Engineering
Centrifugal Pump & Fan Affinity Laws: Flow (N), Head (N^2) & Cube Power (N^3)
Explore ideal speed scaling for flow, head and shaft power at corresponding centrifugal-pump operating points.
Separate affinity scaling from electricity-bill savings
At constant impeller geometry and fluid density, corresponding operating points approximately scale as Q proportional to speed, H to speed squared and shaft power to speed cubed when efficiency is unchanged. The example compares 1800 and 1440 RPM: the predicted shaft-power ratio is 0.512.
This is not a measurement of motor electrical input or guaranteed energy savings. Motor and drive efficiency, operating hours and delivered duty also matter. Actual system operating points depend on the pump curve and system curve, including static head; scaling a single duty point does not solve that intersection.
Head is energy per unit weight, reported here in meters, not pressure in Pa. Pressure conversion requires density and gravity. The same ideal scaling can be useful for fans under similarity assumptions, but this workbook uses pump units.
The 2160 RPM scenario is an arithmetic comparison, not permission to exceed a rated speed. Real changes require manufacturer operating limits, efficiency and cavitation checks. Baseline speed is positive; target speed and baseline quantities must be nonnegative. Zero target speed is only the model's zero-speed limit, not a prediction of all system flows or electrical standby use.
| Affinity Law Quantity | Scaling Equation | 1,800 -> 1,440 RPM (0.80x) | Overspeed 2,160 RPM (1.20x) |
|---|---|---|---|
| Baseline Rated Duty | B2=1800, Q1=100, H1=40, P1=15 | VFD Target B6 = 1,440 RPM | VFD Target = 2,160 RPM |
| Law 1: Flow Rate Q2 | Q1 * (N2 / N1) | 80.00 m^3/h (-20%) | 120.00 m^3/h (+20%) |
| Law 2: Dynamic Head H2 | H1 * (N2 / N1)^2 | 25.60 m (-36%) | 57.60 m (+44%) |
| Law 3: Shaft Power P2 | P1 * (N2 / N1)^3 | 7.68 kW (48.8% lower modeled shaft power) | 25.92 kW (72.8% higher modeled shaft power) |
Cube-Law VFD Shaft Power P2 = P1 * (N2 / N1)^3 in kW
The output estimates shaft power under the stated similarity assumptions. It does not report real-time measured motor demand.
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<0,B4<0,B5<0,B6<0),"Invalid inputs",ROUND(B5 * ((B6 / B2) ^ 3), 2)))Scaled Volumetric Flow Q2 and Total Head H2
The head formula uses the square of the speed ratio; the separately shown flow formula uses its first power.
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<0,B4<0,B5<0,B6<0),"Invalid inputs",ROUND(B4 * ((B6 / B2) ^ 2), 2)))Scaled volumetric flow in m^3/h
This is the corresponding-point flow under the same assumptions, not a solved system duty.
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<0,B4<0,B5<0,B6<0),"Invalid inputs",ROUND(B3 * (B6 / B2), 2)))Try the example
Use the original values or change them to test your understanding.