Pump & Fan Affinity Laws Pumps & Fans
One operating point and a speed (or impeller) change, scaled by the affinity laws. Flow tracks the ratio, head tracks its square, and power tracks its cube — which is the whole reason a VFD slowing a pump saves so much more energy than it gives up in flow. By speed is the VFD case; By impeller diameter is the trim case, exact for speed and approximate for small trims of the same casing.
Input
Speeds take any consistent unit (RPM, Hz, %). Flow, head, and power pass through unchanged, so the outputs carry whatever units you entered — head (ft) for pumps, static pressure (in. w.c.) for fans, power in bhp or kW.
Output — at N₂
Worked example
A VFD slows a pump from 60 Hz to 50 Hz — 83 % speed — at an operating point of 100 GPM, 50 ft head, 10 bhp.
- Ratio:
50 ÷ 60 = 0.833. - Flow:
100 × 0.833 = 83.3 GPM(linear). - Head:
50 × 0.833² = 34.7 ft(square). - Power:
10 × 0.833³ = 5.8 bhp(cube).
A 17 % speed cut gives back 17 % of flow but nearly halves the power. The cube law is the energy case for variable-speed pumping — and the reason the savings shrink fast once you stop riding the curve down (see Pump Control).
Input
Diameters take any consistent unit (in, mm). The diameter laws hold only for modest trims of the same casing — past roughly 10–15 % the efficiency falls off and the simple ratios drift.
Output — at D₂
The laws
| Quantity | By speed (N) | By diameter (D) |
|---|---|---|
| Flow Q | ∝ N₂/N₁ | ∝ D₂/D₁ |
| Head / pressure H | ∝ (N₂/N₁)² | ∝ (D₂/D₁)² |
| Power P | ∝ (N₂/N₁)³ | ∝ (D₂/D₁)³ |
Speed is the exact, reversible case — slow a pump and speed it back up and it returns to the same point. Impeller trim is permanent metal and only approximate: good for a one-time match of an oversized pump to its duty, not for control.