Affinity Laws and Variable Speed

6.6.1Three exponents, one similarity assumption

For the same impeller handling the same liquid at geometrically similar points, the affinity laws give Q₂/Q₁ = N₂/N₁, H₂/H₁ = (N₂/N₁)², and P₂/P₁ = (N₂/N₁)³. Flow follows speed, head its square, and shaft power its cube.

speed ratio r = 1,350/1,800 = 0.75
Q₂ = 100 L/s × 0.75 = 75.0 L/s
H₂ = 40.0 m × 0.75² = 22.5 m
P₂ = 60.0 kW × 0.75³ = 25.31 kW

Reduce 1,800 rpm to 1,350 rpm

These relations move corresponding points on the pump curve. They do not guarantee the system will operate at 75 L/s, because the new pump curve must still intersect the unchanged system curve. Efficiency, motor/drive losses, minimum speed, cooling, vibration, and NPSH also limit useful range.

6.6.2Static head creates a speed floor

A system curve can be written Hs(Q) = Hstatic + kQ². Speed changes the pump curve; Hstatic remains the elevation difference between water surfaces. If the reduced-speed shutoff head falls below static lift, the pump has no forward-flow intersection at all.

For a full-speed shutoff head of 40.0 m and static lift of 25.0 m, the theoretical minimum speed ratio is √(25/40) = 0.7906. Below 79.1% speed, even shutoff head cannot reach the receiving surface. Real controls require margin above that limit and must respect the check valve and transient behaviour; the equality is a boundary, not an operating setpoint.

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6.6.3When a VFD saves energy

In a friction-dominated loop, reducing flow removes most of the required head, so the cubic power trend can be approached. In a static-dominated lift station, most head remains at low flow; speed has little room to fall before the curve loses its intersection. The US Department of Energy’s Variable Speed Pumping guide makes this system distinction explicit.

The correct comparison is annual energy at the expected operating distribution, including drive and motor efficiency, against throttling, on/off storage, staging, or a better-sized impeller. Include minimum scouring velocity for wastewater, motor cooling at low speed, harmonic and bypass requirements, control instrumentation, and the cost of added failure modes.

6.6.4A defensible variable-speed decision

1. Draw the minimum, normal, and maximum system curves, separating static and friction head.
2. Generate speed curves from certified data and find each operating intersection.
3. Check every point against preferred region, NPSH margin, motor power, cooling, and minimum useful speed.
4. Integrate wire power over the load-duration distribution, including drive losses.
5. Compare against fixed-speed staging and impeller selection; state the control and bypass philosophy.