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Fan Affinity Laws

Predict new operating point from a known one when speed, impeller diameter, or fluid density changes. Same equations work for centrifugal pumps and fans. Used for VFD energy savings analysis, fan trim, and altitude correction.

cfm or m³/h or gpm
in. wg, kPa, or ft of fluid
hp or kW
RPM or in/m or kg/m³
RPM or in/m or kg/m³
—— same units as Q1
—— same units as P1
—— same units as BHP1
—% (BHP_2 / BHP_1)

Defaults: 10,000 cfm at 3 in. w.g. and 7.5 hp, slow from 1750 to 1450 RPM (≈ 17% slowdown). Power drops ~43% — the cube law payoff for VFD speed control.

Speed change at constant diameter and density:
$$ \frac{Q_2}{Q_1} = \frac{N_2}{N_1}, \quad \frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{\!2}, \quad \frac{BHP_2}{BHP_1} = \left(\frac{N_2}{N_1}\right)^{\!3} $$
Diameter change at constant speed and density:
$$ \frac{Q_2}{Q_1} = \left(\frac{D_2}{D_1}\right)^{\!3}, \quad \frac{P_2}{P_1} = \left(\frac{D_2}{D_1}\right)^{\!2}, \quad \frac{BHP_2}{BHP_1} = \left(\frac{D_2}{D_1}\right)^{\!5} $$
Density change (e.g., altitude correction):
$$ Q_2 = Q_1, \quad \frac{P_2}{P_1} = \frac{\rho_2}{\rho_1}, \quad \frac{BHP_2}{BHP_1} = \frac{\rho_2}{\rho_1} $$
Q volumetric flow · P total pressure (or head for pumps) · BHP brake horsepower · N rotational speed · D impeller diameter · ρ fluid density.

The cube law and VFD economics

Power scales with the cube of speed. Cutting fan speed in half cuts power to 1/8. This is why variable-frequency-drive (VFD) control on fans and pumps almost always pays back, especially when the system actually varies its load. A constant-speed fan throttling 20% of its flow with a damper might use 90% of full power. The same fan on a VFD doing the same flow uses 50% (= 0.8³) of full power.

For continuous service, the energy savings often pay for the VFD in 1–3 years. Add: smaller starting current, soft starts, less mechanical stress on belts and bearings, and the VFD becomes the standard solution for any motor > 5 hp on variable-load service.

Affinity laws assume constant efficiency

The pure affinity laws assume η_pump = constant across the speed range. This is a good approximation when the operating point traces a "system curve" through the BEP region. Far from BEP, efficiency drops and the affinity laws over-predict performance by 5–15%.

For wide-range speed control, plot the family of fan curves at multiple speeds (each shifted by affinity laws) and find the actual duty point where each crosses the system curve. AMCA Pub. 200 has worked examples.

Diameter trim — limits

Centrifugal pumps and fans can have impellers trimmed to reduce capacity without speed change. Trim is limited to about 75–80% of the maximum diameter — beyond that, the curve flattens, efficiency drops sharply, and the relationships above no longer apply. Trim is permanent (have to swap impeller back for design changes) so VFD is preferred for variable service.

Density correction (altitude, temperature)

Fans see ρ_2/ρ_1 directly proportional to pressure rise and BHP. A fan rated at sea level (ρ = 1.20 kg/m³) installed at 5,000 ft altitude (ρ = 1.05) delivers the same flow at 87% of rated pressure and 87% of rated BHP. Specify fan and motor with this correction at the actual installation density, not at standard conditions.

What's NOT in affinity laws

Reference: AMCA Publication 200 (2010). Air Systems. ASHRAE Handbook (2024), Fundamentals, ch. 21. Hydraulic Institute (2021). HI Pump Standards.

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