The affinity laws are three equations that describe how centrifugal pump performance changes with speed. They are arguably the most important relationships in plant engineering because they explain why variable frequency drives produce such dramatic energy savings. A small reduction in pump speed yields a cubic reduction in power consumption.
These laws apply to all centrifugal machines — pumps, fans, and blowers — and they hold reasonably well for impeller diameter changes within about 10 to 15 percent of the original size. Beyond that range, the geometry changes enough that the laws become approximations rather than reliable predictions.
This guide covers each affinity law, demonstrates the calculations with real numbers, explains impeller trimming, and shows how to read and shift pump curves to match system requirements.
The Three Affinity Laws
The first law states that flow is directly proportional to speed: Q2 = Q1 x (N2 / N1). If you reduce pump speed by 20 percent (from 1750 to 1400 RPM), flow drops by 20 percent. This is a linear relationship and the easiest to understand.
The second law states that head (pressure) varies with the square of speed: H2 = H1 x (N2 / N1)². That same 20 percent speed reduction cuts head by 36 percent (0.8² = 0.64). This is why throttling a pump with a valve is wasteful — the pump still produces full head and the valve simply burns it off as friction and noise.
The third law states that power varies with the cube of speed: P2 = P1 x (N2 / N1)³. The 20 percent speed reduction drops power consumption by 49 percent (0.8³ = 0.512). This cubic relationship is the entire economic justification for VFDs on centrifugal pump applications. A pump running at 80 percent speed uses roughly half the energy.
All three laws assume the system curve remains unchanged and that the pump is operating at or near its best efficiency point. They do not account for changes in static head, which remains constant regardless of speed.
VFD Savings and Payback Calculations
To calculate VFD savings, compare the power consumed at reduced speed to the power consumed at full speed with throttling. A 50 HP pump running at full speed with a partially closed discharge valve might deliver 70 percent of rated flow while still drawing close to 45 HP. The same pump on a VFD at 70 percent speed would draw roughly 50 x 0.7³ = 17.2 HP — a savings of about 28 HP continuously.
At an electricity cost of $0.10 per kWh, 28 HP saved equals 20.9 kW, which translates to roughly $18,300 per year for a pump running 8,760 hours. A VFD for a 50 HP motor typically costs $5,000 to $8,000 installed, producing a payback period well under six months.
The savings are less dramatic when significant static head is present. Static head is the pressure required just to lift the fluid to the discharge elevation, regardless of flow rate. In a system where static head accounts for 60 percent or more of total head, the cubic power savings diminish considerably because the pump cannot reduce speed below the point needed to overcome static head.
For systems with high static head, a VFD still saves energy but the payback period is longer. Always model the actual system curve, including static and friction components, before quoting savings numbers to management.
Impeller Trimming and Diameter Changes
The affinity laws also apply to changes in impeller diameter, substituting D2/D1 for N2/N1 in each equation. Trimming an impeller (machining it to a smaller diameter) permanently reduces the pump's maximum flow and head, moving the entire pump curve downward and to the left.
Impeller trimming is a one-time, low-cost alternative to a VFD when the required operating point is fixed and will not change. It is commonly used when a pump was oversized at installation or when system conditions have permanently changed. The maximum recommended trim is typically 10 to 15 percent of the original diameter. Beyond that, the vane exit angle changes enough that the affinity law predictions become unreliable and pump efficiency drops off.
To determine the required trim diameter, plot the system curve on the pump curve and identify the desired operating point. Then use the affinity laws to back-calculate the diameter ratio: D2 = D1 x sqrt(H2 / H1) for the head requirement, or D2 = D1 x (Q2 / Q1) for the flow requirement. If the two calculations give different answers, use the larger diameter (less trim) and accept slightly more flow or head than the minimum target.
After trimming, the impeller should be dynamically balanced and the pump performance tested against the predicted values. Document the final trim diameter on the pump's maintenance record for future reference.
Reading and Shifting Pump Curves
A pump performance curve plots head (in feet or meters) on the vertical axis against flow (in GPM or m³/h) on the horizontal axis. The curve slopes downward from left to right: at zero flow (shutoff), the pump produces maximum head; at maximum flow (runout), head drops to near zero. Superimposed on this are efficiency contours, power curves, and NPSH required curves.
The best efficiency point (BEP) is the flow rate at which the pump operates most efficiently, typically 80 to 87 percent for well-designed process pumps. Operating within 80 to 110 percent of BEP flow is the preferred operating window. Below 70 percent of BEP, recirculation inside the impeller causes vibration, noise, and accelerated wear. Above 120 percent of BEP, cavitation and shaft deflection become concerns.
When you change pump speed with a VFD, the entire curve shifts. Each point on the original curve maps to a new point using the affinity laws: the flow coordinate scales linearly with speed ratio, and the head coordinate scales with the square of the speed ratio. Connecting these shifted points produces the new pump curve at the reduced speed.
The system curve — which plots the head required by the piping system against flow — is a parabola starting at the static head value and curving upward. The pump operating point is where the pump curve intersects the system curve. Reducing pump speed shifts the pump curve down, moving the intersection point to a lower flow rate along the same system curve.
Limitations and Common Mistakes
The affinity laws assume an ideal centrifugal machine. Real-world deviations occur due to mechanical losses, seal friction, and changes in Reynolds number at different speeds. Below about 30 percent of rated speed, most centrifugal pumps lose efficiency rapidly and may not produce useful head. VFD applications should generally not reduce speed below 30 to 40 percent of nameplate RPM.
A common mistake is applying the affinity laws to positive displacement pumps. PD pumps (gear, piston, diaphragm) deliver flow proportional to speed regardless of pressure, and their power consumption is proportional to flow times pressure, not the cube of speed. The affinity laws do not apply to PD pumps.
Another error is ignoring the minimum flow requirement. Every centrifugal pump has a minimum continuous stable flow below which internal recirculation causes damage. When using a VFD to throttle flow, the control system must include a minimum speed setpoint or a recirculation line to protect the pump.
Finally, applying the affinity laws to systems with high static head without accounting for it leads to overestimated savings. The cubic power law applies only to the friction component of the system head. Static head is constant and requires a fixed amount of power regardless of speed, which compresses the available savings range.