# Hydrodynamic calculation Spherical valve

## Values for calculation

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

## 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 rotating body in the axis x
Coefficient of hydraulic force on a rotating body 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 rotating body in axis x
Forces on rotating body in axis y
Forces on rotating body
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
$α\mathrm{[°]}$

$\text{if }\ α=90$
$$F_x=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$$
$\text{else}$
$$F_x=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$$
$$F_y=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_y$$
$$F=\sqrt{F_x^2+F_y^2}$$

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

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

$F_{by} \mathrm{[kN]}$
 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 rotating body
Force perpendicular to the axis of the rotating body
Moment from eccentricity
Hydraulic torque
$α$ $M$ $F_{e1}$ $F_{e2}$ $M_e$ $M_H$
$\mathrm{°}$ $\mathrm{kNm}$ $\mathrm{kN}$ $\mathrm{kN}$ $\mathrm{kNm}$ $\mathrm{kNm}$
No data

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

$$M=\cfrac{D^3\cdot ρ\cdot g\cdot H_v\cdot K_m}{10^{12}}$$
$$M_e=\cfrac{\left(F_{e1}+F_{e2}\right)\cdot e}{10^3}$$

$F_{e1} \mathrm{[kN]}$
 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
$α\mathrm{[°]}$

$$M_H=M+M_e$$
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

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

$p_{air} \mathrm{[Pa]}$
 No data
$α\mathrm{[°]}$

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

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

$\text{if }\ \text{n}= \text{no}$
$$Q_{air}=NAN$$
$\text{else if }\ p_{air}<\cfrac{p_{air}}{2}$
$$Q_{air}=\min\left(Q_{max}-Q, 0.2\cdot Q\right)$$
$\text{else}$
$$Q_{air}=\max\left(Q_{max}-Q, 0.2\cdot Q\right)$$
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
$α\mathrm{[°]}$

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

$A_{air} \mathrm{[m^2]}$
 No data
$α\mathrm{[°]}$

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

$A_{air-pipe} \mathrm{[m^2]}$
 No data
$α\mathrm{[°]}$

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