The fan laws are the affinity laws with different labels on the axes. $\dfrac{Q_2}{Q_1} = \dfrac{N_2}{N_1}$ for airflow, $\dfrac{SP_2}{SP_1} = \left(\dfrac{N_2}{N_1}\right)^{2}$ for static pressure, and $\dfrac{P_2}{P_1} = \left(\dfrac{N_2}{N_1}\right)^{3}$ for shaft power. $Q$ is the airflow in m³/s, $SP$ the fan static pressure in pascals, $P$ the shaft power in kW and $N$ the wheel speed in rpm — and the subscripts mean what they meant on the water side: 1 is where you are, 2 is where you are going.

The power itself is friendlier in SI than in any other units: $P_{fan} = \dfrac{Q \, \Delta p}{\eta}$ — P-fan equals Q delta-p over eta. $Q$ in cubic metres per second times $\Delta p$ in pascals is watts exactly, with no trade constant at all, and $\eta$ — the fan efficiency, a bare fraction — divides on the way to the shaft just as it did for the pump. The imperial trade writes the same thing as $\dfrac{Q \cdot SP}{6356\eta}$ in cfm and inches of water; the 6356 is nothing but unit conversion, and SI simply does not need it.

Where this bites in the plant room is the re-sheave. A belt-driven fan's speed is set by pulley diameters, and a balancer chasing design airflow will change one. Ten per cent more speed is 10 % more air — and 21 % more static pressure, which the flex connections feel, and 33 % more shaft power, which the motor feels. The cube is not a warning printed on the side of the calculation; it IS the calculation. Check the nameplate before the sheave, every time.

And the same exponent, read the other way, is why variable-air-volume systems save what they do. A fan spending its shoulder season at 80 % speed is drawing barely half its design power, all day, without anyone touching a damper.