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Projected discharge from a conical diffuser

Projected discharge from a conical diffuser d d E α
Projected discharge from a conical diffuser

Values for calculation

$ \{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_E\in\mathbb{R}\mid d_E>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_E $ $ \mathrm{mm} $
$ \{α\in\mathbb{R}\mid α>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.
$ α $ $ \mathrm{°} $
$ \{f\in\mathbb{R}\mid f>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.
$ f $

Calculation

Diameter ratio

$$β=\cfrac{d}{d_E}$$
$β=\cfrac{d}{d_E}$
Equations in LaTeX code
β=\cfrac{d}{d_E}

Loss coefficient

$\text{if }\ α\le 20$
$\text{if }\ α\le 20$
Equations in LaTeX code
\text{if }\ α\le 20
$$ζ=8.3\cdot\left[\tan{\left(α/2\right)}\right]^{1.75}\cdot\left(1-β^2\right)^2+\cfrac{f\cdot\left(1-β^4\right)}{8\cdot\sin{\left(α/2\right)}}+β^4$$
$ζ=8.3\cdot\left[\tan{\left(α/2\right)}\right]^{1.75}\cdot\left(1-β^2\right)^2+\cfrac{f\cdot\left(1-β^4\right)}{8\cdot\sin{\left(α/2\right)}}+β^4$
Equations in LaTeX code
ζ=8.3\cdot\left[\tan{\left(α/2\right)}\right]^{1.75}\cdot\left(1-β^2\right)^2+\cfrac{f\cdot\left(1-β^4\right)}{8\cdot\sin{\left(α/2\right)}}+β^4
$\text{else if }\ α\le 60\wedgeβ<0.5$
$\text{else if }\ α\le 60\wedgeβ<0.5$
Equations in LaTeX code
\text{else if }\ α\le 60\wedgeβ<0.5
$$ζ=\left\{1.366\cdot\sin{\left[\cfrac{2\cdot π\cdot\left(α-15°\right)}{180}\right]}^{1/2}-0.17-3.28\cdot\left(0.0625-β^4\right)\cdot\sqrt{\cfrac{α-20°}{40°}}\right\}\cdot\left(1-β^2\right)^2+\cfrac{f\cdot\left(1-β^4\right)}{8\cdot\sin{\left(α/2\right)}}+β^4$$
$ζ=\left\{1.366\cdot\sin{\left[\cfrac{2\cdot π\cdot\left(α-15°\right)}{180}\right]}^{1/2}-0.17-3.28\cdot\left(0.0625-β^4\right)\cdot\sqrt{\cfrac{α-20°}{40°}}\right\}\cdot\left(1-β^2\right)^2+\cfrac{f\cdot\left(1-β^4\right)}{8\cdot\sin{\left(α/2\right)}}+β^4$
Equations in LaTeX code
ζ=\left\{1.366\cdot\sin{\left[\cfrac{2\cdot π\cdot\left(α-15°\right)}{180}\right]}^{1/2}-0.17-3.28\cdot\left(0.0625-β^4\right)\cdot\sqrt{\cfrac{α-20°}{40°}}\right\}\cdot\left(1-β^2\right)^2+\cfrac{f\cdot\left(1-β^4\right)}{8\cdot\sin{\left(α/2\right)}}+β^4
$\text{else if }\ α\le 60$
$\text{else if }\ α\le 60$
Equations in LaTeX code
\text{else if }\ α\le 60
$$ζ=\left\{1.366\cdot\sin{\left[\cfrac{2\cdot π\cdot\left(α-15°\right)}{180}\right]}^{1/2}-0.17\right\}\cdot\left(1-β^2\right)^2+\cfrac{f\cdot\left(1-β^4\right)}{8\cdot\sin{\left(α/2\right)}}+β^4$$
$ζ=\left\{1.366\cdot\sin{\left[\cfrac{2\cdot π\cdot\left(α-15°\right)}{180}\right]}^{1/2}-0.17\right\}\cdot\left(1-β^2\right)^2+\cfrac{f\cdot\left(1-β^4\right)}{8\cdot\sin{\left(α/2\right)}}+β^4$
Equations in LaTeX code
ζ=\left\{1.366\cdot\sin{\left[\cfrac{2\cdot π\cdot\left(α-15°\right)}{180}\right]}^{1/2}-0.17\right\}\cdot\left(1-β^2\right)^2+\cfrac{f\cdot\left(1-β^4\right)}{8\cdot\sin{\left(α/2\right)}}+β^4
$\text{else if }\ α\le 180\wedgeβ<0.5$
$\text{else if }\ α\le 180\wedgeβ<0.5$
Equations in LaTeX code
\text{else if }\ α\le 180\wedgeβ<0.5
$$ζ=\left[1.205-3.28\cdot\left(0.0625-β^4\right)-12.8\cdot β^6\cdot\sqrt{\cfrac{α-60°}{120°}}\right]\cdot\left(1-β^2\right)^2+β^4$$
$ζ=\left[1.205-3.28\cdot\left(0.0625-β^4\right)-12.8\cdot β^6\cdot\sqrt{\cfrac{α-60°}{120°}}\right]\cdot\left(1-β^2\right)^2+β^4$
Equations in LaTeX code
ζ=\left[1.205-3.28\cdot\left(0.0625-β^4\right)-12.8\cdot β^6\cdot\sqrt{\cfrac{α-60°}{120°}}\right]\cdot\left(1-β^2\right)^2+β^4
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$ζ=\left(1.205-0.2\cdot\sqrt{\cfrac{α-60°}{120°}}\right)\cdot\left(1-β^2\right)^2+β^4$$
$ζ=\left(1.205-0.2\cdot\sqrt{\cfrac{α-60°}{120°}}\right)\cdot\left(1-β^2\right)^2+β^4$
Equations in LaTeX code
ζ=\left(1.205-0.2\cdot\sqrt{\cfrac{α-60°}{120°}}\right)\cdot\left(1-β^2\right)^2+β^4

Discharge coefficient

$$μ=\cfrac{1}{\sqrt{ζ+1}}$$
$μ=\cfrac{1}{\sqrt{ζ+1}}$
Equations in LaTeX code
μ=\cfrac{1}{\sqrt{ζ+1}}
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