Necessity of Surge Tank
In general, a long pressure tunnel, the headrace / tailrace of pumped storage power plant needs a surge tank. The necessity of surge tank is judged whether length of pressure tunnel exceeds 500m or not. A surge tank has following functions.
- To absorb abnormal pressure increase in the tunnel by water hammering when load rejection or input power rejection.
- To absorb the abnormal pressure decrease in the tunnel by supplying water when rapid increase of discharge.
Surge tanks are categorized in following 4 types.
- Simple surge tank
- Differential surge tank
- Restricted orifice surge tank (An orifice of a surge tank is called a ''port'')
- Surge tank with chamber
These days Restricted Orifice Surge Tanks are usually adopted. This type can effectively attenuate amplitude of the water level in the tank and has comparatively simple design. In addition, in the case that a surge tank is constructed deep under the ground, such as a tailrace surge tank, the diameter of vertical shaft can be reduced by adding upper chamber.
Requirements for Hydraulic Design of Restricted Orifice Surge Tank
The following conditions are required for the design of Restricted Orifice Surge Tank.
- The maximum amplitude of water level by surging is between the elevation of top and bottom of the tank.
- To meet the required stability conditions (dynamic and static conditions) of water level amplitude.
- Maximum discharge does not exceed the following critical discharge.
Critical Discharge
- Critical Discharge is defined as discharge when amount of water rise at partial load rejection exceeds that at full load rejection in a restricted orifice surge tank.
- If the maximum discharge is smaller than the critical discharge, the amount of water rise in the tank at full load rejection is the largest.
- Although it is quit the thing to confirm the highest level of water rise at the critical discharge in the case that the maximum discharge exceeds the critical discharge, in practical, a surge tank is designed so that maximum discharge does not exceed critical discharge.
Fundamental Differential Equations
Equation of Motion
Euler's equation of fluid motion in one-dimensional problem is as follows.
where | $v$ | : Flow velocity | $t$ | : Time | |
$\rho$ | : Density of the water | $p$ | : Pressure | ||
$x$ | : Position | $h$ | : Water head | ||
$g$ | : Acceleration of gravity |
When the flow direction from reservoir to surge tank is positive, and the direction of water surface movement from up to dawn on the basis of reservoir water level in a surge tank is positive, positive gradient of water head is upward movement of water level in the surge tank.
where | $z$ | : Variation of the water level in surge tank |
(positive is the downward as standard at reservoir water level) | ||
$L$ | : Length of the pressure tunnel | |
$v$ | : Flow velocity of the pressure tunnel |
If head loss $\Delta h$ exists, the acceleration of the water in the pressure tunnel is reduced by $\Delta h$ .
Because friction loss ( $h=c*v*v$ ) in the pressure tunnel and resistance loss of the port ( $k=v_p*v_p/2/g$ ) are caused in the case of Restricted Orifice Surge Tank, the equation of motion is given as follows taking into consideration the flow direction.
Equation of Continuity
When all flow direction of tunnel discharge, discharge in a surge tank and discharge of turbine is assumed positive, equation of continuity is given as follows.
where | $Q$ | : Discharge of the turbine |
$f$ | : Area of the pressure tunnel | |
$F$ | : Area of the surge tank shaft |
Resistance of Port
Because the port velocity ( $v_p$ ) is what discharge in a surge tank is divided by the effective area of the port, the resistance of the port ( $k$ ) can be given as follows, considering the flow direction.
Fundamental differential equations
$v_p$ | : Flow velocity of the port |
$F_p$ | : Area of the port |
$C_d$ | : Discharge coefficient of the port |
$v$ | : Flow velocity of the pressure tunnel (positive is from reservoir to surge tank) |
$z$ | : Variation of the water level in surge tank |
$g$ | : Acceleration of gravity |
$c$ | : Head loss coefficient |
$L$ | : Length of the pressure tunnel (from reservoir to the port) |
$f$ | : Area of the pressure tunnel |
$F$ | : Area of the shaft |
$Q$ | : Discharge (in the interception current of the time) |
Equations for the basic design of Restrict Orifice Surge Tank
Formula to calculate the maximum water level in the tank
Vogt-Forchheimer's Formulas
Vogt-Forchheimer's Formulas are used to calculate the highest rising water level in the basic design stage. This is the one rewritten to be convenient equation after solving the fundamental differential equation on the condition of instant intercept of initial discharge.
Vogt-Forchheimer's formulas
$z_m$ | : Maximum water level in the tank |
(Positive is the downward as astandard at reservoir level) | |
$h_0$ | : Total head loss of the pressure tunnel |
$k_0$ | : Resistance of the port |
$v_0$ | : Flow velocity of the pressure tunnel |
$c$ | : Head loss coefficient ($h_0=c\cdot v_0{}^2$) |
$Q_0$ | : Maximum discharge |
$F_p$ | : Area of the port |
$C_d$ | : Discharge coefficient of the port |
$L$ | : Length of the pressure tunnel (from reservoir to the port) |
$f$ | : Area of the pressure tunnel |
$F$ | : Area of the shaft |
$g$ | : Acceleration of gravity |
To calculate ( $z_m$ ) in the formula above, Newton-Raphson Method is leveraged. $z$ which makes $f(z)=0$ is the maximum water level ( $z_m$ ) in the function of $f(z)$ and $f'(z)$ . Here, $f'(z)$ is a derived function of $f(z)$ .
Calculation of the below equation is iterated until $f(z_i+1)$ becomes nearly equal $0$ .
Initial value $z_0$ is defined as below so that the value in the logarithm paragraph of $f(z)$ can be positive.
Required conditions to calculate water level of reservoir and head loss
To calculate water level, following issues are considered.
- To adopt the most rigid condition of water level of the reservoir corresponding to the discharge conditions.
- To set the roughness small in the case of load rejection or input power rejection, and to set the roughness large in the case of rapid increase of discharge on the safe side.
Conditions of the water level and roughness | |||
---|---|---|---|
Discharge | Water level and roughness | Headrace surge tank | Tailrace surge tank |
Total Load Interception | Reservoir water level | HWL of Upper Res. | LWL of Lower Res. | Checked water level | Upper surge W.L. | Down surge W.L. | Variation of roughness | $-0.0015$ | $-0.0015$ |
Rapid Load Increase | Reservoir water level | LWL of Upper Res. | HWL of Lower Res. | Checked water level | Down surge W.L. | Upper surge W.L. | Variation of roughness | $+0.0015$ | $+0.0015$ |
Total Input Interception | Reservoir water level | LWL of Upper Res. | HWL of Lower Res. | Checked water level | Down surge W.L. | Upper surge W.L. | Variation of roughness | $-0.0015$ | $-0.0015$ |
(Note) Variation of roughness is for the concrete lining. |
(Supplemental Explanation)
- The roughness of concrete to calculate surging water level are set by adding the value above or subtracting it from the normal value of 0.013 $\sim$ 0.0125.
- In the case of steel lining, 0.001, and in the case of no lining, 0.003 is added or subtracted from the normal value to set the roughness respectively.
Requirement for Stability of Water Level Vibration}
Thoma-Schuller's formulas
\begin{align}
&\text{Static stability conditions : } &\quad& h_0<\frac{H_g}{3}\sim\frac{H_g}{6} &\qquad (5)\\
&\text{Dynamic stability conditions : } &\quad& F>\cfrac{L f}{c(1+\eta)g H_g}\sim\cfrac{L f}{2 c g(H_g-z_m)}&\qquad (6)
\end{align}
\begin{equation}
\eta=\frac{k_0}{h_0} \qquad \qquad h_0=c\cdot {v_0}^2 \qquad k_0=\frac{1}{2 g}\left(\frac{Q_0}{C_d F_p}\right)^2
\end{equation}
$H_g$ | : Gross head |
$z_m$ | : Maximum water level as a standard at reservoir water level |
Equation of Critical Discharge
To calculate the critical discharge $Q_c$ , Calame-Garden equation is leveraged.
Procedure of Basic Design of Surge Tank
Correction of Head Loss
Head loss coefficient ($c$) is calculated by using the head loss of intake and headrace for a headrace surge tank, the head loss of tailrace and outlet for a tailrace surge tank, taking into consideration correction of head loss according to above table. The representative velocity of headrace / tailrace can be used as $v_0$ in the calculation.
Check of Static Stability
Requirement of static stability is determined by only total water head $H_g$ as shown in the equation (5). Therefore, if this requirement is not satisfied, head loss should be reduced by increasing of tunnel diameter and so on.
Set Targeted Movement Range of Water Level
The targeted movement range of water level is set corresponding to the low water level of the reservoir and bottom elevation of the tank. Since surging is attenuating vibration, even though full load rejection while reservoir water level is low water level is taken place, the drawdown depth does not exceed the rise up depth. However, in practical, the targeted movement range of water level is set as both depths are the almost same.
Relationship between Port Diameter and Shaft Diameter
The available ranges of the port diameter and the shaft diameter, which are calculated from the requirement of dynamic stability and critical discharge, are to be set. Relationship between the port diameter and shaft diameter is given by following expressions.
(Supplementary Explanation)
- Necessary ranges of the port diameter and shaft diameter are found by equation (8) and (10).
- Minimum value of the shaft diameter is found by substituting the targeted movement range of water level to zm in the equation (9).
- Since requirement of equation (9) is usually more rigid than that of equation (8), the shaft diameter needs to be larger than the value found by equation (9).
Relationship between Port Diameter, Shaft Diameter and Maximum Water Level
The correlation between the port diameter and maximum upper water level by the equation (4) is calculated by making shaft diameter a parameter within the range which meets the requirement of dynamic stability. The correlation between the port diameter and resistance loss of port at the load rejection and input power rejection is calculated.
Selection of the shaft and port diameter
The optimal port diameter is found so that the resistance loss of the port $|z'rm|$ at the load rejection is equal to the maximum upper water level $|z_m|$ .
Therefore, the shaft diameter and the port diameter are determined to meet the above condition and requirement of within the targeted fluctuation range.
(Supplementary Explanation)
The tank water level before interception is lowered only head loss $h_0$ of the pressure tunnel from the reservoir water level. If discharge $Q_0$ before interception tends to flow in in a tank from the port at the moment of interception, the velosity head at the port tends to become $k_0=\frac{1}{2g}(Q_0/C_d/F_p)^2$ , and tends to push up the tank water level with this energy (where $C_d$ is the discharge coefficient of the port, and $F_p$ is the area of the port). Therefore, the amount of the water level rise from the reservoir water level becomes $|z'_{rm}|=k_0-h_0$ . The absolute value is taken because ''positive is the downward'' on the basis of the reservoir water level.
Here, evaluation of the highest rise water level is performed as follows.
$|z_{m}|<|z'_{rm}|$ | : diameter of the port is estimated as a small state |
because of large resistance of the port. | |
$|z_{m}|=|z'_{rm}|$ | : diameter of the port is the optimal. |
$|z_{m}|>|z'_{rm}|$ | : diameter of the port is estimated as a large state |
because of small resistance of the port. | |
where $|z_m|$ is calculated value from the Vogt-Forchheimer formula. |
In addition, the pressure head of the bottom of surge tank is denoted by the sum of the pressure head of water in the tank, and resistance of the port $k$ . For this reason, in the structural design of the bottom of surge tank, it is necessary to carry out in consideration of the additional pressure head $k$.
Check of Dynamic Stability
It should be checked that the selected shaft diameter and port diameter meet the requirement of dynamic stability and critical discharge, by plotting selected value on the figure of correlation between dynamic stability and critical discharge.
Arrangement of Chamber
Excavation volume of the shaft can be reduced by adding a chamber for a surge tank which is constructed in deep underground, such as a tailrace surge tank. The figure of the chamber is designed so that the capacity of chamber can absorb the water volume of upper surge over the level of chamber bottom which is calculated by equation (4) in the case of shaft type surge tank.
Simple shaft tipe | Chamber type |
---|---|
Design concept of Chamber type |
Surging analysis
The design of the surge tank is checked by surging analysis. Chronological movement of water level in the tank is found by solving differential equations of (1), (2), (3).
Examples of Surging Analysis
Conditions for Analyses of Headrace surge tank and Tailrace surge tank
Conditions of the calculation for the stability and maximum water lebel | |||||
---|---|---|---|---|---|
Items | Headrace surge tank | Tailrace surge tank | |||
Load Interception | Input Interception | Load Interception | Input Interception | ||
$H_g$ | (m) | 713 | 713 | ||
$L$ | (m) | 2553.370 | 2167.752 | ||
$d_0$ | (m) | 8.2 | 8.2 | ||
$C_d$ | -- | 0.9 | 0.9 | ||
$z_m$ (target value) | (m) | 35 | 65 | ||
$Q_0$ | (m$^3$/s) | 340 | 240 | 340 | 240 |
$c$ | -- | 0.179 | 0.185 | 0.166 | 0.160 |
Decided Shaft dia. | (m) | 17.0 | 10.0 | ||
Decided Port dia. | (m) | 4.6 | 4.9 | ||
(Note) $d_0$ is the diameter of the pressure tunnel. |
$\displaystyle c=\frac{h_0}{v_0^2}$ | $c$: Head loss coefficient $h_0$: Total head loss of the pressure tunnel $v_0$: Flow velocity of the pressure tunnel |
Conditions for the surging analysis | ||||||
---|---|---|---|---|---|---|
Items | Headrace surge tank | Tailrace surge tank | ||||
EL. (m) | Area (m2) | Shape (m) | EL. (m) | Area (m2) | Shape (m) | |
Top of Surge tank | 1566.000 | 226.980 | $\phi$17.0 | 865.000 | 520.000 | $\square$13x40 |
Bottom of Chamber | -- | -- | -- | 854.050 | 520.000 | $\square$13x40 |
Bottom of Surge tank | 1461.600 | 226.980 | $\phi$17.0 | 727.600 | 778.540 | $\phi$10.0 |
Port | -- | 16.619 | $\phi$4.6 | -- | 18.857 | $\phi$4.9 |
Case | Case-1 | Case-2 | Case-3 | Case-1 | Case-2 | Case-3 |
R.W.L (m) | 1527 | 1500 | 1500 | 814 | 844 | 844 |
$c$ (loss coefficient) | 0.179 | 0.253 | 0.185 | 0.166 | 0.277 | 0.160 |
Discharge (m$^3$/s) | 340 to 0 | 170 to 340 | -240 to 0 | -340 to 0 | -170 to -340 | 240 to 0 |
Interception time (sec) | 8.0 | 40.0 | 5.6 | 8.0 | 40.0 | 5.6 |
Case-1 : Load interception (4 units) | ||||||
Case-2 : Load rapidly increase (4 units) | ||||||
Case-3 : Input interception (4 units) | ||||||
"R.W.L" is the reservoir water level. | ||||||
Discharge "+" is the direction from Reservoir to Surge tank. |
Stability conditions and water level of Headrace surge tank
Load interception | Input interception |
---|---|
Stability conditions for Headrace surgetank |
Load interception | Input interception |
---|---|
Maximum of water head fractuation magnitude of Headrace surgetank |
Stability conditions and water level of Tailrace surge tank
Load interception | Input interception |
---|---|
Stability conditions for Tailrace surgetank |
Load interception | Input interception |
---|---|
Maximum of water head fractuation magnitude of Tailrace surgetank |
Surging analyses of Headrace surge tank and Tailrace surge tank
Mode | Headrace surgetank (without Chamber) | Tailrace surgetank (with Chamber) |
---|---|---|
a. Total load interception |
||
b. Rapid load increase |
||
c. Total input interception |
||
Result of Surging analysis |
Programs
Stability and Maximum or Minimum water Level
Filename | Description |
---|---|
a_f90.txt | Shell script for execution |
a_gmt_SC.txt | GMT command for Drawing (1) |
a_gmt_VF.txt | GMT command for Drawing (2) |
f90_surgeSCVF.f90 | Stability calculation |
inp_KN1H_Input.csv | Input data sample (Headrace S.T.-1) |
inp_KN1H_Load.csv | Input data sample (Headrace S.T.-2) |
inp_KN1T_Input.csv | Input data sample (Tailrace S.T.-1) |
inp_KN1T_Load.csv | Input data sample (Tailrace S.T-2) |
f90_surgeSCVF.f90
- This is a program for Basic design of Restricted Orifice Surge Tank.
- Dynamic stability condition and maximum water level of shaft can be known using this program .
- Thoma-Schüller's formula is used to evaluate dynamic stability of shaft.
- Vogt-Forchheimer's formula is used to calculate maximum water level of shaft.
- In this program, function of drawing by gnuplot is included. You can see the drawings after execution of this program if you have installed gnuplot in your PC and you can see 'eps' format file.
- Following data is the sample for Headrace Surge Tank at load interception in Pumped Storage Power Plant.
- Head loss coefficient of pressure tunnel c is calculated by equation c=h0 / (v0)2 , where h0 is total head loss of the pressure tunnel, v0 is flow velocity of the pressure tunnel before interception.
- When you have no idea about selected diameters xc and yc, you can set all values to 1.0 as initial values. After this calculation, you can select these diameters xc and yc, and you can confirm that selected values are proper or not.
Sample of input data (file: inp_KN1H_Load.csv)
KN No.1 Headrace ST (Load interception) Hg,Q0,L,d0,c,Cd,zm,xc,yc 713,340,2553.370,8.2,0.179,0.9,35,4.6,17
Hg | Total head (m) |
Q0 | Maximum discharge (m3/s) |
L | Length of the pressure tunnel (m) |
d0 | Diameter of the pressure tunnel (m) |
c | Head loss coefficient |
Cd | Discharge coefficient of the port |
zm | Target maximum water level given by designer (m) |
xc | Selected diameter of the port (m) |
yc | Selected diameter of the shaft (m) |
Command for execution
gfortran -o f90_surgeSCVF f90_surgeSCVF.f90 ./f90_surgeSCVF input_data.csv fig_SC.eps fig_VF.eps
Surging Analysis
Filename | Description |
---|---|
a_gmt_surge.txt | GMT command for Drawing |
a_surge.txt | Shell script for execution |
f90_surge.f90 | Surging analysis |
inp_KN1_Ha.csv | Input data sample (Headrace S.T.-a) |
inp_KN1_Hb.csv | Input data sample (Headrace S.T.-b) |
inp_KN1_Hc.csv | Input data sample (Headrace S.T.-c) |
inp_KN1_Ta.csv | Input data sample (Tailrace S.T.-a) |
inp_KN1_Tb.csv | Input data sample (Tailrace S.T.-b) |
inp_KN1_Tc.csv | Input data sample (Tailrace S.T.-c) |
f90_SURGE.f90
- This is a program for Surging Analysis for Restricted Orifice Surge Tank.
- Runge-Kutta method is used to solve the fundamental differential equations for surging.
- This program can evaluate the damped oscillation between surge tank and reservoir through the pressure tunnel.
- This program can evaluate the effect of chambers in surge tank.
- This program can not evaluate the water hummer pressure between surge tank and pump/turbine.
Sample of input data (file: inp_KN1_Ta.csv)
- Following data is for Tailrace Surge Tank at load interception in Pumped Storage Power Plant.
- Since initial velocity of pressure tunnel is in direction from Surge Tank to Lower Reservoir, discharge QTQ(1) is negative value. If initial velocity of pressure tunnel is in direction from Reservoir to Surge Tank, discharge QTQ(1) becomes positive value.
- Although the blank (space) can be used in variable 'Title,' it cannot be used in variable SBL(i). In variable SBL(i), the blank (space) must be changed to '_' (under-bar). When figure is printed out by gnuplot, under-bar ('_') is changed to the blank (space).
- Variable TNC (Head loss coefficient of pressure tunnel) is calculated by equation TNC=h0 / (v0)2 , where h0 is total head loss of the pressure tunnel, v0 is flow velocity of the pressure tunnel before interception.
KN No.1 Tailrace ST (Load interception: 4 units) #: Title 1,70.0,210.0 #: ICT,AFCA,AFCT 600.0,0.01,0.1 #: TMAX,dt,DTWR 18.857,0.9,0.9 #: PAA,PCI,PCO 814.000,2167.752,52.810,0.166 #: RWL,TNL,TNA,TNC 3 #: NST 520.000,865.000,Top_of_Surge_Tank #: SAA(1),SEL(1),SLB(1) 520.000,854.050,Bottom_of_Chamber #: SAA(2),SEL(2),SLB(2) 78.540,727.600,Bottom_of_Surge_Tank #: SAA(3),SEL(3),SLB(3) 3 #: NQT -340,0 #: QTQ(1),QTI(1) 0,8 #: QTQ(2),QTI(2) 0,9999 #: QTQ(3),QTI(3)
Title | Title |
ICT | Calculation case (1: normal, 2: AFC) |
AFCA | Discharge of half amplitude of AFC (m3/s) |
AFCT | Period of AFC (s) |
TMAX | End time for calculation (s) |
dt | Time increment for calculation (s) |
DTWR | Time increment for print out (s) |
PAA | Section area of port (m2) |
PCI | Discharge coefficient of port for in-flow |
PCO | Discharge coefficient of port for out-flow |
RWL | Water level at reservoir (EL.m) |
TNL | Length of pressure tunnel (m) |
TNA | Section area of pressure tunnel (m2) |
TNC | Head loss coefficient of pressure tunnel (s2/m) |
NST | Number of sections |
SAA(i) | Section area of specified level in the shaft (m2) |
SEL(i) | Level definition of sections of shaft (EL.m) |
SLB(i) | Label for section (text: for drawing) |
NQT | Number of discharge input points |
QTQ(i) | Discharge at specified time point (m3/s) |
QTI(i) | Time at specified time point (s) |