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Hydrodynamic calculation Needle valve closing flow against the direction of movement of the piston

Needle valve closing flow against the direction of movement of the piston D Q max -F +F 0.912 D 0.975 D 0.4D L
Needle valve closing flow against the direction of movement of the piston
needle-valve-2 0.133 D - 4 d 1 0.183 D - 8 d 1 0.26 D - 12 d 1 0.35 D - 16 d 1 0.45 D - 20 d 1 0.56 D - 24 d 1 0.666 D - 28 d 1 0.773 D - 32 d 1 0.975 D d 1 =0.03 D
Needle

Values for calculation

$D$ $\mathrm{mm}$
$Q_{max}$ $\mathrm{m^3/s}$
$H$ $\mathrm{m}$
$g$ $\mathrm{m/s^2}$
$T$ $\mathrm{°C}$
$ρ$ $\mathrm{kg/m^3}$
$P_{SV}$ $\mathrm{Pa}$
$ΔP$ $\mathrm{m}$
$h$ $\mathrm{m}$
$ρ_{air}$ $\mathrm{kg/m^3}$
$p_{air}$ $\mathrm{Pa}$
$t$ $\mathrm{s}$
$L$ $\mathrm{m}$

Calculation

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

Effective closing time factor

$$c_{ef}=0.1/\max_{i=1}^{10}{\left(Q_p[i]-Q_p[i+1]\right)}$$
$c_{ef}=0.1/\max_{i=1}^{10}{\left(Q_p[i]-Q_p[i+1]\right)}$
Equations in LaTeX code
c_{ef}=0.1/\max_{i=1}^{10}{\left(Q_p[i]-Q_p[i+1]\right)}
$$c_{ef}\le 1$$
$c_{ef}\le 1$
Equations in LaTeX code
c_{ef}\le 1

Under-pressure behind the valve

$$P_{u}=\max\left(-\cfrac{L\cdot v_{max}}{g\cdot t\cdot c_{ef}}, -\cfrac{p_{air}}{ρ\cdot g}\right)$$
$P_{u}=\max\left(-\cfrac{L\cdot v_{max}}{g\cdot t\cdot c_{ef}}, -\cfrac{p_{air}}{ρ\cdot g}\right)$
Equations in LaTeX code
P_{u}=\max\left(-\cfrac{L\cdot v_{max}}{g\cdot t\cdot c_{ef}}, -\cfrac{p_{air}}{ρ\cdot g}\right)

$\mathrm{[-]}$
Hydraulic profile of the needle valve

Stroke from open position
First stage of cavitation
Second stage of cavitation
Fully developed cavitation
$s$ $σ_1$ $σ_2$ $σ_{min}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Stroke from open position First stage of cavitation Second stage of cavitation Fully developed cavitation
$s$ $σ_1$ $σ_2$ $σ_{min}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position;First stage of cavitation;Second stage of cavitation;Fully developed cavitation;
s;σ_1;σ_2;σ_{min};
\%; ; ; ;

$σ_1 \mathrm{[-]}$
$σ_2 \mathrm{[-]}$
$σ_{min} \mathrm{[-]}$
No data
$s\mathrm{[\%]}$
Stage of cavitation

Stroke from open position First stage of cavitation Second stage of cavitation Fully developed cavitation
$s$ $σ_1$ $σ_2$ $σ_{min}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position; First stage of cavitation; Second stage of cavitation; Fully developed cavitation;
s; σ_1; σ_2; σ_{min};
\%; ; ; ;
Stroke from open position
Flow coefficient for $ σ_1 $
Flow coefficient for $ σ_2 $
Flow coefficient for $ σ_{min} $
$s$ $K_{Q-σ_1}$ $K_{Q-σ_2}$ $K_{Q-σ_{min}}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Stroke from open position Flow coefficient for $ σ_1 $ Flow coefficient for $ σ_2 $ Flow coefficient for $ σ_{min} $
$s$ $K_{Q-σ_1}$ $K_{Q-σ_2}$ $K_{Q-σ_{min}}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position;Flow coefficient for $ σ_1 $ ;Flow coefficient for $ σ_2 $ ;Flow coefficient for $ σ_{min} $ ;
s;K_{Q-σ_1};K_{Q-σ_2};K_{Q-σ_{min}};
\%; ; ; ;

$K_{Q-σ_1} \mathrm{[-]}$
$K_{Q-σ_2} \mathrm{[-]}$
$K_{Q-σ_{min}} \mathrm{[-]}$
No data
$s\mathrm{[\%]}$
Flow coefficient for stages of cavitation

Stroke from open position Flow coefficient for $ σ_1 $ Flow coefficient for $ σ_2 $ Flow coefficient for $ σ_{min} $
$s$ $K_{Q-σ_1}$ $K_{Q-σ_2}$ $K_{Q-σ_{min}}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position; Flow coefficient for $ σ_1 $ ; Flow coefficient for $ σ_2 $ ; Flow coefficient for $ σ_{min} $ ;
s; K_{Q-σ_1}; K_{Q-σ_2}; K_{Q-σ_{min}};
\%; ; ; ;
Stroke from open position
Coefficient of hydraulic force on a needle in the axis x for $ σ_1 $
Coefficient of hydraulic force on a needle in the axis x for $ σ_2 $
Coefficient of hydraulic force on a needle in the axis x for $ σ_{min} $
$s$ $K_{x-σ_1}$ $K_{x-σ_2}$ $K_{x-σ_{min}}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Stroke from open position Coefficient of hydraulic force on a needle in the axis x for $ σ_1 $ Coefficient of hydraulic force on a needle in the axis x for $ σ_2 $ Coefficient of hydraulic force on a needle in the axis x for $ σ_{min} $
$s$ $K_{x-σ_1}$ $K_{x-σ_2}$ $K_{x-σ_{min}}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position;Coefficient of hydraulic force on a needle in the axis x for $ σ_1 $ ;Coefficient of hydraulic force on a needle in the axis x for $ σ_2 $ ;Coefficient of hydraulic force on a needle in the axis x for $ σ_{min} $ ;
s;K_{x-σ_1};K_{x-σ_2};K_{x-σ_{min}};
\%; ; ; ;

$K_{x-σ_1} \mathrm{[-]}$
$K_{x-σ_2} \mathrm{[-]}$
$K_{x-σ_{min}} \mathrm{[-]}$
No data
$s\mathrm{[\%]}$
Coefficient of hydraulic force on a needle in the axis x for stages of cavitation

Stroke from open position Coefficient of hydraulic force on a needle in the axis x for $ σ_1 $ Coefficient of hydraulic force on a needle in the axis x for $ σ_2 $ Coefficient of hydraulic force on a needle in the axis x for $ σ_{min} $
$s$ $K_{x-σ_1}$ $K_{x-σ_2}$ $K_{x-σ_{min}}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position; Coefficient of hydraulic force on a needle in the axis x for $ σ_1 $ ; Coefficient of hydraulic force on a needle in the axis x for $ σ_2 $ ; Coefficient of hydraulic force on a needle in the axis x for $ σ_{min} $ ;
s; K_{x-σ_1}; K_{x-σ_2}; K_{x-σ_{min}};
\%; ; ; ;
Stroke from open position
Flow coefficient
Coefficient of hydraulic force on a needle in the axis x
$s$ $K_Q$ $K_x$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$
No data
Stroke from open position Flow coefficient Coefficient of hydraulic force on a needle in the axis x
$s$ $K_Q$ $K_x$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position;Flow coefficient;Coefficient of hydraulic force on a needle in the axis x;
s;K_Q;K_x;
\%; ; ;

$K_Q \mathrm{[-]}$
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;
\%; ;
$\text{if }\ σ> σ_1$
$\text{if }\ σ> σ_1$
Equations in LaTeX code
\text{if }\ σ> σ_1
$$K_Q=K_{Q-σ_1}$$
$K_Q=K_{Q-σ_1}$
Equations in LaTeX code
K_Q=K_{Q-σ_1}
$\text{else if }\ σ> σ_2$
$\text{else if }\ σ> σ_2$
Equations in LaTeX code
\text{else if }\ σ> σ_2
$$K_Q=K_{Q-σ_2}$$
$K_Q=K_{Q-σ_2}$
Equations in LaTeX code
K_Q=K_{Q-σ_2}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$K_Q=K_{Q-σ_{min}}$$
$K_Q=K_{Q-σ_{min}}$
Equations in LaTeX code
K_Q=K_{Q-σ_{min}}

$K_x \mathrm{[-]}$
No data
$s\mathrm{[\%]}$
Coefficient of hydraulic force on a needle in the axis x

Stroke from open position Coefficient of hydraulic force on a needle in the axis x
$s$ $K_x$
$\mathrm{\%}$ $\mathrm{ }$
No data
Unrounded data
Stroke from open position; Coefficient of hydraulic force on a needle in the axis x;
s; K_x;
\%; ;
$\text{if }\ σ> σ_1$
$\text{if }\ σ> σ_1$
Equations in LaTeX code
\text{if }\ σ> σ_1
$$K_x=K_{x-σ_1}$$
$K_x=K_{x-σ_1}$
Equations in LaTeX code
K_x=K_{x-σ_1}
$\text{else if }\ σ> σ_2$
$\text{else if }\ σ> σ_2$
Equations in LaTeX code
\text{else if }\ σ> σ_2
$$K_x=K_{x-σ_2}$$
$K_x=K_{x-σ_2}$
Equations in LaTeX code
K_x=K_{x-σ_2}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$K_x=K_{x-σ_{min}}$$
$K_x=K_{x-σ_{min}}$
Equations in LaTeX code
K_x=K_{x-σ_{min}}
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
Forces on the needle
$s$ $H_L$ $H_v$ $σ$ $F_x$
$\mathrm{\%}$ $\mathrm{m}$ $\mathrm{m}$ $\mathrm{ }$ $\mathrm{kN}$
No data
Stroke from open position Loss of pressure on the valve Pressure on the valve Cavitation number Forces on the needle
$s$ $H_L$ $H_v$ $σ$ $F_x$
$\mathrm{\%}$ $\mathrm{m}$ $\mathrm{m}$ $\mathrm{ }$ $\mathrm{kN}$
No data
Unrounded data
Stroke from open position;Loss of pressure on the valve;Pressure on the valve;Cavitation number;Forces on the needle;
s;H_L;H_v;σ;F_x;
\%;m;m; ;kN;

$H_L \mathrm{[m]}$
$H_v \mathrm{[m]}$
No data
$s\mathrm{[\%]}$
Loss of height on valve and pressure height on Needle 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
$$H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot\left(ΔP-P_{u}\right)$$
$H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot\left(ΔP-P_{u}\right)$
Equations in LaTeX code
H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot\left(ΔP-P_{u}\right)

$σ \mathrm{[-]}$
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}

$F_x \mathrm{[kN]}$
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;
$$F_x=\cfrac{π\cdot D}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$$
$F_x=\cfrac{π\cdot D}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$
Equations in LaTeX code
F_x=\cfrac{π\cdot D}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x
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