Rotating Equipment

Pump Affinity Laws: Speed & Impeller Trim Fundamentals

Scale a centrifugal pump's flow, head and power when you change its speed or impeller diameter. Q ∝ r, H ∝ r², P ∝ r³ — with the trim limits and VFD-vs-trim trade-offs that matter in practice.

Launch Calculator Impeller trim limits

Flow

Q ∝ r

Flow scales directly with the speed or diameter ratio.

Head

H ∝ r²

Head scales with the square of the ratio.

Power

P ∝ r³

The cube law — halving speed cuts power to one-eighth.

Contents

Use this guide when you need to:

  • Re-rate a pump to a new speed or impeller diameter.
  • Predict the power change before changing speed.
  • Decide between a VFD and an impeller trim.

1. Overview & Key Concepts

The affinity laws (also called the pump similarity or pump scaling laws) predict how a centrifugal pump's flow, head and power change when you change its rotational speed or its impeller diameter. They let you take one measured operating point and project a whole new performance curve without re-testing the pump.

They follow from dimensional analysis of geometrically similar machines: for a fixed impeller geometry, the dimensionless flow, head and power coefficients are constant, so the dimensional quantities scale with simple powers of a single ratio r.

The scaling ratio r

Change Ratio r Held constant
Speed change r = N2 / N1 Impeller diameter (same impeller)
Impeller trim r = D2 / D1 Rotational speed (same motor/gear)
One idea, two uses: whether you slow the pump down with a variable-frequency drive or machine the impeller smaller, you are multiplying the impeller's "effective size or speed" by the same ratio r — and flow, head and power respond to r in exactly the same way.

Variables

Parameter Symbol Units Scales as
Flow rateQgpmr
Total headHftr²
Brake powerPbhpr³
NPSH requiredNPSHrftr² (speed only, approx.)
Efficiencyη%≈ constant (assumed)

2. The Affinity Laws

Speed change (impeller diameter fixed)

When speed changes and the impeller is unchanged, the laws are essentially exact because the machine geometry has not changed at all:

r = N2 / N1 Q2 / Q1 = (N2 / N1) = r H2 / H1 = (N2 / N1)² = r² P2 / P1 = (N2 / N1)³ = r³ NPSHr2 ≈ NPSHr1 × r² (approximate)

Impeller trim (speed fixed)

When the impeller outside diameter is machined down while speed is held constant, the same powers of r apply, with r now the diameter ratio:

r = D2 / D1 Q2 / Q1 = (D2 / D1) = r H2 / H1 = (D2 / D1)² = r² P2 / P1 = (D2 / D1)³ = r³
NPSHr and trims: a diameter trim does not change the impeller inlet (eye) geometry, so NPSHr stays essentially unchanged — only a speed change moves NPSHr (roughly with r²). This is why the calculator only scales NPSHr in speed mode.

Why head goes as r² and power as r³

Impeller tip speed is proportional to r, so the head it can develop (which depends on velocity squared, per Euler's pump equation) goes as r². Hydraulic power is proportional to the product of flow and head:

P ∝ Q × H ∝ r × r² = r³ Halving speed (r = 0.5): Q→50%, H→25%, P→12.5%

That cube-law sensitivity of power is the single most important reason variable-speed pumping saves energy compared with throttling a valve.

3. Worked Examples

Example A — Slowing a pump with a VFD

A pump runs at 1800 rpm delivering 1000 gpm at 150 ft, drawing 50 bhp. You slow it to 1500 rpm with a VFD.

r = N2/N1 = 1500 / 1800 = 0.8333 Q2 = 1000 × 0.8333 = 833 gpm H2 = 150 × 0.8333² = 104 ft P2 = 50 × 0.8333³ = 28.9 bhp

A 17% speed reduction cut the power demand by about 42% — the cube law at work.

Example B — Trimming an impeller

A pump with a 10.0 in impeller delivers 500 gpm at 100 ft drawing 21.7 bhp. The duty point is over-performing, so you trim the impeller to 8.48 in.

r = D2/D1 = 8.48 / 10.0 = 0.848 (a 15.2% trim) Q2 = 500 × 0.848 = 424 gpm H2 = 100 × 0.848² = 71.9 ft P2 = 21.7 × 0.848³ = 13.2 bhp
Watch the limit: this 15.2% trim is right at the edge of the practical ~10–15% guideline. The affinity prediction tends to slightly over-state the delivered head for a trim this large — verify against the vendor trim curve before committing.

4. Impeller Trim Limits

Trimming the impeller OD is a cheap, permanent way to move a pump's curve down to the duty point. But the affinity laws assume geometric similarity, and a trimmed impeller is no longer geometrically similar to the original — the blade exit angle and the clearance to the casing/volute change. Within a modest trim these effects are small; past it they grow quickly.

Trim (1 − r) Ratio r Accuracy & effect
0–10%0.90–1.00Affinity laws hold well; efficiency loss negligible.
10–15%0.85–0.90Acceptable; small efficiency loss; verify against trim curve.
> 15%< 0.85Not recommended; flow no longer fills the casing, head falls below prediction, efficiency drops sharply.
Why large trims under-deliver: as the impeller shrinks relative to the volute, the gap between blade tip and casing grows, recirculation increases, and the actual head comes in below the r² estimate. Manufacturers publish trim-corrected curves for exactly this reason — for big trims, trust the curve, not the law.

Practical notes: trim in small increments and re-test where possible; never trim below the minimum diameter on the vendor curve; and remember that a trim is irreversible — if the future duty might rise, leave material on or use a VFD instead.

5. VFD vs Impeller Trim

Both a variable-frequency drive and an impeller trim use the affinity laws to bring a pump down to its duty point. Choosing between them is an engineering and economic trade-off.

Aspect Variable-Frequency Drive (speed) Impeller Trim (diameter)
Affinity accuracyEssentially exact (geometry unchanged)Good to ~10–15%, then drifts
ReversibilityFully reversible; any speed on demandPermanent — metal removed
Variable dutyIdeal — track a changing system curveFixed single operating point
Energy savingsBest (power follows r³ continuously)One-time reduction to the trimmed point
Capital costHigher (drive, harmonics, enclosure)Low (machine shop labor)
Other effectsSoft start, reduced water hammer; needs min-speed check for bearings/coolingNo electrical complexity; slightly lower peak efficiency
Rule of thumb: use a VFD when the duty varies, when future re-rates are likely, or when energy is the main cost driver. Trim when the over-performance is fixed and modest (within ~10–15%) and a low first cost matters most.

Key Standards & References

  • Affinity laws – Pump similitude / dimensional analysis of geometrically similar rotodynamic machines
  • ANSI/HI 1.3 – Rotodynamic (Centrifugal) Pumps for Design and Application
  • API 610 – Centrifugal Pumps for Petroleum, Petrochemical and Natural Gas Industries (rated point & performance curves)
  • Cameron Hydraulic Data – Industry reference handbook (affinity laws & impeller trim)

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Frequently Asked Questions

What are the pump affinity laws?

The affinity laws scale a centrifugal pump's performance when speed or impeller diameter changes: flow varies directly with the ratio (Q2/Q1 = r), head with the square (H2/H1 = r²), and power with the cube (P2/P1 = r³). For a speed change r = N2/N1; for an impeller trim r = D2/D1. Efficiency is assumed constant between the two points.

Why does pump power follow a cube law?

Hydraulic power is proportional to flow times head. Flow scales with r and head with r², so their product scales with r³. Halving the speed cuts flow in half, head to one-quarter, and power to one-eighth — which is why variable-speed operation saves so much energy on throttled pumps.

How much can a pump impeller be trimmed?

Keep impeller trims within about 10–15% of the original diameter. Within that band the affinity laws predict performance well; beyond it the discharge no longer fully fills the casing, efficiency falls, and actual head drops below the r² prediction. Confirm large trims against the manufacturer's published trim curve.