Hydrodynamic calculation vertical lift gates

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

$$A=\cfrac{s_0\cdot b}{10^6}$$

Area of the horizontal projection of the top seal

$$A_s=\cfrac{B\cdot a_2}{10^6}$$

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

$$A_1=a_1\cdot b$$

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

$$A_2=a_2\cdot b$$

Coefficient $K_T$

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

Velocity before the gate

$$v_{max}=\cfrac{10^6\cdot Q_{max}}{s_0\cdot b}$$

Theoretical pressure in the gate at full opening

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

Effective closing time factor

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

Under-pressure behind the gate

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

Pressure parameter

$$p=\cfrac{Δ_h}{H}$$

$$0 < p \le 1$$
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

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

$\text{if }\ s/s_0= 0$
$$K_Q=1E-100$$
$\text{else}$
$$K_Q=s/s_0\cdot C_c$$

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

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

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

$$s=s/s_0\cdot s_0$$
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

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

$$ζ=\cfrac{1-K_Q^2}{K_Q^2}$$

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

$$f_r=\cfrac{K_Q}{\max\left(K_Q\right)}$$
$$Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}$$

$Q \mathrm{[m^3/s]}$
 No data
$s/s_0\mathrm{[-]}$

$\text{if }\ K_Q\cdot\sqrt{2\cdot g\cdot H} \le v_{max}$
$$Q=K_Q\cdot A\cdot\sqrt{2\cdot g\cdot H}$$
$\text{else}$
$$Q=Q_p\cdot Q_{max}$$

$v \mathrm{[m/s]}$
 No data
$s/s_0\mathrm{[-]}$

$$v=\cfrac{10^6\cdot Q}{s_0\cdot b}$$

$v_j \mathrm{[m/s]}$
 No data
$s/s_0\mathrm{[-]}$

$\text{if }\ s/s_0= 0$
$$v_j=0$$
$\text{else}$
$$v_j=\cfrac{Q}{K_Q\cdot A}$$
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

$H_L \mathrm{[m]}$
$H_v \mathrm{[m]}$
 No data
$s/s_0\mathrm{[-]}$

$$H_L=\cfrac{v^2}{2\cdot g}\cdot ζ$$
$$H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot \left(ΔP-P_{u}\right)$$

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

$$σ=\cfrac{\cfrac{p_{air}-P_{SV}}{ρ\cdot g}+H-H_L}{H_v}$$

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

$\text{if }\ s/s_0= 0$
$$W=\cfrac{ρ\cdot g\cdot H_v\cdot B\cdot s_s}{10^6}$$
$\text{else}$
$$W=\cfrac{ρ\cdot g\cdot H_v\cdot B\cdot\left(s_0-s\right)}{10^6}$$
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

$P_1 \mathrm{[kN]}$
 No data
$s/s_0\mathrm{[-]}$

$$P_1=\left(K_T-K_B\right)\cdot B\cdot d\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^9}$$

$P_2 \mathrm{[kN]}$
 No data
$s/s_0\mathrm{[-]}$

$$P_2=K_T\cdot A_s\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^3}$$

$P_3 \mathrm{[kN]}$
 No data
$s/s_0\mathrm{[-]}$

$$P_3=K_T\cdot B\cdot d^´\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^9}$$

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

$$P=P_1+P_2+P_3$$
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

$h_c \mathrm{[m]}$
 No data
$s/s_0\mathrm{[-]}$

$$h_c=\cfrac{K_Q\cdot s_0}{10^3}$$

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

$\text{if }\ s/s_0= 0$
$$F_c=0$$
$\text{else}$
$$F_c=\sqrt{\cfrac{2\cdot\left(H-h_c\right)}{h_c}}$$

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

$\text{if }\ s/s_0= 0$
$$β=0$$
$\text{else}$
$$β=0.03\cdot\left(F_c-1\right)^{1.06}$$
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

$f_{air} \mathrm{[-]}$
 No data
$s/s_0\mathrm{[-]}$

$p_{air} \mathrm{[Pa]}$
 No data
$s/s_0\mathrm{[-]}$

$$p_{air}=-\min\left(p_{air}, f_{air}\cdot\cfrac{v^2}{2\cdot g}\cdot ρ+\left(1-Q_p\right)\cdot\min\left(\cfrac{L\cdot v_{max}\cdot ρ}{t\cdot c_{ef}}, p_{air}\right)\right)$$

$Q_{air} \mathrm{[m^3/s]}$
 No data
$s/s_0\mathrm{[-]}$

$\text{if }\ p_{air}< \cfrac{p_{air}}{2}$
$$Q_{air}=\min\left(Q_{max}-Q, β\cdot Q\right)$$
$\text{else}$
$$Q_{air}=\max\left(Q_{max}-Q, β\cdot Q\right)$$

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

$$v_{air}=\min\left(0.7\cdot\sqrt{-\cfrac{2\cdot p_{air}}{ρ_{air}}}, 250\right)$$

$A_{air} \mathrm{[m^2]}$
 No data
$s/s_0\mathrm{[-]}$

$\text{if }\ v_{air}= 0$
$$A_{air}=0$$
$\text{else}$
$$A_{air}=\cfrac{Q_{air}}{v_{air}}$$

$A_{air-pipe} \mathrm{[m^2]}$
 No data
$s/s_0\mathrm{[-]}$

$\text{if }\ v_{air}> 50$
$$A_{air-pipe}=\cfrac{Q_{air}}{50}$$
$\text{else}$
$$A_{air-pipe}=A_{air}$$