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Hydrodynamic calculation vertical lift gates

Vertical lift gate Q max H s 0 b B s s d' θ a 1 a 2 d r e L s Q air +W +P
Vertical lift gate

Values for calculation

$s_0$ $\mathrm{mm}$
$b$ $\mathrm{mm}$
$B$ $\mathrm{mm}$
$s_s$ $\mathrm{mm}$
$d^´$ $\mathrm{mm}$
$θ$ $\mathrm{°}$
$e/d$
$a_1$ $\mathrm{mm}$
$a_2$ $\mathrm{mm}$
$d$ $\mathrm{mm}$
$r$ $\mathrm{mm}$
$e$ $\mathrm{mm}$
$Q_{max}$ $\mathrm{m^3/s}$
$H$ $\mathrm{m}$
$ΔP$ $\mathrm{m}$
$g$ $\mathrm{m/s^2}$
$T$ $\mathrm{°C}$
$ρ$ $\mathrm{kg/m^3}$
$P_{SV}$ $\mathrm{Pa}$
$h$ $\mathrm{m}$
$ρ_{air}$ $\mathrm{kg/m^3}$
$p_{air}$ $\mathrm{Pa}$
$t$ $\mathrm{s}$
$L$ $\mathrm{m}$

Calculation

Cross-sectional area of the conduit

$$A=\cfrac{s_0\cdot b}{10^6}$$
$A=\cfrac{s_0\cdot b}{10^6}$
Equations in LaTeX code
A=\cfrac{s_0\cdot b}{10^6}

Area of the horizontal projection of the top seal

$$A_s=\cfrac{B\cdot a_2}{10^6}$$
$A_s=\cfrac{B\cdot a_2}{10^6}$
Equations in LaTeX code
A_s=\cfrac{B\cdot a_2}{10^6}

Minimum cross-sectional area between upstream face of the gate and upstream wall of the gate chamber

$$A_1=a_1\cdot b$$
$A_1=a_1\cdot b$
Equations in LaTeX code
A_1=a_1\cdot b

Cross-sectional area of the contracted jet issuing from the gap between the downstream face of the gate and the downstream wall of the gate chamber

$$A_2=a_2\cdot b$$
$A_2=a_2\cdot b$
Equations in LaTeX code
A_2=a_2\cdot b

Coefficient $ K_T $

$$K_T=\cfrac{1}{1+\left(\cfrac{A_2}{A_1}\right)^2}$$
$K_T=\cfrac{1}{1+\left(\cfrac{A_2}{A_1}\right)^2}$
Equations in LaTeX code
K_T=\cfrac{1}{1+\left(\cfrac{A_2}{A_1}\right)^2}

Velocity before the gate

$$v_{max}=\cfrac{10^6\cdot Q_{max}}{s_0\cdot b}$$
$v_{max}=\cfrac{10^6\cdot Q_{max}}{s_0\cdot b}$
Equations in LaTeX code
v_{max}=\cfrac{10^6\cdot Q_{max}}{s_0\cdot b}

Theoretical pressure in the gate 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)

Effective closing time factor

$$c_{ef}=0.1/\max_{i=1}^{10}{\left(Q_p[i]-Q_p[i+1]\right)}$$
$c_{ef}=0.1/\max_{i=1}^{10}{\left(Q_p[i]-Q_p[i+1]\right)}$
Equations in LaTeX code
c_{ef}=0.1/\max_{i=1}^{10}{\left(Q_p[i]-Q_p[i+1]\right)}

Under-pressure behind the gate

$$P_{u}=\max\left(-\cfrac{L\cdot v_{max}}{g\cdot t\cdot c_{ef}}, -\cfrac{p_{air}}{ρ\cdot g}\right)$$
$P_{u}=\max\left(-\cfrac{L\cdot v_{max}}{g\cdot t\cdot c_{ef}}, -\cfrac{p_{air}}{ρ\cdot g}\right)$
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)

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
Gate position
Flow coefficient
Coefficient $ K_B $
Coefficient of contraction
Gate opening
$s/s_0$ $K_Q$ $K_B$ $C_c$ $s$
$\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{mm}$
No data
Gate position Flow coefficient Coefficient $ K_B $ Coefficient of contraction Gate opening
$s/s_0$ $K_Q$ $K_B$ $C_c$ $s$
$\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{mm}$
No data
Unrounded data
Gate position;Flow coefficient;Coefficient $ K_B $ ;Coefficient of contraction;Gate opening;
s/s_0;K_Q;K_B;C_c;s;
; ; ; ;mm;

$K_Q \mathrm{[-]}$
No data
$s/s_0\mathrm{[-]}$
Flow coefficient

Gate position Flow coefficient
$s/s_0$ $K_Q$
$\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Gate position; Flow coefficient;
s/s_0; K_Q;
; ;
$\text{if }\ s/s_0= 0$
$\text{if }\ s/s_0= 0$
Equations in LaTeX code
\text{if }\ s/s_0= 0
$$K_Q=1E-100$$
$K_Q=1E-100$
Equations in LaTeX code
K_Q=1E-100
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$K_Q=s/s_0\cdot C_c$$
$K_Q=s/s_0\cdot C_c$
Equations in LaTeX code
K_Q=s/s_0\cdot C_c

$K_B \mathrm{[-]}$
No data
$s/s_0\mathrm{[-]}$
Coefficient $ K_B $

Gate position Coefficient $ K_B $
$s/s_0$ $K_B$
$\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Gate position; Coefficient $ K_B $ ;
s/s_0; K_B;
; ;

$C_c \mathrm{[-]}$
No data
$s/s_0\mathrm{[-]}$
Coefficient of contraction

Gate position Coefficient of contraction
$s/s_0$ $C_c$
$\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Gate position; Coefficient of contraction;
s/s_0; C_c;
; ;

$s \mathrm{[mm]}$
No data
$s/s_0\mathrm{[-]}$
Gate opening

Gate position Gate opening
$s/s_0$ $s$
$\mathrm{ }$ $\mathrm{mm}$
No data
Unrounded data
Gate position; Gate opening;
s/s_0; s;
; mm;
$$s=s/s_0\cdot s_0$$
$s=s/s_0\cdot s_0$
Equations in LaTeX code
s=s/s_0\cdot s_0
Gate position
Loss coefficient
Reduced free flow area in the throttle control system
Relative flow
Flow of water before the gate
Water velocity before the gate
Velocity in the contracted jet issuing from underneath the gate
$s/s_0$ $ζ$ $f_r$ $Q_p$ $Q$ $v$ $v_j$
$\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{m^3/s}$ $\mathrm{m/s}$ $\mathrm{m/s}$
No data
Gate position Loss coefficient Reduced free flow area in the throttle control system Relative flow Flow of water before the gate Water velocity before the gate Velocity in the contracted jet issuing from underneath the gate
$s/s_0$ $ζ$ $f_r$ $Q_p$ $Q$ $v$ $v_j$
$\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$ $\mathrm{m^3/s}$ $\mathrm{m/s}$ $\mathrm{m/s}$
No data
Unrounded data
Gate position;Loss coefficient;Reduced free flow area in the throttle control system;Relative flow;Flow of water before the gate;Water velocity before the gate;Velocity in the contracted jet issuing from underneath the gate;
s/s_0;ζ;f_r;Q_p;Q;v;v_j;
; ; ; ;m^3/s;m/s;m/s;

$ζ \mathrm{[-]}$
No data
$s/s_0\mathrm{[-]}$
Loss coefficient

Gate position Loss coefficient
$s/s_0$ $ζ$
$\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Gate position; Loss coefficient;
s/s_0; 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}

$f_r \mathrm{[-]}$
$Q_p \mathrm{[-]}$
No data
$s/s_0\mathrm{[-]}$
Coefficients

Gate position Reduced free flow area in the throttle control system Relative flow
$s/s_0$ $f_r$ $Q_p$
$\mathrm{ }$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Gate position; Reduced free flow area in the throttle control system; Relative flow;
s/s_0; f_r; Q_p;
; ; ;
$$f_r=\cfrac{K_Q}{\max\left(K_Q\right)}$$
$f_r=\cfrac{K_Q}{\max\left(K_Q\right)}$
Equations in LaTeX code
f_r=\cfrac{K_Q}{\max\left(K_Q\right)}
$$Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}$$
$Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}$
Equations in LaTeX code
Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}

$Q \mathrm{[m^3/s]}$
No data
$s/s_0\mathrm{[-]}$
Flow of water before the gate

Gate position Flow of water before the gate
$s/s_0$ $Q$
$\mathrm{ }$ $\mathrm{m^3/s}$
No data
Unrounded data
Gate position; Flow of water before the gate;
s/s_0; Q;
; m^3/s;
$\text{if }\ K_Q\cdot\sqrt{2\cdot g\cdot H} \le v_{max}$
$\text{if }\ K_Q\cdot\sqrt{2\cdot g\cdot H} \le v_{max}$
Equations in LaTeX code
\text{if }\ K_Q\cdot\sqrt{2\cdot g\cdot H} \le v_{max}
$$Q=K_Q\cdot A\cdot\sqrt{2\cdot g\cdot H}$$
$Q=K_Q\cdot A\cdot\sqrt{2\cdot g\cdot H}$
Equations in LaTeX code
Q=K_Q\cdot A\cdot\sqrt{2\cdot g\cdot H}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$Q=Q_p\cdot Q_{max}$$
$Q=Q_p\cdot Q_{max}$
Equations in LaTeX code
Q=Q_p\cdot Q_{max}

$v \mathrm{[m/s]}$
No data
$s/s_0\mathrm{[-]}$
Water velocity before the gate

Gate position Water velocity before the gate
$s/s_0$ $v$
$\mathrm{ }$ $\mathrm{m/s}$
No data
Unrounded data
Gate position; Water velocity before the gate;
s/s_0; v;
; m/s;
$$v=\cfrac{10^6\cdot Q}{s_0\cdot b}$$
$v=\cfrac{10^6\cdot Q}{s_0\cdot b}$
Equations in LaTeX code
v=\cfrac{10^6\cdot Q}{s_0\cdot b}

$v_j \mathrm{[m/s]}$
No data
$s/s_0\mathrm{[-]}$
Velocity in the contracted jet issuing from underneath the gate

Gate position Velocity in the contracted jet issuing from underneath the gate
$s/s_0$ $v_j$
$\mathrm{ }$ $\mathrm{m/s}$
No data
Unrounded data
Gate position; Velocity in the contracted jet issuing from underneath the gate;
s/s_0; v_j;
; m/s;
$\text{if }\ s/s_0= 0$
$\text{if }\ s/s_0= 0$
Equations in LaTeX code
\text{if }\ s/s_0= 0
$$v_j=0$$
$v_j=0$
Equations in LaTeX code
v_j=0
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$v_j=\cfrac{Q}{K_Q\cdot A}$$
$v_j=\cfrac{Q}{K_Q\cdot A}$
Equations in LaTeX code
v_j=\cfrac{Q}{K_Q\cdot A}
Gate position
Loss of pressure on the gate
Pressure on the gate
Cavitation number
Force on gate
$s/s_0$ $H_L$ $H_v$ $σ$ $W$
$\mathrm{ }$ $\mathrm{m}$ $\mathrm{m}$ $\mathrm{ }$ $\mathrm{kN}$
No data
Gate position Loss of pressure on the gate Pressure on the gate Cavitation number Force on gate
$s/s_0$ $H_L$ $H_v$ $σ$ $W$
$\mathrm{ }$ $\mathrm{m}$ $\mathrm{m}$ $\mathrm{ }$ $\mathrm{kN}$
No data
Unrounded data
Gate position;Loss of pressure on the gate;Pressure on the gate;Cavitation number;Force on gate;
s/s_0;H_L;H_v;σ;W;
;m;m; ;kN;

$H_L \mathrm{[m]}$
$H_v \mathrm{[m]}$
No data
$s/s_0\mathrm{[-]}$
Loss of height on gate and pressure height on gate

Gate position Loss of pressure on the gate Pressure on the gate
$s/s_0$ $H_L$ $H_v$
$\mathrm{ }$ $\mathrm{m}$ $\mathrm{m}$
No data
Unrounded data
Gate position; Loss of pressure on the gate; Pressure on the gate;
s/s_0; 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
$s/s_0\mathrm{[-]}$
Cavitation number

Gate position Cavitation number
$s/s_0$ $σ$
$\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Gate position; Cavitation number;
s/s_0; 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}

$W \mathrm{[kN]}$
No data
$s/s_0\mathrm{[-]}$
Force on gate

Gate position Force on gate
$s/s_0$ $W$
$\mathrm{ }$ $\mathrm{kN}$
No data
Unrounded data
Gate position; Force on gate;
s/s_0; W;
; kN;
$\text{if }\ s/s_0= 0$
$\text{if }\ s/s_0= 0$
Equations in LaTeX code
\text{if }\ s/s_0= 0
$$W=\cfrac{ρ\cdot g\cdot H_v\cdot B\cdot s_s}{10^6}$$
$W=\cfrac{ρ\cdot g\cdot H_v\cdot B\cdot s_s}{10^6}$
Equations in LaTeX code
W=\cfrac{ρ\cdot g\cdot H_v\cdot B\cdot s_s}{10^6}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$W=\cfrac{ρ\cdot g\cdot H_v\cdot B\cdot\left(s_0-s\right)}{10^6}$$
$W=\cfrac{ρ\cdot g\cdot H_v\cdot B\cdot\left(s_0-s\right)}{10^6}$
Equations in LaTeX code
W=\cfrac{ρ\cdot g\cdot H_v\cdot B\cdot\left(s_0-s\right)}{10^6}
Gate position
Downpull resulting from the difference between the pressures acting on the top and bottom surfaces of the gate
Downpull resulting from the pressure differential acting on the horizontal protrusions of the gate
Downpull resulting from the lip
Downpull force
$s/s_0$ $P_1$ $P_2$ $P_3$ $P$
$\mathrm{ }$ $\mathrm{kN}$ $\mathrm{kN}$ $\mathrm{kN}$ $\mathrm{kN}$
No data
Gate position Downpull resulting from the difference between the pressures acting on the top and bottom surfaces of the gate Downpull resulting from the pressure differential acting on the horizontal protrusions of the gate Downpull resulting from the lip Downpull force
$s/s_0$ $P_1$ $P_2$ $P_3$ $P$
$\mathrm{ }$ $\mathrm{kN}$ $\mathrm{kN}$ $\mathrm{kN}$ $\mathrm{kN}$
No data
Unrounded data
Gate position;Downpull resulting from the difference between the pressures acting on the top and bottom surfaces of the gate;Downpull resulting from the pressure differential acting on the horizontal protrusions of the gate;Downpull resulting from the lip;Downpull force;
s/s_0;P_1;P_2;P_3;P;
;kN;kN;kN;kN;

$P_1 \mathrm{[kN]}$
No data
$s/s_0\mathrm{[-]}$
Downpull resulting from the difference between the pressures acting on the top and bottom surfaces of the gate

Gate position Downpull resulting from the difference between the pressures acting on the top and bottom surfaces of the gate
$s/s_0$ $P_1$
$\mathrm{ }$ $\mathrm{kN}$
No data
Unrounded data
Gate position; Downpull resulting from the difference between the pressures acting on the top and bottom surfaces of the gate;
s/s_0; P_1;
; kN;
$$P_1=\left(K_T-K_B\right)\cdot B\cdot d\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^9}$$
$P_1=\left(K_T-K_B\right)\cdot B\cdot d\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^9}$
Equations in LaTeX code
P_1=\left(K_T-K_B\right)\cdot B\cdot d\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^9}

$P_2 \mathrm{[kN]}$
No data
$s/s_0\mathrm{[-]}$
Downpull resulting from the pressure differential acting on the horizontal protrusions of the gate

Gate position Downpull resulting from the pressure differential acting on the horizontal protrusions of the gate
$s/s_0$ $P_2$
$\mathrm{ }$ $\mathrm{kN}$
No data
Unrounded data
Gate position; Downpull resulting from the pressure differential acting on the horizontal protrusions of the gate;
s/s_0; P_2;
; kN;
$$P_2=K_T\cdot A_s\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^3}$$
$P_2=K_T\cdot A_s\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^3}$
Equations in LaTeX code
P_2=K_T\cdot A_s\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^3}

$P_3 \mathrm{[kN]}$
No data
$s/s_0\mathrm{[-]}$
Downpull resulting from the lip

Gate position Downpull resulting from the lip
$s/s_0$ $P_3$
$\mathrm{ }$ $\mathrm{kN}$
No data
Unrounded data
Gate position; Downpull resulting from the lip;
s/s_0; P_3;
; kN;
$$P_3=K_T\cdot B\cdot d^´\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^9}$$
$P_3=K_T\cdot B\cdot d^´\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^9}$
Equations in LaTeX code
P_3=K_T\cdot B\cdot d^´\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^9}

$P \mathrm{[kN]}$
No data
$s/s_0\mathrm{[-]}$
Downpull force

Gate position Downpull force
$s/s_0$ $P$
$\mathrm{ }$ $\mathrm{kN}$
No data
Unrounded data
Gate position; Downpull force;
s/s_0; P;
; kN;
$$P=P_1+P_2+P_3$$
$P=P_1+P_2+P_3$
Equations in LaTeX code
P=P_1+P_2+P_3
Gate position
Depth of water at vena contracta
Froude number
Aerated coefficient
$s/s_0$ $h_c$ $F_c$ $β$
$\mathrm{ }$ $\mathrm{m}$ $\mathrm{ }$ $\mathrm{ }$
No data
Gate position Depth of water at vena contracta Froude number Aerated coefficient
$s/s_0$ $h_c$ $F_c$ $β$
$\mathrm{ }$ $\mathrm{m}$ $\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Gate position;Depth of water at vena contracta;Froude number;Aerated coefficient;
s/s_0;h_c;F_c;β;
;m; ; ;

$h_c \mathrm{[m]}$
No data
$s/s_0\mathrm{[-]}$
Depth of water at vena contracta

Gate position Depth of water at vena contracta
$s/s_0$ $h_c$
$\mathrm{ }$ $\mathrm{m}$
No data
Unrounded data
Gate position; Depth of water at vena contracta;
s/s_0; h_c;
; m;
$$h_c=\cfrac{K_Q\cdot s_0}{10^3}$$
$h_c=\cfrac{K_Q\cdot s_0}{10^3}$
Equations in LaTeX code
h_c=\cfrac{K_Q\cdot s_0}{10^3}

$F_c \mathrm{[-]}$
No data
$s/s_0\mathrm{[-]}$
Froude number

Gate position Froude number
$s/s_0$ $F_c$
$\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Gate position; Froude number;
s/s_0; F_c;
; ;
$\text{if }\ s/s_0= 0$
$\text{if }\ s/s_0= 0$
Equations in LaTeX code
\text{if }\ s/s_0= 0
$$F_c=0$$
$F_c=0$
Equations in LaTeX code
F_c=0
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$F_c=\sqrt{\cfrac{2\cdot\left(H-h_c\right)}{h_c}}$$
$F_c=\sqrt{\cfrac{2\cdot\left(H-h_c\right)}{h_c}}$
Equations in LaTeX code
F_c=\sqrt{\cfrac{2\cdot\left(H-h_c\right)}{h_c}}

$β \mathrm{[-]}$
No data
$s/s_0\mathrm{[-]}$
Aerated coefficient

Gate position Aerated coefficient
$s/s_0$ $β$
$\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Gate position; Aerated coefficient;
s/s_0; β;
; ;
$\text{if }\ s/s_0= 0$
$\text{if }\ s/s_0= 0$
Equations in LaTeX code
\text{if }\ s/s_0= 0
$$β=0$$
$β=0$
Equations in LaTeX code
β=0
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$β=0.03\cdot\left(F_c-1\right)^{1.06}$$
$β=0.03\cdot\left(F_c-1\right)^{1.06}$
Equations in LaTeX code
β=0.03\cdot\left(F_c-1\right)^{1.06}
Gate position
Coefficient of under-pressure of aerated hole
Under-pressure in the aerated pipeline
Air flow
Air velocity
The flow area of the aerated hole
The flow area of the aerated pipeline
$s/s_0$ $f_{air}$ $p_{air}$ $Q_{air}$ $v_{air}$ $A_{air}$ $A_{air-pipe}$
$\mathrm{ }$ $\mathrm{ }$ $\mathrm{Pa}$ $\mathrm{m^3/s}$ $\mathrm{m/s}$ $\mathrm{m^2}$ $\mathrm{m^2}$
No data
Gate position Coefficient of under-pressure of aerated hole Under-pressure in the aerated pipeline Air flow Air velocity The flow area of the aerated hole The flow area of the aerated pipeline
$s/s_0$ $f_{air}$ $p_{air}$ $Q_{air}$ $v_{air}$ $A_{air}$ $A_{air-pipe}$
$\mathrm{ }$ $\mathrm{ }$ $\mathrm{Pa}$ $\mathrm{m^3/s}$ $\mathrm{m/s}$ $\mathrm{m^2}$ $\mathrm{m^2}$
No data
Unrounded data
Gate position;Coefficient of under-pressure of aerated hole;Under-pressure in the aerated pipeline;Air flow;Air velocity;The flow area of the aerated hole;The flow area of the aerated pipeline;
s/s_0;f_{air};p_{air};Q_{air};v_{air};A_{air};A_{air-pipe};
; ;Pa;m^3/s;m/s;m^2;m^2;

$f_{air} \mathrm{[-]}$
No data
$s/s_0\mathrm{[-]}$
Coefficient of under-pressure of aerated hole

Gate position Coefficient of under-pressure of aerated hole
$s/s_0$ $f_{air}$
$\mathrm{ }$ $\mathrm{ }$
No data
Unrounded data
Gate position; Coefficient of under-pressure of aerated hole;
s/s_0; f_{air};
; ;

$p_{air} \mathrm{[Pa]}$
No data
$s/s_0\mathrm{[-]}$
Under-pressure in the aerated pipeline

Gate position Under-pressure in the aerated pipeline
$s/s_0$ $p_{air}$
$\mathrm{ }$ $\mathrm{Pa}$
No data
Unrounded data
Gate position; Under-pressure in the aerated pipeline;
s/s_0; p_{air};
; Pa;
$$p_{air}=-\min\left(p_{air}, f_{air}\cdot\cfrac{v^2}{2\cdot g}\cdot ρ+\left(1-Q_p\right)\cdot\min\left(\cfrac{L\cdot v_{max}\cdot ρ}{t\cdot c_{ef}}, p_{air}\right)\right)$$
$p_{air}=-\min\left(p_{air}, f_{air}\cdot\cfrac{v^2}{2\cdot g}\cdot ρ+\left(1-Q_p\right)\cdot\min\left(\cfrac{L\cdot v_{max}\cdot ρ}{t\cdot c_{ef}}, p_{air}\right)\right)$
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
$s/s_0\mathrm{[-]}$
Air flow

Gate position Air flow
$s/s_0$ $Q_{air}$
$\mathrm{ }$ $\mathrm{m^3/s}$
No data
Unrounded data
Gate position; Air flow;
s/s_0; Q_{air};
; m^3/s;
$\text{if }\ p_{air}< \cfrac{p_{air}}{2}$
$\text{if }\ p_{air}< \cfrac{p_{air}}{2}$
Equations in LaTeX code
\text{if }\ p_{air}< \cfrac{p_{air}}{2}
$$Q_{air}=\min\left(Q_{max}-Q, β\cdot Q\right)$$
$Q_{air}=\min\left(Q_{max}-Q, β\cdot Q\right)$
Equations in LaTeX code
Q_{air}=\min\left(Q_{max}-Q, β\cdot Q\right)
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$Q_{air}=\max\left(Q_{max}-Q, β\cdot Q\right)$$
$Q_{air}=\max\left(Q_{max}-Q, β\cdot Q\right)$
Equations in LaTeX code
Q_{air}=\max\left(Q_{max}-Q, β\cdot Q\right)

$v_{air} \mathrm{[m/s]}$
No data
$s/s_0\mathrm{[-]}$
Air velocity

Gate position Air velocity
$s/s_0$ $v_{air}$
$\mathrm{ }$ $\mathrm{m/s}$
No data
Unrounded data
Gate position; Air velocity;
s/s_0; v_{air};
; m/s;
$$v_{air}=\min\left(0.7\cdot\sqrt{-\cfrac{2\cdot p_{air}}{ρ_{air}}}, 250\right)$$
$v_{air}=\min\left(0.7\cdot\sqrt{-\cfrac{2\cdot p_{air}}{ρ_{air}}}, 250\right)$
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
$s/s_0\mathrm{[-]}$
The flow area of the aerated hole

Gate position The flow area of the aerated hole
$s/s_0$ $A_{air}$
$\mathrm{ }$ $\mathrm{m^2}$
No data
Unrounded data
Gate position; The flow area of the aerated hole;
s/s_0; A_{air};
; m^2;
$\text{if }\ v_{air}= 0$
$\text{if }\ v_{air}= 0$
Equations in LaTeX code
\text{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
$s/s_0\mathrm{[-]}$
The flow area of the aerated pipeline

Gate position The flow area of the aerated pipeline
$s/s_0$ $A_{air-pipe}$
$\mathrm{ }$ $\mathrm{m^2}$
No data
Unrounded data
Gate position; The flow area of the aerated pipeline;
s/s_0; A_{air-pipe};
; m^2;
$\text{if }\ v_{air}> 50$
$\text{if }\ v_{air}> 50$
Equations in LaTeX code
\text{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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