Bernoulli, friction and pumps

Along a streamline in a smooth flow with no friction, energy is just traded between three forms: pressure, speed and height. That's Bernoulli's equation: P + ½ρv² + ρgh stays the same. Speed up and the pressure drops, which is how a venturi meter measures flow and why a narrowed pipe can suck in air.

One quick use: liquid draining from a hole h below the surface of an open tank comes out at about v = √(2gh), the same speed as something dropped from that height. From 5 m down: √(2 × 9.81 × 5) = 9.9 m/s.

Real fluids lose energy to friction, so engineers add a head loss term, h_f. For a straight pipe the Darcy–Weisbach equation gives h_f = f × (L/D) × v²/(2g), where L is the length and f a friction factor (about 0.02 for turbulent flow in ordinary steel pipe; for laminar flow it's 64 ÷ Re). Valves, bends and fittings add their own losses. Friction grows with v², so doubling the flow through the same pipe roughly quadruples the loss.

A pump adds the head the system needs: lift (height), pressure difference and friction losses together. The power the fluid receives is ρ × g × Q × H, with H the total head in metres. Pumping 0.01 m³/s of water up 20 m of total head takes 1000 × 9.81 × 0.01 × 20 = 1962 W. Pumps aren't perfect, so divide by the efficiency: at 70% the motor supplies about 2.8 kW.

A centrifugal pump gives less head as the flow rises (its curve), while the pipe system needs more (the system curve). Where the two curves cross is where it will actually run. If the pressure at the pump inlet falls too low, the liquid boils there and the bubbles collapse violently: cavitation, which sounds like gravel and eats impellers.