STABILITY OF SURGE TANKS

According to Thoma automatically governed turbines, the surge tank will be

hydraulically stable only if the horizontal cross sectional area of the surge tank

exceeds a certain minimum value, otherwise, the instability will result. The water

level in the surge tank begins to oscillate & these oscillations are stable or unstable

depending upon the parameters of the power plant and the type and magnitude of the

disturbance.

For practical purposes, oscillations are said to be stable if they dampen to the final

steady state in a reasonable time and unstable if their magnitude increases with time.

In addition to oscillating instability, a condition called tank drainage has to be

avoided in the surge tank design. In this case, when the flow is demanded due to

increase in load, it should not accelerate to an extent of sucking the air from

atmosphere into tunnel / penstock.

There are four cases of change in the turbine flow are of interest. viz.,

(i) constant flow

(ii) constant gate opening

(iii) constant power

(iv) constant power combined with full gate opening.

There are three governing equations ,viz., the dynamic equation, continuity equation

and governor equation which may be applied to any general case. These equations

can be expressed in single form as

( ) ( ) 0ztdtdzt

dtzd2

2

=++ ζφ

Governor equation which is function of Q and z is represented by φ(Q, z)=0

Q=constant function represents a straight line II in fig 3.7. The discharge Q is given

by eq..

( )H

zHQQ o+

=

2

This function may be represented by the curve I as shown in fig 3.7. If the above

equation is approximated by a series and neglecting the higher order terms, reduces

to

+=

H2z1QQ o ( 4.39 )

III

II

I

Qb

Ib

II

+Z

CIV

I Eq. 3.23

II Q = constant

III Instability occurs in this case

IV Governer equation 3.25

Governor characteristics

Similarly, the governor equation for constant power output will be

( )zHHQ

= Q ooo

+ηη

( 4.40 )

in which

η, ηo are efficiency at instant of time and rated efficiency of turbine

respectively

Qo is the rated discahrge .

H, Ho are head at any instant of time and head respectively.

The above equation is represented by a falling curve such as IV in Fig 3.7

For constant discharge ,the oscillations will be stable if friction is taken into account.

Oscillations will also be stable whatever the area of cross section of the surge tank if

3

the governor equation is represented by the rising curve. Instability will occur when

it is falling curve such as curve III or IV.

Governing equations for stability of simple surge tank.

Using the Eqn 3.10 and 3.3 along with Eqn. 3.23

( ) CzHQ P

=+=ηγ

( 4.43 )

in which Qt is the flow through the tunnel, L is the ength of the tunnel, Qtur is

thedischarge through the turbine,As , area of the surge tank At and At are area of

cross section of tunnel and surge tank respectively, Qosteady state discharge, Ho is

the steady state head.

c= coefficient of friction loss

η efficiency of the turbine

γ=unit weight of water.

P=power out put of the turbine after sudden change and is assumed

to remain constant.Differentiating equations (4.38) and (4.39) and equating and after

rearranging the equations can be written as

( ) ( )2ts

2

s

t2

st

2

t

s2

2

zHALAgcCz

LAgA

)zH(AC

zHLAgcC2

dtdz

dtdz

LAgcA

dtzd

+−=+

+

−+

+

+ (4.44 )

Equation ( 4.44 ) represents an oscillation. The axis about which the oscillation

occurs may be found by substituting steady conditions, i.e. z= zo= constant and dzdt

= 0 and d zdt

2

2 =0. The equation ( 4.43 ) reduces to

4

At2 zo (H+zo)2 + c (QoHo)2=0 ( 4.45 )

C=QoHo=At (H+zo)z

co−

( 4.46 )

or zo= 2

t

o

AQc

− is the limit z will approach and which applies to the steady

state conditions giving constant output of power P

As shown in the fig the position of the axis of oscillation may be determined

graphically by intersection of the curves representing the simultaneously equations -

cQ2 =z(parabola for dynamic equation) and QoHo=constant (hyperbola ,for

governor equation ).

Under steady condition both must be satisfied ,and this will lead to two solutions I

and II. Also from the figure there is curve Cmax corresponding to maximum

possible power output. from the system. which is tangent to the curve of friction

loss,(dynamic equation) The two axes I and II will coincide to form a single axis

passing through the tangent point. The co-ordinates of the tangent points are

obtained by differentiating equation ( 4.46),with respect to z, and equating to zero

we get z=−H3

From above relation the it may be concluded that the maximum power output

of the system will be obtained when the friction loss between reservoir and turbine

amounts to zo = z=H/3, although in practice it will be found uneconomical to run a

scheme with such large friction loss.

For further analysis of equation [4.44] it will be convenient to move the horizontal

axis by zo=-cQo2 such that s=z-zo

d sdt

a dsdt

bs2

2 2 0+ + = ( 4.47 )

5

equation [4.47] is linear differential equation whose solution depends on the roots of

the second degree equation., the roots are = − ± −a a b2

Table -Roots of the equation

for b>0 and a2<b [a periodic oscillation]

a>0 a=0

a<0

a damped oscillation a pure sinusoidal

oscillation a forced oscillation.

Table Aperiodic oscillation

fora2>b [aperiodic oscillation]

a> b a= b

a damped oscillation a pure sinusoidal

oscillation

If the oscillation is to be damped ,b>0 and a≥0.

The condition in equation (4.44)] means that

H zH zo

++

3 > 0 and H+zo > 0

-zo = cQo2 ≤ 1

2(H+zo) ( 4.48 )

This limitation of the friction losses means that certain minimum power output must

not be exceeded .

Similarly the condition a>0 leads to

6

or AV

g cQ H cQth

o

o o

≥−

2

2 22

LA t

( ) ( 4.49 )

This is Thoma's condition for stability of surge tanks with oscillations of small

amplitudes. and Thomas's least areas of cross section of a surge tank.

In the recent past the equations are normalised and solved by phase plane method.

The characteristic of these roots lead to the determination of singularity as node ,

saddle , vortex and focus [C1]

In the above equation following assumption are made.:

• Turbine governor maintains constant power output.

• The surge tank oscillations are small

• Turbine efficiency is constant.

• Head losses in penstock are neglected.

• Head losses in tunnel under steady state condition is valid for

transient flow conditions.

• Tunnel velocity head is neglected

• The power station is isolated one

Stability of surge tank for large amplitude of oscillation:

In case of large amplitudes the function φ(t) and ζ(t) of equation [4.47] are

complicated functions of time t. The complex equation are simplified by taking

mean values of φ(t) and ζ(t) and rewriting equation as below

d zdt

a dzdt

bz2

2 2 0+ + = ( 4.50 )

On integrating the above equation between limits 0 to t,rearranging the terms ,

(J1 )obtained the expression for oscillations from which the stability is obtained.

+

+== ∗∗

oTh

s

zHz482.01

AAn ( 4.51 )

7

in which z*=VLAgAo

t

s is amplitude of oscillation due to sudden rejection

of load ,neglecting friction and the period is= 2π Lg

AA

s

t

Defining ( )22o

22o

Qcz= and ,

HcQ ∗= εβ ( 4.52 )

β

βεβ

βε−

=

−

+==∗

12

1482.01

AAn

Th

s ( 4.53 )

or ∗∗ −

−

+−

== n = n for ,0=21

482.011A

An *

*

*

*

Th

s

εββε

ββ ( 4.54 )

To ensure stability β=β*.

The figure 2. indicate the curves β vs ε , and also compared with the value of ε

calculated directly by graphical method by Frank and also with the Thoma 's

expressions.For β<β* ,the surge collapses , i.e., water level in the tank reaches the

axis of oscillation II and surge tank emptied without the possibility of water level

rising again in the same oscillation.

4.6.4 STABILITY OF ORIFICE SURGE TANK:

Govening Equations for orifice surge tank are aleready written viz. eqn 4.1 and 4.2

Turbine boundary condition:

Qo ( Ho - hf ) = ( Qo- Qs ) ( Ho - z - c Q Qorf s s ) ( 4.57)

The above equations can be modified and can be rewritten as

8

dzdt

M Q zs= ( , ) ( 4.58)

dQdt

N Q zss= ( , ) ( 4.59 )

Linearizing its governing equations about a given equilibrium state is a most

fundamental and generally useful technique. The proceedure facilitates judging the

stability without actually solving the equations. To that end the equilibrium of the

system should be first identified. This is done by letting M Q zs( , ) =0 and

N Q zs( , ) =0.

From the above step three solutions are found ,in which

Qs=0, i.e., corf Qs Qs =0 ( 4.60 )

Consequently the prescence of orifice has no effect on singularities and throttled

chambers have the same sigular points as unthrottled ones. The locations of singular

points on ( Q, z ) plane are shown in table.

Table Location of singular points

( )11 y,x ( , )Q zo o

. ( )22 y,x . ( )[ ]of21oo1 Mh1bz,Qb −+

. ( )33 y,x ( )[ ]of22oo2 Mh1bz,Qb −+

in which Mo=1; b1= ( )[ ]134 5.0 −−ξ , b2= ( )[ ]134 5.0 +−ξ and ξ = H hf

The third singularity is virtual one and will not be discussed. The first one is steady

state hydraulic grade line with head loss in the tunnel equal to hf, and surge

oscillations in actual situations take plcace about it. The stability of equations (4.58)

9

and (4.59) depend whether this point is stable or not.For this to be asymptotically

stable requires

3 hf < H ( 4.64 )

As > ATh ( 4.65 )

in which ATh= LgA cHt o2

The second sigular point lies below the first , where the drainage of surge chamber is

likely to occur. This can be formulated as

3 hf > H ( 4.66 )

4 2 412 2 2b h M A Hhf o t f+ − < 0 ( 4.67 )

The condition ( 4.66 ) implies taht the second singularity can only be stable if the

head loss exceeds 1/3 of gross head. It would be highly uneconomical if power plant

is designed withsuch an high head loss in tunnel. The stability of this singularity,

there fore has little practical importance.