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Hydrodynamic calculation Spherical valve

Spherical valve D D s Q max e e +M H L 0.0875D 0.512D α +F y +F x +F by +F bx Q air Q air +F Φ
Spherical valve

Values for calculation

$ \{D\in\mathbb{R}\mid D>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ D $ $ \mathrm{mm} $
$ \{D_s\in\mathbb{R}\mid D_s>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ D_s $ $ \mathrm{mm} $
$ \{Q_{max}\in\mathbb{R}\mid Q_{max}>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ Q_{max} $ $ \mathrm{m^3/s} $
$ \{H\in\mathbb{R}\mid H>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ H $ $ \mathrm{m} $
$ \{g\in\mathbb{R}\mid g>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ g $ $ \mathrm{m/s^2} $
$ \{T\in\mathbb{R}\mid T\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ T $ $ \mathrm{°C} $
$ ρ $ $ \mathrm{kg/m^3} $
$ P_{SV} $ $ \mathrm{Pa} $
$ \{ΔP\in\mathbb{R}\mid ΔP>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ ΔP $ $ \mathrm{m} $
$ \{h\in\mathbb{R}\mid h\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ h $ $ \mathrm{m} $
$ ρ_{air} $ $ \mathrm{kg/m^3} $
$ p_{air} $ $ \mathrm{Pa} $
$ n $
$ \{e\in\mathbb{R}\mid e\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ e $ $ \mathrm{mm} $
$ \{t\in\mathbb{R}\mid t>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ t $ $ \mathrm{s} $
$ \{L\in\mathbb{R}\mid L\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ 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 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
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
Unrounded data
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;
°; ; ; ; ; ; ;

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

Angle from open position 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
$ 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 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;
α; K_x; K_y; K_{bx}; K_{by};
°; ; ; ; ;

$ K_Q\ [-] $
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\ [-] $
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\ [-] $
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\ [-] $
$ Q_p\ [-] $
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 Spherical 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\ [-] $
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 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
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
Unrounded data
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};
°;kN;kN;kN;kN;kN;

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

Angle from open position Forces on rotating body in axis x Forces on rotating body in axis y Forces on rotating body
$ α $ $ F_x $ $ F_y $ $ F $
$ \mathrm{°} $ $ \mathrm{kN} $ $ \mathrm{kN} $ $ \mathrm{kN} $
No data
Unrounded data
Angle from open position; Forces on rotating body in axis x; Forces on rotating body in axis y; Forces on rotating body;
α; F_x; F_y; F;
°; kN; kN; kN;
$\text{if }\ α=90$
$\text{if }\ α=90$
Equations in LaTeX code
\text{if }\ α=90
$$F_x=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$$
$F_x=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$
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
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$F_x=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$$
$F_x=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$
Equations in LaTeX code
F_x=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x
$$F_y=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_y$$
$F_y=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_y$
Equations in LaTeX code
F_y=\cfrac{π\cdot D^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;
$\text{if }\ α=90$
$\text{if }\ α=90$
Equations in LaTeX code
\text{if }\ α=90
$$F_{bx}=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$$
$F_{bx}=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$
Equations in LaTeX code
F_{bx}=\cfrac{π\cdot D_s^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$F_{bx}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$$
$F_{bx}=\cfrac{π\cdot D^2}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_{bx}$
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 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
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
Unrounded data
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;
°;kNm;kN;kN;kNm;kNm;

$ M\ [\mathrm{kNm}] $
$ M_e\ [\mathrm{kNm}] $
No data
$ α\ [\mathrm{°}] $
Torque of the rotating body

Angle from open position Hydraulic torque without eccentricity Moment from eccentricity
$ α $ $ M $ $ M_e $
$ \mathrm{°} $ $ \mathrm{kNm} $ $ \mathrm{kNm} $
No data
Unrounded data
Angle from open position; Hydraulic torque without eccentricity; Moment from eccentricity;
α; M; M_e;
°; kNm; kNm;
$$M=\cfrac{D^3\cdot ρ\cdot g\cdot H_v\cdot K_m}{10^{12}}$$
$M=\cfrac{D^3\cdot ρ\cdot g\cdot H_v\cdot K_m}{10^{12}}$
Equations in LaTeX code
M=\cfrac{D^3\cdot ρ\cdot g\cdot H_v\cdot K_m}{10^{12}}
$$M_e=\cfrac{\left(F_{e1}+F_{e2}\right)\cdot e}{10^3}$$
$M_e=\cfrac{\left(F_{e1}+F_{e2}\right)\cdot e}{10^3}$
Equations in LaTeX code
M_e=\cfrac{\left(F_{e1}+F_{e2}\right)\cdot e}{10^3}

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

Angle from open position Force parallel to the axis of the rotating body
$ α $ $ F_{e1} $
$ \mathrm{°} $ $ \mathrm{kN} $
No data
Unrounded data
Angle from open position; Force parallel to the axis of the rotating body;
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 rotating body

Angle from open position Force perpendicular to the axis of the rotating body
$ α $ $ F_{e2} $
$ \mathrm{°} $ $ \mathrm{kN} $
No data
Unrounded data
Angle from open position; Force perpendicular to the axis of the rotating body;
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;
$$M_H=M+M_e$$
$M_H=M+M_e$
Equations in LaTeX code
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
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
Unrounded data
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};
°; ;Pa;m^3/s;

$ f_{air}\ [-] $
No data
$ α\ [\mathrm{°}] $
Coefficient of under-pressure of aerated hole

Angle from open position Coefficient of under-pressure of aerated hole
$ α $ $ f_{air} $
$ \mathrm{°} $ $ \mathrm{\ } $
No data
Unrounded data
Angle from open position; Coefficient of under-pressure of aerated hole;
α; f_{air};
°; ;

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

Angle from open position Under-pressure in the aerated pipeline
$ α $ $ p_{air} $
$ \mathrm{°} $ $ \mathrm{Pa} $
No data
Unrounded data
Angle from open position; Under-pressure in the aerated pipeline;
α; p_{air};
°; Pa;
$\text{if }\ \text{n}= \text{no}$
$\text{if }\ \text{n}= \text{no}$
Equations in LaTeX code
\text{if }\ \text{n}= \text{no}
$$p_{air}=NAN$$
$p_{air}=NAN$
Equations in LaTeX code
p_{air}=NAN
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$p_{air}=-\min\left(p_{air}, f_{air}\cdot\cfrac{v^2}{2\cdot g}\cdot ρ+\left(1-Q_p\right)\cdot\min\left(\cfrac{L\cdot v_{max}\cdot ρ}{t\cdot c_{ef}}, p_{air}\right)\right)$$
$p_{air}=-\min\left(p_{air}, f_{air}\cdot\cfrac{v^2}{2\cdot g}\cdot ρ+\left(1-Q_p\right)\cdot\min\left(\cfrac{L\cdot v_{max}\cdot ρ}{t\cdot c_{ef}}, p_{air}\right)\right)$
Equations in LaTeX code
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

Angle from open position Air flow
$ α $ $ Q_{air} $
$ \mathrm{°} $ $ \mathrm{m^3/s} $
No data
Unrounded data
Angle from open position; Air flow;
α; Q_{air};
°; m^3/s;
$\text{if }\ \text{n}= \text{no}$
$\text{if }\ \text{n}= \text{no}$
Equations in LaTeX code
\text{if }\ \text{n}= \text{no}
$$Q_{air}=NAN$$
$Q_{air}=NAN$
Equations in LaTeX code
Q_{air}=NAN
$\text{else if }\ p_{air}<\cfrac{p_{air}}{2}$
$\text{else if }\ p_{air}<\cfrac{p_{air}}{2}$
Equations in LaTeX code
\text{else if }\ p_{air}<\cfrac{p_{air}}{2}
$$Q_{air}=\min\left(Q_{max}-Q, 0.2\cdot Q\right)$$
$Q_{air}=\min\left(Q_{max}-Q, 0.2\cdot Q\right)$
Equations in LaTeX code
Q_{air}=\min\left(Q_{max}-Q, 0.2\cdot Q\right)
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$Q_{air}=\max\left(Q_{max}-Q, 0.2\cdot Q\right)$$
$Q_{air}=\max\left(Q_{max}-Q, 0.2\cdot Q\right)$
Equations in LaTeX code
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
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
Unrounded data
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};
°;m/s;m^2;m^2;

$ v_{air}\ [\mathrm{m/s}] $
No data
$ α\ [\mathrm{°}] $
Air velocity

Angle from open position Air velocity
$ α $ $ v_{air} $
$ \mathrm{°} $ $ \mathrm{m/s} $
No data
Unrounded data
Angle from open position; Air velocity;
α; v_{air};
°; m/s;
$\text{if }\ \text{n}= \text{no}$
$\text{if }\ \text{n}= \text{no}$
Equations in LaTeX code
\text{if }\ \text{n}= \text{no}
$$v_{air}=NAN$$
$v_{air}=NAN$
Equations in LaTeX code
v_{air}=NAN
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$v_{air}=\min\left(0.7\cdot\sqrt{-\cfrac{2\cdot p_{air}}{ρ_{air}}}, 250\right)$$
$v_{air}=\min\left(0.7\cdot\sqrt{-\cfrac{2\cdot p_{air}}{ρ_{air}}}, 250\right)$
Equations in LaTeX code
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

Angle from open position The flow area of the aerated hole
$ α $ $ A_{air} $
$ \mathrm{°} $ $ \mathrm{m^2} $
No data
Unrounded data
Angle from open position; The flow area of the aerated hole;
α; A_{air};
°; m^2;
$\text{if }\ \text{n}= \text{no}$
$\text{if }\ \text{n}= \text{no}$
Equations in LaTeX code
\text{if }\ \text{n}= \text{no}
$$A_{air}=NAN$$
$A_{air}=NAN$
Equations in LaTeX code
A_{air}=NAN
$\text{else if }\ v_{air}=0$
$\text{else if }\ v_{air}=0$
Equations in LaTeX code
\text{else if }\ v_{air}=0
$$A_{air}=0$$
$A_{air}=0$
Equations in LaTeX code
A_{air}=0
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$A_{air}=\cfrac{Q_{air}}{v_{air}}$$
$A_{air}=\cfrac{Q_{air}}{v_{air}}$
Equations in LaTeX code
A_{air}=\cfrac{Q_{air}}{v_{air}}

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

Angle from open position The flow area of the aerated pipeline
$ α $ $ A_{air-pipe} $
$ \mathrm{°} $ $ \mathrm{m^2} $
No data
Unrounded data
Angle from open position; The flow area of the aerated pipeline;
α; A_{air-pipe};
°; m^2;
$\text{if }\ \text{n}= \text{no}$
$\text{if }\ \text{n}= \text{no}$
Equations in LaTeX code
\text{if }\ \text{n}= \text{no}
$$A_{air-pipe}=NAN$$
$A_{air-pipe}=NAN$
Equations in LaTeX code
A_{air-pipe}=NAN
$\text{else if }\ v_{air}>50$
$\text{else if }\ v_{air}>50$
Equations in LaTeX code
\text{else if }\ v_{air}>50
$$A_{air-pipe}=\cfrac{Q_{air}}{50}$$
$A_{air-pipe}=\cfrac{Q_{air}}{50}$
Equations in LaTeX code
A_{air-pipe}=\cfrac{Q_{air}}{50}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$A_{air-pipe}=A_{air}$$
$A_{air-pipe}=A_{air}$
Equations in LaTeX code
A_{air-pipe}=A_{air}
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