The fan laws are a set of equations that describe how changes in the operating conditions of a fan, such as changes in flow rate, pressure, or speed, affect the performance of the fan. There are three main fan laws: the law of affinity, the law of volume, and the law of pressure.

The Law of Affinity
The law of affinity states that if the speed of a fan is changed, the flow rate and pressure will change in proportion to the change in speed. This is expressed mathematically as:

(Q2/Q1) = (N2/N1) (P2/P1) = (N2/N1)^2

Where Q is the flow rate, P is the pressure, and N is the fan speed. The subscripts 1 and 2 denote the initial and final operating conditions, respectively. These equations are derived from the fact that the power required to drive a fan varies with the cube of the speed, while the flow rate varies with the speed, and the pressure varies with the square of the speed.

The Law of Volume
The law of volume states that if the volume of air handled by a fan is changed, the flow rate and pressure will change in proportion to the change in volume. This is expressed mathematically as:

(Q2/Q1) = (V2/V1) (P2/P1) = (V2/V1)^2

Where V is the volume of air handled by the fan. The subscripts 1 and 2 denote the initial and final operating conditions, respectively.

The Law of Pressure
The law of pressure states that if the pressure produced by a fan is changed, the flow rate and power required will change in proportion to the change in pressure. This is expressed mathematically as:

(Q2/Q1) = (P2/P1)^0.5 (N2/N1) = (P2/P1)^0.5

Where P is the pressure produced by the fan. The subscripts 1 and 2 denote the initial and final operating conditions, respectively.

These fan laws are important for understanding how changes in operating conditions affect the performance of fans, and can be used to optimize fan systems for different applications. For example, the law of affinity can be used to calculate the performance of a fan at a different speed, which can be useful for controlling the flow rate in a duct system. The law of volume can be used to determine the required fan size for a given volume of air, while the law of pressure can be used to calculate the power required to produce a desired pressure in a system.

Overall, the fan laws are an important tool for HVAC engineers and designers, and understanding how they work can help to optimize the performance and efficiency of fan systems in a wide range of applications.