RESEARCH NOTE

ANALYSIS AND DESIGN OF A SIMPLE SURGE TANK

M. Azhdary Moghaddam

Civil Engineering Department, University of Sistan and BaluchestanZahedan, Iran, mazhdary@hamoon.usb.ac.ir

(Received: Decembcr 9,2003 - Acccpted in Revised Form: October 14, 2004)

Abstract In a hydroelectric power plant or in a pumping station in order to avoid sudden largeincrease of pressure due to instantaneous valve closure sometimes a surge tank is installed. The heightof surge tank is designed by the highest possible water level during the operation. The theoreticaltreatment of oscillation in a surge tank is difficult because of the non-linearity of friction tern1 in thegoverning differential equation of the system. The present study attempts to find a general solution forthe surge oscillation in a simple surge tank in terms of non-dimensional parameters. Equations for thehighest and the lowest water level in the tank, which are very important in the design of a surge tankhave been found.

Key Words Surge Tank, Pressure, Surge Oscillation, Non-Linear Differential Equation

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1. INTRODUCTION

When the valve in a hydroelectric power plant issuddenly completely closed, because of its smallinertia the water in the penstock stops almost atonce [1,3]. The water in the pipeline, with largeinertia retards slowly. The difference in flowsbetween pipeline and penstock causes a rise in thewater level in the surge tank. The water level risesabove the static level of the reservoir water,producing a counter-pressure so that water in thepipeline flows towards the reservoir and the levelof water in the surge tank drops [4].

In the absence of damping, osciJIation wouldcontinue indefinitely with the same amplitude. Theextent of damping is governed by roughness condition,restricted orifice, and so on. The flow into the surgetank and water level in the tank at any time during theosciHation depends on the dimension of the pipelineand tank and on the type of valve movement. The

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main functions of a surge tank are [5]:1. It reduces the amplitude of pressure fluctuationsby reflecting the incoming pressure waves;2. It improves the regulation characteJistic of ahydraulic turbine.Depending upon its configuration, a surge tank may beclassified as simple, orifice, differential, or closed. Asimple surge tank is defined as a tank or shaft ofconstant horizontal cross sectional area that connectsthe conduit of a hydroelectric power plant forpreventing the pressure surges entering into it (FigureJ). The maximum amplitude of water level (maximumsurge) can be observed when a fuIJ load is suddenlyrejected [6].

2. MAIN CONSIDERATIONS IN THE DESIGNOF A SURGE TANK

in order to accomplish its mission most effectively,

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RESERVOIR

- - -----1~

x~

H..

~--- - --~y.

I

z.,

SIMPLE SURGE TANK

Dr

Figure 1. Simple surge tank system.

the surge tank dimensions and location are basedon the following considerations [6], [7]:I. The surge tank should be located as close to thepower or pumping plant as possible;2. The surge tank should be of sufficient height toprevent overflow for all conditions of operation;3. The bottom of surge tank should be low enoughthat during its operation the tank is drained out andadmit air into the turbine penstock or pumpingdischarge line; and4. The surge tank must have sufficient crosssectional area to ensure stability.

The height of a surge tank is governed by thehighest possible water level that can be anticipatedduring its operation. All available methods arebased on a linearized resistance relationship, sincethe resistance law flow varies as Reynolds numberand relation roughness [2]. These equations describeapproximate values of peaks and downsurges.

3. ANALYSIS OF SURGES IN SIMPLESURGE TANK

In a simple surge tank, there is very little head loss

340 - Vol. 17, No. I, January 2005

between the surge tank and the pipeline, also thereservoir is considered so large that its levelremains constant [8].

3.1. Derivation of Governing Equation Tosimplify the derivation of dynamic and continuityequations that describe the oscillations of the waterlevel in the tank, it has been assumed that(i) the conduit walls are rigid;(ii) the water is incompressible; and(iii) the effect of entrance loss in comparisonwith the friction loss has been neglected.The equation of motion is written as [9], [12] :

av av ay( S )-+V-+g-=g So- f

at Ox Ox(l.a)

av + V av + g ay =-g(az+ ahf

)at Ox Ox ax ax (l.b)

Integration (l.b) with respect to x between the limitsx=O,x=L (see Figure I) and simplifying, one gets

dVL - + gh f + gy = 0

dt(2)

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From continuity condition between tank and pipe,it can be shown that

V=(

DT

)2 dy

Dp dt(3)

That with use of 3 in 2 the following equation canbe found:

)

2D d2

L(--.l -.r + gh + gy = 0D de fp

(4)

With initial condition at t=O

y = -hfo (5.a)

dy =(

Dp

J2Vo

dt DT(5.b)

Equation 4 is the governing differential equationfor surge oscillation in a simple surge tank.

3.2. Adopted Methods for the Solution Dependingon the nature of friction loss, the followingparticular cases arise:

3.2.1. Frictionless Flow In this case hf = -hro= O.Thus Equation 4 changes to

d2y+gD~ =0dt2 D2LYT

(6)

Solving 6 for the initial condition 5.a, 5.b one gets

V DT

If.

Y= 0 - - SIlltDp g

(7)

Equation 7 describes sinusoidal oscillations[ 10].

3.2.2. Laminar and Turbulent Flow In thiscase head loss is expressed by Darcy-Weisbachequation

hf = fLV22gD2p

(8)

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Combining 3 and 8 one gets

fLV2 fLD2

(d

)

2

hf = 2gD~ = 2gD1 d~(9.a)

With the change of flow direction, the directionof

friction also changes. Hence (~~)2 occurring in

. dy

l

dy

l

9.a should be splIt to- - . Thus 9.a changes todt dt

h = fLDi dyI

dylf 2gD: dt dt

(10)

In which 8[11]

0.12

(

64

)8 9.5 -

f-I - + ~-'vv- Re

lLnl~+ ~~;j-l~ j j J

Dp e (11)

Substituting 10 in 4 one gets

d2y + fDi dyI

dyl

+ gD~ - 0dt2 2D~ dt dt DiL y-

(12)

The initial conditions prescribed on 12 are at t=O

y=-h - foLV2fO - 2..2gD2p

(13.a)

dy =(~

J2 Vo

dt DT(13.b)

In order to reduce the number of parameters, thefollowing dimensionless groups are formed:

y.=y~ rgDpVo~L

(14.a)

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TABLE 1. Coefficients of Equation 20.

t, = t Dp rgDTVL

(l4.b)

The conduit velocity may be expressed as

v =V dy,0-

dt,(15)

Substituting 15 in 11 one gets

R = R dy*e 0-

dt,(16)

Ro = VoDpv

(17)

in which Ro is the initial Reynolds numberdefined as using non-dimensional variables 12reduces to:

dZy, +~h ,dy'l

dY,, + ,=0

dt; fo fO dt* dt, y(18)

The initial conditions for sudden valve closure att=O

dy, = 1dt,

(l9.a)

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and the initial conditions for sudden valve

opening are

dy, =-1dt,

(l9.b)

Being non-linear it wil1 be very difficult to find aclosed form solution of 18. A numerical solution,using a fourth order Runga Kutta method, can beattempted. From this numerical solution one cansee that the various properties of the solution like

Ypl', tpl" Ytl', and ttl' are functions ofRo, ~, andDp

hro',

With varying 0::; ~ ::;0.02 the followingDp

empirical formula for finding the mentioned valueshas been suggested:

[ [

, kZ

]

k3

S = C 1+ 01~~,J(20)

In which the values of S, C, kt, kz, k3 can be foundfrom the Table I.

('

[ J

OOI44]

] .85! 1+0.35 -~-_O I

" Dp)k)= (i~32:'"R~oiT--

(20.a)

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S.No S C kl kz k3 Max.Error %

I Ypl* 1 (20.a) (20.b) (20.c) 4

2 tpl* 7r/2 (20.d) (20.e) 0.587 3

3 YtI* -1 (20.t) (20.g) (20.h) 5

4 ttl* 37r/2 1.9 1.49 0.653 4.2

5 YtOl* -1 (20.1) 0.914 -0.946 4.4

6 !pz* 57r/ 2 3.0 1.362 0.7 4.6

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0.6

0.4

0.2

0

«>. -0.2

-0.4

-0.6

-0.8

-1

kz = 1.896R:.OIl

k3 =0.946R:.oI

/~'~~~

1'1 2 3 4-~~~~==--=-

1 Eq. 20. I- - - - Eq. 21.a

Time

Figure 2. Surge Oscillation in Simple Surge Tank.

(20.b)

(20.c)

k, =0.786+

( (& y:o; (20.d)1+ Dp (lOs+Ro)

kz = 1.73+

(

. 7:~

)1+19.6{;,)' (3162+R:')(20.e)

( ( J

O.OI6

)- 1.36 1+0.05 ;kl- p

1.5 + R 0.035°

k - 0.462(1.92 + R 0.OZ4 )Z -

( J

O~83

1+ 0.6 ~Dp

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(20.t)

(20.g)

( J

O.Z87

k, =1.66 + 1.227 ;p (20.h)

k3 = 0.57(1+ 1.6R:.04' )

( J

O.037

1+ 0.054 ;p

20.i)

Figure 2 shows a typical plot of surge oscillationfor sudden valve closure obtained by solutionEquation 18 using fourth order Runga Kuttamethod. Analysis of a large number of surgeoscillationcurves suggest the following empiricalequation for h fO'< 1.5:

y. = h:o. (exp(-at.)+exp(-r3t.))sin(ro.t. -~)2sm~

(21.a)

in which

21t(2 1.b)

ro. = tpz' - tpl'

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1t 5t 1* - tp2*~= p

2 tp2* - tpl*

(2 1.c)

Putting t* = tpl* in Equation 21.a and equating it

to Ypl* one gets:

exp(- atpl*)+ exp(- r3tpl*)= 2y pl*sin ~hfo*

(22)

Also by putting t* = tll* in Equation 21.a and

equating it to y IJ*one gets:

exp(-attl*)+exp(-~ttl*)= 2YII*sin~hfo*

(23)

The quantities of a and [3 can be obtained by

solving Equations 22 and 23, simultaneously [1].Figure 2 shows the result obtained by solvingEquation 21.a and compares it with the resultobtained by numerical solution of differentialEquation 18 by Runga Kutta method. The Y*

versus t * curve can be converted to a y versus t

curve via using Equations 14.a and 14.b.

4. CONCLUSION

In the present study a general solution for surgeoscillation in a simple surge tank and an optimaldesign of system has been discussed. The mainconclusions are as follows:

1. Equations for maximum surge height andcorresponding time of occurrence have beenobtained.

Equations for minimum downsurge and thecorresponding time of occurrence have beendeveloped.An equation for the occurrence of a secondpeak of the surge oscillation has beenobtained.An equation for surge oscillation has beenobtained.

2.

3.

4.

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Dp

Dpo

DT

DTs

f

fo

g

hi

h2

hf

hjV

hj{}*

Ho

HT

Kp

KTL

m

Po

QoR

Rot

1-

tpl

t pl*

t p2*

tll

t tI *

vVOYy*

ypl

Y pl*

Ytl

Y 11*

ZL

Zo

5. NOT AnONS

Diameter of Conduit

Minmum Conduit Diameter

Diameter of Simple Surge TankStable Diameter of TankFriction Factor

Friction Factor for Initial VelocityGravitational AccelerationFree Board in TankCushion Level in TankHead Loss

Initial Head LossNon-Dimensional Initial Head LossDesired Head

Height of TankCost Parameter for Conduit

Cost of Tank per Unit AreaLength of ConduitCost Parameter for ConduitDesired Power

Initial DischargeReynolds NumberInitial Reynolds Numbertime

Non-Dimensional Time ParameterThe Occurrence Time of First Peak

Non-Dimensional Time of OccurrenceFirst PeakNon-Dimensional Time of OccurrenceSecond Peak

The Occurrence Time of First DownsurgeNon-Dimensional Time of First DownsurgeVelocityInitial Velocity

Height of Surge

Non-Dimensional Height of SurgeHeight of First Peak

Non-Dimensional Height of First PeakHeight of First Downsurge

Non-Dimensional Height of First DownsurgeBottom Level of TankLevel of Reservoir

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6. REFERERENCE

1. Azhdary Moghaddam, M., "Simple Surge Tank Analysis andDesign", M. E. Thesis presented at Univ. of R06rkee,Roorkee U.P. India, in Partial Fulfilment of Required for theMaster of Engineering in Civil Engineering, (1992).

2. Chaudhary, M. H., "Applied Hydraulic Transients", VanNostrand Reinhold Co., New York, USA, (1987).

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3. Jakobsen, B. F., "Surge Tanks", Trans. ASCE, (85), (1956),1357-1382.

4. Jeager, c., "Engineering Fluid Mechanics", Blakie and SonLtd., London, U.K., (1956).

5. Marris, A. W., "Large Water Displacement in Simple SurgeTank", J. Basic Engrg. ASME, (81), (1959), 446-454.

6. Parmakian, J., "Water Hammer Analysis", Prentice-Hall Inc.,New York, U.S.A., (1955).

7. Pickford, J., "Analysis of Surges", Macmillan and Co. Ltd.,bath, U.K., (1969).

8. Rich, G. R., "Hydraulic Transient", Mc Graw-Hill Book Co.Inc., New York, USA, (1951).

9. Rouse, H., "Engineering Hydraulics", John Wiley & SonsInc., New York, USA, (1950).

10. Sutton, B. A., "Series Solution of Some Surge TankProblems", Proc. Instn. Civil Engrs., (16), (1960), 225-234.

11. Swamee, P. K., Jain, A. K., "Explicit Equation for PipeflowProblems", J. Hyd. Div. ASCE, 102(5), (1976), 657-664.

12. Wylie, E. B., Streeter, L., "Fluid Transients", Mc Graw HillInc., New York, USA., (1978).

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E Roughness of Conduit

110 Steady State Efficiency

Viscosity of Water

v Kinematic Viscosity of Water

p Mass Density of Water

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