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Elliptical rounded contraction

Elliptical rounded contraction d 2 d 1 r 1 r 2
Elliptical rounded contraction

Values for calculation

$ \{d_1\in\mathbb{R}\mid d_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.
$ d_1 $ $ \mathrm{mm} $
$ \{d_2\in\mathbb{R}\mid d_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.
$ d_2 $ $ \mathrm{mm} $
$ \{r_1\in\mathbb{R}\mid r_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.
$ r_1 $ $ \mathrm{mm} $
$ \{r_2\in\mathbb{R}\mid r_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.
$ r_2 $ $ \mathrm{mm} $

Calculation

Radius

$$r=\sqrt[3]{r_1^2\cdot r_2}$$
$r=\sqrt[3]{r_1^2\cdot r_2}$
Equations in LaTeX code
r=\sqrt[3]{r_1^2\cdot r_2}

Diameter ratio

$$β=\cfrac{d_2}{d_1}$$
$β=\cfrac{d_2}{d_1}$
Equations in LaTeX code
β=\cfrac{d_2}{d_1}

Jet contraction ratio

$$λ=1+0.622\cdot\left(1-0.3\cdot\sqrt{\cfrac{r}{d_2}}-0.7\cdot\cfrac{r}{d_2}\right)^4\cdot\left(1-0.215\cdot β^2-0.785\cdot β^5\right)$$
$λ=1+0.622\cdot\left(1-0.3\cdot\sqrt{\cfrac{r}{d_2}}-0.7\cdot\cfrac{r}{d_2}\right)^4\cdot\left(1-0.215\cdot β^2-0.785\cdot β^5\right)$
Equations in LaTeX code
λ=1+0.622\cdot\left(1-0.3\cdot\sqrt{\cfrac{r}{d_2}}-0.7\cdot\cfrac{r}{d_2}\right)^4\cdot\left(1-0.215\cdot β^2-0.785\cdot β^5\right)

Loss coefficient

$\text{if }\ r/d_2\geq 1$
$\text{if }\ r/d_2\geq 1$
Equations in LaTeX code
\text{if }\ r/d_2\geq 1
$$ζ=0.03\cdot\left(1-β\right)\cdot\left(1-β^4\right)$$
$ζ=0.03\cdot\left(1-β\right)\cdot\left(1-β^4\right)$
Equations in LaTeX code
ζ=0.03\cdot\left(1-β\right)\cdot\left(1-β^4\right)
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$ζ=0.0696\cdot\left(1-0.569\cdot\cfrac{r}{d_2}\right)\cdot\left(1-\sqrt{\cfrac{r}{d_2}}\cdot β\right)\cdot\left(1-β^5\right)\cdot λ^2+\left(λ-1\right)^2$$
$ζ=0.0696\cdot\left(1-0.569\cdot\cfrac{r}{d_2}\right)\cdot\left(1-\sqrt{\cfrac{r}{d_2}}\cdot β\right)\cdot\left(1-β^5\right)\cdot λ^2+\left(λ-1\right)^2$
Equations in LaTeX code
ζ=0.0696\cdot\left(1-0.569\cdot\cfrac{r}{d_2}\right)\cdot\left(1-\sqrt{\cfrac{r}{d_2}}\cdot β\right)\cdot\left(1-β^5\right)\cdot λ^2+\left(λ-1\right)^2

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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