Pump Affinity Laws for Variable-Speed Hydronic Systems: A Worked Example
A practical engineering guide to scaling a centrifugal pump curve, checking the system curve and avoiding common variable-speed hydronic design errors.
pump affinity laws
Pump Affinity Laws for Variable-Speed Hydronic Systems: A Worked Example
Original MEPFlow engineering guide
Prepared to help mechanical designers understand the calculation, assumptions and review checks. Examples are original and educational; verify the governing code, project criteria and equipment data before using a result for construction or permit documents.
The pump affinity laws provide a fast way to estimate how a rotodynamic pump's flow, head and power change when rotational speed changes. They are especially useful when reviewing a variable-frequency drive, comparing duty points or checking whether a proposed speed range can reach the required hydronic operating point.
The equations are simple, but they do not replace the pump curve or the system curve. A pump does not independently choose both flow and head. Its actual operating point is where its speed-specific pump curve intersects the resistance curve of the connected system. That distinction is the difference between a useful estimate and a misleading one.
The three pump affinity laws
For the same rotodynamic pump, with the same impeller diameter and approximately similar hydraulic efficiency, flow varies directly with rotational speed, head varies with the square of speed, and power varies with the cube of speed. The Hydraulic Institute and U.S. Department of Energy present these relationships as tools for estimating pump performance when speed changes.
Use a consistent speed basis: revolutions per minute, hertz when motor speed follows frequency closely, or percent speed. Because every equation uses a ratio, the units cancel as long as the same unit is used for both conditions.
Q₂/Q₁ = N₂/N₁; H₂/H₁ = (N₂/N₁)²; P₂/P₁ = (N₂/N₁)³
Q is flow, H is pump head, P is pump input or brake power on a consistent basis, and N is rotational speed. These equations scale corresponding points on the pump curve; they do not by themselves solve the connected system's new operating point.
| Speed ratio | Estimated flow | Estimated head | Estimated power |
|---|---|---|---|
| 1.00 | 100% | 100% | 100% |
| 0.90 | 90% | 81% | 72.9% |
| 0.80 | 80% | 64% | 51.2% |
| 0.70 | 70% | 49% | 34.3% |
Worked example: reducing a hydronic pump from 1,750 to 1,400 rpm
Consider a centrifugal pump operating at 1,750 rpm. At one point on its published curve it delivers 600 US gpm at 55 ft of head and requires 18.5 hp. The proposed variable-speed condition is 1,400 rpm. The speed ratio is 1,400 divided by 1,750, or 0.80.
First estimate flow: Q₂ = 600 × 0.80 = 480 gpm. Then estimate head: H₂ = 55 × 0.80² = 35.2 ft. Finally estimate power: P₂ = 18.5 × 0.80³ = 9.47 hp. Rounded for a preliminary check, the corresponding point becomes 480 gpm at 35 ft with about 9.5 hp.
The power result is appealing—a 20% speed reduction produces an estimated 48.8% reduction at the corresponding curve point—but it is not an energy guarantee. Motor and drive efficiency vary with load, pump efficiency can shift, and the connected system may move to a different point than the simple 480 gpm estimate.
1. Establish the original point
Record the original speed, flow, head and power from a manufacturer curve or verified operating data.
2. Calculate the speed ratio
Divide the proposed speed by the original speed: 1,400 ÷ 1,750 = 0.80.
3. Scale flow
Multiply original flow by the first power of the speed ratio: 600 × 0.80 = 480 gpm.
4. Scale head
Multiply original head by the square of the ratio: 55 × 0.64 = 35.2 ft.
5. Scale power
Multiply original power by the cube of the ratio: 18.5 × 0.512 = 9.47 hp.
6. Check the actual operating point
Plot or obtain the reduced-speed pump curve and intersect it with the system curve before selecting equipment or control limits.
Why the system curve controls the real result
A closed hydronic distribution system is commonly dominated by pipe, fitting, valve, coil and equipment resistance. Turbulent-flow friction is often represented approximately by a relationship in which head varies with flow squared. When both the pump curve and the system resistance behave in this manner, the operating point can track the affinity-law estimate reasonably well.
Systems with a meaningful fixed head do not follow a pure square-law system curve. Examples include open condenser-water systems with elevation between basin and discharge, pressure-maintenance objectives, and processes with a required differential pressure. The pump must satisfy that fixed portion before additional head produces flow through the variable-resistance portion.
H_system = H_static + KQ²
H_static represents the fixed head or pressure requirement, while KQ² approximates flow-dependent resistance. In a closed hydronic loop, building elevation is not normally added as operating friction head, but fixed control or equipment requirements may still affect the curve.
Do not cube the whole system blindly
The cube relationship estimates pump power at corresponding pump-curve points under its assumptions. A system with static head, bypass flow, minimum differential pressure or changing valve positions can produce a different operating path.
Applying the laws to variable-speed hydronic design
Start with the design flow and critical-circuit head. The design point should come from connected terminal loads, pipe and fitting losses, valves, coils and equipment—not from a generic pump allowance. Use the manufacturer's curve to choose a pump whose design point falls within an appropriate operating region, then examine reduced-load conditions.
For a variable-primary or secondary distribution pump, the control objective matters. A differential-pressure sensor near the hydraulically remote circuit can allow the speed to fall as two-way valves close. A sensor at the pump may preserve more pressure than the remote system needs. Resetting the differential-pressure target can reduce excess valve pressure, but the sequence must still protect minimum equipment flow and the authority of control valves.
- Verify the design duty point on the manufacturer's pump curve.
- Review the allowable speed range and every required minimum-flow condition.
- Check motor and drive limits at maximum speed, fluid density and worst operating condition.
- Evaluate net positive suction head available against the manufacturer's requirement.
- Confirm how boilers, chillers, heat exchangers and terminal devices behave as flow changes.
- Document the differential-pressure sensor location and control sequence.
Affinity laws versus trimming an impeller
Speed changes and impeller-diameter changes are related design tools, but they are not automatically interchangeable. Published manufacturer data may use affinity-style relationships to estimate an impeller trim, yet changes in clearances and hydraulic geometry can cause efficiency and performance to depart from a simple similarity calculation.
For an actual selection, use the manufacturer's curve for the available impeller diameter and speed whenever possible. Treat a calculated trim as a screening estimate, especially near the edge of a pump's published range.
Common calculation and selection mistakes
Most affinity-law errors are not arithmetic errors. They come from applying the equations outside their assumptions or confusing a corresponding curve point with the system operating point.
- Scaling head directly with speed instead of with speed squared.
- Scaling power with speed squared instead of with speed cubed.
- Assuming pump efficiency, motor efficiency and drive efficiency remain constant at every load.
- Ignoring fixed head, minimum differential pressure or equipment flow requirements.
- Using a duty point that omits the return path, valves, coils or heat exchangers.
- Treating building height as operating pump head in a filled closed loop.
- Selecting from the affinity calculation without checking the published pump curve, operating region and NPSH.
A practical review sequence
The affinity laws work best inside a connected hydronic workflow. Calculate terminal flow from the heating or cooling load and design temperature difference. Accumulate those flows through the pipe network. Size the pipes, identify the critical circuit, and calculate its required head. Only then use the pump curve and affinity laws to study alternate speeds and part-load operation.
MEPFlow's hydronic workspace can help keep loads, flows, pipe sizes and circuit losses connected on the plan. Pump selection, controls, equipment limits and project requirements still need engineering review using current manufacturer data.
Frequently asked questions
What are the three pump affinity laws?
For the same rotodynamic pump and impeller, flow is proportional to speed, head is proportional to speed squared, and power is proportional to speed cubed, subject to approximate similarity and efficiency assumptions.
What happens when pump speed is reduced by 20%?
At a speed ratio of 0.80, corresponding pump-curve estimates are 80% flow, 64% head and 51.2% power. The connected system curve determines the actual operating point.
Can affinity laws predict variable-speed pump energy savings?
They provide a useful preliminary power estimate, but actual energy depends on the system curve, operating hours, controls, pump efficiency, motor and drive efficiency, minimum-flow requirements and fixed head.
Do pump affinity laws apply to closed hydronic systems?
Yes, they can scale the curve of a rotodynamic pump used in a closed loop. The system operating point still requires the closed-loop resistance curve and equipment or control requirements.
Should building height be included in closed-loop pump head?
Normally not as operating friction head for a filled closed loop. The pump overcomes circuit and component resistance, while static fill pressure and expansion control are separate design considerations.
Continue the engineering workflow
Put this guide into practice with MEPFlow
Move from the calculation method into a connected browser-based design workflow while keeping the assumptions visible for engineering review.
Explore hydronic design softwarePrimary references
Use the edition and method accepted for your project. These authoritative resources provide further context; this article is educational and is not a code-compliance determination.