Affinity Laws

Overview

Engineering reference data for Affinity Laws in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Speed Change Effects

When increasing pump speed by 10%, the following changes occur:

• Volume Flow: Increases with 10%
• Head (Pressure): Increases with 21%
• Power Consumption: Increases with 33%

This means increasing flow capacity by 10% requires a 33% increase in power supply.

Specific Pump Affinity Laws Formulas

These laws apply to a specific centrifugal pump when changing speed (rpm) or impeller diameter.

Constant Impeller Diameter

When the wheel diameter (d) is constant, the affinity laws simplify to:

$\frac{q_1}{q_2} = \frac{n_1}{n_2} \quad \text{(1a)}$

$\frac{dp_1}{dp_2} = \left(\frac{n_1}{n_2}\right)^2 \quad \text{(2a)}$

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

Constant Wheel Velocity

When the wheel velocity (n) is constant, the laws simplify to:

$\frac{q_1}{q_2} = \frac{d_1}{d_2} \quad \text{(1b)}$

$\frac{dp_1}{dp_2} = \left(\frac{d_1}{d_2}\right)^2 \quad \text{(2b)}$

$\frac{P_1}{P_2} = \left(\frac{d_1}{d_2}\right)^3 \quad \text{(3b)}$

Geometrically Similar Pumps

For a family of geometrically similar pumps, the relationships account for both speed and diameter changes simultaneously:

$\frac{q_1}{q_2} = \left(\frac{n_1}{n_2}\right)\left(\frac{d_1}{d_2}\right)^3 \quad \text{(4)}$

$\frac{dp_1}{dp_2} = \left(\frac{n_1}{n_2}\right)^2\left(\frac{d_1}{d_2}\right)^2 \quad \text{(5)}$

$\frac{P_1}{P_2} = \left(\frac{n_1}{n_2}\right)^3\left(\frac{d_1}{d_2}\right)^5 \quad \text{(6)}$

Worked Examples

Example: Changing Pump Speed

Given: Constant impeller, speed increases from 1750 rpm to 3500 rpm. Initial conditions: q₁ = 100 gpm, dp₁ = 100 ft, P₁ = 5 bhp.

Results:

• Final flow: q₂ = 100 gpm × (3500/1750) = 200 gpm
• Final head: dp₂ = 100 ft × (3500/1750)² = 400 ft
• Final power: P₂ = 5 bhp × (3500/1750)³ = 40 bhp

Example: Changing Impeller Diameter

Given: Constant speed (1750 rpm), diameter reduced from 8 in to 6 in. Initial conditions: q₁ = 100 gpm, dp₁ = 100 ft, P₁ = 5 bhp.

Results:

• Final flow: q₂ = 100 gpm × (6/8) = 75 gpm
• Final head: dp₂ = 100 ft × (6/8)² = 56.3 ft
• Final power: P₂ = 5 bhp × (6/8)³ = 2.1 bhp

Important Notes

• Affinity laws for fans are not identical to those for pumps.
• The laws assume fluid density remains constant.
• These are approximation laws based on theoretical analysis of centrifugal machines.

References

• Original Source

Quick Reference: Speed Change Effects

When the wheel diameter remains constant and only speed changes, the following relationships apply:

Quick Reference: Diameter Change Effects

When the wheel velocity remains constant and only impeller diameter changes:

Speed Change Calculator

This calculator determines the new operating conditions when pump speed changes at constant impeller diameter. Enter the initial performance parameters and both speeds to calculate the resulting flow, head, and power.