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Hydrodynamic calculation Butterfly valve lenticular disc s/D=0.16

Butterfly valve (lenticular disc s/D=0.16) L 0.5D D Q max Q air s R +M H +F y +F x +F 0.3D +F by +F bx α Φ
Butterfly valve (lenticular disc s/D=0.16)

Values for calculation

D \mathrm{mm}
s \mathrm{mm}
R \mathrm{mm}
Q_{max} \mathrm{m^3/s}
H \mathrm{m}
g \mathrm{m/s^2}
T \mathrm{°C}
ρ \mathrm{kg/m^3}
P_{SV} \mathrm{Pa}
ΔP \mathrm{m}
h \mathrm{m}
ρ_{air} \mathrm{kg/m^3}
p_{air} \mathrm{Pa}
n
t \mathrm{s}
L \mathrm{m}

Calculation

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

Effective closing time factor

c_{ef}=\cfrac{1}{18}/\max_{i=1}^{18}{\left(Q_p[i]-Q_p[i+1]\right)}

c_{ef}\le 1

Under-pressure behind the valve

P_{u}=\max\left(-\cfrac{L\cdot v_{max}}{g\cdot t\cdot c_{ef}}, -\cfrac{p_{air}}{ρ\cdot g}\right)
Angle from open position
Flow coefficient
Coefficient of hydraulic force on a disc in the axis x
Coefficient of hydraulic force on a disc in the axis y
Coefficient of hydraulic force on body in the axis x
Coefficient of hydraulic force on body in the axis y
Hydraulic torque coefficient
α K_Q K_x K_y K_{bx} K_{by} K_m
\mathrm{°} \mathrm{\ } \mathrm{\ } \mathrm{\ } \mathrm{\ } \mathrm{\ } \mathrm{\ }
No data

K_x\ [-]
K_y\ [-]
K_{bx}\ [-]
K_{by}\ [-]
No data
α\ [\mathrm{°}]
Coefficient of force

K_Q\ [-]
No data
α\ [\mathrm{°}]
Flow coefficient

K_m\ [-]
No data
α\ [\mathrm{°}]
Hydraulic torque coefficient

Angle from open position
Loss coefficient
Angle between pipe axis and hydraulic force
Reduced free flow area in the throttle control system
Relative flow
Flow of water in the pipeline
Water velocity in pipeline
α ζ φ f_r Q_p Q v
\mathrm{°} \mathrm{\ } \mathrm{°} \mathrm{\ } \mathrm{\ } \mathrm{m^3/s} \mathrm{m/s}
No data

ζ\ [-]
No data
α\ [\mathrm{°}]
Loss coefficient

ζ=\cfrac{1-K_Q^2}{K_Q^2}

φ\ [\mathrm{°}]
No data
α\ [\mathrm{°}]
Angle between pipe axis and hydraulic force

\text{if }\ K_x= 0
φ=0
\text{else}
φ=\tan^{-1}\left(\cfrac{K_y}{K_x}\right)\cdot\cfrac{180}{π}

f_r\ [-]
Q_p\ [-]
No data
α\ [\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
α\ [\mathrm{°}]
Flow and speed of water in the pipeline

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

H_L\ [\mathrm{m}]
H_v\ [\mathrm{m}]
No data
α\ [\mathrm{°}]
Loss of height on valve and pressure height on Butterfly 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\left(ΔP-P_{u}\right)

σ\ [-]
No data
α\ [\mathrm{°}]
Cavitation number

σ=\cfrac{\cfrac{p_{air}-P_{SV}}{ρ\cdot g}+H-H_L}{H_v}
Angle from open position
Forces on disc in axis x
Forces on disc in axis y
Forces on disc
The force at the valve axis x
The force at the valve axis y
α F_x F_y F F_{bx} F_{by}
\mathrm{°} \mathrm{kN} \mathrm{kN} \mathrm{kN} \mathrm{kN} \mathrm{kN}
No data

F_x\ [\mathrm{kN}]
F_y\ [\mathrm{kN}]
F\ [\mathrm{kN}]
No data
α\ [\mathrm{°}]
Forces on the disc

F_x=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x
F_y=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_y
F=\sqrt{F_x^2+F_y^2}

F_{bx}\ [\mathrm{kN}]
No data
α\ [\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}

F_{by}\ [\mathrm{kN}]
No data
α\ [\mathrm{°}]
The force at the valve axis y

F_{by}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{by}
Angle from open position
Hydraulic torque without eccentricity
Hydraulic torque
α M M_H
\mathrm{°} \mathrm{kNm} \mathrm{kNm}
No data

M\ [\mathrm{kNm}]
No data
α\ [\mathrm{°}]
Hydraulic torque without eccentricity

M=\cfrac{D^3\cdot ρ\cdot g\cdot H_v\cdot K_m}{10^{12}}

M_H\ [\mathrm{kNm}]
No data
α\ [\mathrm{°}]
Hydraulic torque

M_H=M
Angle from open position
Coefficient of under-pressure of aerated hole
Under-pressure in the aerated pipeline
Air flow
α f_{air} p_{air} Q_{air}
\mathrm{°} \mathrm{\ } \mathrm{Pa} \mathrm{m^3/s}
No data

f_{air}\ [-]
No data
α\ [\mathrm{°}]
Coefficient of under-pressure of aerated hole

p_{air}\ [\mathrm{Pa}]
No data
α\ [\mathrm{°}]
Under-pressure in the aerated pipeline

\text{if }\ \text{n}= \text{no}
p_{air}=NAN
\text{else}
p_{air}=-\min\left(p_{air}, f_{air}\cdot\cfrac{v^2}{2\cdot g}\cdot ρ+\left(1-Q_p\right)\cdot\min\left(\cfrac{L\cdot v_{max}\cdot ρ}{t\cdot c_{ef}}, p_{air}\right)\right)

Q_{air}\ [\mathrm{m^3/s}]
No data
α\ [\mathrm{°}]
Air flow

\text{if }\ \text{n}= \text{no}
Q_{air}=NAN
\text{else if }\ p_{air}<\cfrac{p_{air}}{2}
Q_{air}=\min\left(Q_{max}-Q, 0.2\cdot Q\right)
\text{else}
Q_{air}=\max\left(Q_{max}-Q, 0.2\cdot Q\right)
Angle from open position
Air velocity
The flow area of the aerated hole
The flow area of the aerated pipeline
α v_{air} A_{air} A_{air-pipe}
\mathrm{°} \mathrm{m/s} \mathrm{m^2} \mathrm{m^2}
No data

v_{air}\ [\mathrm{m/s}]
No data
α\ [\mathrm{°}]
Air velocity

\text{if }\ \text{n}= \text{no}
v_{air}=NAN
\text{else}
v_{air}=\min\left(0.7\cdot\sqrt{-\cfrac{2\cdot p_{air}}{ρ_{air}}}, 250\right)

A_{air}\ [\mathrm{m^2}]
No data
α\ [\mathrm{°}]
The flow area of the aerated hole

\text{if }\ \text{n}= \text{no}
A_{air}=NAN
\text{else if }\ v_{air}=0
A_{air}=0
\text{else}
A_{air}=\cfrac{Q_{air}}{v_{air}}

A_{air-pipe}\ [\mathrm{m^2}]
No data
α\ [\mathrm{°}]
The flow area of the aerated pipeline

\text{if }\ \text{n}= \text{no}
A_{air-pipe}=NAN
\text{else if }\ v_{air}>50
A_{air-pipe}=\cfrac{Q_{air}}{50}
\text{else}
A_{air-pipe}=A_{air}
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