## Hydrodynamic calculation Howell-Bunger valve

Fig. 1 - Howell-Bunger valve

$S$ $=0.6D$

$R$ $=0.5555D$

$a$ $=0.1583D$

$b$ $=0.1361D$

$c$ $=0.5694D$

$e$ $=0.6861D$

$f$ $=0.1222D$

$g$ $=0.0666D$

The Howell-Bunger valve today is one of the most widely used valves of dam outlets. Its construction is simple and the operating forces during opening are relatively small. Other advantages compared to other types of valves include efficient effluent damping as well as good aeration of water, which can often be critical to improving its quality.
Incorrect hydraulic design of the discharge object, or even in the case of improper design of the shutter itself, can result in dangerous shocks and vibrations, which can cause a crash of the Howell-Bunger valve.
The main components are the cylindrical valve body, the outlet cone with radial ribs, the sliding cylinder and the drive. The outflow cone forms an annular opening with the cylindrical valve body. The outflow of water creates a hollow cone which is directed in the outflow chamber into the drain channel. By vigorously swirling water and air in the effluent chamber, the effluent energy of the water is significantly dampened in a relatively small space.
The sliding cylinder is the only sliding part of the valve that allows continuous flow control. The driving forces of the sliding cylinder are small and practically unchanged over the entire stroke.

Fig. 2 - model no.1

Fig. 3 - model no.2

Fig. 4 - model no.3

Fig. 5 - model no.1

Fig. 6 - model no.2

Fig. 7 - model no.3

### 1. Calculation of pressure on the Howell-Bunger valve during its rapid closure:

To calculate the pressure on the Howell-Bunger valve, we need to know the rated net head at the zero flow (closed valve). Increase pressure on water hammer must be calculated before the hydrodynamic calculation of the Howell-Bunger. The pressure loss before the valve must be defined without the Howell-Bunger valve (to diameter $D$).
To calculate the pressure on the Howell-Bunger valve because of the ignorance of the piping system, a calculation for a simple serial connection of the control valves (variable and constant resistance) will be used.

Relative flow:

$Q_p$ $=\cfrac{f_r}{\sqrt{p+f_r^2(1-p)}}$

where:
 $Q_p$ relative flow $\mathrm{ }$ $f_r$ reduced free flow area in the throttle control system $\mathrm{ }$ $p$ pressure parameter $\mathrm{ }$

Reduced free flow area in the throttle control system:

$f_r$ $=\cfrac{K_Q}{K_{Qmax}}$

where:
 $f_r$ reduced free flow area in the throttle control system $\mathrm{ }$ $K_Q$ flow coefficient $\mathrm{ }$ $K_{Qmax}$ flow coefficient max $\mathrm{ }$

Pressure parameter:

$p$ $=\cfrac{Δ h}{h_0}$

$0 < p\le1$

where:
 $p$ pressure parameter $\mathrm{ }$ $Δ h$ theoretical pressure in the valve at full opening $\mathrm{m}$ $h_0$ rated net head (max 6500m) $\mathrm{m}$

Theoretical pressure in the valve at full opening:

$Δ h$ $=\cfrac{v_0^2}{2g}\cdot\left(ξ+1\right)$

where:
 $Δ h$ theoretical pressure in the valve at full opening $\mathrm{m}$ $v_0$ valve speed $\mathrm{m/s}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $ξ$ loss coefficient $\mathrm{ }$

Valve speed:

$v_0$ $=\cfrac{4\cdot{10}^6\cdot Q_{max}}{π\cdot D^2}$

where:
 $v_0$ valve speed $\mathrm{m/s}$ $Q_{max}$ flow $\mathrm{m^3/s}$ $D$ valve diameter $\mathrm{mm}$

Flow of water in the pipeline:

$Q$ $=Q_p\cdot Q_{max}$

where:
 $Q$ flow of water in the pipeline $\mathrm{m^3/s}$ $Q_p$ relative flow $\mathrm{ }$ $Q_{max}$ flow $\mathrm{m^3/s}$

Water velocity in pipeline:

$v$ $=\cfrac{4Q\cdot 10^6}{π\cdot D^2}$

where:
 $v$ water velocity in pipeline $\mathrm{m/s}$ $Q$ flow of water in the pipeline $\mathrm{m^3/s}$ $D$ valve diameter $\mathrm{mm}$

Loss of pressure on the valve:

$H_L$ $=\cfrac{v^2}{2g}ξ$

where:
 $H_L$ loss of pressure on the valve $\mathrm{m}$ $v$ water velocity in pipeline $\mathrm{m/s}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $ξ$ loss coefficient $\mathrm{ }$

Loss coefficient:

$ξ$ $=\cfrac{1-K_Q^2}{K_Q^2}$

where:
 $ξ$ loss coefficient $\mathrm{ }$ $K_Q$ flow coefficient $\mathrm{ }$

Pressure on the valve:

$H_v$ $=h_0-P_0+\cfrac{v^2}{2g}-\cfrac{v^2}{2g}Σξ+\left(1-Q_p\right)\cdotΔ P$

where:
 $H_v$ pressure on the valve $\mathrm{m}$ $h_0$ rated net head (max 6500m) $\mathrm{m}$ $v$ water velocity in pipeline $\mathrm{m/s}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $Q_p$ relative flow $\mathrm{ }$ $Δ P$ increasing pressure on water hammer $\mathrm{m}$ $P_0$ under-pressure behind the valve $\mathrm{m}$ $ξ$ loss coefficient $\mathrm{ }$

### 2. Guideline for the use of the hydrodynamic characteristics of the Howell-Bunger valve:

In the annex section of Fig. 8 to 55, charts of dimensionless coefficients are constructed. These graphs are the basis for constructing the hydrodynamic characteristics of the Howell-Bunger valve for the projected water dam.

Cavitation number:

$\sigma=\cfrac{10-0,1+h_0-H_{L}}{H_{v}}$
where:
 $\sigma$ cavitation number $\mathrm{-}$ $h_0$ rated net head $\mathrm{m}$ $H_{L}$ the pressure loss in the pipeline $\mathrm{m}$ $H_{v}$ pressure on the Howell-Bunger valve $\mathrm{m}$

Under-pressure behind the valve:

$P_0=max\left\{K_{P0}\cdot H_v;\ -10\right\}$
where:
 $P_0$ under-pressure behind the valve $\mathrm{m}$ $K_{P0}$ coefficient of under-pressure $\mathrm{-}$ $H_{v}$ pressure on the Howell-Bunger valve $\mathrm{m}$

Forces in axis x:

$F_x=\cfrac{\pi D^2}{4}\cdot\rho\cdot g\cdot H_v\left(K_x\pm a_x\right)$
where:
 $F_x$ forces in axis x $\mathrm{kN}$ $D$ valve diameter $\mathrm{mm}$ $ρ$ density of liquid $\mathrm{kg/m^3}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $H_{v}$ pressure on the Howell-Bunger valve $\mathrm{m}$ $K_{x}$ coefficient of hydraulic force in the axis x $\mathrm{-}$ $a_{x}$ the amplitude of the hydraulic force to the axis x $\mathrm{-}$

Energy before the valve:
For model no.1 a 2

$T_x=h_0+\cfrac{v^2}{2g}+2,5D-\cfrac{v^2}{2g}\Sigma\xi$
For model no.3
$T_x=h_0+\cfrac{v^2}{2g}+1,5D-\cfrac{v^2}{2g}\Sigma\xi$
where:
 $T_x$ energy before the valve $\mathrm{m}$ $h_0$ rated net head $\mathrm{m}$ $v$ the water speed in the pipeline $\mathrm{m/s}$ $D$ valve diameter $\mathrm{mm}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $\Sigma\xi$ pressure loss before valve without Howell-Bunger valve $\mathrm{-}$

Under-pressure in hole 1:

$P_{1AIR}=max\left\{K_{P1AIR}\cdot H_v;\ -10\right\}$
where:
 $P_{1AIR}$ under-pressure in hole 1 $\mathrm{m}$ $K_{P1AIR}$ under-pressure coefficient in hole 1 $\mathrm{-}$ $H_{v}$ pressure on the Howell-Bunger valve $\mathrm{m}$

Under-pressure in hole 2:

$P_{2AIR}=max\left\{K_{P2AIR}\cdot H_v;\ -10\right\}$
where:
 $P_{2AIR}$ under-pressure in hole 2 $\mathrm{m}$ $K_{P2AIR}$ under-pressure coefficient in hole 2 $\mathrm{-}$ $H_{v}$ pressure on the Howell-Bunger valve $\mathrm{m}$

### 3. Dimensioning aerated hole:

To reduce valve vibration, pulsation of hydrodynamic forces and erosion effects of cavitation by aerating the area behind the valve. The aerated hole should be large enough for air flow to reach according to research results.

Air flow:

$Q_a=\cfrac{Q\cdot\beta}{100}$
where:
 $Q_a$ air flow $\mathrm{m^3/s}$ $Q$ flow in pipeline $\mathrm{m^3/s}$ $\beta$ aerated coefficient $\mathrm{-}$
Air velocity in hole 1 and 2:
Air velocity in the narrowest cross section
$v_{a1}=min\left\{0,7\cdot\sqrt{\cfrac{2{\cdot g\cdot P}_{1AIR}\cdot\rho}{\rho_{air}}};250\right\}$
$v_{a2}=min\left\{0,7\cdot\sqrt{\cfrac{2{\cdot g\cdot P}_{2AIR}\cdot\rho}{\rho_{air}}};250\right\}$
where:
 $v_{a1}$ air velocity in hole 1 $\mathrm{m/s}$ $v_{a2}$ air velocity in hole 2 $\mathrm{m/s}$ $P_{1AIR}$ under-pressure in hole 1 $\mathrm{m}$ $P_{2AIR}$ under-pressure in hole 2 $\mathrm{m}$ $ρ_{air}$ air density $\mathrm{kg/m^3}$ $ρ$ density of liquid $\mathrm{kg/m^3}$ $g$ gravitational acceleration $\mathrm{m/s^2}$

Hole area 1 and 2:

$S=\cfrac{Q_a}{v_{a1}+v_{a2}}$
where:
 $S$ hole area 1 and 2 $\mathrm{m^2}$ $Q_a$ air flow $\mathrm{m^3/s}$ $v_{a1}$ air velocity in hole 1 $\mathrm{m/s}$ $v_{a2}$ air velocity in hole 2 $\mathrm{m/s}$

Diameter of the hole 1 and 2:

$D_a=\sqrt{\cfrac{4S}{\pi}}$
where:
 $D_a$ diameter of the hole 1 and 2 $\mathrm{m}$ $S$ hole area 1 and 2 $\mathrm{m^2}$

### 4. Conclusion:

The most effective is the No.3 design, the design is simple and has very good hydraulic parameters, mainly the effective aeration of the space before and after the outflow cone, very good damping of the water and low force loading of the Howell-Bunger valve with small pulsations of hydrodynamic forces and pressures.

Literature:
- Miroslav Žajdlík: Výskum priehradových výpustov s rozstrekovacími uzáverami 1980.
- Miroslav Nechleba: Vodní turbíny jejich konstrukce a příslušenství 1954.
- V. Kolář, St. Vinopal: Hydraulika průmyslových armatur 1963.

-●-$K_Q$ for $h_D=\min \mathrm{[-]}$
-●-$K_Q$ for $h_D=1.2 \mathrm{[-]}$
-●-$K_Q$ for $h_D=2.5 \mathrm{[-]}$
-●-$K_Q$ for $h_D=3.5 \mathrm{[-]}$
-●-$K_Q$ for $h_D=4.5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 8 - Coefficient of discharge - model no.1

-●-$K_x$ for $h_D=\min \mathrm{[-]}$
-●-$K_x$ for $h_D=1.2 \mathrm{[-]}$
-●-$K_x$ for $h_D=2.5 \mathrm{[-]}$
-●-$K_x$ for $h_D=3.5 \mathrm{[-]}$
-●-$K_x$ for $h_D=4.5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 9 - Coefficient of force - model no.1

-●-$a_x$ for $h_D=\min \mathrm{[-]}$
-●-$a_x$ for $h_D=1.2 \mathrm{[-]}$
-●-$a_x$ for $h_D=2.5 \mathrm{[-]}$
-●-$a_x$ for $h_D=3.5 \mathrm{[-]}$
-●-$a_x$ for $h_D=4.5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 10 - Amplitude coefficient - model no.1

-●-$K_{P_0}$ for $h_2=\min \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=2 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=2.5 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=3 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=3.5 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=3.6 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=3.9 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=4 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=4.2 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=4.5 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 11 - Under-pressure coefficient - model no.1

-●-$K_{P_{1-AIR}}$ for $h_2=\min \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=2 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=2.5 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=3 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=3.5 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=3.6 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=3.9 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=4 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=4.2 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=4.5 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 12 - Under-pressure coefficient in hole 1 - model no.1

-●-$K_{P_{2-AIR}}$ for $h_2=\min \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=2 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=2.5 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=3 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=3.5 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=3.6 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=3.9 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=4 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=4.2 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=4.5 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 13 - Under-pressure coefficient in hole 2 - model no.1

-●-$h_2/D$ for $T_x/D=10 \mathrm{[-]}$
-●-$h_2/D$ for $T_x/D=20 \mathrm{[-]}$
-●-$h_2/D$ for $T_x/D=30 \mathrm{[-]}$
-●-$h_2/D$ for $T_x/D=40 \mathrm{[-]}$
-●-$h_2/D$ for $T_x/D=50 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 14 - Ratio water level to valve diameter - model no.1

-●-$L_1$ for $T_x/D=10 \mathrm{[m]}$
-●-$L_1$ for $T_x/D=20 \mathrm{[m]}$
-●-$L_1$ for $T_x/D=30 \mathrm{[m]}$
-●-$L_1$ for $T_x/D=40 \mathrm{[m]}$
-●-$L_1$ for $T_x/D=50 \mathrm{[m]}$
$S/D\mathrm{[-]}$
Fig. 15 - Length $L_1$ - model no.1

-●-$L_2$ for $T_x/D=10 \mathrm{[m]}$
-●-$L_2$ for $T_x/D=20 \mathrm{[m]}$
-●-$L_2$ for $T_x/D=30 \mathrm{[m]}$
-●-$L_2$ for $T_x/D=40 \mathrm{[m]}$
-●-$L_2$ for $T_x/D=50 \mathrm{[m]}$
$S/D\mathrm{[-]}$
Fig. 16 - Length $L_2$ - model no.1

-●-$K_Q$ for $h_D=\min \mathrm{[-]}$
-●-$K_Q$ for $h_D=1.2 \mathrm{[-]}$
-●-$K_Q$ for $h_D=2.5 \mathrm{[-]}$
-●-$K_Q$ for $h_D=3.5 \mathrm{[-]}$
-●-$K_Q$ for $h_D=4.5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 17 - Coefficient of discharge β - model no.1

-●-$K_x$ for $h_D=\min \mathrm{[-]}$
-●-$K_x$ for $h_D=1.2 \mathrm{[-]}$
-●-$K_x$ for $h_D=2.5 \mathrm{[-]}$
-●-$K_x$ for $h_D=3.5 \mathrm{[-]}$
-●-$K_x$ for $h_D=4.5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 18 - Coefficient of force β - model no.1

-●-$a_x$ for $h_D=\min \mathrm{[-]}$
-●-$a_x$ for $h_D=1.2 \mathrm{[-]}$
-●-$a_x$ for $h_D=2.5 \mathrm{[-]}$
-●-$a_x$ for $h_D=3.5 \mathrm{[-]}$
-●-$a_x$ for $h_D=4.5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 19 - Amplitude coefficient β - model no.1

-●-$β$ for $h_D=\min \mathrm{[-]}$
-●-$β$ for $h_D=1.2 \mathrm{[-]}$
-●-$β$ for $h_D=2.5 \mathrm{[-]}$
-●-$β$ for $h_D=3.5 \mathrm{[-]}$
-●-$β$ for $h_D=4.5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 20 - Aerated coefficient β - model no.1

-●-$K_{P_0}$ for $h_2=\min \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=2 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=2.5 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=3 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=3.5 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=3.6 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=3.9 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=4 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=4.2 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=4.5 \mathrm{[-]}$
-●-$K_{P_0}$ for $h_2=5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 21 - Under-pressure coefficient β - model no.1

-●-$K_{P_{1-AIR}}$ for $h_2=\min \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=2 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=2.5 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=3 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=3.5 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=3.6 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=3.9 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=4 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=4.2 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=4.5 \mathrm{[-]}$
-●-$K_{P_{1-AIR}}$ for $h_2=5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 22 - Under-pressure coefficient in hole 1 β - model no.1

-●-$K_{P_{2-AIR}}$ for $h_2=\min \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=2 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=2.5 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=3 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=3.5 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=3.6 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=3.9 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=4 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=4.2 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=4.5 \mathrm{[-]}$
-●-$K_{P_{2-AIR}}$ for $h_2=5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 23 - Under-pressure coefficient in hole 2 β - model no.1

-●-$h_2/D$ for $T_x/D=10 \mathrm{[-]}$
-●-$h_2/D$ for $T_x/D=20 \mathrm{[-]}$
-●-$h_2/D$ for $T_x/D=30 \mathrm{[-]}$
-●-$h_2/D$ for $T_x/D=40 \mathrm{[-]}$
-●-$h_2/D$ for $T_x/D=50 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 24 - Ratio water level to valve diameter β - model no.1

-●-$L_1$ for $T_x/D=10 \mathrm{[m]}$
-●-$L_1$ for $T_x/D=20 \mathrm{[m]}$
-●-$L_1$ for $T_x/D=30 \mathrm{[m]}$
-●-$L_1$ for $T_x/D=40 \mathrm{[m]}$
-●-$L_1$ for $T_x/D=50 \mathrm{[m]}$
$S/D\mathrm{[-]}$
Fig. 25 - Length $L_1$ β - model no.1

-●-$L_2$ for $T_x/D=10 \mathrm{[m]}$
-●-$L_2$ for $T_x/D=20 \mathrm{[m]}$
-●-$L_2$ for $T_x/D=30 \mathrm{[m]}$
-●-$L_2$ for $T_x/D=40 \mathrm{[m]}$
-●-$L_2$ for $T_x/D=50 \mathrm{[m]}$
$S/D\mathrm{[-]}$
Fig. 26 - Length $L_2$ β - model no.1

-●-$K_Q$ for $h_D=\min \mathrm{[-]}$
-●-$K_Q$ for $h_D=1.2 \mathrm{[-]}$
-●-$K_Q$ for $h_D=2.5 \mathrm{[-]}$
-●-$K_Q$ for $h_D=3.5 \mathrm{[-]}$
-●-$K_Q$ for $h_D=4.5 \mathrm{[-]}$
$S/D\mathrm{[-]}$
Fig. 27 - Coefficient of discharge - model no.2