Pump Power Calculation: Formula and Worked Examples

Chemical Engineering · Engineering Calculations

Sizing a pump motor comes down to three numbers stacked on top of each other: the power the fluid itself needs (hydraulic power), the power the pump has to draw at the shaft to deliver that (shaft power), and the power the motor has to supply to turn that shaft (motor power). Each step loses a bit to inefficiency, which is why a pump moving water at a modest flow and head can still need a noticeably larger motor than the "theoretical" number suggests.

Short on time? Use the free Pump Power Calculator. Enter flow, head, density and efficiencies and it returns hydraulic, shaft and motor power.

The Formula

Where ρ is fluid density (kg/m³), g is 9.81 m/s², Q is flow rate (m³/s), H is total developed head (m), and η_pump / η_motor are the pump and motor efficiencies as fractions (e.g. 0.65 for 65%).

Quick Formula: Pump Power from m³/h

On the plant floor, flow is usually read in m³/h, not m³/s. Folding the constants together gives a shortcut that is easy to remember:

SG is the specific gravity of the liquid (1.0 for water). The 367 comes from 3,600 × 1,000 / (1,000 × 9.81). To get kW in horsepower, divide by 0.7457 (1 hp = 0.7457 kW).

Variables and Units

Total head H isn't just the static lift. It's static lift plus friction losses in the piping plus any pressure the destination vessel is already holding, all converted to an equivalent height of fluid. Getting H wrong (usually by forgetting friction losses) is the single most common source of an undersized pump.

Worked Example

Take water (ρ = 1,000 kg/m³) at a flow of 50 m³/hr (0.0139 m³/s) against a total head of 30 m, with a pump efficiency of 65% and motor efficiency of 92%:

In practice you'd round up to the next standard motor frame size above 6.84 kW (commonly a 7.5 kW motor) rather than specifying the exact calculated value, to leave margin for wear and off-curve operation. The quick formula gives the same hydraulic power: 50 × 30 × 1.0 / 367 ≈ 4.09 kW.

Worked Example 2: Solvent into a Pressurised Vessel

Now a case closer to a pharma plant: a solvent with density 870 kg/m³ (SG 0.87) is transferred at 20 m³/hr into a receiver held at 2 bar(g). The static lift is 12 m and the line friction loss is 5 m. Pump efficiency is 60% and motor efficiency 90%.

First, convert the receiver pressure to head of this liquid:

Then the power:

Notice that the receiver pressure is more than half of the total head here. Leave it out and the motor would be undersized by more than half.

In the Plant

Pump efficiency isn't constant. It peaks at the best-efficiency point (BEP) on the pump curve and drops off the further you run from it. A pump selected purely on this formula using catalog efficiency, but actually operated well off its BEP, will draw more real power than this calculation predicts.

Common Mistakes

Using static head alone and skipping friction losses; using density from a different fluid than the one actually being pumped; and applying a single "safety factor" on top of an already-conservative efficiency estimate, which can lead to an oversized motor running inefficiently at partial load.

Frequently Asked Questions

What is the formula for pump power?

Hydraulic power (kW) = ρ × g × Q × H / 1,000, with ρ in kg/m³, Q in m³/s and H in m. Divide by pump efficiency for shaft power and by motor efficiency for motor input power.

What is the quick formula for pump power in kW from m³/h?

Hydraulic power (kW) = Q (m³/h) × H (m) × SG / 367. For 50 m³/h of water at 30 m head this is about 4.1 kW.

What is the difference between hydraulic power and shaft power?

Hydraulic power is the energy actually given to the fluid. Shaft power (also called brake power) is what the pump draws at its shaft. It is always higher because of losses inside the pump: shaft power = hydraulic power / pump efficiency.

How do I convert pressure to head?

Head (m) = ΔP (Pa) / (ρ × 9.81). For water, 1 bar is about 10.2 m of head. For a lighter liquid the same pressure is more metres of head, which is why the density matters.