Affinity laws

The affinity laws (also known as the "Fan Laws" or "Pump Laws") for pumps/fans are used in hydraulics, hydronics and/or HVAC to express the relationship between variables involved in pump or fan performance (such as head, volumetric flow rate, shaft speed) and power. They apply to pumps, fans, and hydraulic turbines. In these rotary implements, the affinity laws apply both to centrifugal and axial flows.

The laws are derived using the Buckingham π theorem. The affinity laws are useful as they allow the prediction of the head discharge characteristic of a pump or fan from a known characteristic measured at a different speed or impeller diameter. The only requirement is that the two pumps or fans are dynamically similar, that is, the ratios of the fluid forced are the same. It is also required that the two impellers' speed or diameter are running at the same efficiency.

Essential to understanding the affinity laws requires understanding the pump discharge and head coefficient dimensionless numbers. For a given pump, one can compute the discharge and head coefficients as follows:

${\displaystyle {C_{d}}={Q \over nD^{3}}}$

${\displaystyle {C_{h}}={gH \over n^{2}D^{2}}}$

The coefficient for a given pump is considered to be constant over a range of input values. Therefore, you can estimate the impact of changing one variable while keeping the others constant. When determining the ideal pump for a given application we are regularly changing the motor (i.e. altering the pump speed), or milling down the impeller diameter to tune the pump to operate at the flowrate and head needed for our system. The following laws are derived from the two coefficient equations by setting the coefficient for one operating condition (e.g. Q₁, n₁, D₁) equal to the coefficient for a different operating condition (e.g. Q₂, n₂, D₂).