Affinity laws

The affinity laws for pumps/fans are used in hydraulics 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 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.

Law 1. With impeller diameter (D) held constant:

Law 1a. Flow is proportional to shaft speed:^[1]

${ Q_1 \over \ Q_2} = { \left ( {N_1 \over N_2} \right )}$

Law 1b. Pressure or Head is proportional to the square of shaft speed:

${H_1 \over H_2} = { \left ( {N_1 \over N_2} \right )^2 }$

Law 1c. Power is proportional to the cube of shaft speed:

${P_1 \over P_2} = { \left ( {N_1 \over N_2} \right )^3 }$

Law 2. With shaft speed (N) held constant: ^[1]

Law 2a. Flow is proportional to impeller diameter:

${ Q_1 \over \ Q_2} = { \left ( {D_1 \over D_2} \right ) }$

Law 2b. Pressure or Head is proportional to square of impeller diameter:

${H_1 \over H_2} = { \left ( {D_1 \over D_2} \right )^2 }$

Law 2c. Power is proportional to the cube of impeller diameter:

${P_1 \over P_2} = { \left ( {D_1 \over D_2} \right )^3 }$

where

• $Q$ is the volumetric flow rate (e.g. CFM, GPM or L/s),
• $D$ is the impeller diameter (e.g. in or mm),
• $N$ is the shaft rotational speed (e.g. rpm),
• $H$ is the pressure or head developed by the fan/pump (e.g. psi or Pascal), and
• $P$ is the shaft power (e.g. W).^[2]

These laws assume that the pump/fan efficiency remains constant i.e. $\eta_1 = \eta_2$, which is rarely exactly true, but can be a good approximation when used over appropriate frequency or diameter ranges (i.e., a fan will not move anywhere near 1000 times as much air when spun at 1000 times its designed operating speed, but the air movement may be increased by 99% when the operating speed is only doubled). 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. When applied to pumps the laws work well for constant diameter variable speed case (Law 1) but are less accurate for constant speed variable impeller diameter case (Law 2).

References

1. "Basic Pump Parameters and the Affinity Laws" (PDF). PDH Online.
2. "Pump Affinity Laws". Retrieved 18 November 2014.