Pump Affinity Laws Calculator
Scale pump flow, head, and power when speed or impeller diameter changes, using the affinity laws.
What it does
A centrifugal pump has a performance curve that ties flow, head, power, and efficiency together at one speed. Change the speed, and the whole curve moves. The affinity laws describe how it moves without a new test for every operating point. They are among the most useful shortcuts in pumping, because a small speed reduction can cut power use by far more than the flow reduction suggests.
The three laws use the same ratio. If the new speed is 80 percent of the old speed, flow becomes 80 percent, head becomes 80 percent squared, or 64 percent, and shaft power becomes 80 percent cubed, or 51.2 percent. The same exponents apply when an impeller is trimmed, with the diameter ratio replacing the speed ratio, as long as the trim is modest and the pump geometry stays similar.
The power law is the reason variable speed drives pay for themselves in many systems. A pump that runs at 80 percent speed delivers 80 percent flow and uses about half the power. A pump at 60 percent speed delivers 60 percent flow and uses about 22 percent of the power. The saving is real only when the system does not need the old pressure, so the operating point has to be checked on the actual system curve.
The laws are approximations. They assume the pump stays geometrically similar, efficiency stays close to the original value, and the system curve does not change shape. A system with a large static head, such as a tall building or a deep well, will not follow the cube law over the whole range. At very low speeds, efficiency falls, and the motor may not have enough cooling.
Use the calculator for a first estimate, then confirm the new point on the manufacturer's curve and check NPSH, motor load, shaft stress, and minimum flow. The result is a planning number, not a guarantee. For a large change, a pump specialist should review the curve and the system together.
How to use
- Choose imperial or metric units before entering the current values.
- Select whether the change comes from shaft speed or impeller diameter.
- Enter the current flow, head, and shaft power at the known operating point.
- Enter the current and new speed, or the current and new impeller diameter.
- Read the new flow, head, power, and the percentage power change.
- Plot the new point on the pump curve and check the system curve, NPSH, and motor limits.
Formula
- Q
- Volumetric flow, in gpm or m3/h.
- H
- Pump head, in ft or m.
- P
- Shaft power, in hp or kW.
- N
- Shaft speed, in rpm.
- D
- Impeller diameter, in in or mm.
- 1 and 2
- Subscripts for the current and new operating points.
The affinity laws come from dimensional analysis of geometrically similar pumps. Flow scales with the characteristic velocity times the characteristic area, head scales with velocity squared, and power scales with flow times head. The cube law for power therefore follows from the first two laws when efficiency is constant. Impeller diameter and speed are not interchangeable in every case: a trimmed impeller changes the gap between the impeller and the casing, so the diameter laws are less exact than the speed laws, especially for large trims.
Worked example
Example: 500 gpm at 1750 rpm slowed to 1400 rpm
A pump delivers 500 gpm at 120 ft of head and draws 30 hp at 1750 rpm. The drive is set to 1400 rpm.
- Find the speed ratio: 1400 / 1750 = 0.80.
- New flow: 500 x 0.8 = 400 gpm.
- New head: 120 x 0.8 squared = 120 x 0.64 = 76.8 ft.
- New power: 30 x 0.8 cubed = 30 x 0.512 = 15.36 hp.
- Power saved: 30 - 15.36 = 14.64 hp, or 48.8 percent.
- Metric mirror: 113.6 m3/h becomes 90.9 m3/h, 36.58 m becomes 23.41 m, and 22.37 kW becomes 11.45 kW.
- Check the 1400 rpm curve and the system curve before running at the new point.
Reference table
| Ratio | Flow factor | Head factor | Power factor |
|---|---|---|---|
| 0.6 | 60 % | 36 % | 21.6 % |
| 0.7 | 70 % | 49 % | 34.3 % |
| 0.8 | 80 % | 64 % | 51.2 % |
| 0.9 | 90 % | 81 % | 72.9 % |
| 1.0 | 100 % | 100 % | 100 % |
Limits
- The laws assume geometric similarity and nearly constant efficiency.
- A large static head changes the operating point, so the cube law can overstate savings.
- Very low speeds reduce efficiency and may starve the motor cooling fan.
- The calculation does not check NPSH, cavitation, shaft stress, or motor service factor.
Common mistakes
- Applying the cube law to head instead of power.
- Using the flow ratio to estimate power.
- Ignoring static head in a system that lifts water to a fixed height.
- Mixing gpm with m3/h, or ft with m, in one calculation.
- Trimming an impeller far beyond the manufacturer's recommended range.
Try it yourself
Quick checks. Answers sit tucked under each question.
A pump runs at 800 gpm, 100 ft, and 40 hp at 1200 rpm. What happens at 900 rpm?
Ratio = 900 / 1200 = 0.75. Flow = 600 gpm, head = 56.25 ft, power = 16.88 hp. Power is 57.8 percent lower.
A metric pump runs at 200 m3/h, 30 m, and 15 kW at 1450 rpm. What happens at 1160 rpm?
Ratio = 0.8. Flow = 160 m3/h, head = 19.2 m, power = 7.68 kW. Power is 48.8 percent lower.
An impeller is trimmed from 10 in to 9 in. What happens to flow, head, and power?
Ratio = 0.90. Flow is 90 percent, head is 81 percent, and power is 72.9 percent of the original values.
Questions
What are the pump affinity laws?
Three scaling rules: flow follows the speed or diameter ratio, head follows its square, and power follows its cube.
Do the affinity laws work for fans?
Yes, the same relationships apply to fans and blowers, with the same assumptions about similarity and efficiency.
Why is power proportional to the cube?
Power is flow times head. Flow scales with the ratio and head scales with the ratio squared, so their product scales with the ratio cubed.
What if the system has a large static head?
The operating point will not follow the cube law exactly. Plot the pump curve and the system curve to find the real flow and power.
Can I change speed and impeller diameter at the same time?
Yes, multiply the two ratios together and apply the combined ratio to flow, its square to head, and its cube to power.
How accurate are the results?
They are close for modest changes in geometrically similar pumps. Large changes, high static head, or viscosity differences need a manufacturer's curve.
Sources
- US Department of Energy: Adjustable speed pumping applications
- University of Hong Kong: Fans and Pumps lecture notes