Pump affinity laws
What a speed change does to a centrifugal pump: flow tracks speed, head tracks speed squared, power tracks speed cubed. The cube on power is why slowing a pump down saves so much energy.
Centrifugal pump affinity laws · speed vs. flow, head & power
Q ∝ N · H ∝ N² · P ∝ N³
What this gives you
When you change the speed of a centrifugal pump — by fitting a VFD, swapping a sheave, or re-motoring it — the pump does not just move less fluid. Flow, head, and power all shift together, but on different curves. The affinity laws tie them to the speed ratio r = N₂ ÷ N₁: flow scales with the ratio (Q₂ = Q₁ · r), head with the square (H₂ = H₁ · r²), and power with the cube (P₂ = P₁ · r³). Enter the pump's known duty point at one speed and the new speed, and this tool gives you the duty point you should expect at the new speed.
Speed change or impeller trim
The same laws cover the two common ways of re-rating a pump. Switch the tool to Speed change (VFD) and the ratio is r = N₂ ÷ N₁ from the old and new shaft speeds. Switch it to Impeller trim and the ratio is r = D₂ ÷ D₁ from the original and machined-down impeller diameters — trimming an impeller behaves, to a good first approximation, like slowing the pump by the same proportion. Enter flow in either gpm or m³·h⁻¹; the ratio math is unit-agnostic, so the flow column simply carries whichever unit you pick.
Why the cube on power matters
The exponents are the whole story. Because power follows the cube of speed, a small speed reduction returns a large energy saving: slowing a pump to 80 percent speed drops flow to 80 percent but drops shaft power to about 51 percent (0.80³). That cube law is the reason a VFD trimming a throttled pump back to its actual required flow pays for itself so quickly — you are buying back energy on a cubic curve while giving up flow only linearly. It also works the other way: pushing a pump 15 percent faster to chase more flow raises the power draw by roughly 52 percent, which is how motors get overloaded after a “quick” sheave change.
Field note — the laws assume the same system curve
The affinity laws describe the pump, and they hold exactly only when the pump rides along the same operating point on its curve at both speeds. In a real system the operating point is set by where the pump curve crosses the system curve. If the system is nearly all friction (long pipe, throttled), the prediction tracks well. But if there is a lot of static head — lifting fluid to a fixed elevation or against a set pressure — the pump cannot follow the square-law head curve all the way down, and it will simply stop producing flow (dead-head) at some minimum speed before the math says it should. Treat the numbers as the pump's capability, then check them against the static head the system actually demands.
Worked example
A pump runs 100 gpm at 150 ft while drawing 10 hp at 1750 rpm. Put it on a VFD and slow it to 1450 rpm: the ratio is 1450 ÷ 1750 = 0.829. Flow falls to 100 × 0.829 = 82.9 gpm, head to 150 × 0.829² = 103 ft, and shaft power to 10 × 0.829³ = 5.69 hp. You gave up 17 percent of the flow and got back nearly 43 percent of the power — the cube law at work.