# Multi-stage valve

## Values for calculation

$Q_{max}$ $\mathrm{m^3/s}$
$D$ $\mathrm{mm}$
$P_1$ $\mathrm{Pa}$
$P_2$ $\mathrm{Pa}$
$T$ $\mathrm{°C}$
$ρ$ $\mathrm{kg/m^3}$
$P_{SV}$ $\mathrm{Pa}$
$μ$
$d$ $\mathrm{mm}$

## Calculation

### Velocity in valve

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

### Number of stages

$\text{if }\ \left(P_2-0.6\cdot P_{SV}\right)/0.4>P_1$
$$n_{s}=1$$
$\text{else if }\ \left(\left(P_2-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4>P_1$
$$n_{s}=2$$
$\text{else if }\ \left(\left(\left(P_2-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4>P_1$
$$n_{s}=3$$
$\text{else if }\ \left(\left(\left(\left(P_2-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4>P_1$
$$n_{s}=4$$
$\text{else if }\ \left(\left(\left(\left(\left(P_2-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4>P_1$
$$n_{s}=5$$
$\text{else if }\ \left(\left(\left(\left(\left(\left(P_2-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4>P_1$
$$n_{s}=6$$
$\text{else if }\ \left(\left(\left(\left(\left(\left(\left(P_2-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4>P_1$
$$n_{s}=7$$
$\text{else if }\ \left(\left(\left(\left(\left(\left(\left(\left(P_2-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4-0.6\cdot P_{SV}\right)/0.4>P_1$
$$n_{s}=8$$
$\text{else}$
$$n_{s}=9$$
Stage
Maximum pressures between stages
Pressure difference between the stages
The flow area
Number of holes
$S$ $P_{max-stages}$ $ΔP_{stages}$ $A$ $I$
$\mathrm{ }$ $\mathrm{Pa}$ $\mathrm{Pa}$ $\mathrm{m^2}$ $\mathrm{ }$
No data

$P_{max-stages} \mathrm{[Pa]}$
 No data
$S\mathrm{[-]}$

$\text{if }\ S=1$
$$P_{max-stages}=P_1$$
$\text{else if }\ S=2$
$$P_{max-stages}=\left(P_1\cdot 0.4+0.6\cdot P_{SV}\right)$$
$\text{else if }\ S=3$
$$P_{max-stages}=\left(\left(P_1\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)$$
$\text{else if }\ S=4$
$$P_{max-stages}=\left(\left(\left(P_1\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)$$
$\text{else if }\ S=5$
$$P_{max-stages}=\left(\left(\left(\left(P_1\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)$$
$\text{else if }\ S=6$
$$P_{max-stages}=\left(\left(\left(\left(\left(P_1\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)$$
$\text{else if }\ S=7$
$$P_{max-stages}=\left(\left(\left(\left(\left(\left(P_1\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)$$
$\text{else}$
$$P_{max-stages}=\left(\left(\left(\left(\left(\left(\left(P_1\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)\cdot 0.4+0.6\cdot P_{SV}\right)$$

$ΔP_{stages} \mathrm{[Pa]}$
 No data
$S\mathrm{[-]}$

$\text{if }\ S=n_{s}$
$$ΔP_{stages}=P_{max-stages}-P_2+\left(n_{s}-1\right)\cdot\left(\left(P_2-0.6\cdot P_{SV}\right)/0.4-\left(P_{max-stages}\right)[n_{s}]\right)/n_{s}$$
$\text{else}$
$$ΔP_{stages}=P_{max-stages}-P_{max-stages}[i+1]-\left(\left(P_2-0.6\cdot P_{SV}\right)/0.4-\left(P_{max-stages}\right)[n_{s}]\right)/n_{s}$$

$A \mathrm{[m^2]}$
 No data
$S\mathrm{[-]}$

$$A=\cfrac{Q_{max}}{μ\cdot \sqrt{2\cdot \cfrac{ΔP_{stages}}{ρ}}}$$

$I \mathrm{[-]}$
 No data
$S\mathrm{[-]}$

$$I=\left\lfloor\cfrac{4\cdot A}{π\cdot d^2\cdot 10^{-6}}\right\rfloor$$

## Requirements

$$\cfrac{D}{50}\geq d$$$$\cfrac{4\cdot 10^6\cdot \max\left(A\right) }{π\cdot D^2}< 0.5$$