# Multi-hole 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

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

### Allowable maximum number of holes in one row

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

### Number of rows

$$i=\left\lfloor\cfrac{L}{d}\right\rfloor$$

### The flow area

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

### Total number of holes

$$I_{total}=\left\lfloor\cfrac{4\cdot 10^6 \cdot A}{π\cdot d^2}\right\rfloor$$
Stroke from open position
Reduced free flow area in the throttle control system
Number of holes
$s$ $f_r$ $I$
$\mathrm{\%}$ $\mathrm{ }$ $\mathrm{ }$
No data

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

$\text{if }\ \text{flow }$$\text{characteristic}= \text{linear}$$f_r=1-\cfrac{s}{100}$$\text{else if }\ \text{flow }$$\text{characteristic}= \text{modified linear}\wedge 1-\cfrac{s}{100}\le 0.5$
$$f_r=\left(2\cdot\left(1-\cfrac{s}{100}\right)\right)^{2}\cdot 0.5$$
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{modified linear}$$f_r=\left(\left(1-\cfrac{s}{100}-0.5\right)\cdot 2\cdot\left(2-\left(1-\cfrac{s}{100}-0.5\right)\cdot 2\right)\right)\cdot0.5+0.5$$\text{else if }\ \text{flow }$$\text{characteristic}= \text{parabolic}$
$$f_r=\left(1-\cfrac{s}{100}\right)^{2}$$
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{equal percentage}\wedge 1-\cfrac{s}{100}= 0$$f_r=0$$\text{else if }\ \text{flow }$$\text{characteristic}= \text{equal percentage}\wedge 1-\cfrac{s}{100}= 1$
$$f_r=1$$