Fluid Mechanics and Fluid Machinery: Bernoulli, Losses and Turbomachines
Fluid statics and flow
Pressure at depth p = ρgh; absolute pressure = gauge + atmospheric. Continuity gives A₁V₁ = A₂V₂, and Bernoulli along a streamline gives p/ρg + V²/2g + z = constant. Reynolds number Re = ρVD/μ = VD/ν decides laminar (Re < 2000 in pipes) or turbulent flow.
Losses and machinery
- Darcy–Weisbach head loss h_f = fLV²/(2gD); f = 64/Re for laminar pipe flow, Moody chart for turbulent.
- Minor losses h = kV²/2g for fittings, bends and sudden expansions.
- Pump power P = ρgQH/η; specific speed classifies impellers; cavitation occurs when NPSH available falls below NPSH required.
- Turbines: Pelton for high head and low discharge, Francis for medium, Kaplan for low head and high discharge; the draft tube recovers exit kinetic energy.
Worked example
A pump delivers 0.02 m³/s against a total head of 20 m with overall efficiency 75%. Hydraulic power = ρgQH = 1000 × 9.81 × 0.02 × 20 = 3924 W. Shaft power = 3924/0.75 = 5232 W ≈ 5.2 kW.
Traps
- Using gauge pressure where absolute pressure is required (cavitation, gas laws).
- Mixing the pump and turbine specific-speed formulae; they differ in the √H placement.
- Forgetting that NPSH available falls as suction lift or water temperature rises.
- Using pipe diameter instead of hydraulic diameter in non-circular ducts.