# Hydrodynamic calculation Butterfly valve for pump operation lattice disc

## Values for calculation

$D$ $\mathrm{mm}$
$L_D$ $\mathrm{mm}$
$a$ $\mathrm{mm}$
$b$ $\mathrm{mm}$
$c$ $\mathrm{mm}$
$D_s$ $\mathrm{mm}$
$Q_{max}$ $\mathrm{m^3/s}$
$H$ $\mathrm{m}$
$g$ $\mathrm{m/s^2}$
$T$ $\mathrm{°C}$
$ρ$ $\mathrm{kg/m^3}$
$P_{SV}$ $\mathrm{Pa}$
$ΔP$ $\mathrm{m}$
$h$ $\mathrm{m}$
$ρ_{air}$ $\mathrm{kg/m^3}$
$p_{air}$ $\mathrm{Pa}$
$e$ $\mathrm{mm}$
$t$ $\mathrm{s}$
$L$ $\mathrm{m}$

## 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
$α\mathrm{[°]}$

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

$K_m \mathrm{[-]}$
 No data
$α\mathrm{[°]}$

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
$α\mathrm{[°]}$

$$ζ=\cfrac{1-K_Q^2}{K_Q^2}$$

$φ \mathrm{[°]}$
 No data
$α\mathrm{[°]}$

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

$f_r \mathrm{[-]}$
$Q_p \mathrm{[-]}$
 No data
$α\mathrm{[°]}$

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

$Q \mathrm{[m^3/s]}$
$v \mathrm{[m/s]}$
 No data
$α\mathrm{[°]}$

$$Q=Q_p\cdot Q_{max}$$
$$v=Q_p\cdot v_{max}$$
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
$α\mathrm{[°]}$

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

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

$$σ=\cfrac{\cfrac{p_{air}-P_{SV}}{ρ\cdot g}+H-H_L}{H_v}$$
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
$α\mathrm{[°]}$

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

$F_{bx} \mathrm{[kN]}$
 No data
$α\mathrm{[°]}$

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

$F_{by} \mathrm{[kN]}$
 No data
$α\mathrm{[°]}$

$$F_{by}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{by}$$
Angle from open position
Hydraulic torque without eccentricity
Force parallel to the axis of the disc
Force perpendicular to the axis of the disc
Moment from the axis of the trunnion to the axis of the disc
Moment from eccentricity
Hydraulic torque
$α$ $M$ $F_{e1}$ $F_{e2}$ $M_{LD}$ $M_e$ $M_H$
$\mathrm{°}$ $\mathrm{kNm}$ $\mathrm{kN}$ $\mathrm{kN}$ $\mathrm{kNm}$ $\mathrm{kNm}$ $\mathrm{kNm}$
No data

$M \mathrm{[kNm]}$
$M_{LD} \mathrm{[kNm]}$
$M_e \mathrm{[kNm]}$
 No data
$α\mathrm{[°]}$

$$M=\cfrac{D_s^3\cdot ρ\cdot g\cdot H_v\cdot -K_m}{10^{12}}$$
$$M_{LD}=\cfrac{F_{e1}\cdot 0.12\cdot D-F_{e1}\cdot L_D}{10^{3}}$$
$$M_e=\cfrac{-F_{e2}\cdot e}{10^3}$$

$F_{e1} \mathrm{[kN]}$
 No data
$α\mathrm{[°]}$

$$F_{e1}=F\cdot \sin\left(\left(90-α-φ\right)\cdot\cfrac{π}{180}\right)$$

$F_{e2} \mathrm{[kN]}$
 No data
$α\mathrm{[°]}$

$$F_{e2}=F\cdot \cos\left(\left(90-α-φ\right)\cdot\cfrac{π}{180}\right)$$

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

$$M_H=M+M_{LD}+M_e$$