Hydraulic power vs shaft pump power
How rho g Q H gives hydraulic power, why efficiency raises it to shaft power, and how total dynamic head and kW to hp fit in.
The hydraulic power equation
Hydraulic power is the useful power a pump adds to the fluid, given by rho times g times Q times H. Here rho is density in kilograms per cubic metre, g is 9.80665 metres per second squared, Q is volumetric flow in cubic metres per second and H is head in metres. The product comes out in watts. For water at 1000 kg/m3 moving 0.05 m3/s against 20 metres of head, that is 9806.65 watts, or about 9.81 kilowatts. This figure is the theoretical output and ignores every loss inside the machine.
From hydraulic to shaft power
A real pump never turns all its input into fluid energy. Bearing friction, internal leakage and turbulence mean the shaft must supply more power than the fluid receives. Shaft power equals hydraulic power divided by efficiency, so a pump that is 70 percent efficient moving the duty above needs 9806.65 divided by 0.7, about 14009.5 watts at the shaft. Efficiency is a fraction from 0 to 1, and centrifugal pumps commonly land between 0.6 and 0.85. The lower the efficiency, the larger the gap between the two power figures.
Total dynamic head
Head H in the formula is the total dynamic head, not just the vertical lift. It sums the static lift between source and delivery, the friction losses in pipes and fittings, and any pressure or velocity head at the outlet. Friction rises sharply with flow, so a system that looks easy at low flow can demand far more head when pushed. Estimating total dynamic head honestly matters more than any other input, because power scales directly with it. Undersizing the head is a frequent cause of pumps that cannot meet their duty.
Kilowatts, horsepower and motor sizing
Motors are often specified in horsepower, so shaft power in watts is worth converting. One mechanical horsepower is 745.7 watts, so divide watts by 745.7 to get horsepower, which makes the 9806.65 watt example about 13.15 hp. When choosing a motor, size it above the shaft power to leave margin for the worst operating point and for motor efficiency, which is separate from pump efficiency. A drive that runs continuously near its limit wastes energy and shortens its life. Adding a service factor of 10 to 25 percent is common practice.