Design of Port of Restricted Orifice Surge Tank (2021.11.28)

Introduction

In the hydraulic design of restricted orifice surge tank, the discharge coefficient of port is important parameter. For the short port structure, the discharge coefficient of port is usually set to nearly 0.9 and this value is almost true. However, the discharge coefficient of long port sometimes exceeds 1.0. Therefore, it is important for the proper hydraulic design of restricted orifice surge tank to estimate the discharge coefficient of the port by appropriate method. In this report, the estimation method of port resistance and discharge coefficient of a restricted orifice surge tank established by Central Research Institute of Electric Power Industry (CRIEPI) is introduced.

Theorem

Required parameters for design

f:id:damyarou:20211128112355j:plain:w400 — Figure-1 Explanation of symbols

$\begin{align*} &n & & \text{Manning's roughness coeficient of port} \\ &D_s & & \text{shaft diameter} \\ &D_p & & \text{port diameter} \\ &d & & \text{waterway diameter} \\ &\ell & & \text{length of port} \\ &r & & \text{radius of lower bevel} \\ &r' & & \text{radius of upper bevel} \\ &Q & & \text{maximum discharge} \\ \end{align*}$

Head loss of port

The head loss of port of $h$ can be expressed as follow.

$\begin{equation} h=\mu\times (h_b+h_s)+hf=\zeta\times \cfrac{v_p{}^2}{2g} \end{equation}$

$\begin{align*} &h & & \text{head loss of port} \\ &h_b & & \text{head loss due to right angle branch} \\ &h_s & & \text{head loss due to section suddenly changed} \\ &h_f & & \text{head loss due to friction} \\ &\mu & & \text{interference factor} \\ &\zeta & & \text{head loss coefficient} \\ &v_p & & \text{flow velocity of port} \\ &g & & \text{gravity acceleration} \end{align*}$

Interference factor

An interference factor for flow-in of $\mu_{in}$ and that for flow-out $\mu_{out}$ can be defined using Figure-2depending on parameters of $\delta$ and $\psi$ . Where,

$\begin{equation} \psi=\left(\cfrac{D_p}{D_s}\right)^2 \qquad \delta=\cfrac{\ell}{D_p} \end{equation}$

f:id:damyarou:20211128112525j:plain — Figure-2 Interference factors for in-flow and out-flow

f:id:damyarou:20211128112544j:plain — Figure-2 Interference factors for in-flow and out-flow

Head loss due to right angle branch

The head loss due to right angle branch of $h_b$ can be estimated using following equations.

(a) Flow-in

$\begin{equation} \phi^2\cdot f_{b,in}=(\phi^2-0.1 \phi+0.4)\cdot (1-0.9 \sqrt{\rho / \phi}) \end{equation}$

(b) Flow-out

$\begin{equation} \phi^2\cdot f_{b,out}=0.4\cdot\{(1-2.5\sqrt{\rho})\phi^2 + 2\} \end{equation}$

Where,

$\begin{equation} \phi=\left(\cfrac{D_p}{d}\right)^2 \qquad \rho =\cfrac{r}{d} \quad \text{(waterway side)} \end{equation}$

Using above,

$\begin{equation} h_b= \begin{cases} \phi^2\cdot f_{b,in}\times \cfrac{v_p{}^2}{2g} \quad \text{(flow-in)} \\ \phi^2\cdot f_{b,out}\times \cfrac{v_p{}^2}{2g} \quad \text{(flow-out)} \end{cases} \end{equation}$

f:id:damyarou:20211128112728j:plain:w400 — Figure-3 (Reference) Head loss of right angle branch

Head loss due to section suddenly changed

The head loss due to section suddenly changed of $h_s$ can be estimated using following equations.

(a) Flow-in (sudden expansion)

$\begin{equation} f_{se}=(1-\psi)^2 \end{equation}$

(b) Flow-out (sudden reduction)

$\begin{gather} f_{c}=f_r \cdot \left(\cfrac{1}{C_c}-1\right)^2 \\ f_{r}=\cfrac{0.055}{\sqrt{0.003+\rho'}} \qquad \text{(effect of bevel)} \\ \left(\cfrac{1}{C_c}-1\right)^2=0.5\cdot (1-1.144 \psi)\cdot (1+0.740 \psi) \qquad \text{(Weisbach's equation)} \end{gather}$

Where,

$\begin{equation} \psi=\left(\cfrac{D_p}{D_s}\right)^2 \qquad \rho'=\cfrac{r'}{D_s} \quad \text{(surge tank side)} \end{equation}$

Using above,

$\begin{equation} h_s= \begin{cases} f_{se}\times \cfrac{v_p{}^2}{2g} \quad \text{(flow-in)} \\ f_{c} \times \cfrac{v_p{}^2}{2g} \quad \text{(flow-out)} \end{cases} \end{equation}$

Head loss due to friction

The head loss due to friction of $h_f$ can be estimated using following equations.

$\begin{gather} f'=\delta\cdot \cfrac{124.5 n^2}{D_p{}^{1/3}} \qquad \delta=\cfrac{\ell}{D_p} \\ h_f=f'\times \cfrac{v_p{}^2}{2g} \end{gather}$

Summary of head loss calculation

From above, the head loss of port of $h$ can be expressed as follow.

$\begin{equation} h=\zeta\times \cfrac{v_p{}^2}{2g} \end{equation}$

$\begin{equation} \zeta= \begin{cases} \mu_{in} \cdot (\phi^2\cdot f_{b,in}+f_{se})+f' \qquad \text{(flow-in)} \\ \mu_{out}\cdot (\phi^2\cdot f_{b,out}+f_{c})+f' \qquad \text{(flow-out)} \end{cases} \end{equation}$

Calculation of discharge coefficient

The port discharge $Q$ can be expressed as follow using the port discharge coefficient of $C_d$ and port resistance of $k$ .

$\begin{equation} Q=C_d\times \cfrac{\pi D_p{}^2}{4}\times \sqrt{2 g k} \rightarrow v_p=C_d\cdot \sqrt{2 g k} \end{equation}$

When Bernoulli's theorem is applied between waterway and surge shaft after interception referring Figure-4, following can be established.

$\begin{equation} Z_1+\cfrac{V_1{}^2}{2g}=Z_2+\cfrac{V_2{}^2}{2g}\pm h \qquad \text{($+$: flow-in, $-$: flow out)} \end{equation}$

Where,

$\begin{align*} &Z_1 & & \text{pressure head of waterway} \\ &V_1 & & \text{flow velocity of waterway} \\ &Z_2 & & \text{water head in surge shaft} \\ &V_2 & & \text{flow velocity in surge shaft} \\ &h & & \text{head loss of port} \end{align*}$

f:id:damyarou:20211128112813j:plain:w400 — Figure-4 Explanation of symbols

Since $Z_1 > Z_2$ during flow-in and $Z_2 > Z_1$ during flow-out, the port resistance of $k$ can be expressed as follow.

$\begin{equation} k= \begin{cases} Z_1-Z_2 \quad \text{(flow-in)} \\ Z_2-Z_1 \quad \text{(flow-out)} \end{cases} \end{equation}$

Therefore,

$\begin{equation} h=k\pm \left(\cfrac{V_1{}^2}{2g}-\cfrac{V_2{}^2}{2g}\right) =k\pm \cfrac{v_p{}^2}{2g}\cdot\left\{ \left(\cfrac{D_p{}^2}{d^2}\right)^2-\left(\cfrac{D_p{}^2}{D_s{}^2}\right)^2 \right\} =k\pm \cfrac{v_p{}^2}{2g}\cdot(\phi^2-\psi^2) \end{equation}$

$\begin{equation} k=h\mp \cfrac{v_p{}^2}{2g}\cdot (\phi^2-\psi^2) =\left\{\zeta\mp (\phi^2-\psi^2)\right\}\times \cfrac{v_p{}^2}{2g} =\cfrac{1}{C_d{}^2}\times \cfrac{v_p{}^2}{2g} \end{equation}$

From above, the port discharge of $C_d$ can be expressed as follow.

$\begin{equation} C_d=\cfrac{1}{\sqrt{\zeta\mp (\phi^2-\psi^2)}} \qquad \text{($-$: flow-in, $+$: flow-out)} \end{equation}$

Procedure of calculation

Input parameters

$\begin{align*} &n & & \text{Manning's roughness coefficient of port} \\ &D_s & & \text{shaft diameter} \\ &D_p & & \text{port diameter} \\ &d & & \text{waterway diameter} \\ &\ell & & \text{length of port} \\ &r & & \text{radius of lower bevel} \\ &r' & & \text{radius of upper bevel} \\ &Q & & \text{maximum discharge} \\ \end{align*}$

Parameters for calculation

$\begin{align} &\psi=(D_p / D_s)^2 \\ &\phi=(D_p / d)^2 \\ &\delta=\ell / D_p \\ &\rho =r /d \qquad \text{(waterway side)} \\ &\rho'=r' / D_s \qquad \text{(surge tank side)} \end{align}$

Calculation of port resistance

Interference factor

Define the interference factor for flow-in of $\mu_{in}$ and that for flow-out $\mu_{out}$ from Figure-2 depending on the parameters of $\delta$ and $\psi$ .

Head loss coefficient due to right angle branch

$\begin{align} &\phi^2\cdot f_{b,in}=(\phi^2-0.1 \phi+0.4)\cdot (1-0.9 \sqrt{\rho / \phi}) & & \text{(flow-in)} \\ &\phi^2\cdot f_{b,out}=0.4\cdot\{(1-2.5\sqrt{\rho})\phi^2 + 2\} & & \text{(flow-out)} \end{align}$

Head loss coefficient due to section suddenly changed

$\begin{align} &f_{se}=(1-\psi)^2 & & \text{(flow-in)} \\ &f_{c}= \cfrac{0.055}{\sqrt{0.003+\rho'}}\times 0.5\cdot (1-1.144 \psi)\cdot (1+0.740 \psi) & & \text{(flow-out)} \end{align}$

Head loss coefficient due to friction

$\begin{equation} f'=\delta\cdot \cfrac{124.5 n^2}{D_p{}^{1/3}} \end{equation}$

Summary of head loss coefficient

$\begin{align} &\zeta_{in}=\mu_{in}\cdot (\phi^2 \cdot f_{b,in}+f_{se})+f' & & \text{(flow-in)} \\ &\zeta_{out}=\mu_{out}\cdot (\phi^2 \cdot f_{b,out}+f_{c})+f' & & \text{(flow-out)} \end{align}$

Port resistance

$\begin{equation} v_p=\cfrac{4 Q}{\pi D_p{}^2} \qquad \text{(flow velocity in port)} \end{equation}$

$\begin{equation} k= \begin{cases} \left\{\zeta_{in}-(\phi^2-\psi^2)\right\}\times \cfrac{v_p{}^2}{2 g} \qquad \text{(flow-in)} \\ \left\{\zeta_{out}+(\phi^2-\psi^2)\right\}\times \cfrac{v_p{}^2}{2 g} \qquad \text{(flow-out)} \end{cases} \end{equation}$

Calculation of discharge coefficient

$\begin{equation} C_{d}= \begin{cases} \cfrac{1}{\sqrt{\zeta_{in}-(\phi^2-\psi^2)}} \qquad \text{(flow-in)} \\ \cfrac{1}{\sqrt{\zeta_{out}+(\phi^2-\psi^2)}} \qquad \text{(flow-out)} \end{cases} \end{equation}$

Example of calculation

f:id:damyarou:20211128112914j:plain:w400 — Figure-5 Model of port

(1) Input parameters

Roughness coefficient of port	$n$	0.013
Shaft diameter	$D_s$	12.500 (m)
Port diameter	$D_p$	2.500 (m)
Waterway diameter	$d$	4.600 (m)
Port length	$\ell$	1.500 (m)
Radius of lower bevel	$r$	0.800 (m)
Radius of upper bevel	$r'$	0.500 (m)
Maximum discharge	$Q$	69.000 (m$^3$/s)
Gravity acceleration	$g$	9.8 (m/s$^2$)

(2) Parameters for calculation

$\begin{align*} &\psi=(D_p / D_s)^2 = (2.5 / 12.5)^2 = 0.040 \\ &\phi=(D_p / d)^2 = (2.5 / 4.6)^2 = 0.295 \\ &\delta=\ell / D_p = 1.5 / 2.5 = 0.600 \\ &\rho =r / d = 0.8 / 4.6 = 0.174 \qquad \text{(waterway side)} \\ &\rho'=r' / D_s = 0.5 / 12.5 = 0.040 \qquad \text{(surge tank side)} \end{align*}$

(3) Interference factor

From Figure-2, the interference factors can be defined as follows.

$\begin{align*} &\mu_{in} =1.230 \\ &\mu_{out}=1.190 \end{align*}$

(4) Head loss coefficient due to right angle branch

$\begin{align*} &\phi^2\cdot f_{b,in}& &=(\phi^2-0.1 \phi+0.4)\cdot (1-0.9 \sqrt{\rho / \phi}) \\ & & &=(0.295^2-0.1\cdot 0.295+0.4)\cdot (1-0.9 \sqrt{0.174 / 0.295}) =0.141 \\ &\phi^2\cdot f_{b,out}& &=0.4\cdot\{(1-2.5\sqrt{\rho})\phi^2 + 2\} \\ & & &=0.4\cdot\{(1-2.5\sqrt{0.174})\cdot 0.295^2 + 2\} =0.799 \end{align*}$

(5) Head loss coefficient due to section sudden changed

$\begin{align*} &f_{se}& &=(1-\psi)^2=(1-0.040)^2=0.922 \\ &f_{c}& &=\cfrac{0.055}{\sqrt{0.003+\rho'}}\times 0.5\cdot (1-1.144 \psi)\cdot (1+0.740 \psi) \\ & & &=\cfrac{0.055}{\sqrt{0.003+0.040}}\times 0.5\cdot (1-1.144\cdot 0.040)\cdot (1+0.740 \cdot 0.040) =0.130 \end{align*}$

(6) Head loss coefficient due to friction

$\begin{equation*} f'=\delta\cdot \cfrac{124.5 n^2}{D_p{}^{1/3}} =0.600\cdot \cfrac{124.5\cdot 0.013^2}{2.500{}^{1/3}} =0.009 \end{equation*}$

(7) Summary of head loss coefficient

$\begin{align*} &\zeta_{in}=\mu_{in}\cdot (\phi^2 \cdot f_{b,in}+f_{se})+f' =1.230\cdot (0.141+0.922)+0.009 =1.316 \\ &\zeta_{out}=\mu_{out}\cdot (\phi^2 \cdot f_{b,out}+f_{c})+f' =1.190\cdot (0.799+0.130)+0.009 =1.115 \end{align*}$

(8) Resistance of port

$\begin{equation*} v_p=\cfrac{4 Q}{\pi D_p{}^2}=\cfrac{4\cdot 69.0}{\pi\cdot 2.5{}^2}=14.057 \end{equation*}$

$\begin{align*} &k_{in}=\left\{\zeta_{in}-(\phi^2-\psi^2)\right\}\times \cfrac{v_p{}^2}{2 g} =\left\{1.316-(0.295^2-0.040^2)\right\}\times \cfrac{14.057{}^2}{2 g} =12.406 \\ &k_{out}=\left\{\zeta_{out}+(\phi^2-\psi^2)\right\}\times \cfrac{v_p{}^2}{2 g} =\left\{1.115+(0.295^2-0.040^2)\right\}\times \cfrac{14.057{}^2}{2 g} =12.102 \end{align*}$

(9) Discharge coefficient of port

$\begin{align*} &C_{d,in}=\cfrac{1}{\sqrt{\zeta_{in}-(\phi^2-\psi^2)}} =\cfrac{1}{\sqrt{1.316-(0.295^2-0.040^2)}}=0.901 \\ &C_{d,out}=\cfrac{1}{\sqrt{\zeta_{out}+(\phi^2-\psi^2)}} =\cfrac{1}{\sqrt{1.115+(0.295^2-0.040^2)}}=0.913 \end{align*}$

Program

Sample code

import numpy as np

g=9.8   # gravity acceleration


def inp_saku():
    n=0.013 # Manning's roughness coefficient of port
    ds=12.5 # shaft diameter
    dp=2.5  # port diameter
    dd=4.6  # waterway diameter
    ell=1.5 # port length
    r1=0.8  # radius of lower part of port
    r2=0.5  # radius of upper part of port
    qq=69   # maximum discharge
    return n,ds,dp,dd,ell,r1,r2,qq


def inp_bacai():
    n=0.011 # Manning's roughness coefficient of port
    ds=12.5 # shaft diameter
    dp=3.0  # port diameter
    dd=5.5  # waterway diameter
    ell=26.6 # port length
    r1=0.5  # dradius of upper part of port
    r2=0.5  # radius of lower part of port
    qq=174   # maximum discharge
    return n,ds,dp,dd,ell,r1,r2,qq


def main():
    # (1) data input
    n,ds,dp,dd,ell,r1,r2,qq=inp_saku()
    #n,ds,dp,dd,ell,r1,r2,qq=inp_bacai()
    # (2) parameters
    psi=np.round((dp/ds)**2, decimals=3)
    phi=np.round((dp/dd)**2, decimals=3)
    delta=np.round(ell/dp, decimals=3)
    rho1=np.round(r1/dd, decimals=3)
    rho2=np.round(r2/ds, decimals=3)
    # (3) interference factor
    mu_i=1.230
    mu_o=1.190
    if 6.0 <= delta: mu_i=1.0
    if 6.0 <= delta: mu_o=1.0
    # (4) head loss coefficient due to section suddenly changed
    fse=np.round((1-psi)**2, decimals=3)
    fc=np.round(0.055/np.sqrt(0.003+rho2)*0.5*(1-1.144*psi)*(1+0.740*psi), decimals=3)
    # (5) head loss coefficient due to right angle branch
    cfb_i=np.round((phi**2-0.1*phi+0.4)*(1-0.9*np.sqrt(rho1/phi)), decimals=3)
    cfb_o=np.round(0.4*((1-2.5*np.sqrt(rho1))*phi**2+2), decimals=3)
    # (6) head loss due to friction
    df=np.round(delta*124.5*n**2/dp**(1/3), decimals=3)
    # (7) summary of head loss coefficient
    zeta_in=np.round(mu_i*(cfb_i+fse)+df, decimals=3)
    zeta_ot=np.round(mu_o*(cfb_o+fc)+df, decimals=3)
    # (8) head loss of port
    vp=np.round(4*qq/np.pi/dp**2, decimals=3) # velocity of port
    k_in=np.round((zeta_in-(phi**2-psi**2))*vp**2/2/g, decimals=3) # for flow-in
    k_ot=np.round((zeta_ot+(phi**2-psi**2))*vp**2/2/g, decimals=3) # for flow-out
    # (9) discharge coefficient of port
    cd_in=np.round(1/np.sqrt(zeta_in-(phi**2-psi**2)), decimals=3) # for flow-in
    cd_ot=np.round(1/np.sqrt(zeta_ot+(phi**2-psi**2)), decimals=3) # for flow-out

    # Print out of result
    print('# (1) Input parameters')
    print('Roughness coefficient n  ={0:6.3f}'.format(n))
    print('Shaft diameter        ds ={0:6.3f} (m)'.format(ds))
    print('Port diameter         dp ={0:6.3f} (m)'.format(dp))
    print('Waterway diameter     dd ={0:6.3f} (m)'.format(dd))
    print('Port length           ell={0:6.3f} (m)'.format(ell))
    print('Radius of lower bevel r1 ={0:6.3f} (m)'.format(r1))
    print('Radius of upper bevel r2 ={0:6.3f} (m)'.format(r2))
    print('Maximum discharge     qq ={0:6.3f} (m3/s)'.format(qq))
    print('# (2) Parameters for calculation')
    print('psi          ={0:6.3f}'.format(psi))
    print('phi          ={0:6.3f}'.format(phi))
    print('delta        ={0:6.3f}'.format(delta))
    print('rho1 (lower) ={0:6.3f}'.format(rho1))
    print('rho2 (upper) ={0:6.3f}'.format(rho2))
    print('# (3) Interference factor')
    print('mu_i   ={0:6.3f}'.format(mu_i))
    print('mu_o   ={0:6.3f}'.format(mu_o))
    print('# (4) Head loss coefficient due to section sudden changed')
    print('fse    ={0:6.3f}'.format(fse))
    print('fc     ={0:6.3f}'.format(fc))
    print('# (5) Head loss coefficient due to right angle branch')
    print('cfb_i  ={0:6.3f}'.format(cfb_i))
    print('cfb_o  ={0:6.3f}'.format(cfb_o))
    print('# (6) Head loss coefficient due to friction')
    print('df     ={0:6.3f}'.format(df))
    print('# (7) Summary of head loss coefficient')
    print('zeta_in={0:6.3f}'.format(zeta_in))
    print('zeta_ot={0:6.3f}'.format(zeta_ot))
    print('# (8) Resistance of port')
    print('vp     ={0:6.3f} (m/s)'.format(vp))
    print('k_in   ={0:6.3f} (m)'.format(k_in))
    print('k_ot   ={0:6.3f} (m)'.format(k_ot))
    print('# (9) Discharge coefficient of port')
    print('cd_in  ={0:6.3f}'.format(cd_in))
    print('cd_ot  ={0:6.3f}'.format(cd_ot))


#---------------
# Execute
#---------------
if __name__ == '__main__': main()

Example of output

# (1) Input parameters
Roughness coefficient n  = 0.013
Shaft diameter        ds =12.500 (m)
Port diameter         dp = 2.500 (m)
Waterway diameter     dd = 4.600 (m)
Port length           ell= 1.500 (m)
Radius of lower bevel r1 = 0.800 (m)
Radius of upper bevel r2 = 0.500 (m)
Maximum discharge     qq =69.000 (m3/s)
# (2) Parameters for calculation
psi          = 0.040
phi          = 0.295
delta        = 0.600
rho1 (lower) = 0.174
rho2 (upper) = 0.040
# (3) Interference factor
mu_i   = 1.230
mu_o   = 1.190
# (4) Head loss coefficient due to section sudden changed
fse    = 0.922
fc     = 0.130
# (5) Head loss coefficient due to right angle branch
cfb_i  = 0.141
cfb_o  = 0.799
# (6) Head loss coefficient due to friction
df     = 0.009
# (7) Summary of head loss coefficient
zeta_in= 1.316
zeta_ot= 1.115
# (8) Resistance of port
vp     =14.057 (m/s)
k_in   =12.406 (m)
k_ot   =12.102 (m)
# (9) Discharge coefficient of port
cd_in  = 0.901
cd_ot  = 0.913

Reference

技術研究報告 NO.65003 基本形ポートの水頭損失に関する実験的研究　1965.4　電力中央研究所技術研究所
奥清津第二発電所の調圧水槽のサージングおよび水撃圧の検討　藤野・橋本・芳賀　No.263 電力土木　1996.5

以　上