AtlasCalcsUtility Calculation Suite
2026-07-30 10 min

Pump Affinity Laws Calculator

Apply pump affinity (similarity) laws to predict performance changes with speed and impeller diameter. Essential for pump operation optimization and variable speed drive analysis.

Calculator

Unit System
Impeller Trim Type
Initial Flow Q₁ (m³/h)
Initial Head H₁ (m)
Initial Power P₁ (kW)
Initial Speed n₁ (RPM)
New Speed n₂ (RPM)

Results

Normal

New Flow (Q₂)

82.76m³/h

Flow rate at new operating condition

New Head (H₂)

34.24m

Total head at new operating condition

New Power (P₂)

11.34kW

Power required at new operating condition

Speed Ratio (n₂/n₁)

0.8276-

Ratio of new to initial speed

Diameter Ratio (D₂/D₁)

0.9000-

Ratio of new to initial impeller diameter

Energy Savings

43.3%

Power reduction from operating change

Visualizations

Results Overview

82.8m³/hNew Flow
34.2m³/hNew Head
11.3m³/hNew Power
0.8m³/hSpeed Ratio
0.9m³/hDiameter Ratio
43.3m³/hEnergy Savings

Formula

Pump affinity (similarity) laws relate flow, head, and power to changes in speed and impeller diameter.

Q₂/Q₁ = n₂/n₁ | H₂/H₁ = (n₂/n₁)² | P₂/P₁ = (n₂/n₁)³

Alternative Form

Q₂/Q₁ = (D₂/D₁)³ | H₂/H₁ = (D₂/D₁)² | P₂/P₁ = (D₂/D₁)⁵

Variables

SymbolNameUnitDescription
Q₁, Q₂Initial and New Flow Ratem³/h / gpmVolumetric flow rate before and after change
H₁, H₂Initial and New Headm / ftTotal dynamic head before and after change
P₁, P₂Initial and New PowerkW / hpPower consumption before and after change
n₁, n₂Initial and New SpeedRPMRotational speed before and after change
D₁, D₂Initial and New Impeller DiametermmImpeller diameter before and after trim

Explanation

The affinity laws (also called fan laws or pump laws) describe the relationship between pump operating parameters when speed or impeller diameter changes. Flow varies linearly with speed, head varies with the square of speed, and power varies with the cube of speed. For impeller diameter changes, the exponents are 3, 2, and 5 respectively. These relationships assume geometric similarity and constant efficiency.

Derivation

From dimensional analysis and Buckingham Pi theorem: The pump performance parameters (Q, H, P) are functions of speed (n), diameter (D), fluid density (ρ), and viscosity (μ). Under dynamic similarity (same Reynolds number and specific speed), the affinity laws emerge: Q ∝ nD³, H ∝ n²D², P ∝ ρn³D⁵. The exponents reflect the underlying physics of momentum transfer in the impeller.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Enter the initial pump operating parameters: flow Q₁, head H₁, and power P₁.

Flow (Q₁):100.00 m³/h
Head (H₁):50.00 m
Power (P₁):20.00 kW
Result: Initial: Q₁=100.00, H₁=50.00, P₁=20.00
Step 2

Apply the speed affinity law to compute new performance parameters.

Speed Ratio:0.8276
Diameter Ratio:0.9000
Result: Ratio: n₂/n₁=0.8276, D₂/D₁=0.9000
Step 3

Calculate resulting flow, head, and power at the new operating condition.

Q₂ = Q₁ × (n₂/n₁) ; H₂ = H₁ × (n₂/n₁)² ; P₂ = P₁ × (n₂/n₁)³
Q₂:82.76 m³/h
H₂:34.24 m
P₂:11.34 kW
Intermediate: Q₂=100.00×0.8276=82.76, H₂=50.00×0.6849=34.24
Result: New operating point: Q₂=82.76, H₂=34.24, P₂=11.34
Final Answer
82.76

Worked Examples

See how the calculator works with different scenarios.

Scenario

A pump operating at 1450 RPM delivers 100 m³/h at 50 m head consuming 20 kW. Speed is reduced to 1200 RPM to match reduced system demand.

Inputs

Q₁:100 m³/h
H₁:50 m
P₁:20 kW
n₁:1450 RPM
n₂:1200 RPM

Calculation

Speed ratio = 1200/1450 = 0.8276 Q₂ = 100 × 0.8276 = 82.76 m³/h H₂ = 50 × 0.8276² = 34.24 m P₂ = 20 × 0.8276³ = 11.34 kW

Result

Q₂ = 82.8 m³/h, H₂ = 34.2 m, P₂ = 11.3 kW, Energy Savings = 43.3%

Interpretation

A 17% speed reduction reduces power by 43% due to the cubic relationship, demonstrating the dramatic energy savings from variable speed control.

Understanding Your Result

Your Result: New Flow (Q₂): 82.76 m³/h

Typical range: Speed ratios 0.5-1.2, diameter ratios 0.7-1.1, specific speed 500-5000 (US units). Affinity laws apply within ±20% of the design point for best accuracy.

What This Means

The affinity laws provide a mathematical relationship between pump speed, impeller diameter, and the resulting flow, head, and power. They allow engineers to predict how a pump will perform when operating conditions change.

Why It Matters

These laws are fundamental to pump energy optimization. Since power varies with the cube of speed, even modest speed reductions yield substantial energy savings, making VFD (variable frequency drive) applications highly cost-effective.

Professional Interpretation

The affinity laws assume constant efficiency, which is an approximation. In practice, efficiency changes slightly with speed and diameter. For precise analysis, use actual pump curves. The laws are most accurate for geometrically similar pumps operating at the same specific speed. When trimming impellers, maintain at least 70% of the original diameter to avoid excessive efficiency loss.

Recommendations

  • •Use VFD speed control for variable flow applications - the cubic power relationship maximizes savings
  • •Trim impellers only when permanent flow reduction is needed and speed control is impractical
  • •Verify affinity law predictions with actual pump curve data for critical applications
  • •Consider minimum speed limits: pump head must still overcome system static head
  • •Account for motor and VFD efficiency when calculating actual energy savings

Educational Content

Introduction

Pump affinity laws (also called fan laws or similarity laws) are mathematical relationships that predict how pump performance parameters change when operating speed or impeller diameter is altered. They are essential tools for pump system optimization and energy management.

Concept Explanation

The affinity laws derive from dimensional analysis and the principle of dynamic similarity. When two pumps (or the same pump at different conditions) are geometrically similar and operate at the same efficiency point, their performance parameters scale according to specific power-law relationships. Flow scales linearly with speed, head scales with speed squared, and power scales with speed cubed. For impeller diameter changes, the exponents become 3, 2, and 5 respectively. These relationships reflect the underlying momentum transfer physics in the centrifugal impeller.

Applications

▸Variable speed drive (VFD) energy savings analysis
▸Pump selection and matching to system requirements
▸Impeller diameter trimming for permanent capacity adjustment
▸Performance prediction for pump operation at off-design conditions
▸Multi-pump system optimization with parallel or series operation
▸Troubleshooting pump performance issues

Benefits

  • •Enables accurate prediction of pump performance changes without retesting
  • •Quantifies the substantial energy savings from speed reduction
  • •Guides impeller trim decisions for permanent flow adjustment
  • •Supports system optimization and efficiency improvement programs
  • •Essential for VFD business case development and ROI calculation

Limitations

  • •Assumes constant pump efficiency, which is an approximation
  • •Most accurate within ±20% of the design operating point
  • •Does not account for changes in system curve interaction
  • •Impeller trim reduces efficiency if diameter change exceeds ~30%

Best Practices

  • •Use actual pump curves for final design decisions, not just affinity laws
  • •Apply affinity laws within the recommended operating range
  • •Verify speed ratio stays above 0.5 to maintain adequate head
  • •Consider motor and drive efficiency in total energy calculations
  • •Document baseline measurements before and after modifications

Common Misconceptions

Myth: Affinity laws are exact for all operating conditions.
Truth: They are approximations that assume geometric and dynamic similarity. Actual pump performance may deviate, especially at extreme speed or diameter changes.
Myth: Power savings from speed reduction are proportional to speed reduction.
Truth: Power varies with the cube of speed, so a 20% speed reduction yields approximately 49% power reduction, not 20%.
Myth: Impeller diameter can be trimmed by any amount.
Truth: Excessive trimming (>30% reduction) causes significant efficiency loss due to flow pattern disruption and increased clearance losses.

Practical Tips

  • •A 10% speed reduction reduces power by approximately 27% (0.9³ = 0.729)
  • •Use VFDs instead of throttling valves - throttling wastes energy, speed control does not
  • •Verify motor can handle the new power requirement before changing speed or diameter
  • •For impeller trimming, machine the trailing edge to maintain blade geometry
  • •Speed increases above rated RPM may overload the motor - check power limits first
  • •Affinity laws apply to fans and compressors too, not just pumps
  • •Log pump performance data regularly to validate affinity law predictions

Common Mistakes

Assuming affinity laws apply across the entire pump curve
Affinity laws apply at a specific point on the curve, not to the entire curve. They predict how a given operating point shifts with speed/diameter changes.
Confusing diameter ratio exponents (3, 2, 5) with speed ratio exponents (1, 2, 3)
Remember: diameter change has higher exponents because it affects both the flow path area and the impeller tip speed simultaneously.
Applying diameter affinity laws to impeller trimming when pump geometry changes significantly
Trimming violates geometric similarity. Limit diameter changes to <20% for reasonable accuracy with affinity laws.
Neglecting motor and drive efficiency in energy savings calculations
Actual energy savings are less than predicted by affinity laws alone because motor and VFD efficiencies decrease at reduced loads.

Assumptions

This calculator makes the following assumptions to provide estimates:

Geometric Similarity: The pump geometry remains unchanged except for diameter scaling. Impeller trim approximates this but deviates at large reductions.
Constant Efficiency: Pump efficiency is assumed unchanged when speed or diameter changes. In reality, efficiency varies slightly with operating point.
Dynamic Similarity: The Reynolds number and flow regime remain similar across the change, ensuring the same dimensionless performance.
Incompressible Flow: The fluid is treated as incompressible, which is valid for liquids but requires modification for gas compressors.

Limitations

Be aware of these limitations when using this calculator:

Accuracy Range: Affinity laws are most accurate within ±20% of the design speed or diameter. Beyond this range, deviations from assumed similarity reduce accuracy.
System Interaction: The laws predict pump performance alone, not the interaction with the system curve. Actual operating point depends on both pump and system characteristics.
Efficiency Variation: The assumption of constant efficiency introduces error, typically 1-5% for moderate speed changes and more for extreme changes.
Geometric Limitations: Diameter trim changes blade angles, tip clearance, and flow path geometry, violating the geometric similarity assumption.

Frequently Asked Questions

Answers to common questions about this calculator.

What are pump affinity laws?
Pump affinity laws (also called similarity laws or fan laws) are mathematical relationships that predict how pump flow, head, and power change with variations in operating speed or impeller diameter. They are derived from dimensional analysis of centrifugal pump performance.
How do affinity laws help save energy?
Since power varies with the cube of speed (P ∝ n³), reducing pump speed by just 20% reduces power consumption by approximately 49%. This makes variable speed drive (VFD) control far more efficient than throttling valves or bypass recirculation.
What is the difference between speed change and diameter change formulas?
For speed: Q ∝ n, H ∝ n², P ∝ n³. For diameter: Q ∝ D³, H ∝ D², P ∝ D⁵. Diameter changes have higher exponents because altering the impeller diameter changes both the tip speed and the flow passage geometry simultaneously.
How accurate are the affinity laws?
Affinity laws are typically accurate within 5-10% for speed changes up to ±20% of the design point. For diameter changes, accuracy decreases because trimming alters the impeller geometry. Always verify with actual pump curves for critical applications.
Can I apply affinity laws to any pump type?
Affinity laws apply best to centrifugal pumps. Positive displacement pumps have different characteristics and do not follow these laws. Axial flow pumps follow modified versions where power may decrease at reduced flow.
What is the maximum recommended impeller trim?
Most pump manufacturers recommend a maximum trim of 20-30% of the original diameter. Excessive trimming causes significant efficiency loss, increased clearance effects, and potential vibration issues. Some pumps have minimum diameter limits specified on the curve sheet.
Do affinity laws apply to pump curves or just operating points?
Affinity laws apply to individual operating points. To generate a new pump curve at reduced speed, apply the laws to multiple points along the original curve. The resulting curve shifts downward and to the left.
What is the affinity parabola?
When pump speed changes, the operating point moves along a parabolic curve called the affinity parabola (H ∝ Q²). This curve represents points of equal efficiency as predicted by the affinity laws.
How does system curve interaction affect affinity law predictions?
The actual operating point is the intersection of the pump curve with the system curve. When speed changes, both the pump curve and the operating point shift. The new operating point depends on the system curve shape (static head vs friction head).
Can I use affinity laws to increase pump speed?
Yes, but check motor power limits, maximum operating speed of the pump, NPSH requirements at higher speed, and shaft stress limits. Speed increases above rated conditions may require motor replacement.
What is specific speed and why does it matter?
Specific speed (Ns) is a dimensionless parameter that characterizes pump geometry and performance. Pumps with the same specific speed maintain dynamic similarity. Affinity laws are most accurate when applied within the same specific speed range.
How do affinity laws differ for fans vs pumps?
The same laws apply to fans (fan laws), with the same exponents. However, fan power is often expressed in terms of static pressure rather than head, and compressibility effects may need consideration for high-pressure fans.
What is the effect of viscosity on affinity laws?
The affinity laws assume constant Reynolds number, which changes with viscosity. For viscous fluids, pump performance deviates from affinity law predictions, and correction factors from standards like HI 9.6.7 should be applied.
How do I calculate energy savings from VFD installation?
Calculate power at each operating point using affinity laws, multiply by operating hours, sum across all operating conditions, and subtract VFD and motor losses. Compare to the baseline (throttling or constant speed) to determine annual savings and payback period.
What is the difference between affinity laws and the pump curve?
The pump curve shows head vs flow at a fixed speed. Affinity laws describe how the entire curve shifts when speed or diameter changes. Together, they provide a complete picture of pump performance under varying conditions.
Do affinity laws apply to parallel or series pump operation?
Yes, the laws apply to individual pumps. For parallel operation, add flows at the same head. For series operation, add heads at the same flow. Then apply affinity laws to each pump individually.
What is the minimum speed for a pump?
The minimum speed depends on the system static head. At low speeds, pump head may drop below the static head required, causing the pump to deadhead (no flow). Check the system curve to determine the minimum practical speed.
How does impeller trim affect pump efficiency?
Trimming an impeller reduces efficiency because the exit angle, blade length, and flow path geometry change. Most manufacturers provide efficiency correction curves for trimmed impellers. Expect 1-3% efficiency loss per 10% diameter reduction.
Can I use affinity laws to predict pump performance with different fluids?
Affinity laws primarily address speed and diameter changes, not fluid properties. For different fluids, consider density (affects power), viscosity (affects head and efficiency), and vapor pressure (affects NPSH).
What is the relationship between pump speed and NPSH?
NPSHr varies approximately with the square of speed (NPSHr ∝ n²). Increasing pump speed significantly increases NPSH requirements, which may cause cavitation if the available NPSH is insufficient.
How do I determine the new operating point on a system curve?
Plot the pump curve at the new speed (using affinity laws) and find its intersection with the system curve. The system curve is typically parabolic: H_sys = H_stat + k × Q², where H_stat is the static head.
What is the cage frequency and how does it relate to affinity laws?
Cage frequency analysis uses affinity laws to identify flow and head changes from minor speed variations. It is a useful diagnostic tool for condition monitoring of pump performance over time.
Do affinity laws apply to positive displacement pumps?
No, positive displacement pumps have fundamentally different characteristics. Flow is proportional to speed (Q ∝ n) but head is determined by system resistance, making the power-speed relationship approximately linear, not cubic.
What is the difference between the 3:2:1 and 5:2:3 exponents?
The 1:2:3 exponents apply to speed changes (Q∝n¹, H∝n², P∝n³). The 3:2:5 exponents apply to diameter changes (Q∝D³, H∝D², P∝D⁵). The higher diameter exponents reflect the additional area change effect.
How does pump specific speed change with impeller trim?
Specific speed (Ns = n × Q⁰·⁵ / H⁰·⁷⁵) changes with trim because Q and H change at different rates. The specific speed of the trimmed impeller may differ slightly from the original, indicating a shift in optimal operating characteristics.
What is the practical speed range for VFD pump control?
Most VFD-controlled pumps operate between 30% and 100% of rated speed. Below 30%, motor cooling, bearing lubrication, and pump efficiency become concerns. Consult manufacturer recommendations for minimum speed limits.
How do I account for slip in induction motors when using affinity laws?
Induction motor slip varies with load. At reduced load (lower power), slip decreases and actual motor speed increases slightly. For accurate affinity law calculations, use measured shaft speed rather than synchronous speed.
What is the effect of speed on pump vibration?
Pump vibration levels generally decrease with speed reduction (vibration velocity ∝ speed²). However, operating at certain speeds may excite structural resonances. Variable speed drives should be programmed to skip problematic speed ranges.
Can I use affinity laws to predict pump wear progression?
Indirectly, yes. By tracking performance changes over time and comparing to affinity law predictions, you can detect wear. A pump that deviates from expected performance likely has worn internal clearances or impeller damage.
What are the limitations of using affinity laws for pump selection?
Affinity laws should not replace pump curve analysis for final selection. They provide estimates of performance changes but cannot capture efficiency variations, system interactions, or mechanical constraints. Always verify with manufacturer data.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Affinity Law
Mathematical relationship predicting how pump performance (Q, H, P) scales with changes in speed or impeller diameter under dynamic similarity.
Affinity Parabola
A parabolic curve (H ∝ Q²) representing operating points of equal efficiency as pump speed changes.
Dynamic Similarity
Condition where two pumps have the same dimensionless performance parameters, enabling affinity law application.
Impeller Trim (Rebate)
Machining down the impeller outer diameter to reduce pump head and flow output.
Specific Speed (Ns)
Dimensionless pump parameter characterizing impeller geometry and optimal operating range.
System Curve
Graphical representation of system head requirement as a function of flow rate.
Variable Frequency Drive (VFD)
Electronic device that varies motor speed by adjusting input frequency, enabling pump speed control.
Deadhead
Operating condition where pump runs against a closed valve with zero flow, converting all power to heat.
Tip Speed
Tangential velocity of the impeller outer diameter, proportional to n × D.
Cubic Power Relationship
The relationship P ∝ n³, meaning power changes dramatically with small speed changes.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

HI 9.6.7ISO 9906ASME B73.1ANSI/HI 1.3API 610ISO 5199

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Page Information

Last Updated:

2026-07-30

Version:

2.0.0

License:

Free to use for educational and professional purposes.