| $ $ | model number for Howell-Bunger valve | $\mathrm{ }$ |
| $ $ | flow characteristic | $\mathrm{ }$ |
| $A$ | the flow area | $\mathrm{m^2}$ |
| $A_{air-pipe}$ | 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 | $\mathrm{mm}$ |
| $D_c$ | inner diameter of the cage | $\mathrm{mm}$ |
| $D_p$ | 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 | $\mathrm{kN}$ |
| $H$ | static head | $\mathrm{m}$ |
| $H_L$ | loss of pressure on the valve | $\mathrm{m}$ |
| $H_v$ | pressure on the valve | $\mathrm{m}$ |
| $I$ | number of holes | $\mathrm{ }$ |
| $K$ | 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 | $\mathrm{ }$ |
| $K_{Q-hydraulic-cylinder}$ | flow coefficient $ K_{Q-hydraulic-cylinder} $ | $\mathrm{ }$ |
| $K_{Q-valve[0]}$ | flow coefficient $ K_{Q-valve[0]} $ | $\mathrm{ }$ |
| $K_{Q-valve[10]}$ | flow coefficient $ K_{Q-valve[10]} $ | $\mathrm{ }$ |
| $K_{Q-valve[20]}$ | flow coefficient $ K_{Q-valve[20]} $ | $\mathrm{ }$ |
| $K_{Q-valve[30]}$ | flow coefficient $ K_{Q-valve[30]} $ | $\mathrm{ }$ |
| $K_{Q-valve[40]}$ | flow coefficient $ K_{Q-valve[40]} $ | $\mathrm{ }$ |
| $K_{Q-valve[50]}$ | flow coefficient $ K_{Q-valve[50]} $ | $\mathrm{ }$ |
| $K_{Q-valve[60]}$ | flow coefficient $ K_{Q-valve[60]} $ | $\mathrm{ }$ |
| $K_{Q-valve[70]}$ | flow coefficient $ K_{Q-valve[70]} $ | $\mathrm{ }$ |
| $K_{Q-valve[80]}$ | flow coefficient $ K_{Q-valve[80]} $ | $\mathrm{ }$ |
| $K_{Q-valve[90]}$ | flow coefficient $ K_{Q-valve[90]} $ | $\mathrm{ }$ |
| $K_{Q-valve[s+10]}$ | flow coefficient $ K_{Q-valve[s+10]} $ | $\mathrm{ }$ |
| $K_{Q-valve[s-10]}$ | flow coefficient $ K_{Q-valve[s-10]} $ | $\mathrm{ }$ |
| $K_{Q-valve}$ | flow coefficient $ K_{Q-valve} $ | $\mathrm{ }$ |
| $K_{Qmax}$ | 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 | $\mathrm{m}$ |
| $L$ | cage length | $\mathrm{mm}$ |
| $L$ | distance between the axis of rotation of the valve and the axis of the hydraulic cylinder | $\mathrm{mm}$ |
| $L$ | 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 | $\mathrm{mm}$ |
| $L_d$ | damping phase | $\mathrm{\%}$ |
| $M$ | 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 | $\mathrm{ }$ |
| $S_S$ | stroke in the pivot position | $\mathrm{mm}$ |
| $S_S$ | the distance between the axis of the hydraulic cylinder and the axis of the eye of the hydraulic cylinder | $\mathrm{mm}$ |
| $S_T$ | percentage of hydraulic cylinder stroke at a given time | $\mathrm{\%}$ |
| $S_{S\%}$ | stroke percentage | $\mathrm{\%}$ |
| $S_{max}$ | stroke | $\mathrm{mm}$ |
| $T$ | height $ T $ | $\mathrm{mm}$ |
| $T$ | the water temperature | $\mathrm{°C}$ |
| $T_c$ | total closing time | $\mathrm{s}$ |
| $T_d$ | damping time | $\mathrm{s}$ |
| $T_s$ | time value | $\mathrm{s}$ |
| $T_x$ | energy before the valve | $\mathrm{m}$ |
| $a$ | dimension $ a $ | $\mathrm{mm}$ |
| $a$ | speed pressure waves in the pipe | $\mathrm{m/s}$ |
| $a$ | distance to hydraulic cylinder $ a $ | $\mathrm{mm}$ |
| $a_1$ | length $ a_1 $ | $\mathrm{mm}$ |
| $a_2$ | length $ a_2 $ | $\mathrm{mm}$ |
| $b$ | dimension $ b $ | $\mathrm{mm}$ |
| $b$ | distance to hydraulic cylinder $ b $ | $\mathrm{mm}$ |
| $c$ | dimension $ c $ | $\mathrm{mm}$ |
| $c_{ef}$ | effective closing time factor | $\mathrm{ }$ |
| $d$ | diameter of the hole | $\mathrm{mm}$ |
| $d$ | inner diameter | $\mathrm{mm}$ |
| $d_1$ | dimension $ d_1 $ | $\mathrm{mm}$ |
| $e$ | thickness of the pipe wall | $\mathrm{mm}$ |
| $e$ | dimension $ e $ | $\mathrm{mm}$ |
| $e$ | eccentricity | $\mathrm{mm}$ |
| $f$ | 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 $ | $\mathrm{mm}$ |
| $l$ | rod length | $\mathrm{mm}$ |
| $m$ | dimension $ m $ | $\mathrm{mm}$ |
| $n$ | aeration | $\mathrm{ }$ |
| $n$ | dimension $ n $ | $\mathrm{mm}$ |
| $n_{max}$ | allowable maximum number of holes in one row | $\mathrm{ }$ |
| $o$ | 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 $ | $\mathrm{mm}$ |
| $r$ | lever arm length | $\mathrm{mm}$ |
| $s$ | stroke from open position | $\mathrm{\%}$ |
| $s$ | dimension $ s $ | $\mathrm{mm}$ |
| $t$ | 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 | $\mathrm{°}$ |
| $α$ | lever angle in closed position | $\mathrm{°}$ |
| $β$ | aerated coefficient | $\mathrm{ }$ |
| $β$ | swing angle | $\mathrm{°}$ |
| $β_S$ | angle rotation of the rocking motion | $\mathrm{°}$ |
| $β_{S[s-10]}$ | angle rotation of the rocking motion | $\mathrm{°}$ |
| $ζ$ | loss coefficient | $\mathrm{ }$ |
| $ζ$ | valve loss coefficient | $\mathrm{ }$ |
| $μ$ | 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 | $\mathrm{°}$ |
| $𝛾$ | the angle between the axis of the hydraulic cylinder and the imaginary line between the axis of the closure and the pivot axis of the hydraulic cylinder | $\mathrm{°}$ |
| $𝛿$ | the angle between the lever axis and the imaginary line between the valve axis and the pivot axis of the hydraulic cylinder | $\mathrm{°}$ |
| $𝛿_{[0]}$ | the angle between the axis of the lever and the imaginary line between the axis of the valve and the pivot axis of the hydraulic cylinder in the open position | $\mathrm{°}$ |
