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

Hydraulic analysis of unsteady flow in pipe networks

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

Simple waterhammer theory

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

Hydraulic analysis of unsteady flow in pipe networks

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.

Simple waterhammer theory

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

Hydraulic analysis of unsteady flow in pipe networks

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

Simple waterhammer theory

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)

Figure 1.4 (continued)

Simple waterhammer theory

J

13

L

:4

""reservo1r

X

~reduced pipe diameter

pressure head plot

(f)

Figure 1.4 (continued)

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)

Figure 1.4 (continued)

reduced pipe d1ameter

14

Hydraulic analysis of unsteady flow in pipe networks

~

"reservoir

L

-·

pressure head plot

I= 4L

c

(h)

Figure 1.4 (continued)

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

Simple waterhammer theory

h

valve closure

occurs

J/ L

h,

_{ c

(b)

Figure 1.5 (continued)

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)

Figure 1.5 (continued)

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

Simple waterhammer theory

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

Hydraulic analysis of unsteady flow in pipe networks

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

Simple waterhammer theory

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

Hydraulic analysis of unsteady flow in pipe networks

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

Simple waterhammer theory

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

Hydraulic analysis of unsteady flow in pipe networks

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

Hydraulic analysis of unsteady flow in pipe networks

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

Hydraulic analysis of unsteady flow in pipe networks

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

Hydraulic analysis of unsteady flow in pipe networks

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

Analytic and graphical methods

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-

Hydraulic analysis of unsteady flow in pipe networks

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

Analytic and graphical methods

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

Hydraulic analysis of unsteady flow in pipe networks

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

Hydraulic analysis of unsteady flow in pipe networks

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

Hydraulic analysis of unsteady flow in pipe networks

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

Analytic and graphical methods

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

Hydraulic analysis of unsteady flow in pipe networks

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.

Analytic and graphical methods

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

Analytic and graphical methods

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

Hydraulic analysis of unsteady flow in pipe networks

/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.

Analytic and graphical methods

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

Analytic and graphical methods

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

Hydraulic analysis of unsteady flow in pipe networks

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

Hydraulic analysis of unsteady flow in pipe networks

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>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

Boundary conditions for use with graphical methods

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

Hydraulic analysis of unsteady flow in pipe networks

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

Boundary conditions for use with graphical methods

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

Hydraulic analysis of unsteady flow in pipe networks

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

Boundary conditions for use with graphical methods

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

Hydraulic analysis of unsteady flow in pipe networks

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.

Boundary conditions for use with graphical methods

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

Hydraulic analysis of unsteady flow in pipe networks

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 -4prn

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

Hydraulic analysis of unsteady flow in pipe networks

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.

Boundary conditions: pumps

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

Hydraulic analysis of unsteady flow in pipe networks

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.

Boundary conditions: pumps

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

Hydraulic analysis of unsteady flow in pipe networks

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

Boundary conditions: pumps

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

Other boundary conditions

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

Hydraulic analysis of unsteady flow in pipe networks

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)

Other boundary conditions

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

Hydraulic analysis of unsteady flow in pipe networks

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

Other boundary conditions

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

Hydraulic analysis of unsteady flow in pipe networks

~~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)

Other boundary conditions

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

Hydraulic analysis of unsteady flow in pipe networks

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

Other boundary conditions

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

Hydraulic analysis of unsteady flow in pipe networks

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

Other boundary conditions

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.

Other boundary conditions

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

Hydraulic analysis of unsteady flow in pipe networks

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

Other boundary conditions

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

Hydraulic analysis of unsteady flow in pipe networks

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

Hydraulic analysis of unsteady flow in pipe networks

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

Hydraulic analysis of unsteady flow in pipe networks

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

Hydraulic analysis of unsteady flow in pipe networks

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

Hydraulic analysis of unsteady flow in pipe networks

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

Hydraulic analysis of unsteady flow in pipe networks

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

Impedance methods of pipeline analysis

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

Hydraulic analysis of unsteady flow in pipe networks

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

Impedance methods of pipeline analysis

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

Hydraulic analysis of unsteady flow in pipe networks

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

Impedance methods of pipeline analysis

Figure 9.1

r

and

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