Pump Affinity Laws Calculator

Calculate how changes in pump speed or impeller diameter affect flow rate, head, and power using the pump affinity laws.

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How It Works

The Pump Affinity Laws (also known as fan laws or similarity laws) describe the mathematical relationship between pump speed, impeller diameter, flow rate, head, and power. These laws allow engineers to predict how a pump will perform when operating conditions change.

The Three Affinity Laws

The affinity laws relate the change in speed (N) or impeller diameter (D) to flow rate (Q), head (H), and power (P):

First Law — Flow Rate:

Q₂ / Q₁ = N₂ / N₁ (or D₂ / D₁)

Second Law — Head:

H₂ / H₁ = (N₂ / N₁)² (or (D₂ / D₁)²)

Third Law — Power:

P₂ / P₁ = (N₂ / N₁)³ (or (D₂ / D₁)³)

These relationships assume the pump efficiency remains approximately constant across the change in operating conditions.

When to Use Affinity Laws

The affinity laws are commonly applied in the following scenarios:

  • • VFD Speed Control: Predicting pump performance at different motor speeds when using a Variable Frequency Drive to match system demand and save energy.
  • • Impeller Trimming: Estimating the effect of reducing impeller diameter to match a pump to system requirements without replacing the entire pump.
  • • Energy Audits: Calculating potential energy savings from reducing pump speed instead of throttling flow with valves.
  • • System Design: Sizing pumps for different operating points on the system curve.

Limitations

The affinity laws have important limitations to keep in mind:

  • • They are only valid for geometrically similar conditions and the same pump.
  • • Accuracy decreases for speed or diameter changes greater than about 20–25%.
  • • They assume pump efficiency remains constant, which is an approximation.
  • • They do not account for changes in fluid viscosity or density.
  • • Impeller trimming laws are less accurate than speed change laws, especially for large diameter reductions.
  • • They apply to centrifugal pumps operating away from shutoff and runout conditions.

FAQ

Here you will find the answers to the frequently asked questions about pump affinity laws.

Frequently Asked Questions

What are the pump affinity laws?

The pump affinity laws are a set of three mathematical relationships that describe how changes in pump speed or impeller diameter affect flow rate, head, and power. Flow varies linearly with the ratio, head varies with the square of the ratio, and power varies with the cube of the ratio. These laws are derived from dimensional analysis and are fundamental to centrifugal pump engineering.

How do VFDs use the affinity laws to save energy?

Variable Frequency Drives (VFDs) adjust motor speed to match pump output to system demand. Because power varies with the cube of speed, even a small reduction in speed yields significant energy savings. For example, reducing pump speed by 20% reduces power consumption by approximately 49%. This is far more efficient than throttling with a valve, which wastes energy as pressure drop.

How does impeller trimming differ from speed change?

While both methods use the same affinity law equations, impeller trimming is a permanent physical modification that reduces the impeller outer diameter. Speed change via a VFD is adjustable and reversible. The affinity laws for impeller trimming are slightly less accurate because reducing the diameter changes the impeller geometry, and corrections may be needed for large trims exceeding 10-15% of the original diameter.

What energy savings can I expect from reducing pump speed?

Energy savings follow the cube law. A 10% speed reduction saves approximately 27% power. A 20% reduction saves approximately 49% power. A 50% reduction saves approximately 87.5% power. However, the pump must still meet the system's minimum flow and head requirements. The actual savings depend on the system curve and the pump's operating point.

What are the limitations of the affinity laws?

The affinity laws assume geometrically similar conditions and constant efficiency, which are approximations. They become less accurate for speed or diameter changes exceeding 20-25%. They do not account for viscosity effects, cavitation, or operation near the pump's shutoff or runout conditions. For critical applications, manufacturer pump curves at different speeds should be used for more precise performance prediction.

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