Pump/Fan Affinity Laws

Scale flow, head/pressure, and power when a centrifugal pump or fan changes speed or impeller diameter.

InputQ₂/Q₁ = r , H₂/H₁ = r² , P₂/P₁ = r³ where r = N₂/N₁ or D₂/D₁

Original flow Q₁ (m³/h)or enter gpm — converts to m³/hOriginal head/pressure H₁ (m)or enter ft — converts to mOriginal power P₁ (kW)or enter hp — converts to kWChange typeOriginal speed/diameter ((N or D))New speed/diameter ((N or D))

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The engineering

The affinity laws let you predict a centrifugal pump or fan at a new operating point from one known point: flow tracks the speed (or diameter) ratio directly, head tracks its square, and shaft power tracks its cube. Reach for them when trimming an impeller or dropping a VFD to a new RPM and you need the new duty and motor load.

The cube on power is the headline: a 10% speed reduction drops power to ~0.73 — nearly a 27% energy saving — which is why VFDs pay off fast on variable-flow HVAC loops. The gotcha is that diameter scaling is only accurate for modest trims (roughly within 10–15% of full diameter) since blade geometry and efficiency drift, and the laws assume constant efficiency and no cavitation or system-curve shift.

Where this math comes from

The affinity laws fall straight out of dimensional analysis of turbomachinery — the same head, flow, and power coefficients that Rayleigh and Buckingham formalized around 1914. Pump and fan builders had been using the proportionalities empirically well before that, matching impellers to duties by scaling test data across speeds and sizes.

They became standard engineering shorthand through the mid-20th-century handbooks. Igor Karassik, longtime chief engineer at Worthington and author of the Pump Handbook (first edition 1976), codified the speed and diameter versions as the everyday tool bench engineers still use for impeller trims and speed changes.

1914Lord Rayleigh / Edgar BuckinghamDimensional analysis (Π-theorem) grounds the flow/head/power coefficients.
1937A. J. StepanoffCentrifugal-pump texts formalize similarity scaling for design use.
1976Igor Karassik et al.Pump Handbook codifies the affinity laws as standard practice.
2004Hydraulic Institute (ANSI/HI)Standards pin speed and diameter affinity relations for rating pumps.

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