## Hydrodynamic calculation Spherical valve

Fig. 1 - Spherical valve

The spherical valve is a relatively old and proven type of valve that is most often used to close the water dam pipeline.
In the fully open position, the spherical valve is very advantageous from a hydraulic point of view. It forms a direct continuation of the pipeline, the water flows through the spherical valve without changing the direction and without changing the flow cross section. The spherical valve in the fully open position causes virtually no energy loss.
From the spherical valve, initially, it was only required to withstand the static water pressure and seal perfectly in the closed position.
Today it is required from the spherical valve to be able to safely close the piping at maximum flow. When closing from an open position, the body resists the current flowing through not only the internal rotating body but also through the obstruction of the outside. The flow is very complex in this case and in certain opening positions the valve is extensively stressed by hydrodynamic forces. The dynamic effects of the flow are manifested by a pulsating force acting in the front of the rotating body eccentrically with respect to the axis of rotation. As a result, it acts on the rotating body of the hydrodynamic torque in the sense of closing. The resulting hydrodynamic force in the spherical valve may, in some positions, be greater than the force which causes a static water pressure on the valve in the closed position.
In order to reduce the water hammer in the piping, it is very important to solve the most appropriate course of closure of the spherical valve. A very important basis for calculating the water hammer is the characteristic flow and pressure losses in the valve.
The dynamic effects of the water stream are substantially dependent on the hydraulic ratios in the pipeline and the valve. The flow of water without cavitation is different from the flow with cavitation or aeration of the space behind the valve. In the stage of fully developed cavitation there is a strong vibration, noise, great pulsation of hydrodynamic forces and torque. Shocks and vibrations of the spherical valve can be transferred to other constructions parts (vibrations of foundations). In adverse cases, resonance of different origins can occur, and thus a serious threat to safety.
We need to know not only the mean values of the hydrodynamic forces and torque values, but also the pulsation, the maximum amplitude and the mean frequency, when designing the spherical valve and the connecting pipe. We also need to fully understand the effect of cavitation, the effect of aeration and the effect of the different placement of the valve in the pipeline on its hydraulic and dynamic characteristics.

### 1. Calculation of pressure on the spherical valve during its rapid closure:

To calculate the pressure on the spherical valve, we need to know the rated net head at the zero flow (closed valve). Increase pressure on water hammer must be calculated before the hydrodynamic calculation of the spherical valve. The maximum flow rate must be defined in the open position, which must be closed safely. A must be a defined aerated space behind the valve due to the under-pressure behind the valve.
To calculate the pressure on the spherical valve because of the ignorance of the piping system, a calculation for a simple serial connection of the control valves (variable and constant resistance) will be used.

Relative flow:

$Q_p$ $=\cfrac{f_r}{\sqrt{p+f_r^2(1-p)}}$

where:
 $Q_p$ relative flow $\mathrm{ }$ $f_r$ reduced free flow area in the throttle control system $\mathrm{ }$ $p$ pressure parameter $\mathrm{ }$

Reduced free flow area in the throttle control system:

$f_r$ $=\cfrac{K_Q}{K_{Qmax}}$

where:
 $f_r$ reduced free flow area in the throttle control system $\mathrm{ }$ $K_Q$ flow coefficient $\mathrm{ }$ $K_{Qmax}$ flow coefficient max $\mathrm{ }$

Pressure parameter:

$p$ $=\cfrac{Δ h}{h_0}$

$0 < p\le1$

where:
 $p$ pressure parameter $\mathrm{ }$ $Δ h$ theoretical pressure in the valve at full opening $\mathrm{m}$ $h_0$ rated net head (max 6500m) $\mathrm{m}$

Theoretical pressure in the valve at full opening:

$Δ h$ $=\cfrac{v_0^2}{2g}\cdot\left(ξ+1\right)$

where:
 $Δ h$ theoretical pressure in the valve at full opening $\mathrm{m}$ $v_0$ valve speed $\mathrm{m/s}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $ξ$ loss coefficient $\mathrm{ }$

Valve speed:

$v_0$ $=\cfrac{4\cdot{10}^6\cdot Q_{max}}{π\cdot D^2}$

where:
 $v_0$ valve speed $\mathrm{m/s}$ $Q_{max}$ flow $\mathrm{m^3/s}$ $D$ valve diameter $\mathrm{mm}$

Flow of water in the pipeline:

$Q$ $=Q_p\cdot Q_{max}$

where:
 $Q$ flow of water in the pipeline $\mathrm{m^3/s}$ $Q_p$ relative flow $\mathrm{ }$ $Q_{max}$ flow $\mathrm{m^3/s}$

Water velocity in pipeline:

$v$ $=\cfrac{4Q\cdot 10^6}{π\cdot D^2}$

where:
 $v$ water velocity in pipeline $\mathrm{m/s}$ $Q$ flow of water in the pipeline $\mathrm{m^3/s}$ $D$ valve diameter $\mathrm{mm}$

Loss of pressure on the valve:

$H_L$ $=\cfrac{v^2}{2g}ξ$

where:
 $H_L$ loss of pressure on the valve $\mathrm{m}$ $v$ water velocity in pipeline $\mathrm{m/s}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $ξ$ loss coefficient $\mathrm{ }$

Loss coefficient:

$ξ$ $=\cfrac{1-K_Q^2}{K_Q^2}$

where:
 $ξ$ loss coefficient $\mathrm{ }$ $K_Q$ flow coefficient $\mathrm{ }$

Pressure on the valve:

$H_v$ $=H_L+\cfrac{v^2}{2g}+(1-Q_p)\cdot (Δ P+P_{atm})$

where:
 $H_v$ pressure on the valve $\mathrm{m}$ $H_L$ loss of pressure on the valve $\mathrm{m}$ $v$ water velocity in pipeline $\mathrm{m/s}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $Q_p$ relative flow $\mathrm{ }$ $Δ P$ increasing pressure on water hammer $\mathrm{m}$ $P_{atm}$ under-pressure behind the valve $\mathrm{m}$

Relative flow:
To calculate the increase in water hammer pressure.

Valve position Relative flow
$s$ $Q_p$
$\mathrm{\%}$ $\mathrm{ }$
00.978
100.893
200.718
300.561
400.42
500.3
600.212
700.138
800.079
900.036
1000

### 2. Accuracy of measurements:

Line pressure was measured on vertical U-tubes with fill, limit relative error in pressure measurement:
$\delta_p\le\pm1\%$
When measuring water flow using a calibrated Venturi tube and using a calibrated Thomson Overflow, the limit relative error was not higher as:
$\delta_Q\le\pm 1.5 \%$
When measuring the flow of air through a calibrated gas meter, the limit relative error was not higher as:
$\delta_{Qa}\le\pm2\%$
The surface of the pipeline was made with a limit relative error:
$\delta_F=\pm 1.2 \%$
Limit relative medium speed error $v$:
$\delta_v=\delta_Q+\delta_F=\pm( 1.5 + 1.2 )=\pm 2.7 \%$
Limit relative error of the flow coefficient $\mu$:
$\delta_\mu=\delta_Q+\delta_F+\cfrac{1}{2}\delta_v=\pm( 1.5 + 1.2 +\cfrac{1}{2}\cdot 2.7 )=\pm 3.2$
Limit relative error of cavitation factor $\sigma$:
$\delta_\sigma=\pm 2.7 \%$
Pressure diaphragm sensors and electrical devices that were used to measure hydrodynamic forces and torques were the limit relative error:
$\delta_{F_x}=\delta_{F_y}=\delta_{F_{bx}}=\delta_{F_{by}}=\delta_m=\pm2\%$
Limit relative pipe diameter error:
$\delta_D=\pm 0.625 \%$
Limit relative torque coefficient error $k_m$:Limit relative error of cavitation factor
$\delta_{k_m}=\delta_m+3\delta_D+\delta_p=\pm(2+3\cdot 0.625 +1)=\pm 4.9 \%$