# Affinity Laws

on . Posted in Classical Mechanics

The affinity laws, also called the fan laws, are a set of mathematical relationships that describe how changes in certain parameters affect the performance of a fan or pump system.  These laws are applicable to centrifugal fans, axial fans, and centrifugal pumps.  The affinity laws are based on the principles of fluid dynamics and can be used to estimate the effects of changes in speed, flow rate, head pressure, and power consumption.  Being able to predict these affects allows the rotating equipment engineer to examine the effects before implementing the changes.

### CONSTANT IMPELLER DIAMETER formulas

$$Q_1\;/\;Q_2 = (n_1\;/\;n_2)^1$$    (Capacity varies directly with impeller diameter and speed)

$$h_1\;/\;h_2 = (n_1\;/\;n_2)^2$$     (Head varies directly with the square of impeller diameter and speed)

$$BHP_1\;/\;BHP_2 = (n_1\;/\;n_2)^3$$     (Horsepower varies directly with the cube of impeller diameter and speed)

Symbol English Metric
$$BHP$$ = brake horsepower $$lbf-ft\;/\;sec$$ $$J\;/\;s$$
$$Q$$ = capacity $$gal\;/\;min$$ $$l\;/\;min$$
$$n$$ = pump speed $$rpm\;/\;min$$ $$rpm\;/\;min$$
$$h$$ = total head $$ft$$ $$m$$

### CONSTANT PUMP SPEED formulas

$$Q_1\;/\;Q_2 = (d_1\;/\;d_2)^1$$     (Capacity varies directly with impeller diameter and speed)

$$h_1\;/\;h_2 = (d_1\;/\;d_2)^2$$     (Head varies directly with the square of impeller diameter and speed)

$$BHP_1\;/\;BHP_2 = (d_1\;/\;d_2)^3$$     (Horsepower varies directly with the cube of impeller diameter and speed)

Symbol English Metric
$$BHP$$ = brake horsepower $$lbf-ft\;/\;sec$$ $$J\;/\;s$$
$$Q$$ = capacity $$gal\;/\;min$$ $$l\;/\;min$$
$$d$$ = impeller diameter $$in$$ $$mm$$
$$h$$ = total head $$ft$$ $$m$$

## RULE OF THUMB

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

### NPSHr formula

$$NPSH_r1\;/\;NPSH_r2 = d_1\;/\;d_2$$     (Net Positive Suction Head Required by the pump varies directly with the impeller diameter)
Symbol English Metric
$$NPSH_r$$ = net positive suction head required $$lbf\;/\;in^2$$ $$N\;/\;m^2$$
$$d$$ = impeller diameter $$in$$ $$mm$$

### shaft deflection formula

$$\theta_1 \;/\; \theta_2 = d_1\;/\;d_2$$     (Shaft Deflection (runout) measured prior to changing the impeller size varies with the impeller diameter)
Symbol English Metric
$$\theta$$  (Greek symbol theta) = shaft deflection $$in$$ $$mm$$
$$d$$ = impeller diameter $$in$$ $$mm$$