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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\in\mathbb{R}\mid D>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ D $ $ \mathrm{mm} $
$ \{D_{needle}\in\mathbb{R}\mid D_{needle}>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ D_{needle} $ $ \mathrm{mm} $
$ \{d\in\mathbb{R}\mid d>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ 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\in\mathbb{R}\mid H>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ H $ $ \mathrm{m} $
$ \{g\in\mathbb{R}\mid g>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ g $ $ \mathrm{m/s^2} $
$ \{T\in\mathbb{R}\mid T\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ T $ $ \mathrm{°C} $
$ ρ $ $ \mathrm{kg/m^3} $
$ P_{SV} $ $ \mathrm{Pa} $
$ \{ΔP\in\mathbb{R}\mid ΔP>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ ΔP $ $ \mathrm{m} $
$ \{Σζ\in\mathbb{R}\mid Σζ>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ Σζ $
$ \{h\in\mathbb{R}\mid h\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ 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\ [-] $
$ K_{x-upstream}\ [-] $
$ K_{bx}\ [-] $
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\ [-] $
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\ [-] $
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\ [-] $
$ Q_p\ [-] $
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\ [-] $
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;
$\text{if }\ α=90$
$\text{if }\ α=90$
Equations in LaTeX code
\text{if }\ α=90
$$F_{bx}=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$$
$F_{bx}=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$
Equations in LaTeX code
F_{bx}=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$F_{bx}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$$
$F_{bx}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$
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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