# Hydrodynamic calculation Howell-Jet valve

## Values for calculation

$D$ $\mathrm{mm}$
$D_{needle}$ $\mathrm{mm}$
$d$ $\mathrm{mm}$
$a$ $\mathrm{mm}$
$b$ $\mathrm{mm}$
$c$ $\mathrm{mm}$
$e$ $\mathrm{mm}$
$f$ $\mathrm{mm}$
$g$ $\mathrm{mm}$
$h$ $\mathrm{mm}$
$i$ $\mathrm{mm}$
$j$ $\mathrm{mm}$
$k$ $\mathrm{mm}$
$l$ $\mathrm{mm}$
$m$ $\mathrm{mm}$
$n$ $\mathrm{mm}$
$o$ $\mathrm{mm}$
$p$ $\mathrm{mm}$
$q$ $\mathrm{mm}$
$r$ $\mathrm{mm}$
$t$ $\mathrm{mm}$
$u$ $\mathrm{mm}$
$v$ $\mathrm{mm}$
$w$ $\mathrm{mm}$
$x$ $\mathrm{mm}$
$y$ $\mathrm{mm}$
$z$ $\mathrm{mm}$
$d_1$ $\mathrm{mm}$
$H$ $\mathrm{m}$
$g$ $\mathrm{m/s^2}$
$T$ $\mathrm{°C}$
$ρ$ $\mathrm{kg/m^3}$
$P_{SV}$ $\mathrm{Pa}$
$ΔP$ $\mathrm{m}$
$Σζ$
$h$ $\mathrm{m}$
$ρ_{air}$ $\mathrm{kg/m^3}$
$p_{air}$ $\mathrm{Pa}$

## Calculation

### Flow

$$Q_{max}=\cfrac{1}{\sqrt{1+Σζ+\min\left(ζ\right)}}\cdot\cfrac{π\cdot D^2}{4\cdot 10^6}\cdot\sqrt{2\cdot g\cdot H}$$

### Velocity in valve

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

### Theoretical pressure in the valve at full opening

$$Δ_h=\cfrac{v_{max}^2}{2\cdot g}\cdot \left({\min\left(ζ\right)}+1\right)$$

### Pressure parameter

$$p=\cfrac{Δ_h}{H}$$

$$0 < p \le 1$$
Stroke from open position
Flow coefficient
Coefficient of hydraulic force on a needle in the axis x
Coefficient of hydraulic force on a needle upstream in the axis x
Coefficient of hydraulic force on body in the axis x
$s$ $K_Q$ $K_x$ $K_{x-upstream}$ $K_{bx}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data

$K_x \mathrm{[-]}$
$K_{x-upstream} \mathrm{[-]}$
$K_{bx} \mathrm{[-]}$
 No data
$s\mathrm{[\%]}$

$K_Q \mathrm{[-]}$
 No data
$s\mathrm{[\%]}$

Stroke from open position
Loss coefficient
Reduced free flow area in the throttle control system
Relative flow
Flow of water in the pipeline
Water velocity in pipeline
$s$ $ζ$ $f_r$ $Q_p$ $Q$ $v$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{m^3/s}$ $\mathrm{m/s}$
No data

$ζ \mathrm{[-]}$
 No data
$s\mathrm{[\%]}$

$$ζ=\cfrac{1-K_Q^2}{K_Q^2}$$

$f_r \mathrm{[-]}$
$Q_p \mathrm{[-]}$
 No data
$s\mathrm{[\%]}$

$$f_r=\cfrac{K_Q}{K_{Qmax}}$$
$$Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}$$

$Q \mathrm{[m^3/s]}$
$v \mathrm{[m/s]}$
 No data
$s\mathrm{[\%]}$

$$Q=Q_p\cdot Q_{max}$$
$$v=Q_p\cdot v_{max}$$
Stroke from open position
Loss of pressure on the valve
Pressure on the valve
Cavitation number
$s$ $H_L$ $H_v$ $σ$
$\mathrm{\%}$ $\mathrm{m}$ $\mathrm{m}$ $\mathrm{ }$
No data

$H_L \mathrm{[m]}$
$H_v \mathrm{[m]}$
 No data
$s\mathrm{[\%]}$

$$H_L=\cfrac{v^2}{2\cdot g}\cdot ζ$$
$$H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot ΔP$$

$σ \mathrm{[-]}$
 No data
$s\mathrm{[\%]}$

$$σ=\cfrac{\cfrac{p_{air}-P_{SV}}{ρ\cdot g}+H-H_L}{H_v}$$
Stroke from open position
Forces on the needle
The force at the valve axis x
$s$ $F_x$ $F_{bx}$
$\mathrm{\%}$ $\mathrm{kN}$ $\mathrm{kN}$
No data

$F_x \mathrm{[kN]}$
 No data
$s\mathrm{[\%]}$

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

$F_{bx} \mathrm{[kN]}$
 No data
$s\mathrm{[\%]}$

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