## Hydrodynamic calculation Howell-Jet valve

Fig. 1 - Howell-Jet valve

$a$ $=1.1597D$

$b$ $=0.5D$

$c$ $=0.5D$

$e$ $=0.1797D$

$f$ $=0.0068D$

$g$ $=0.0555D$

$h$ $=0.2533D$

$i$ $=0.0777D$

$j$ $=0.1116D$

$k$ $=0.0287D$

$l$ $=0.2927D$

$m$ $=0.1346D$

$n$ $=1.2083D$

$o$ $=0.1875D$

$p$ $=0.186D$

$q$ $=0.138D$

$r$ $=0.25D$

$t$ $=0.0538D$

$u$ $=0.8D$

$v$ $=0.6198D$

$w$ $=0.1146D$

$x$ $=3.7083D$

$y$ $=0.0208D$

$z$ $=0.0026D$

$d_1$ $=0.02D$

The Howell-Jet is a type of valve used for dam outlets. The valve uses a flow-shaped body which moves in the direction of the longitudinal axis inside the valve. The water jet leaving the valve has the shape of a hollow cylinder. Unlike the Howell-Bunger closure, the Howell-Jet has a more complex design that must provide control of the movement of the bypassed element outside the valve.
When designing a Howell-Jet valve, it is necessary to recognize several important parameters that express the hydraulic conditions and dynamic effects of flow in the valve. The basic parameters are the cavitation coefficient, the pressure loss, the flow rate and the resulting hydrodynamic forces acting on the movable part of the valve. You need to know the values of these parameters depending on the opening of the valve.

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

To calculate the pressure on the Howell-Jet 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-Jet. The pressure loss before the valve must be defined without the Howell-Jet valve (to diameter $D$).
To calculate the pressure on the Howell-Jet 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:

$H_v$ $=H_L+\cfrac{v^2}{2g}+(1-Q_p)\cdot (Δ P+P_0)$

where:
 $H_v$ pressure on the valve $\mathrm{m}$ $H_L$ loss of pressure on the valve $\mathrm{m}$ $v$ water velocity in pipeline $\mathrm{m/s}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $Q_p$ relative flow $\mathrm{ }$ $Δ P$ increasing pressure on water hammer $\mathrm{m}$ $P_0$ under-pressure behind the valve $\mathrm{m}$

Relative flow:
To calculate the increase in water hammer pressure.

Valve position Relative flow
$s$ $Q_p$
$\mathrm{\%}$ $\mathrm{ }$
00.743
100.706
200.657
300.612
400.546
500.478
600.414
700.337
800.246
900.166
1000.09

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

In the annex section of Fig. 2 to 3, charts of dimensionless coefficients are constructed. These graphs are the basis for constructing the hydrodynamic characteristics of the Howell-Jet valve for the projected water dam.

Cavitation number:

$σ$ $=\cfrac{10-0.1+h_0-H_L}{H_v}$

where:
 $σ$ cavitation number $\mathrm{ }$ $h_0$ rated net head (max 6500m) $\mathrm{m}$ $H_L$ loss of pressure on the valve $\mathrm{m}$ $H_v$ pressure on the valve $\mathrm{m}$

Forces on the needle:

$F_x$ $=\cfrac{π\cdot ρ\cdot g\cdot H_v}{4\cdot{10}^9}\cdot\left(D^2\cdot K_x-D^2(1.1162^2-0.04)K_{x-upstream}+\left(D_{needle}^2-d^2\right)\cdot K_{x-upstream}\right)$

where:
 $F_x$ forces on the needle $\mathrm{kN}$ $D$ valve diameter $\mathrm{mm}$ $ρ$ density $\mathrm{kg/m^3}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $H_v$ pressure on the valve $\mathrm{m}$ $K_x$ coefficient of hydraulic force on a needle in the axis x $\mathrm{ }$ $K_{x-upstream}$ coefficient of hydraulic force on a needle upstream in the axis x $\mathrm{ }$ $D_{needle}$ diameter needle $\mathrm{mm}$ $d$ inner diameter $\mathrm{mm}$

The force at the valve axis x:

$F_{bx}$ $=\cfrac{π D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$

where:
 $F_{bx}$ the force at the valve axis x $\mathrm{kN}$ $D$ valve diameter $\mathrm{mm}$ $ρ$ density $\mathrm{kg/m^3}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $H_v$ pressure on the valve $\mathrm{m}$ $K_{bx}$ coefficient of hydraulic force on body in the axis x $\mathrm{ }$

### 3. Conclusion:

The Howell-Jet valve must be hydraulically positioned in a straight diameter $D$ . The effect of cavitation on pressure losses, flow and dynamic effects of the water stream does not occur immediately in the initial stage (when the first steam bubbles are formed), but only with a fully developed cavitation.

Literature:
- Miroslav Nechleba: Vodní turbíny jejich konstrukce a příslušenství 1954.
- V. Kolář, St. Vinopal: Hydraulika průmyslových armatur 1963.
- Hydraulic Laboratory Report No. Hyd-446: Stilling basin for high head outlet works utilizing Hollow-Jet valve control (basin VIII).

-●-$K_x \mathrm{[-]}$
-●-$K_{x-upstream} \mathrm{[-]}$
-●-$K_{bx} \mathrm{[-]}$
$s\mathrm{[\%]}$
Fig. 2 - Coefficient of force

-●-$K_Q \mathrm{[-]}$
$s\mathrm{[\%]}$
Fig. 3 - Flow coefficient