# Hydrodynamic calculation Needle valve

## Values for calculation

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

## Calculation

### 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

$$p=\cfrac{Δ_h}{H}$$

$$0 < p \le 1$$

### Effective closing time factor

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

$$c_{ef}\le 1$$

### Under-pressure behind the valve

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

$\mathrm{[-]}$

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

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

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

$K_{Q-σ_1} \mathrm{[-]}$
$K_{Q-σ_2} \mathrm{[-]}$
$K_{Q-σ_{min}} \mathrm{[-]}$
 No data
$s\mathrm{[\%]}$

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

$K_{x-σ_1} \mathrm{[-]}$
$K_{x-σ_2} \mathrm{[-]}$
$K_{x-σ_{min}} \mathrm{[-]}$
 No data
$s\mathrm{[\%]}$

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

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

$\text{if }\ \text{n}= \text{yes, hole A}$
$$K_Q=\left\{0.680, 0.660, 0.630, 0.590, 0.530, 0.440, 0.332, 0.232, 0.144, 0.073, 1E-100\right\}$$
$\text{else if }\ \text{n}= \text{yes, hole B}$
$$K_Q=\left\{0.660, 0.640, 0.600, 0.536, 0.450, 0.372, 0.289, 0.200, 0.128, 0.069, 1E-100\right\}$$
$\text{else if }\ \text{n}= \text{yes, hole A+B}$
$$K_Q=\left\{0.650, 0.635, 0.595, 0.528, 0.450, 0.364, 0.272, 0.195, 0.128, 0.069, 1E-100\right\}$$
$\text{else if }\ σ> σ_1$
$$K_Q=K_{Q-σ_1}$$
$\text{else if }\ σ> σ_2$
$$K_Q=K_{Q-σ_2}$$
$\text{else}$
$$K_Q=K_{Q-σ_{min}}$$

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

$\text{if }\ \text{n}= \text{yes, hole A}$
$$K_x=\left\{-0.020, -0.040, -0.080, -0.160, -0.263, -0.215, -0.090, -0.020, -0.079, -0.006, 0.000\right\}$$
$\text{else if }\ \text{n}= \text{yes, hole B}$
$$K_x=\left\{-0.020, -0.030, -0.050, -0.108, -0.079, -0.090, -0.090, -0.072, -0.068, -0.052, 0.000\right\}$$
$\text{else if }\ \text{n}= \text{yes, hole A+B}$
$$K_x=\left\{-0.020, -0.027, -0.026, -0.022, -0.030, -0.031, -0.025, -0.028, -0.030, -0.013, 0.000\right\}$$
$\text{else if }\ σ> σ_1$
$$K_x=K_{x-σ_1}$$
$\text{else if }\ σ> σ_2$
$$K_x=K_{x-σ_2}$$
$\text{else}$
$$K_x=K_{x-σ_{min}}$$
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

$ζ \mathrm{[-]}$
 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
Forces on the needle
$s$ $H_L$ $H_v$ $σ$ $F_x$
$\mathrm{\%}$ $\mathrm{m}$ $\mathrm{m}$ $\mathrm{ }$ $\mathrm{kN}$
No data

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

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

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

$$σ=\cfrac{\cfrac{p_{air}-P_{SV}}{ρ\cdot g}+H-H_L}{H_v}$$

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

$$F_x=\cfrac{π\cdot D}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$$
Stroke from open position
Aerated coefficient
Coefficient of under-pressure of aerated hole
Under-pressure in the aerated pipeline
Air flow
$s$ $β$ $f_{air}$ $p_{air}$ $Q_{air}$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{Pa}$ $\mathrm{m^3/s}$
No data

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

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

$p_{air} \mathrm{[Pa]}$
 No data
$s\mathrm{[\%]}$

$\text{if }\ \text{n}= \text{no}$
$$p_{air}=NAN$$
$\text{else}$
$$p_{air}=-\min\left(p_{air}, f_{air}\cdot\cfrac{v^2}{2\cdot g}\cdot ρ+\left(1-Q_p\right)\cdot\min\left(\cfrac{L\cdot v_{max}\cdot ρ}{t\cdot c_{ef}}, p_{air}\right)\right)$$

$Q_{air} \mathrm{[m^3/s]}$
 No data
$s\mathrm{[\%]}$

$\text{if }\ \text{n}= \text{no}$
$$Q_{air}=NAN$$
$\text{else if }\ p_{air}<\cfrac{p_{air}}{2}$
$$Q_{air}=\min\left(Q_{max}-Q, β\cdot Q\right)$$
$\text{else}$
$$Q_{air}=\max\left(Q_{max}-Q, β\cdot Q\right)$$
Stroke from open position
Air velocity
The flow area of the aerated hole
The flow area of the aerated pipeline
$s$ $v_{air}$ $A_{air}$ $A_{air-pipe}$
$\mathrm{\%}$ $\mathrm{m/s}$ $\mathrm{m^2}$ $\mathrm{m^2}$
No data

$v_{air} \mathrm{[m/s]}$
 No data
$s\mathrm{[\%]}$

$\text{if }\ \text{n}= \text{no}$
$$v_{air}=NAN$$
$\text{else}$
$$v_{air}=\min\left(0.7\cdot\sqrt{-\cfrac{2\cdot p_{air}}{ρ_{air}}}, 250\right)$$

$A_{air} \mathrm{[m^2]}$
 No data
$s\mathrm{[\%]}$

$\text{if }\ \text{n}= \text{no}$
$$A_{air}=NAN$$
$\text{else if }\ v_{air}=0$
$$A_{air}=0$$
$\text{else}$
$$A_{air}=\cfrac{Q_{air}}{v_{air}}$$

$A_{air-pipe} \mathrm{[m^2]}$
 No data
$s\mathrm{[\%]}$

$\text{if }\ \text{n}= \text{no}$
$$A_{air-pipe}=NAN$$
$\text{else if }\ v_{air}>50$
$$A_{air-pipe}=\cfrac{Q_{air}}{50}$$
$\text{else}$
$$A_{air-pipe}=A_{air}$$