Affinity laws

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.

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:

${\displaystyle C_d=\frac{Q}{n{D}^3}}$

${\displaystyle C_h=\frac{gH}{n^2{D}^2}}$

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).

Fan affinity laws

The equations below are the fan affinity laws: (see for more info)

Volume flow rate:

${\displaystyle \frac{Q_1}{Q_2}=(\frac{n_1}{n_2})(\frac{D_1}{D_2}{)}^3}$

Head or pressure gain:

${\displaystyle \frac{\mathrm{\Delta }{P}_1}{\mathrm{\Delta }{P}_2}=(\frac{{\rho }_1}{{\rho }_2})(\frac{n_1}{n_2}{)}^2(\frac{D_1}{D_2}{)}^2}$

Power consumption:

${\displaystyle \frac{{\dot{W}}_1}{{\dot{W}}_2}=(\frac{{\rho }_1}{{\rho }_2})(\frac{n_1}{n_2}{)}^3(\frac{D_1}{D_2}{)}^5}$

where

• ${\displaystyle Q}$ is the volumetric flow rate (units length^3/time)
• ${\displaystyle D}$ is the impeller diameter (units of length)
• ${\displaystyle n}$ is the shaft rotational speed (units of 1/time)
• ${\displaystyle \rho }$ is the fluid density (units of mass/length^3)
• ${\displaystyle \mathrm{\Delta }P}$ is the pressure or head developed by the fan/pump (units of pressure)
• ${\displaystyle \dot{W}}$ is the shaft power (units of power, or energy/time).

These laws assume that the pump/fan efficiency remains constant i.e. ${\displaystyle {\eta }_1={\eta }_2}$, which is rarely exactly true, but can be a good approximation when used over appropriate frequency or diameter ranges. The exact relationship between speed, diameter, and efficiency depends on the particulars of the individual fan or pump design. Product testing or computational fluid dynamics become necessary if the range of acceptability is unknown, or if a high level of accuracy is required in the calculation. Interpolation from accurate data is also more accurate than the affinity laws.