be duplicated to ensure certainty of operation. This is important because if the surge tank loses all its air it ceases to be effective and full pressure transients will be generated wltich may severely damage the pipeline. Variations in design of surge tanks are frequently encountered. A common variation of the simple surge tank design is the use of variable cross sections and horizontal galleries to increase the storage of the tank, as shown in figure 3.8.

I I I

Figure 3.8

The analysis of surge tanks falls into two parts:

(1) The analysis of the ability of the surge tank to minimise the transmission of transients into the upstream pipeline. (2) The analysis of the mass oscillation of the fluid in the tank and pipeline. 3.6 Transient analysis of surge tanks At the instant that the turbine (or other hydraulic control) operates, the situation will be as illustrated in figure 3.9. During the very short time that pressure transients of significant size exist, the fluid level in the tank will not alter very much so L 2 can be treated as a constant without much error. The problem thus reduces to the analysis of a three-way junction. The pressure at the top of the surge tank must be constant at atmospheric pressure so it can be assumed that the surface acts as a reservoir. The problem can then be analysed in the way described for the three-way junction pp. 46 to 48 in chapter 2. Although pressure transients so transmitted may be small it is possible that they may be large enough to excite 'organ piping' or resonance in the upstream pipeline which can cause high pressures at its nodes and these may be capable of fracturing the pipe.

66


Figure 3.9

3.7 Mass oscillation of surge tanks In this type of analysis it is assumed that velocity changes are so slow that their effects are propagated throughout the pipe length in a relatively negligible time. In other words the instantaneous velocities at all points in the pipe are assumed to be the same, i.e. 'rigid column' theory applies (chapter 1). In chapter 6 a technique of analysis will be described in which this assumption need not be made but, as this depends upon the use of the computer, the following analysis may be found useful when a computer is not available.

't -

~~c-'- t -~

eswc

- - : .J L

Figure 3.10

At an instant t seconds after shut down of the downstream flow control the situation is as illustrated in figure 3.1 0. The velocity in the pipeline is then v, the level of the surface in the surge tank is Z below the reservoir static water level (RSWL) and the friction head loss is (4fL/d)V.vlj(2g), i.e. Ollvl where C = 4fL/(2gd), d being the diameter of the pipeline). (Note Z is measured positively upwards.) If the flow were steady at timet, at velocity v, the value of Z would equal -Cv 2 but as the surface in the tank is higher than this, a deceleradve


67

head is applied to the pipeline, i.e. a head of Cvz- ( -Z) (remember, Z has negative magnitude), so

Ldv Cv2 +Z=--g dt

(3.19)

so

~; = -f (Cvlvl + Z): the dynamic equation.

(3.20)

The use of Cv lv I instead of Cv2 allows for the reversal of the frictional head with flow reversal. The continuity equation is equally simple to derive: let the flow through the hydraulic control at time t be Q1 dZ av=Ad1 +Q1

(3.21)

dZ av-Q1 h . . . dt = A : t e conhnmty equation

(3.22)

then so

These two equations can be combined into a second order differential equation but this equation has no analytic solution. If Q1 = 0 and friction is ignored the equation of simple harmonic motion results, i.e.

~~ and

=

-f Z from equation 3.19

v = ~ ddZ from equation 3.22 a

t

(3.23) (3.24)

Differentiating equation 3.24 with respect to time and substituting for dvjdt from equation 3.23 gives: d2Z+ag Z=O dt 2 AL

(3.25)

The angular velocity il of this SHM isJII. so the period is: T=

21TJ[f-

(3.26)

The amplitude is then readily obtained as follows. As ril = Vs1 where Vs is the initial steady state velocity in the pipeline and r is the amplitude

r = Vsa/A

~

(3.27)

~AL

(3.28)

68


The result for the period is very good and produces good results even when friction is present. The value given for the amplitude is only approximately correct and becomes less and less accurate with increasing friction. However, it overestimates the peak surge and can be used to obtain a quick initial estimate of its magnitude. The equations for choke-ring surge tanks can be derived by modifying the dynamic equation to include for local losses created at the choke ring's orifices by increasing the friction term by kvrlvr lj2g, where Vr is the velocity in the riser and equal to (av- Qt)/ar· If the choke ring is equipped with orifices of different sizes, then the k value must be adjusted when the surge direction changes. The continuity equation for the choke ring tank is the same as that for simple tanks. The Johnson differential tank has to be analysed differently for its six different modes of operation during a surge. At first the water is rising in the central riser and flow enters the annular tank through the orifices in the base. When the level rises to the top of the riser, flow spills into the annular tank as a weir flow. When the level in the annular tank rises to equal that in the riser the level across the cross section rises uniformly with almost negligible flow through the base orifices. Flow through the base orifices in either direction involves local losses and these must be included in the analysis. When the surge starts to fall the level across the entire cross section falls uniformly and during this phase no flow occurs through the base orifices. When it reaches the level of the top of the riser, the level in the riser will drop rapidly and flow out of the base orifices into the riser will occur, dropping the level in the annular portion of the tank. Once this happens the analyst will have to consider the annular section and the riser separately. The analysis of all types of surge tanks is based on finite difference methods so it is simple to write the groups of dynamic and continuity equations applicable to the various phases of the tank's operation. These are used, as appropriate, as the analysis passes through each phase.

3.8 Pressurised surge tanks or air vessels In steady state the value of the pressure in the tank

(3.29) Pairs must be absolute so Pa- the atmospheric pressure- mvst be included. In this case Z 5 i=- Cv5 lv5 l in steady state. Z 5 is measured relative to the reservoir static water level and is taken as positive upwards. After t seconds from the closure of the hydraulic control the level will have risen in the tank. As Z increases, the air will be compressed to Pairt


s

69

z

Figure 3.11

according to a polytropic process and it is usually assumed that the index is 1.2 so

Pairs v.airs1·2 -- pairt v.airt1.2

(3.30)

Thus

(3.31)

where Vair denotes the volume of air in the tank. If the tank has a uniform cross section Vair is proportional toy (see figure 3.11 ). :.

Pairt = (

y:

y

)1.2

(3.32)

Pairs

P.. -P. Now atrt a= the height of a water column equivalent to the gauge w

pressure in the tank = hairt whairt + Pa = (ysfYt { :.

hairt = (ysfYt)

12 (

2

(3.33)

Pairs

-(Zs + CvsiVsi)

+ ha) -ha

(3.34)

where ha is the head equivalent to atmospheric pressure. Remember that in the convention used, Z is negative downwards (S is also taken negatively downwards), but

Ys=S -Zs

(3.35)

and

Yt=S-Zt

(3.36)

so

hairt =

G=~;)

12 • (-(Zs

+ Cvslvsl) + ha)-ha

(3.37)

The dynamic equation thus becomes

(3.38)

70


and the continuity equation (as before) is dZ av-Qt -=--dt A

(3.39)

So the equations of mass oscillation of an air vessel are: dZ=av-Qt dt A

(3.40) (3.41)

hairt =

(~ =~:)

12 • (

-(Z5 + Cv5 lv5 1) + ha)- ha

(3.42)

Although the index 1.2 is in common use, different analysts have suggested slightly different values. The value 1.0 gives an isothermal process and 1.4 gives an isentropic process but the actual process must be polytropic.

3.9 Methods of integrating the surge tank equations There is no analytic solution to the equations. Numerical methods are the only available techniques. Such methods can be performed by hand or preferably by computer. Consider, for example, the integration of the pressurised surge tank equations. At a time t the velocity v will be Vt, the Z value will be Zt, the hair value will be hairt' etc, and these values will have been established from previous steps of integrations (3.43) so

Zt+~t = (avt~ Qt)~t + Zt ~v g At =- L( Zt + Cvtlvtl + hairt)

so and

Vt+~t =- g

At

L

(Zt + Cvtlvtl + hairt) + Vt

(3.44) (3.45) (3.46) (3.47)

This method is the simplest form of initial value integration. There are many more refined methods but this one will work quite well if ~tvalues are made sufficiently small and a computer is used to perform the arithmetic processes. Mid-value iterative methods can be used but although these permit larger ~t intervals to be used they may involve as much computer time.


71

Corrector-predictor methods and Runge-Kutta techniques can be applied to this problem. Such refinements are beyond the scope of this book and the reader should study the mathematical literature if he wishes to employ such techniques. If the integration is to be performed by hand it is best done using a tabular layout.

4 The method of characteristics

4.1 Introduction The Schnyder-Bergeron technique has, until recently, been considered the best method of solving transient problems. It is very limited and subject to error, however. The work involved in analysing networks containing a significant number of pipes, say more than eight or nine, is unacceptable in any engineering circumstance. However, there are more complex reasons for considering graphical methods to be inadequate.

(I) Almost all liquids encountered in engineering practice contain small volumes of air in bubble form so, when pressures in the liquid increase, the bubbles decrease in volume and when the pressures fall they expand. In effect, the bulk modulus of the liquid changes with pressure and as the wavespeed depends upon the bulk modulus the wavespeed varies with pressure. Such variations in wavespeed can be very great. For example, the wavespeed may fall from 1300 m s- 1 to as low as 100m s- 1 due to the presence of an air bubble content of 0.01% solely due to pressure changes. This means that the eagre lines on a graphical analysis instead of being straight should really be complex curves. These curves cannot readily be calculated so the author cannot see how the variable wavespeed effect can be included in a graphical analysis. (2) Not only do liquids contain free bubbles, but such liquids as oil and water, can carry significant volumes of dissolved gases. Water can carry 2% by volume of dissolved air per atmosphere of pressure and oil can contain very much larger volumes of dissolved gases depending upon its geological place of origin. During the passage of a low pressure transient this dissolved gas may come out of solution in the form of free bubbles so greatly increasing the quantity of bubbles in free form and decreasing the wavespeed even more. Again, the graphical method cannot include this effect. (3) At the present time complex servocontrolled valves are being fitted into oil pipelines and designers of long water pipelines are also using such devices. The description of the behaviour of such equipment in a graphical

72

The method of characteristics

73

method is extremely difficult due partly to the difference in time scale between the time of operation of such valves and the pipe period and partly to the difficulty in satisfactorily describing the multi-variate nature of such a valve's operation graphically. ( 4) The run up to steady state of a pumped network may not cause as large transients as those created by pump trip but in a complex network in which complicated valve operations may occur or in which dump tanks empty or fill, the pump trip case may be of only relatively minor interest. In the operation of oil pipelines pump trip is a rare event and the study of events caused by changes of pump speed (including speed increases), operation of valves, permitting flow from or to dump tanks, etc, are the events which are of importance. To study such pipelines many analyses are necessary and the excessive labour involved in graphical analyses of so many cases (even if possible at all) may well be totally unacceptable. (5) Many types of boundary conditions can be mathematically modelled but cannot be graphically represented - an example of such a boundary condition is that of a sewage ejector which can be graphically represented only if gross simplifications are made (see (3) above). (6) The requirement that pipe lengths in a pipe network must all be in relatively simple ratios is very limiting and can lead to considerable error. (7) In complex networks there may be junctions at which as many as fifteen pipes meet, i.e. a manifold. If the network is not to be simplified the analysis of a fifteen-way junction must be faced. Very few graphical analysts would be prepared to contemplate such a problem. For the above reasons it has been necessary to find an analytic method of solving the waterhammer problem which offers fewer constraints than the graphical methods. To do this it is necessary to return to the fundamental equations of water hammer which were presented on p. 26 ff. They are requoted here: continuity equation and dynamic equation These equations can be integrated directly by finite difference methods but great care must be taken in doing so as instability problems can arise. The papers of Lax 7 and Lax and Wendrof 8 are relevant in this context and the reader is recommended to them if he wishes to employ such methods. The author is convinced that the finite difference integration of the characteristic forms of these partial differential equations is preferable. This is known as the method of characteristics. Before this can be demonstrated it is necessary to derive these characteristic equations.

74


4.2 Method of deriving the characteristic forms of the waterhammer equations The waterhammer equations are a pair of quasi-linear hyperbolic partial differential equations. There are a variety of ways of obtaining the characteristic forms of such a pair of equations. The method given here is a modification of a method presented by Lister 9• A different method is used in chapter 10 of this book to illustrate another technique. Considering any pair of partial differential equations of the type shown below:

(4.1) and L2

au

au

av

av

=A2 ax+ B2 ay + C2 ax+ D2 ay + E2 = o

(4.2)

where u and v are dependent variables and x andy are independent vari· abies; A" A 2, B,., B 2, C., C2, D" D 2, £ 1 and E 2 are all continuous known functions ofu, v,x andy. The condition that

A1 _ B1 _ C1 _ D1 A2 - B2 - C2 - D2 in part or in total is prohibited. Consider a combination of L 1 and L 2 such that L =L 1 + XL 2

(4.3)

Then

ov av (C1 +X C2) ox+ (D1 + XD2) oy + £1 + XE2

Let y = y (x) be the equation of a curve of which :

(4.4)

is the tangent slope.

If u = u(x, y) and v = v(x, y) and these are solutions ofL 1 and L 2 then

au au ~=~b+~~

~~

ov ov dv=-8x +-8y ox oy

(4.6)

and

75

The method of characteristics Now,

(au

au)

au au B 1 + A.B2 (A 1 + A.A2) ax+ (B1 + A.B2) ay =(At + A.A2) 3.x + A 1 + A.A 2 ay

(4.7a)

so if

B1 + A.B2 _ Dt + A.D2 _ dy At +A.A2- C1 +A.C2- dx

(4.8)

Rearranging equation 4.8 gives A.=A 1 dy-B 1 dx =C1dy-D,dx B 2 dx -A 2 dy D 2 dx - C2 dy

( 4 .IO)

hence

p(dy) 2 + qdxdy + r(dx) 2 = 0

(4.11)

where

p=A 1C2 -A2Ct

(4.12)

q = A2D1 + B2C1- A1D2- B, C2

(4.13)

r = B 1D 2 - B2D1

(4.14)

If the roots of this quadratic equation are real and different the original pair of partial differential equations are hyperbolic. If the roots are real and equal the original equations are parabolic and if they are complex the equations are elliptic, i.e.

q 2 -4pr >O

hyperbolic

q 2 -4pr = 0

parabolic

q 2 -4pr
elliptic

Thus by substituting the values of A" B" C1 , D" E" A 2, B 2, C2 , D2, £ 3 , and solving the quadratic, two values of dy/dx can be obtained if the equations are hyperbolic. Denote dy/dx by~ (Greek letter zeta). The quadratic is then p~ 2 +q~+r=O

and the two values

of~ are~+ and~-·

Thus two directions are defined in an

x- y space. So, at a point x, y in the x - y space there must be two lines of gradients dy/dx =~+and dy/dx =~-passing through the point and these conform to the differential equations. The lines tend to become colinear when


76

~+-+~_which arises in hypersonic flows in gases and in highly super critical cases of free surface flow, and then the equations tend to become parabolic. Having calculated the values of~+ and~_ two total differential equations have been established, i.e.

(4.15a) dy = ,_ dx )-

and Back substitution of these values

of~

(4.15b)

into equation 4.10 gives

X= At~±- Bt = Ct ~±- Dt B 2 -A 2 ~±

D 2 -C 2 ~±

Substitution for X in equation 4.9 gives

Simplifying gives:

(AtBz- AzBt)du

+(CtBz- CzBt + (CzAt- CtAzn±) dv + (BzEt- BtEz + (EzA 1 -

Substituting C2 A 1 - C1 A 2

where

E 1 Az)~±)

dx

=0

(4.16)

=p from equation 4.12 gives:

Fdu + (P~+- G) dv + (K~+- H) dx = 0

(4.17a)

Fdu + (P~-- G)dv+

(4.17b)

(K~_-H)dx =

0

F= A 1Bz- A 2Bt

( 4.18)

G=B1C2 -BzCt

(4.19)

K=A1E2 -AzEt

{4.20)

H=B 1E 2 -B 2E1

(4.21)

Thus four equations are available:

Fdu + (P~±- G) dv + (K~±- H)dx = 0

[4.17a and b)*

and [4.15a and b]

* Square brackets indicate that the equation with this number was introduced earlier.


77

The two lines in the x - y space specified by dyfdx =~±are called characteristic lines (hence the name method of characteristics) and along these lines the equations Fdu + (P~±- G) dv + (K~±- H) dx = 0

apply. These equations are solved by finite difference methods. 4.3 The characteristic forms of the waterhammer equations The equations are:

az

av at

av g ax

ah ah c v-+-+--+O x- - v - =0 2

ax

at

ax

and

(see chapter 2). Comparing these with the general forms previously quoted, x in the general case is equivalent to X, y tot, A, to v, B, to 1' c, to c2 /g, D, to 0, £, to- v az ax 2ftilvl A2 to 1, B 2 to 0, C2 to v/g, D 2 to 1/g, £ 2 to ---g;r-• u to hand v to v. p =A, C2- A2C1 = (v 2 -c 2 )/g

So q

= A 2 D 1 + B 2 C1 r = B1D 2

A 1D 2 -

B1 C2

=-2v/g

B 2 D 1 = 1/g

(4.22) (4.23) (4.24)

~± =- ip±J(ipY -~

As

after substitution and reduction 1 v±c

~±=-

(4.25)

F= A 1 B 2 - A 2 B 1 = -1

(4.26)

G = B1 C2 - B 2 C1 = vfg

(4.27)

A 2 E 1 = 2fti 2 lvl/(gd) + v ~ ax H =B1 E 2 - B 2 E 1 = 2ftilvlf(gd)

K = A 1E 2

-

P=(v 2 -c 2 )/g

(4.28) (4.29) (4.30)

substituting these values and reducing gives -dh

+£. dv + £. ( 2fvlvl g

g

d

dx +gvdz ~) = 0 (4.31a and b) v±c cdxv±c

78


and

dx=(v±c)dt

(4.32a and b)

±g_dh + dv + 2jVivi + gv dz = 0 edt dt d c dx

so and

(4.33a and b)

dx -=v±c dt

(4.34a and b)

The~ :. term is small and is usually omitted. These are the characteristic forms of the water hammer equations. 4.4 The zone of influence and the domain of dependency At point P(x 1 , t 1) assume that the velocity of the flow v and the celerity of the wave c are both known. Through P two lines can be drawn for which dt I dt and -d = -I - · Note that the supplement of tan_1( -I -) has a dx = -+v c x v-c v-c value of tan- 1

(-I-) · c-v

These lines are tangent to the characteristic lines of course, and they are drawn on figure 4.1, the forward characteristic C+, i.e. RPW and the backward characteristic c_, i.e. SPV. range of influence of P

w

X

domain of dependency of P

Figure 4.1

If at P a disturbance is initiated, i.e. at location x, and at timet, the disturbance will travel as a wave downstream at velocity v and c and upstream at velocity c- v. In other words, in an x - t space a characteristic line is a plot of the movement of a wave travelling either upstream or downstream as ( :

t

= v ±c.

Any point lying within the shaded area on figure 4.1 (i.e. VPW) will have experienced the effect of the disturbance so this area is called the zone of influence of P.

79


Consider what happens to disturbances which originate from points lying between RandS at time zero: say X and Y. The forward characteristic from X will intersect the backward characteristic from Y at Z. Point P, which is located at a later time than Z, lies within the zone of influence of Z so events occurring at Pare affected by events occurring at points lying between R and S. Thus R -Sis called the domain of dependency of P. Any events lying outside the R - S segment cannot affect events at Pin any way. 4.5 The zone of quiet The zone of quiet is that area in an x - t space in which no effects of any disturbance can be experienced. If the disturbance of flow originates at the upstream end the zone of quiet is as illustrated in figure 4.2a, if at the downstream end it is as illustrated in figure 4.2b and if disturbances originate from both ends simultaneously it is as shown in figure 4.2c. The characteristics may be curved or straight. They are straight if v and care

(a)

X

X

(b)

zone of quiet

(c)

Figure 4.2

constant for all points on the characteristic and curved if they are not constant. The concepts discussed above give considerable insight into the nature of unsteady flow, and the reader is strongly recommended to consider them very carefully. 4.6 The integration of the characteristic equations Consider a forward characteristic of slope tan- 1( -1- ) originating from

v +c,

R (in figure 4.3) intersecting at P a backward characteristic of slope

1 -)originating from S. Assume that t:.t and t:.x are so small that tan- 1( -

v-c

80


6t

ore ton

(v~c)

X

6x Figure 4.3

the short segments of the characteristics RP and SP can be treated as straight. The lines RP and SP satisfy the equations dx

dt=--

[4.34a and b]

v ±c

so RP and SP have been constructed so that along them the other two characteristic equations will apply:

+! dh + dv + 2fvlvl = 0 - c dt

dt

d

[4.33a and b]

Therefore along RP equation .!_dh + dv + 2[vlvl edt dt d

=0

[4.33a]

will apply, and along SP equation _[dh + dv + 2[vlvl c dt dt d

=0

[4.33b] will apply. The first of these two equations applies at both R and P and the second applies at Sand P. Thus at P they both apply simultaneously. The equations must be cast into finite difference form if anything further is to be done with them AlongRP

(4.35) and alongSP g(h P - h ) ( s + Vp - vs ) + 2fsvslvsl~t -_ 0 c d

--

(4.36)

If hR, vR, hs and vs are known, vp and hp can be obtained by solving equations 4.35 and 4.36 simultaneously. By this process a pair of values of head and velocity have been obtained at a time ~t seconds after the last known value so this technique provides


81

a basis for calculating heads and velocities in unsteady flows at as many points in a pipe as required including frictional effects, convective accelerations and instantaneous head gradients all of which were omitted completely or in part in earlier methods. A complete technique can be built up on the basis described above. Assume that the x axis lies along the centre line of a pipeline and that at the point x = 0 an upstream control exists (see figure 4.4). Next, assume that at & points along the pipe, heads and velocities are known at time t = 0. Points R and Scan be established, R from X and Y and S from Y and Z by methods already described. Then P can be established from points R and S. This process can be generalised over the entire x - t space. At x = 0 values of hand v or a relationship between h and v must be known or specified in some way.

I 6x I 6x I 6x

X

Figure 4.4

4.7 Boundary conditions At a boundary there is only one characteristic available - a backward characteristic at an upstream boundary and a forward characteristic at a downstream boundary (see figure 4.5). It is from the boundaries that unsteadiness is initiated so the analysis of boundary conditions is extremely important. As only one characteristic is available it is necessary to express the operation of the boundary condition in mathematical form so that it can be solved simultaneously with the characteristic equation to give solutions for hp( or hQ) and vp (or vQ). The details of this process will be discussed more fully later in the chapter. The method described above was developed long before modern computing methods were available. It involved graphical constructions and an unacceptably heavy computing load and consequently was rarely employed. Computer programs can be written which duplicate the graphical process and perform the solution of the simultaneous equations rapidly

82

Hydraulic analysis of unsteady flow in pipe networks p

0

s

R X

Figure 4.5

so the method could be used today without difficulty. However, it is not used because as velocities change with time and with distance, the slopes of the characteristic lines change their slopes, and consequently the intersection points are not located at regular intervals of x and t so the results obtained from the curved network require interpolation to give values on a regular rectangular x - t grid. Such values can be interpreted by examining results at a fixed x distance and considering their variation with time or alternatively examining results at a fixed time and considering their variation with x. The original irregular grid is awkward to use for this purpose. A modification of this method is now in common use.

4.8 The method of the regular rectangular grid The method is based upon the concept of deciding initially that results are to be obtained on a regular rectangular grid and to then devise a method of obtaining values that apply at the nodes of the grid. An interpolative process is still necessary but this is much simpler than the one required for the conversion of the results of the Hartree method to a rectangular grid basis. Choose a value of flt which is less than !lxf(v +c). In waterhammer calculations the author has found that a value of flt equal to 0. 95 x !:J.xjc works well. (cis the maximum wavespeed occurring anywhere in the network). If a value of flt equal to flxjc were chosen, it would follow that ifv were positive then R would lie upstream of M (see figure 4.6) and this could lead to numerical instability. In the author's experience, the value of the constant 0.95 has been found to provide an adequate margin to avoid this happening. The requirement that flt < flx/(v +c) is known as the Courant and Lewy 10 stability criterion. The criterion so specified automatically satisfies the other required condition that flt < flx/(c- v). Some authors of papers on this subject have suggested that the criterion should be that flt = flx/c ignoring the v term as it is felt that vis much less than c, so the excursion outside of the domain of dependency will be small. For pure water with no contaminating air flowing at low velocities in steel pipes this is true. Typical values of v and c are 2 m s- 1 and 1300


83

X

Figure 4.6

m s- 1 respectively. If vis as high as 15 m s- 1 which is common in the flexible hoses of tanker loading facilities, and cis as low as 100 m s- 1 due to the presence of air or gas bubbles in the flow or due to high distensibility of the pipe itself this assumption could be dangerous. There is a considerable incentive to place the intersection of the characteristic from P back to the previous time level, as close to the grid point as possible, i.e. RM should be made as small a fraction of OM as possible, consistent with R not falling outside of MO with the consequent risk of numerical instability. The reason for this is that if RM is a large fraction of ~x there will be a tendency to disperse the wave front, steep waves will gradually flatten and highly curved sections of the wave will become progressively less curved. The author has not heard of any analysis in which this occurred which produced erroneous maximum magnitudes of heads and velocities, however, providing that a reasonably large number of ~x intervals was used in the pipe length. He believes that the use of the constant 0.95 produces a satisfactory compromise between the need for the characteristic to intersect the previous time line as near to a grid point as possible and yet leave a margin for a v and c variation. If the ~t = ~xfc criterion is used the interpolative procedure referred to earlier will not be necessary but the author is persuaded that the use of interpolation is worthwhile. Referring to figure 4.6, assume that the first time level of integration has been completed for all points in the pipe length. Heads and velocities at M, 0 and N will then be available. Construct two lines through P of 1 - and dtfdx = - 1-- respectively. These will pass slopes dt/dx = - Vo -co Vo +co through R and S. It may be thought that the slope values would have been

dt . . better approximated by either dx = v

P

!c

P

1

VR + CR

dt 1 dt and-d =---or dx = X

VS - CS

and dxdt = - -1-but at this stage none of the variables VR, vs, cR, vp-cp

cs, vp or cp has yet been calculated. Next, linearly interpolate between M and 0 to obtain VR and cR and between 0 and N to obtain vs and cs.

84

Hydraulic analysis of unsteady flow in pipe networks The characteristic equations can now be applied, i.e. g (h p - h R) + Vp- VR 2fRvRivRIM _ O + c0 d

[4.15]

-

so and

h p_ h R - c-0 ( Vp-VR + 2fRvRivRIM) g d

(4.37)

g (h p- h ) s +vp-vs+ 2fsvsivsiM -_ 0 c0 d

[4.36]

--

h _h

so

p-

S+

gco ( Vp -

Vs +

2/svslvsiM) d

(4.38)

co ( 2fRvRivRIM) _ h co ( 2/svslvsiM\ h R-g vp-vR+ d - s+g- Vp-vs+ d f

(4.39) so

c0 ( ) co (2fRvRivRIM + 2/svslvsiM) h R- h S +VR + VS - d g g vp= (4.40) 2c0 g

vp =...L (hR _ hs) + vR + vs _ _!_ (2fRvRivRIM + 2/vslvsiM)

2c0

2

d

2

d

(4.41)

hp can then be simply obtained from equation 4.37 or equation 4.38. Elsewhere in this text cR has been used in place of CQ in equation 4.15 and cs in place of co in equation 4.36. Both techniques are valid and produce only slightly different answers. Remembering that the initial calculation of the position of R and S was based upon the values of v0 and c0 it may be felt that these should now be recalculated using slopes based upon mean values, i.e. dt dx

VR

+ Vp CR + Cp

-2-+

2

to establish a new position of R and dt dx

VS + Vp _ CS + Cp

2

2

to establish S again and the entire process iterated until two successive values of vp and cp respectively are insignificantly different. This is simple to do but the author has not found it necessary, providing that a sufficient number of Llx lengths has been used. The only way to be sure that this number has been chosen sufficiently large is to run the computer program


85

over and over again using progressively smaller .::lx values. When two successive values of D.x are found to give acceptably similar answers, the larger of the Llx values can be used. In fact, analysts gain considerable experience in judging the value of D.x to be used and do not need to use this multi-running technique. It must always be borne in mind that for the case of a simple pipe, making .::lx equal to the pipe length gives results which will be at least as good as a Schnyder graphical analysis using one throttle at the upstream end, so even if only a small number of D.x lengths are used the result will be considerably better than that produced by a graphical analysis. The author typically uses ten .::lx lengths but in long pipelines he uses more. The problem arising in using a large number of D.x lengths is that of the run time on the computer. The run time is approximately proportional to the square of the number of D.x lengths so increasing this number rapidly increases the cost of a run. Achieving run cost economy whilst still maintaining a high level of accuracy is very much a matter of experience. Some analysts use a programmed version of the graphical technique. What such analysts are doing is using a different presentation of the waterhammer equations. The equations in use are the same as those used in developing the graphical technique and these are limited versions of the full water hammer equations and, as such, they are not completely correct. The convective acceleration term v

~~

is omitted from the dynamic

equation and the v ~~term is omitted from the continuity equation. This involves small errors if wavespeeds are high but these become larger as wavespeeds fall. As wavespeeds vary and can become relatively small in flows containing free gas bubbles this effect may not be negligible. The argument that programmed Schnyder is equivalent to the method of characteristics is true if the above limitations are accepted, as the problem is one in four dimensions, the variables being h, v, x and t. SchnyderBergeron methods look at the problem on the h - v (or h - q) plane whilst the method of characteristics looks at the problem on the x- t plane. In this sense there is little difference between the two methods but the method of characteristics is far simpler to use, facilitates the mathematical description of boundary conditions and does not require the elimination of any terms in the basic waterhammer equations. For these reasons the author is convinced that the method of characteristics is better than the programmed Schnyder method and he believes that in the course of time programmed Schnyder methods will be superseded by the method of characteristics.

4.9 Other fmite difference methods It is possible to use direct finite difference methods of integration of the fundamental partial differential equations of waterhammer.

86

Hydraulic analysis of unsteady flow in pipe networks Consider point E (see figure 4.7): examining the continuity and dynamic equations

-f

+

~

.f

+E

+

A

8

H

F

!:::.t

I

6.x

c X

!:::.x

Figure 4.7

and ah +!:!. av +_!_ av + 2f
at

g

ax

gat

gd

( ah\ =hH- hB, {~~) at }E 2!:::.t \ax E

(av) at

=vH-vB E

2 !:J '

(av) ax

=hr- ho 2!:::.x

=Vf-VD E

2/:::,x

As ho, hr, vo, Vf, hB and VB are known, substitution of the above expressions for the partial differentials will provide two simultaneous expressions for VH and hH. The values of b.x and tlt, however, will still have to be related by the Courant and Lewy criterion of stability, i.e. dt -=--

dx

v +c

There are many variations of this method, such as the l..ax-Wendroff method, the leap frog method, Amein's four point method, Liggett and Woolhiser's method, all of which are better than the one illustrated above, but the author is of the opinion that the regular grid characteristic method previously illustrated is superior to all others for the waterhammer problem11.

5 Variable parameters in unsteady flow

5.1 Variation of wavespeed The variation of wavespeed has been mentioned in earlier chapters. Wavespeed is dependent upon the effective bulk modulus of the fluid and this has already been shown to be dependent upon the distensibility of the pipe, i.e.

for bubble free liquid. Where K' is the effective bulk modulus, d, T and E are the internal diameter, wall thickness and the Young modulus of the pipe wall material respectively, and K is the bulk modulus of the fluid. If gas bubbles are present in the liquid the effective bulk modulus is reduced greatly, the effect of gas bubbles becoming greater than the pipe distensibility term d/TE at low pressures. As the volume of gas in free bubble form in any ax length depends upon the absolute pressure in this length, the effective bulk modulus and hence the wavespeed must be different for each and every Ax segment. Transients can generate very different pressures at different parts of a pipeline at the same time instant so it is possible to have a waves peed of 1000 m s- 1 at one point of a pipeline and at another a wavespeed as low as 10m s- 1• To ignore this effect can only lead to error and the author therefore believes that its inclusion is vital if an analysis is to bear any similarity to reality. At the moment of writing, the subject of wavespeed variability is receiving a great deal of attention throughout the world and it is being researched in many universities. The flow of gas/liquid mixtures is an example of twophase flow and little work has yet been done on the subject of unsteady two-phase flow. The understanding of steady two-phase flow is still incomplete although much effort has been expended upon research into it so the problem of unsteady two-phase flow is even less adequately comprehended.

87

88


5.2 Gas evolution The problem is further complicated by the evolution of additional gas bubbles from the dissolved gas present in all liquids during the passage of negative transients. When a liquid which was originally saturated with gas at some relatively high pressure, i.e. atmospheric pressure for water, is subjected to a low pressure for a long time some of its gas content will be released into bubble form and these bubbles will rise and escape from the liquid at the free surface. The saturated gas content of the liquid depends upon its absolute pressure. Henry's law states that the volume of dissolved gas depends directly upon the pressure, i.e. Vg = kp where k is a constant which reduces with temperature. For water the saturated air content is about 2% by volume at atmospheric pressure. Thus if water at atmospheric pressure containing 2% by volume of air is later subjected to a pressure of one half an atmosphere, I% by volume reduced to atmospheric pressure will be released from the water in bubble form, providing that sufficient time is allowed to elapse. This phenomenon is a commonplace of everyday life. When a bottle of beer is first opened it froths violently but gas (C0 2 ) bubbles continue to be evolved for a relatively long time. When they finally cease being evolved the beer is said to have 'gone flat'. If a liquid is subjected to a high pressure it will absorb gas across its free surface but, again, this process is highly time dependent. If the high pressure is applied by bubbling gas into the liquid the process of absorption is much faster because of the greatly increased surface area through which gas can diffuse. In the case of transient low pressures, the time available for gas release and reabsorption is very much smaller than the time required to establish equilibrium so the amount of gas generated by the passage of a low pressure transient cannot be calculated by the use of Henry's law. A method of modelling the process of bubble evolution will be suggested here. It should be appreciated that the model proposed is not complete and is based upon assumptions that cannot be fully validated. Consider a bubble of radius r, as shown in figure 5.1. The pressure inside the bubble will be greater than that outside of it by an amount IJ.p because of the surface tension acting within the bubble, and the excess pressure can be calculated by considering the force balance across a bubble's diametral plane. i.e. so

/J.prrr 2

= 2rrrr 2T

IJ.p =T

(5.1)

where T is the coefficient of the liquid gas interfacial surface tension. Thus, the excess pressure in a bubble increases hyperbolically as radius decreases. If it is assumed that the excess pressure /J.p causes gas to diffuse outwards

Variable parameters in unsteady flow

89

T

Figure 5.1

from the bubble and the magnitude of p decides how much gas must pass into the bubbles present in the flow, it will be appreciated that at any particular pressure (less than saturation pressure) and bubble radius, gas will be attempt· ing to diffuse into the bubble but, due to the high pressure within the bubble, gas molecules will be moving so fast that many will be leaving the bubble. At a critical bubble radius, gas will be leaving the bubble at a rate equal to that at which it is diffusing into it. At any radius less than this critical radius b.p will be even larger and gas will leave the bubble faster than it is coming in so the bubble will diminish in radius; this will cause b.p to increase still further and the bubble will implode. So, for every pressure there will be a critical value of bubble radius below which no bubble can exist. If a critical radius exists below which bubbles implode, how can bubbles ever come into existence? The answer to this question is simply that they must come into existence with a radius greater than the critical radius. For this to happen a mechanism must exist within the fluid which generates such bubbles. Some authorities have hypothesised the existence within the fluid of small particles called micronuclei upon which a film of gas is adsorbed. If a micronucleus has a suitable geometry and size, bubbles can form which are of a radius greater than the critical value. Another competing hypothesis has recently been advanced which suggests that either upon the surface of the containing vessel or upon the surface of suspended particles there are fine cracks or surface roughnesses within or upon which gas is trapped. Gas can cross the gas/liquid interface because the interface has a radius of curvature which is greater than the critical value. A bubble can generate at the crack and be released when it reaches an adequate size for its buoyancy to tear it free (see figure 5.2). The crack is then available to generate another bubble. This hypothesis is attractive as any drinker of carbonated drinks will realise. In a glass of liquid supersaturated with carbon dioxide streams of bubbles can be seen rising, originating from defined sites on the glass surface. It seems probable that both mechanisms operate, the micronucleus mechanism during the initial frothing phase and the surface crack mechanism continuing to operate when the degree of supersaturation has dropped to a point at which the micronucleus mechanism has ceased to function. (Micronuclei are thought to be approximately eight microns in diameter.) For gas to evolve rapidly from a liquid, either micronuclei must operate as coalescence points or microcracks must perform this function. In either

90

Hydraulic analysis of unsteady flow in pipe networks successive gas/liquid interfacial shapes

Figure 5.2 case, such active devices will not be recruited until the pressure has fallen significantly below the original saturation pressure as the unusually large micronuclei (or microcracks) necessary to generate bubbles at pressures just below the saturation pressure are unlikely to exist in normal liquids. It seems probable that there will be a spectrum of particle/crack sizes present and only a few of these will be very small and conversely only a few of them will be large. The addition of sugar crystals to a solution saturated with carbon dioxide (e.g. beer or soda water) produces rapid gas evolution which seems to support this hypothesis. The main band of particle/crack sizes present within a liquid will evolve bubbles when the pressure falls to a value which lies within a band of pressure values. This implies the existence of a 'gas release head' band which the author suggests can be modelled as a unique 'gas release head'. For water saturated with air at atmospheric pressure this gas release head seems to be about 2.4 metres (8 feet) absolute. If this concept is valid, a rapid reduction of the pressure of'a water mass below atmospheric pressure will evolve negligible volumes of gas from solution until the pressure drops below the gas release head. At this point most of the dissolved gas will come out of solution. Once the gas has come out of solution it is difficult to force it back into solution by a subsequent pressure rise to atmospheric pressure but, as a negative transient is followed by a corresponding positive pressure transient, the pressure usually rises far above atmospheric pressure and much of the gas will return into solution. For this reason the author suggests that, in the absence of a better theory, gas may be assumed to come out of solution when the pressure falls below gas release head and re-dissolve· when it rises above it again. This assumption is not correct but it leads to pressure predictions which are higher than those that will actually occur and is therefore safe. Turbulence also has a powerful potentiating effect upon bubble release. If a bottle of beer is shaken and then rapidly opened its dissolved gas will be released in an almost explosive manner. Almost all pipelines operate in turbulent flow so this effect is nearly always present. It is fully appreciated that in many ways the foregoing concept is naive, but it does permit an attempt to be made to describe the phenomenon. S .3 The magnitude of variable wavespeeds and the inclusion of gas release The effective bulk modulus is given this chapter.

by~,

=} + :E as stated earlier in

91


Consider a mass of liquid containing a fractional volume of gas in free bubble forme, the volume of the gas plus liquid being V. The volume of liquid is (1- e) Vand the free gas volume Vg is eV. Apply a pressure increment to the liquid !J.p. The liquid volume will change to V1 where V1 = (1 - t:.pjK)(l- e)V. The gas volume is assumed to be distributed in small bubble form and to have an unchanging temperature equal to that of the water. Any gas volume changes will therefore be isothermal

(5.2)

peV= (p + Ap)V~

so

Vg = p

so

r

(5.3)

tJ.p eV

where v~ is the fractional gas volume at pressure p + t:.p. The volume of the gas/liquid mixture will therefore become (approximating (1 + Apfpr 1 to (1- Apjp))

VT=

(1-'i) (1-e)V+ (1-tJ.;)ev

(5.4)

where VT is the total volume of gas plus liquid at the pressure p + Ap. Expanding

!J.p) !J.pe tJ.p VT=V ( 1---e+ --+e-e p K K

(5.5)

The term !J.:e will be very small and can be ignored.

~ = (1- !J.p(~ +~)) The volumetric strain (V- VT )/V = I - VT/V= b.p(I/K + e/p) so

1 !J.p - l _ VT - _!_ + :.._ V K p

K'-

(5.6) (5.7)

(5.8)

Including the distensibility of the pipe

d 1 e 1 -,=-+- +KT K p TE

(5.9)

Where KT' is the effective bulk modulus of the gas/liquid mixture including the pipe distensibility effect. So

(5.1 0)

92


but w =we(! -E) neglecting the weight of the free gas so

c=Jwe(l-E) g

(5.11)

(___!_+~+~) K

p

TE

Pearsall 12 .

This result was published by The value ofp used must be the absolute pressure. Equation 5.11 is only valid for small values of € as, when values of E are large fractions of unity, the flow becomes a frothing flow which may separate into open channel flow with a gas flow over the top of it, or it may become a slug flow in which large bubbles possibly of diameter equal to that of the pipe are interspersed with liquid. Again, such flows may consist of a ftlm of fluid flowing around the periphery of the pipe with a central core of gas flowing at high speed through it. For this reason, the wavespeed equation quoted above should only be applied to two-phase flows for which the assumption that gas bubbles constitute a small fraction of the flow and are homogeneously distributed throughout it is applicable. The wavespeed is related to the gas/liquid volume ratio as shown in figure 5.3. The curve shown is pressure dependent, a different curve would

c 1372m s·'

0

/o gas

0

100

Figure 5.3

be produced for a different pressure, but the end values at 0% gas and 100% gas are not pressure dependent. Equation 5.11 is only valid for the low end of the gas percentage range. It is obviously incorrect at the high end as at 100% gas the equation predicts an infinite wavespeed. As the wavespeed curve has a wide and relatively flat minimum it is reasonable to suggest that once the value of c ralculated from equation 5.11 falls to this minimum value the minimum value is used. Figure 5.4

93


indicates how the minimum wavespeed varies with pressure (absolute). The results are for air/water mixtures and are based upon results quoted in a paper by Karplus 13 . No such similar results are known to the author 150

V>

E 100 "C

a.> a.>

&1-

a.>

> 3:

0

50

250

500

pressure N cm- 2

Figure 5.4

for other gas/liquid mixtures but as the liquid's bulk modulus has almost no effect upon the minimum wavespeed and as the gas bubble expansion/ compression is isothermal, results for air/water mixtures should be usable for other gas/liquid mixtures without significant error. It is felt that the results obtained from an analysis in which local pressures fall well below gas release head, so greatly expanding gas evolved from solution, will be erroneous. The reason for such errors is that the 2% gas evolved will be about 8% by volume at gas release head and correspondingly more at lower pressures. Such large gas percentages invalidate a number of assumptions made in the derivation of the waterhammer equations and in the development of the wavespeed equation, for example the four now listed.

(I) In the development of the waterhammer equation in chapter 2, second order terms were ignored, in particular the term Au ap(ax ax at. However, ap(ax may not be negligible when large volumes of gas are present in the flow. (2) In the same development the term pfp 0 was approximated to unity but this will not be true when large volumes of gas are present. (3) When developing the waterhammer equation the substitution of h for the pjw + z term was made. Normally w can be taken as the usual liquid specific weight value without significant error but when E is large this assumption is far from true. (4) In the development of equation 5.11 the term t:.pE/K was neglected. This term is undoubtedly small but its omission may produce some error when Eisa large fraction of unity. An analysis which produced results in which sections of pipe were at very low pressures would be regarded by most engineers as suggesting that the surge needed some type of suppression since such low pressure domains cannot usually be accepted in engineering conditions. The analysis, although inaccurate, would ihus be seen as of diagnostic value. Subsequent

94


analyses would then be performed to ensure that the surge suppressing device operated to prevent the development of low pressure domains. Near vacuum conditions in pipeline segments are not acceptable for reasons mentioned earlier, i.e. high transmural pressures causing pipe flattening, the possible development of running buckles, fatigue of the pipe wall material, diametral pulsing causing high local stresses at stress raiser points such as stones in the pipe trench or other pipes wrongly located in the same trench. In water supply pipe networks it is not usual to accept subatmospheric pressures in buried pipes because of the risk of pollution from groundwater drawn into the pipeline through pipe joints. 5.4 The use of the variable wavespeed equation A subroutine can be written to calculate wavespeeds at the pressures existing at the interpolation points. If the wavespeed calculated from equation 5.11 should be less than the minimum wavespeed taken from figure 5.4 then it must be assigned the valu~ of this minimum wave speed. Thus by using the subroutine to calculate cR and cs, complete variability of wavespeed can be included in the analysis. S.S Vaporous cavitation If the local pressure falls to vapour pressure the liquid will boil at the ambient pressure. Vapour bubbles will appear within the fluid and it will be impossible for the local pressure to fall any further. If the pressure drop is very rapid it may be that a large bubble may appear in the top section of the pipe. This large bubble will be filled with a mixture of vapour and gas. The bubble(s) will collapse when flow reverses and will disappear leaving behind a faint haze of fine gas bubbles which will only return into solution slowly and at relatively high pressure. This phenomenon is thought, by some, to occur by the opening of two vertical liquid faces and after their re-meeting it is thought that a very large pressure transient is initiated. This is called column separation. The formation of large long bubbles can occur in long pipelines but, in the author's laboratory, the closure of such bubbles has been shown to occur by a surface wave mechanism without the development of pressure transients of any significant magnitude. Even large bubbles close by a process of wave action on a free surface flow, and the small pressure rises that may be generated are further minimised by the suppressant effect of the compressibility of gas bubbles present in the liquid which were previously recruited from the dissolved state during the previous low pressure. The process of vapour filled bubble flow, like gas filled bubble flow, is that of a two-phase flow. One difference between vaporous and gaseous cavitation is that whilst pressures can fall below the gas release head by expansion of the limited quantity of gas present, this is not true of va-


95

porous cavitation. In vaporous cavitation the liquid's vapour is freely available to expand any bubble indefinitely without changing the pressure to a measurable extent. The phenomenon of vaporous cavitation can thus be dealt with by calculating the local pressure head from the local absolute potential head. If this is below the vapour head it must then be set to the vapour head and the absolute potential head recalculated by adding the elevation of the point above the datum level to the vapour head. This technique may not model the behaviour of the bubble closure by surface wave action. At a closed end or at a closed valve a bubble may form; this bubble may be filled with vapour and gas or, if the pressure is not low enough, gas only. If the valve closure causing the transient occurs in a very short time and the 2Ljc value is also small then such a bubble will not close by surface wave action as one of the two necessary waves is not present. The changing volume of such a bubble, and the pressure rise that results when it finally closes must be calculated, the flow velocity in the liquid then being calculated (see J. Swaffieid 18 ). The speed at which surface waves travel is relatively very small, being of the order of I to 6 m s-1 , depending on the flow depth. An analysis by Marsden and Fox 19 demonstrated that by combining the transient analysis of the water hammer phenomenon with a surface wave analysis of the bubble it was possible to obtain a highly accurate solution of the problem for bubble closures occurring by surface wave action, i.e. in pipelines of normal engineering size. The analysis takes a very long time on a computer and so is not economic as a routine method. The use of the surface wave celerity of approximately 1-6m s- 1 will also produce errors in the type of analysis proposed here, however, for the following reasons. In this type of analysis the formation of a bubble is assumed to occur over a complete .:ix length when, in fact, it would be only a small fraction of such a lu in length, perhaps a few metres long. Therefore, the closure of a bubble of .:ix length at surface wavespeed would take an excessively long time and this time would be completely erroneous. The author therefore suggests that the wavespeed to be used when vaporous cavitation occurs should be the minimum value relevant to the vapour pressure taken from figure 5.4, e.g. for water, the vapour pressure of which is about 0.1 M absolute (i.e. about 10 N cm- 2 ), the wavespeed should be 20 m s- 1 • For very large .:ix values this value would have to be considerably increased if the timing of the bubble closure were to be accurately predicted, however. For such a case, much smaller .:ix lengths should be used. Much more research into this subject is needed.

5.6 Calculation of friction In the characteristic equation, the Darcy (Fanning)fvalue is required at the interpolated points. Undoubtedly, the Colebrook-White formula is the best available but unfortunately [is given in an implicit form, i.e.

_1 _ _

YJ-

[ 2.51 k/d 41ogto 2Y2JRe + 3.71

J

96


(Note that the f value commonly used in the USA is four times larger than the Darcy [value used here.) To obtain an [value from the Colebrook-White equation an iterative method of solution must be used First a reasonably good approximation for [must be substituted for fin the right-hand side. A new value off can then be calculated. This new value can be back substituted in the RHS and another [value obtained. The process must then be repeated until two successively obtained values differ by an acceptably small value. The 'reasonably good approximation' for [which is initially needed can be obtained from the Moody formula for[, i.e.

t= o.ooi375

[1 + (20 oooj+ ~~ 6 ) ' 13]

(5.12)

This is accurate to within 5% of the Colebrook-White f for Reynolds numbers between 4000 and 10 000 000 and k/d values up to 0.01, so it is certainly a good approximation. It may be thought that the Moody formula is adequately accurate in its own right. The iterative method should not need to be iterated more than five times if a percentage difference between two successive iterations of 0.1% is acceptable. Of course, it is necessary to check that the Reynolds number is greater than 2300. If it is below 2300 the laminar flow expression[= 16/Re should be used instead. It is fully accepted that steady state/values are not applicable to unsteady flow but the errors produced by using steady state/values in unsteady conditions do not appear to produce significant errors and no satisfactory method of calculatingfvalues in unsteady flow is available. In unsteady flow, boundary layers are thicker in the presence of adverse pressure gradients than they are in steady flow at the same Reynolds number. If the adverse pressure gradient is sufficiently large the boundary layer may separate from the boundary with a very large increase in energy loss. Conversely, boundary layers are thinner in the presence of a negative pressure gradient than they are in steady flow at the same Reynolds number. This is the reason for the failure of steady state [values to accurately describe the frictional conditions of unsteady state. In gas flows this effect can be large but in liquids the effect is usually small.

5.7 The use of variable f values

As with the variable wavespeed (5.4) a subroutine/procedure must be written which evaluates[ at every interpolation point. The parameters that must be supplied are the interpolated velocity, the relevant pipe roughness and the relevant pipe diameter. The subroutine/procedure must first calculate the Reynolds number. This must be tested to check whether

97


it is above or below the critical value of 2300. If the velocity is zero the Reynolds number will be zero so the f value will become infinite, therefore a cut off value must be supplied. It is suggested that if Re is less than 0.1 the/value is assigned the value of 160. If Re is greater than 0.1 and less than 2300 then!= 16/Re. If it is greater than 2300 then a value [ 1 must be calculated from the Moody formula (see equation 5.12). Usingfi then

f=

(~ 4

[ 2.51 k/d 2 og 10 2V2!;Re + 3.71)

I

(5.13)

A test must then be made for the difference between fi and f. If the difference is greater than the acceptable percentage error of 0.01% then fi must be assigned the calculated value off, the program control returned to the f calculation and the process repeated until the accuracy test is satisfied.

5.8 Interpolation Providing that the .:lx value is sufficiently small, a linear interpolative process can be used. A subroutine/procedure should be written to perform the interpolations at every .:lx point and .:lt stage.

6x

6x

X

Figure 5.5

First, it is necessary to calculate RO and OS values (see figure 5.5) . .:lt will have been calculated or read in and will be an available value. The

wavespeed at the pressure head current at 0 must next be evaluated (using the subroutine/procedure previously developed for calculating wavespeeds ). Then

RO = (v0 + c0 ) M

and

OS= (c 0

-

Vo)M

At this point it is wise to check that RO and OS are both smaller than .:lx. If either of them become greater than .:lx a suitable warning message should

98


be printed and the run terminated. If this occurs during a run then either there must have been a slip in preparing data for the run or !:lt has been assigned an excessively large value. Then the velocities and the potential heads at R and S must be calculated. VR = v0 -(v 0

and

-

RO vM) !:lx

RO hR = ho - (ho - hM) !:lx OS vs) !:lx

similarly

vs = v0

and

OS hs = ho- (ho- hs) !:lx

-

(v0

-

(5.14) (5.15) (5.16) (5.17)

As pointed out earlier in chapter 4 this process should, strictly, be iterated once the velocity and head at P has been calculated, using a velocity and wavespeed which is the mean between that at P and R to recalculate the slope of the forward characteristic and to relocate R, and using a velocity and wavespeed which is the mean between that at P and S, to recalculate the slope of the backward characteristic and relocateS. However, this iteration should not be necessary if the !:lx value chosen is sufficiently small.

S.9 The calculation of the free bubble content The free bubble content at atmospheric head must be supplied initially and the expansion or contraction of these bubbles together with that of any bubbles evolved from solution during low pressure phases must then be calculated. First, it is necessary to calculate the local pressure head. The value of head in the characteristic equation is an absolute potential head and consists of the local absolute pressure head plus the local elevation of the pipe centre line. So the pressure head must be calculated by subtracting the elevation from the potential head, i.e.

p = (h -z)w

(5.18)

The value of the fractional bubble content e1 must then be increased by the amount that comes out of solution, i.e. e 2 • For water e2 is about 0.02 and a reasonably typical value for e1 is about 0.001. e 2 is only evolved if the local head pfw falls below the gas release head hG. The total gas in bubble form must then be obtained, i.e.

e = E1 + (if p ~ whG then e 2 else 0) so and

€ €

=

€1

if p

> WhG

= €1 + €2 if p ~

W X

(5.19) (5.20)

hG

(5.21)

99


e must then be multiplied by whafp to obtain the actual fractional gas volume expanded (or contracted) appropriately to match the local absolute pressure p. The gas process is assumed to be isothermal. (ha is the height of the liquid barometer or the atmospheric head, i.e. I 0.3m or 34ft approximately for water.) The way in which the total fractional gas volume is computed above involves a number of assumptions. (I) If the local absolute pressure falls below the gas release head all of the dissolved gas is assumed to be evolved. (2) If the local absolute pressure rises above the gas release head all that portion of the gas that was originally dissolved but was evolved in bubble form is assumed to redissolve. The justification of these assumptions was given earlier on pp. 89-90. The calculation of e is needed for use in the wavespeed calculation. 5.10 Evaluation of velocities and potential heads at internal points in a pipe length It is suggested that a subroutine/procedure called midstream be written to calculate the potential heads and velocities at all internal points in each pipeline. This can be called at every internal point. The equations for velocity and head already derived are requoted below.

[4.37] Before these equations can be evaluated uR, us, hR and hs must be calculated from the interpolation subroutine/procedure, e must be calculated from the fractional gas volume subroutine/procedure, c0 from the wavespeed subroutine/procedure and fR and fs from the friction subroutine/procedure. Up must then be evaluated and this value can be used to calculate hp.

6 Boundary conditions· pumps

6.1 Introduction

The writing of a computer program to perform unsteady network analysis depends upon an accurate description of wave transmission (described in chapter 5) which is relatively simple to do, but the problem of describing the boundary conditions that either generate the unsteadiness or reflect (totally or partially) pressure transients already generated is more difficult. 6.2 Pumps equipped with a nonreturn valve (forward flow only permitted) The method of dealing with pumps differs from analyst to analyst. The

author has evolved a technique which has certain defects but has definite advantages. It will be shown below that a centrifugal pump's characteristic curve can be described by the equation H = AN 2 + BNQ - CQ 2

(6.1)

where His the head developed by the pump, N is its running speed in rev min- 1 and Q is the flow through it. A, Band Care constants applicable to the particular pump. Only pumps with radial flow impellers can be accurately described by equation 6.1 and as the impeller type becomes progressively more and more axial the equation becomes less and less accurate. This is the major defect in the analytic technique to be described here. In figure 6.1 a the characteristic curve designated A is for a pump equipped with a highly efficient volute, e.g. a free vortex volute or a variable velocity volute and curve B is for a pump with a less efficient (but much cheaper) volute. In figure 6.1 b curve Cis for a true axial flow pump and Dis the curve which most closely approximates to it and which can be described by equation 6.1. Providing that curveD is so chosen that it passes through the expected steady state point of the network, only very trivial error will result from its use as the starting transient will be only slightly wrong

100

Boundary conditions: pumps

101

h

h

q Pumps with 1adial flow type impellers

q Pumps with axial flow type impellers

(a)

(b)

Figure 6.1

but some advantage can be gained as shown below. The derivation of the

H ~ Q (equation 6.1) is necessary as, from this derivation, the pump's

behaviour when operating as a turbine or dynamometer can be deduced. The ability of this technique to describe turbining of the pump in the case of forward flow is the advantage mentioned in section 6.2.

6.3 The derivation of the pump's characteristic equation u is the peripheral velocity of the blades of the impeller at the outer radius. Vis the absolute velocity of the fluid leaving the blades. V w is the tangential component of the absolute velocity, i.e. the velocity of whirl. Vr is the velocity of the fluid relative to the blading of the impeller. Vr is the radial component of the absolute velocity, i.e. the velocity of flow. Then assuming radial entry to the impeller

H = Vwu- k ~ - k Vr 2 g v2g r2g

u

Figure 6.2

102


where kv

~ is the energy loss due to friction and local turbulence in the

volute and kr

~2 is the friction loss between the fluid and the impeller

blading. See Fox 14 for a more detailed presentation of this theory.

Q

rrDN

u = - - and Vr=60 Ae

where Ae is the flow area of the blading at exit, and Dis the external diameter of the impeller. Vw = u - V[cot "f

Vr V2

:. V

2

:. V 2

=

Vr cosec 'Y

= v~ + v( = (u-

=u 2 -

Vr cot r) 2 + V( 2u Vr cot 'Y + V( cosec 2 'Y

H= (2 -kv)u 2 -(i- kv)2u Jif·cot 'Y -kv Vr2 cosec 2 "f -krVl cosec 2 'Y 2g

so H = (2- kv )u 2 + 2(kv- I)u Jif· cot "f _ (kr + kr) V( cosec 2 "f 2g 2g 2g

substituting for u and

Vr H = AN 2 + BNQ - CQ 2

where A= B

2-kv (rrD) 2g 60

2

= (kv- I) cot "f....!!!!_ 60Ae

g

(6.2) (6.3)

2

C = (k + k ) cosec 'Y v

r

2gAi

(6.4)

Hence equation 6.1 is valid. 6.4 Dynamometer/turbine operation of a pump with forward flow

It is possible for the flow through a pump to become so large that the head across it becomes negative. In other words, the operating point on the pump H- Q characteristic curve is far over to the right with a large flow and negative head. In such a circumstance the pump is causing a reduction


103

in the energy of the flow and so is acting as a dynamometer rather than as a pump. For the case of a pump in a normally designed network, a transient which could cause a negative head across it whilst the pump is still being driven could only be generated by the operation of another hydraulic control such as a valve. It is suggested that the head- flow equation H =AN 2 + BNQ- CQ 2 is capable of describing the pump's behaviour whilst operating in the dynamometer mode since, in this condition, the only difference between its behaviour as a pump and as a dynamometer is that at high delivery the friction loss in the volute and the impeller becomes larger than the no flow head plus the head regained in the volute. Thus the way in which it is working as a dynamometer does not differ essentially from the way in which it is working as a pump. Almost certainly the kv value will not be a constant and for an accurate representation of the pump acting as a dynamometer greater knowledge of its variation would be needed. Even so, the assumption of a constant kv value will give an adequate representation of the pump as a dynamometer at a level of accuracy sufficient for most engineering purposes. Research on this point is continuing in the author's laboratory. When pump trip occurs, the head across the pump may become negative, i.e. the potential head upstream of the pump may become larger than that downstream of it. This may be due to the development of transients in the upstream and downstream pipes, or it may simply be due to the pump being required to act as a booster to an existing flow in which case the head upstream of the pump will be greater than that downstream once the pump has been switched off. Under these conditions, with no power being supplied to the pump from the motor, the pump will absorb energy from the flow and act as a turbine. In effect, if negative heads occur across the pump after pump trip the run down of pump speed will be slower and in the case of the booster pump the pump may not come to rest if a steady flow occurs through it. Of course, in the circumstance of a pump which is acting as a turbine with forward flow through it, the flow is in the opposite direction to that which is normal for an inward flow radial flow turbine. A simple analytic method can be developed for the case of the turbining pump. Assume that the running speed of the pump at any instant in its run down is N rev min- 1, that it is passing a forward flow of Q m 3 sec- 1 and the head across the pump under these conditions isH where His negative. The amount of energy transferred to the pump impeller per unit weight of flow will be Vwu/g and kr the impeller and kv volute. Then

~2 units of energy will be lost in friction in

~g2 units of energy will be lost in turbulence in the

104


also as before

Vw = u - Vr cot 'Y

v;. = Vj·cot 'Y V 2 = v~ + v? Vwu _

k (u 2 - 2u Vj· cot 'Y + V l cosec 2 -y) v 2g

kr Vl cosec 2 'Y

g- -H---'--~-2g

Vwu = _ H -(k + k ) V? cosec 2 'Y _ kv(u 2 - 2u Vj· cot -y) g r v 2g 2g

(6.5) From equations 6.2 and 6.3

)2 k =2-2g( 60 v rrD A

(6.6a)

cot 'Y _ 60gB ~- (kv- l)rrD

(6.6b)

and

The values of A, Band C will be available by direct calculation from the Q curve supplied by the pump manufacturer. Using equations 6.5, 6.6a and 6.6b the value of Vwu/g can be obtained. The power supplied to the impeller is partly consumed in overcoming bearing losses, disc friction and windage losses in the motor. If P0 is the power consumed by the pump at zero flow at steady running speed N~ then these bearing, disc friction and windage losses at the current speed N can be represented by the term P0 (N/N5 ) 3 . Thus the power P available to accelerate the impeller will be 3 wQVwu . g Ns

H

~

-Po(!!_)

Then

H2 =P

(6.7)

T= frl.

(6.8)

where rl. is the angular velocity, rl. is the angular acceleration and I is the moment of inertia . T p so (6.9) rl. =I= rl.f 2rr(N2

-

Nd _

60~t

P

-~

60NI

(6.10)

Boundary conditions: pumps N _ N + 3600Pflt 2 1 4rr 2 N 1 I

105 ( 6.11)

So N 2 can be calculated and may be larger or smaller than N 1 according to the magnitudes of P0

(ZJ

3

and V;u, i.e. according to whether Pis positive

or negative.

6.5

Pump efficiency

The pump efficiency is involved in the problem during the pump run down phase only when the pump is not turbining. The output power of the pump or waterhorse power is simply

(6.12) Assume that the input power can be described by the following equation:

= w(Def/ 2 Q + EeNQ 2 + f""efl 3 )

~

(6.13)

where De, Ee and Fe are constants. To justify this equation consider the following argument. The input power is given by

Pi= wQ(V;u) +Po

(6.14)

where P0 is the power required to overcome the zero flow torque, i.e. it is the disc friction power. Now the disc friction, no flow power is proportional to N 3 so R I

= wQ (u 2 -

uVrcot -y) +F. g

I\T 3

e''

Pi= w(DeN 2 Q + E"'ef/Q 2 + FeN 3 )

thus equation 6.8 is justified. Therefore the efficiency of a pump can be calculated from the following equation: AN 2Q + BNQ 2 - CQ 3 (6.15) Eff =DeNz Q + EeNQ2 + FeN3 A Band C can be calculated from the head~ flow curve and these values can then be used to calculate the De E'e and Fe values from the efficiency curve.

6.6 Pump power The pump power calculation is needed in the calculation of pump run down. The run down may occur in two phases as mentioned before: (I) the normal pump run down and (2) in turbine mode. In both phases

106


the pump's power is needed to calculate the reduction or increase of the pump speed occurring in the t:.t interval. In phase 1 the pump's speed is reducing because the flow is still being pumped and therefore energy is being absorbed from the rotational energy of the pump. The power absorbed by the flow in phase 1 is thus:

Pwr =- ~~~

(6.16)

(The negative sign implies that the pump is absorbing energy from the flow rather than supplying energy to it. Remember also that when the pump is operating as a dynamometer the value of Eff will be negativ~ The H value is given by the head equation and the efficiency by the efficjency equation. Before this equation is used a check must be made that the suction side head is less than the delivery head. In phase 2 power will be supplied to the pump by the flow through it. The power supplied will be given by V. u Pwr=wQ~ g

-P0

fNN3) -.-

3

(6.17)

and the expression for Vwu given in equation 6.5 should be used here. g

Again, before this expression is used a check must be made that the suction head is greater than the delivery head.

6. 7 Pump start up Pump start up is difficult to simulate because of the many and varied ways in which different types of electrical starter gear start and run up the electric motor and pump to speed. If it were possible to specify the action of the control gear it would be possible to simulate its action, but to make provision for the large number of possible ways of arranging the starting gear in a general program is virtually impossible (but see p. 120). However, if the pump is assumed to start instantaneously at its full running speed the starting transient will be the largest it can possibly be, but even normal start up sequences can cause rapid run up to speed and may well produce as large a start up transient as would be given by an instantaneous start up. The start up transient can only reach a maximum equal to the closed valve head (i.e. no flow head) of the pump and so will only rarely be dangerous. However, by including a start up in the analysis it is possible to run it on to steady state before tripping the pump. The check on steady state that this gives is valuable to the designer. In the author's experience steady states so obtained usually agree within 1% of that of a steady state analysis and, as the frictional formula used in the program is better than those in common use in most steady state programs, the unsteady result for steady state can be considered to be the better value.


107

6.8 Pump run down The method of analysing the pump's behaviour consequent upon cut off of power is given on pp. 55-6. An outline only, without the discussion, will be presented here. The methods used must be mathematically based if they are to be used in a computer program employing the method of characteristics, not graphically based as in chapter 3. The pump run down is described by the equations used to describe pump turbining, i.e. equations 6.7, 6.8, 6.9, 6.10 and 6.11. The only difference is that the power P to be used in equation 6.11 should be the value of Pwr calculated in section 6.6. In a computer program it is possible to employ a better method than the simple initial value finite difference method of calculating N 2 given previously. Even the simple predictor method described below will give better results. Any rotating mass which is not driven will slow down according to an exponential law. In the circumstance of a pump impeller running down, yet still doing some pumping while it does so, the precise nature of its run down curve will not be a perfect exponential, yet for a short time interval t:.t, it will be far better approximated by a short segment of an exponential curve than by a short straight line of gradient equal to that prevailing at the speed at the beginning of the time interval. At the beginning of the next l::.t interval a new exponential decay curve can be calculated and the process repeated until the entire speed decay curve has been developed. This technique has been investigated and the author has found that it gives results which are far superior to those obtainable from the initial value finite difference method. This method can be expressed mathematically as follows. Denote the gradient of the pump speed- time curve by Gr. Then G _

r- +

3600 x Pwr

4rr2NI

Assuming that N = ae bt- a general exponential law- then dN=abebt dt but so

Now

dN 3600 x Pwr dt = + 4rr 2N/

Gr

dN

dt

abebt N =~=b

N 1 =ae

b=Gr N

Gr t N 1

(6.18)

108

Hydraulic analysis of unsteady flow in pipe networks N 2 =ae

and

~r (t+At) 1

(6.19) This method has the advantage of calculating the speed relevant at the time instant when the pump is running down or when it is turbining. It is noniterative and is well adapted to the use of relatively large tl.t intervals. However, if the pump speed is increasing, as it may in turbine mode, the simple initial valve method outlined earlier is more suitable as the exponential law does not then apply. 6.9 The in-line pump boundary condition Points M and N denote the upstream and downstream tl.x points on the suction and delivery pipes. a1 and a2 are the points immediately upstream and downstream of the pump. R and S are interpolation points, R being

Figure 6.3

interpolated back from the P1 point based upon v0 and c 0 values, and S being interpolated from~ based upon v02 and c02 vaiues (se~ figure 6.4). As the heads across the pump (and the velocities also if pipe diameters are not equal) are not the same, the points a and P have to be separately designated, i.e. a1 and a2 , }\ and ~Values of heads and velocities will be known at all grid points at the t level and the interpolated values at RandS can be calculated from the interpolation subroutine/procedure mentioned earlier. . . 2/RvRivRIAt 2fsvslvsltl.t The fncbon terms d and must then be calcu1

d2

lated; denote these terms by FR and Fs. Then writing the forward characteristic equation:

(6.20)

109


M

N

X

Figure 6.4

The backward characteristic equation is:

-L(hp2 - hs) + (Vp 2 - vs) + Fs = 0 cs hp 2 - hp1 = AN2 + BNQ- CQ 2

(6.21) (6.22) (6.23)

where d 1 =diameter of suction pipe, d 2 =diameter of delivery pipe. Adding the two characteristic equations

hp!- hp2- hR + hs + CR Vp + cs Vp g

g

I

2

-

(CR VR + cs vs) + g

g

( ~R FR + ; Fs)

=0

Substituting for hp1 - hp2 and equating Vp1 tog_ and Vp 2 tog_ where ap1 ap 2 ap1 is the cross sectional area of the delivery pipe then

-AN 2 - BNQ + CQ 2 -hR + hs + Q(~+___:§_) -(·~VR + cs vs) ap1g

ap~

g

g

+ CR FR + csFs = O g

Let

-AN2- (cR vR + cs vs) + (cRFR + csFs) -hR + hs g

g

.

g

g

="' (6.24)

Let Let

C=a

(6.25) (6.26)

110


The equation then reduces to

and so

aQ 2 +/3Q+-y=O

(6.27)

Q = t3 + v't32- 4a-y 2a

(6.28)

If Q is less or equal to zero the nonreturn valve will close and negative flows will therefore be prevented, so it is necessary to write an algolrithm into the program to the effect that if Q ~ 0 then Q = 0. Having calculated Q, Vp1 and Vp 2 can be simply calculated from Vp

1

and Vp =JL. =JL 2 ap2 apl

hp1 and hp 2 can then be calculated by back substitution into equations 6.20 and 6.21. The surd in equation 6.28 should always be positive and failures in attempting to take the square root of a negative number should not occur but it is wise to arrange to check the possibility. The author has never known the value of /3 2 - 4a-y to become negative however. If an account of time is kept sot= i x flt, i.e. i is a counter of the number of flt intervals that have elapsed since the commencement of pump run, a test can be made to see if this time is less than the time at which the pump is to be switched off. If it is less, then the pump speed to be used in the next time level calculation must be the original pump steady speed and, if it is larger than the time of pump trip initiation, then the pump run down calculation must be used to obtain the pump speed applicable to the next time level.

6.10 Suction well pumps Pumps, pumping straight from a suction well, can easily be analysed by. setting the suction side potential head to the elevation of the suction well level above the datum level of the scheme plus atmospheric head. It is best to use an absolute potential as this simplifies the calculation of the absolute pressures (by deducting the elevation of any point under consideration above the datum level) which are needed in the calculation of e bubble fractions and in checking whether local pressures have fallen to vapour pressure or not. As only the backward characteristic is needed to solve a suction well pump the equations for the in-line pump can be redeveloped in a simpler form. The suction well pump is an important case as it is in common use in sewage pumping schemes. The pump equation is: (6.29) where Z'NI is the elevation of the base of the suction well above datum level, dsw is the depth in the suction well, ha is the height of the liquid barometer, i.e. atmospheric head. The Cvalue used should be the value appropriate to the curve obtained by subtracting the pump station losses (including losses in the suction pipe) from the ordinates of the pump curve.


111

The backward characteristic equation is:

-~ (h -hs)+ v -vs + 2fsvslvsl.1t = 0 CR

p

p

d

(6.30)

As was done for the in-line pump, these two equations can be solved simul-

taneously for Q(vp

=a~

where ap is the cross sectional area of the pipe).

A quadratic solution for Q results. Having obtained Vp the value of hp can be found from the backward characteristic equation. The value of dsw• which is slowly changing, can then be updated. There may be an inflow Qi to the suction well and there will be an outflow, i.e. Q, so

(6.31) where dsw, is the depth in the well at the end of the .1t time interval, dsw 1 is the depth at the beginning and Asw is the cross sectional area of the well. If the cross sectional area varies with depth this variation can be taken into account but it may be thought that a mean area over the depth range will produce a sufficiently accurate result. The variation of suction well depth is important because in most sewage pumping schemes suction well level switches are used to start and stop the pump. The static head of a system may be a large or small fraction of the pumping head. In a long flat main, friction head will be large and static head will be small, and in such a case the variation in suction well level from high to low suction well level will be a large fraction of the static head. The pump delivery will be large at high suction well level, yet if power failure occurred, the pump might turbine, the flow reduction would then occur slowly and the consequent pressure transient would hence be minimal. Conversely, flow delivered by the pump is minimal when the suction well level is at its lowest value as then the static head is at its maximum value. Turbining is less likely to occur in such circumstances so a much larger pressure transient may be generated by pump trip. It is not possible to predict what sort of pump behaviour will occur without performing a complete analysis. It is therefore important that pump switch on at a high electrode level, followed by pump trip at a low electrode level, together with pump trip at any level for the case of power failure, should all be simuiable in any program.

6.11 Four quadrant pump operation A pump is normally equipped with a nonreturn or reflux valve but for very large installations it is not possible to construct an efficient reflux valve. Such very large pumps are used to supply the cooling water of thermal

112


power stations and large scale gas liquefaction plants, for example. The reflux valve is normally fitted to prevent reverse flow through the pump when the power supply to the pump is cut off. If it is possible to fit are· flux valve it is advisable to do so. When the pump starts up after a stop it is very desirable that it should deliver its discharge into an already full pipeline since, if a pump has to deliver into an empty pipeline, it will experience very little opposing head. It will therefore deliver a very large flow and will swing down its characteristic curve, and its power demand will be relatively very large. When the high speed flow that it generates arrives at a hydraulic control a very large transient may be generated. If the pipeline into which such a pump delivers has a sinuous longitudinal profile, flow may occur through pipe segments of large negative gradients in an intermittent manner because large air bubbles will rise upwards against the flow in the pipe sections of negative slope and it may take a long time to clear the pipeline of air. Empty pipelines of complex topography do not fill in a simple manner. Long sections of pipe may contain air whilst others run full. Air, entrapped in a long leg of a pipe network, can be com· pressed and so store a great amount of energy. Upon pump trip, such large volumes of compressed air can rapidly expand causing liquid in other sections of the network to move at high speeds. Such motions can interact to generate pressure transients of surprisingly large magnitudes. Large trapped bubbles can be set into vibration which resonate with flapping reflux valves and then standing waves can be set up in the network. Standing waves of this type are known to have been the cause of catastrophic pipe bursts. It is therefore clear that pumping into empty or part-empty pipe networks should be avoided if at all possible unless allowance has been specifically made for this event. In large pump installations it is usual to trip the pump and then to shut a downstream valve (usually ofbuttert1y type) to avoid emptying the system. In cooling water systems it is not unusual to pump from a water reservoir through heat exchangers and, after cooling, back into the original reservoir. The static head on such a system will not be large but providing that the local absolute pressure does not fall below the gas release head, there will be no tendency to drain the system. However, if local pressures fall below gas release head during the transient phase it is possible that gas will be released and this will collect at high points. This gas will not redissolve readily and will cause problems when the pump starts again. A large pump delivering into a very large diameter main is unlikely to be fitted with a reflux valve and so reverse flow back through the pump will be permitted after pump trip, a downstream valve must then be slowly closed and this will generate transients. Depending upon the time allowed between the pump trip and the valve closure, the pump may continue running forward at a gradually reducing speed and may continue to deliver flow in a forward direction or flow may reverse even while the pump is still running forward. In one of these cases the pump is in its pumping mode ~zone Pl (see figures 6.5a and b). With small negative head and


113

forward flow the pump acts as a dynamometer- zone Dl. If the pump head becomes sufficiently negative the forward flow may start to drive the pump and it then operates in a turbine mode - zone Tl. Once flow has reversed, another series of operational modes arises. With reverse flow and a forward rotation of the pump's impeller another dynamometer phase occurs- zone D2. With reverse flow but with reversed pump direction and a higher downstream pressure than that in the suction pipe the pump operates as a turbine- zone T2. With the same conditions but with large negative speeds it operates as a dynamometer- zone D3. With forward flow and large reversed pump speed the pump operates as a pump zone P2 and with forward flow but with rather smaller reversed pump speed it operates as a dynamometer again - zone D4. It is most unusual for a pump to operate in the fourth quadrant, however.

+N +ve

torqu~ ~~ t \:~~ ~~:e

+ve head normal pumping ---r-~

~

+ve torque - ve head

Positive torque, positive head, positive Nand positive 0 are a II as for normal pumping (a)

Figure 6.5

There are eight different modes of behaviour that a pump can exhibit spread over the four quadrants of theN- Q graph. If a standard N- Q plot is used, four quadrant pump characteristics are complex and difficult to use in any computer model of a pump's behaviour. Suter 15 produced a dimensionless representation which is very well suited to computer use. The following presentation is taken from this paper.

114


0 Q* 200%

HIS head

0 IS flaw N 1s speed • denotes pump duty values

(b)


If dimensionless variables WH and WT are used where WH is given by WH

= sign(H)

H/H* (N/N*)2 + (Q/Q*)2

(6.32)

WT

= sign(T)

T/T* (N/N*) 2 + (Q/Q*) 2

(6.33)

and WT by

where affix * denotes steady state pumping conditions, His the head across the pump, Q is the flow, N the rotational speed in rev min- 1 , Tis


115

the torque on the impeller shaft then graphs can be constructed of WH and WT against 8 where 8 is given by N ·Q*) e = arctan (N* Q

(6.34)

i.e. 1.2

-1.2

Figure 6.6

These two graphs provide a complete representation of the very much more complex complete set of characteristic curves of a pump over its four quadrants of operation. It is relatively simple to read in to the computer, say, 64 points on the WH and WT curves and to achieve a quite accurate representation of them. By interpolation a particular value of WT or WH corresponding to a particular value can be readily obtained. From the equations quoted before (6.32 and 6.33) these values of WT and WH can be converted into T and H values. The torque value T can be used to calculate the running speed of the pump at a D.t time period later and the value H, i.e. the head across the pump, can be used in the boundary condition calculation representing the pump. Before describing these calculations the problems a network analyst will encounter in obtaining the necessary data concerning four quadrant pump operation are discussed. Very few pump manufacturers are prepared to supply the four quadrant characteristics of their pumps. It is most unlikely that anything other than the normal H ~ Q, P ~ Q and E ~ Q curves of the pump in normal pumping mode will be available, so it is necessary to devise a reasonable approximation to the Suter curve for the other seven modes of pump operation. Three pumps were tested by Donsky (see. Bibliography (2)) in all modes of operation. These three have a very wide range of specific speeds between them so the analyst should find that the specific speed of the pump he is proposing to use lies within the bracket of specific speeds covered by these pumps. The specific speeds of the three pumps involved are 35 m 3 s- 1 units

e

116


(radial flow), 147m3 s- 1 units (mixed flow) and 261m 3 f 1 units (axial flow). The Suter diagrams for these pumps are given in figure 6.7 and the data is shown in table 6.1. The portion of the characteristic curves on the Suter diagram that will be obtainable from the pump manufacturer's H ~ Q curve will lie in the first quadrant, i.e. fore between rr/4 and rr/2. If this very limited piece of information is plotted onto the Suter diagrams it will be found to lie between two of the specified WH curves. Table 6.1

I I

N 5= 261

N 5= 147

N 5 = 35

Radian

Wfi

WT

WH

WT

WH

WT

0 0.168 0.318 0.464 0.588 0.695 0.785 0.876 0.983 1.107 1.249 1.406 1.571 1.736 1.893 2.034 2.159 2.266 2.356 2.447 2.554 2.678 2.820 2.976 3.142 3.307 3.463 3.605 3.730 3.836 3.927 4.018 4.124 4.249 4.391 4.547 4.712

-0.728 -0.639 -0.445 -0.179 +0.398 +0.576 +0.707 +0.806 +0.904 +0.992 +1.069 +1.120 +1.136 +1.129 +1.102 +1.107 +1.039 +1.010 +0.997 +0.979 +0.947 +0.930 +0.901 +0.876 +0.831 +0.789 +0.754 +0.727 +0.710 +0.709 +0.711 +0.721 +0.740 +0.764 +0.788 +0.801 +0.794

-0.548 -0.394 +0.095 +0.400 +0.545 +0.644 +0.707 +0.745 +0.772 +0.785 +0.771 +0.725 +0.663 +0.608 +0.585 +0.587 +0.606 +0.661 +0.721 +0.777 +0.831 +0.885 +0.926 +0.940 +0.927 +0.887 +0.828 +0.743 +0.654 +0.565 +0.480 +0.376 +0.263 -0.155 -0.379 -0.600 -0.819

-1.249 -1.048 -0.789 -0.529 +0.186 +0.555 +0.707 +0.791 +0.881 +0.984 +1.094 +1.216 +1.400 +1.450 +1.479 +1.505 + 1.536 +1.573 +1.624 +1.674 + 1.703 +1.725 +1.700 + 1.620 +1.473 +1.247 +0.996 +0.785 +0.644 +0.528 +0.624 +0.335 +0.204 -0.310 -0.502 -0.669 -0.819

-1.249 -0.951 -0.651 -0.297 +0.447 +0.630 +0.707 +0.761 +0.807 +0.853 +0.939 +1.071 +1.217 +1.240 +1.244 +1.274 +1.308 +1.381 +1.442 +1.535 + 1.594 +1.650 +1.658 + 1.580 +1.450 +1.235 +1.018 +0.815 +0.622 +0.428 0 -0.414 -0.564 -0.709 -0.843 -1.030 -1.225

-0.707 -0.935 -0.828 -0.632 -0.276 +0.468 +0.707 +0.896 +1.043 +1.187 +1.348 +1.506 +1.652 +1.784 +1.864 +1.891 +1.873 +1.803 +1.809 +1.689 +1.576 +1.4 70 +1.350

-0.748 -0.776 -0.736 -0.559 +0.144 +0.550 +0.707 +0.787 +0.861 +0.951 +1.102 + 1.275 +1.400 + 1.520 +1.627 +1.713 +1.741 +1.716 +1.660 +1.596 +1.477 +1.342 +1.201

+1.040 +0.887 +0.839 +0.785 +0.680 +0.510 +0.255 -0.407 -0.645 -0.829 -1.013 -1.228 -1.480

I

+0.818 +0.646 +0.644 +0.710 +0.610 +0.326 -0.274 -0.570 -0.763 -0.938 -1.082 -1.240 -1.526

I

117

Boundary conditions: pumps 2.0

-2.0 (a)

2.0

(b)

2.0

¥

0+---~------~----------~--~~,--.--~

/

~

f

N

~

-~

~H Wr

-2.0 (c)

Figure 6.7

The torque - flow curve can be deduced from the power and efficiency curves supplied by the manufacturer and this information can also be plotted onto the Suter diagram and will be found to lie between the WT curves of the specified pumps. It is suggested that these short segments of the WH and WT curves of the pump under examination can be completed for the entire 0 - 2n range of e by interpolating between the two adjacent WH curves and WT curves. This process is not free from risk. It is possible for two pumps to have the same specific speeds yet have very different impeller geometries so it would not be reasonable to expect the Suter diagrams of two such

118


pumps to be identical even though their specific speeds are the same. The only way to be completely sure that the interpolated section of the Suter diagram is correct is to obtain further information about the pump's behaviour in another range of e but this may well be very difficult to do. If no other information is available then the analyst may be able to find a full set of characteristic curves for a different pump which has a similar geometry and this information 'can be used to plot another set of Suter diagrams. The shape of these curves can be used to guide the con· struction of the required Suter curves for the pump under examination. Some sympathy is felt for the pump manufacturer in this situation. The operation of pumps in four quadrant conditions usually only occurs in the case of large pumps. The test of a pump operating in four quadrant circumstances requires two identical pumps; flows and powers are very large and the test rig is very big. For a manufacturer to supply a full set of pump characteristics in such circumstances involves him in the expenditure of a considerable amount of money and his reluctance to do so is understandable. However, the analyst's position is not enviable. If the manufacturer were prepared to perform four quadrant tests on model pumps of the same specific speed as the prototype (which will probably be included in his selling range anyway) and to then apply standard scaling laws to the results of such tests, an acceptable solution to this problem might be possible. 6.12 The use of the Suter curves The way in which the Suter diagrams are used will now be described. First the two curves, the WH and the WT against e curves, for the pump must be represented in the computer by two arrays (sequences of numbers). It is suggested that the coordinates of sixty-four points on each curve should be used and a simple interpolation process be used to establish intermediate values. At steady stateN= Ns and Q = Qs so e =*,i.e. arctan( I). The then current values of WH and WT can be obtained from the stored arrays. Obviously H will equal Hs and T = Ts, the motor torque Tm will suddenly become zero so in the torque equation

Tm- Ts = /Q so

Tm will equal 0 at pump trip

-Ts = JQ n = -TJI 21T 60 (~ -Ns) _Ts b.t I

So

N = N. _ 60 7'sb.t I s 21T I

{6.35)

119


At the beginning of the pump run down the head across the pump will beH=H5 • The forward characteristic equation can then be applied to the last dx length of the suction pipe and the backward characteristic equation to the first dx length of the delivery pipe i.e.

2fRvRivRIM_ 0 ) CR( (h p1 - h) d R +- Vp1 - VR + CR ~ su g

(6.36)

2fsvslvsl~t 0

(6.37)

cs( (h P - h s) -Vp

and

g

2

2

-vs ) -cs

gddel

where dsu = suction pipe diameter and dctel = delivery pipe diameter. Now hp 2 - hp 1 =H so by subtracting equation 6.36 from 6.3 7 then substituting

*

the known value of H for hp2 - hp1 , substituting Q 1/( Q 1I

d~)

for Vp1 and

(~dJe1) for Vp 2 gives the value of Q 1•

M

R

5 N

0 pump

I

/':,.x

/':,.x

Figure 6.8

Next (} can be calculated from

Qs)

Nt 01 = arctan ( Ns Q 1

(6.38)

From the stored arrays of WT and WH the values of WH, and WT, relevant to 81 can be obtained by interpolation. From equations 6.32 and 6.33

and Using

11

11 = Ts w~ ( (Z~r + (g~r)

(6.39)

hl =Hs wJ, ((Z~r + ( g~r)

(6.40)

N2 = N1 _ 60 T1 ~t 2rr I

(6.41)

Using /ft, the value of Q2 can be found and the above process repeated as often as required.

120


6.13 Pump run up to steady pumping speed The approach described in the preceding sections is applicable to the four quadrant operation of a pump after its motor has been switched off (i.e. pump trip). If it is wished to deal with the case of the run up to steady speed of a pump starting from rest, it is necessary to possess the torque (Tm) ~speed (N) curve of the motor. This torque~ speed curve must be stored in the computer as an array and an interpolation technique used to calculate the torque supplied by the motor at any particular speed. The torque available to accelerate the pump impeller and the rotating parts of the motor will then be Tm- Tp where Tm is the torque applied by the motor and Tp is the torque of the pump at the particular running speed. Tp is the torque obtained from the Suter diagram.

Tm- Tp =/D.

Then

(6.42)

n= Tm- Tp

so

I

2rr(N2 - N 1 ) 60.M N2

= Tm- Tp I

30 (Tm- Tp)ill

=N, +n

I

(6.43)

The method of dealing with the analysis is essentially the same as that used for four-quadrant pump trip except for the substitution of equation 6.43 for equation 6.41. 6.14 Pumps with by-pass valves In-line pumps are often equipped with a by-pass reflux valve (see figure 6.9). The advantage of fitting such a by-pass reflux valve is that in a long pipeline there may be a number of pump stations at roughly equidistant points apart. When the demand on the pipeline is low some of the pumps may not be used whilst the others remain operating to supply the required flow. When such pumps are off-line the flow automatically passes through the by-pass but the reflux valve is fitted to prevent backwards flow through the by-pass when the pump is operational. Alternatively a gravity flow system may require a booster pump to provide peak demand. Such a pump will not be operational for much of the time and flow will then pass through the by-pass under the gravity head. The by-pass has a further advantage. When a pump trip occurs, there will be a drop in head downstream of the pump. If this head falls below the upstream head the by-pass valve will open and supply flow. The head downstream of the pump will then not fall to such a low level as it would have done if no by-pass had been present and the consequent waterhammer


-

121

-

Figure 6.9

will be greatly reduced. The effect is similar to that which occurs when a pump acts as a turbine during the low pressure phase following upon pump trip. The analysis of this circumstance is relatively simple. During the analysis of the pump's behaviour (see section 6.12) a check should be made that the upstream head remains less than the downstream head. If it becomes greater than the downstream head then, whilst the pump speed run down calculations should be continued, the head- flow calculations should be replaced by calculations which ignore the presence of the pump and treat it as if it were a simple joint in the pipeline, i.e. the two characteristic equations should be written and solved for head and velocity as if no pump were present. If the full open reflux valve can create a local energy loss this can be included by either lumping the loss into friction or, more accurately, writing it as a separate head loss term, e.g. (6.44)

_ _!_ (hp -hs) +(up -us)+ 2/suslusiM = 0

cs

2

ds

2

(6.45)

2

hp -hp =K~ 1

2

2g

(6.46)

K% is the head loss caused by the open by-pass reflux valve. 2

Now These equations can be reduced to a quadratic equation in Q which can readily be solved. hp 1 and hp2 can then be obtained by back substitution into equations 6.44 and 6.45. This approach neglects the flow through the pump after pump trip but this is usually small.

6.15 Pump stations It is unusual for only one pump to be installed in a pumping station. Usually there are at least two, one being held in reserve for use if the other should

122


break down. More often there are more than two pumps, e.g. three or four, and usually a number of these operate with, say, two on standby. When modelling a group of pumps they can be treated as a single pump. The characteristic of the group has to be calculated so that this can be used as if it were that of a single pump. Groups of pumps in a station usually operate in parallel and in this case they should be identical if stable, satisfactory operation is to be obtained.

Pumps in parallel Identical pumps operating at the same speed should have identical H ~ Q and Eff ~ Q characteristics. The flow Q that the group produces is the sum of that produced by the individual pumps. If there are n pumps in the group then the equation of the equivalent pump is:

He =A'N2 + B'NQ- C'Q 2

(6.47)

A'=A B'=B/n

where

C'=

C/n 2

The inertia of the pump group can also be calculated. As far as an individual pump is concerned it is delivering its normal rated flow at steady state and is unaffected by the operation of the other pumps in the group. When the pump group is switched off the speed of the individual pump will decay in exactly the same manner as if it were operating alone but each one is only pumping 1/nth of the fluid. The inertia value to be used to represent that of the group must, therefore, be that of an individual pump multiplied by the number of pumps in the group. The efficiency of the group will be the same as that of the individual pumps and the efficiency equation quoted previously can be used but, for the group, the flow Q will be replaced by Q/n, i.e.

_ A 'N 2 + B'NQ- C'Q 2 Effe -

Q

DeN\1 +EeN

so

D~ =De/n

and

E; F;

Q)3 (Q)2 n +Fe(n

(6.48)

= Ee/n 2 = Fe/n 3

where De, Ee, Fe are the constants for an individual pump and n;, E~ and F; are the constants of the pump equivalent to the pump group.

Pumps in series If the pumps in a pump group are arranged to operate in series the total flow is the same as that through any one pump but the head is the sum


123

of the heads of all the pumps. The pumps do not need to be identical for satisfactory series operation. Thus the H- Q equation for a series arranged pump group is: (6.49)

where "f.ApNp2 =A 1 ~ 2 + A2Nl + A3Nl + ...

=B1 N1 + B2N2 + B~3 + ... r.cp = Ci + C2 + c3 + ...

"f.BpNp

If the pumps are running at different speeds (an unusual arrangement) the equivalent pump which represents the pump group will store kinetic energy within its equivalent rotating mass so where

! I'N2 = r,t:fpNp2 r.tip~ =t(/,~2 + I2Nl + hNl + ... )

I'N 2 = "f.IpNp2 I'= "f.Ipfp2 N

so

(6.50)

where I' is the equivalent pump's moment of inertia and N is the equivalent pump's running speed. A value of N can be chosen arbitrarily, say the speed of the fastest pump. Then the equivalent head- Q equation will be: He =A'N 2 +B'NQ-C'Q 2

where

A'=

"f.ApN~ N2

B' = "f.BpNp N

and

C'= "f.Cp

The efficiency of a series pump group is given by Effe =He

L it· l->n

P

Compound pump arrangement It is possible to arrange a group of pumps in parallel and to then connect this group to another group which is arranged in series. By obtaining the equivalent pump to the parallel group and the equivalent pump to the series group, and then combining these two groups in series, an equivalent pump to the compound group can be obtained.

124


Tripping of a subgroup of pumps in a pumping station It is possible to write a pump station subroutine/proc edure in which the equivalent pump's characteristic curve, its inertia and its efficiency curve are changed suddenly (when the time counter i in the expression t = i x dt exceeds a preset value) due to the trip of a subgroup of pumps within the main group. It will not be necessary to model the run down in speed of the tripped subgroup because the continuing delivery from the untripped pumps will prevent the development of transients of any significance. Writing such a subroutine/proc edure is quite simple, the values of A', B', C', n;, E;, F~ and!' being calculated for the entire group if i is less than the preset value and for the group of running pumps only if i is greater than the preset value. Alternatively the group of pumps in the station can be divided into two subgroups each of which is treated as a pump station, see figure 6.1 0. The pipe network is slightly more complicated due to the introduction of

Figure 6.10

the additional .:::lx lengths. If station 1 represents the group of pumps which will continue running and station 2 represents the group that is to be tripped at a preset time then calculations of the characteristic curve constants, etc, of the equivalent pumps can be performed without complication and station 2 can be tripped at the appropriate time. The run down of the equivalent pump will be modelled by this method but it may be thought that the introduction of the additional .:::lx lengths offsets the advantage that this represents. 6.16 Surge suppression of transients generated by pump trip Two main methods of surge suppression are available:

(1) Fitting flywheels to the pump. (2) Fitting air vessels or surge tanks to the pipe just downstream of the pump.


125

By fitting a flywheel onto the shaft of the pump, the inertia of the pumpset can be greatly increased. This means that the pump run down in speed will occur over a longer time so the decrease in the pump delivery will take longer also. If this reduction in the pump's delivery can be made to occur over a sufficiently long time without requiring an excessively large flywheel a practical method of surge suppression is available. The pump delivers for the period during which the closed valve head of the pump, AN 2 at the current reducing value of N, is greater than the static head on the system. If this period is greater than the pipe period of the delivery pipe (2L/c) some surge suppression will be obtained. Obviously the larger the flywheel the longer the delivery period and the greater the surge suppression resulting. Flywheels are expensive and there is a limit to how large they can be made economically so this method of surge suppression is only useful when pipes are relatively short with correspondingly short pipe periods. The delivery period during the pump run down can thus be made large relative to the pipe period, only if the pipe is short. To analyse a flywheel case is easy however, as it is only necessary to increase the pump set's inertia by that of the flywheel and then carry out another computer run. It has been the author's experience that it is necessary to do about four runs using an initial value of the flywheel's inertia of zero and increasing this up to the largest practicable size over the next three runs. By plotting the maximum (and, if critical, the minimum) pressure head in the pipeline against the flywheel inertia the following graph results.

E :J

·~

"E flywheel inertia

Figure 6.11

The horizontal section of the graph in figure 6.11 is due to the fact that up to a certain critical flywheel inertia value the delivery period during pump run down is Jess than the pipe period so it has no effect upon the maximum transient pressure head. This is similar to the difference between sudden and slow valve closures. From figure 6.11 the maximum acceptable pressure head determines the necessary flywheel inertia. The maximum pipe length which can be economically surge suppressed by the use of flywheels is about 1-2 km, depending upon the pipe's distensibility. For long pipelines it may be found necessary to fit an air vessel or surge tank. Surge tanks can only be used if the delivery main is relatively flat and

126


the peak head at the location of the tank is not excessively large. A rough guide is that the top of a surge tank located just upstream of the pump will have to be at a height above Ordnance datum equal to the height of the liquid surface in the suction well above Ordnance datum plus the closed valve head of the pump if the tank is not to spill. In practice this height can be reduced somewhat but this rule allows the designer to determine if a surge tank can be considered at all. Air vessels have been discussed in previous chapters but in this context it is necessary to point out that they are fairly expensive, being required to withstand internal pressures which may be quite large and if located at an elevated point they may also be required to withstand subatmospheric pressures. An air vessel which is subjected to internal subatmospheric pressures may need to be internally braced if it is not to fail in a buckling mode. They must be equipped with compressors and the entire installation must be regularly maintained so that air dissolved by the water is replaced regularly. Whereas flywheel surge suppression always works, air vessel surge suppression only operates if the liquid level in the vessel is set correctly by automatic devices or at frequent maintenance inspections. A third method of obtaining surge suppression is to fit a by-pass around the pump. This by-pass pipe is usually arranged to permit water to flow from a point just downstream of the pump via an electrically controlled valve back into the suction well. Normally, the valve is closed during pumping but when pump trip occurs a solenoid operates to open the valve. If the pressure drops sufficiently after pump trip, water will be drawn through the by-pass into the pipe so minimising the size of the pressure reduction. The initial transient generated by pump trip will return to the pump after reflection at the downstream end and this transient would normally reflect positively if the by-pass valve were closed. If the valve is open, however, the transient will be greatly reduced in magnitude and a reverse flow will be generated; the slow closure of the valve will arrest this flow without development of significant transients providing that the closure is slow enough. This type of valve would be used more frequently if it were 'fail safe'. However, depending as it does upon an electricity supply it is not 'fail safe' as pump trip occurs due to a failure of the electricity supply as well as to normal pipeline operating procedures. However, a valve of this type, actuated by compressed air, is available and should be virtually 'fail safe' if regularly maintained. Subroutines/procedures written to describe the operation of hydraulic controls can only be used at the beginning or end of a pipe length, the smallest value of which is the ~x value chosen. Thus if a pump is to be equipped with a surge tank or air vessel it would normally be necessary to interpose a ~x length between the pump and the air vessel (see figure 6.12). In many cases where the ~x value is small this method would be acceptable but if the system is long, say 10 km, and the ~x length is to be 1 km, the use of a length of 1 km to represent a distance of perhaps 10 m would lead to very bad modelling.


127

air vessel

Actual pump/air vessel arrangement

Simulation of pump/air vessel arrangement

Figure 6.12

A case can therefore be made out for developing a subroutine/procedure which would model the pump and air vessel as one unit. The methods already described for modelling pumps can be combined with the method given in the next chapter for modelling air vessels to give a description of the behaviour of the pump/air vessel combination. In long pipelines in which booster pumps are fitted it may be desirable to fit a suction side and a delivery side air vessel. These can be modelled by the method outlined in section 7.3 for the pump/delivery side air vessel. Surge tanks are really only a special case of air vessels. As will be described later, the gas compression/expansion process in the air vessel is usually assumed to be polytropic, the index being taken as 1.2. If this index is given the value zero, the equation

p V 1.2

constant

p

constant

becomes

which is the case in the gas above the surface of a surge tank, the pressure there being atmospheric pressure. If the height of the air vessel is made very large no compression or expansion of the gas is possible so, either making the polytropic index of the gas process zero, or making the air vessel's height very large, or both, will give the required conditions for describing a surge tank of simple type. 6.17 Line pack and attenuation When a hydraulic control such as a valve operates at the downstream end of a pipeline causing the flow there to fall rapidly to zero, a steep wave

128


front travels in the upstream direction. As it reduces the velocity of successive layers of fluid, the pressure of these layers rises and this rise is transmitted downstream through the near-stationary fluid maintaining its elevated pressure. However, the pressure of the fluid before it experiences the effect of the wave is higher than the initial pressure of layers of fluid downstream of it due to friction, and this higher pressure is also transmitted downstream, together with the rise caused by the momentum effect of the velocity change. Therefore, as the wave progresses upstream the pressure at the downstream control continues to rise even after the initial steep fronted rise has been generated. The fluid in the pipe between the wave and the downstream control thus experiences a progressively increasing pressure which causes it to compress and the pipe wall to distend, so, even though the velocity immediately downstream of the wave is zero at the control, it increases as the wave travels upstream in order to supply the fluid that must continue to flow into the downstream section to fill the space made available by the fluid compression and pipe distension. The velocity change across the wave is therefore reduced so the wave height itself decreases. The phenomenon of pressure increase after the wave has passed is called line pack and is commonly found in long pipelines, especially in oil pipelines. The phenomenon of the reduction of the pressure wave magnitude as the wave travels upstream is called attenuation.

6.18 Lock in If a pipeline has a valve at its downstream end and a pump equipped with a reflux valve at its upstream end a phenomenon called lock in occurs when the downstream valve is closed. The valve closure causes a positive wave to travel upstream towards the pump. When it reaches the pump the progressive flow reduction causes the pump's operating point to move up along the pump's H- Q characteristic, the pressure rising as it does so. A reflection will thus be generated due to the interaction of the pump with the incident wave and this will travel downstream to the closed valve where it will positively reflect and travel back upstream again to the pump. Throughout the time of travel of the wave down to the valve and back again the pump will still be delivering a reduced flow into the pipe which is closed at its downstream end. Eventually, after one or more pipe periods the magnitude of the pressure downstream of the pump will become greater than the no-flow head of the pump and the reflux valve will then close. The complex wave system in the pipe will then travel backward and forward along the pipe attenuating as it does so. During this period the pressure at the pump may fall below its no-flow head for short times and the pump will deliver more fluid into the pipe. Eventually flow will cease at all points in the pipe and a pressure head will exist throughout the pipe


129

which may be significantly higher than the no-flow head of the pump. If the reflux valve and the downstream valve do not leak, this pressure will remain in the pipe and this is the lock-in pressure. The lock-in pressures that can occur in long pipelines can be large. The author has encountered lock-in heads as large as 1.5 times the no-flow head and in long pipelines this can be a dangerously high value. If the downstream valve does not seal perfectly, lock-in pressures of dangerous magnitude can be avoided as only small quantities of fluid need be leaked to reduce pressures sufficiently. Such a leaking valve can be modelled using methods illustrated elsewhere in this text (see page 136) by reducing the K value of the closed valve from a very high to a lower, but still high, value. The phenomenon of lock in is automatically analysed by the method of characteristics technique but not by the Schnyder Bergeron graphical method. These phenomena can all occur in any pipeline. Obviously, line pack will only be detectable in relatively long pipelines and lock in is also unlikely to be a matter of concern in short pipelines. They will be predicted by the characteristic method even so.

7 Other boundary conditions

7 .1 Junctions As stated earlier it is vitally necessary to have a subroutine/procedure which is capable of describing an 'n way' junction if a program is to have any generality. An 'n way' junction is one in which n pipes join at a point in the network. It may be thought that it is only possible physically to join a few pipes at one point, say four or five, but often pipes are joined at a manifold. The author has encountered a circumstance in which 22 such pipes joined at a manifold. Also pipes may join at a simple junction, followed by another junction connected to the first by a short length of pipe, and so on. None of the short lengths between such junctions may be long enough to be considered as a Llx length so it is logical to combine these distributed junctions into one large junction and this will give the best modelling. It is usual to ignore local losses at junctions. Their effect can be allowed for by increasing pipe lengths slightly or by increasing pipe roughness but in liquid flows their effect is generally small. If they are included, the junction procedure becomes very much more complex and expensive in computer run time. At a junction, continuity of flow must be maintained and the head must be the same for all pipes joining there. The continuity equation can be simply written, i.e.

(7.1) where aPa is the cross sectional area of the ath pipe and vPa is the velocity at the junction end of the ath pipe at the end of the Llt interval, and n is the number of pipes joining at the junction. A convention has been adopted here, i.e. flow towards the junction is positive and flow away from it is negative. Next consider the dynamic circumstances at a junction. If the junction

130


131

is located at the downstream end of a pipe (i.e. flow in the pipe is assumed to be towards the junction) then a forward characteristic line can be drawn from some point in the last ~x interval in the pipe length and, similarly, for pipes for which the junction is located upstream, a backward characteristic can be drawn towards the junction from some point in the down· stream ~x length (see figure 7 .I). In figure 7 .I if the pipes associated with the junction are numbered with positive sign if flow in the pipe has been assumed to be towards the junction and negative sign if flow has been assumed to be away from the junef

axis

No pipe may be assigned a zero number

Figure 7.1

tion, then an integer numbers can be obtained from this numbering convention which will either take the value +I if flow is towards the junction or -1 if it is away from the junction; sis calculated from the following equation s = sign(a) where a is the signed pipe number and s is the required integer number. This s value can now be used to define the characteristic directions to be used for each pipe i.e.

(7.2) where a denotes the ath pipe and Va, Ca, ha, fa represents the interpolated values in the ath pipe, vPa denotes the velocity at the junction end of the ath pipe at the end of the ~t interval and dais the diameter of the ath pipe.

From equation 7 .I LSaPa vPa = 0 (s has been introduced in this equation to preserve a positive value of

132


vPa in a pipe carrying fluid away from a junction, and remember that s 2 soL saPavPa = 0 1-->n

As hp is the

L

=

saPaVa- L~a aPahP + L;a aPaha

1-->n

sar.~e

=1)

1-->n

1-->11

for all pipes

(7.3)

This equation is very well suited to computer solution. All values on the right hand side are known and hp can be readily calculated. Then values of vPa- one for each pipe - can then be found by back substitution into equation 7 .2. It can be seen that there is no limitation on the number 11. The junction has therefore been solved. 7.2 Joints Joints consist of two pipes joining end to end. A joint is obviously a twoway junction. However there is a good reason for writing a separate joint subroutine/procedure. If two pipes meet end to end, 'two-way junction' can be used but if there should be a reflux valve interposed between the two ends, reverse or forward flows will be preverted according to the direction in which the reflux valve is fitted. Also the use of 'two-way junction' would be rather more expensive in computer run time than a straightforward 'joint' procedure as it involves slow processes such as 'for' clauses (in Algol) or 'do' loops (in Fortran). As there are many joints in most networks there is a good case for using a joint procedure in any transient analysis program. Assume that the reflux valve causes trivial losses when flow is occurring. through it. The forward characteristic equation describes conditions along RP

(7.4)


133

-N=*--------kJ~---

M

R

5

JOint

Figure 7.2

N

Figure 7.3

and along SP the backward characteristic equation applies:

-L(h -h )+v -v + 2fsvs lvs l.0.t=O ds s p, s cs p,

( 7 _5 )

Note vp, may not equal vp, if the pipe diameters are not equal, i.e. dR =I= ds but hp is the same for both pipes if local losses can be ignored and vp, is not zero. From equation 7.4 CR 2fR VRivRiflt CR hp =hR--(vp -vR)--x~"--d:.:-;:_:__R g I g ' and from equation 7.5

h p -_ h S + -cs ( Vp - VS ) + cs 2

,

g

g

X

ivs lilt 2fsvs --=--==:.....:;-------"-ds

(7.6)

cs2fsvs I vs I .0.t gds CR 2fR VRIVRI!lt

gdR 2fsvs ivs llltcs h -h + cRvR + csvs gds g S R

g

2fR vR lvR llltcR

-~~~~-~

gdR

(7.7)

134


At this point in the calculation it is necessary to check what type, if any, of reflux valve is located at the joint. If none the calculation of vp, stands. If a reflux valve is fitted that only permits forward flow then a test must be made, i.e. if Vp, < 0 then Vp, = 0. lf a reflux valve that only permits reverse flow is fitted then the following test must be made: If vp, > 0 then vp = 0 After the value of vp, has been so adjusted Vp 2 can be calculated i.e. v

Then and

h

p,

=h

h

P2

R

p2

=

a

aP2p, v p,

_ CR (v _ v ) _ 2fR VRivRILHcR g p, R gdR

= h

s

+ cs(v g

P2

_ v ) + 2fsvs Ius l.::ltcs s gds

(7.8) (7.9)

Although hp1 will equal hp 2 if the reflux valve is open hp, will not equal hp 2 if the valve IS closed so both values of hp, and hp 2 must be calculated separately in the latter case. 7.3 Air vessels A facility must be provided in the program that permits any number of air vessels to be located anywhere in the pipe network. The basic method of analysing an air vessel used in chapter 3 to demonstrate how mass oscillation of air vessels is performed will be used here. There is a difference, however, in that in this presentation an account of water hammer will be kept in the pipes leading to and from the tank although the propagation of waves up through the tank itself will not be included. This limitation is unavoidable if run times are to be kept within reasonable bounds. The path length of a wave from the junction of the air vessel with the pipe to the free surface in the air vessel is very small, much smaller than .::lx. If .::lx is reduced to approximately equal this distance the run time of the program will become unacceptably large. The analytic technique offered below is thus a combination of quasi-steady analysis of the tank with a full waterhammer treatment within the pipe network. The passage of transients through the tank will not be demonstrated by this method but these are heavily attenuated by the tank anyway and are trivial in magnitude. However, they can be calculated using developments of techniques already demonstrated. In figure 7 .4/t is the longitudinal length of the tank, dt is its internal diameter, hb is the height of the base of the tank above the centre line of the pipe, Zt is the elevation above datum of the centre line of the main pipeline at the point of its junction with the tank.

135


Figure 7.4

At the beginning of the analysis the initial height of the water in the tank and the initial potential' head at the junction must be known. Denote these two values by hq andhpj· The pressure in the air in the tank expressed in height of an equivalent water column can be calculated. Let this equivalent water column height be denoted by hwr Then hwi =hpj- Zt- hb- htj

This will be the absolute pressure head of the air because hPi is the absolute potential head. Applying a forward characteristic equation to the Ax segment upstream of the tank

_!_(h p -h)+( CR

R

Vpl

_ VR ) + 2fRvRivRIA!_ 0 d I

and a backward characteristic equation to the Ax segment downstream of the tank g (h -cs - h s ) + (v p 2 -vs ) + 2fsvsd2lvs IAt -_ 0 p

The subscripts R and Shave the significance ascribed to them throughout this book. hp will be the same value for both the Ax segments but vp, will not equal Vp 2 because of the flow into (or out of) the air vessel. By continuity

(7 .l 0)

so where At is the cross sectional area of the air vessel and equals

i" dt.

136


~~5 -~~'--------l~----""---::-"-'~IDI

pipe I ----:::-'-'

M

I

pipe 2

-R-=D.-'--x--+1\+-------"D.='--x--------10

\

1--1

.

1unct•on leading to air vessel

Figure 7.5

The depth of liquid in the air vessel at the end of the b.t interval therefore is given by ht2 = ht, + b.ht where h 1 is the depth in air vessel at the end of the b.t interval and h 1, is the deptfi in the air vessel at the beginning of the b.t interval. The new pressure head in the air in the air vessel will be:

l - h

h w2 =( [ t t

)n

h t2lj.

X

h w,.

(7 .11)

where n is about 1.2

(7 .12)

so

By back substitution of hp into the forward and backward characteristic equations values of Vp and Vp 2 can be easily obtained. From the values of Vp, and Vp 2 the flow into the air vessel at the end of b.t interval is readily calculated, the potential head at the junction has already been obtained and the change in level in the air vessel has been predicted. The calculation is an initial value integration and could be improved by iteration if this is felt necessary. A horizontally positioned cylindrical air vessel can be analysed by a similar method but the more complex geometry of the circular cross section makes the problem marginally more difficult.

7.4 The motorised valve A motorised valve may be fitted on any pipe in a network. To analyse such a situation it is necessary to split the pipe into two sections one upstream and one downstream of the valve. It may be assumed initially that the valve is driven by an actuator which moves the valve spindle at a fixed speed. The head loss across the valve can be taken as being given by the equation

v2

h =Kv

2g

(7 .13)


137

where v is the velocity in the upstream pipeline. The value of K can be obtained from manufacturers' catalogues. The value of K so obtained is only valid for the circumstances in which the manufacturer performed the test, i.e. in the situation of the valve fitted into a pipe of a specified internal diameter. Before the valve K quoted can be used in a pipe of different diameter it must be adjusted to suit the different conditions. The loss created by a valve is almost entirely caused by the expansion of the flow downstream of the partly closed valve and is caused by the boundary layer separation occurring there.

-

A, v,

Figure 7.6 The loss is closely approximated by h =(vv-v2)2

(7.14)

2g

v

There is a small loss occurring in the convergent section of the flow, i.e. in the change of velocity from V1 in the pipe to Vv which occurs at the point of the contracted stream emerging from the valve, but this has almost no effect upon the validity of the argument that follows since this loss is also very nearly proportional to Now A 1 V1 = Av Vv= V: where A 1 =area of upstream pipe, A 2 =area of downstream pipe, Av =area of contracted section. Then

vt

A2

A.

Vv =-v.

Av

AI

and so

v2=A2vl

h h

v

v

=(A1 _A1.) Av

A2

2

~ 2g

2 =A•2g2(A2-Av) A2Av v

2

1

The value of K is thus given by

(7.15a)

138


and if A2

=A1 (7.15b)

Denote this value of K by Km, t!1c subscript m denoting that Km is the manufacturer's value. If the valve were fitted into a different size of pipe from that used in the manufacturer's test the following analysis shows how Km .should be modified.

,----. l y Lll I I

I

I

(

.-lh

actuator

\

Figure 7.7 Again the head loss can be regarded as being caused by the flow expansion downstream of the valve.

So

h

y

- v4 ) =(vy 2g

2

as before, but in this case v4 is much smaller than it was in the pipe used in the manufacturer's test and hv is correspondingly larger. Vv is the same as that obtained in the manufacturer's test for the same valve setting. Now for the same flow

so

hv =

(~r(A4 A4

1

Av

-I)

2 v32

2g

using appropriate values: A 3 and A 4 in place of A 1 and A 2 in equation 7.15a. Then


139

In all cases of which the author is aware the manufacturer tests his valves in a pipeline of the same diameter upstream as that of the downstream pipe, so A 1 = A 2

~2

4 1 Avhv = {_A3)2 A ( A 2_ 1 \A4 Av

hv =

so

Km v/ 2g

(A3)2 {_A4 - Av)2 Km v3 2 A4

~A2 - Av.

2g

The value of K for the modified pipe diameters is

But

Av

= (ffrn + 1)

from equation 7 .15b

so

and then

K=

(A3) 2(ffrn + 1 - A2/A4) A2

vKm

2

Km

(7.16)

Now, usually the effect produced by a partly closed valve is only significant if K (and hence Km) has become two orders of magnitude greater than (A 2 /A 4 ) and at least one order of magnitude greater than unity so it may well bt: thought that this result can be simplified to

K=(~:r Km or

K = ( d act ) dtest

4

Km

(7.17)

where dact is the diameter of the upstream pipe to be used and dtest is the diameter of the pipe used in the manufacturer's test. However, the result given in equation 7.16 could be used with no particular difficulty if great accuracy is required~ It may be argued that the treatment of the downstream enlargement of pipe diameter as being of sudden type will introduce an error in cases where a reversed taper section is used but it should be remembered that the angle of divergence of the taper must be less than 30° before it will reduce the loss by a factor of less than 15% for an area ratio of the taper of 1 :4 and by a factor of 33% for an area ratio of the taper of 1 :9. As it is unusual to fit valves of such relatively small size into pipelines the error will be less than 10% in most cases and will tend to overestimate the K

140


value. It is possible to include the effect of a taper by introducing a taper coefficient into the analysis, i.e.

hv = cT(AA4 (AAv 1) 3) 2

4-

2

!!:1__ 2g

(7.18)

and for a particular taper CT can be obtained from manufacturer's catalogues. The previous development will then be modified to

(7.19) Having obtained the multiplier to be used to convert the manufacturer's values, an array of K values can be obtained for stroke positions ranging from fully open to fully closed. (If the valve is to be opened instead of closed this array must be read in reverse order.) The stroke positions could be at 1% or 5% of stroke intervals. Consider the case of a valve closing (or opening) at a fixed speed, i.e. the stroke increasing (or decreasing) at a constant rate with time. Initially the valve is full open (or closed) and the stroke position stays at this value, as does its associated K value, until the number of !:lts for which the integration process has been performed exceeds a previously chosen value. The stroke then commences to decrease (or increase) at a previously chosen rate. The stroke position is thus known at every subsequent f:lt value and the associated K values for these !:lt values can be obtained by interpolation between the appropriately arrayed K values already supplied. If a sufficiently large array has been used, a linear interpolation process will be found to work well. The author uses 21 values in the K array giving K values at 5% intervals throughout the valve stroke. If the valve is to close from an initially part-closed position the array values supplied must run from the K value appropriate to the part-closed valve position to the fully closed value. (Note that the fully closed valve must always be infinity but as a computer cannot store such a value it should be assigned a very large one such as 10 20 .) The valve closure rate can be calculated by taking the fraction of the stroke over which the closure is to occur and dividing this by the difference between the time of closure completion and closure commencement. The circumstance can arise in which a two or three-speed actuator is used. Such actuators are employed to provide a rapid closure of the valve over the first portion of the closure (say 80% of the stroke) and then to provide a slow closure of the valve over the last portion of the stroke. It should be remembered in this context that during a very large fraction of the closure of a valve, relatively small reductions of flow occur with the development of only small transients; the effective portion of the closure occurs in the last 10% of the valve stroke. It is possible to describe this circumstance by setting the K values appropriate to 5% time increments, instead of 5% stroke increments, using

141


the K values corresponding to the stroke's increments which are current at these time intervals. In effect this means that the multispeed actuator's operation can be represented by a distorted presentation of the K- stroke curve of the valve. Similarly, the action of an actuator that produces a stepwise valve closure can be modelled by the same technique. The representation of the last step of valve closure constitutes a special problem. The value of Kin the penultimate position in the array will be very large, say 10 000 or 100 000, but the last value will be extremely large (I 0 20 ). This means that an attempt to linearly interpolate between the penultimate and the final value will be extremely unrealistic. It is suggested that for stroke values lying in this stroke range an exponential interpolation process should be used. At the penultimate value it is simple to calculate the gradient of the K curve and this can be used to set one of the constants in the exponential curve. The value of Kat its closed position (10 20 ) can be used to calculate the other constant. The author's firm has also used a logarithmic representation of the entire K array most successfully. By careful choice of the K value at the valve's closed position a leaking valve can be accurately represented. This facility can be important as a leaking valve can be used to prevent lock in. (See p. 128 for a description of this phenomenon.) Having established a technique for finding K at any position of a closing valve it is possible to calculate heads upstream and downstream of the valve together with velocities upstream and downstream of it. These vela-

+

+

~

---------"'::-5-+ -+--..,R.&.~------1 N,

N2

Figure 7.8

cities are not necessarily the same since pipe diameters either side of the valve are not necessarily equal. (7.20) g --(h -hs)+(v p, -vs)+ cs P2

2f:svs Ius ldt =0 ds

2

also

(v ) . hp, - h p2 -- K Vp, 2g stgn 11

but

CR (2/R VR I VR ) CR ( VR . - hp -hR-- Up dR g I g '

(7.21)

(7.22)

I

dt)

142 and

Hydraulic analysis of unsteady flow in pipe networks hp = hs + ~(v _ vs) + cs(2fsvs I vs I dt) , g p, g ds .

_ 2£:RfR vR I vR I dt _ 2csfsvs I vs I dt gdR gds 2 . Kvp ( -1 sign v )-h - R - h s - (cRvp 1 +csvp) 1 + cRvR+csvs __:_:____:_:______::::..__::: 2g PI g g

_ (2fR vR 1 vR 1 cRdt + 2/svs I vs I csdt) ( 7 .23 ) gdR gds 1r

Now so

2

4dR Vp 1 cR "•• +csvp, • g

1r

=4ds

t•

2

Vp 1

+cs g

(~f)vp,

Avp, 2 + Bvp, + C =0

(7.24) (7 .25)

where A= K sign (vp.) 2g

Then

v p, =

and

-B + yB'l- 4AC 2A

Vp 1 = (dRf dS Vp 1

(7.26) (7.27)

By back substitution into equations 7.20 and 7.21 hp and hp can be calculated. VR, Vs, fR,fs, CR and cs are calculated by ~ethods ~lready described. 7.5 Servocontrolled valves This is a most important type of valve. As the current methods of operating pipelines develop, this type of valve will be used more and more frequently.


143

It is a valve which is of exactly the same type as the motorised valve described earlier but it is controlled by a servomechanism which is operated by pressure transducers located elsewhere in the system. Frequently these transducers are located immediately upstream or downstream of the valve but quite often the transducer is located on another pipe and may be far away from the valve it is controlling. Usually the valve is required to commence closing/opening if the pre sure at the transducer rises above (or falls below) a certain critical value. If the pressure continues to rise (or fall) the valve will continue its movement in an attempt to regulate pressure fluctuations but at another critical pressure the valve will be completely closed (or full open). The valve thus opens or closes as the pressure at the transducer fluctuates within the defined band of pressure. If the fluctuations are slow the valve will move in sympathy with the pressure fluctuations but if the pressure fluctuations are rapid it may not be able to do so and then the valve will be altering its stroke in a manner which will be out of phase with the pressure transient. (Note that such behaviour can cause resonance see chapter 9 for a description of this effect.) Let the valve's stroke at any instant be Sj. Let the pressure head sensed by the transducer at that instant be htr- The servomechanism will calculate that the valve's stroke for such a pressure should be:

(7.28) where Sreq is the required stroke, c 5 is a constant which converts the amount by which the transducer head exceeds the lower critical pressure head of the valve into the required valve stroke. If Sreq is not the same as Sj the actuator will start and the valve stroke will increase or reduce at the valve's stroke rate in an attempt to reach the required stroke Sreq· However, during the 11t interval it may or may not not be possible for the valve to reach the required position so the stroke at the end of the dt interval will become either Sreq or Si ± Sr x dt where sr is the amount by which the stroke can be altered per second by the actuator, i.e. the stroke rate. Denoting this value by Sf it is now possible to calculate the relevant K for the valve and then to solve the head upstream and downstream of it together with the velocities in exactly the same manner as for the motorised valve. The ± sign in the expression quoted above for the value of Sf appears because the valve might be opening in which case the + sign is applicable, or closing, in which case the -sign applies. This means that if the value of cs(htr - hcrit) is greater than Si the +sign should be used and if it is less than si the negative sign should be employed.

So

Sf= Sj +sign (htr- her it) x Sr x

11t

(7.29)

unless this value is greater than Sreq if the valve is opening, or less than Sreq if the valve is closing, in either of which cases Sf= Sreq = cs(htr- hcrit). Of course, the value of Sf so calculated becomes the next value of si for the subsequent 11t period and so on.

144


7.6 Reservoirs The reservoir constitutes a very simple boundary condition but reservoirs may be equipped with any of the following different types of valve: (I) a control valve which will be either open or closed, (2) a reflux valve which only permits flow into the reservoir, (3) a reflux valve which only permits flow out of the reservoir, ( 4) a part-closed valve. It is probably uneconomic to include the part-closed valve case in the reservoir analysis as it can easily be represented by a motorised valve, for which the associated K array has all its values set to the same value equal to that corresponding to the part-closed valve position, and to insert this value one 6-x away from the reservoir. The valve's operating times must then be set at zero for the time of commencement of the valve's operation, and a time greater than the simulated time of the computer run for the time of finish of the valve's operation. This method of representing the part-closed reservoir valve is slightly erroneous and a little expensive in computer run time as it involves K array fetching at every time level, but it does save program size and the associated file store costs. On balance the author favours this type of approach. For a reservoir at which no valve is fitted the potential head can be treated as constant if its plan area is relatively large and slowly varying if it is not. Then if the reservoir is located at the upstream/downstream end of a pipe a backward/forward characteristic must be used.

p~~--+-""'-----+p 5

N

+ _.!_

(hp -

cs;R

As hp then

= hres:

R

Figure 7.9

N

hs/R) + (Up - US/R) + 2/s;R us;RI us;RI M = 0 (7 .30) ds;R

the reservoir head, _

Up -

US/R

±-

g (h

cs;R

res

_h

S/R

) _ 2/s;R us;RI us;RI 6. t ds;R

(7 .31)

The minus sign is used for an upstream and the plus sign for a downstream reservoir. If a forward reflux valve is fitted this calculation is correct and can be accepted as it stands but if the flow should ever attempt to reverse and Up take a negative value this answer will be wrong because the reflux valve will close under these circumstances and Up must be zero. Therefore it is now necessary to test what type of reflux valve, if any, If it is of forward type and Up is positive, then Up can be acceppresent. is


145

ted but if it is negative it must be set to zero. If the reflux valve is of backward type (i.e. it only permits flow in the negative direction- an inflow to an upstream reservoir and an outflow from a downstream reservoir) then the calculated velocity can be accepted if it is negative but otherwise must be set to zero. If the valve is of the control type it may be fully open or fully closed. If it is fully open the calculation can be accepted but if it is fully closed the value of Vp must be set to zero irrespective of its size or direction. It is next necessary to calculate hp again as, if the valve is closed, the head at the valve is no longer equal to the reservoir head, SO

hp

2fstR vs/RI vstRI A t~ = hs(R ±cs/R - - [Vp- VS(R +-....c...._..:__.:___ g

d~R

The minus sign is used for an upstream and the plus sign for a downstream reservoir. Note that the subscript S/R means that Sis to be used for an upstream reservoir case and R for a downstream reservoir case. If the transient case is the only case of interest the above approach is adequate as reservoir levels cannot change significantly during the very small periods of time in which transient conditions are present. If it wished to study the cases that can arise when the downstream boundary condition is not a simple reservoir but may be a sewer outfall to a manhole, a reservoir of small area, a small reservoir equipped with an overflow weir to a channel, a rock stratum into which water is to be injected, a badly defined exit point from the network into another network, in which there is storage capacity plus a great deal of looping, a further facility can be built in to the reservoir analysis. Assume that the exit condition at the reservoir can be represented as shown in figure 7.10. plan area A

l

\~

00

I

p1pe ex1f

Figure 7.10

Essentially this consists of a tank of plan area A into which the pipe discharges. Once the level in this tank has risen to the sill height of the weir an overflow occurs and discharge is then into a tank of infinite plan area. The depth in the first tank rises according to the equation df= d· + (Qp- Qo) At I A

146


If the level of the liquid in the tank is higher than sill level Q 0 will be given by Qo= kB(di- hsin)notherwise Q 0 = 0, where dris the depth in the first tank at the end of the time interval D.t, di is the depth in the first tank at the beginning of the time interval D.t, Qp is the flow in the pipe, hsmis the height of the sill above the pipe centre line, A is the plan area of the first tank, B is the breadth of the weir, k is the weir constant, n is the weir power index. Then the absolute potential head of the flow at the pipe exit will be hp=dr+z+ha

where z is the elevation of the pipe centreline above datum and ha is the atmospheric head in height of liquid. This model is extraordinarily flexible. By making A very large a normal reservoir circumstance can be modelled. By making A small, B large and n large an initial rise in reservoir level followed by a fixed reservoir level can be modelled. By choosing realistic values a reservoir with an overflow weir can be modelled. By taking the head-flow characteristic curve of almost any downstream boundary condition and employing a little ingenuity in picking appropriate values of k, B, n, hsill and A, it is possible to duplicate this curve almost exactly. A realistic representation of many boundary conditions is possible using this device. 7.7 Bends Fully fixed bends in a pipeline do not cause any reflection of an incident transient. However, if the bend is not fully fixed, the increase in the force acting on the bend caused by the passage of a pressure transient through it will make the pipes leading to and from the bend extend and the bend will move correspondingly. This movement will generate a partial negative pressure reflection. As the amount of movement is controlled by the effectiveness of the bend's anchor block it is not possible to calculate the magnitude of this reflection without modelling the block's behaviour. To do this, information is required about the dynamic forces created between the anchor block and the soil and this will not usually be available. The negative reflections created are of the order of 10% of the incident wave but it must again be emphasised that the magnitude of this fraction depends upon many factors. Usually, the effect of bends is to diminish the maximum pressures experienced by the section of the pipe on the side of the bend opposite to that in which the transient was initiated.

8 Unsteady flow in gas networks

8.1 Introduction

The methods presented for analysing unsteady flow in pipe networks transporting liquids need modification before they can be applied to highly compressible fluids such as gases. The fundamental equations are essentially similar to those applicable to liquids.

8.2 Basic equations The continuity equation The rate of increase of mass of an element of gas of length Ax equals the difference between the rates of entry of gas to the element and exit of gas from the element. So

a~ (pAox) = pAv- (pA v +a: (pAv) ox) a

a

at (pA) +ax (pAv) = 0 ap aA ap aA av Aa-r+par-+Avax +pvax +Apax=O ap p aA

ap

pv aA ax

at+ A at+ v ax+ A

av

+p ax= O

In pipelines transporting gases the pressure fluctuations are not sufficiently large to cause any pipe distension of significance so aAjat is very small and can be neglected. The continuity equation therefore reduces to ap ap av pv aA - + v - + p - + - -=0 at ax ax A ax

147

{8.1)

148


The dynamic equation Applying Newton's second law of motion to an element of gas of length ox the net force acting in the direction of flow (i.e. in the direction of x increasing) equals the rate of change of mom en tum of the element. dv ) aA a pA- (pA +ax (pA) ox +pax ox- Fox= pAoxdt

The p

~:ox term is the longitudinal force acting upon the projection

of the increment of area in the longitudinal direction. F is the frictional force acting per unit length of pipe, dv aA a -ax (pA) +pax- F= pA dt av av ap A ax + F + pAv ax+ pA at= 0 av F av ap ax + pv ax + P at+ A= 0

(8.2)

The equation of state of the gas It is not necessary to involve an equation of state for a liquid but, for a gas, an equation of state is necessary because of the large changes of density and temperature caused by the compression or rarefaction of the gas when a transient pressure passes through it. It may be thought sufficient to use an adiabatic, an isothermal or polytropic process according to circumstance but the best method is to use the equation of energy. This can be derived as follows: For a perfect gas the internal energy per unit masse is e=C T=-1-!!... r-Ip v where Cv is the specific heat at constant volume, also where Cp is the specific heat at constant pressure. Tis the absolute temperature. Let the perimeter of the gas element be sand the rate of heat inflow to the system be q units of heat per unit area. Then heat inflow to the element =qsox

Unsteady flow in gas networks

149

The rate of increase of the energy of the element with time

The rate at which energy leaves the element axially

=a:

(pAv(e + v; )) 5x

The rate of work done by the element against pressure forces

a

= - (pAv)5x

ax

By the first law of thermodynamics: rate of heat inflow= rate of increase of stored energy +net rate of energy outflow+ net rate of work done by the system. Combining this equation with the continuity and dynamic equation:

(8.3)

8.3 Characteristic equations By a similar method to that employed before, the characteristic equations can be obtained.

dv c dp -+--=£1 dt 'YP dt

dx along dt = v + c

{8.4)

dv c dp_E dt- 2 dt-

dx along dt = v - c

(8.5)

dx along dt = v

(8.6)

w

dp -c2 dp= £3 dt dt where E = r-1 1

!1.. + .f_[v(r-1)

c pm

pA

c

1]

-!!.E.. aA A ax vc aA A ax

+--

(8.7) (8.8) (8.9)

m =hydraulic mean radius of the duct. aA{ax is the rate of increase of

area of the duct. In a parallel sided pipe

~~ = 0.

150

Hydraulic analysis of unsteady flow in pipe networks p

s Figure 8.1

Along RP equation 8.4 applies and is a forward characteristic. Along SP equation 8.5 applies and is a backward characteristic. MP is a particle path and along the particle path equation 8.6 applies. Combining the finite difference forms of equations 8.4 and 8.5 and eliminating Vp gives:

Pp = (

CR

PR

c)~ L Vs +_!_(cR'Y

)lvR-

'Y

+~ Ps

+ (E1 - £ 2 ) dt

(8.10)

Substitutingp p back into the finite difference form of equation 8.4

vp=vR+ cR (pR-pp)+E 1 dt 'YPR

(8.11)

Usingpp in the finite difference form of equation 8.6

=PM+~

[pp- PM- E3dt) (8.12) CM Relevant values at R, M and S must be obtained by interpolation as demonstrated in chapter 4. Values at P can be calculated using equations 8.1 0, 8.11 and 8.12. Any mid-pipe point can be established in this manner and the technique is very similar to that illustrated earlier for liquids but because of the need to use three characteristics the process takes rather longer and costs rather more. The above presentation follows closely the relevant portions of a paper by Edgell 16 . PP

8.4 The value of q

q is the rate of heat inflow per unit area of pipe wall. In gas mains it may be considered sufficient to calculate q on a quasisteady heat flow basis. This approach assumes that at any instant the steady flow of heat through the pipe wall appropriate to the temperature difference across it is a sufficiently accurate estimate of q. This ignores the heat required to raise (or lower) the temperature of the pipe-wall material itself. Should it be felt that this quasi-steady heat flow method is acceptable then q may be estimated as follows.

Unsteady flow in gas networks inside of p1pe

151

external environment

pipe wall

Temperature distribution across the pipe wall

Figure 8.2

Newton's law of heat transfer can be used, i.e.

q=CD.Te where D. Te is the temperature difference. In steady state

= G.v ( Tw 2

2

- Tw 3 ) = Cw 3 ( Tw 3 - Te)

(8.I3)

where Cr is the coefficient of heat transfer from the fluid to the wall, Cw1 is the coefficient of heat transfer through the wall, Cw 2 is the coefficient of heat transfer through the lagging, and Cw is the coefficient of heat transfer from the lagging to the external envi~onment.

Cqf =TJ.-TwI -

q

CwI

Cq

w,

=Twl- Tw.

= Tw,- Te

q (....!.._ + _I_ + _I_ + _I_) Cr

Cw I

_ q- I

Cw 2

TJ.-Te

Cw 3

=

I

I

I

Cw1

Cw 2

Cw,

T1 - Te

-+-+-+-Cr

(8.I4)

152


where Cc is the compound coefficient of heat transfer, i.e.

q = Cc(T1- Te)· It now only remains to calculate Cf, Cw,, Cw, and Cw 3 •

(8.15)

The transfer of heat from the gas to the pipe wall In laminar flow:

K ( 11 ) o.14 (pvd C!J. d )o .33 -- ·- ·, 11 K L

Cf=1.86- d !lw

(8.16)

where Cr= heat transfer coefficient in watt per metre squared per kelvin. K =thermal conductivity of fluid in W m- 1 K- 1 • 1K=1°C 1 Js- 1 = 1 N m s- 1 = 1 W c =specific heat per unit mass in J kg- 1 K- 1 !lw = dynamic viscosity of fluid in kg m -I s-1 units at the pipe wall tern· perature Tw 11 =dynamic viscosit'y of fluid in kg m- 1 s- 1 d =internal pipe diameter in m L =pipe length in m v = mean flow velocity p =mass density of fluid in kg m- 3 In turbulent flow (Re > 2300):

K(dv) o.s (C!J.) -

Cf= 0.027-d V

0.33 (

K

!1 ) !lw

-

0.14

(8.17)

where dis the internal diameter of the pipe, vis the kinematic viscosity of the fluid at fluid temperature.

The transfer of heat through the pipe wall =

C

w,

2rrK

loge(~::)

heat units per second per metre per kelvin (8.18)

where K is the thermal conductivity of the pipe wall material in W m- 1 K- 1 dw, is the internal diameter of the pipe dw, is the external diameter of the pipe.

The transfer of heat through the pipe insulation This is essentially the same case as for the pipe wall.

=

C

w,

2rrK

loge(~)

(8.19)

Unsteady flow in gas networks

153

where K is the thermal conductivity of the insulation, dw 3 is the external diameter of the insulation, dw 2 is the internal diameter of the insulation. Note that Cw and Cw are the heat losses in watt per metre length of pipe per kelvin. They are different from Cr above and Cw 3 immediately below which are heat losses in watt per square metre of pipe wall surface per kelvin.

The transfer of heat to the external environment If the outside environment is air:

(8.20) where Cw 3 =heat transfer coefficient in W m- 2 K- 1 dw3 =outside diameter of the insulation K= thermal conductivity of air in W m- 1 K- 1 . Re = Reynolds number Vd/v where Vis the wind speed in m s-1 . If the external environment is soil (i.e. a buried pipeline):

(8.21)

where Z is the depth in metres to the pipe centreline dw is the ex temal diameter of the insulation. K i~ the soil thermal conductivity in W m- 1 K- 1. If flow in the pipe is laminar (a most unlikely circumstance in a gas pipeline) the value of Cw, will depend upon the temperature difference 11 - Tw, and as Tw, is not known an iterative method of solution will have to be used. If this quasi-steady heat flow approach is not regarded as sufficiently accurate then a more accurate method may be used. This is based on the heat diffusion equation, i.e.

(8.22) where

e

K pC

~=--

and is the temperature difference. This equation can be solved by finite difference methods but this will lead to a great increase in the program size and complication and may well be regarded as an unacceptable method.

154


8. S Boundary conditions Boundary conditions can be solved by methods similar to those described for liquids but almost invariably the equations produced are of much greater complexity and usually have to be solved by iterative methods such as perturbation or Newton-Raphson techniques. The boundary conditions that are of interest in gas pipelines include the following.

(1) Upstream or downstream reservoir: this is a tank from which gas may flow or into which it is delivered. (2) Turbo-blowers or compressors: these act in the same manner as pumps in liquid networks and can be handled in a surprisingly similar way. The energy per unit weight supplied to the gas is given by the equation H

=AN 2 + BNQ -

CQ 2

where Q is the flow in m3 s- 1 at inlet pressure and temperature. The H- Q curve is supplied by the blower manufacturer in the same way that the manufacturers of pumps supply the pump curves. Once H has been determined from the flow value the pressures can be calculated from: n-1

(8.23) where n is the polytropic index, p2 is the discharge pressure and p 1 is the inlet pressure. 1i is the temperature at inlet (absolute). n can be estimated from: {8.24) where ep is the polytropic efficiency of the blower. The polytropic efficiency must be known or estimated from previous experience. (3) Junctions: these can be analysed by the same method as that demonstrated for the case of liquids but using the gas characteristic equations.

9 Impedance methods of pipeline analysis

9.1 Introduction

It is possible for resonance to occur in pipe networks. Organ pipes and other musical wind instruments operate by generating such a resonance. Small pressure or flow fluctuations applied at one end of a pipeline can superimpose upon one another if their frequency matches a simple multiple of the pipeline's period leading to the development of a standing wave of considerable magnitude. This phenomenon has caused some major catastrophes: pipelines have ruptured when the forcing vibration has been of trivial magnitude. One of the circumstances which will generate small fluctuations of pressure or flow is the oscillating valve. A very common example of this is the vibrating ball valve in a domestic water supply. As the level in the cistern rises it raises the ball which progressively closes the valve. If the valve is very nearly closed, a small further closing movement shuts off all flow causing a pressure wave to travel down the supply pipe. The pressure rise tends to force the valve off its seat and to let flow recommence so initiating a negative pressure wave which starts off down the pipe following the initial positive wave. The valve is forced back onto its seat by the upthrust upon the ball. The rubber valve seat and the mass of the moving parts of the valve ball and its arm constitute a mass and spring circumstance which has a natural frequency of vibration. If the pipeline has a harmonic (i.e. a multiple of its fundamental frequency C/4L) which matches this mass-spring frequency, a resonant vibration will be generated. This can be heard as a low thrumming throughout the house and, if it is allowed to continue for a sufficiently long time, the pipe may burst. To stop it, it is only necessary to arrest the ball's vibration by touching it but to prevent it from occurring it is necessary to either alter the hydraulic parameters or the mechanical parameters of the ball valve. Changing the valve's rubber seat alters its stiffness so changing the spring constant in the mass-spring circumstance of the ball valve. By thus changing the forcing frequency the resonance may be prevented but usually the rubber alters its stiffness after some period of use and the problem reappears. By ftxing a square plate to the base of the

155

156


ball in a horizontal position two mechanical parameters are changed: (a) the mass of the ball and arm, (b) the frictional damping of its motion is increased. This often solves the problem. The pipe can be replaced with one of larger diameter in which velocities and velocity variations are smaller. This involves the generation of smaller pressure transients so reducing the magnitude of the forcing vibration. Frictional damping already present in the system may then be sufficient to prevent the development of resonance. This simple example of the problem illustrates many of the features that occur in very large scale pipelines. In hydroelectric installations, for example, various mechanisms can generate resonance. In the reservoir supplying the scheme, wind driven surface waves of amplitude 1.5-3 metres can be generated. These can have a frequency which may coincide with a harmonic or the fundamental of the pipe system and resonance can thus be generated. In large pipes it is usual to shut down a flow by the use of a butterfly valve; such valves do not usually seal perfectly and a rubber seal located around the periphery of the valve disc is inflated with oil (or water from the pipeline) after closure has been completed. If for any reason such a seal leaks, and hence the valve's sealing becomes imperfect, leakage over the seal may cause it to flutter, generating small pressure waves travelling up the pipe. As described earlier this can cause resonance and can cause pipe bursts under apparent no-flow conditions. Servocontrolled valves may also cause resonance if they have a natural frequency which matches that of the pipeline and are inadequately damped. Governor controlled spear valves of Pelton Wheels and the governor controlled guide vanes of Francis Turbines are examples of such valves. It must be emphasised that the assessment of a resonance risk is not an academic exercise but is of real practice importance. If there is the slightest possibility of a resonance occurring, a resonance analysis should be performed to assess the risks involved and to ensure that palliative measures are effective. Resonance analysis can be performed by the characteristics method as described in previous chapters and if no better method were available this would be the one of choice. However, an analytic technique exists which is adequately accurate and which can be used to provide a computer solution in a very small fraction of the time taken by a characteristics technique.

9.2 The analogy between electrical and hydraulic impedance The method is based upon an analogy between electrical and hydraulic flow. In the theory of the transmission of alternating current over transmission lines two equations are used which bear considerable similarity to those of waterhammer. This can be seen by comparison of equations 9.1 and 9.3 and equations 9.2 and 9.4.

Impedance methods of pipeline analysis

157

Transmission line:

(9.1)

av

Pipeline:

ai

.

-+L-+R 1z=O ax at e

(9.2)

ah ah c 2 au -+u-+--= 0 at ax g ax

(9.3)

(9.4) where C in the transmission line equation is the capacitance/unit length, L is the inductance/unit length, Rei is the resistance/unit length of the transmission line, Vis voltage and i is the current. The variables in the waterhammer equation are as defined elsewhere in this book. If the potential head is seen as analogous to the voltage V and the flow Av as analogous to the current i and, if the v

ah term in equation 9.3 and ax

the~aav term in equation 9.4 can be ignored, the analogy becomes exact g X 2jVIvl . d. prov1.d.1ng t h at t he d - term can be 1·1neanse The v large as

~~ term is small compared with the ~~term if the wavespeed is

~~ ~ (v +c)~~· So if it is assumed that c is constant and large this

term can be neglected in equation 9.3; similarly in equation 9.4 the

E. aav g

X

term is small compared with the other terms if c is large and constant,

so it can be neglected under such circumstances. Equations 9.3 and 9.4 can now be rewritten:

ah+~ aq = 0 at gA ax

(9.5) (9.6)

9.3 The linearisation of the waterhammer equations The remaining question concerns the linearisation of the 2..:{ ~~I term. This term cannot be linearised as it stands but if the flow q is regarded


158

as made up of a steady component plus an oscillating component the term can be linearised. Thus let q =q + q' where q is the steady state flow and q' is the unsteady flow component.

aq aq aq' -=-+ax ax ax

Then By

definition~; is zero as lj is constant throughout the length of the

pipe so

aq _ aq' ax- ax aq aq aq' at =-ar+ar

Similarly and

~; = 0 as the flow q, being steady, cannot vary with time,

aq- aq '

a-r-ar-

so

Similarly the head h can be split up into a steady head and an oscillating component, i.e.

h=h+h' The value

~~is the hydraulic gradient and so equals- 2{d~1 1 •

The value~~= 0 as the steady head component cannot change with time. Thus equation 9.5

a(Ji+h') c 2 a(q+q') ax + gA at

O

becomes

ah' c2 aq' -+--=0 at gA ax

(9.7)

and equation 9.6

a(Ti. +h') 1 a(-+ q q')1 q q')IC+ q q') + 2/C+ ---'-:,....----'-+gdA at gA ax

=0

(9.8)

-2fqlql +~ + _!_ aq' + 2f(l! + q')l(q + q')l = 0 gdA 2 gdA 2 ax gA at

(9.9)

becomes


159

If the analogy is to be drawn between friction and resistance the f value must be treated as a constant / 0 and for this to be done the terms

2fqlql gdH 2

2f(q + q')l(q + q')l gdA

and

must be rewritten 2/oQn -gdAn

and

where n takes a value between 1. 7 5 and 2.0 according to the roughness of the pipe. 2!( (if+ ')n Then the term 0 q can be expanded by the use of the Binomial . g, equation

dAn

2fo(q+q')n _ 2/o ('-n+ -n-1 '+n·(n-1) n-2 •2 ) gdA n - gdA n q nq q 1• 2 q q ··· If q' is small compared with q, second order terms can be ignored so 2/o (q + q')n

2/oQn + 2nfoQn-lq 1

gdAn

gdAn

gdAn

so equation 9.9 becomes -2~'-n

---'J:..::(O_,_Q_

gdAn

ah' 1 a, 2~'-n 2nr-n-l , + _ + _ _!!__ + ...l!!!L_ + JoQ q =0

ax

gA at

gdAn

gdAn

Therefore it reduces to a , 2nr-n-l , -ah' + _1 _!]_+ JOQ q

ax

Denote

gA at

2nfoqn-l gdAn

gdAn

(9.10)

byR

Then the hydraulic equations are ah' c 2 aq' -+--=0

at

gA ax

~+_l_aq' +R ax

gA at

q

'=o

[9. 7]

*

(9 .11)

Comparing these with the electrical equations av+_!_ai=o

[9.1]

aV + Lj!_ + Reti = 0

[9.2]

at

ax

cax

at

*Square brackets indicate that the equation with this number was introduced earlier.

160


it can be seen that the hydraulic equivalent to the capacitance/unit length Cis gA/c 2 , the hydraulic equivalent to the inductance/unit length L is 1/gA and the hydraulic equivalent to the resistance/unit length Rei is R 2nfolfn-l 32v which equals gdA n · If the flow is laminar R becomes gd 2 A from the Hagen Poiseuille formula. The [ 0 value used in this development is the British form not the American form although/0 is known in the USA under the name of the Fanning[. The form commonly used in the USA is equal to four times the British form as shown below: USA British

[Lv 2 hr=-2gd

hr= 4[Lv2 2gd

As a matter of interest, the reason that the British form retains the 4 in the numerator is that it is derived from the basic Darcy-Weisbach form,

[Lv 2 hr = '2gm where m is the hydraulic mean radius. As m = d/4 for a circular pipe, the British form results immediately and the f values apply to pipes of any cross section. 9.4 The solution of the linearised waterhammer equations First differentiate equation 9.7 with respect to time

a2hl c2 a2ql at2 =- gA axat a2 A a2 h ~--L axat - c 2 at 2

1

I

Rearranging equation 9.11

a..!!...= -gA (ah -+Rq at ax 1

I

1)

Differentiating with respect to x

but

a;a =-~at I

A i1h 1

from equation 9.7


a2 h' RA ah' 1 a2 h' ax 2 -g7at-c2 at 2 =O

so

161 (9.12)

This is the wave equation ( cf equation 2.18 in which friction was ignored). If the wavespeed is assumed to be constant then, although the wave may attenuate with distance along the pipe, an oscillatory wave will not alter its amplitude at any particular point on the pipeline. Thus, for the case of a sinusoidal pressure head applied at the upstream end of a pipeline, a solution of equation 9.12 must be:

h'=Heint where His the amplitude of the wave i.e. atx = 0 at x =x 1

h~~ = Hx,. eint (Note eint =cos

n = 2rrf

n t + i sin n t where i = v'-1

where f is the frequency of the oscillation)

a h' a H · ----e'.nt 2 2

2

ax

2 -

ax

ah' -rm int e ar=' a2h'

.

2 He'.nt --=-n 2

at

Substituting these values into equation 9.12 · ·.n Q 2 RA a2H ·m -g-iDHe 1 t+-He 1.nt=o --e' 2 2 2

ax

c

c

so (9.13) denote (9.14)

then In transmission line theory 'Y is called the propagation constant. Equation 9.14 is well known and is called the harmonic equation. A solution of equation 9.14 is of the form

H=C1emx

162


a2

H --=Cm2emx I 3x2

Then

Cim 2emx =

So

:.

'Y2

Cl emx

=r2 :.m =±r m2

The required solution is therefore H = ae -yx +be --yx where a and b are constants. So h' = (ae'Yx + be-'Yx) ei.nt

(9.15)

To obtain the solution for q' differentiate equation 9.15 ah' =in (ae'YX + be-'YX) ei.nt

at

but from equation 9.7

a ' A ah' _!f_=_G!!__ c 2 at ax q' = -~ inei.nt

J

(ae'Yx + be-'Yx)dx

q' = g~n eintlae-yx_ be_'YJ ] ~ ZC 2 'Y

(9.16)

This is the general solution of the wave equation for any sinusoidal oscillation.

9.5 The evaluation of 'Y 'Y is complex as can be seen from its definition (equations 9.13 and 9.14) so it will have the form

r=a+i(3

Consider a de Moivre diagram (figure 9.1). If a and (3 are both positive real numbers, 'Y must lie in the first quadrant. Now 2_

n2

'Y - - - 2 +

c

iAgnR

c

2

The real part of r 2 is thus negative and the imaginary part is positive so r 2 must lie in the second quadrant. r 2 =(a+ i(3) 2 = r'Y2 eiop'Y 2 Now


Figure 9.1

r

and

0: + i(3 = r..,ei
=

r..,• = (r..,) 2 and
and

= 2
from the simple theory of complex numbers

r.., =Jcx..,2+(3/ r..,• =JOG{ + (3~·

n2

0:-y2 = -~

Now and so r • 'Y

and

Now and but

=

+ R2 Ag.Qj(R)2 gA c2

= rr- tan- 1 ((3-y•\ 0:-y .,

= rr- tan- 1 (R~)

=~-ttan-I(R~)

r-y=~=H ((~f +R2Y'4

163

164


but

cos (~-A)

= sin

A

a~ =R[(g~r +R2r14 sin(~tan-l(R;A)) (3~ = r~

sin 1{1~

J

f3~ =~ [(g~r + R 2

1 \os

(~tan-l(R~A))

(9.17)

(9.18)

9.6 The impedance concept The risk of resonance occurring in an electrical network can be assessed by investigating the way in which its impedance varies with the frequency of the voltage of the applied signal. This method can be applied to oscillatory flow circumstances in a pipe network. The hydraulic concept of impedance is based on the analogy between electrical and hydraulic variables, i.e. voltage and head, current and flow. Electrical impedance is defined as the ratio voltage/current so hydraulic impedance is defined ash' fq'. As in electrical theory the D.C. components of voltage and current are ignored so, in hydraulic impedance theory, the D.C. components of head h and flow q are also ignored. Thus Z(x) = h'lq'

Z(x) The term

=ic 2 -y (ae~x + be-~x) gAn

ae~x-be

~x

(9.19)

-ic 2 -y

c 2 -y gAn = igAU

has the dimensions of impedance. It is called the characteristic impedance and is denoted by Zc. At the end of a very long line the values of h' and q' would both tend to infinity unless a is zero. Thus the impedance of an infinitely long line is given by Zoo=-Zc

so -Zc is the impedance of an infinitely long pipeline extenC:ing in the x direction and +Zc is the impedance of a pipeline extending in the -x direction. In an infinite pipeline no reflected wave can be travelling in the opposite direction to the incident wave so the characteristic impedance can only be used in connection with heads and flows of a wave travelling in one direction. (9.20)


165

9.7 Receiving and sending ends Following the practice used in transmission line theory the sending end will be assumed to be at the x = 0 position and the receiving end at x = L position. This assumes that the forcing oscillation of pressure head is located at the x = 0 position. In some other texts this practice is reversed and readers are warned of this difference should they consult such texts. Hs

L

s

HR

I•

Q(x) H(x)

R

X

Figure 9.2

S denotes sending end and R denotes receiving end. 9.8 The equation of impedance The change in impedance from one end of a pipe to the other will be derived next. Referring to figure 9.2, at x the oscillatory head component is given by

and the flow by q~ = Qxeintwhere Hx is the amplitude of the head oscillation at x and Qx is the amplitude of the corresponding flow oscillation atx. At the sending end where x = 0 h~ =Hoeint=Hseint q~ =Qoeint=Qseint

From equation 9.15 h~ =Hseint =(ae-rxo+be--rxo)eint

Hs =a+ b

(9.21)

Similarly for q' from equation 9.16 q' =Q eint=~Ail (ae-rxo _ be--yxo) eint 0

S

IC 2'Y

gAil

Qs=~(a-b) IC

'Y

(9.22)

166


Adding equation 9.22 to equation 9.21

ic2r 2a=Hs + - Qs gAD

! (Hs - ZcQs) b = ! (Hs + ZcQs) a=

so h~

= eint (HHs- ZcQs)e'Yx + ~(Hs + ZcQs)e-'Yx = eint

(Hs (

-e--yx))

(c'Yx e'Yx + e--yx) _ 2 ZcQs 2

but

e'Yx +e--yx =cosh (rx) 2

and

e'Yx - e--yx 2

sinh (rx)

h~ = eint (Hs cosh (rx)- ZcQs sinh (rx)) Similarly

q~ = eint (-~:sinh (rx) + Qs cosh (rx)) _ h~ _ Hs cosh (rx)- ZcQs sinh (rx) Zx - ----, - ----:-;;---'----____::___ __ -Hs q x -sinh (rx) + Qs cosh (rx) Zc

Hs- ZcQs tanh ('yx) Hs Qs --tanh (rx) Zc

z

Zs- Zc tanh (rx) Zs I - - tanh (rx) Zc

=-----X

At x = 0 tanh (rx) = 0 and the above result reduces to a simple equality

Zo = Zs Atx =L ZL = ZR =

Zs- Zc tanh (rL) Zs I --tanh (rL) Zc

(9.23)

Re-expressing Zs in terms of ZR by manipulating equation (9.23)

_:______.:. _ _____:(rL) __::___tanh _:_:__+ Zc z s =ZR ZR I +-tanh (rL) Zc

(9.24)


167

From these results for Zs and ZR the change of impedance from one end of a pipe to the other end can be calculated.

9.9 Boundary conditions In previous chapters, boundary conditions were described and the way in which they interact with incident waves producing reflected waves was discussed. In a very similar way it is necessary to describe the boundary conditions in a resonant network so that the impedance at a pipe end may be specified and to link the impedances of constituent pipes in the network to one another to obtain the impedance of the entire network. Reservoirs

At a reservoir the head is rigidly controlled and must equal the height of the reservoir surface above the datum. Thus h equals the height and h' must be zero. Therefore at a reservoir the ratio h'/q' = 0 irrespective of flow.

Z=O

(9.25)

A blank end

At a blank end of a pipe the flow must be zero irrespective of head so 7j and q' must both equal zero. h'

Z=----;=oo q

(9.26)

A junction

For a junction the head is the same for all pipes joining there and also ~Qin

= ~Qout

As the head is the same for all pipes joining at the junction

,

,

h'

h'

~ Qin= ~ Qout

1

I

i.e.~-=~--

Zin

Zout

For example consider a four-way junction (see figure 9.3).

168


®

Figure 9.3

Suppose that Zs 3 , Zs 4 and ZR 2 are known by calculation from their other ends then 1 ( ZR = 1

1 I z;-+ Zs +Zs1) l

4

3

A joint This is a two way junction (see figure 9.4) and so

=

CD _ __ . r - - ® Figure 9.4

Series sections of a network are merely a series of pipes separated by joints and so are simply dealt with as illustrated in figure 9.5.

Figure 9.5

Now ZR 3 = 0 as R 3 is at a reservoir. The characteristic impedance of pipe 3 is given by equation 9.20, requoted here. (9.27)


169

and a 3 are calculated from equations 9 .I7 and 9 .I8 so Z c 3 can be evaluated from equation 9.27

~3

= O =Zs 3 -

ZR

Zs

3

z

I -

so

Zc 3 tanh ('yL) 3

c3

tanh ('yL)

and Zs 3 can be evaluated. From the joint boundary condition

ZR, = +Zs3 then

= Zs, -

ZR 2

Zc 2 tanh ("yL 2 )

z I - -r:- tanh ('yL2) c,

As ZR is known, Zs can be obtained once Zc has been calculated. Again ZR 1 = Zs 2 an"J in the same manner as fo; pipe 2 Zs 1 can be found.

Loops

Zc can be calculated for every pipe in the network (see figure 9.6).

® Figure 9.6

Then

so Zs 4 can be calculated. At the junction

1 -) -I= ( -I+ -

Zs •

and

ZR 2

ZR 3

170

Hydraulic analysis of unsteady flow in pipe networks Z

and

R3

_ Zs 3 - Zc 3 tanh (·Y3L3) 1

At the upstream junction

z

-zc3s3 tanh (-y3L3) 1 )

( 1

1

+ Zs3

ZRI = Zs2

Additionally by the reciprocal rule for adding impedances in parallel 1 1 1 + (9.29) --- = Zs 4 -ZR I ZR 2 -Zs 2 ZR 3 -Zs 2 ZR

and

I

=Zs

I

- Zc tanh (-y1Ld I

z

1--j; tanh(-y 1Ld cl

Sufficient equations have been obtained to solve all the impedance values. When two loops overlap as shown in figure 9.7 a circumstance called second order looping exists.

1

1 1 ' - - - -- -

I Second order looping

First order looping

Figure 9.7

The author is not aware of any method of analysing second order looped networks.

9.10 The impedance of a network By starting from a point in a network at which the impedance is known, e.g. a reservoir or a blank end it is possible to work back up the network calculating impedances en route until the sending end is reached. blank end~

R3

®53

®

® s

Figure 9.8

Impedance methods of pipeline analysis The process is illustrated below by reference to figure 9.8.

zR, =oo

zR. =zR. =o Then

Zs = s

ZR + Zc tanh (rsLs) s

1

z

s

tanh(rsLs) +2 c,

As this reduces to:

Zs s = Zc s tanh (rsLs) Similarly

and

171

172


Values of the impedance of the downstream network at every node have now been calculated. It must be appreciated that all values of impedance are complex. The programming language Fortran has the facility of specifying variables as complex and has the necessary libraries for performing complex algebra. The language Algol 60 does not possess these facilities so despite its many advantages over Fortran it is not as well suited as Fortran to perform resonance analyses and the programmer must be prepared to write his own procedures for performing complex algebra - a comparatively simple task. 9.11

Harmonic analysis

The result of performing the calculations for the network impedance illustrated above is a series of complex numbers, the magnitudes of which depend upon the angular velocity n, so the moduli of these numbers varies with the value of n. By calculating the modulus of the value of Z for any point in the network for a range of values of n, incrementing n in, say, 0.01 steps, a curve such as that illustrated in figure 9.9 results.

z f\

I I

......-

/

I

I

\

Figure 9.9

From the diagram the risk of a resonant condition being developed in the network, downstream of the point under consideration, by the application of a forcing oscillation at that point can be assessed. If the frequencies which cause resonance cannot occur, the network is safe but the likelihood of their development must be assessed with great care. The actual magnitudes of the amplitude of the oscillatory pressure depends upon the amplitude of the forcing oscillation and unless this can be specified the magnitude of the peak pressures cannot be calculated. However, it may be sufficient to estimate dangerous frequencies and to ensure that they cannot occur. The method of determining the magnitude of the forcing oscillation is discussed in the next section. It is possible to perform a harmonic analysis of the type described above by using characteristic methods. Each run of a characteristic program would cost more than a complete run of an impedance program. A harmonic analysis would require many runs so the characteristic approach is not suitable, for economic reasons.


173

9.12 The forcing oscillation Resonance occurs due to the interaction of waves generated by the forcing oscillation with those reflected from distant parts of the network. For resonance to develop, the frequency of the forcing oscillation must match a harmonic or the fundamental of the network. The fundamental frequency of the network cannot be easily specified unless it is of very simple type. The frequency of natural vibration of the network is not simply related to the pipe periods of the pipes making up the network, the situation being greatly complicated by the partial transmission and reflection of waves at junctions. Even in a simple branched network no simple relationship between the frequency of the fundamental and that of the pipes in the network exists because different branches cause reflections to arrive back from their downstream ends to their upstream junction at different times. The harmonic analysis described in section 9.11 is the only economic way of calculating the frequencies of the fundamental and its harmonics. To calculate actual pressure heads in the network it is necessary to supply the amplitude of the forcing oscillation. Sometimes this is very simple to do, as in the case of a network connected to a reservoir upon the surface of which waves are generated by wind and these waves constitute a forcing oscillation. The wave forms may not be simple sine waves but may be made up of a number of sine waves of different frequencies, but these can be determined by Fourier analysis of the wave form. By performing the calculations to obtain the pressure head amplitude at various points in the network for each constituent sine wave, superposition of the results will give the resultant wave amplitude at all points required. To demonstrate how the amplitude of a pressure wave is calculated, consider a simple pipeline. At the upstream (i.e. sending) end assume that a forcing oscillation is present of known amplitude H 0 , i.e. so As Zs will have already been evaluated in the impedance analysis 1

h~

qo= Zs

so

Qs =Hs/Zs

At a point x downstream

h~ = e int (Hs cosh('yx)- ZcQs sinh( rx )) so

Hx = Hs cosh(r.x)- ZcQs sinh(r.x)

and similarly

Hs. Qx= Qs cosh(r.x) --smh(r.x) Zc

174


The pressure head amplitude and the amplitude of flow fluctuation can thus be determined anywhere in the network. As mentioned earlier, the forcing oscillation can be generated by a variety of mechanisms. As an example of the method of dealing with such a forcing oscillation the technique of analysing an oscillating valve will be demonstrated in the next section. 9.13 The oscillating valve The head drop across a valve due to flow expansion after passing the valve orifice is

Kv 2

h=-

(9.30)

2g

where K is a constant which varies from low values to high values as the valve closes from its fully open to its fully closed position ap

q=7KV2ih

(9.31)

where ap is the cross sectional area of the pipe. Denote the value

a }K by ae the 'effective' valve area.

q =aeVfih

(9.32)

As before q=

q + q' = ae ..}2g(Ti + h')

This equation must be linearised if it is to be used in impedance theory. Consider the term (ii + h' ). Now

)2 h' h' 2 (l + h' 2h = l + h + 4fi2

h'2 Providing that the term 47i 2 is small in comparison with the term h'jii,

which will be true if h' is small relative to h, then

(l + ~~r ~ h;

So

Then

h'


175

Expanding and remembering that q ="iie .J2ih gives

q

I

_hi

a~ _

h1

=aeV2ifl + q -+-:::-q27i ae 2n I

If ae0 is the effective area of the valve when fully open ql

ae aeo -

=ae ~o~.J)ih +q[~ +a~ ~ ae ae 0 ae aeo

hJ

ae0 ae 2hJ

2h

I

r::;;y

Denote- by € and -bye and remember that q = llev 2gh. 1

ql=q

Then

h') I] h1 (I+-=~ [-+ 2h €

2Ti

(9.33)

If the valve's oscillatory amplitude is small this approximates to

q

1

-[h

1

1

€ =q ili+"f

(9.34)

]

Note € denotes the ratio of the valve's steady flow area to its fully open area and € 1 denotes the ratio of the oscillatory component of the valve's effective area to that of the full open valve. Thus

z =--qr= -(h € q --=+h1

h1 1

2h

1

(9.35) )

€

Usually a valve's position depends upon the head at the valve. Valves which can be set into the oscillatory motion described here are controlled by a servomotor which causes them to start opening further if the head upstream becomes larger than a certain set value or to start closing if the head downstream exceeds a certain set value according to which pipeline (the upstream or the downstream one) is to be protected against excessive pressure. The precise way in which the valve behaves depends upon the servomechanism used but usually the valve's movement will lag behind the ap· plied head by a phase angle
where

is the amplitude of the oscillation of the valve oscillation, is its frequency of oscillation and <{! is the phase angle of the valve's motion. Assuming that these values <~;re known and h 1 is given by the usual oscillatory head equation h 1 = H 0 e 1n.t ~

n = 2rrf where f

176 then

Hydraulic analysis of unsteady flow in pipe networks h' Z =q'

H eint

= -(Ho_:i.nt + tei(.nt-.p)) q

z"

t

2h

~r~+_;v" J ~ Ho

(9.36)

If t = 0, i.e. there is zero valve amplitude, the problem reduces to that of a fixed valve opening or orifice. This case is thus defined by

Note

Z=

2}j q

(9.37)

Now Zs can be calculated from consideration of the downstream network independently and when Z = Zs a matched impedance condition is achieved, then H 0 can be solved and the forcing head amplitude generated by the valve is obtained. As before, this value can then be used to calculate pressure head amplitudes anywhere else in the network. 9.14 A network in which resonance can be excited by forcing oscillations located at different points in the network Examples of such networks occur when, for example, a network has intakes from two (or more) reservoirs upon the surfaces of which surface waves - perhaps of different type and frequency - occur. Again, a group of turbines in a hydroelectric scheme may be supplied by an asymmetric pipe network, and each of these turbines could generate a forcing oscilla· tion. An in-line pump can cause a discharge side forcing oscillation, which the downstream portion of the network will experience, and an inlet side forcing oscillation, which the upstream side of the network will experience. If the network is looped in a manner which joins the upstream and downstream portions of the network, the network will be experiencing two forcing oscillations. Such examples occur in many circumstances. In networks such as these there must be as many impedance analyses performed as there are sources of forcing vibrations. Because the analysis is of a linearised type, superposition is permissible and the risk to the network can then be assessed from the graph resulting from the superposition of all the harmonic analyses performed. Students of the phenomenon of resonance are strongly recommended to study the relevant section of Streeter's book: Hydraulic Transients.

10 Unsteady flow in open channels

10.1 Introduction Flow in open channels is normally considered in steady state conditions only. The methods of analysis of unsteady flow conditions in pipe networks can be applied to open channel networks with only trivial modification however. Nonrectangular channel sections can be dealt with, the channel need not be of prismoidal type, variable friction can be incorporated into the analysis and the results obtained from such analyses are usually of high accuracy. The main limitation upon the unidimensional method described below is that no sudden significant changes of depth must occur within the channel cross section. A

A section suitable lor analysis

A section unsuitable lor analysis (b)

(a)

Figure 10.1

The flow in the main channel to the left of line A -A in figure 10.1 b will be at higher speed than that in the remainder of the cross section. Consequently there will be a shear plane at A -A across which heavy turbulence will be generated. The wavespeed in the left-hand part of the section will be much higher than in the right-hand part and so a unidimensional analysis cannot be accurate for such a cross section. Bi-characteristic methods exist which could be used but they are considerably more complex and involve much more computer run time so the analyst may well decide to simplify the section by considering only the main channel and ignoring the flood plain.

177

178


Unsteady flow occurs in almost every channel. A river is never in steady state - it is either in flood or it is on a part of its recession curve. Flows in sewers are constantly changing. Flows in the head and tail races of hydropower schemes are constantly changing as the load on the power station changes. In some cases the consequential depth changes in the channel are unimportant but in others unexpectedly large changes occur and, frequently, travelling surges or bores are created which may overtop the channel's sides. To ignore these effects can have most serious consequences. A strong case can be made for investigating the unsteady flow conditions of open channel networks and it is to be hoped that in the future such investigations will be undertaken in the same way that unsteady flow investigations in pipe networks are carried out routinely today. 10.2 The equations of unsteady flow in open channels Consider the flow into and out of the element depicted in figure 10.2. water surface

,,,.n~ Figure 10.2

Net volume entering the element in time

ot =

[Av-(A+~~ox) (v+~~ox)]ot Storage within the element =

ad BX ox xatot where B is the mean surface breadth of the element

aAvox- A~ox] l_ ax [ ax so

0t = B adoxot

at

v aA + A av + Bad= 0 ax ax at

This ignores fluid compression within the element and this is reasonable in a free surface flow. Now

aA aAad ax ad ax

179

Unsteady flow in open channels

aA=B

aa

and

vB ad+ A av + B ad= O at ax ax

(10.1)

Next consider the forces acting upon the element and the momentum change that these cause. Force in left to right direction=

wzA -

w(:z +~! 8x)

G+~~ 8x) +wA8x

sin(i)- Pr8x

Tis the frictional shear stress acting over the perimeter of the flow, z is the depth of the centroid of the upstream cross section of the element and P is the mean wetted perimeter of the element. Force= w (-A

az- z aA + Ai- Pr\8x w) ax

ax

where second order small quantities have been ignored. Also sine(i) has been approximated to i so the usual limitation of small bed slope applies to this analysis. Now taking moments about the surface at the downstream end of the element (see figure 10.3):

(A+~~8x) (z+~!8x) =A(:z+~:8x) +~(~:8xf 2 B(aa aa +a:z =Az +A -;-8x _ aA +A -;-8x -8x) Az_ +z~8x 2ax uX

uX

uX

surface at

aA 8x

ax

-r--.--------.----.;,-1

Figure 10.3

ignoring second order small quantities.

. PT) 8x ad Az-Force= w (-A-+ ax

w

x + 8x

180


The rate of change of momentum that this force causes = dv (, av av) pAfJx dt =pAfJx ~ ax + at w(-A ad+ Ai- Pr\fJx ' ax w}

=pAfJx lv av + av) ~ ox

at

ad v av 1 av . PT -+--+---z + -=0 ax gax gat wA Pr _ ....!__ vlvl _. wA - wm - C2 m- I

Now

where Cis the Chezy C and j is the energy loss per unit wt of fluid per unit length of channel. ad v ov 1 ov . . ox + ox + ot +1 -' = 0

K'

'i

(10.2)

Equations 10.1 and 10.2 are called the St Venant equations i.e.

v od +4 av + od= 0 ox B ax ot

[10.1]

od VoV 1ov . . ox + ox+ ot +1 -' = 0

[10.2]

K'

'i

In the development of the continuity equation an assumption is implied, i.e. that the channel is prismoidal. This means that 6A = B6d. If the channel is nonprismoidal this is not true as can be seen from figure 10.4.

Nonprismoidal trian31e-like channel

8A~88d+ ll~x

8+88

Nonprismoidal rectangle-like channel

8A =88d+d88

Figure 10.4

Unsteady flow in open channels so

~A

= B~d prismoidal channel

~A :!:: BM +!!._ aaB ~x 2

181

X

~A =B~d + d ~: ~z

nonprismoidal triangle-like cross section nonprismoidal rectangle-like cross section

If the channel is nonprismoidal an additional term will appear in the continuity equation, i.e.

ad av ad vB-+A- +B-=0 ax ax at

prismoidal

and

ad aB av ad . . vB ax+ Otd ax v +A ax+ Bat= 0 nonpnsm01dal where Ot lies between 0.5 and 1.0 depending upon the nature of the nonprismoidality of the channel reach. aB IS . a constant 1"or . aB IS . a constant. Le Now ax a gtven reach so Ot ax t

daB OtB ax= A (Note that Ais a function of depth as B changes with depth.) Then ad A av ad v-+Av+ --+-=0 ax B ax at

{lO.la)

10.3 The characteristic forms of the open channel equations The equations can be cast into their characteristic forms in exactly the same way used for the waterhammer equations, but a different, simpler technique will be demonstrated here. First re-express the equations replacing the depth d by the small wave celerity c using the relating equation.

c2 g

d =-

(see equation 10.24a)

Then

ad

2c ac

ax =-g ax ad 2c ac at =g-at

182


The characteristic forms of the unsteady flow equations for a nonprismoidal channel are more complicated than those of the prismoidal channel and can be obtained using the Lister method presented in chapter 4. The prismoidal forms will be developed here. Assuming that A/B = d which is exactly true for a rectangular prismoidal channel, and closely approximately true for a broad channel, then equation 10.1 becomes:

c

2cv ac + av + 2c ac = 0 g ax g ax g at

(10.1b)

and the dynamic equation becomes:

. ") = 0 av av ( J-z ac 2c -+v-+-+g ax

ax

at

(1 0.3)

Dividing equation 10.1 b by cjg and adding equation 10.3

. . av av ac ac 2(v +c) ax+ 2 at+ (v +c) ax+ at+ g(J -z) = 0 Then

2((v+c)a~ +;)c+ ((v+c)a~ +a~)v + g(j-i) = 0

[(v+c)a~ +:tJ(v+2c)+E=O

(10.4)

where E = g(j - i). Next, divide equation 10.1 b by cjg and then subtract equation 10.3

.. avav ac ac 2(v-c)ax +2at-(v-c)ax -at-g(J-z)=O Multiply through by -1

r(v -c)..£_+ l_J(v-2c) + E= 0 ax at L So

[(v ±c) a~ + :t}v ± 2c) + E =0

Now, comparing this equation with the definition of a total differential i.e.

siP=

alP ox+ a~Pst at

ax

Then

ax alP

diP_ alP ---+-

dt

ax dt

at

183


a+a) at '{J

d'{J_(dx dt- dt ax

i.e.

If E is slowly varying and can be treated as a constant for the 8 t period and if dt 1 -=-dx v±c I(J=v± 2c +Et

then

dt 1 . 1 Thus m an x-t space dx = -+- defines two hnes, one of slope - - and 0

v-c

v+c

the other of slope - 1-; along these lines d'{J = 0 that is, '{J is a constant. In v-c dt fact, ddxt = - 1 - defines a positive characteristic line along which v + 2c + Et

v+c

0 ch aractensttc 0 0 10me aIong whotc h constant an d dt dx = -1- defimes a negative v-c v- 2c + Et is constant. Summarising the characteristic equations of free surface flow: 0 ts

dt 1 -=--

(lO.Sa and b)

d dt (v ± 2c + Et) = 0

(1 0.6a and b)

dx

v± c

As will be seen there is a very great similarity between these equations and those ofwaterhammer. There is also a great similarity in the way in which they are manipulated. The concepts of domain of dependency, zone of influence and zone of quiet described in chapter 4 apply equally to unsteady flow in open channels. The method of solving the value of c and v at any point on an x - t grid is exactly similar to that used in chapter 4 for the calculations of h and v values. Once c has been obtained, the depth d is immediately available, as d = c 2/g (see figure 10.5).

Figure l0o5

184


As in chapter 4 Vx and Cx and Vy and Cy must be calculated by linear interpolation of values at M and Nor Nand R, using the relation and Once XN and NY have been calculated, values of c and v at X and Y are simple to evaluate. Then and

Vx

+ 2cx +Ext= Vp + 2cp + Ep(t + Llt)

(10.7a)

=Vp- 2cp + Ep(t + Llt)

(10.7b)

Vy- 2cy + Eyt

Now Ep is unknown as it depends upon Vp and dp. It is given by Ep = J.vP~vpl -

"\Cp mp

i\ J

However, E varies slowly and will change very little in a Llt time interval. It is usual to approximate Ep to En (which is known of course). Some analysts go even further and approximate Ex, Ey and Ep to EN and this does not produce large errors if Llx and hence Llt are both small. Solving equations 10.7a and 10.7b simultaneously gives values ofvp and cp without difficulty and d is then given by dp = c~/g. The choice of Llt is more difficult to make than in the case of waterhammer analysis. As c = .Jid, and d can vary considerably during the passage of a flood surge or other large wave, c can also vary significantly. v can be a large fraction of c in subcritical flow and is greater than c in supercritical flow. Ll Ll Thus for a given value of Llx the Llt value is either~ or~ . v+c

v-c

If the negative characteristic were chosen the value of Llt could be very large but would be unsuitable for use with the positive characteristic. Using

the positive characteristic the Llt value becomes Llx . However, if a v+c

regular grid is to be used - as in the case of waterhammer - the Llt value must be fixed. Both v and c vary throughout the duration of the analysis so a Llt value must be chosen which will always ensure that points X and Y lie within the segment MR. A suggested value for Llt is Llx/(2c) where c in this instance equals the wave celerity calculated from the depth in the channel at the commencement of the analysis. This depth must be known. Either it may be assumed that the flow in the channel is in an initial steady state or that the flow is zero, the velocity everywhere being zero and the water surface being horizontal. The first of these alternatives requires the calculation of the surface proflle in gradually varied steady state using the usual equation, i.e.


185

where FN is the Froude number and equals Q2B

v2

A 3g or go where 8 = ~ and B is the surface breadth. Calculations of this type, as usually performed, are not really sufficiently accurate to be used as an initial steady state but if the unsteady analytic method is applied to this slightly erroneous steady state until it has become truly steady, the required unsteady condition can be applied and a satisfactory solution obtained. The second initial condition of zero flow avoids the need to perform a gradually varied flow analysis. The steady condition actually applicable to the circumstance must be applied as if it were an unsteady state and the analysis then performed for a sufficient time for the true steady state to develop. After this has been established, the required unsteady condition can be initiated and the unsteady analysis then performed. The second initial condition of zero flow requires a longer run time but saves the effort of calculating the initial gradually varied steady state. If v and/or c change so as to cause the domain of dependency criterion to be violated, the analysis becomes invalid. It is therefore necessary to provide an Algolrithm which outputs a warning message and aborts the run. The program should then be re-run using a smaller !JJ value. Unsteady supercritica/ flow

In supercritical flow the value of c is less than v. This implies that the slope of the negative characteristic becomes positive as illustrated in figure 10.6. In supercritical flow, information from downstream cannot affect conditions upstream as the local velocity vis greater than the wavespeed c,

M~l---x------~~~x--------~ Figure 10.6

and information travelling at wavespeed cannot be carried upstream through a fluid travelling downstream at a greater speed because waves travelling upstream relative to the liquid are washed downstream by the current. Consequently the downstream Ax length NR contains no information that effects conditions at P and the domain of dependency XY lies entirely within the upstream Ax length MN. Only one Ax length need be considered. Of course if supercritical flow occurs in the negative direction the relevant Ax length becomes NR.

186


In the preceding paragraph it was stated that in supercritical flow a wave cannot travel in the upstream direction. This is true for small waves. If a large shock wave develops, its celerity may be larger than the local velocity and under this condition it can travel upstream. The method of dealing with this situation is described later in this chapter. As flow becomes progressively more and more supercritical both the positive and the negative characteristics slope more and more downstream, tending to the horizontal and becoming progressively more parallel. This implies that the basic partial differential equations are tending to become more and more parabolic in type although they always, in fact, remain hyperbolic. However, the !::.t values that must be used become smaller and smaller as the flow becomes more supercritical so the run time costs increase for any given simulated time. In conditions such as these it becomes possible for the chosen At value to be too large. It is wise therefore to provide a facility for aborting a run and outputting a warning message to the effect that the value of XN has become greater than Ax (see figures 10.7a, 10.7b and 10.7c).

't ~~ /::,.x

I

X

Characteristics in a highly super critical flow

(a)

Crilical flows: wilh positive v

wi lh negative v

(b)

(c)

Figure 10.7

Boundary conditions As with waterhammer, unsteadiness of flow is generated from a boundary condition. There are a number of boundary conditions to be considered, some of which are listed here.

(1) A downstream estuary in which the level fluctuates with the tides. (2) An upstream catchment in which flows vary with rainfall producing flood or storm surges. (3) Upstream or downstream weirs. (4) Upstream or downstream sluice gates.

Unsteady flow in open channels (5) (6) (7) (8)

187

Downstrea m reservoirs in which the level fluctuates slowly. As (5) but with discharge over a spillway. An upstream spillway. A junction.

Tidal estuary The level history in a tidal estuary can be measured and is often available from records. Such a history can be read in as data and values of depth at the end of the channel at At intervals can be interpolate d from the readin array. The tidal level variation occurs at the downstream end of the channel so a forward characteris tic can be combined with tidal level dT at any At time level to obtain a solution for the local velocity (see figure

10.8).

p

~M~-X~------~N

Figure 10.8

Cp

= YidT

Vx + 2cx +Ext= Vp + 2cp + Ep(t +At) Vx and Cx can be obtained by interpolatio n. Ex can be calculated and Ep can be approxima ted to EN. Then Vp = Vx + 2cx +Ext- 2cp- EN(t +At) Once

up

has been found the boundary condition is solved.

Flood or storm surge

The parameter that must be known in this circumstan ce is the flow at the upstream end of the channel. It will vary with time and so a flow array must be fJ.iled by read-in values, from which upstream flows at At time intervals can be interpolate d. Then if the channel is broad or rectangular in cross section Qt+~t

so

= bdt+~tVt+M

Qt+~t

vt+~t = v = - - (as dp = dt+~t) p bdp

188

Hydraulic analysis of unsteady flow in pipe networks dp = c~fg

as

v =gQt+~t P

(10.8)

be~

Combining this result with a backward characteristic equation (see figure l 0.9) Vy- 2cy + Eyt = Vp- 2cp + Ep(t + ilt)

Substituting for Vp from equation 10.8 and equating Ep to EN

Qt+~t--2cp+EN ( t+ ilt) vy-2cy+Eyt=g-2 bcp This gives a cubic equation in Cp which can be readily solved by, for example, the Newton-Raphson method. Once Cp has been obtained vp can be calculated by back-substitution of cp into equation l 0.8. This solves the boundary condition. p

y

R

Figure 10.9

Upstream or downstream weir

Consider the case of a downstream weir first. The weir formula is: (10.9) 2

where Eu = du + ~;which is the specific energy just upstream of the weir, is the depth, and Vu the velocity just upstream of the weir and hw is the height of the weir crest above channel bed level. Use a positive characteristic (a negative characteristic for an upstream weir)

du

Vx + 2cx +Ext= Vp+ 2cp + EN(t + ilt)

Now Q = bdpvp for a rectangular channel so bdpvp =1.71b (dp- hw

As

and

+ ~J"s c2

d u =dp =~ g

(10.10)

189

Unsteady flow in open channels it follows that b

2

( 2

2)1.5

~ = 1.71 ~-h +2. g g w 2g

2)1.5

2

1.71 (~ -hw +2_ Vx + 2cx +Ext=

~

+ 2cp +EN (t + At)

2g

2

~ g

(I 0.11)

In this equation the term v~ appears. Upstream of a weir v is small and v~ is therefore also small. Initially v~ can be ignored. This feaves a cubic equation in Cp to be solved. Once this has been done v0 can be calculated. By substituting this Vp value back into equation 10.11 a second estimate of Cp can be obtained and the process repeated until two successive values of Cp are insignificantly different. In the case of an upstream weir a negative characteristic must be used in place of the positive characteristic. The Q value will have been obtained during the analysis of the weir as a downstream weir at the end of the reach upstream of the reach under consideration. Q will thus be known. The solution is thus identical to that for case 2, the flood or storm surge. Sluice gates

Consider a sluice gate located at the downstream end of a reach. The sluice gate equation is:

where cd is the coefficient of discharge, dco is the depth at the vena contracta immediately downstream of the sluice and equals Cc x d 5 where d 5 is the height of the opening below the sluice, Cc is the coefficient of contraction of the sluice, and Eu is the specific energy of the flow just up·

v2

stream of the sluice and equals du + 2;. Now du and Vu equal dp and Vp respectively, so (10.12) If the sluice is undrowned dd = dco but if it is drowned dd is the depth of the flow just downstream of the sluice. Using a positive characteristic Vx + 2cx +Ext= Vp+ 2cp+ EN(t +At)

and

(c

2 \ bc 2 v I v2 bdpvp=T=Cdbdcoy2g f+~-ddJ

190

so


=

V

Cdbdco~g(c~+ v~-dd\ g g 1) bc_ _ P 2

P

g

(10.13) As for the weir, the v~ term can be initially ignored and the resulting cubic equation in cp solved. Vp can then be solved and the v~/2g term evaluated and inserted into the equation which can then be solved again for cp, the process then being repeated until two successive values are insignificantly different. The solution of an upstream sluice is handled similarly to that of an upstream weir, the Q value having been obtained during the solution of the sluice when it was considered as a downstream sluice in the upstream reach, so the depth downstream of a sluice can be found by the method demonstrated for the upstream weir. The method illustrated for the sluice can be simply adapted to solve a Venturi-flume also. A reservoir in which the level varies slowly If the level fluctuation is very slow it will usually be possible to treat it as a constant level point for the duration of the unsteady flow analysis, i.e. dp = dreservoir so Cp = will be a constant. The solution is then trivial. Using a positive characteristic for a downstream reservoir (a negative characteristic for an upstream reservoir), then

..Jid;

Vx + 2cx +Ext= Vp + 2cp+ EN(t + and so

~t)

Vp = Vx + 2cx +Ext- 2cp- EN (t + ~t)

(10.14)

A downstream reservoir equipped with a spillway The water surface in the reservoir is assumed to be flat and horizontal. Assume that the plan area of the reservoir at a time t is At and the discharge equation of the spillway is Q5 = kh~'. Then let the flow from the channel entering the reservoir= Qc, so

dh Qc=At dt + Qs

i.e.

bdv =A{dp-dN)+kh 1"5 p p

t\

~~

sp


191

where hsp = height of the reservoir surface above the spillway crest. Thus bcJ,vP=A g t

(c~ic~) +kht.s g t

sp

using a positive characteristic Vx

+ 2cx +Ext= Vp + 2cp A

Now

Vp

t

+EN (t

+ ~t)

(c~gM - c ~) + kh t.s sp bCp g

This gives a cubic in Cp and once solved Vp can be obtained by back substitution. hsp must be increased by an amount of dp- dN before the next time level is analysed. The method of routing the flood through the reservoir assumes that no wave action occurs on the reservoir and that the level increments occur at the same time at all points on the reservoir surface. The method is commonly used for normal flood routing but this does not usually involve the unsteady analysis of the flow in the channel as described here. An upstream spillway

The upstream spillway case anises when a reservoir discharges into its downstream river or channel. The Q of the spillway discharge will have been calculated in the downstream reservoir example, so will be available, and the method demonstrated in the flood or storm surge and upstream weir can then be applied. A junction

At a junction the depth is the same for all channels joining there. The inflow to the junction must equal the outflow from it. The situation is thus the same as that described for waterhammer at pipe junctions. (For flows at other than small Froude numbers it will be necessary to include local losses and kinetic energy terms, however. If the depth is constant for all channels joining at the junction the value of the wave celerity Cp must also be the same for these channels.

192


Using the same methods as those used for the waterhammer case, continuity gives: (10.16) Where subscript a denotes the ath channel. As in the section on junctions in chapter 7, define the flow direction in any channel by the use of an integers which takes the value of+ 1 if the channel is transporting water towards the junction and -1 if it is transporting water away from it. Let the channel number 'a' be signed, 'a' being positive if the channel is transporting water to the junction and negative if away; then s = sign(a) Then for each channel joining at the junction, the characteristic equation will be (10.17) where Va and Ca are the interpolated velocity and wavespeed respectively on the t time level and vPa and cPa are the junction velocity and wavespeed on the t + !::.t level respectively, appropriate to the channel numbered a. (Note In equation 10.17 Ea is the value of Eat the junction end of the !::.x length.) Then (10.18)

Vpa = Va- 2s(cpa- Ca)- Ea!::.t substituting this value into equation 10.16 c• Lsba ;\va- 2s(cPa- Ca)- Ea!::.t)

=0

As cPa is the same for all channels, the c~afg term can be cancelled through. Then 'E.sbaVz- 'E.2baCpa + 'E.2baCa- 'E.sbaEa!::.t so

c

Pa

=

'E.sbaVa + 2 'E.baCa- ill 'E.sbaEa Z 'E.ba

=0 (10.19)

The right-hand side of equation 10.19 contains known values only so cPa can be evaluated. dp at the junction then equals c ~a/g. vPa can then be calculated for each channel using equation 10.18. Boundary conditions other than those already described can be modelled by similar methods.

10.4 The travelling surge The unsteady analysis described up to this point is based on equations which are accurate only when the flow is gradually varied and the stream-


193

lines are everywhere curved to only a small degree. If the small waves in the characteristics approach can run over one another, the flow through the resulting large wave will not be even approximately gradually varied but will be rapidly varied. For such cases a separate approach must be used. Consider a travelling surge, a large steep-fronted wave often with a roller front (see figure IO.IO).

~v1

~Vw

CD

f·· @

Figure 10.10

By applying a velocity Vw to the system as illustrated in figure IO.IO the free surface boundary can be brought to rest (figure I O.II ). Applying the continuity equation at sections I and 2 (v1 - Vw)At = (v2- Vw)A2

'1 t+··-~ ~0

E--~ ®

CD Figure 10.11

Applying the force equation to sections I and 2 wA 1z 1 - wA 2z 2 = w A 2 (v 2 - Vw) 2 -~A 1 (v 1 - Vw) 2 g g

A.z. -A2z2

=: (cv.- vw)~~f 2

-:•

Cv1- vw) 2 =(A~_A 1) (vl-Vw) 2 A2

A2 (A1z1 -A2z2) ~...,..:.----:--=---=:..:... (A1- A2)

g-

A1

g

(10.20)

where z is the depth of the centroid of the flow cross section below the free surface. A is the area of the cross section. This equation can be solved, although sometimes with difficulty, for any regular cross section.

194 i.e.


If v1 is zero Vw becomes the velocity of the wave through still water, Cw,

so

The plus sign applies if d1 > d 2 and the minus sign if d1 < d 2 • Of course, this celerity is larger than it would be for a small wave travelling over the fluid at the smaller depth, as will be seen from the following sections. A1

-A 2 ~-Bh

where h is the height of the wave 2 A 2Z2- ~A tZtA I (-Zt + h) -2-Bh 2 ~ A I h - Bh A tZt2-

Also

(10.21) Denote Atfb by 8 the mean depth based upon the surface breadth b. (10.22) If h is small compared with 8

Vw~Vt ± Jg((i +%h)

(10.23)

and if it is very small (10.24) i.e. for a small wave c =.Ji8 and for a rectangular channel 8

c =.Jid

=d

so (10.24a)

Of course, all of the above results are approximate compared with that quoted in equation 10.20. Commonly channels are of rectangular section or are effectively broad and then equation 10.20 reduces to: (10.25) So The result of this analysis of the travelling surge is exactly the same as


195

the dt = - 1 -result from the characteristics method and this is to be exdx v±c pected as small and large waves must obey this relationship. In this result, however, the value of cw is different from c in the characteristics equations. 10.5 The profJle of a free surface flow when a travelling surge is present The phenomenon of the travelling surge is not as simple to analyse as the simple equation derived above would appear to indicate. After a surge has passed a point in the channel the depth at that point does not stay constant. The proftle downstream of the surge changes with time due to storage and frictional effect, as illustrated in figure I 0.12. (This corresponds to line pack in pipelines as described earlier.)

Figure 10.12

Consider the sequence of events illustrated in figure I 0.12. If a travelling surge moving in an upstream direction is generated by a closing movement of the downstream sluice gate the wave will travel upstream as indicated, i.e. the wave will move through positions I, 2, 3 and 4 sequentially. As it does so the surface downstream of it will rise and progressively flatten and the wave itself will diminish in height. The amount by which the surface will rise may be much larger than the wave height so if this effect is ignored the result could be very erroneous. The analysis of the phenomenon is best performed by the characteristics method but a separate account must be kept of the surge wave as the characteristics method is not valid across a shock of significant size. 10.6 The method of analysis of an unsteady free surface flow in which travelling surges are present Various types of wave motion are possible as illustrated in figures I 0.13a, b, c and d. The negative wave illustrated in figure 10.13d cannot propagate with unchanged shape. Such a wave will rapidly flatten because the deeper portions of the wave will travel more rapidly than the shallow portions as their celerities are greater. The negative wave flattens as shown in figures I 0.14a, I 0.14b and 10.14c.

196


---- ------- ---~

~

(b)

(a)

~

~

(c)

(d)

Figure 10.13

The type of wave illustrated in figure 10.13a is sometimes called a rejection surge as it is generated by a reduction of the downstream flow. That illustrated in figure 10.13b is called a demand surge and is generated by an increase in downstream flow. The wave illustrated in figure 10.13c is called a flood surge and is caused by an increase in upstream flow. A wave that looks very similar to that in figure 10.13a is called an ebb surge and is caused by a reduction in upstream flow. It does not follow that increases or decreases in flows upstream or downstream necessarily cause a travelling surge. For a travelling surge to be generated it is necessary that the Froude number taken relative to the surge, i.e. (v1 - Vw) 2 /{gd1 ), must be either greater than or less than unity, whilst that downstream (v2 - Vw) 2 /{gd2 ) must be less than or greater than unity ( Vw is taken positive in the direction in which x is increasing) respectively. In other words the relative Froude number change across the surge must be such that the two relative Froude numbers must span the value of unity. If both relative Froude numbers are greater than unity or if both are less than unity no surge will be generated. Denoting the relative Froude numbers by FR 1 and FR 2 (the absolute Froude numbers are denoted by FN 1 and FN;) then a suitable test for the presence of a surge would be that (FR 1 - l)/~FR 2 -1) < 0.

(a)

(b)

(c)

Figure 10.14

197


Now if a surge wave is present the absolute Froude number on one side of the surge wave must be less than unity and that on the other may be either less than or greater than unity. If both absolute Froude numbers are less than unity, i.e. flow is sub· critical on both sides of the surge, both forward and backward characteristics may be available for upstream and downstream points on either side of the surge (see figure 10.15).

X

Figure 10.15

The velocity of the surge wave is given by

Vw=v1±

gd2(d1+d2\ 2d1

J

(10.26)

The plus sign is used when d1 > d 2 and the negative sign is used when d1 Vw > v2 + ..../i(i; for the case illustrated in figure 10.13c. Therefore the surge always travels faster than a small wave on the shallow flow and slower than a small wave on the deep flow. In figure 10.13a a small wave can run upstream on the deep flow, and overtake the surge which is travelling upstream, becoming incorporated into it. In figure 1O.l3b a small wave can travel downstream on the shallow flow and will catch up with the surge, whilst a wave travelling downstream on the deep flow will travel faster than the surge and so can not be overtaken by it. In figure 10.13c a wave on the deep flow will overtake the surge and become incorporated into it and the surge will overtake small waves on the shallow flow incorporating them into itself as it does so. It should now be clear that the flow upstream of the surge may be super or subcritical and that downstream the flow must be subcritical. Consider the conditions at the point at the upstream end of the ..:lx length and the conditions at the downstream end of the ..:lx length in which a surge wave is present. In figure l 0.16b characteristic PS crosses the wavefront and this invalidates the basic assumptions upon which the characteristic method is premised. Even so a simple technique can be evolved.

198

Hydraulic analysis of unsteady flow in pipe networks A slope

R M

N

v

T

u

w

o

__!_

v.

X

Upstream flow supercritical Downstream flow subcril ical (a)

X

x,

x2

Upstream flow subcritical Downstream flow subcritical (b)

Figure10.16

The wavespeed is assumed to be known at X (from a previous step of integration); see figure 10.17. Using this waves peed, point Y is readily located. From Y, three characteristics can be constructed YA, YB and YC. A must always lie upstream of Beven when the upstream flow is subcritical (as illustrated). When it is supercritical the point A is even further upstream. Depths (and hence celerities) and velocities at A, Band C can be obtained by interpolating appropriately between M and Nor Nand 0. Thus three characteristic equations can be written

AJongAY:

VA+ 2c A+ EAt= VVl + 2cY1 + Ey1 (t +

Along BY:

= vy2 + 2cY2 + EY2 (t + M) vc- 2cc + Ect = vy2- 2cy2 + EY2(t + ~t)

AlongCY:

VB+ 2cn + Ent

~t)

(10.27) (10.28) (10.29)

EYl can be approximated to EN and Ey2 to Eo. The continuity equation can also be written for conditions across the wave:

i.e.

(10.30)

Solving equations 10.28 and 10.29 simultaneously gives vy2 and cy2.

199

Unsteady flow in open channels wave path

/ _, L______ slope ..L v,. 0

M slope

-f.-w,

2

X

Figure 10.17

The wave equation I 0.26 is

+j!<~

V. _

w - Vt -

As before, use the positive sign if d 1 Cy22

2

(dt + dz) 2dt '

> d 2 and the negative if d 1 < d 2 •

Cy}2

Substituting-- for d 2 and-- for d 1 g g gives Vw y

=Vy}

±

(I 0.31)

substituting already calculated values for cy2 and vy2, substituting the expression for Vwy into equation I 0.30 and then solving the resulting equation simultaneously with equation 10.27 gives solutions for vy1 and CYl· Vwy can then be calculated by substitution of the values ofvy1, cy1, and cy2 into equation I 0.31. This value of Vwy2 gives the new slope of the wave path, i.e., I I Vwy and using this the next time step can be solved similarly. For a more extensive treatment of the topic of waves in open channels, Water Waves by Stoker 17 is recommended. If the backward characteristic toP crosses the wave path XY proceed as follows. Calculate the length AR and linearly interpolate for the values of VR and CR from VA and c A and vy and cy. Then use the forward characteristic equation toP and the backward equation from R to solve Vp and cp. 10.7 Other methods of analysis The method of characteristics has many adherents and it still is considered by most analysts to be the best method. However, a number of different methods have been evolved in recent years all of which are based upon

200


the integration of the St Venant equations using various finite difference schemes. The leading methods are listed below:

(l) The leap frog method. (2) The two step Lax-Wendroff explicit method. (3) Amein's four point implicit method. (4) Six point implicit methods- used by Liggitt and Woolhiser, Abbot and lonescu. It is claimed that these methods permit increases in l:lt values without significant increases in error or instability, giving a worthwhile reduction in computer run time. A paper by Price 11 outlines methods 1, 2 and 3 and describes the technique ofthe fixed characteristic grid used in this chapter. The paper suggests that Amein's four point method is superior to all others. These methods can also be applied to the analysis of water hammer but the author is not convinced that they offer any advantages over the fiXed mesh characteristic method used for waterhammer analysis in this book. Lack of space precludes the presentation of these other methods but the reader can study them in the papers listed in the Bibliography.

11 Global programming

11.1 Introduction

A global program is one which provides the facility of being able to describe the topology of any pipe network by the use of numbers, which are read in as data. Additionally the presence, or absence, of any particular hydraulic control, such as a pump, turbine, reservoir, air vessel at any location must also be capable of specification by the use of numbers. Such a program can be used to solve any network without amendment and if the facility is not available a new program will have to be developed for every network to be analysed. Obviously global programming increases the value of the technique of analysis many times. There are many ways of writing a global program but only one such method is presented here. In the literature the nodal method is usually described as being the method of choice but the author would prefer to present his own method which he believes to possess significant advantages over the nodal technique.

11.2 The route or link method of global programming

First a schematic of the network must be prepared. On this schematic all hydraulic controls must be entered and all controls of the same type must be numbered sequentially. For example, if the network contains ten pumps each pump must be assigned a unique number between 1 and 10. Similarly if there are 15 valves of the same type (but not necessarily possessing the same parameters) then each valve must be sequentially numbered from I to 15. Reservoirs, air vessels, surge tanks, etc, must be treated in this manner also. Next, arrows must be inserted onto the schematic to indicate the direction in which flow in each pipe is expected. If the arrow is inserted pointing in the wrong direction the steady state velocity for such a pipe will be found to be negative so it does not really matter which way the arrow points as long as it is remembered that a negative steady state velocity only im·

201

202


plies that the flow is in a direction opposite to that assigned by the arrow's direction. Next the pipes making up the network must be numbered. Here advantage can be taken of the fact that if numbering is always sequential across a joint, a pump, an air vessel, a surge tank, etc, less data need be supplied. To retain complete flexibility, however, then way junction must retain total freedom of numbering and no pre-determined sequences of numbering should be employed around a junction (except for a two-way junction, i.e. a joint). An array called route should be created. It should be two dimensional and should contain one row per pipe. Each row must contain n numbers where n is the total of numbers necessary to specify the conditions at the upstream end, the downstream end plus any other additional numbers associated with these boundary conditions. Considering a hypothetical route array, each row could contain the following numbers: In the 1st position: 1 - defines an upstream reservoir. (Upstream boundary 2 - defines a pump fitted with a free surface suction well. condition) 3 - defines an in-line booster pump. 4 - defines a joint. 5 - defines an air vessel and also a surge tank. In the 2nd position 6- defines a downstream reservoir. (downstream boun- 7 - defines a motorised valve. day condition) 8 - defines a servocontrolled valve. 9 - defines an n way junction. 10- etc 11 - etc In the 3rd position: a - the number of the upstream pump. If no pump is fitted then 0 must be supplied. In the 4th position: b 1 - the number of the upstream reservoir. If no reservoir exists on the upstream end of the pipe then 0 must be supplied. In the 5th position: b2- the number of the downstream reservoir. If no downstream reservoir is connected to the pipe than 0 must be supplied. In the 6th position: c - the number of pipes joining at a downstream c way junction, i.e. 0 must be supplied if no junction is present. In the 7th position: + pipe number transporting fluid to the junction 7th+ 1 position: + pipe number transporting fluid to the junction 7th+ 2 position: + pipe number transporting fluid to the junction 7th+ 3 position: -pipe number transporting fluid from the junction 7th+ 4 position: -pipe number transporting fluid from the junction 7th + c position: etc

Global programming

203

If c is 0, the 8th position should be assigned the value 0 also. In the 8th+ c position: d- the number of the motorised valve. It should be 0 if no valve is present. In the 9th+ c position: e - the number of the servocontrolled valve. It should be 0 if no valve is present. Any other boundary hydraulic control number can then be inserted in the lOth+ c, 11th+ c etc, position. It will now be realised that array 'route' must be dimensioned from 1 to 11 + c (if no other hydraulic controls other than those listed above are present) and 1 to netno. In Algol, dynamic arrays are permitted so n (where n = 11 + the largest value of c in the network) and netno (netno = the number of pipes in the network) can be read in and used to dimension the route array.

11.3 Pipe description Next an array 'net' must be created and supplied with values. This array must be dimensioned from 1 to 5 and 1 to netno (i.e. it is a two dimensional array). Each row of the array must carry five variables applicable to an individual pipe in the network. These are:

(I) Pipe length (2) (3) (4) (5)

Pipe diameter Pipe wall thickness Pipe roughness Elastic modulus of the pipe wall material

11.4 Longitudinal profiles Next an array 'neti' must be created. This array must be of type integer. In this array two numbers must be stored in each row and there must be one row per pipe in the network. As it is not possible to model the pipe length exactly it is necessary to calculate the number of Ax lengths that most nearly represents the pipe length and store this number in the 1st place in the row. In the 2nd place the number of points at which the elevation of the pipe centre line is to be supplied must be read in. (This information is necessary as by deducting the elevation of any point from the potential head at that point, the pressure head is obtained.) Two arrays must next be created, one called Xz and the other called Z. These arrays are two dimensional and are dimensioned from I to the largest value of neti [no, 2] (no is the pipe number) of any pipe in the network. In the first of these two arrays the distances from the beginning of each pipe to the points at which each elevation of the pipe centre line is to be supplied, is stored. One complete set of elevation point distances

204


must be supplied for every pipe in the network. In array Z (dimensioned identically to that of array Xz) the elevations of each point must be stored similarly.

11.5 Upstream reservoirs An array Ust (short for upstream) must be declared. In this array the level of the water surface in each upstream reservoir is stored. This array is dimensioned from I to the total number of upstream reservoirs present in the network.

11.6 Downstream reservoirs An array Dnst (short for downstream) must then be created. In this array sets of five numbers must be stored: one set per downstream reservoir in the network. A set of five numbers consists of the following:

(1) The level of the water surface in the downstream reservoir, at the beginning of a D.t interval. (2) The area of the reser10ir free surface. (3) The width of the weir at the end of the downstream reservoir (see section 7 .6). ( 4) The weir constant of the weir in the downstream reservoir. (5) The index in the weir equation.

11.7 Pump description A further array called puma must be created. This is a two dimensional array and there must be a row of 36 variables per pump in the network. The 36 variables contain values describing the pumps' characteristic curves, i.e. the H- Q curve and the Efficiency- Q curve, and allocated spaces in which values calculated can be stored. By supplying heads and corresponding flow values the A, B and C values in the H- Q equation can be calculated. By supplying efficiency values corresponding to the flow rates the constants in the efficiency - flow equation can also be calculated. Once calculated the constants are then stored back into the unassigned places in the puma array. To perform the computation of the H- Q equations' constants A, Band C and the Efficiency- Q equations' constants it is probably best to write a subroutine or procedure to calculate them. A call of this can be made which will perform the calculations and then make the necessary assignments to the puma array for each pump present.

Global programming 11.8 Pipe longitudinal profile at

~x

205

intervals

An array called Zd is required. This is a two dimensional array dimensioned from 1 to the largest value of neti (no, 1) of any pipe in the network and from 1 to netno. In this array, values of pipe centreline elevations at ~x intervals are stored. These values are obtained by linear interpolation between appropriate values taken from the Xz and Z arrays. 11.9 Calls of procedures Having read in all the necessary data and made their assignment to the appropriate positions in the various arrays it becomes necessary to call the relevant procedures or subroutines at the correct point in the control sequence of the program run. This is done in the following way. At any time level the entire network is scanned in pipe numbering sequence. At every ~x distance point the local condition is examined to see what procedure/subroutine should be employed to calculate the potential head and velocity at the point at a time a ~t interval later. To decide which procedure/subroutine is to be used a simple algorithm is used, e.g. if the position under examination is located at the upstream end of a pipe and route(no, 1) = l then call the procedure/subroutine for the upstream reservoir. If route(no, 1) = 2 then call the subroutine for the suction well pump. If route(no, 1) = 3 call the subroutine for the in-line booster pump. If the point under examination is located at the downstream end of a pipe then: if route(no, 2) = 6 call the subroutine for the downstream reservoir, if it equals 7 call the motorised valve subroutine, if it equals 8 call the servocontrolled valve subroutine and if it equals 9 call then way junction subroutine. For midstream points the midstream subroutine should be used. The subroutine, 'upstream reservoir' needs a parameter which specifies the reservoir number- this it obtains from route(no, 4). It uses this to obtain the relevant surface level from the array Ust. Similarly the downstream reservoir subroutine requires the reservoir number and it obtains this from route(no, 5). Using this number it obtains the five relevant data items from array Dnst. Other subroutines obtain their data similarly; in particular, subroutines 'suction well pump' and 'inline booster pump' obtain the parameters defining their numbers from route(no, 3) and use this to obtain the parameters from array puma which are needed in the computations that they perform. The pipe numbers used in 'n way junction' are obtained from route(no, 7), route(no, 8) etc, up to route(no, 7) + route(no, 6). The method of turning a specific program into a 'global' program should now be obvious. The procedures/subroutines described in previous chapters should all be included and suitable output sections should be added. The writer of

206


such a program is warned that iflarge amounts of output are expected it is best to produce the output directly via a graph plotter. However, some line-printer output is recommended as it is difficult to pinpoint errors from graph plotter output alone. 11.1 0 Time level scanning

Having completed one scan of the network the counter i which determines the simulated time T of the run obtained from T = i x tlt must be increased by 1 and the scanning process repeated. This should be repeated until the ·total simulated time equals or just exceeds the specified run time. As tlt is usually small and tlx is usually between one tenth and one twentieth of a pipe line length it may be thought desirable to inhibit output by only outputting every 5th or lOth tlt level and every 2nd or 4th tlx point. Integer type parameters can be read in as data to enable choice of the amount of output to be made. Graph plotter output can also be controlled similarly and this can be important as output storage must be kept to reasonable proportions.

References

1. Allievi, L. Teoria generale del moto perturbato dell' acqua nei tubi in pressione, Milan 1903. Translated into English by E. E. Halmos, The Theory of waterhammer. Anz. Soc. Civil Eng., 1925. 2. Schnyder, 0. Druckstosse in Pumpensteigleitungen. Schweiz Bauztg., 94, Nos. 22 and 23, 1929. 3. Bergeron, L. Etudes des variations de regime dans les conduites d'eau. Rev. gen. Hydraulique, Nos. 1 and 2 (1935). 4. Zienkiewicz, 0. C. and Hawkins, P. Transmission of waterhammer pressures through surge tanks. Proc. Inst. Mech. Eng., 68, No. 25 (1954). 5. Angus, R. W. Waterhammer in pipes, including those supplied by centrifugal pumps; graphical treatment. Proc. Inst. Mech. Eng., pp. 136 and 245 (1937). 6. Angus, R. W. Waterhammer pressures in compound and branched pipes. Trans. Anz. Soc. Civ. Eng., pp. 104 and 340 (1939). 7. Lax, P. and Wendroff, B. Systems of conservation laws, Comm. Pure Appl. Maths, XII, 217-37 (1960). 8. Lax, P. Weak solutions of nonlinear hyperbolic equations and their numerical computations, Comnz. Pure Appl. Maths, VII, 159-93

(1954).

9. Lister, M. The numerical solution of hyperbolic partial differential equations by the method of characteristics, in Mathematical Methods for Digital Computers (ed. Wilf, A and Ralston, H. S.) Wiley, New York (1960). 10. Courant, R., Friedrichs, K. and Lewy, H. On the Partial Differential Equations of Mathematical Physics, New York University Institute of Mathematics, translated by P. Fox (1956). 11. Price, R. K. Comparison of four numerical methods for flood routing, J. Hydr. Div., Anz. Soc. Civ. Eng., July (1974). 12. Pearsall, I. The velocity of waterhammer waves in Symposium on Surges in Pipelines, Inst. Mech. Eng., 180 (1965-66). 13. Karplus, M. B. The velocity of sound in a liquid containing gas bubbles, Armour Research Earth Foundation Report, June (1958). 14. Fox, J. A. An Introduction to Engineering Fluid Mechanics, Macmillan, London (1974).

207

208


15. Suter, P. Representation of pump characteristics for calculation of waterhammer, Sulzer Review, No. 1966. 16. Edgell, G. Pressure Transients in Tunnels; Extension of Theory to !"eversible, Nonadiabatic Flow. Limited publication by Leeds University, Department of Civil Engineering, April (1974). 17. Stoker, J. J. Water Waves, Pure and Applied Mathematics, vol. 4, The Institute of Mathematical Sciences, New York University (1957) 18. Swaffield, J. A. A Study of column separation following valve closure in a pipeline carrying aviation kerosine, Proc. lnst. Mech Eng., No. 23 (1969). 19. Marsden, N. and Fox, J. A. An alternative approach to the problem of column separation in an elevated section of pipeline, Proceedings of 2nd International Symposium on Pressure Surges, September 1976.

Bibliography

I. Angus, R. W. Simple graphical solution for pressure rise in pipes and pump discharge lines, J. Inst. Ca1Ulda, February, 72-81 (1935). 2. Donsky, B. Complete pump characteristics and the effects of specific speeds on hydraulic transients, J. Basic Eng., December, 685-99

(1961).

3. Fox, J. A. The use of the digital computer in the solution of waterhammerproblems,Proc. Inst. Civ. Eng., 39,127-31 (1968) 4. Fox, J. A. and Henson, D. A. The prediction of the magnitudes of pressure transients generated by a train entering a single tunnel, Proc. /nst. Civ. Eng., 49, 53-69 (1971). 5. Henson, D. A. and Fox, J. A. Transient flows in tunnel complexes of the type proposed for the channel tunnel, Proc. Inst. Mech. Eng., 188, No. I 5, I 53-67 (1974). (Two papers). 6. Jaeger, C. Engineering Fluid Mechanics, Blackie, London (1956). 7. Knapp, R. T. Complete characteristics of centrifugal pumps and their use in prediction of transient behaviour, Trans, Am Soc. Civ. Eng., 59, 683-9 (I 939). 8. Lax, P. D. and Richtmyer, R. D. Survey of the stability of finite difference equations, Comm Pure. Appl. Maths, ix, 267-93 (I 956). 9. Parmakian, J. Waterhammer A1Ullysis, Dover, New York (1963). I 0. Pickford, J. A1Ullysis of Surge, Macmillan, London ( 1969). 11. Rich, G. Hydraulic Transients, Dover, New York (1963) 12. Stepanoff, A. J. Centrifugal and Axial Flow Pumps, Wiley, New York

(1948). 13. Stoker, J. 1. Water Waves, Pure and Applied Mathematics, vol. 4, The Institute of Mathematical Sciences, New York University (1957). 14. Streeter, V. and Wylie, E., Hydraulic Transients, McGraw-Hill, New York (1967). 15. Symposium on surges in pipelines, Proc. Inst. Mech. Eng., 180, part E (1965-66). 16. Wilf, H. S. and Ralston, A. Mathematical Methods for Digital Computers, Wiley, New York (1960).

209

Index

Buckling mode I 25 Abbot and Ionescu method 200 Bursts 112 Acceleration angular 56 convective 81, 8 5 Actuators 136, 140 Capacitance 157, 160 Adiabatic 148 Catchments 186, 187 Air vessels 39, 64, 68, 124, I 25, Cavity 50 126, 127,134, 135, 136, Celerity 78, 181, 184, 186, 191, 201, 202 194, 195, 198 Allievi expression 17 Channels, open I 77 Allievi interlocking equation 21, 25, equations for unsteady flow in 29, 35, 36 178 Allievi pipe characteristic 35 nonprismoidal 180, 182 Amein's four-point method 86, 200 nonrectangular I 77 Analytical techniques I, 23 prismoidal 180, 182 Anchor block 146 rectangular I 77 Arrays 118,119,120,140,187, supercf"itical flow in I 85 202, 203 Characteristic 79, 80, 81 for global programming 203 Characteristic equations 73, 74, 80, Attenuation 13, 15, 16, 63, 64, 127, 84,129,149,181 128,134,161 in gas flow 148 Characteristic line 7 8, 131, 13 2 Choke, see throttle Coefficient of discharge 6, 24 Bends 146 Bernoulli equation 6, 25 Coefficient of heat transfer !51, Bi-Characteristic method 177 !52, 153 Colebrook-White equation 95, 96 Boiling I 7, 94 Bore 178 Column separation 16, 17, 50, 95 Boundary conditions 36, 39, 55, 73, Compressor 65, 125, 154 81, 108, 130, 154, 167, Computerised methods 85, 201 Continuity equation 26, 29, 30, 46, 186 Boundary layer 96 66, 67, 68, 70, 73, 85, 147, 178, 179, 193, 198 Bubble 17, 72, 87, 88, 89, 90, 91, Corrector-predictor method 71 93, 94, 95, 99,110, 112 Courant and Lewy stability evolution of 88 criterion 82 Buckle, running 50, 94

211

212

Index

Crack, surface 89 Crack, micro- 89, 90 Current 157 Darcy-Weisbach equation 4, 29, 95, 160 Differential equation, hyperbolic partial quasi-linear, 29, 73, 74 Dimensional analysis of rotodynamic machines 57 Dispersion of wave 83 Distensible pipe-elastic fluid theory 2, 9 Domain of dependency 78, 183, 185 Dump tank 39, 73 Dynamic equation 2, 3, 26, 28, 30, 67' 68, 69, 73, 85, 148, 179, 193 Dynamometer mode of pump operation 62, 100, 102, 103, 113 Eagre lines 36, 38, 39, 40, 42, 45, 46, 47, 48,49, 53 plotting of 43 Edgell, G., 150 Efficiency, pump 56, 57, 60 Elasticity effects 9 Elasticity of fluid 6, 9 Elasticity of pipe wall material 6 Elliptic partial differential equations 75 Ends, blank 167 receiving 16 5 sending 165, 173 Energy, conservation 13 kinetic 14, 46, 55, 191 strain 6, 13 velocity 13 Equation of state 148 Estuaries 186, 187 Euler equation 3 Expansion joints in pipes 18 Fanning equation, see DarcyWeisbach equation Finite difference methods 7, 9, 68, 85, 107, 199

Flood plain 177 Flow, two phase 50, 87, 92 Flywheels 124, 125, 126 Fourier analysis 173 Francis turbine 156 Frequencies, dangerous 172 Frequency 173 Friction 4, 13, 23, 28, 29, 36, 49, 66, 67, 68, 96, 99, 108, 121, 128,148 variable 96 Frictional damping 156 Froude number 185, 191, 197 relative 196 Fundamental 156, 173 Galleries 65 Gas 81, 89, 91,92 bubbles 72, 87, 88 calculation of free content 98 compressed 112 evolution 88 release 90 release head 93, 95, 96, 98, 99, 112 Global programming 201 Governor and mechanism 63 Graphical method 36, 50 Graphical techniques 36, 55, 72 Grid, regular rectangular 82, 86 Hagen Poiseuille formula 160 Harmonic 155, 17 3 Harmonic analysis 172, 173 Hartree method 82 Head frictional 2 9, Ill inertia 10, 15 maximum 56 no flow 55 pump 56 Heat flow 150 transfer 152 Henry's Law 88 Hydraulic controls 1, 36, 37, 62, 112,201 Hydraulic radius 28 Hydroelectric installations 156 Hyperbolic partial differential equations 7 5 Hypersonic 7 6

Index Impedance 172, 173, 176 characteristic 164, 16 8 concept 164 equation 165 hydraulic 164 loops 169 methods !55 networks 170 Impeller, pump 55, 56, 57, 100, 101, 103, 104, 117 roughness 57 Inductance !57, 160 Inertia, moment of 56, 60, I 04, 122, 123, 124, 125 Interpolation 83, 97, 99, 115, 118 120,140,184,187,198 Isothermal 148 Iterative methods 70, 84, 96 Joints 132, 202 impedance theory of 168 method of dealing with 45 Joukowsky 17 Junctions 14, 39, 46, 48, 65, 73, 167 impedance theory of 167, 169, 170, 173 in open channels 187, 191, 192 n way, 130, 202 Korteweg 17

Lax- Wendroff method 86, 199 Leap frog method 86, 199 Ligget and Woolhiser method 86, 200 Line pack 127, 128, 129 Link method see Route method Lister method 7 4, 18 2 Local losses 5, 46, 68, 130, 133, 154, 191 Lock in 128, 129 Loops in networks 169, 170, 176 Method of characteristics 74, 77, 85, 107 Micronuclei 89, 90

213

Modulus 172 bulk 18, 20, 72, 87, 90, 91 Young's 18, 20, 87 Moens 17 Momentum II Motor I 03, II 8, I 20 Motor armature 56 Network 36, 73, 94, I 00, I 03, 112, 124, 130, 136, 145, 168, 169,172,173,176,178, 201, 206 gas 147 open channel 178 resonant I 6 7 Newtonian theory I Newton- Raphson 154, 188 Newton's 2nd law of motion 2, 28, 148 Nodal method 20 I Nodes 65 Nonprismoidal channel 180, 181, 182 Open channels 177 nonrectangular I 77 prismoidal 177 rectangular 177 Organ pipes !55 Organ piping 65 Orifice 49, 63, 68, 176, see throttle Oscillating component of flow 158, 165 of head 158, 165 Oscillating valve 155, 174 Oscillation forcing 166,172,173,174,176 frequency of 161 mass 62, 65, 66, 134 sinusoidal 162 Oscillatory wave 161 Parabolic partial differential equations 7 5 Pelton wheel !56 Perimeter, wetted 28 Period, pipe 22, 23, 25, 41, 60, 125 Pipe bends 146 blank end of 167

214

Index

Pipe (contd.) by-pass 126 description for global programming 203, 205 joints 45, 13 2, 168 junctions 46, 48, 73, 130, 154, 167 lengths 73 loops 169 period 22, 23, 25, 41, 60, 125 Poisson's ratio 20 Polytropic efficiency !54 Polytropic index 69, 127 Polytropic process 70, I 27, 148 Potential head equation 84, 99 Power 55 Pressure head amplitude 173 Pressure head, calculation at t:.t time intervals 41 Pressure rise, after instantaneous valve closure 21 Pressure transients, see Transients and waterhammer Price 200 Prismoidal channels 180, 181, 182 Procedure calls, in global programming 205 Propagation constant 161 evaluation of 162 Pump by-pass 120, 1 26 characteristic equation 55, 56, 58, 59, 60, 62, 100, 101 102 , compound arrangement 123 description for global programming 204 dynamometer behaviour 102 113 , efficiency 56, 58, 59, 60, 105, 122, 123, 124, 202 four quadrant operation 60, 111, 113, 120 impeller 55, 56, 57 in-line 108, 176 in parallel 1 22 in series 122 power 104, I 05, 106, 107 run-down 107 run-up 120 start-up 106 stations 120, 1 21

suction-well 11 0 torque 56 trip 55, 56, 73, 103, 110, 111, 112, 121,124, 126 turbine behaviour 1 03, 111, 113 Pumps 55, 56, 57, 100, 201, 202, 204, 205 Reflection I 0, 13, 3 2, 3 6, 55, 1 00, 128, 173 negative 14, 22, 23, 25, 146 partial 14 Regular rectangular grid method 82, 184 Reimann equations 30, 32, 37 Reservoir50 characteristic 39 42 45 , , , Reservoirs 39, 40, 50, 55, 65, 144, 145, 146, 154, 167, 173, 187, 190, 191, 201,204, 205 static water level 62, 63, 66, 68 Resistance, electrical 157, !59, 160 Resonance 65, 15 5, 156, 164, 172, 173, 176 Reynolds number 95, 96 Rigid pipe-incompress ible fluid theory 2, 66 Riser 63, 64, 68 Route method, in global programming 201, 202 Schnyder-Berger on graphical method 36, 49, 72, 85, 129 Separation 16 Shock wave 186 Simple harmonic motion 67 Sluice gates 186, 189, 190 Solenoid 126 Sound 1 Specific energy 188, 189 Specific speed 115, 117, 118 Speed, rotational 56 Steady flow component 158 Steady head component 158 Stoker, J. 199 Strain circumferential 18 diametral 18 volumetric 91

Index Strain energy 6 Streeter, V. 176 Stress, hoop 18 St. Venant equations 180, 199 Subcritical free surface flow 197, 198 Supercritical free surface flow 76, 185, 186, 197, 198 Superposition 17 6 Surface tension 88 Surge 197 demand 196 ebb 196 floodorstorm 186,187,189, 196 rejection 196 suppression 124 travelling 17 8, 192, 1 93, 19 5 Surge tanks 47, 62, 68, 124, 125, 126, 127,201,202 choke ring 63, 68 equation integration 70 Johnson differential 63, 68 mass oscillation 66 pressurised 64, 65, 68, 125, 134 simple 63 transient analysis 65 Suter diagrams 116, 117, 118 Thermodynamics, first law of 149 Throttles 85 causing friction 49, 50, 52, 53, 54 multiple 52 Time level scanning, in global programming 206 Torque, pump 56,115,117,118, 120 Transducer, pressure 143 Transients I, 10, 16, 17, 22, 29, 36, 42, 50, 55, 60,63, 65, 72, 73, 87, 88, 90, 100, 103, 106, 111, 112, 125, 146, 156 Transmission line equation 157 Transmission line theory 161 Turbine gate 62, 63 Turbine mode of operation of a pump 62, 100, 102, 103, 107, 111, 113 Turbo-blowers 154

215

Unidimensional flow in channels 177 Vacuum 94 Valve 103, 129, 201 ball 155 butterfly 112, 156 by-pass reflux 1 20, 126 characteristic 3 9, 40, 41, 4 2, 52 control 144 downstream closing 112 opening 4 effective area 17 5 fractional opening 35, 39 instantaneous closure 10, 21 linear closure 8 motorised 136, 137, 138, 139, 140, 141, 205 nonreturn or reflux 55, 100, 111, 112,120,121,128,129, 132, 134, 144, 145 part closed 144 reflux, see nonreturn servocontrolled 72, 142, 156, 175 slow closure 6, 17, 22, 23, 45, 125 slow opening 42 slow partial closure 43 stepwise closure 23 sudden closure 21, 125 Vapourous cavitation 94, 95 Vapour pressure 16, 50, 94, 110 Venturi-flume 190 Voltage 157 Volute 100 Waterhammer 1, 9, 28, 36, 60, 73, 77, 78, 85, 86, 93, 121, 134, 156, 157, 160, 181, 183, 184, 186 Waterhammer equations 73, 74, 77 integration of 79 linearisation of 157, 160 Waterhammer theory 1 history of 2 Waves 10, 11, 12, 13, 14, 15, 16, 21, 173 compressional 2

216

Index

Waves (contd.) [type 24, 25, 31, 32, 33 F type 24, 25, 31, 32, 33 formation I 0 propagation I 0 shape 15 Wave form 64, 173 Waverider 38, 40, 41, 42, 45, 46, 47, 53 Wavespeed 18, 42, 72, 85, 97 ·-99,

151,161,177,185,198, 199 equation 84, 99, 161 magnitude of variability 90 minimum 93 variation in 87,94 Weirs 64, 145, 146, 186, 188, 190 Zone, of influence 78, 183 of quiet 79, 183

Hydraulic Analysis of Unsteady Flow in Pipe Networks

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