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Hydrodynamic calculation Howell-Jet valve

Howell-Jet valve Q max D a b c e f g h i j k l m n o p r q t ELLIPSE s VANES 33° B u 8 d 1 d v w x 10° y VANES ALTERNATE AT 45° DETAIL "B" 33° 40° z +F -F D needle
Howell-Jet valve

Values for calculation

D \mathrm{mm}
D_{needle} \mathrm{mm}
d \mathrm{mm}
a \mathrm{mm}
b \mathrm{mm}
c \mathrm{mm}
e \mathrm{mm}
f \mathrm{mm}
g \mathrm{mm}
h \mathrm{mm}
i \mathrm{mm}
j \mathrm{mm}
k \mathrm{mm}
l \mathrm{mm}
m \mathrm{mm}
n \mathrm{mm}
o \mathrm{mm}
p \mathrm{mm}
q \mathrm{mm}
r \mathrm{mm}
t \mathrm{mm}
u \mathrm{mm}
v \mathrm{mm}
w \mathrm{mm}
x \mathrm{mm}
y \mathrm{mm}
z \mathrm{mm}
d_1 \mathrm{mm}
H \mathrm{m}
g \mathrm{m/s^2}
T \mathrm{°C}
ρ \mathrm{kg/m^3}
P_{SV} \mathrm{Pa}
ΔP \mathrm{m}
Σζ
h \mathrm{m}
ρ_{air} \mathrm{kg/m^3}
p_{air} \mathrm{Pa}

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}

Velocity in valve

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)

Pressure parameter

p=\cfrac{Δ_h}{H}

0 < p \le 1
Stroke from open position
Flow coefficient
Coefficient of hydraulic force on a needle in the axis x
Coefficient of hydraulic force on a needle upstream in the axis x
Coefficient of hydraulic force on body in the axis x
s K_Q K_x K_{x-upstream} K_{bx}
\mathrm{\%} \mathrm{\ } \mathrm{\ } \mathrm{\ } \mathrm{\ }
No data

K_x\ [-]
K_{x-upstream}\ [-]
K_{bx}\ [-]
No data
s\ [\mathrm{\%}]
Coefficient of force

K_Q\ [-]
No data
s\ [\mathrm{\%}]
Flow coefficient

Stroke from open position
Loss coefficient
Reduced free flow area in the throttle control system
Relative flow
Flow of water in the pipeline
Water velocity in pipeline
s ζ f_r Q_p Q v
\mathrm{\%} \mathrm{\ } \mathrm{\ } \mathrm{\ } \mathrm{m^3/s} \mathrm{m/s}
No data

ζ\ [-]
No data
s\ [\mathrm{\%}]
Loss coefficient

ζ=\cfrac{1-K_Q^2}{K_Q^2}

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

f_r=\cfrac{K_Q}{K_{Qmax}}
Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}

Q\ [\mathrm{m^3/s}]
v\ [\mathrm{m/s}]
No data
s\ [\mathrm{\%}]
Flow and speed of water in the pipeline

Q=Q_p\cdot Q_{max}
v=Q_p\cdot v_{max}
Stroke from open position
Loss of pressure on the valve
Pressure on the valve
Cavitation number
s H_L H_v σ
\mathrm{\%} \mathrm{m} \mathrm{m} \mathrm{\ }
No data

H_L\ [\mathrm{m}]
H_v\ [\mathrm{m}]
No data
s\ [\mathrm{\%}]
Loss of height on valve and pressure height on Howell-Jet valve

H_L=\cfrac{v^2}{2\cdot g}\cdot ζ
H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot ΔP

σ\ [-]
No data
s\ [\mathrm{\%}]
Cavitation number

σ=\cfrac{\cfrac{p_{air}-P_{SV}}{ρ\cdot g}+H-H_L}{H_v}
Stroke from open position
Forces on the needle
The force at the valve axis x
s F_x F_{bx}
\mathrm{\%} \mathrm{kN} \mathrm{kN}
No data

F_x\ [\mathrm{kN}]
No data
s\ [\mathrm{\%}]
Forces on the needle

F_x=\cfrac{π\cdot ρ\cdot g\cdot H_v}{4\cdot 10^9}\cdot \left(D^2\cdot K_x-D^2\cdot\left(1.1162^2-0.04\right)\cdot K_{x-upstream}+\left(D_{needle}^2-d^2\right)\cdot K_{x-upstream}\right)

F_{bx}\ [\mathrm{kN}]
No data
s\ [\mathrm{\%}]
The force at the valve axis x

F_{bx}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}