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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}\in\mathbb{R}\mid Q_{max}>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ Q_{max} $ $ \mathrm{m^3/s} $
$ \{D\in\mathbb{R}\mid D>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ D $ $ \mathrm{mm} $
$ \{D_c\in\mathbb{R}\mid D_c>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ D_c $ $ \mathrm{mm} $
$ \{P_1\in\mathbb{R}\mid P_1>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ P_1 $ $ \mathrm{Pa} $
$ \{P_2\in\mathbb{R}\mid P_2>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ P_2 $ $ \mathrm{Pa} $
$ \{T\in\mathbb{R}\mid T\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ T $ $ \mathrm{°C} $
$ ρ $ $ \mathrm{kg/m^3} $
$ P_{SV} $ $ \mathrm{Pa} $
$ \{μ\in\mathbb{R}\mid μ>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ μ $
$ \{d\in\mathbb{R}\mid d>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ d $ $ \mathrm{mm} $
$ \{L\in\mathbb{R}\mid L>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ 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
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
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
Unrounded data
Stroke from open position;Reduced free flow area in the throttle control system;Number of holes;
s;f_r;I;
\%; ; ;

$ f_r\ [-] $
No data
$ s\ [\mathrm{\%}] $
Reduced free flow area in the throttle control system

Stroke from open position Reduced free flow area in the throttle control system
$ s $ $ f_r $
$ \mathrm{\%} $ $ \mathrm{\ } $
No data
Unrounded data
Stroke from open position; Reduced free flow area in the throttle control system;
s; f_r;
\%; ;
$\text{if }\ \text{flow }$$\text{characteristic}= \text{linear}$
$\text{if }\ \text{flow }$$\text{characteristic}= \text{linear}$
Equations in LaTeX code
\text{if }\ \text{flow }$$\text{characteristic}= \text{linear}
$$f_r=1-\cfrac{s}{100}$$
$f_r=1-\cfrac{s}{100}$
Equations in LaTeX code
f_r=1-\cfrac{s}{100}
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{modified linear}\wedge 1-\cfrac{s}{100}\le 0.5$
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{modified linear}\wedge 1-\cfrac{s}{100}\le 0.5$
Equations in LaTeX code
\text{else if }\ \text{flow }$$\text{characteristic}= \text{modified linear}\wedge 1-\cfrac{s}{100}\le 0.5
$$f_r=\left(2\cdot\left(1-\cfrac{s}{100}\right)\right)^{2}\cdot 0.5$$
$f_r=\left(2\cdot\left(1-\cfrac{s}{100}\right)\right)^{2}\cdot 0.5$
Equations in LaTeX code
f_r=\left(2\cdot\left(1-\cfrac{s}{100}\right)\right)^{2}\cdot 0.5
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{modified linear}$
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{modified linear}$
Equations in LaTeX code
\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$$
$f_r=\left(\left(1-\cfrac{s}{100}-0.5\right)\cdot 2\cdot\left(2-\left(1-\cfrac{s}{100}-0.5\right)\cdot 2\right)\right)\cdot0.5+0.5$
Equations in LaTeX code
f_r=\left(\left(1-\cfrac{s}{100}-0.5\right)\cdot 2\cdot\left(2-\left(1-\cfrac{s}{100}-0.5\right)\cdot 2\right)\right)\cdot0.5+0.5
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{parabolic}$
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{parabolic}$
Equations in LaTeX code
\text{else if }\ \text{flow }$$\text{characteristic}= \text{parabolic}
$$f_r=\left(1-\cfrac{s}{100}\right)^{2}$$
$f_r=\left(1-\cfrac{s}{100}\right)^{2}$
Equations in LaTeX code
f_r=\left(1-\cfrac{s}{100}\right)^{2}
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{equal percentage}\wedge 1-\cfrac{s}{100}= 0$
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{equal percentage}\wedge 1-\cfrac{s}{100}= 0$
Equations in LaTeX code
\text{else if }\ \text{flow }$$\text{characteristic}= \text{equal percentage}\wedge 1-\cfrac{s}{100}= 0
$$f_r=0$$
$f_r=0$
Equations in LaTeX code
f_r=0
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{equal percentage}\wedge 1-\cfrac{s}{100}= 1$
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{equal percentage}\wedge 1-\cfrac{s}{100}= 1$
Equations in LaTeX code
\text{else if }\ \text{flow }$$\text{characteristic}= \text{equal percentage}\wedge 1-\cfrac{s}{100}= 1
$$f_r=1$$
$f_r=1$
Equations in LaTeX code
f_r=1
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{equal percentage}$
$\text{else if }\ \text{flow }$$\text{characteristic}= \text{equal percentage}$
Equations in LaTeX code
\text{else if }\ \text{flow }$$\text{characteristic}= \text{equal percentage}
$$f_r=\cfrac{0.25}{100}\cdot\exp{1}^{6\cdot\left( 1-\cfrac{s}{100}\right)}$$
$f_r=\cfrac{0.25}{100}\cdot\exp{1}^{6\cdot\left( 1-\cfrac{s}{100}\right)}$
Equations in LaTeX code
f_r=\cfrac{0.25}{100}\cdot\exp{1}^{6\cdot\left( 1-\cfrac{s}{100}\right)}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$f_r=\left(1-\cfrac{s}{100}\right)\cdot\left(2-\left(1-\cfrac{s}{100}\right)\right)$$
$f_r=\left(1-\cfrac{s}{100}\right)\cdot\left(2-\left(1-\cfrac{s}{100}\right)\right)$
Equations in LaTeX code
f_r=\left(1-\cfrac{s}{100}\right)\cdot\left(2-\left(1-\cfrac{s}{100}\right)\right)

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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