Affinity Laws Calculator

Fundamental Principles

The affinity laws are derived from dimensional analysis and similarity principles in fluid dynamics. They relate pump performance parameters (flow rate, head, and power) to operating conditions (speed and diameter). These relationships are based on the principle that geometrically similar pumps operating at equivalent points on their performance curves will have predictable performance ratios.

The laws assume: • Constant efficiency between operating points • Geometric similarity of impellers • Dynamic similarity (same Reynolds number regime) • Incompressible fluid with constant properties • Same system resistance characteristics

Speed Affinity Laws

When the impeller diameter remains constant and only the rotational speed changes, the performance parameters scale according to the following relationships:

**Flow Rate (First Law):** The volumetric flow rate is directly proportional to the rotational speed. This relationship reflects that faster rotation moves more fluid per unit time.

**Head (Second Law):** The pump head varies with the square of the speed ratio. This quadratic relationship comes from the kinetic energy imparted to the fluid being proportional to the square of the velocity.

**Power (Third Law):** The required power varies with the cube of the speed ratio. This cubic relationship results from power being the product of head (speed²) and flow (speed).

Mathematically: (Q_2)/(Q_1) = (N_2)/(N_1) (H_2)/(H_1) = ((N_2)/(N_1))^2 (P_2)/(P_1) = ((N_2)/(N_1))^3

Where: • Q = volumetric flow rate (m³/s, GPM, L/s) • H = total dynamic head (m, ft) • P = brake horsepower or shaft power (W, HP) • N = rotational speed (rpm, rad/s) • Subscripts 1 and 2 denote initial and new conditions

(Q_2)/(Q_1) = (N_2)/(N_1) \\ (H_2)/(H_1) = ((N_2)/(N_1))^2 \\ (P_2)/(P_1) = ((N_2)/(N_1))^3

Diameter Affinity Laws

When the rotational speed remains constant and only the impeller diameter changes (through trimming or replacement), the performance parameters follow similar scaling relationships:

**Flow Rate:** The flow rate is directly proportional to the diameter ratio, reflecting the change in swept volume.

**Head:** The head varies with the square of the diameter ratio, similar to the speed relationship, due to changes in peripheral velocity.

**Power:** The power requirement varies with the cube of the diameter ratio.

Mathematically: (Q_2)/(Q_1) = (D_2)/(D_1) (H_2)/(H_1) = ((D_2)/(D_1))^2 (P_2)/(P_1) = ((D_2)/(D_1))^3

Where: • D = impeller diameter (mm, in) • Other variables as defined above

**Important:** Impeller trimming should generally be limited to 10-15% reduction to maintain reasonable accuracy.

(Q_2)/(Q_1) = (D_2)/(D_1) \\ (H_2)/(H_1) = ((D_2)/(D_1))^2 \\ (P_2)/(P_1) = ((D_2)/(D_1))^3

Combined Speed and Diameter Changes

When both speed and diameter change simultaneously, the affinity laws can be combined:

(Q_2)/(Q_1) = (N_2)/(N_1) · (D_2)/(D_1) (H_2)/(H_1) = ((N_2)/(N_1))^2 · ((D_2)/(D_1))^2 (P_2)/(P_1) = ((N_2)/(N_1))^3 · ((D_2)/(D_1))^3

These combined relationships are useful when both parameters are modified or when converting between different pump sizes and speeds.

(Q_2)/(Q_1) = (N_2)/(N_1) · (D_2)/(D_1) \\ (H_2)/(H_1) = ((N_2)/(N_1) · (D_2)/(D_1))^2 \\ (P_2)/(P_1) = ((N_2)/(N_1) · (D_2)/(D_1))^3

Specific Speed and Pump Selection

The specific speed (N_s) is a dimensionless parameter that characterizes pump geometry and performance:

N_s = (N√(Q))/(H^3/4)

Specific speed remains constant for geometrically similar pumps and helps in: • Selecting the appropriate pump type • Predicting efficiency characteristics • Comparing different pump designs

Typical ranges: • Radial flow (centrifugal): N_s = 500-1500 • Mixed flow: N_s = 1500-4000 • Axial flow: N_s = 4000-10000

N_s = (N√(Q))/(H^3/4)

System Curve and Operating Point

The pump operates where its performance curve intersects the system curve. The system head is described by:

H_system = H_static + K · Q^2

Where: • H_static = elevation difference + pressure head (constant) • K = system resistance coefficient • Q^2 term represents friction losses

When pump speed changes, the new operating point shifts along the system curve. Understanding this interaction is crucial for proper system design and energy optimization.

H_system = H_static + K · Q^2

Efficiency Considerations

While affinity laws assume constant efficiency, in practice:

• Efficiency varies with operating point • Peak efficiency occurs at the Best Efficiency Point (BEP) • Operating far from BEP reduces efficiency and increases wear • Reynolds number effects can cause efficiency changes • Very low speeds may experience increased viscous losses

The Moody equation provides efficiency correction: (1-\eta_2)/(1-\eta_1) = ((D_1)/(D_2))^1/4

Where \eta is the pump efficiency.

(1-\eta_2)/(1-\eta_1) = ((D_1)/(D_2))^1/4

Practical Applications

The affinity laws are used extensively in:

**Variable Speed Drives (VSD):** VSDs control pump speed to match system demand, providing significant energy savings compared to throttling valves.

**Pump Trimming:** Reducing impeller diameter allows fine-tuning of pump performance to match system requirements without replacing the entire pump.

**Scaling Studies:** Predicting full-scale pump performance from model tests or vice versa.

**Energy Analysis:** The cubic relationship between speed and power means that small speed reductions yield large energy savings. For example, reducing speed by 20% reduces power by approximately 49%.

**Troubleshooting:** Diagnosing pump performance issues by comparing measured values to predicted affinity law relationships.

Limitations and Validity

The affinity laws have important limitations:

**Valid When:** • Changes are moderate (±20% speed, ±15% diameter) • Pumps are geometrically similar • Operating in turbulent regime (Re > 10⁵) • Same fluid properties (density, viscosity) • Similar relative operating points

**Invalid When:** • Large speed or diameter changes (>20-25%) • Significant Reynolds number effects (laminar or transitional flow) • Cavitation occurs • Compressibility effects are significant • Different impeller geometries • Operating far from BEP

**Accuracy:** Typically ±5% for flow and head, ±10% for power within valid ranges. Accuracy decreases with larger changes and deviations from assumptions.

Worked Example

Inputs

• originalFlow: 0.1
• originalHead: 20
• originalPower: 2000
• originalSpeed: 1750
• newSpeed: 2000

Result: New Flow: 0.114 m³/s, New Head: 26.1 m, New Power: 2986 W, Speed Ratio: 1.143, Energy Increase: 49.3%

Explanation

**Example 1: Speed Change Analysis**

A centrifugal pump operates at the following conditions: • Original flow rate (Q₁) = 0.1 m³/s • Original head (H₁) = 20 m • Original power (P₁) = 2000 W • Original speed (N₁) = 1750 rpm • New speed (N₂) = 2000 rpm

**Step 1: Calculate Speed Ratio** \textSpeed Ratio = (N_2)/(N_1) = (2000)/(1750) = 1.143

**Step 2: Apply First Affinity Law (Flow)** Q_2 = Q_1 × (N_2)/(N_1) = 0.1 × 1.143 = 0.114 \text m^3\text/s

The flow increases by 14.3% proportionally to the speed increase.

**Step 3: Apply Second Affinity Law (Head)** H_2 = H_1 × ((N_2)/(N_1))^2 = 20 × (1.143)^2 = 20 × 1.306 = 26.1 \text m

The head increases by 30.6% (square of speed ratio).

**Step 4: Apply Third Affinity Law (Power)** P_2 = P_1 × ((N_2)/(N_1))^3 = 2000 × (1.143)^3 = 2000 × 1.493 = 2986 \text W

The power increases by 49.3% (cube of speed ratio).

**Key Observations:** • A 14.3% speed increase results in a 49.3% power increase • This demonstrates the cubic relationship and importance of speed control for energy efficiency • Operating at lower speeds when possible provides significant energy savings

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**Example 2: Impeller Trimming**

Consider trimming an impeller from 250 mm to 225 mm diameter: • Original diameter (D₁) = 250 mm • New diameter (D₂) = 225 mm • Original conditions: Q₁ = 50 L/s, H₁ = 30 m, P₁ = 20 kW

\textDiameter Ratio = (225)/(250) = 0.9

Q_2 = 50 × 0.9 = 45 \text L/s (10% reduction) H_2 = 30 × (0.9)^2 = 24.3 \text m (19% reduction) P_2 = 20 × (0.9)^3 = 14.6 \text kW (27% reduction)

This 10% diameter reduction yields 27% power savings while maintaining 90% of original flow.

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**Example 3: Variable Speed Drive (VSD) Energy Savings**

A pump runs 24/7 at 1800 rpm consuming 50 kW. System analysis shows flow can be reduced to 80% during off-peak hours:

(Q_2)/(Q_1) = 0.8 = (N_2)/(N_1) \implies N_2 = 1800 × 0.8 = 1440 \text rpm

P_2 = P_1 × (0.8)^3 = 50 × 0.512 = 25.6 \text kW

Power savings = 50 - 25.6 = 24.4 kW (48.8% reduction)

If off-peak hours are 16 hours/day: • Peak power (8 hrs): 50 kW × 8 = 400 kWh • Off-peak power (16 hrs): 25.6 kW × 16 = 410 kWh • Total daily consumption: 810 kWh • Without VSD: 50 kW × 24 = 1200 kWh • Daily savings: 390 kWh (32.5%) • Annual savings: ~142,000 kWh

Second Scenario

Inputs

• originalFlow: 0.075
• originalHead: 20
• originalPower: 2000
• originalSpeed: 1750
• newSpeed: 2000

Result: New Flow: 0.114 m³/s, New Head: 26.1 m, New Power: 2986 W, Speed Ratio: 1.143, Energy Increase: 49.3%

Explanation

This scenario uses different inputs (originalFlow = 0.075, originalHead = 20, originalPower = 2000, originalSpeed = 1750, newSpeed = 2000) to show how changing one variable affects the affinity laws result. Run the calculator above with these values to get the exact updated output with step-by-step work.