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Multi-hole cage

Cage L d D c
Cage
Developed view of the cage d d d d L 1 2 3 4 min(3d)
Developed view of the cage

Values for calculation

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

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}

Allowable maximum number of holes in one row

$$n_{max}=\left\lfloor\cfrac{π\cdot D_c}{\sqrt{\left(9\cdot d\right)^2-\left(3\cdot d\right)^2}}\right\rfloor$$
$n_{max}=\left\lfloor\cfrac{π\cdot D_c}{\sqrt{\left(9\cdot d\right)^2-\left(3\cdot d\right)^2}}\right\rfloor$
Equations in LaTeX code
n_{max}=\left\lfloor\cfrac{π\cdot D_c}{\sqrt{\left(9\cdot d\right)^2-\left(3\cdot d\right)^2}}\right\rfloor

Number of rows

$$i=\left\lfloor\cfrac{L}{d}\right\rfloor$$
$i=\left\lfloor\cfrac{L}{d}\right\rfloor$
Equations in LaTeX code
i=\left\lfloor\cfrac{L}{d}\right\rfloor

The flow area

$$A=\cfrac{Q_{max}}{μ\cdot \sqrt{2\cdot \cfrac{P_1-P_2}{ρ}}}$$
$A=\cfrac{Q_{max}}{μ\cdot \sqrt{2\cdot \cfrac{P_1-P_2}{ρ}}}$
Equations in LaTeX code
A=\cfrac{Q_{max}}{μ\cdot \sqrt{2\cdot \cfrac{P_1-P_2}{ρ}}}

Total number of holes

$$I_{total}=\left\lfloor\cfrac{4\cdot 10^6 \cdot A}{π\cdot d^2}\right\rfloor$$
$I_{total}=\left\lfloor\cfrac{4\cdot 10^6 \cdot A}{π\cdot d^2}\right\rfloor$
Equations in LaTeX code
I_{total}=\left\lfloor\cfrac{4\cdot 10^6 \cdot A}{π\cdot d^2}\right\rfloor

Requirements

$$\cfrac{P_1-P_2}{0.6\cdot\left(P_1-P_{SV}\right)}\le1$$
$\cfrac{P_1-P_2}{0.6\cdot\left(P_1-P_{SV}\right)}\le1$
Equations in LaTeX code
\cfrac{P_1-P_2}{0.6\cdot\left(P_1-P_{SV}\right)}\le1
$$\cfrac{4\cdot 10^6\cdot A}{π\cdot D^2}< 0.5$$
$\cfrac{4\cdot 10^6\cdot A}{π\cdot D^2}< 0.5$
Equations in LaTeX code
\cfrac{4\cdot 10^6\cdot A}{π\cdot D^2}< 0.5
$$\cfrac{D}{50}\geq d$$
$\cfrac{D}{50}\geq d$
Equations in LaTeX code
\cfrac{D}{50}\geq d
$$n_{max}\geq \max\left(I\right)$$
$n_{max}\geq \max\left(I\right)$
Equations in LaTeX code
n_{max}\geq \max\left(I\right)
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