Hydraulic Formulas Reference
Complete collection of hydraulic and fluid power engineering formulas. All equations use consistent units with conversion factors provided.
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Cylinder Formulas
Piston Area
A = (pi / 4) x D^2
A = Piston area (in^2 or cm^2)
D = Bore diameter (in or cm)
D = Bore diameter (in or cm)
Annular Area (Rod Side)
A_ann = (pi / 4) x (D^2 - d^2)
A_ann = Annular area (in^2 or cm^2)
D = Bore diameter (in or cm)
d = Rod diameter (in or cm)
D = Bore diameter (in or cm)
d = Rod diameter (in or cm)
Cylinder Force - Extension (Push)
F = P x A
F = Force (lbf or N)
P = Pressure (psi or bar)
A = Piston area (in^2 or cm^2)
US: F (lbf) = P (psi) x A (in^2)
Metric: F (N) = P (bar) x A (cm^2) x 10
P = Pressure (psi or bar)
A = Piston area (in^2 or cm^2)
US: F (lbf) = P (psi) x A (in^2)
Metric: F (N) = P (bar) x A (cm^2) x 10
Cylinder Force - Retraction (Pull)
F = P x A_ann
F = Force (lbf or N)
P = Pressure (psi or bar)
A_ann = Annular area (in^2 or cm^2)
P = Pressure (psi or bar)
A_ann = Annular area (in^2 or cm^2)
Cylinder Speed - Extension
v = Q / A
v = Velocity (in/s or m/s)
Q = Flow rate (in^3/s or m^3/s)
A = Piston area (in^2 or m^2)
US: v (in/min) = Q (GPM) x 231 / A (in^2)
Metric: v (m/min) = Q (L/min) / (A (cm^2) x 10)
Q = Flow rate (in^3/s or m^3/s)
A = Piston area (in^2 or m^2)
US: v (in/min) = Q (GPM) x 231 / A (in^2)
Metric: v (m/min) = Q (L/min) / (A (cm^2) x 10)
Cylinder Speed - Retraction
v = Q / A_ann
v = Velocity
Q = Flow rate
A_ann = Annular area
Q = Flow rate
A_ann = Annular area
Flow Required for Cylinder Speed
Q = A x v
US: Q (GPM) = A (in^2) x v (in/min) / 231
Metric: Q (L/min) = A (cm^2) x v (m/min) x 10
Metric: Q (L/min) = A (cm^2) x v (m/min) x 10
Cylinder Area Ratio
R = A / A_ann = D^2 / (D^2 - d^2)
R = Area ratio (dimensionless)
Typical values: 1.25 to 2.0
Typical values: 1.25 to 2.0
Rod Buckling (Euler Column)
F_cr = (pi^2 x E x I) / (K x L)^2
F_cr = Critical buckling load (lbf or N)
E = Modulus of elasticity (psi or Pa)
I = Moment of inertia = pi x d^4 / 64
K = End fixity factor (0.5 to 2.0)
L = Stroke length (in or m)
E = Modulus of elasticity (psi or Pa)
I = Moment of inertia = pi x d^4 / 64
K = End fixity factor (0.5 to 2.0)
L = Stroke length (in or m)
Pump Formulas
Pump Theoretical Flow
Q_th = D x N
Q_th = Theoretical flow
D = Displacement per revolution
N = Rotational speed (RPM)
US: Q (GPM) = D (in^3/rev) x N (RPM) / 231
Metric: Q (L/min) = D (cc/rev) x N (RPM) / 1000
D = Displacement per revolution
N = Rotational speed (RPM)
US: Q (GPM) = D (in^3/rev) x N (RPM) / 231
Metric: Q (L/min) = D (cc/rev) x N (RPM) / 1000
Pump Actual Flow
Q_act = Q_th x eta_v
Q_act = Actual flow rate
Q_th = Theoretical flow rate
eta_v = Volumetric efficiency (typically 0.85-0.95)
Q_th = Theoretical flow rate
eta_v = Volumetric efficiency (typically 0.85-0.95)
Pump Displacement from Flow
D = Q x 231 / (N x eta_v) [US]
D = Displacement (in^3/rev or cc/rev)
Q = Flow rate (GPM or L/min)
N = Speed (RPM)
eta_v = Volumetric efficiency
Q = Flow rate (GPM or L/min)
N = Speed (RPM)
eta_v = Volumetric efficiency
Hydraulic Power (Fluid)
P_hyd = Q x p
US: HP = Q (GPM) x P (psi) / 1714
Metric: kW = Q (L/min) x P (bar) / 600
Metric: kW = Q (L/min) x P (bar) / 600
Pump Input Power
P_in = P_hyd / eta_o
P_in = Input shaft power (HP or kW)
P_hyd = Hydraulic power output
eta_o = Overall efficiency = eta_v x eta_m
US: HP_in = Q (GPM) x P (psi) / (1714 x eta_o)
Metric: kW_in = Q (L/min) x P (bar) / (600 x eta_o)
P_hyd = Hydraulic power output
eta_o = Overall efficiency = eta_v x eta_m
US: HP_in = Q (GPM) x P (psi) / (1714 x eta_o)
Metric: kW_in = Q (L/min) x P (bar) / (600 x eta_o)
Pump Torque
T = (D x P) / (2 x pi x eta_m)
US: T (lb-in) = D (in^3/rev) x P (psi) / (2 x pi x eta_m)
Metric: T (Nm) = D (cc/rev) x P (bar) / (62.83 x eta_m)
Metric: T (Nm) = D (cc/rev) x P (bar) / (62.83 x eta_m)
Volumetric Efficiency
eta_v = Q_actual / Q_theoretical
eta_v = Volumetric efficiency (decimal)
Typical: 0.85 - 0.95 for gear pumps
Typical: 0.92 - 0.98 for piston pumps
Typical: 0.85 - 0.95 for gear pumps
Typical: 0.92 - 0.98 for piston pumps
Mechanical Efficiency
eta_m = T_theoretical / T_actual
eta_m = Mechanical efficiency (decimal)
Typical: 0.90 - 0.95
Typical: 0.90 - 0.95
Overall Efficiency
eta_o = eta_v x eta_m = P_hyd / P_in
eta_o = Overall efficiency (decimal)
Typical: 0.80 - 0.90
Typical: 0.80 - 0.90
Hydraulic Motor Formulas
Motor Speed
N = Q / D
US: N (RPM) = Q (GPM) x 231 / D (in^3/rev)
Metric: N (RPM) = Q (L/min) x 1000 / D (cc/rev)
Metric: N (RPM) = Q (L/min) x 1000 / D (cc/rev)
Motor Theoretical Torque
T_th = (D x P) / (2 x pi)
US: T (lb-in) = D (in^3/rev) x P (psi) / (2 x pi)
Metric: T (Nm) = D (cc/rev) x P (bar) / 62.83
Metric: T (Nm) = D (cc/rev) x P (bar) / 62.83
Motor Actual Torque
T_act = T_th x eta_m
T_act = Actual output torque
T_th = Theoretical torque
eta_m = Mechanical efficiency
T_th = Theoretical torque
eta_m = Mechanical efficiency
Motor Output Power
P_out = T x N x (2 x pi) / 60
US: HP = T (lb-ft) x N (RPM) / 5252
Metric: kW = T (Nm) x N (RPM) / 9549
Metric: kW = T (Nm) x N (RPM) / 9549
Motor Displacement from Requirements
D = T x 2 x pi / (P x eta_m)
D = Required displacement
T = Required torque
P = Available pressure
eta_m = Mechanical efficiency
T = Required torque
P = Available pressure
eta_m = Mechanical efficiency
Pressure Loss Formulas
Reynolds Number
Re = (v x D) / nu = (rho x v x D) / mu
Re = Reynolds number (dimensionless)
v = Fluid velocity (ft/s or m/s)
D = Pipe inside diameter (ft or m)
nu = Kinematic viscosity (ft^2/s or m^2/s)
rho = Fluid density (slug/ft^3 or kg/m^3)
mu = Dynamic viscosity (lb-s/ft^2 or Pa-s)
Re < 2000: Laminar flow
2000 < Re < 4000: Transition
Re > 4000: Turbulent flow
v = Fluid velocity (ft/s or m/s)
D = Pipe inside diameter (ft or m)
nu = Kinematic viscosity (ft^2/s or m^2/s)
rho = Fluid density (slug/ft^3 or kg/m^3)
mu = Dynamic viscosity (lb-s/ft^2 or Pa-s)
Re < 2000: Laminar flow
2000 < Re < 4000: Transition
Re > 4000: Turbulent flow
Darcy-Weisbach Equation
dP = f x (L/D) x (rho x v^2 / 2)
dP = Pressure drop (psi or Pa)
f = Darcy friction factor (dimensionless)
L = Pipe length (ft or m)
D = Pipe inside diameter (ft or m)
rho = Fluid density (slug/ft^3 or kg/m^3)
v = Fluid velocity (ft/s or m/s)
f = Darcy friction factor (dimensionless)
L = Pipe length (ft or m)
D = Pipe inside diameter (ft or m)
rho = Fluid density (slug/ft^3 or kg/m^3)
v = Fluid velocity (ft/s or m/s)
Laminar Flow Friction Factor
f = 64 / Re
f = Darcy friction factor
Re = Reynolds number
Valid for Re < 2000
Re = Reynolds number
Valid for Re < 2000
Turbulent Flow Friction Factor (Colebrook-White)
1/sqrt(f) = -2 x log10((e/D)/3.7 + 2.51/(Re x sqrt(f)))
f = Darcy friction factor
e = Surface roughness (same units as D)
D = Pipe diameter
Re = Reynolds number
Typical roughness: Steel pipe e = 0.0018 in (0.046 mm)
e = Surface roughness (same units as D)
D = Pipe diameter
Re = Reynolds number
Typical roughness: Steel pipe e = 0.0018 in (0.046 mm)
Simplified Pressure Drop (Hydraulic Hose)
dP = 0.0273 x (Q^1.75 x mu^0.25 x L x SG) / D^4.75
dP = Pressure drop (psi)
Q = Flow rate (GPM)
mu = Kinematic viscosity (cSt)
L = Length (ft)
SG = Specific gravity
D = Inside diameter (in)
Q = Flow rate (GPM)
mu = Kinematic viscosity (cSt)
L = Length (ft)
SG = Specific gravity
D = Inside diameter (in)
Fluid Velocity
v = Q / A = 4 x Q / (pi x D^2)
US: v (ft/s) = Q (GPM) x 0.4085 / D^2 (in)
Metric: v (m/s) = Q (L/min) x 21.22 / D^2 (mm)
Metric: v (m/s) = Q (L/min) x 21.22 / D^2 (mm)
Recommended Velocities
Pressure lines: 10-15 ft/s (3-4.5 m/s)
Return lines: 5-10 ft/s (1.5-3 m/s)
Suction lines: 2-4 ft/s (0.6-1.2 m/s)
Fitting Pressure Loss
dP = K x (rho x v^2) / 2
dP = Pressure drop
K = Loss coefficient (varies by fitting type)
rho = Fluid density
v = Fluid velocity
Typical K values:
90 deg elbow: 0.9
45 deg elbow: 0.4
Tee (branch): 1.8
Tee (run): 0.6
Gate valve (open): 0.2
Check valve: 2.5
K = Loss coefficient (varies by fitting type)
rho = Fluid density
v = Fluid velocity
Typical K values:
90 deg elbow: 0.9
45 deg elbow: 0.4
Tee (branch): 1.8
Tee (run): 0.6
Gate valve (open): 0.2
Check valve: 2.5
Accumulator Formulas
Isothermal Process (Slow Discharge)
P1 x V1 = P2 x V2 = constant
P1 = Pre-charge pressure
V1 = Total gas volume at pre-charge
P2 = System pressure
V2 = Gas volume at P2
Use for: Leakage compensation, slow volume storage
V1 = Total gas volume at pre-charge
P2 = System pressure
V2 = Gas volume at P2
Use for: Leakage compensation, slow volume storage
Adiabatic Process (Fast Discharge)
P1 x V1^n = P2 x V2^n = constant
n = Polytropic exponent (1.4 for nitrogen)
Use for: Emergency functions, shock absorption,
pulsation dampening
Use for: Emergency functions, shock absorption,
pulsation dampening
Accumulator Sizing (Isothermal)
V0 = dV x P2 x P1 / (P0 x (P2 - P1))
V0 = Required accumulator volume
dV = Volume of oil to be stored/delivered
P0 = Pre-charge pressure
P1 = Minimum operating pressure
P2 = Maximum operating pressure
dV = Volume of oil to be stored/delivered
P0 = Pre-charge pressure
P1 = Minimum operating pressure
P2 = Maximum operating pressure
Accumulator Sizing (Adiabatic)
V0 = dV / [(P0/P1)^(1/n) - (P0/P2)^(1/n)]
V0 = Required accumulator volume
dV = Volume of oil required
P0 = Pre-charge pressure
P1 = Minimum pressure
P2 = Maximum pressure
n = 1.4 for nitrogen
dV = Volume of oil required
P0 = Pre-charge pressure
P1 = Minimum pressure
P2 = Maximum pressure
n = 1.4 for nitrogen
Pre-charge Pressure Guidelines
Energy storage: P0 = 0.90 x P1
Pulsation dampening: P0 = 0.60 x P_mean
Shock absorption: P0 = 0.25 x P_max
P0 = Pre-charge pressure
P1 = Minimum system pressure
P_mean = Mean operating pressure
P_max = Maximum shock pressure
P1 = Minimum system pressure
P_mean = Mean operating pressure
P_max = Maximum shock pressure
Volume Ratio
Compression ratio = V1 / V2
Volume ratio = P2 / P1 (isothermal)
Maximum compression ratio: 4:1 (bladder)
Maximum compression ratio: 10:1 (piston)
Maximum compression ratio: 10:1 (piston)
Valve Formulas
Valve Flow Coefficient (Cv)
Q = Cv x sqrt(dP / SG)
Q = Flow rate (GPM)
Cv = Flow coefficient (GPM at 1 psi drop, SG=1)
dP = Pressure drop across valve (psi)
SG = Specific gravity of fluid
Cv = Flow coefficient (GPM at 1 psi drop, SG=1)
dP = Pressure drop across valve (psi)
SG = Specific gravity of fluid
Cv to Kv Conversion
Kv = Cv x 0.865
Cv = Kv x 1.156
Cv = US flow coefficient (GPM)
Kv = Metric flow coefficient (m^3/hr at 1 bar drop)
Kv = Metric flow coefficient (m^3/hr at 1 bar drop)
Orifice Flow
Q = Cd x A x sqrt(2 x dP / rho)
Q = Flow rate
Cd = Discharge coefficient (0.6-0.65 for sharp edge)
A = Orifice area
dP = Pressure drop
rho = Fluid density
US: Q (GPM) = 29.8 x Cd x d^2 (in) x sqrt(dP (psi) / SG)
Metric: Q (L/min) = 1.9 x Cd x d^2 (mm) x sqrt(dP (bar) / SG)
Cd = Discharge coefficient (0.6-0.65 for sharp edge)
A = Orifice area
dP = Pressure drop
rho = Fluid density
US: Q (GPM) = 29.8 x Cd x d^2 (in) x sqrt(dP (psi) / SG)
Metric: Q (L/min) = 1.9 x Cd x d^2 (mm) x sqrt(dP (bar) / SG)
Relief Valve Heat Generation
Heat = Q x dP
US: Heat (BTU/hr) = Q (GPM) x dP (psi) x 1.48
Metric: Heat (kW) = Q (L/min) x dP (bar) / 600
Metric: Heat (kW) = Q (L/min) x dP (bar) / 600
Counterbalance Valve Pilot Ratio
PR = (P_load x A_ratio + P_back) / P_pilot
PR = Pilot ratio (typically 3:1 to 10:1)
P_load = Load-induced pressure
A_ratio = Cylinder area ratio
P_back = Back pressure
P_pilot = Pilot pressure
P_load = Load-induced pressure
A_ratio = Cylinder area ratio
P_back = Back pressure
P_pilot = Pilot pressure
Unit Conversion Factors
Pressure
| From | To | Multiply By |
|---|---|---|
| psi | bar | 0.06895 |
| bar | psi | 14.504 |
| psi | kPa | 6.895 |
| kPa | psi | 0.1450 |
| bar | MPa | 0.1 |
| MPa | bar | 10 |
| psi | kg/cm^2 | 0.0703 |
| kg/cm^2 | psi | 14.22 |
Flow Rate
| From | To | Multiply By |
|---|---|---|
| GPM | L/min | 3.785 |
| L/min | GPM | 0.2642 |
| GPM | in^3/s | 3.850 |
| L/min | cm^3/s | 16.67 |
| GPM | m^3/hr | 0.2271 |
| m^3/hr | GPM | 4.403 |
Length / Distance
| From | To | Multiply By |
|---|---|---|
| inch | mm | 25.4 |
| mm | inch | 0.03937 |
| foot | m | 0.3048 |
| m | foot | 3.281 |
Force
| From | To | Multiply By |
|---|---|---|
| lbf | N | 4.448 |
| N | lbf | 0.2248 |
| lbf | kgf | 0.4536 |
| kgf | lbf | 2.205 |
| kN | ton (US) | 0.1124 |
Torque
| From | To | Multiply By |
|---|---|---|
| lb-ft | Nm | 1.356 |
| Nm | lb-ft | 0.7376 |
| lb-in | Nm | 0.1130 |
| Nm | lb-in | 8.851 |
Power
| From | To | Multiply By |
|---|---|---|
| HP | kW | 0.7457 |
| kW | HP | 1.341 |
| HP | BTU/hr | 2545 |
| kW | BTU/hr | 3412 |
Volume
| From | To | Multiply By |
|---|---|---|
| gallon (US) | liter | 3.785 |
| liter | gallon (US) | 0.2642 |
| in^3 | cm^3 (cc) | 16.39 |
| cm^3 (cc) | in^3 | 0.06102 |
| gallon (US) | in^3 | 231 |
Viscosity
| From | To | Multiply By |
|---|---|---|
| cSt | SUS @ 100F | 4.632 |
| SUS @ 100F | cSt | 0.216 |
| cSt | mm^2/s | 1.0 |
| cP | cSt | divide by SG |
| cSt | cP | multiply by SG |
Temperature
| From | To | Formula |
|---|---|---|
| F | C | (F - 32) x 5/9 |
| C | F | (C x 9/5) + 32 |
| C | K | C + 273.15 |
| F | R | F + 459.67 |