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Hydrodynamic calculation Howell-Bunger valve

Howell-Bunger valve D 90° 48° R 72° S 25° 40° a b c e f g +F x
Howell-Bunger valve

Values for calculation

$D$ $\mathrm{mm}$
$S$ $\mathrm{mm}$
$R$ $\mathrm{mm}$
$a$ $\mathrm{mm}$
$b$ $\mathrm{mm}$
$c$ $\mathrm{mm}$
$e$ $\mathrm{mm}$
$f$ $\mathrm{mm}$
$g$ $\mathrm{mm}$
$H$ $\mathrm{m}$
$g$ $\mathrm{m/s^2}$
$T$ $\mathrm{°C}$
$ρ$ $\mathrm{kg/m^3}$
$P_{SV}$ $\mathrm{Pa}$
$n$
$ΔP$ $\mathrm{m}$
$Σζ$
$h$ $\mathrm{m}$
$ρ_{air}$ $\mathrm{kg/m^3}$
$p_{air}$ $\mathrm{Pa}$
$h_j$ $\mathrm{m}$
Hydraulic jump behind the valve - model no.1 D 2.5 D L 1 L 2 h l h j
Hydraulic jump behind the valve - model no.1
Dimensional sketch - model no.1 x D 1.25 D 1 D 45° 2.5 D 2.5 D 30° 3 D 6 D 0.4D 2.76D 0.39D 0.86D 1.36D 1.39D 2.77 D 3.08 D 45° P u P u P 1 u-air P 2 u-air P 1 u-air P 2 u-air
Dimensional sketch - model no.1
Hydraulic jump behind the valve - model no.2 D 2.5 D L 1 L 2 h l h j
Hydraulic jump behind the valve - model no.2
Dimensional sketch - model no.2 D 2.82 D 3.33 D 45° 1.36D 3 D 6 D 30° 0.39D 2.56D 1.94D 0.39D 0.58D 45° 0.64D 2.5 D 1.11D 0.6D 0.44 D 0.1 D 4.36D 1.5 D 1.5 D P u P u P 1 u-air P 1 u-air P 2 u-air P 2 u-air
Dimensional sketch - model no.2
Hydraulic jump behind the valve - model no.3 h l h j D 1.2 D L 1 L 2
Hydraulic jump behind the valve - model no.3
Dimensional sketch - model no.3 D 1.82 D 2.33 D 45° 1.36D 3 D 6 D 30° 0.39D 2.56D 1.94D 45° 1.2 D 1.11D 0.6D 0.1 D 4.36D 1.5 D 1.5 D P u P u P 1 u-air P 1 u-air P 2 u-air P 2 u-air
Dimensional sketch - model no.3

Calculation

Flow

$$Q_{max}=\cfrac{1}{\sqrt{1+Σζ+\min\left(ζ\right)}}\cdot\cfrac{π\cdot D^2}{4\cdot 10^6}\cdot\sqrt{2\cdot g\cdot H}$$
$Q_{max}=\cfrac{1}{\sqrt{1+Σζ+\min\left(ζ\right)}}\cdot\cfrac{π\cdot D^2}{4\cdot 10^6}\cdot\sqrt{2\cdot g\cdot H}$
Equations in LaTeX code
Q_{max}=\cfrac{1}{\sqrt{1+Σζ+\min\left(ζ\right)}}\cdot\cfrac{π\cdot D^2}{4\cdot 10^6}\cdot\sqrt{2\cdot g\cdot H}

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}

Theoretical pressure in the valve at full opening

$$Δ_h=\cfrac{v_{max}^2}{2\cdot g}\cdot \left({\min\left(ζ\right)}+1\right)$$
$Δ_h=\cfrac{v_{max}^2}{2\cdot g}\cdot \left({\min\left(ζ\right)}+1\right)$
Equations in LaTeX code
Δ_h=\cfrac{v_{max}^2}{2\cdot g}\cdot \left({\min\left(ζ\right)}+1\right)

Pressure parameter

$$p=\cfrac{Δ_h}{H}$$
$p=\cfrac{Δ_h}{H}$
Equations in LaTeX code
p=\cfrac{Δ_h}{H}
$$0 < p \le 1$$
$0 < p \le 1$
Equations in LaTeX code
0 < p \le 1
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