Hydrodynamic calculation vertical lift gates
Values for calculation
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\ [-] $
| No data |
$ s/s_0\ [-] $
$$K_Q=1E-100$$
$$K_Q=s/s_0\cdot C_c$$
$ K_B\ [-] $
| No data |
$ s/s_0\ [-] $
$ C_c\ [-] $
| No data |
$ s/s_0\ [-] $
$ s\ [\mathrm{mm}] $
| No data |
$ s/s_0\ [-] $
$$s=s/s_0\cdot s_0$$
|
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\ [-] $
| No data |
$ s/s_0\ [-] $
$$ζ=\cfrac{1-K_Q^2}{K_Q^2}$$
$ f_r\ [-] $
$ Q_p\ [-] $
$ Q_p\ [-] $
| No data |
$ s/s_0\ [-] $
$$f_r=\cfrac{K_Q}{\max\left(K_Q\right)}$$
$$Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}$$
$ Q\ [\mathrm{m^3/s}] $
| No data |
$ s/s_0\ [-] $
$$Q=K_Q\cdot A\cdot\sqrt{2\cdot g\cdot H}$$
$$Q=Q_p\cdot Q_{max}$$
$ v\ [\mathrm{m/s}] $
| No data |
$ s/s_0\ [-] $
$$v=\cfrac{10^6\cdot Q}{s_0\cdot b}$$
$ v_j\ [\mathrm{m/s}] $
| No data |
$ s/s_0\ [-] $
$$v_j=0$$
$$v_j=\cfrac{Q}{K_Q\cdot A}$$
|
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}] $
$ H_v\ [\mathrm{m}] $
| No data |
$ s/s_0\ [-] $
$$H_L=\cfrac{v^2}{2\cdot g}\cdot ζ$$
$$H_v=H_L+\cfrac{v^2}{2\cdot g}+\left(1-Q_p\right)\cdot \left(ΔP-P_{u}\right)$$
$ σ\ [-] $
| No data |
$ s/s_0\ [-] $
$$σ=\cfrac{\cfrac{p_{air}-P_{SV}}{ρ\cdot g}+H-H_L}{H_v}$$
$ W\ [\mathrm{kN}] $
| No data |
$ s/s_0\ [-] $
$$W=\cfrac{ρ\cdot g\cdot H_v\cdot B\cdot s_s}{10^6}$$
$$W=\cfrac{ρ\cdot g\cdot H_v\cdot B\cdot\left(s_0-s\right)}{10^6}$$
|
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\ [-] $
$$P_1=\left(K_T-K_B\right)\cdot B\cdot d\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^9}$$
$ P_2\ [\mathrm{kN}] $
| No data |
$ s/s_0\ [-] $
$$P_2=K_T\cdot A_s\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^3}$$
$ P_3\ [\mathrm{kN}] $
| No data |
$ s/s_0\ [-] $
$$P_3=K_T\cdot B\cdot d^´\cdot ρ\cdot\cfrac{v_j^2}{2\cdot10^9}$$
$ P\ [\mathrm{kN}] $
| No data |
$ s/s_0\ [-] $
$$P=P_1+P_2+P_3$$
|
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\ [-] $
$$h_c=\cfrac{K_Q\cdot s_0}{10^3}$$
$ F_c\ [-] $
| No data |
$ s/s_0\ [-] $
$$F_c=0$$
$$F_c=\sqrt{\cfrac{2\cdot\left(H-h_c\right)}{h_c}}$$
$ β\ [-] $
| No data |
$ s/s_0\ [-] $
$$β=0$$
$$β=0.03\cdot\left(F_c-1\right)^{1.06}$$
|
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}\ [-] $
| No data |
$ s/s_0\ [-] $
$ p_{air}\ [\mathrm{Pa}] $
| No data |
$ s/s_0\ [-] $
$$p_{air}=-\min\left(p_{air}, f_{air}\cdot\cfrac{v^2}{2\cdot g}\cdot ρ+\left(1-Q_p\right)\cdot\min\left(\cfrac{L\cdot v_{max}\cdot ρ}{t\cdot c_{ef}}, p_{air}\right)\right)$$
$ Q_{air}\ [\mathrm{m^3/s}] $
| No data |
$ s/s_0\ [-] $
$$Q_{air}=\min\left(Q_{max}-Q, β\cdot Q\right)$$
$$Q_{air}=\max\left(Q_{max}-Q, β\cdot Q\right)$$
$ v_{air}\ [\mathrm{m/s}] $
| No data |
$ s/s_0\ [-] $
$$v_{air}=\min\left(0.7\cdot\sqrt{-\cfrac{2\cdot p_{air}}{ρ_{air}}}, 250\right)$$
$ A_{air}\ [\mathrm{m^2}] $
| No data |
$ s/s_0\ [-] $
$$A_{air}=0$$
$$A_{air}=\cfrac{Q_{air}}{v_{air}}$$
$ A_{air-pipe}\ [\mathrm{m^2}] $
| No data |
$ s/s_0\ [-] $
$$A_{air-pipe}=\cfrac{Q_{air}}{50}$$
$$A_{air-pipe}=A_{air}$$