## Affinity

Consuming 10 W of power, a fan with a 1.5 m diameter impeller rotates at 500 rpm to cool a warehouse in the summer. If the same fan were to increase its rotational speed to 3,000 rpm, how much power is the fan now using? Assume the density of air is 101 kPa.

##
__
__**Expand Hint**

**Expand Hint**

$$$\left ( \frac{\dot{W}}{\rho N^3D^5} \right )_2= \left ( \frac{\dot{W}}{\rho N^3D^5} \right )_1$$$

where
$$\dot{W}$$
is the power,
$$\rho$$
is the fluid density,
$$N$$
is the rotational speed, and
$$D$$
is the impeller diameter.

##
__
__**Hint 2**

**Hint 2**

Since both diameter and density are constant, the equation becomes:

$$$\left ( \frac{\dot{W}}{N^3} \right )_2= \left ( \frac{\dot{W}}{N^3} \right )_1$$$

Using the scaling/affinity laws:

$$$\left ( \frac{\dot{W}}{\rho N^3D^5} \right )_2= \left ( \frac{\dot{W}}{\rho N^3D^5} \right )_1$$$

where
$$\dot{W}$$
is the power,
$$\rho$$
is the fluid density,
$$N$$
is the rotational speed, and
$$D$$
is the impeller diameter.

Since both diameter and density are constant, the equation becomes:

$$$\left ( \frac{\dot{W}}{N^3} \right )_2= \left ( \frac{\dot{W}}{N^3} \right )_1$$$

$$$\left ( \frac{\dot{W_1}}{N_{1}^{3}} \right )= \left ( \frac{\dot{W_2}}{N_{2}^{3}} \right )$$$

$$$\left ( \frac{10W}{(500rpm)^{3}} \right )= \left ( \frac{\dot{W_2}}{(3,000rpm)^{3}} \right )$$$

$$$\dot{W_2}=\frac{10W}{(500rpm)^{3}}\cdot (3,000rpm)^{3}=2,160\:W$$$

2,160 W

Similar Problems from FE Sub Section:

199. Affinity Law

477. Scaling Law

Similar Problems from FE Section:

182. Pump Power

199. Affinity Law

213. Centrifugal Fan

297. Pressurized Pump Power

323. Pump Work

348. Centrifugal Pump

366. Pump Efficiency

477. Scaling Law

514. A Pumpâ€™s Efficiency

**Performance of Components**199. Affinity Law

477. Scaling Law

Similar Problems from FE Section:

**Fluid Flow Machinery**182. Pump Power

199. Affinity Law

213. Centrifugal Fan

297. Pressurized Pump Power

323. Pump Work

348. Centrifugal Pump

366. Pump Efficiency

477. Scaling Law

514. A Pumpâ€™s Efficiency