## Hydrodynamic calculation Butterfly valve for pump operation (lattice disc)

Fig. 1 - Lattice-disc

Fig. 2 - Lattice-disc

$a$ $=0.35D$

$b$ $=0.35D$

$c$ $=0.13D$

The butterfly valve is a relatively old and proven type of valve that is most often used to close the water dam pipeline.
From the butterfly 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 butterfly valve to be able to safely close the piping at maximum flow. When closing from an open position, the disc resists the current flowing. 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 disc eccentrically with respect to the axis of rotation. As a result, it acts on the disc of the hydrodynamic torque in the sense of closing. The resulting hydrodynamic force in the butterfly 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 butterfly 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 butterfly 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, when designing the butterfly 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 butterfly valve during its rapid closure:

To calculate the pressure on the butterfly 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 butterfly 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 butterfly 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
$s$ $Q_p$
$\mathrm{\%}$ $\mathrm{ }$
00.931
100.873
200.751
300.577
400.426
500.305
600.213
700.128
800.081
900.043
1000

### 2. Guideline for the use of the hydrodynamic characteristics of the butterfly valve:

In the annex section of Fig. 3 to 5, charts of dimensionless coefficients are constructed. These graphs are the basis for constructing the hydrodynamic characteristics of the butterfly valve for the projected water dam.

Angle between pipe axis and hydraulic force:

$ϕ$ $=\tan^{-1}\cfrac{K_y}{K_x}\cdot \cfrac{180}{π}$

where:
 $ϕ$ angle between pipe axis and hydraulic force $\mathrm{°}$ $K_x$ coefficient of hydraulic force on a disc in the axis x $\mathrm{ }$ $K_y$ coefficient of hydraulic force on a disc in the axis y $\mathrm{ }$

Cavitation number:

$σ$ $=\cfrac{10-0.1+h_0-H_L}{H_v}$

where:
 $σ$ cavitation number $\mathrm{ }$ $h_0$ rated net head (max 6500m) $\mathrm{m}$ $H_L$ loss of pressure on the valve $\mathrm{m}$ $H_v$ pressure on the valve $\mathrm{m}$

Forces on disc in axis x:

$F_x$ $=\cfrac{π D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$

where:
 $F_x$ forces on disc in axis x $\mathrm{kN}$ $D_s$ max diameter of the disc seal $\mathrm{mm}$ $ρ$ density $\mathrm{kg/m^3}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $H_v$ pressure on the valve $\mathrm{m}$ $K_x$ coefficient of hydraulic force on a disc in the axis x $\mathrm{ }$

Forces on disc in axis y:

$F_y$ $=\cfrac{π D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_y$

where:
 $F_y$ forces on disc in axis y $\mathrm{kN}$ $D_s$ max diameter of the disc seal $\mathrm{mm}$ $ρ$ density $\mathrm{kg/m^3}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $H_v$ pressure on the valve $\mathrm{m}$ $K_x$ coefficient of hydraulic force on a disc in the axis x $\mathrm{ }$

Forces on disc:

$F$ $=\sqrt{F_x^2+F_y^2}$

where:
 $F$ forces on disc $\mathrm{kN}$ $F_x$ forces on disc in axis x $\mathrm{kN}$ $F_y$ forces on disc in axis y $\mathrm{kN}$

The force at the valve axis x:

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

where:
 $F_{bx}$ the force at the valve axis x $\mathrm{kN}$ $D$ valve diameter $\mathrm{mm}$ $ρ$ density $\mathrm{kg/m^3}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $H_v$ pressure on the valve $\mathrm{m}$ $K_{bx}$ coefficient of hydraulic force on body in the axis x $\mathrm{ }$

The force at the valve axis y:

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

where:
 $F_{by}$ the force at the valve axis y $\mathrm{kN}$ $D$ valve diameter $\mathrm{mm}$ $ρ$ density $\mathrm{kg/m^3}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $H_v$ pressure on the valve $\mathrm{m}$ $K_{by}$ coefficient of hydraulic force on body in the axis y $\mathrm{ }$

Hydraulic torque without eccentricity:

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

where:
 $M$ hydraulic torque without eccentricity $\mathrm{kNm}$ $D_s$ max diameter of the disc seal $\mathrm{mm}$ $ρ$ density $\mathrm{kg/m^3}$ $g$ gravitational acceleration $\mathrm{m/s^2}$ $H_v$ pressure on the valve $\mathrm{m}$ $K_m$ hydraulic torque coefficient $\mathrm{ }$

Moment from the axis of the trunnion to the axis of the disc:

$M_{LD}$ $=\cfrac{-F_{e1}\cdot0.12D+F_{e1}\cdot L_D}{10^3}$

where:
 $M_{LD}$ moment from the axis of the trunnion to the axis of the disc $\mathrm{kNm}$ $F_{e1}$ force parallel to the axis of the disc $\mathrm{kN}$ $L_D$ length from the axis of rotation to the outer edge of the disc $\mathrm{mm}$ $D$ valve diameter $\mathrm{mm}$

Force parallel to the axis of the disc:

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

where:
 $F_{e1}$ force parallel to the axis of the disc $\mathrm{kN}$ $F$ forces on disc $\mathrm{kN}$ $α$ angle from open position $\mathrm{°}$ $ϕ$ angle between pipe axis and hydraulic force $\mathrm{°}$

Moment from eccentricity:

$M_e$ $=\cfrac{F_{e2}\cdot e}{10^3}$

where:
 $M_e$ moment from eccentricity $\mathrm{kNm}$ $F_{e2}$ force perpendicular to the axis of the disc $\mathrm{kN}$ $e$ eccentricity $\mathrm{mm}$

Force perpendicular to the axis of the disc:

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

where:
 $F_{e2}$ force perpendicular to the axis of the disc $\mathrm{kN}$ $F$ forces on disc $\mathrm{kN}$ $α$ angle from open position $\mathrm{°}$ $ϕ$ angle between pipe axis and hydraulic force $\mathrm{°}$

Hydraulic torque:

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

where:
 $M_H$ hydraulic torque $\mathrm{kNm}$ $M$ hydraulic torque without eccentricity $\mathrm{kNm}$ $M_{LD}$ moment from the axis of the trunnion to the axis of the disc $\mathrm{kNm}$ $M_e$ moment from eccentricity $\mathrm{kNm}$

### 3. Conclusion:

The butterfly valve must be hydraulically positioned in a straight diameter $D$ . The effect of cavitation on pressure losses, flow and dynamic effects of the water stream does not occur immediately in the initial stage (when the first steam bubbles are formed), but only with a fully developed cavitation.

Literature:
- Miroslav Nechleba: Vodní turbíny jejich konstrukce a příslušenství 1954.
- V. Kolář, St. Vinopal: Hydraulika průmyslových armatur 1963.

-●-$K_x \mathrm{[-]}$
-●-$K_y \mathrm{[-]}$
-●-$K_{bx} \mathrm{[-]}$
-●-$K_{by} \mathrm{[-]}$
$α\mathrm{[°]}$
Fig. 3 - Coefficient of force

-●-$K_Q \mathrm{[-]}$
$α\mathrm{[°]}$
Fig. 4 - Flow coefficient

-●-$K_m \mathrm{[-]}$
$α\mathrm{[°]}$
Fig. 5 - Coefficient of hydraulic torque