Effective closing time factor

Reduced free flow area in the throttle control system:

$f_r=\cfrac{Q}{Q_{max}}$
where:
$f_r$ reduced free flow area in the throttle control system $\mathrm{-}$
$Q$ flow coefficient $\mathrm{-}$
$Q_{max}$ max. flow coefficient $\mathrm{-}$

Relative Flow:

$Q_p=\cfrac{f_r}{\sqrt{p+f_r^2(1-p)}}$
where:
$Q_p$ relative Flow $\mathrm{-}$
$f_r$ reduced free flow area in the throttle control system $\mathrm{-}$
$p$ pressure parameter $\mathrm{-}$

Pressure parameter:

$p=\cfrac{∆h}{h_0}$
where:
$p$ pressure parameter $\mathrm{-}$
$∆h$ theoretical pressure in the closure at full opening $\mathrm{m}$
$h_0$ rated net head $\mathrm{m}$

Valve speed:

$v_0=\cfrac{Q_0}{\cfrac{\pi\cdot D_0^2}{4}}$
where:
$v_0$ valve speed $\mathrm{m/s}$
$Q_0$ flow $\mathrm{m^3/s}$
$D_0$ valve diameter $\mathrm{mm}$

Effective closing time factor:

$$c_{ef}=\min\left[\lim_{n \to 90}{\cfrac{0,1}{[Q_{pn}-Q_{pn+10}]}}\right]>0,125$$
where:
$c_{ef}$ effective closing time factor $\mathrm{-}$
$Q_{pn}$ relative flow in position n $\mathrm{-}$

Flow characteristics:
The flow characteristic is the dependence of the actual flow rate on the position of the control actuator of the control system.
The flow characteristics of Fig. 1-7 are the dependence of the actual flow rate on the position of the valve of the control system.

Note: The flow characteristics (depending on the valve manufacturer) that are dependent on the position of the valve must be recalculated for dependence on the position of the actuator. Because before the actuator may be a member (crank mechanism, ...) that is not linear.

$K_Q\mathrm{[-]}$ -●-
$s/s_{max}\mathrm{[\%]}$
Fig. 1 - Flow characteristic Lattice disc Butterfly valve

$K_Q\mathrm{[-]}$ -●-
$s/s_{max}\mathrm{[\%]}$
Fig. 2 - Flow characteristic Spherical valve

$K_Q\mathrm{[-]}$ -●-
$s/s_{max}\mathrm{[\%]}$
Fig. 3 - Flow characteristic Knife gate valve

$K_Q\mathrm{[-]}$ -●-
$s/s_{max}\mathrm{[\%]}$
Fig. 4 - Flow characteristic Needle valve

$K_Q\mathrm{[-]}$ -●-
$s/s_{max}\mathrm{[\%]}$
Fig. 5 - Flow characteristic Hollow-Jet

$K_Q\mathrm{[-]}$ -●-
$s/s_{max}\mathrm{[\%]}$
Fig. 6 - Flow characteristic Howell-Bunger

$K_Q\mathrm{[-]}$ -●-
$s/s_{max}\mathrm{[\%]}$
Fig. 7 - Flow characteristic Tainter gate

Example:
We have to determine the difference between the effective shut-off time for the flow characteristic of the valve and the flow characteristic of the control actuator (hydraulic cylinder) according to Fig. 8 shut-off butterfly valve $DN300$ with the following parameters:
$h_0 = 33\ \mathrm{m}$; $Q_0 = 0,314\ \mathrm{m^3/s}$; $\xi = 0,106$

$s/s_{max}$ $Q_1$ $Q_2$
00,9510,951
100,8850,888
200,7330,755
300,5310,583
400,3730,433
500,2490,306
600,1830,218
700,1230,155
800,0730,102
900,0350,047
10000

$Q_1\mathrm{[-]}$ -●- $Q_2\mathrm{[-]}$ -●-
$s/s_{max}\mathrm{[\%]}$
Fig. 8 - Flow characteristic

where:
$Q_1$ flow characteristic of the valve $\mathrm{-}$
$Q_2$ flow characteristic of the control actuator $\mathrm{-}$

Valve speed:

$v_0=\cfrac{Q_0}{\cfrac{\pi\cdot D_0^2}{4}}=\cfrac{0,314}{\cfrac{\pi\cdot{0,3}^2}{4}}=4,44\ \mathrm{m/s}$

Theoretical pressure in the closure at full opening:

$∆h=\cfrac{v_0^2}{2g}\cdot\left(\xi+1\right)=\cfrac{{4,44}^2}{2\cdot9,807}\cdot\left(0,106+1\right)=1,11\ \mathrm{m}$

Pressure parameter:

$p=\cfrac{∆h}{h_0}=\cfrac{1,11}{33}=0,034$

$s/s_{max}$ $Q_{p1}$ $Q_{p2}$
011
100,9970,998
200,9890,99
300,9640,973
400,9180,941
500,8270,879
600,7290,787
700,5770,667
800,3850,505
900,1960,259
10000

$Q_{p1}\mathrm{[-]}$ -●- $Q_{p2}\mathrm{[-]}$ -●-
$s/s_{max}\mathrm{[\%]}$
Fig. 9 - Relative flow characteristic

where:
$Q_{p1}$ relative flow characteristic of the valve $\mathrm{-}$
$Q_{p2}$ relative flow characteristic of the control actuator $\mathrm{-}$
$c_{ef1}=0,511$
$c_{ef2}=0,386$
where:
$c_{ef1}$ effective closing time factor (valve) $\mathrm{-}$
$c_{ef2}$ effective closing time factor (control actuator) $\mathrm{-}$
$x=100-\cfrac{c_{ef2}}{c_{ef1}}\cdot100=100-\cfrac{0,386}{0,511}\cdot100=24,5\%$

Literature:
- F. Strohmer: Investigating the characteristics of shutoff valves by model tests. Water Power & Dam Construction 1977.
- By G. L. Beichley, M. ASCE and A. J. Peterka, F. ASCE: Hydraulic design of Hollow-Jet valve stilling basins. Journal of the hydraulics division 1961.
- Stanislav Kratochvil: Vodní nádrže a přehrady. 1961.
- Б. И. ЯНЬШИН: ЗАТВОРЫ И ПЕРЕХОДЫ ТРУБОПРОВОДОВ. 1962.