Fan & Pump Affinity Laws
The affinity laws predict how a centrifugal machine — fan, blower or pump — responds to a change in speed, impeller diameter or fluid density. They follow from geometric similarity and constant efficiency, and they are identical in form for fans and pumps. The power law is cubic, which is the entire economic case for variable-speed drives.
Speed Change (same impeller, same fluid)
| Quantity | Relation | Ratio form |
|---|---|---|
| Flow — Q (cfm or gpm) | Q ∝ N | Q₂/Q₁ = N₂/N₁ |
| Head or pressure — H, Δp | H ∝ N² | H₂/H₁ = (N₂/N₁)² |
| Shaft power — P | P ∝ N³ | P₂/P₁ = (N₂/N₁)³ |
| Efficiency — η | ≈ constant | assumed unchanged |
What the Cube Law Actually Buys
| Speed | Flow | Head / pressure | Power | Power saved |
|---|---|---|---|---|
| 100% | 100% | 100% | 100% | — |
| 90% | 90% | 81% | 72.9% | 27% |
| 80% | 80% | 64% | 51.2% | 49% |
| 70% | 70% | 49% | 34.3% | 66% |
| 60% | 60% | 36% | 21.6% | 78% |
| 50% | 50% | 25% | 12.5% | 87.5% |
Read the 80 percent row: giving up a fifth of the flow cuts power roughly in half. This is why throttling a damper or a valve to trim flow is so wasteful compared with slowing the machine — throttling moves you up the head curve instead of down the power curve.
Impeller Diameter Change (same speed)
| Quantity | Relation | Ratio form |
|---|---|---|
| Flow | Q ∝ D | Q₂/Q₁ = D₂/D₁ |
| Head | H ∝ D² | H₂/H₁ = (D₂/D₁)² |
| Power | P ∝ D³ | P₂/P₁ = (D₂/D₁)³ |
Density Change (fans and blowers)
| Quantity | Relation | Practical effect |
|---|---|---|
| Volumetric flow | Q ∝ ρ0 | Unchanged — a fan moves the same cfm regardless of density |
| Static pressure | Δp ∝ ρ | Falls with altitude and with hot air |
| Power | P ∝ ρ | Falls with density — motor sized at sea level is conservative at altitude |
| Mass flow | ṁ ∝ ρ | The quantity that actually matters for heat transfer and combustion |
Fan curves are published at standard air, 0.075 lb/ft³ (roughly 70°F at sea level). At 5,000 ft or in a 400°F flue-gas duct the delivered pressure is materially lower, and selections made straight off the catalogue curve will fall short.
The Trap That Invalidates the Laws
Assumptions Behind the Laws
| Assumption | Where it fails |
|---|---|
| Geometric similarity | Trimmed impellers; different casing |
| Constant efficiency across the change | Large speed turndown moves you off the best-efficiency point |
| Dynamically similar flow (same Reynolds regime) | Very low speeds; viscous fluids |
| Incompressible flow | Blowers and compressors above roughly 7 percent pressure rise |
| No cavitation | NPSH available must still exceed NPSH required at the new point |
Sources: Hydraulic Institute Standards (ANSI/HI 14.6) for centrifugal pump affinity relations. AMCA Publication 201, Fans and Systems. ASHRAE Handbook — HVAC Systems and Equipment, fan chapter. Karassik et al., Pump Handbook. The relations are exact consequences of similarity and constant efficiency; the caveats above are where the underlying assumptions, not the algebra, give way.
Related cheat sheets and tools
The affinity laws only tell you how the machine moves — you still need the system curve. Build it from Hazen-Williams or Darcy-Weisbach roughness, add fittings from minor loss K values, and total it with total dynamic head. For wastewater duty points see lift station sizing, and for the electrical side motor full-load amps and voltage drop. For pumped stormwater systems modelled end to end, see HydroComplete, the SaaS sister product to PE-Calc.