Centrifugal Pump Affinity Laws Calculator
Scale the operating point of a centrifugal pump to a changed speed or a trimmed impeller diameter. Enter the base point Q1, H1, P1 and choose the target n2 or D2 – the calculator returns flow rate Q2, head H2 and power P2 from the affinity laws, live with every input.
Affinity Laws Calculator (Centrifugal Pump)
Model: centrifugal pump affinity laws under geometrically similar flow and approximately constant efficiency. Impeller trimming is an approximation for small trims. For reliable sizing use the manufacturer pump curve; a static head component of the system is not considered here.
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Formulas and fundamentals
The affinity laws relate two similar operating points of a centrifugal pump through the ratio k of the governing quantity. For a speed change k = n2/n1, for impeller trimming approximately k = D2/D1. Flow rate scales linearly, head quadratically and shaft power cubically: Q2 = Q1·k, H2 = H1·k² and P2 = P1·k³. Because H scales with k² and Q with k, both operating points lie on the same affinity parabola through the origin, H = c·Q².
The speed law is exact as long as the pump geometry is unchanged and the efficiency stays approximately constant over the speed range. Impeller trimming, in contrast, is an approximation: only the outer contour of the impeller is reduced while the channel geometry remains – so the simple k = D2/D1 relation holds only for small trims of roughly 10 to 20 percent, beyond which the efficiency drops and the approximation becomes unreliable.
The cubic power dependence explains the large saving potential of variable speed drives: halving the speed drops the power to one eighth in theory. In practice the system resistance is rarely a pure parabola through the origin – a static head component shifts the real operating point, so the actual saving is smaller than the ideal cubic curve suggests.
Worked example
A centrifugal pump delivers Q1 = 50 m³/h against H1 = 30 m at n1 = 1450 rpm, requiring P1 = 5 kW. The speed is raised to n2 = 1740 rpm, giving a ratio k = 1740/1450 = 1.2.
The flow rate grows linearly to Q2 = 50·1.2 = 60 m³/h, the head quadratically to H2 = 30·1.2² = 43.2 m and the power cubically to P2 = 5·1.2³ = 8.64 kW.
So the power rises by 73 percent while the flow rate increases by only 20 percent. This is the core message of the affinity laws: small speed changes affect the shaft power disproportionately – the lever of variable speed control for energy saving.
Frequently asked questions
Why does power scale with the third power?
Hydraulic power is the product of flow rate and head (P ~ Q·H). Because Q scales linearly with k and H quadratically with k², power scales with the third power: P ~ k·k² = k³. That is why even a small speed change has a strong effect on the power demand.
Do the laws apply equally to speed and impeller diameter?
The exponents are identical (Q ~ k, H ~ k², P ~ k³), but the speed law is exact while trimming is only approximate. Trimming reduces only the outer diameter, the channel geometry stays – so the simple k = D2/D1 relation is reliable only for small trims.
How far may an impeller be trimmed?
Typical trims reach about 10 to 20 percent of the diameter. Within that range the approximation with roughly constant efficiency is usable. For larger trims the efficiency drops noticeably and the manufacturer curve should be used.
Does the efficiency really stay constant?
Approximately yes, as long as the change is small. For speed changes the efficiency is nearly constant over a wide range; for trimming and large jumps it deviates increasingly. The calculator assumes constant efficiency and shows a note for large ratios.
Why is the real energy saving less than k³?
The ideal cubic curve only holds if the system curve is a pure parabola through the origin. A static head component shifts the operating point, so at reduced speed the pump needs more power than the pure affinity calculation predicts.
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