Pump & Fan Affinity Laws Calculator
See how a pump or fan’s flow, head and power change with speed or impeller size. The affinity laws scale flow with speed, head with its square and power with its cube, and the VFD mode works out the energy a variable-speed drive saves.
How to Use
- Pick a mode: a speed change, an impeller trim, both together, the speed needed for a target flow, or the energy saved by slowing down with a variable-speed drive (VFD).
- Enter the original speed (rpm, or the drive frequency in Hz) and the new one, or the original and trimmed impeller diameters.
- Enter the duty point you know: flow, head or fan pressure, and power, each in your own unit. The results come back in the same units.
- Read the new flow, head and power, and see them on the chart of the three laws. For a VFD, add the running hours and the price per kWh.
- Press a preset to load an example, and open Show Work for each ratio and step.
Worked Example
Slowing a pump. A pump gives 100 m³/h at 40 m of head and takes 15 kW at 1,450 rpm. At 1,200 rpm the speed ratio is 1,200 ÷ 1,450 = 0.8276. Flow: 100 × 0.8276 = 82.76 m³/h. Head: 40 × 0.8276² = 40 × 0.6849 = 27.40 m. Power: 15 × 0.8276³ = 15 × 0.5668 = 8.502 kW, 43.32% less. If its NPSH required was 3 m, it falls to about 3 × 0.6849 = 2.05 m.
A fan on a variable-speed drive. A 30 kW fan turned down from 50 Hz to 40 Hz runs at 80% speed, so it moves 80% of the air for 0.8³ = 51.2% of the power: 15.36 kW. Over 6,000 hours a year the 14.64 kW saved is 87,840 kWh, worth 13,176 at 0.15 per kWh.
The common mistake: scaling power with speed, or with speed squared. For the pump above, power ∝ N would give 15 × 0.8276 = 12.41 kW, and power ∝ N² would give 10.27 kW; the cube law gives 8.502 kW. The opposite mistake is using the cube law on a pump that lifts water to a fixed height, where the saving is much smaller than the cube law says.
Show Work
Formulas
From Euler’s Turbine Equation to the Variable-Speed Drive
The affinity laws are a case of dynamic similarity. Leonhard Euler showed in 1754 that the head a rotating impeller gives depends on the speed of its blade tips and how much it turns the flow, so a pump that is only run faster, or a geometrically similar pump made bigger, keeps the same velocity triangles at its blades. Flow then follows the tip speed (N × D) times the flow area, head follows the tip speed squared, and power is their product, which gives the exponents 1, 2 and 3. The same similarity reasoning lets engineers predict a full-size pump, fan or turbine from tests on a small model.
For most of the twentieth century pumps and fans ran at one speed and flow was controlled with valves and dampers, which throw energy away as pressure drop. Power-transistor inverters made variable-frequency drives affordable from the 1980s, and because the cube law turns a small cut in speed into a large cut in power, drives on fans and circulating pumps became one of the standard energy-saving measures in buildings and industry.
About This Tool
This calculator applies the affinity laws to a centrifugal pump or fan: a change of speed, a trim of the impeller, or both, with an optional change of fluid density for fans handling hot or thin air. It can also find the speed for a target flow and estimate the energy and money a variable-speed drive saves by the cube law, and it scales NPSH required as a rough guide. Results come back in the units you entered, and the chart shows where the new point sits on the flow, head and power curves. The laws describe a single pump moving along a system curve through zero; with static lift, or for large impeller trims, use the maker’s performance curves.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Pipe Pressure Drop Calculator, Fluid Flow Rate & Reynolds Number Calculator, and Hydraulic Cylinder Calculator.
Frequently Asked Questions
What are the pump affinity laws?
For the same pump at a different speed N: flow scales with N, head with N² and power with N³. Slowing a pump delivering 100 m³/h at 40 m and 15 kW from 1,450 to 1,200 rpm (ratio 0.8276) gives 82.76 m³/h, 27.40 m and 8.502 kW: 17% less flow for 43% less power.
How much energy does a VFD save on a fan?
Power falls with the cube of speed, so running at 80% speed takes 0.8³ = 51.2% of full power. A 30 kW fan slowed from 50 to 40 Hz draws 15.36 kW, saving 14.64 kW; over 6,000 hours a year that is 87,840 kWh, or 13,176 at 0.15 per kWh.
What about trimming the impeller?
Cutting the impeller diameter D scales flow, head and power by D, D² and D³ in the same way, but only approximately, because the blade shape and clearances change too; trims beyond about 10–20% stray from the laws. Trimming 250 mm to 230 mm (0.92) takes 100 m³/h, 40 m and 15 kW to 92 m³/h, 33.86 m and 11.68 kW.
Do the fan laws work the same way?
Yes: airflow ∝ speed, static pressure ∝ speed², power ∝ speed³, and pressure and power also scale with air density. Speeding a fan from 1,000 to 1,200 rpm takes 10,000 cfm at 1.5 in. w.g. and 5 hp to 12,000 cfm, 2.16 in. w.g. and 8.64 hp, a 72.8% jump in power for 20% more air.
When do the affinity laws not apply?
They move the pump along a parabola through zero, which only matches a system with no static lift. If the system needs, say, 30 m just to raise the water, the pump above slowed to 1,200 rpm cannot reach its scaled point of 82.76 m³/h at 27.40 m, so the real flow drops much more than 17%. NPSH required also rises roughly with speed squared, so speeding a pump up can bring on cavitation.
How do I use the Pump & Fan Affinity Laws Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Energy audits
A 30 kW fan run at 40 Hz instead of 50 Hz for 6,000 h a year saves 87,840 kWh, 48.8% of its energy.
Matching a pump to a lower flow
To bring a 200 gpm, 1,750 rpm pump down to 150 gpm, run it at 1,312.5 rpm; head falls to 56.25 ft and power from 10 hp to 4.219 hp.
Impeller trimming
Trimming a 250 mm impeller to 230 mm cuts head from 40 m to 33.86 m and power by 22.13%, a fixed fix for an oversized pump.
Ventilation upgrades
Asking 20% more airflow from a 5 hp fan needs 8.64 hp, so the motor usually has to be changed too.
Hot-air and altitude corrections
Air 20% less dense (ratio 0.8) at the same speed gives the same 10,000 cfm but only 1.2 in. w.g. and 4 hp.
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