Pump Affinity Laws and Speed Variation

Overview

The pump affinity laws describe how flow rate, head, and power change when a centrifugal pump's rotational speed or impeller diameter is varied. They are fundamental to pump selection, variable-speed drive sizing, and predicting off-design performance.

There are two sets of affinity laws: one set for a specific pump when rotational speed changes, and one set for a family of geometrically similar pumps when impeller diameter changes.

Key Formulas

Speed Variation

When only the rotational speed changes (impeller diameter held constant):

$\frac{Q_2}{Q_1} = \frac{n_2}{n_1}$

$\frac{H_2}{H_1} = \left(\frac{n_2}{n_1}\right)^2$

$\frac{P_2}{P_1} = \left(\frac{n_2}{n_1}\right)^3$

$\frac{NPSH_{a2}}{NPSH_{a1}} \approx \left(\frac{n_2}{n_1}\right)^2$

Impeller Diameter Variation

When only the impeller diameter changes (speed held constant):

$\frac{Q_2}{Q_1} = \frac{D_2}{D_1}$

$\frac{H_2}{H_1} = \left(\frac{D_2}{D_1}\right)^2$

$\frac{P_2}{P_1} = \left(\frac{D_2}{D_1}\right)^3$

Hydraulic Power

$P = \frac{Q \cdot H \cdot \rho \cdot g}{\eta}$

NPSH Available

$NPSH_a = \frac{P_s}{\rho g} + \frac{v_s^2}{2g} - \frac{P_v}{\rho g}$

Variables

Speed Variation Reference

Speed Change Calculator

Impeller Diameter Change Calculator

Unit Converter

Interactive Affinity Curves

Restored Original Source Tables

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Original Source Images

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Engineering Notes

• Affinity laws are approximate for large changes. Deviations grow as speed departs far from the original design point because system curves shift and efficiency is not perfectly preserved. A ±20% speed change is generally reliable; beyond that, verify with manufacturer curves.
• Impeller diameter laws use the ratio $D$, not the trimmed width. Some texts distinguish $D^1$ (fan-law exponent) from the empirical fan-law exponent of ~1.0 for diameter variation on centrifugal pumps. The classic $Q \propto D$ relationship is most accurate for small trims.
• NPSH rises with speed squared, so reducing speed often improves suction conditions, while increasing speed can push a pump into cavitation. Always check NPSH requirements at the new speed.
• Motor and VFD sizing. Because power scales with the cube of speed, small reductions in speed yield large savings. However, the motor and variable-frequency drive must still handle the full-speed power rating during startup or transient spikes.
• System curve interaction. The affinity laws describe the pump curve shift; actual operating point depends on the intersection with the system curve. A speed reduction shifts the pump curve inward and downward, changing the duty point.
• Not applicable to positive-displacement pumps. These laws are derived from centrifugal pump geometry. Positive-displacement pumps follow different speed-flow relationships with near-constant flow per revolution.

References

• Engineering ToolBox — Pump Affinity Laws