Formula Sheet: Pumps And Turbines

Fundamental Pump Concepts

Head and Energy

Total Dynamic Head (TDH): \[H = h_s + h_d + h_f + h_v\]

• H = total dynamic head (ft or m)
• h_s = static suction head/lift (ft or m)
• h_d = static discharge head (ft or m)
• h_f = friction head loss (ft or m)
• h_v = velocity head (ft or m)

Total Head (Alternative Form): \[H = \frac{p_2 - p_1}{\gamma} + (z_2 - z_1) + \frac{V_2^2 - V_1^2}{2g}\]

• p₁, p₂ = pressure at points 1 and 2 (lb/ft² or Pa)
• γ = specific weight of fluid (lb/ft³ or N/m³)
• z₁, z₂ = elevation at points 1 and 2 (ft or m)
• V₁, V₂ = velocity at points 1 and 2 (ft/s or m/s)
• g = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)

Velocity Head: \[h_v = \frac{V^2}{2g}\]

• h_v = velocity head (ft or m)
• V = velocity (ft/s or m/s)
• g = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)

Pump Power

Water Horsepower (Hydraulic Power): \[WHP = \frac{Q \times H \times \gamma}{550}\] US Customary Units:

• WHP = water horsepower (hp)
• Q = flow rate (ft³/s)
• H = total head (ft)
• γ = specific weight (lb/ft³)
• 550 = conversion factor (ft·lb/s per hp)

Water Horsepower (Alternative Form): \[WHP = \frac{Q \times H \times SG}{3960}\]

• Q = flow rate (gal/min)
• H = total head (ft)
• SG = specific gravity (dimensionless)
• 3960 = conversion factor

SI Units - Hydraulic Power: \[P_h = \frac{\rho \times g \times Q \times H}{1000}\]

• P_h = hydraulic power (kW)
• ρ = fluid density (kg/m³)
• g = gravitational acceleration (9.81 m/s²)
• Q = flow rate (m³/s)
• H = total head (m)

Brake Horsepower (Shaft Power): \[BHP = \frac{WHP}{\eta_p}\]

• BHP = brake horsepower (hp)
• WHP = water horsepower (hp)
• η_p = pump efficiency (decimal)

Motor Power: \[P_{motor} = \frac{BHP}{\eta_m}\]

• P_motor = motor power required (hp)
• BHP = brake horsepower (hp)
• η_m = motor efficiency (decimal)

Pump Efficiency

Overall Pump Efficiency: \[η_p = \frac{WHP}{BHP} = \frac{\text{Power out}}{\text{Power in}}\]

• η_p = pump efficiency (decimal or %)
• Typical range: 0.50 to 0.90 for centrifugal pumps

Overall Efficiency (including motor): \[η_{overall} = η_p \times η_m\]

• η_overall = overall system efficiency (decimal)
• η_m = motor efficiency (decimal)

Pump Affinity Laws

Single Pump at Different Speeds

Flow Rate Relationship: \[\frac{Q_1}{Q_2} = \frac{N_1}{N_2}\]

• Q₁, Q₂ = flow rates (gal/min or m³/s)
• N₁, N₂ = pump speeds (rpm)

Head Relationship: \[\frac{H_1}{H_2} = \left(\frac{N_1}{N_2}\right)^2\]

• H₁, H₂ = heads (ft or m)
• N₁, N₂ = pump speeds (rpm)

Power Relationship: \[\frac{P_1}{P_2} = \left(\frac{N_1}{N_2}\right)^3\]

• P₁, P₂ = power (hp or kW)
• N₁, N₂ = pump speeds (rpm)

Geometrically Similar Pumps (Homologous Pumps)

Flow Rate with Different Impeller Diameter: \[\frac{Q_1}{Q_2} = \left(\frac{D_1}{D_2}\right)^3 \times \frac{N_1}{N_2}\]

• D₁, D₂ = impeller diameters (in or mm)
• N₁, N₂ = pump speeds (rpm)

Head with Different Impeller Diameter: \[\frac{H_1}{H_2} = \left(\frac{D_1}{D_2}\right)^2 \times \left(\frac{N_1}{N_2}\right)^2\] Power with Different Impeller Diameter: \[\frac{P_1}{P_2} = \left(\frac{D_1}{D_2}\right)^5 \times \left(\frac{N_1}{N_2}\right)^3\]

Specific Speed and Similarity Parameters

Specific Speed

Pump Specific Speed (US Customary): \[N_s = \frac{N \sqrt{Q}}{H^{0.75}}\]

• N_s = specific speed (dimensionless or US units)
• N = pump rotational speed (rpm)
• Q = flow rate at best efficiency point (gal/min)
• H = head per stage at best efficiency point (ft)
• For multi-stage pumps, use head per stage
• For double-suction pumps, use Q/2

Pump Specific Speed (SI): \[N_s = \frac{N \sqrt{Q}}{H^{0.75}}\]

• N = pump rotational speed (rpm)
• Q = flow rate (m³/s)
• H = head (m)

Pump Type Classification by Specific Speed:

• Radial flow (centrifugal): N_s < 4200="" (us="">
• Mixed flow: 4200 <>_s < 9000="" (us="">
• Axial flow: N_s > 9000 (US units)

Suction Specific Speed

Suction Specific Speed: \[S = \frac{N \sqrt{Q}}{NPSH_R^{0.75}}\]

• S = suction specific speed
• N = pump speed (rpm)
• Q = flow rate (gal/min)
• NPSH_R = required net positive suction head (ft)
• Maximum recommended S ≈ 8500 to 11,000 for low cavitation risk

Net Positive Suction Head (NPSH)

NPSH Available

NPSH Available (General Form): \[NPSH_A = h_{atm} + h_s - h_f - h_{vp}\]

• NPSH_A = net positive suction head available (ft or m)
• h_atm = atmospheric pressure head (ft or m)
• h_s = static suction head (ft or m) (positive if above pump, negative if below)
• h_f = friction loss in suction line (ft or m)
• h_vp = vapor pressure head of liquid (ft or m)

NPSH Available (Pressure Form): \[NPSH_A = \frac{p_{atm} - p_{vp}}{\gamma} + z_s - h_{fs}\]

• p_atm = atmospheric pressure (lb/ft² or Pa)
• p_vp = vapor pressure of liquid (lb/ft² or Pa)
• γ = specific weight of fluid (lb/ft³ or N/m³)
• z_s = elevation of liquid surface above (+) or below (-) pump centerline (ft or m)
• h_fs = friction loss in suction piping (ft or m)

NPSH Available (Closed Tank): \[NPSH_A = \frac{p_{tank} - p_{vp}}{\gamma} + z_s - h_{fs}\]

• p_tank = pressure in closed tank (lb/ft² or Pa)
• Other variables as defined above

NPSH Required

NPSH Requirement: \[NPSH_A > NPSH_R\]

• NPSH_R = required NPSH (provided by pump manufacturer) (ft or m)
• Safety margin typically 2-5 ft (0.6-1.5 m)
• Cavitation occurs when NPSH_A <>_R

Thoma Cavitation Parameter

Thoma Parameter: \[σ = \frac{NPSH_R}{H}\]

• σ = Thoma cavitation parameter (dimensionless)
• H = pump head (ft or m)
• Lower σ indicates better cavitation performance

Pumps in Series and Parallel

Pumps in Series

Total Head (Series): \[H_{total} = H_1 + H_2 + H_3 + ...\]

• Flow rate remains the same: Q_total = Q₁ = Q₂ = ...
• Heads add at constant flow rate
• Used when high head is required

Pumps in Parallel

Total Flow (Parallel): \[Q_{total} = Q_1 + Q_2 + Q_3 + ...\]

• Head remains the same: H_total = H₁ = H₂ = ...
• Flow rates add at constant head
• Used when high capacity is required
• Each pump must overcome the same system head

System Curves and Operating Point

System Head Curve

System Head Equation: \[H_{sys} = H_{static} + K \times Q^2\]

• H_sys = total system head (ft or m)
• H_static = static head (elevation difference) (ft or m)
• K = system resistance coefficient
• Q = flow rate (gal/min or m³/s)

System Resistance Coefficient: \[K = \frac{h_f}{Q^2}\]

• h_f = friction head loss at flow Q (ft or m)
• Accounts for pipe friction and minor losses

Operating Point

Operating Point Condition: \[H_{pump}(Q) = H_{sys}(Q)\]

• Intersection of pump curve and system curve
• Defines actual flow rate and head at operation

Turbine Fundamentals

Turbine Power Output

Theoretical Water Power: \[P_w = \gamma \times Q \times H\]

• P_w = theoretical water power (ft·lb/s)
• γ = specific weight of water (lb/ft³)
• Q = flow rate (ft³/s)
• H = net head (ft)

Theoretical Water Horsepower: \[WHP = \frac{\gamma \times Q \times H}{550}\]

• WHP = water horsepower (hp)
• 550 = conversion factor (ft·lb/s per hp)

Turbine Shaft Power: \[P_{shaft} = η_t \times P_w\]

• P_shaft = shaft power output (ft·lb/s or W)
• η_t = turbine efficiency (decimal)

Brake Horsepower (Turbine): \[BHP = η_t \times WHP\]

• BHP = brake horsepower (hp)
• Actual power delivered to shaft

Turbine Efficiency

Turbine Efficiency: \[η_t = \frac{P_{shaft}}{P_w} = \frac{\text{Power out}}{\text{Power in}}\]

• η_t = turbine efficiency (decimal or %)
• Typical range: 0.80 to 0.95 for large turbines

Net Head for Turbines

Net Head: \[H_{net} = H_{gross} - h_L\]

• H_net = net head available to turbine (ft or m)
• H_gross = gross head (elevation difference) (ft or m)
• h_L = head loss in penstock and piping (ft or m)

Turbine Specific Speed

Specific Speed (Turbines)

Turbine Specific Speed (US Customary): \[N_s = \frac{N \sqrt{P}}{H^{1.25}}\]

• N_s = specific speed
• N = rotational speed (rpm)
• P = shaft power output (hp)
• H = net head (ft)

Turbine Specific Speed (SI): \[N_s = \frac{N \sqrt{P}}{H^{1.25}}\]

• N = rotational speed (rpm)
• P = shaft power output (kW)
• H = net head (m)

Turbine Type Selection by Specific Speed (US units):

• Impulse (Pelton wheel): N_s = 1 to 5
• Francis turbine: N_s = 10 to 100
• Kaplan/Propeller turbine: N_s > 100

Impulse Turbines (Pelton Wheel)

Jet Velocity

Jet Velocity from Nozzle: \[V_j = C_v \sqrt{2gH}\]

• V_j = jet velocity (ft/s or m/s)
• C_v = velocity coefficient (typically 0.94 to 0.99)
• g = gravitational acceleration (32.2 ft/s² or 9.81 m/s²)
• H = net head (ft or m)

Bucket Velocity

Optimal Bucket Velocity: \[U = \frac{V_j}{2}\]

• U = bucket velocity (ft/s or m/s)
• V_j = jet velocity (ft/s or m/s)
• Optimal efficiency occurs when bucket velocity is half jet velocity

Bucket Velocity from RPM: \[U = \frac{\pi D N}{60}\]

• D = pitch diameter of wheel (ft or m)
• N = rotational speed (rpm)

Power and Efficiency

Power Output (Impulse Turbine): \[P = \dot{m} \times (V_j - U) \times U \times (1 + k)\]

• P = power output (ft·lb/s or W)
• ṁ = mass flow rate (lb/s or kg/s)
• k = factor accounting for exit velocity angle (typically k ≈ cos(β) where β is bucket exit angle)
• For ideal case with β = 180° (flow reversal), k = 1

Maximum Power Condition: \[U_{opt} = \frac{V_j}{2}\]

• Maximum power occurs when bucket speed is half jet speed

Pelton Wheel Specific Parameters

Flow Rate through Nozzle: \[Q = A_j \times V_j\]

• Q = flow rate (ft³/s or m³/s)
• A_j = jet cross-sectional area (ft² or m²)
• V_j = jet velocity (ft/s or m/s)

Reaction Turbines

Francis Turbine

Euler Turbine Equation: \[P = \dot{m} \times (U_1 V_{t1} - U_2 V_{t2})\]

• P = power output (ft·lb/s or W)
• ṁ = mass flow rate (lb/s or kg/s)
• U₁, U₂ = blade velocity at inlet and outlet (ft/s or m/s)
• V_t1, V_t2 = tangential component of absolute velocity at inlet and outlet (ft/s or m/s)

Blade Velocity: \[U = \frac{\pi D N}{60}\]

• U = blade velocity (ft/s or m/s)
• D = diameter at point of interest (ft or m)
• N = rotational speed (rpm)

Velocity Triangles

Tangential Velocity Component: \[V_t = V \cos α\]

• V_t = tangential component of absolute velocity (ft/s or m/s)
• V = absolute velocity (ft/s or m/s)
• α = angle between absolute velocity and tangential direction

Relative Velocity Relationship: \[W^2 = V^2 + U^2 - 2VU \cos α\]

• W = relative velocity (velocity relative to blade) (ft/s or m/s)
• V = absolute velocity (ft/s or m/s)
• U = blade velocity (ft/s or m/s)
• α = absolute velocity angle

Degree of Reaction

Degree of Reaction: \[R = \frac{\text{Pressure drop in rotor}}{\text{Total pressure drop}}\]

• R = degree of reaction (dimensionless)
• R = 0 for pure impulse turbines (Pelton)
• R = 0.5 for typical Francis turbines
• R ≈ 1.0 for Kaplan/Propeller turbines

Turbine Regulation and Performance

Speed Regulation

Speed Regulation (Percent): \[SR = \frac{N_{NL} - N_{FL}}{N_{FL}} \times 100\%\]

• SR = speed regulation (%)
• N_NL = no-load speed (rpm)
• N_FL = full-load speed (rpm)

Runaway Speed

Runaway Speed: \[N_{runaway} = (1.8 \text{ to } 2.5) \times N_{rated}\]

• N_runaway = maximum speed with no load and full gate opening (rpm)
• N_rated = rated operating speed (rpm)
• Exact factor depends on turbine type and specific speed

Dimensionless Performance Parameters

Flow Coefficient

Flow Coefficient (Pumps): \[φ = \frac{Q}{ND^3}\]

• φ = flow coefficient (dimensionless)
• Q = flow rate (ft³/s or m³/s)
• N = rotational speed (rev/s)
• D = impeller diameter (ft or m)

Head Coefficient

Head Coefficient (Pumps): \[ψ = \frac{gH}{N^2 D^2}\]

• ψ = head coefficient (dimensionless)
• g = gravitational acceleration (ft/s² or m/s²)
• H = head (ft or m)
• N = rotational speed (rev/s)
• D = impeller diameter (ft or m)

Power Coefficient

Power Coefficient (Pumps): \[λ = \frac{P}{\rho N^3 D^5}\]

• λ = power coefficient (dimensionless)
• P = power (ft·lb/s or W)
• ρ = fluid density (lb/ft³ or kg/m³)
• N = rotational speed (rev/s)
• D = impeller diameter (ft or m)

Positive Displacement Pumps

Flow Rate (Positive Displacement)

Theoretical Flow Rate: \[Q_{th} = V_d \times N\]

• Q_th = theoretical flow rate (gal/min or m³/s)
• V_d = displacement volume per revolution (gal or m³)
• N = rotational speed (rpm or rev/s)

Actual Flow Rate: \[Q_{actual} = η_v \times Q_{th}\]

• Q_actual = actual flow rate (gal/min or m³/s)
• η_v = volumetric efficiency (decimal)
• Accounts for slip and leakage

Volumetric Efficiency

Volumetric Efficiency: \[η_v = \frac{Q_{actual}}{Q_{th}}\]

• η_v = volumetric efficiency (decimal)
• Typically 0.85 to 0.98 for positive displacement pumps

Cavitation and Erosion

Cavitation Index

Cavitation Index (Sigma): \[σ = \frac{NPSH_A}{H}\]

• σ = cavitation index (dimensionless)
• NPSH_A = net positive suction head available (ft or m)
• H = pump head (ft or m)
• Higher σ indicates lower cavitation risk

Critical Sigma

Critical Cavitation Index: \[σ_c = \frac{NPSH_R}{H}\]

• σ_c = critical cavitation index (dimensionless)
• NPSH_R = required NPSH (ft or m)
• Cavitation begins when σ <>_c

Pump Performance Corrections

Viscosity Corrections

Viscosity Correction Factors: \[Q_{viscous} = C_Q \times Q_{water}\] \[H_{viscous} = C_H \times H_{water}\] \[η_{viscous} = C_η \times η_{water}\]

• C_Q, C_H, C_η = correction factors from Hydraulic Institute charts
• All correction factors < 1.0="" for="" viscous="">
• Performance decreases with increasing viscosity

Specific Gravity Corrections

Head (unchanged with specific gravity): \[H_{liquid} = H_{water}\]

• Head in feet or meters is independent of fluid density

Power Correction for Specific Gravity: \[P_{liquid} = SG \times P_{water}\]

• P_liquid = power for liquid with specific gravity SG
• SG = specific gravity of liquid
• P_water = power for water at same Q and H