# Affinity Laws

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

The affinity laws express the mathematical relationship between the several variables involved in pump performance. They apply to all types of centrifugal and axial flow pumps. Being able to predict these affects allows the rotating equipment engineer to examine the effects before implementing the changes.  Transposing a pump curve into an analysis program, such as Microsoft Excel or Open Office Calc provides an excellend visual representation of how varying parameters affects the pump performance.

### CONSTANT IMPELLER DIAMETER Formula

Capacity varies directly with impeller diameter and speed.

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

Head varies directly with the square of impeller diameter and speed.

$$\large{ \frac{h_1}{h_2}=\left(\frac{n_1}{n_2}\right)^2 }$$

Horsepower varies directly with the cube of impeller diameter and speed.

$$\large{ \frac{BHP_1}{BHP_2}=\left(\frac{n_1}{n_2}\right)^3 }$$

Where:

$$\large{ BHP }$$ = brake horsepower

$$\large{ h }$$ = total head

$$\large{ n }$$ = pump speed

$$\large{ NPSH_r }$$ = Net positive suction head required

$$\large{ Q }$$ = capacity

### CONSTANT PUMP SPEED Formula

Capacity varies directly with impeller diameter and speed.

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

Head varies directly with the square of impeller diameter and speed.

$$\large{ \frac{h_1}{h_2}=\left(\frac{D_1}{D_2}\right)^2 }$$

Horsepower varies directly with the cube of impeller diameter and speed.

$$\large{ \frac{BHP_1}{BHP_2}=\left(\frac{D_1}{D_2}\right)^3 }$$

Where:

$$\large{ BHP }$$ = brake horsepower

$$\large{ D }$$ = impeller diameter

$$\large{ h }$$ = total head

$$\large{ Q }$$ = capacity

## RULE OF THUMB

While not an exact representation, the following relationships have been observed with regards to changing impeller diameters.

### NPSHr

Net Positive Suction Head Required by the pump varies directly with the impeller diameter.

$$\large{ \frac{NPSH_r1}{NPSH_r2}=\frac{D_1}{D_2} }$$

Where:

$$\large{ D }$$ = impeller diameter

$$\large{ NPSH_r }$$ = Net positive suction head required

### shaft deflection

Shaft Deflection (runout) measured prior to changing the impeller size varies with the impeller diameter.

$$\large{ \frac{d_1}{d_2}=\frac{D_1}{D_2} }$$

Where:

$$\large{ d }$$ = shaft deflection

$$\large{ D }$$ = impeller diameter