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Hydrodynamic calculation Butterfly valve for pump operation lattice disc

Butterfly valve for pump operation (lattice disc) D s e +F by +F bx -M H D +F y +F x +F Φ α 0.3D 0.5D Q max L
Butterfly valve for pump operation (lattice disc)
Lattice disc c D s a b e 20° L D 140°
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

$$v_{max}=\cfrac{4\cdot 10^6\cdot Q_{max}}{π\cdot D^2}$$
$v_{max}=\cfrac{4\cdot 10^6\cdot Q_{max}}{π\cdot D^2}$
Equations in LaTeX code
v_{max}=\cfrac{4\cdot 10^6\cdot Q_{max}}{π\cdot D^2}

Theoretical pressure in the valve at full opening

$$Δ_h=\cfrac{v_{max}^2}{2\cdot g}\cdot \left({\min\left(ζ\right)}+1\right)$$
$Δ_h=\cfrac{v_{max}^2}{2\cdot g}\cdot \left({\min\left(ζ\right)}+1\right)$
Equations in LaTeX code
Δ_h=\cfrac{v_{max}^2}{2\cdot g}\cdot \left({\min\left(ζ\right)}+1\right)

Pressure parameter

$$p=\cfrac{Δ_h}{H}$$
$p=\cfrac{Δ_h}{H}$
Equations in LaTeX code
p=\cfrac{Δ_h}{H}
$$0 < p \le 1$$
$0 < p \le 1$
Equations in LaTeX code
0 < p \le 1

Effective closing time factor

$$c_{ef}=\cfrac{1}{18}/\max_{i=1}^{18}{\left(Q_p[i]-Q_p[i+1]\right)}$$
$c_{ef}=\cfrac{1}{18}/\max_{i=1}^{18}{\left(Q_p[i]-Q_p[i+1]\right)}$
Equations in LaTeX code
c_{ef}=\cfrac{1}{18}/\max_{i=1}^{18}{\left(Q_p[i]-Q_p[i+1]\right)}
$$c_{ef}\le 1$$
$c_{ef}\le 1$
Equations in LaTeX code
c_{ef}\le 1

Under-pressure behind the valve

$$P_{u}=\max\left(-\cfrac{L\cdot v_{max}}{g\cdot t\cdot c_{ef}}, -\cfrac{p_{air}}{ρ\cdot g}\right)$$
$P_{u}=\max\left(-\cfrac{L\cdot v_{max}}{g\cdot t\cdot c_{ef}}, -\cfrac{p_{air}}{ρ\cdot g}\right)$
Equations in LaTeX code
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
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
Unrounded data
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;
°; ; ; ; ; ; ;

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

Angle from open position 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
$α$ $K_x$ $K_y$ $K_{bx}$ $K_{by}$
$\mathrm{°}$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Angle from open position; 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;
α; K_x; K_y; K_{bx}; K_{by};
°; ; ; ; ;

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

Angle from open position Flow coefficient
$α$ $K_Q$
$\mathrm{°}$ $\mathrm{ }$
No data
Unrounded data
Angle from open position; Flow coefficient;
α; K_Q;
°; ;

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

Angle from open position Hydraulic torque coefficient
$α$ $K_m$
$\mathrm{°}$ $\mathrm{ }$
No data
Unrounded data
Angle from open position; Hydraulic torque coefficient;
α; K_m;
°; ;
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
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
Unrounded data
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;
°; ;°; ; ;m^3/s;m/s;

$ζ \mathrm{[-]}$
No data
$α\mathrm{[°]}$
Loss coefficient

Angle from open position Loss coefficient
$α$ $ζ$
$\mathrm{°}$ $\mathrm{ }$
No data
Unrounded data
Angle from open position; Loss coefficient;
α; ζ;
°; ;
$$ζ=\cfrac{1-K_Q^2}{K_Q^2}$$
$ζ=\cfrac{1-K_Q^2}{K_Q^2}$
Equations in LaTeX code
ζ=\cfrac{1-K_Q^2}{K_Q^2}

$φ \mathrm{[°]}$
No data
$α\mathrm{[°]}$
Angle between pipe axis and hydraulic force

Angle from open position Angle between pipe axis and hydraulic force
$α$ $φ$
$\mathrm{°}$ $\mathrm{°}$
No data
Unrounded data
Angle from open position; Angle between pipe axis and hydraulic force;
α; φ;
°; °;
$\text{if }\ K_x= 0$
$\text{if }\ K_x= 0$
Equations in LaTeX code
\text{if }\ K_x= 0
$$φ=0$$
$φ=0$
Equations in LaTeX code
φ=0
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$φ=\tan^{-1}\left(\cfrac{K_y}{K_x}\right)\cdot\cfrac{180}{π}$$
$φ=\tan^{-1}\left(\cfrac{K_y}{K_x}\right)\cdot\cfrac{180}{π}$
Equations in LaTeX code
φ=\tan^{-1}\left(\cfrac{K_y}{K_x}\right)\cdot\cfrac{180}{π}

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

Angle from open position Reduced free flow area in the throttle control system Relative flow
$α$ $f_r$ $Q_p$
$\mathrm{°}$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Angle from open position; Reduced free flow area in the throttle control system; Relative flow;
α; f_r; Q_p;
°; ; ;
$$f_r=\cfrac{K_Q}{K_{Qmax}}$$
$f_r=\cfrac{K_Q}{K_{Qmax}}$
Equations in LaTeX code
f_r=\cfrac{K_Q}{K_{Qmax}}
$$Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}$$
$Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}$
Equations in LaTeX code
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

Angle from open position Flow of water in the pipeline Water velocity in pipeline
$α$ $Q$ $v$
$\mathrm{°}$ $\mathrm{m^3/s}$ $\mathrm{m/s}$
No data
Unrounded data
Angle from open position; Flow of water in the pipeline; Water velocity in pipeline;
α; Q; v;
°; m^3/s; m/s;
$$Q=Q_p\cdot Q_{max}$$
$Q=Q_p\cdot Q_{max}$
Equations in LaTeX code
Q=Q_p\cdot Q_{max}
$$v=Q_p\cdot v_{max}$$
$v=Q_p\cdot v_{max}$
Equations in LaTeX code
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
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
Unrounded data
Angle from open position;Loss of pressure on the valve;Pressure on the valve;Cavitation number;
α;H_L;H_v;σ;
°;m;m; ;

$H_L \mathrm{[m]}$
$H_v \mathrm{[m]}$
No data
$α\mathrm{[°]}$
Loss of height on valve and pressure height on Butterfly valve

Angle from open position Loss of pressure on the valve Pressure on the valve
$α$ $H_L$ $H_v$
$\mathrm{°}$ $\mathrm{m}$ $\mathrm{m}$
No data
Unrounded data
Angle from open position; Loss of pressure on the valve; Pressure on the valve;
α; H_L; H_v;
°; m; m;
$$H_L=\cfrac{v^2}{2\cdot g}\cdot ζ$$
$H_L=\cfrac{v^2}{2\cdot g}\cdot ζ$
Equations in LaTeX code
H_L=\cfrac{v^2}{2\cdot g}\cdot ζ
$$H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot\left(ΔP-P_{u}\right)$$
$H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot\left(ΔP-P_{u}\right)$
Equations in LaTeX code
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
$α\mathrm{[°]}$
Cavitation number

Angle from open position Cavitation number
$α$ $σ$
$\mathrm{°}$ $\mathrm{ }$
No data
Unrounded data
Angle from open position; Cavitation number;
α; σ;
°; ;
$$σ=\cfrac{\cfrac{p_{air}-P_{SV}}{ρ\cdot g}+H-H_L}{H_v}$$
$σ=\cfrac{\cfrac{p_{air}-P_{SV}}{ρ\cdot g}+H-H_L}{H_v}$
Equations in LaTeX code
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
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
Unrounded data
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};
°;kN;kN;kN;kN;kN;

$F_x \mathrm{[kN]}$
$F_y \mathrm{[kN]}$
$F \mathrm{[kN]}$
No data
$α\mathrm{[°]}$
Forces on the disc

Angle from open position Forces on disc in axis x Forces on disc in axis y Forces on disc
$α$ $F_x$ $F_y$ $F$
$\mathrm{°}$ $\mathrm{kN}$ $\mathrm{kN}$ $\mathrm{kN}$
No data
Unrounded data
Angle from open position; Forces on disc in axis x; Forces on disc in axis y; Forces on disc;
α; F_x; F_y; F;
°; kN; kN; kN;
$$F_x=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$$
$F_x=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$
Equations in LaTeX code
F_x=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x
$$F_y=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_y$$
$F_y=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_y$
Equations in LaTeX code
F_y=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_y
$$F=\sqrt{F_x^2+F_y^2}$$
$F=\sqrt{F_x^2+F_y^2}$
Equations in LaTeX code
F=\sqrt{F_x^2+F_y^2}

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

Angle from open position The force at the valve axis x
$α$ $F_{bx}$
$\mathrm{°}$ $\mathrm{kN}$
No data
Unrounded data
Angle from open position; The force at the valve axis x;
α; F_{bx};
°; kN;
$$F_{bx}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$$
$F_{bx}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$
Equations in LaTeX code
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

Angle from open position The force at the valve axis y
$α$ $F_{by}$
$\mathrm{°}$ $\mathrm{kN}$
No data
Unrounded data
Angle from open position; The force at the valve axis y;
α; F_{by};
°; kN;
$$F_{by}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{by}$$
$F_{by}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{by}$
Equations in LaTeX code
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
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
Unrounded data
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;
°;kNm;kN;kN;kNm;kNm;kNm;

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

Angle from open position Hydraulic torque without eccentricity Moment from the axis of the trunnion to the axis of the disc Moment from eccentricity
$α$ $M$ $M_{LD}$ $M_e$
$\mathrm{°}$ $\mathrm{kNm}$ $\mathrm{kNm}$ $\mathrm{kNm}$
No data
Unrounded data
Angle from open position; Hydraulic torque without eccentricity; Moment from the axis of the trunnion to the axis of the disc; Moment from eccentricity;
α; M; M_{LD}; M_e;
°; kNm; kNm; kNm;
$$M=\cfrac{D_s^3\cdot ρ\cdot g\cdot H_v\cdot -K_m}{10^{12}}$$
$M=\cfrac{D_s^3\cdot ρ\cdot g\cdot H_v\cdot -K_m}{10^{12}}$
Equations in LaTeX code
M=\cfrac{D_s^3\cdot ρ\cdot g\cdot H_v\cdot -K_m}{10^{12}}
$$M_{LD}=\cfrac{F_{e1}\cdot 0.12\cdot D-F_{e1}\cdot L_D}{10^{3}}$$
$M_{LD}=\cfrac{F_{e1}\cdot 0.12\cdot D-F_{e1}\cdot L_D}{10^{3}}$
Equations in LaTeX code
M_{LD}=\cfrac{F_{e1}\cdot 0.12\cdot D-F_{e1}\cdot L_D}{10^{3}}
$$M_e=\cfrac{-F_{e2}\cdot e}{10^3}$$
$M_e=\cfrac{-F_{e2}\cdot e}{10^3}$
Equations in LaTeX code
M_e=\cfrac{-F_{e2}\cdot e}{10^3}

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

Angle from open position Force parallel to the axis of the disc
$α$ $F_{e1}$
$\mathrm{°}$ $\mathrm{kN}$
No data
Unrounded data
Angle from open position; Force parallel to the axis of the disc;
α; F_{e1};
°; kN;
$$F_{e1}=F\cdot \sin\left(\left(90-α-φ\right)\cdot\cfrac{π}{180}\right)$$
$F_{e1}=F\cdot \sin\left(\left(90-α-φ\right)\cdot\cfrac{π}{180}\right)$
Equations in LaTeX code
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

Angle from open position Force perpendicular to the axis of the disc
$α$ $F_{e2}$
$\mathrm{°}$ $\mathrm{kN}$
No data
Unrounded data
Angle from open position; Force perpendicular to the axis of the disc;
α; F_{e2};
°; kN;
$$F_{e2}=F\cdot \cos\left(\left(90-α-φ\right)\cdot\cfrac{π}{180}\right)$$
$F_{e2}=F\cdot \cos\left(\left(90-α-φ\right)\cdot\cfrac{π}{180}\right)$
Equations in LaTeX code
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

Angle from open position Hydraulic torque
$α$ $M_H$
$\mathrm{°}$ $\mathrm{kNm}$
No data
Unrounded data
Angle from open position; Hydraulic torque;
α; M_H;
°; kNm;
$$M_H=M+M_{LD}+M_e$$
$M_H=M+M_{LD}+M_e$
Equations in LaTeX code
M_H=M+M_{LD}+M_e
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