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List of symbols

Symbol Name of the symbol Unit
$ $ model number for Howell-Bunger valve $\mathrm{ }$
$ $ flow characteristic $\mathrm{ }$
$ $ data $\mathrm{ }$
$A$ the flow area $\mathrm{m^2}$
$A$ function $ A $ $\mathrm{ }$
$A_{air-pipe}$ the flow area of the aerated pipeline $\mathrm{m^2}$
$A_{air}$ the flow area of the aerated hole $\mathrm{m^2}$
$B$ function $ B $ $\mathrm{ }$
$C$ function $ C $ $\mathrm{ }$
$D$ 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 $\mathrm{mm}$
$D_{HC}$ diameter of piston in the hydraulic cylinder $\mathrm{mm}$
$D_{needle}$ 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 $\mathrm{kN}$
$F_{HC-flow-with-frictional}$ force to the hydraulic cylinder during the closing at flow rate with frictional resistances $\mathrm{kN}$
$F_{HC-flow}$ force to the hydraulic cylinder during the closing at flow rate without frictional resistances $\mathrm{kN}$
$F_{HC-with-frictional}$ force to the hydraulic cylinder during the closing without flow rate with frictional resistances $\mathrm{kN}$
$F_{HC}$ force to the hydraulic cylinder during the closing without flow rate without frictional resistances $\mathrm{kN}$
$F_{bx}$ 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 $\mathrm{m}$
$H$ geopotential altitude $\mathrm{m}$
$H_L$ loss of pressure on the valve $\mathrm{m}$
$H_b$ lower limit geopotential altitude $\mathrm{m}$
$H_p$ pressure scale height $\mathrm{m}$
$H_v$ pressure on the valve $\mathrm{m}$
$I$ number of holes $\mathrm{ }$
$I$ exponent $ I $ $\mathrm{ }$
$I_{total}$ total number of holes $\mathrm{ }$
$J$ exponent $ J $ $\mathrm{ }$
$J^o$ exponent $ J^o $ $\mathrm{ }$
$K$ 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 $\mathrm{ }$
$K_{Pu-0}$ coefficient of under-pressure for $ \cfrac{h_l}{D}=0 $ $\mathrm{ }$
$K_{Pu-2.5}$ coefficient of under-pressure for $ \cfrac{h_l}{D}=2.5 $ $\mathrm{ }$
$K_{Pu-2}$ coefficient of under-pressure for $ \cfrac{h_l}{D}=2 $ $\mathrm{ }$
$K_{Pu-3.5}$ coefficient of under-pressure for $ \cfrac{h_l}{D}=3.5 $ $\mathrm{ }$
$K_{Pu-3.6}$ coefficient of under-pressure for $ \cfrac{h_l}{D}=3.6 $ $\mathrm{ }$
$K_{Pu-3.9}$ coefficient of under-pressure for $ \cfrac{h_l}{D}=3.9 $ $\mathrm{ }$
$K_{Pu-3}$ coefficient of under-pressure for $ \cfrac{h_l}{D}=3 $ $\mathrm{ }$
$K_{Pu-4.2}$ coefficient of under-pressure for $ \cfrac{h_l}{D}=4.2 $ $\mathrm{ }$
$K_{Pu-4.5}$ coefficient of under-pressure for $ \cfrac{h_l}{D}=4.5 $ $\mathrm{ }$
$K_{Pu-4}$ coefficient of under-pressure for $ \cfrac{h_l}{D}=4 $ $\mathrm{ }$
$K_{Pu-5}$ coefficient of under-pressure for $ \cfrac{h_l}{D}=5 $ $\mathrm{ }$
$K_{Pu-air1-0}$ under-pressure coefficient in hole 1 for $ \cfrac{h_l}{D}=0 $ $\mathrm{ }$
$K_{Pu-air1-2.5}$ under-pressure coefficient in hole 1 for $ \cfrac{h_l}{D}=2.5 $ $\mathrm{ }$
$K_{Pu-air1-2}$ under-pressure coefficient in hole 1 for $ \cfrac{h_l}{D}=2 $ $\mathrm{ }$
$K_{Pu-air1-3.5}$ under-pressure coefficient in hole 1 for $ \cfrac{h_l}{D}=3.5 $ $\mathrm{ }$
$K_{Pu-air1-3.6}$ under-pressure coefficient in hole 1 for $ \cfrac{h_l}{D}=3.6 $ $\mathrm{ }$
$K_{Pu-air1-3.9}$ under-pressure coefficient in hole 1 for $ \cfrac{h_l}{D}=3.9 $ $\mathrm{ }$
$K_{Pu-air1-3}$ under-pressure coefficient in hole 1 for $ \cfrac{h_l}{D}=3 $ $\mathrm{ }$
$K_{Pu-air1-4.2}$ under-pressure coefficient in hole 1 for $ \cfrac{h_l}{D}=4.2 $ $\mathrm{ }$
$K_{Pu-air1-4.5}$ under-pressure coefficient in hole 1 for $ \cfrac{h_l}{D}=4.5 $ $\mathrm{ }$
$K_{Pu-air1-4}$ under-pressure coefficient in hole 1 for $ \cfrac{h_l}{D}=4 $ $\mathrm{ }$
$K_{Pu-air1-5}$ under-pressure coefficient in hole 1 for $ \cfrac{h_l}{D}=5 $ $\mathrm{ }$
$K_{Pu-air1}$ under-pressure coefficient in hole 1 $\mathrm{ }$
$K_{Pu-air2-0}$ under-pressure coefficient in hole 2 for $ \cfrac{h_l}{D}=0 $ $\mathrm{ }$
$K_{Pu-air2-2.5}$ under-pressure coefficient in hole 2 for $ \cfrac{h_l}{D}=2.5 $ $\mathrm{ }$
$K_{Pu-air2-2}$ under-pressure coefficient in hole 2 for $ \cfrac{h_l}{D}=2 $ $\mathrm{ }$
$K_{Pu-air2-3.5}$ under-pressure coefficient in hole 2 for $ \cfrac{h_l}{D}=3.5 $ $\mathrm{ }$
$K_{Pu-air2-3.6}$ under-pressure coefficient in hole 2 for $ \cfrac{h_l}{D}=3.6 $ $\mathrm{ }$
$K_{Pu-air2-3.9}$ under-pressure coefficient in hole 2 for $ \cfrac{h_l}{D}=3.9 $ $\mathrm{ }$
$K_{Pu-air2-3}$ under-pressure coefficient in hole 2 for $ \cfrac{h_l}{D}=3 $ $\mathrm{ }$
$K_{Pu-air2-4.2}$ under-pressure coefficient in hole 2 for $ \cfrac{h_l}{D}=4.2 $ $\mathrm{ }$
$K_{Pu-air2-4.5}$ under-pressure coefficient in hole 2 for $ \cfrac{h_l}{D}=4.5 $ $\mathrm{ }$
$K_{Pu-air2-4}$ under-pressure coefficient in hole 2 for $ \cfrac{h_l}{D}=4 $ $\mathrm{ }$
$K_{Pu-air2-5}$ under-pressure coefficient in hole 2 for $ \cfrac{h_l}{D}=5 $ $\mathrm{ }$
$K_{Pu-air2}$ under-pressure coefficient in hole 2 $\mathrm{ }$
$K_{Pu}$ coefficient of under-pressure $\mathrm{ }$
$K_{Q-hydraulic-cylinder}$ flow coefficient $ K_{Q-hydraulic-cylinder} $ $\mathrm{ }$
$K_{Q-valve}$ flow coefficient $ K_{Q-valve} $ $\mathrm{ }$
$K_{Q-valve}[10]$ flow coefficient $ K_{Q-valve}[10] $ $\mathrm{ }$
$K_{Q-valve}[1]$ flow coefficient $ K_{Q-valve}[1] $ $\mathrm{ }$
$K_{Q-valve}[2]$ flow coefficient $ K_{Q-valve}[2] $ $\mathrm{ }$
$K_{Q-valve}[3]$ flow coefficient $ K_{Q-valve}[3] $ $\mathrm{ }$
$K_{Q-valve}[4]$ flow coefficient $ K_{Q-valve}[4] $ $\mathrm{ }$
$K_{Q-valve}[5]$ flow coefficient $ K_{Q-valve}[5] $ $\mathrm{ }$
$K_{Q-valve}[6]$ flow coefficient $ K_{Q-valve}[6] $ $\mathrm{ }$
$K_{Q-valve}[7]$ flow coefficient $ K_{Q-valve}[7] $ $\mathrm{ }$
$K_{Q-valve}[8]$ flow coefficient $ K_{Q-valve}[8] $ $\mathrm{ }$
$K_{Q-valve}[9]$ flow coefficient $ K_{Q-valve}[9] $ $\mathrm{ }$
$K_{Q-σ_1}$ flow coefficient for $ σ_1 $ $\mathrm{ }$
$K_{Q-σ_2}$ flow coefficient for $ σ_2 $ $\mathrm{ }$
$K_{Q-σ_{min}}$ flow coefficient for $ σ_{min} $ $\mathrm{ }$
$K_{Qmax}$ 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 $\mathrm{ }$
$K_{x-σ_1}$ coefficient of hydraulic force on a needle in the axis x for $ σ_1 $ $\mathrm{ }$
$K_{x-σ_2}$ coefficient of hydraulic force on a needle in the axis x for $ σ_2 $ $\mathrm{ }$
$K_{x-σ_{min}}$ coefficient of hydraulic force on a needle in the axis x for $ σ_{min} $ $\mathrm{ }$
$L$ 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 $\mathrm{mm}$
$L_c$ distance between the centre of gravity of the weight (weight + disc + lever) and the axis of pivot rotation $\mathrm{mm}$
$L_c$ distance between the centre of gravity of the weight (weight + rotating body + lever) and the axis of pivot rotation $\mathrm{mm}$
$L_d$ damping phase $\mathrm{\%}$
$L_{1-10}$ coefficient of lenght $ L_1 $ for $ \cfrac{T_x}{D}=10 $ $\mathrm{ }$
$L_{1-20}$ coefficient of lenght $ L_1 $ for $ \cfrac{T_x}{D}=20 $ $\mathrm{ }$
$L_{1-30}$ coefficient of lenght $ L_1 $ for $ \cfrac{T_x}{D}=30 $ $\mathrm{ }$
$L_{1-40}$ coefficient of lenght $ L_1 $ for $ \cfrac{T_x}{D}=40 $ $\mathrm{ }$
$L_{2-10}$ coefficient of lenght $ L_2 $ for $ \cfrac{T_x}{D}=10 $ $\mathrm{ }$
$L_{2-20}$ coefficient of lenght $ L_2 $ for $ \cfrac{T_x}{D}=20 $ $\mathrm{ }$
$L_{2-30}$ coefficient of lenght $ L_2 $ for $ \cfrac{T_x}{D}=30 $ $\mathrm{ }$
$L_{2-40}$ coefficient of lenght $ L_2 $ for $ \cfrac{T_x}{D}=40 $ $\mathrm{ }$
$L_{2-50}$ coefficient of lenght $ L_2 $ for $ \cfrac{T_x}{D}=50 $ $\mathrm{ }$
$L_{HC}$ distance between the axis of hydraulic cylinder and the axis of valve rotation $\mathrm{mm}$
$M$ hydraulic torque without eccentricity $\mathrm{kNm}$
$M$ air molar mass at sea level $\mathrm{kg\cdot kmol^{-1}}$
$M_B$ friction torque bearing during the closing without flow rate $\mathrm{kNm}$
$M_H$ hydraulic torque $\mathrm{kNm}$
$M_S$ friction torque in main sealing $\mathrm{kNm}$
$M_W$ static torque $\mathrm{kNm}$
$M_e$ moment from eccentricity $\mathrm{kNm}$
$M_{B-flow}$ friction torque bearing during the closing at flow rate $\mathrm{kNm}$
$M_{F-HC}$ friction torque from the hydraulic cylinder $\mathrm{kNm}$
$M_{LD}$ moment from the axis of the trunnion to the axis of the disc $\mathrm{kNm}$
$N_A$ Avogadro constant $\mathrm{kmol^{-1}}$
$P_1$ inlet absolute static pressure $\mathrm{Pa}$
$P_2$ output absolute static pressure $\mathrm{Pa}$
$P_S$ contact pressure of the main sealing $\mathrm{MPa}$
$P_{HC-flow-with-frictional}$ pressure of oil under the hydraulic cylinder piston during the closing at flow rate with frictional resistances $\mathrm{MPa}$
$P_{HC-flow}$ pressure of oil under the hydraulic cylinder piston during the closing at flow rate without frictional resistances $\mathrm{MPa}$
$P_{HC-with-frictional}$ pressure of oil under the hydraulic cylinder piston during the closing without flow rate with frictional resistances $\mathrm{MPa}$
$P_{HC}$ pressure of oil under the hydraulic cylinder piston during the closing without flow rate without frictional resistances $\mathrm{MPa}$
$P_{L-HC}$ pressure loss in the hydraulic cylinder $\mathrm{MPa}$
$P_{SV}$ saturated vapor pressure $\mathrm{Pa}$
$P_{max-stages}$ maximum pressures between stages $\mathrm{Pa}$
$P_{u-air1}$ 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 $ $\mathrm{mm}$
$R$ specific gas constant $\mathrm{J\cdot K^{-1}\cdot kg^{-1}}$
$R$ specific gas constant of ordinary water $\mathrm{J\cdot kg^{-1}\cdot K^{-1}}$
$R^*$ universal gas constant $\mathrm{J\cdot K^{-1}\cdot kmol^{-1}}$
$S$ dimension $ S $ $\mathrm{mm}$
$S$ stage $\mathrm{ }$
$S$ Sutherland's empirical coefficients $ S $ $\mathrm{K}$
$S/D$ 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 $\mathrm{\%}$
$S_f$ safety factor during the closing without flow rate $\mathrm{ }$
$S_{S\%}$ stroke percentage $\mathrm{\%}$
$S_{f-flow}$ safety factor during the closing at flow rate $\mathrm{ }$
$S_{max}$ stroke $\mathrm{mm}$
$T$ height $ T $ $\mathrm{mm}$
$T$ temperature $ T $ $\mathrm{K}$
$T$ the water temperature $\mathrm{°C}$
$T^*$ temperature reducing quantity $\mathrm{K}$
$T_b$ lower limit temperature $\mathrm{K}$
$T_c$ total closing time $\mathrm{s}$
$T_d$ damping time $\mathrm{s}$
$T_s$ time value $\mathrm{s}$
$T_x$ energy before the valve $\mathrm{m}$
$W_D$ weight of the disc $\mathrm{kg}$
$W_L$ weight of the lever $\mathrm{kg}$
$W_R$ weight of the rotating body $\mathrm{kg}$
$W_W$ weight of the weight $\mathrm{kg}$
$\text{Region}$ region $\mathrm{ }$
$a$ dimension $ a $ $\mathrm{mm}$
$a$ speed pressure waves in the pipe $\mathrm{m/s}$
$a$ distance to hydraulic cylinder $ a $ $\mathrm{mm}$
$a$ speed of Sound $\mathrm{m/s}$
$a_1$ 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 $ $\mathrm{mm}$
$c_p$ specific isobaric heat capacity $\mathrm{J\cdot kg^{-1}\cdot K^{-1}}$
$c_{ef}$ effective closing time factor $\mathrm{ }$
$c_ν$ specific isochoric heat capacity $\mathrm{J\cdot kg^{-1}\cdot K^{-1}}$
$d$ 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 $\mathrm{mm}$
$e_S$ width of main seal in contact $\mathrm{mm}$
$f$ dimension $ f $ $\mathrm{mm}$
$f_B$ the bearing factor of the bushings, defined as the sum of the forces in the bushings divided by the load force $\mathrm{ }$
$f_r$ reduced free flow area in the throttle control system $\mathrm{ }$
$f_{air}$ coefficient of under-pressure of aerated hole $\mathrm{ }$
$g$ dimension $ g $ $\mathrm{mm}$
$g$ gravitational acceleration $\mathrm{m/s^2}$
$h$ dimension $ h $ $\mathrm{mm}$
$h$ specific enthalpy $\mathrm{J\cdot kg^{-1}}$
$h$ height above sea level $\mathrm{m}$
$h_j$ water depth behind the hydraulic jump $\mathrm{m}$
$h_l$ water level $\mathrm{m}$
$h_{l-10}$ coefficient of water level for $ \cfrac{T_x}{D}=10 $ $\mathrm{ }$
$h_{l-20}$ coefficient of water level for $ \cfrac{T_x}{D}=20 $ $\mathrm{ }$
$h_{l-30}$ coefficient of water level for $ \cfrac{T_x}{D}=30 $ $\mathrm{ }$
$h_{l-40}$ coefficient of water level for $ \cfrac{T_x}{D}=40 $ $\mathrm{ }$
$h_{l-50}$ coefficient of water level for $ \cfrac{T_x}{D}=50 $ $\mathrm{ }$
$i$ dimension $ i $ $\mathrm{mm}$
$i$ number of rows $\mathrm{ }$
$j$ dimension $ j $ $\mathrm{mm}$
$k$ dimension $ k $ $\mathrm{mm}$
$l$ dimension $ l $ $\mathrm{mm}$
$l$ rod length $\mathrm{mm}$
$l$ mean free path of air particles $\mathrm{m}$
$m$ dimension $ m $ $\mathrm{mm}$
$n$ aeration $\mathrm{ }$
$n$ dimension $ n $ $\mathrm{mm}$
$n$ air number density $\mathrm{m^{-3}}$
$n$ coefficient $ n $ $\mathrm{ }$
$n^o$ coefficient $ n^o $ $\mathrm{ }$
$n_{max}$ allowable maximum number of holes in one row $\mathrm{ }$
$n_{s}$ number of stages $\mathrm{ }$
$o$ dimension $ o $ $\mathrm{mm}$
$p$ pressure parameter $\mathrm{ }$
$p$ dimension $ p $ $\mathrm{mm}$
$p$ the water pressure $\mathrm{Pa}$
$p^*$ pressure reducing quantity $\mathrm{Pa}$
$p_b$ lower limit pressure $\mathrm{Pa}$
$p_{air}$ under-pressure in the aerated pipeline $\mathrm{Pa}$
$p_{air}$ atmospheric pressure air $\mathrm{Pa}$
$q$ dimension $ q $ $\mathrm{mm}$
$r$ dimension $ r $ $\mathrm{mm}$
$r$ lever arm length $\mathrm{mm}$
$r$ nominal earth's radius $\mathrm{m}$
$r_T$ radius of trunnions for bearings $\mathrm{mm}$
$s$ stroke from open position $\mathrm{\%}$
$s$ dimension $ s $ $\mathrm{mm}$
$s$ specific entropy $\mathrm{J\cdot kg^{-1}\cdot K^{-1}}$
$t$ closing time $\mathrm{s}$
$t$ dimension $ t $ $\mathrm{mm}$
$t_{ef}$ effective closing time $\mathrm{s}$
$u$ dimension $ u $ $\mathrm{mm}$
$u$ specific internal energy $\mathrm{J\cdot kg^{-1}}$
$v$ 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 $\mathrm{m/s}$
$v̄$ mean air-particle speed $\mathrm{m/s}$
$w$ dimension $ w $ $\mathrm{mm}$
$w$ speed of sound $\mathrm{m\cdot s^{-1}}$
$x$ dimension $ x $ $\mathrm{mm}$
$y$ dimension $ y $ $\mathrm{mm}$
$z$ dimension $ z $ $\mathrm{mm}$
$ΔP$ water hammer $\mathrm{m}$
$ΔP_{stages}$ pressure difference between the stages $\mathrm{Pa}$
$Δ_h$ 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 $\mathrm{°}$
$α_c$ angle of rotation centre of gravity in open position $\mathrm{°}$
$α_p$ relative pressure coefficient $\mathrm{K^{-1}}$
$α_ν$ isobaric cubic expansion coefficient $\mathrm{K^{-1}}$
$β$ aerated coefficient $\mathrm{ }$
$β$ swing angle $\mathrm{°}$
$β$ temperature gradient $ β $ $\mathrm{K\cdot m^{-1}}$
$β_S$ angle rotation of the rocking motion $\mathrm{°}$
$β_p$ isothermal stress coefficient $\mathrm{kg\cdot m^{-3}}$
$β_s$ Sutherland's empirical coefficients $ β_s $ $\mathrm{kg\cdot m^{-1}\cdot s^{-1}\cdot K^{-1/2}}$
$γ$ 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 $\mathrm{°}$
$γ$ dimensionless Gibbs free energy $\mathrm{ }$
$γ^o$ ideal-gas part $\mathrm{ }$
$γ^o_{ππ}$ second partial derivative of $ γ^o $ with respect to $ π $ $\mathrm{ }$
$γ^o_{πτ}$ cross derivative of $ γ^o $ with respect to $ π $ and temperature $ τ $ $\mathrm{ }$
$γ^o_{ττ}$ second partial derivative of $ γ^o $ with respect to $ τ $ $\mathrm{ }$
$γ^o_π$ derivative of $ γ^o $ with respect to the dimensionless pressure $ π $ $\mathrm{ }$
$γ^o_τ$ partial derivative of $ γ^o $ with respect to $ τ $ $\mathrm{ }$
$γ^r$ residual part $\mathrm{ }$
$γ^r_{ππ}$ second partial derivative of $ γ^r $ with respect to $ π $ $\mathrm{ }$
$γ^r_{πτ}$ cross derivative of $ γ^r $ with respect to $ π $ and temperature $ τ $ $\mathrm{ }$
$γ^r_{ττ}$ second partial derivative of $ γ^r $ with respect to $ τ $ $\mathrm{ }$
$γ^r_π$ derivative of $ γ^r $ with respect to the dimensionless pressure $ π $ $\mathrm{ }$
$γ^r_τ$ partial derivative of $ γ^r $ with respect to $ τ $ $\mathrm{ }$
$γ_{air}$ specific weight air $\mathrm{kg\cdot m^{-2}\cdot s^{-2}}$
$γ_{ππ}$ second partial derivative of $ γ $ with respect to $ π $ $\mathrm{ }$
$γ_{πτ}$ cross derivative of $ γ $ with respect to $ π $ and temperature $ τ $ $\mathrm{ }$
$γ_{ττ}$ second partial derivative of $ γ $ with respect to $ τ $ $\mathrm{ }$
$γ_π$ derivative of $ γ $ with respect to the dimensionless pressure $ π $ $\mathrm{ }$
$γ_τ$ partial derivative of $ γ $ with respect to $ τ $ $\mathrm{ }$
$δ$ the angle between the lever axis and the imaginary line between the valve axis and the pivot axis of the hydraulic cylinder $\mathrm{°}$
$δ$ reduced density $\mathrm{ }$
$δ_{[0]}$ 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 $\mathrm{°}$
$ζ$ loss coefficient $\mathrm{ }$
$ζ$ valve loss coefficient $\mathrm{ }$
$θ$ reduced temperature $\mathrm{ }$
$θ$ transformed temperature $\mathrm{ }$
$κ$ adiabatic index $\mathrm{ }$
$κ_T$ isothermal compressibility $\mathrm{Pa^{-1}}$
$λ$ thermal conductivity $\mathrm{W\cdot m^{-1}\cdot K^{-1}}$
$μ$ discharge coefficient $\mathrm{ }$
$μ$ dynamic viscosity $\mathrm{kg\cdot m^{-1}\cdot s^{-1}}$
$μ_B$ coefficient of friction for bearings $\mathrm{ }$
$μ_S$ coefficient of friction for main sealing $\mathrm{ }$
$ν$ kinematic viscosity $\mathrm{m^2\cdot s^{-1}}$
$ν$ specific volume $\mathrm{m^3\cdot kg^{-1}}$
$π$ reduced pressure $\mathrm{ }$
$ρ$ density $\mathrm{kg/m^3}$
$ρ$ mass density $\mathrm{kg\cdot m^{-3}}$
$ρ^*$ mass density reducing quantity $\mathrm{kg\cdot m^{-3}}$
$ρ_{air}$ density air $\mathrm{kg/m^3}$
$σ$ cavitation number $\mathrm{ }$
$σ$ effective collision diameter of an air molecule $\mathrm{m}$
$σ_1$ first stage of cavitation $\mathrm{ }$
$σ_2$ second stage of cavitation $\mathrm{ }$
$σ_{min}$ fully developed cavitation $\mathrm{ }$
$τ$ inverse reduced temperature $\mathrm{ }$
$φ$ angle between pipe axis and hydraulic force $\mathrm{°}$
$φ$ lever angle in open position $\mathrm{°}$
$φ$ dimensionless Helmholtz free energy $\mathrm{ }$
$φ_{δδ}$ second partial derivative of $ φ $ with respect to $ δ $ $\mathrm{ }$
$φ_{δτ}$ cross derivative of $ φ $ with respect to $ δ $ and temperature $ τ $ $\mathrm{ }$
$φ_{ττ}$ second partial derivative of $ φ $ with respect to $ τ $ $\mathrm{ }$
$φ_δ$ derivative of $ φ $ with respect to the dimensionless density $ δ $ $\mathrm{ }$
$φ_τ$ partial derivative of $ φ $ with respect to $ τ $ $\mathrm{ }$
$ω$ air-particle collision frequency $\mathrm{Hz}$