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List of symbols

Symbol Name of the symbol Unit
$ $model number for Howell-Bunger valve$\mathrm{ }$
$A_{air-pipe}$the flow area of the aerated pipeline$\mathrm{m^2}$
$A_{air}$the flow area of the aerated hole$\mathrm{m^2}$
$D$valve diameter$\mathrm{mm}$
$D_p$internal pipe diameter$\mathrm{mm}$
$D_s$max diameter of the disc seal$\mathrm{mm}$
$D_s$max diameter seal on the rotating body$\mathrm{mm}$
$D_{needle}$diameter needle$\mathrm{mm}$
$E$Young's modulus for pipe$\mathrm{Pa}$
$F$forces on disc$\mathrm{kN}$
$F$forces on rotating body$\mathrm{kN}$
$F_x$forces on disc in axis x$\mathrm{kN}$
$F_x$forces on rotating body in axis x$\mathrm{kN}$
$F_x$forces on the needle$\mathrm{kN}$
$F_x$forces on sliding plate in axis x$\mathrm{kN}$
$F_y$forces on disc in axis y$\mathrm{kN}$
$F_y$forces on rotating body in axis y$\mathrm{kN}$
$F_y$forces on sliding plate in axis y$\mathrm{kN}$
$F_{bx}$the force at the valve axis x$\mathrm{kN}$
$F_{by}$the force at the valve axis y$\mathrm{kN}$
$F_{e1}$force parallel to the axis of the disc$\mathrm{kN}$
$F_{e1}$force parallel to the axis of the rotating body$\mathrm{kN}$
$F_{e2}$force perpendicular to the axis of the disc$\mathrm{kN}$
$F_{e2}$force perpendicular to the axis of the rotating body$\mathrm{kN}$
$H$rated net head$\mathrm{m}$
$H_L$loss of pressure on the valve$\mathrm{m}$
$H_v$pressure on the valve$\mathrm{m}$
$K$volume elastic modulus$\mathrm{Pa}$
$K_Q$flow coefficient$\mathrm{ }$
$K_m$hydraulic torque coefficient$\mathrm{ }$
$K_x$coefficient of hydraulic force on a disc in the axis x$\mathrm{ }$
$K_x$coefficient of hydraulic force on a rotating body in the axis x$\mathrm{ }$
$K_x$Coefficient of hydraulic force on a needle in the axis x$\mathrm{ }$
$K_x$coefficient of hydraulic force on a sliding plate in the axis x$\mathrm{ }$
$K_y$coefficient of hydraulic force on a disc in the axis y$\mathrm{ }$
$K_y$coefficient of hydraulic force on a rotating body in the axis y$\mathrm{ }$
$K_y$coefficient of hydraulic force on a sliding plate in the axis y$\mathrm{ }$
$K_{Pu-air1}$under-pressure coefficient in hole 1$\mathrm{ }$
$K_{Pu-air2}$under-pressure coefficient in hole 2$\mathrm{ }$
$K_{Pu}$coefficient of under-pressure$\mathrm{ }$
$K_{Qmax}$max flow coefficient$\mathrm{ }$
$K_{bx}$coefficient of hydraulic force on body in the axis x$\mathrm{ }$
$K_{by}$coefficient of hydraulic force on body in the axis y$\mathrm{ }$
$K_{x-upstream}$Coefficient of hydraulic force on a needle upstream in the axis x$\mathrm{ }$
$L$pipe length$\mathrm{m}$
$L$pipe length behind valve$\mathrm{m}$
$L_1$lenght $ L_1 $ $\mathrm{m}$
$L_2$lenght $ L_2 $ $\mathrm{m}$
$L_D$length from the axis of rotation to the outer edge of the disc$\mathrm{mm}$
$M$hydraulic torque without eccentricity$\mathrm{kNm}$
$M_H$hydraulic torque$\mathrm{kNm}$
$M_e$moment from eccentricity$\mathrm{kNm}$
$M_{LD}$moment from the axis of the trunnion to the axis of the disc$\mathrm{kNm}$
$P_1$inlet absolute static pressure$\mathrm{Pa}$
$P_2$output absolute static pressure$\mathrm{Pa}$
$P_T$saturated vapor pressure$\mathrm{Pa}$
$P_{u-air1}$under-pressure in hole 1$\mathrm{m}$
$P_{u-air2}$under-pressure in hole 2$\mathrm{m}$
$P_{u}$under-pressure behind the valve$\mathrm{m}$
$Q$flow of water in the pipeline$\mathrm{m^3/s}$
$Q_p$relative flow$\mathrm{ }$
$Q_{air}$air flow$\mathrm{m^3/s}$
$Q_{max}$flow$\mathrm{m^3/s}$
$R$dimension $ R $ $\mathrm{mm}$
$S$dimension $ S $ $\mathrm{mm}$
$S/D$valve position$\mathrm{ }$
$T$the water temperature$\mathrm{°C}$
$T_x$energy before the valve$\mathrm{m}$
$a$speed pressure waves in the pipe$\mathrm{m/s}$
$a$dimension $ a $ $\mathrm{mm}$
$b$dimension $ b $ $\mathrm{mm}$
$c$dimension $ c $ $\mathrm{mm}$
$c_{ef}$effective closing time factor$\mathrm{ }$
$d$inner diameter$\mathrm{mm}$
$d_1$dimension $ d_1 $ $\mathrm{mm}$
$d_i$diameter of the hole$\mathrm{mm}$
$e$thickness of the pipe wall$\mathrm{mm}$
$e$eccentricity$\mathrm{mm}$
$e$dimension $ e $ $\mathrm{mm}$
$f$dimension $ f $ $\mathrm{mm}$
$f_r$reduced free flow area in the throttle control system$\mathrm{ }$
$f_{air}$coefficient of under-pressure of aerated hole$\mathrm{ }$
$g$gravitational acceleration$\mathrm{m/s^2}$
$g$dimension $ g $ $\mathrm{mm}$
$h$dimension $ h $ $\mathrm{mm}$
$h$valve axis position to sea level$\mathrm{m}$
$h_j$water depth behind the hydraulic jump$\mathrm{m}$
$h_l$water level$\mathrm{m}$
$i$dimension $ i $ $\mathrm{mm}$
$j$dimension $ j $ $\mathrm{mm}$
$k$dimension $ k $ $\mathrm{mm}$
$l$dimension $ l $ $\mathrm{mm}$
$m$dimension $ m $ $\mathrm{mm}$
$n$aeration$\mathrm{ }$
$n$dimension $ n $ $\mathrm{mm}$
$o$dimension $ o $ $\mathrm{mm}$
$p$pressure parameter$\mathrm{ }$
$p$dimension $ p $ $\mathrm{mm}$
$p_{air}$under-pressure in the aerated pipeline$\mathrm{Pa}$
$p_{a}$atmospheric pressure$\mathrm{Pa}$
$q$dimension $ q $ $\mathrm{mm}$
$r$dimension $ r $ $\mathrm{mm}$
$s$stroke from open position$\mathrm{\%}$
$t$closing time$\mathrm{s}$
$t$dimension $ t $ $\mathrm{mm}$
$t_{ef}$effective closing time$\mathrm{s}$
$u$dimension $ u $ $\mathrm{mm}$
$v$water velocity in pipeline$\mathrm{m/s}$
$v$dimension $ v $ $\mathrm{mm}$
$v_p$speed in the pipe$\mathrm{m/s}$
$v_{air1}$air velocity in hole 1$\mathrm{m/s}$
$v_{air2}$air velocity in hole 2$\mathrm{m/s}$
$v_{air}$air velocity$\mathrm{m/s}$
$v_{max}$velocity in valve$\mathrm{m/s}$
$w$speed of sound$\mathrm{m/s}$
$w$dimension $ w $ $\mathrm{mm}$
$x$dimension $ x $ $\mathrm{mm}$
$y$dimension $ y $ $\mathrm{mm}$
$z$dimension $ z $ $\mathrm{mm}$
$ΔP$water hammer$\mathrm{m}$
$Δ_h$theoretical pressure in the valve at full opening$\mathrm{m}$
$Σζ$loss before valve$\mathrm{ }$
$α$angle from open position$\mathrm{°}$
$ζ$valve loss coefficient$\mathrm{ }$
$ζ$loss coefficient$\mathrm{ }$
$μ_i$output coefficient$\mathrm{ }$
$ρ$density$\mathrm{kg/m^3}$
$ρ_{air}$air density$\mathrm{kg/m^3}$
$σ$cavitation number$\mathrm{ }$
$ϕ$angle between pipe axis and hydraulic force$\mathrm{°}$