Turbines and Pumps

Turbines take energy from a fluid; pumps give energy to it. Both are analysed with velocity triangles and Euler's equation, and classified by specific speed.

Euler's turbomachine equation

The work done per unit weight of fluid:

H = (V_w1 u₁ − V_w2 u₂)/g (turbines)

where u is the blade speed and V_w the whirl (tangential) component of absolute velocity at inlet (1) and outlet (2). For pumps, the sign reverses: H = (V_w2 u₂ − V_w1 u₁)/g, with V_w1 = 0 for radial inflow.

Hydraulic turbines

(Ranges are in the metric units N√P/H^(5/4) with P in kW and approximate.)

• Pelton: jet velocity V = C_v √(2gH). Maximum efficiency when the bucket speed u = V/2 (for 180° deflection, maximum hydraulic efficiency 100% in theory).
• Draft tube on reaction turbines recovers kinetic energy and lets the turbine sit above tailwater.
• Efficiencies: hydraulic, mechanical and overall η₀ = P/(ρgQH).

Specific speed

• Turbine: N_s = N√P / H^(5/4)
• Pump: N_s = N√Q / H^(3/4)

Machines with the same specific speed are geometrically similar. Unit quantities: N_u = N/√H, Q_u = Q/√H, P_u = P/H^(3/2).

Centrifugal pumps

• Manometric head H_m, manometric efficiency = gH_m/(V_w2 u₂).
• Backward-curved vanes give a stable, falling head–flow curve and are most common.
• Pumps in series add head; in parallel add flow.
• Cavitation: if local pressure falls below vapour pressure, bubbles form and collapse, eroding the impeller. Avoid it by keeping the NPSH available above the NPSH required (lower suction lift, larger suction pipe).

Worked example

A Pelton wheel works under 400 m head with C_v = 0.98. Find the jet velocity and the best bucket speed.

• V = 0.98 √(2 × 9.81 × 400) = 0.98 × 88.59 = 86.8 m/s.
• u = V/2 = 43.4 m/s.

Practice