03 / CHAPTER 05
Power and efficiency
Relate energy rate to force, speed, machines, and losses.
ANCHOR EQUATION
Power
In symbols: P = F·v
Power is the instantaneous rate of mechanical energy transfer.
Power measures how fast energy moves
Two machines can perform the same work but demand different power if one does it faster. Pump sizing is a power question because the water must gain energy every second.
Differentiate work
Average power is P_avg = W/Δt. For an infinitesimal transfer dW = F·dr, divide by dt: P = F·(dr/dt) = F·v. For lifting water steadily, useful power is ṁ g h = ρQgh; efficiency η = P_out/P_in compares hydraulic output with supplied power.
Lifting Q = 0.010 m³/s of water through h = 4 m requires P_h = 1000(0.010)(9.81)(4) = 392 W. At η = 0.65, input power must be about 603 W, before other operating margins.
Power and energy have different units
A 400 W pump running for 10 minutes transfers at most 240 kJ at 100% efficiency. A watt is J/s, not joules; confusing the two produces errors that grow with runtime.
Work it: A nozzle flow of 0.006 m³/s rises 3 m while pressure losses consume 120 W. Find hydraulic lift power, total useful power, and electrical input for η = 0.70.