Affinity laws

The affinity laws are used in hydraulics and HVAC to express the relationship between several 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 various rotary implements, the Affinity Laws apply both to centrifugal and axial flows.

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

Law 1a. Flow is proportional to shaft speed: $\{\; Q\_1\; over\; Q\_2\}\; =\; \{\; left\; (\; \{N\_1\; over\; N\_2\}\; ight\; )\}$ Law 1b. Pressure or Head is proportional to the square of shaft speed: $\{H\_1\; over\; H\_2\}\; =\; \{\; left\; (\; \{N\_1\; over\; N\_2\}\; ight\; )^2\; \}$

Law 1c. Power is proportional to the cube of shaft speed: $\{P\_1\; over\; P\_2\}\; =\; \{\; left\; (\; \{N\_1\; over\; N\_2\}\; ight\; )^3\; \}$

Law 2. With shaft speed (N) held constant:

Law 2a. Flow is proportional to impeller diameter: $\{\; Q\_1\; over\; Q\_2\}\; =\; \{\; left\; (\; \{D\_1\; over\; D\_2\}\; ight\; )\}$ Law 2b. Pressure or Head is proportional to the square of impeller diameter: $\{H\_1\; over\; H\_2\}\; =\; \{\; left\; (\; \{D\_1\; over\; D\_2\}\; ight\; )^2\; \}$

Law 2c. Power is proportional to the cube of impeller diameter : $\{P\_1\; over\; P\_2\}\; =\; \{\; left\; (\; \{D\_1\; over\; D\_2\}\; ight\; )^3\; \}$

where
* $Q$ is the volumetric flow rate (e.g. CFM or GPM),
* $D$ is the impeller diameter (e.g. in),
* $N$ is the shaft rotational speed (e.g. rpm),
* $H$ is the pressure or head developed by the fan/pump, and
* $P$ is the shaft power.

These laws assume that the pump/fan efficiency remains constant. In other words, $eta\_1\; =\; eta\_2$ .