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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}[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_{test} test \mathrm{mm}
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(h) pressure at height (h) \mathrm{Pa}
P_0 pressure at sea level \mathrm{Pa}
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 standard temperature at sea level \mathrm{K}
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}
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}