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Hydrodynamic calculation Howell-Jet valve

Howell-Jet valve Q max D a b c e f g h i j k l m n o p r q t ELLIPSE s VANES 33° B u 8 d 1 d v w x 10° y VANES ALTERNATE AT 45° DETAIL "B" 33° 40° z +F -F D needle
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

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

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

Theoretical pressure in the valve at full opening

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

Pressure parameter

$$p=\cfrac{Δ_h}{H}$$
$p=\cfrac{Δ_h}{H}$
Equations in LaTeX code
p=\cfrac{Δ_h}{H}
$$0 < p \le 1$$
$0 < p \le 1$
Equations in LaTeX code
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
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
Unrounded data
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};
\%; ; ; ; ;

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

Stroke from open position 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
$s$ $K_x$ $K_{x-upstream}$ $K_{bx}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position; 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_x; K_{x-upstream}; K_{bx};
\%; ; ; ;

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

Stroke from open position Flow coefficient
$s$ $K_Q$
$\mathrm{\%}$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position; Flow coefficient;
s; K_Q;
\%; ;
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
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
Unrounded data
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;
\%; ; ; ;m^3/s;m/s;

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

Stroke from open position Loss coefficient
$s$ $ζ$
$\mathrm{\%}$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position; Loss coefficient;
s; ζ;
\%; ;
$$ζ=\cfrac{1-K_Q^2}{K_Q^2}$$
$ζ=\cfrac{1-K_Q^2}{K_Q^2}$
Equations in LaTeX code
ζ=\cfrac{1-K_Q^2}{K_Q^2}

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

Stroke from open position Reduced free flow area in the throttle control system Relative flow
$s$ $f_r$ $Q_p$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position; Reduced free flow area in the throttle control system; Relative flow;
s; f_r; Q_p;
\%; ; ;
$$f_r=\cfrac{K_Q}{K_{Qmax}}$$
$f_r=\cfrac{K_Q}{K_{Qmax}}$
Equations in LaTeX code
f_r=\cfrac{K_Q}{K_{Qmax}}
$$Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}$$
$Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}$
Equations in LaTeX code
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
$s\mathrm{[\%]}$
Flow and speed of water in the pipeline

Stroke from open position Flow of water in the pipeline Water velocity in pipeline
$s$ $Q$ $v$
$\mathrm{\%}$ $\mathrm{m^3/s}$ $\mathrm{m/s}$
No data
Unrounded data
Stroke from open position; Flow of water in the pipeline; Water velocity in pipeline;
s; Q; v;
\%; m^3/s; m/s;
$$Q=Q_p\cdot Q_{max}$$
$Q=Q_p\cdot Q_{max}$
Equations in LaTeX code
Q=Q_p\cdot Q_{max}
$$v=Q_p\cdot v_{max}$$
$v=Q_p\cdot v_{max}$
Equations in LaTeX code
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
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
Unrounded data
Stroke from open position;Loss of pressure on the valve;Pressure on the valve;Cavitation number;
s;H_L;H_v;σ;
\%;m;m; ;

$H_L \mathrm{[m]}$
$H_v \mathrm{[m]}$
No data
$s\mathrm{[\%]}$
Loss of height on valve and pressure height on Howell-Jet valve

Stroke from open position Loss of pressure on the valve Pressure on the valve
$s$ $H_L$ $H_v$
$\mathrm{\%}$ $\mathrm{m}$ $\mathrm{m}$
No data
Unrounded data
Stroke from open position; Loss of pressure on the valve; Pressure on the valve;
s; H_L; H_v;
\%; m; m;
$$H_L=\cfrac{v^2}{2\cdot g}\cdot ζ$$
$H_L=\cfrac{v^2}{2\cdot g}\cdot ζ$
Equations in LaTeX code
H_L=\cfrac{v^2}{2\cdot g}\cdot ζ
$$H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot ΔP$$
$H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot ΔP$
Equations in LaTeX code
H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot ΔP

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

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

$F_x \mathrm{[kN]}$
No data
$s\mathrm{[\%]}$
Forces on the needle

Stroke from open position Forces on the needle
$s$ $F_x$
$\mathrm{\%}$ $\mathrm{kN}$
No data
Unrounded data
Stroke from open position; Forces on the needle;
s; F_x;
\%; kN;
$$F_x=\cfrac{π\cdot ρ\cdot g\cdot H_v}{4\cdot 10^9}\cdot \left(D^2\cdot K_x-D^2\cdot\left(1.1162^2-0.04\right)\cdot K_{x-upstream}+\left(D_{needle}^2-d^2\right)\cdot K_{x-upstream}\right)$$
$F_x=\cfrac{π\cdot ρ\cdot g\cdot H_v}{4\cdot 10^9}\cdot \left(D^2\cdot K_x-D^2\cdot\left(1.1162^2-0.04\right)\cdot K_{x-upstream}+\left(D_{needle}^2-d^2\right)\cdot K_{x-upstream}\right)$
Equations in LaTeX code
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
$s\mathrm{[\%]}$
The force at the valve axis x

Stroke from open position The force at the valve axis x
$s$ $F_{bx}$
$\mathrm{\%}$ $\mathrm{kN}$
No data
Unrounded data
Stroke from open position; The force at the valve axis x;
s; F_{bx};
\%; kN;
$$F_{bx}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$$
$F_{bx}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$
Equations in LaTeX code
F_{bx}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}
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