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Pumps & Rotating Equipment

Pump Affinity Laws Explained

The pump affinity laws relate flow, head, and power to pump speed for the same pump and impeller. Learn the three scaling formulas, a worked example, and key limitations.

TypeEngineering guide — concept explainer

Definition

The pump affinity laws (also called fan laws or similarity laws) are a set of approximate proportional relationships that predict how flow rate, head, and power change when the rotational speed of a centrifugal pump (or fan) changes. They apply to the same pump with the same impeller diameter. Flow scales linearly with speed, head scales with the square of speed, and power scales with the cube of speed.

Why it matters

Variable speed drives (VSDs) are common in modern process plants. The affinity laws let engineers quickly estimate the new flow, head, and power when a pump speed is changed — without re-reading the full pump curve. The cubic power relationship is especially important: a small speed reduction yields a large energy saving, which is why VSD-driven pumps are a primary energy efficiency measure in industry.

Formula

Flow — linear with speed
Q₂ = Q₁ × (N₂ / N₁)
Head — square of speed
H₂ = H₁ × (N₂ / N₁)²
Power — cube of speed
P₂ = P₁ × (N₂ / N₁)³

Units involved

  • •Q — flow rate in m³/h, L/s, gpm, etc.
  • •H — head in metres, feet, etc.
  • •P — power in kW, hp, W, etc.
  • •N — rotational speed in rpm
  • •N₂/N₁ — speed ratio, dimensionless

Concept diagram

N₁Q₁, H₁, P₁speedchangeN₂Q₂, H₂, P₂Q₂ = Q₁ × (N₂/N₁)H₂ = H₁ × (N₂/N₁)²P₂ = P₁ × (N₂/N₁)³same pump, same impellerratio = N₂ / N₁speed ratio (dimensionless)

Worked example

A pump runs at 1,000 rpm delivering 100 m³/h flow, 40 m head, and consuming 10 kW. The speed is increased to 1,200 rpm. What are the new flow, head, and power?

  1. 01Speed ratio = N₂ / N₁ = 1200 / 1000 = 1.2
  2. 02Q₂ = 100 × 1.2 = 120 m³/h
  3. 03H₂ = 40 × 1.2² = 40 × 1.44 = 57.6 m
  4. 04P₂ = 10 × 1.2³ = 10 × 1.728 = 17.28 kW
Result

Q₂ = 120 m³/h; H₂ = 57.6 m; P₂ = 17.28 kW

Common mistakes

  • •Using the affinity laws with different impeller diameters — these formulas assume the same impeller. Impeller trimming has its own set of affinity relationships and is less accurate.
  • •Expecting exact results — the affinity laws are approximate. Real pump performance deviates because efficiency changes with speed, and the system curve interaction shifts the operating point.
  • •Forgetting the cubic power relationship — a 20% speed increase raises power by 73% (1.2³ = 1.728). Conversely, a 20% speed reduction cuts power by 49% (0.8³ = 0.512). The impact on power is much larger than on flow.
  • •Applying the laws to positive displacement pumps — affinity laws apply to centrifugal (and axial) machines. Positive displacement pumps have a roughly linear flow-speed relationship but different head and power behaviour.
  • •Ignoring the system curve — the affinity laws predict the pump curve shift, but the actual operating point is where the shifted pump curve intersects the system curve. If the system curve has a large static head component, the flow change may be less than the affinity law predicts.

When to use the calculator

Use the Pump Affinity Laws calculator when you have a known operating point at one speed and need to predict flow, head, and power at a different speed. Enter the original values and the new speed to get the scaled results instantly.

Pump Affinity Laws Calculator →Pump Hydraulic Power Calculator →Head to Pressure Calculator →

FAQ

Do the affinity laws work for fans as well as pumps?
Yes. The affinity laws apply to all centrifugal turbomachinery — pumps, fans, blowers, and compressors (within the incompressible flow assumption). For fans, Q is volumetric air flow, H is static or total pressure, and P is fan power. The same speed-ratio scaling applies.
Why does power change so much more than flow?
Because power scales with the cube of speed. Hydraulic power is proportional to flow × head. Flow scales linearly with speed (exponent 1) and head scales with the square (exponent 2), so their product — power — scales with the cube (exponent 3). A 10% speed reduction cuts power by about 27%.
Are the affinity laws exact?
No. They are approximate relationships based on dimensional analysis and geometric similarity. They are most accurate for small speed changes (±10–20%) and become less reliable for large speed changes because pump efficiency changes with speed and flow. For critical applications, use the manufacturer pump curve at each speed.
What is the difference between affinity laws and pump power calculation?
The affinity laws predict how pump performance changes with speed — they scale an existing operating point to a new speed. Pump power calculation (P = ρgQH/η) computes the actual power required for a known flow, head, density, and efficiency. They are complementary tools: use affinity laws for speed-change estimation, and pump power for absolute power calculation.