Hydrodynamic calculation Needle valve
Values for calculation
Calculation
Velocity in valve
$$v_{max}=\cfrac{4\cdot 10^6\cdot Q_{max}}{π\cdot D^2}$$
Theoretical pressure in the valve at full opening
$$Δ_h=\cfrac{v_{max}^2}{2\cdot g}\cdot \left({\min\left(ζ\right)}+1\right)$$
Pressure parameter
$$p=\cfrac{Δ_h}{H}$$
$$0 < p \le 1$$
Effective closing time factor
$$c_{ef}=0.1/\max_{i=1}^{10}{\left(Q_p[i]-Q_p[i+1]\right)}$$
$$c_{ef}\le 1$$
Under-pressure behind the valve
$$P_{u}=\max\left(-\cfrac{L\cdot v_{max}}{g\cdot t\cdot c_{ef}}, -\cfrac{p_{air}}{ρ\cdot g}\right)$$
$ [-] $
|
Stroke from open position |
First stage of cavitation |
Second stage of cavitation |
Fully developed cavitation |
|---|---|---|---|
| $ s $ | $ σ_1 $ | $ σ_2 $ | $ σ_{min} $ |
| $ \mathrm{\%} $ | $ \mathrm{\ } $ | $ \mathrm{\ } $ | $ \mathrm{\ } $ |
| No data | |||
$ σ_1\ [-] $
$ σ_2\ [-] $
$ σ_{min}\ [-] $
$ σ_2\ [-] $
$ σ_{min}\ [-] $
| No data |
$ s\ [\mathrm{\%}] $
|
Stroke from open position |
Flow coefficient for $ σ_1 $ |
Flow coefficient for $ σ_2 $ |
Flow coefficient for $ σ_{min} $ |
|---|---|---|---|
| $ s $ | $ K_{Q-σ_1} $ | $ K_{Q-σ_2} $ | $ K_{Q-σ_{min}} $ |
| $ \mathrm{\%} $ | $ \mathrm{\ } $ | $ \mathrm{\ } $ | $ \mathrm{\ } $ |
| No data | |||
$ K_{Q-σ_1}\ [-] $
$ K_{Q-σ_2}\ [-] $
$ K_{Q-σ_{min}}\ [-] $
$ K_{Q-σ_2}\ [-] $
$ K_{Q-σ_{min}}\ [-] $
| No data |
$ s\ [\mathrm{\%}] $
|
Stroke from open position |
Coefficient of hydraulic force on a needle in the axis x for $ σ_1 $ |
Coefficient of hydraulic force on a needle in the axis x for $ σ_2 $ |
Coefficient of hydraulic force on a needle in the axis x for $ σ_{min} $ |
|---|---|---|---|
| $ s $ | $ K_{x-σ_1} $ | $ K_{x-σ_2} $ | $ K_{x-σ_{min}} $ |
| $ \mathrm{\%} $ | $ \mathrm{\ } $ | $ \mathrm{\ } $ | $ \mathrm{\ } $ |
| No data | |||
$ K_{x-σ_1}\ [-] $
$ K_{x-σ_2}\ [-] $
$ K_{x-σ_{min}}\ [-] $
$ K_{x-σ_2}\ [-] $
$ K_{x-σ_{min}}\ [-] $
| No data |
$ s\ [\mathrm{\%}] $
|
Stroke from open position |
Flow coefficient |
Coefficient of hydraulic force on a needle in the axis x |
|---|---|---|
| $ s $ | $ K_Q $ | $ K_x $ |
| $ \mathrm{\%} $ | $ \mathrm{\ } $ | $ \mathrm{\ } $ |
| No data | ||
$ K_Q\ [-] $
| No data |
$ s\ [\mathrm{\%}] $
$$K_Q=\left\{0.680, 0.660, 0.630, 0.590, 0.530, 0.440, 0.332, 0.232, 0.144, 0.073, 1E-100\right\}$$
$$K_Q=\left\{0.660, 0.640, 0.600, 0.536, 0.450, 0.372, 0.289, 0.200, 0.128, 0.069, 1E-100\right\}$$
$$K_Q=\left\{0.650, 0.635, 0.595, 0.528, 0.450, 0.364, 0.272, 0.195, 0.128, 0.069, 1E-100\right\}$$
$$K_Q=K_{Q-σ_1}$$
$$K_Q=K_{Q-σ_2}$$
$$K_Q=K_{Q-σ_{min}}$$
$ K_x\ [-] $
| No data |
$ s\ [\mathrm{\%}] $
$$K_x=\left\{-0.020, -0.040, -0.080, -0.160, -0.263, -0.215, -0.090, -0.020, -0.079, -0.006, 0.000\right\}$$
$$K_x=\left\{-0.020, -0.030, -0.050, -0.108, -0.079, -0.090, -0.090, -0.072, -0.068, -0.052, 0.000\right\}$$
$$K_x=\left\{-0.020, -0.027, -0.026, -0.022, -0.030, -0.031, -0.025, -0.028, -0.030, -0.013, 0.000\right\}$$
$$K_x=K_{x-σ_1}$$
$$K_x=K_{x-σ_2}$$
$$K_x=K_{x-σ_{min}}$$
|
Stroke from open position |
Loss coefficient |
Reduced free flow area in the throttle control system |
Relative flow |
Flow of water in the pipeline |
Water velocity in pipeline |
|---|---|---|---|---|---|
| $ s $ | $ ζ $ | $ f_r $ | $ Q_p $ | $ Q $ | $ v $ |
| $ \mathrm{\%} $ | $ \mathrm{\ } $ | $ \mathrm{\ } $ | $ \mathrm{\ } $ | $ \mathrm{m^3/s} $ | $ \mathrm{m/s} $ |
| No data | |||||
$ ζ\ [-] $
| No data |
$ s\ [\mathrm{\%}] $
$$ζ=\cfrac{1-K_Q^2}{K_Q^2}$$
$ f_r\ [-] $
$ Q_p\ [-] $
$ Q_p\ [-] $
| No data |
$ s\ [\mathrm{\%}] $
$$f_r=\cfrac{K_Q}{K_{Qmax}}$$
$$Q_p=\cfrac{f_r}{\sqrt{p+f_r^2\cdot \left(1-p\right)}}$$
$ Q\ [\mathrm{m^3/s}] $
$ v\ [\mathrm{m/s}] $
$ v\ [\mathrm{m/s}] $
| No data |
$ s\ [\mathrm{\%}] $
$$Q=Q_p\cdot Q_{max}$$
$$v=Q_p\cdot v_{max}$$
|
Stroke from open position |
Loss of pressure on the valve |
Pressure on the valve |
Cavitation number |
Forces on the needle |
|---|---|---|---|---|
| $ s $ | $ H_L $ | $ H_v $ | $ σ $ | $ F_x $ |
| $ \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\ [\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\ [-] $
| No data |
$ s\ [\mathrm{\%}] $
$$σ=\cfrac{\cfrac{p_{air}-P_{SV}}{ρ\cdot g}+H-H_L}{H_v}$$
$ F_x\ [\mathrm{kN}] $
| No data |
$ s\ [\mathrm{\%}] $
$$F_x=\cfrac{π\cdot D}{4\cdot 10^9}\cdot ρ\cdot g\cdot H_v\cdot K_x$$
|
Stroke from open position |
Aerated coefficient |
Coefficient of under-pressure of aerated hole |
Under-pressure in the aerated pipeline |
Air flow |
|---|---|---|---|---|
| $ s $ | $ β $ | $ f_{air} $ | $ p_{air} $ | $ Q_{air} $ |
| $ \mathrm{\%} $ | $ \mathrm{\ } $ | $ \mathrm{\ } $ | $ \mathrm{Pa} $ | $ \mathrm{m^3/s} $ |
| No data | ||||
$ β\ [-] $
| No data |
$ s\ [\mathrm{\%}] $
$ f_{air}\ [-] $
| No data |
$ s\ [\mathrm{\%}] $
$ p_{air}\ [\mathrm{Pa}] $
| No data |
$ s\ [\mathrm{\%}] $
$$p_{air}=NAN$$
$$p_{air}=-\min\left(p_{air}, f_{air}\cdot\cfrac{v^2}{2\cdot g}\cdot ρ+\left(1-Q_p\right)\cdot\min\left(\cfrac{L\cdot v_{max}\cdot ρ}{t\cdot c_{ef}}, p_{air}\right)\right)$$
$ Q_{air}\ [\mathrm{m^3/s}] $
| No data |
$ s\ [\mathrm{\%}] $
$$Q_{air}=NAN$$
$$Q_{air}=\min\left(Q_{max}-Q, β\cdot Q\right)$$
$$Q_{air}=\max\left(Q_{max}-Q, β\cdot Q\right)$$
|
Stroke from open position |
Air velocity |
The flow area of the aerated hole |
The flow area of the aerated pipeline |
|---|---|---|---|
| $ s $ | $ v_{air} $ | $ A_{air} $ | $ A_{air-pipe} $ |
| $ \mathrm{\%} $ | $ \mathrm{m/s} $ | $ \mathrm{m^2} $ | $ \mathrm{m^2} $ |
| No data | |||
$ v_{air}\ [\mathrm{m/s}] $
| No data |
$ s\ [\mathrm{\%}] $
$$v_{air}=NAN$$
$$v_{air}=\min\left(0.7\cdot\sqrt{-\cfrac{2\cdot p_{air}}{ρ_{air}}}, 250\right)$$
$ A_{air}\ [\mathrm{m^2}] $
| No data |
$ s\ [\mathrm{\%}] $
$$A_{air}=NAN$$
$$A_{air}=0$$
$$A_{air}=\cfrac{Q_{air}}{v_{air}}$$
$ A_{air-pipe}\ [\mathrm{m^2}] $
| No data |
$ s\ [\mathrm{\%}] $
$$A_{air-pipe}=NAN$$
$$A_{air-pipe}=\cfrac{Q_{air}}{50}$$
$$A_{air-pipe}=A_{air}$$