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Hydrodynamic calculation Butterfly valve double disc

Butterfly valve (double disc) D s +F by +F bx +M H L 0.5D D Q max +F y +F x +F Φ Q air α 0.2D 0.4D c D s L D
Butterfly valve (double disc)
Double disc c D s L D
Double disc

Values for calculation

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

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

$ K_m\ [-] $
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\ [-] $
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\ [-] $
$ Q_p\ [-] $
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\ [-] $
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

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

$ F_{bx}\ [\mathrm{kN}] $
No data
$ α\ [\mathrm{°}] $
The force at the valve axis x

$\text{if }\ α=90$
$$F_{bx}=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$$
$\text{else}$
$$F_{bx}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$$

$ 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
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
Hydraulic torque
$ α $ $ M $ $ F_{e1} $ $ F_{e2} $ $ M_{LD} $ $ M_H $
$ \mathrm{°} $ $ \mathrm{kNm} $ $ \mathrm{kN} $ $ \mathrm{kN} $ $ \mathrm{kNm} $ $ \mathrm{kNm} $
No data

$ M\ [\mathrm{kNm}] $
$ M_{LD}\ [\mathrm{kNm}] $
No data
$ α\ [\mathrm{°}] $
Torque of the disc

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

$ F_{e1}\ [\mathrm{kN}] $
No data
$ α\ [\mathrm{°}] $
Force parallel to the axis of the disc

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

$ F_{e2}\ [\mathrm{kN}] $
No data
$ α\ [\mathrm{°}] $
Force perpendicular to the axis of the disc

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

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

$$M_H=M+M_{LD}$$
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}\ [-] $
No data
$ α\ [\mathrm{°}] $
Coefficient of under-pressure of aerated hole

$ p_{air}\ [\mathrm{Pa}] $
No data
$ α\ [\mathrm{°}] $
Under-pressure in the aerated pipeline

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

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

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

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

$ A_{air}\ [\mathrm{m^2}] $
No data
$ α\ [\mathrm{°}] $
The flow area of the aerated hole

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

$ A_{air-pipe}\ [\mathrm{m^2}] $
No data
$ α\ [\mathrm{°}] $
The flow area of the aerated pipeline

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