The three laws
For a centrifugal pump operating on the same impeller, with similar dimensionless conditions, scaling speed (N) or impeller diameter (D) predicts:
Q₂ / Q₁ = (N₂ / N₁) or (D₂ / D₁) — flow scales linearly
H₂ / H₁ = (N₂ / N₁)² or (D₂ / D₁)² — head scales as the square
P₂ / P₁ = (N₂ / N₁)³ or (D₂ / D₁)³ — shaft power scales as the cubeA fourth implicit relation: efficiency stays roughly constant. The new operating point lies on a parabola through the origin (Q² ∝ H) when only speed or only diameter changes.
These are the same laws you use to:
- Predict reduced-speed operation under a VFD
- Re-rate a pump after an impeller trim
- Move a pump between 60 Hz and 50 Hz service
- Validate a manufacturer-supplied performance curve at a non-tested speed
Speed change (the cleaner case)
A pump runs at 1,760 rpm and delivers 800 gpm at 90 ft TDH consuming 22 BHP. Drop the speed to 1,400 rpm (about 79.5%):
Q₂ = 800 · (1400/1760) = 636 gpm
H₂ = 90 · (1400/1760)² = 56.9 ft
P₂ = 22 · (1400/1760)³ = 11.1 BHPThis holds if the pump still operates near its original BEP-ratio point. It is exact for the *similar* operating point at the new speed — not for any arbitrary point on the new curve.
In particular: the new pump curve does *not* shift uniformly. Every point on the new curve corresponds to a unique parabola through the origin from a point on the old curve. The curves diverge most strongly at low flow (near shutoff) and high flow (near runout).
Impeller trim (the messy case)
The diameter affinity is an approximation valid for small trims (≤ 15-20%) on radial-flow impellers. For axial-flow or mixed-flow pumps, the diameter relationships break down because the trim changes the discharge geometry, not just the tip speed.
Real-world adjustment: many references recommend the Stepanoff exponent correction — use 2.0 for head and 3.0 for power only when D₂/D₁ > 0.85. For deeper trims, head exponent climbs toward 2.3-2.5 and the result understates the head drop.
In practice: pull the manufacturer's trimmed curves directly when they exist (most manufacturers publish curves at 2-4 trims). Use the affinity laws only when you need to go between published trims or extrapolate slightly outside them.
What the laws don't predict
NPSHr. The Hydraulic Institute gives NPSHr ∝ N^1.8 to 2.0 — usually rounded to N². So at half-speed, NPSHr drops by ~75%. This is actually good for low-suction-pressure systems. The reason it's not exactly 2 is that the cavitation-onset mechanism depends on impeller-eye flow patterns that don't scale perfectly.
Efficiency. Real efficiency drops a few points when you move far from the original operating regime. A pump tested at 78% efficiency at design speed might run 75-77% at 60% speed even at the homologous (similar) point. For deep trims, efficiency can drop 2-5 points.
Minimum continuous flow. Manufacturer-published minimums are usually given in absolute gpm at full speed. At reduced speed the minimum scales with the speed ratio, so a pump rated 100 gpm minimum at 1,760 rpm needs ~80 gpm at 1,400 rpm. Some operators incorrectly hold the absolute minimum and end up flow-throttling unnecessarily.
Mechanical considerations. Bearings, seals, and shaft critical speeds care about absolute rpm, not the affinity laws. Don't run a 1,200-rpm-rated bearing pump at 1,800 rpm just because the affinity laws say flow scales nicely.
Worked example: trimming an impeller
A 5x4-8 end-suction pump, full-trim 8.0" impeller, delivers 600 gpm at 65 ft. Field demand is only 480 gpm at the same TDH. The system curve is fixed; the pump has been operating to the right of BEP and the engineer wants it pulled back.
Estimate the trim needed:
Q_ratio = 480 / 600 = 0.80
H_at_480 ≈ 65 · (Q_ratio)² on the constant-system-curve parabola → not directly the affinity relationHere you have to be careful. The system curve says: if Q drops to 480 from 600 on the same friction-dominant system, head also drops with Q². But that means the new operating point is on the system curve, not on the original pump curve.
Use the affinity laws to find a new pump curve that crosses the system curve at 480 gpm. From similarity:
D_new / D_old ≈ Q_ratio = 0.80 → D_new = 6.4"
H_new(at 480) ≈ 65 · 0.80² = 41.6 ftBut your system curve at 480 gpm gives:
H_system(480) = static + friction · (480/600)²If static is 30 ft and the original pump delivered 65 ft against 35 ft of friction at 600 gpm, the friction component at 480 gpm is 35 · 0.64 = 22.4 ft. System at 480: 30 + 22.4 = 52.4 ft.
The trimmed 6.4" impeller's new curve produces 41.6 ft at 480 gpm. 41.6 < 52.4 — the pump can't make the required head at the new flow. A pure trim isn't sufficient because the system has too much static head.
This is exactly the kind of mistake the affinity laws catch quickly: trims work cleanly for friction-dominant systems and fail for static-dominant ones. Look at the static-head fraction before you propose a trim.
When to use affinity laws vs. real curves
| Situation | Use affinity laws? | |---|---| | VFD trim within manufacturer's curve set | Yes, but verify at slowest speed | | Impeller trim < 10% on radial-flow pump | Yes, within Stepanoff correction | | Impeller trim > 20% | No, get a manufacturer cut curve | | Axial-flow / mixed-flow trim | No, geometry-dependent | | 60 Hz → 50 Hz conversion | Yes, but check NPSHr and motor cooling | | Predicting NPSHr at new speed | Approximation: NPSHr₂ = NPSHr₁ · (N₂/N₁)^1.8-2.0 | | Predicting BHP at runout | Yes, but pump BHP near shutoff or runout is least reliably predicted |
The two-line rule: affinity laws are great for the design operating point and similar duty conditions; they get progressively less accurate as you move toward shutoff or runout, and they don't predict mechanical limits at all.
References
- Hydraulic Institute. *ANSI/HI 1.3 — Rotodynamic Centrifugal Pumps for Design and Application.*
- Stepanoff, A. J. *Centrifugal and Axial Flow Pumps,* 2nd ed. (Krieger reprint, original Wiley.)
- Karassik, I. J., et al. *Pump Handbook,* 4th ed. McGraw-Hill — affinity laws and similarity chapters.