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Hydrodynamic calculation Butterfly valve lenticular disc $s/D=0.26$

Butterfly valve (lenticular disc s/D=0.26) L 0.5D D Q max Q air s R +M H +F y +F x +F 0.5D +F by +F bx α Φ
Butterfly valve (lenticular disc $s/D=0.26$)

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

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

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

$K_m \mathrm{[-]}$
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

$ζ \mathrm{[-]}$
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

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

$f_r \mathrm{[-]}$
$Q_p \mathrm{[-]}$
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

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

$σ \mathrm{[-]}$
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

$$M_H=M$$