Fans
Fans or pumps use a rotating set of blades to push air forward.
These fans are often connected to an electric motor.
The speed that the fan turns at is often specified in RPM.
The flow that a fan provides is often measured in cubic feet per minute or CFM.
A measure of fan efficiency used for airflow is watts per CFM.
Fan Laws
The fan laws (or affinity laws or pump laws) relate the following quantities
- fan diameter
- fan speed
- volumetric flow rate
- pressure difference
- power
Fan laws for same diameter
If you hold the fan diameter constant,
- Flow (Q) is proportional to shaft speed (N)
- The pressure difference or head (H) is proportional to the square of the shaft speed
- The power (P) is proportional to the cube of the shaft speed
Here are these relationships written out using mathematical notation.
{ Q_1 \over Q_2} = { \left ( {N_1 \over N_2} \right )^1}
{H_1 \over H_2} = { \left ( {N_1 \over N_2} \right )^2 }
{P_1 \over P_2} = { \left ( {N_1 \over N_2} \right )^3 }
Fan laws for same pump rotational speed
If we hold the rotational speed constant,
- Flow is proportional to the fan diameter
- The pressure difference is proportional to the square of the fan diameter
- The power is proportional to the fan diameter cubed
{Q_1 \over Q_2} = { \left ( {D_1 \over D_2} \right )^1 }
{H_1 \over H_2} = { \left ( {D_1 \over D_2} \right )^2 }
{P_1 \over P_2} = { \left ( {D_1 \over D_2} \right )^3 }
Speed increase example
Lets say we need to increase the flow rate by 10%. How much will the power need increase?
Since the flow rate is proportional to the speed, we have to increase the speed by 10%.
Let’s call the original speed N_1 and the new speed N_2=1.1 N_1. The increase in power will be the ratio of the new increased power P_2 divided by the original power P_1.
{P_1 \over P_2} = { \left ( {N_1 \over 1.1 N_1} \right )^3 }
{P_1 \over P_2} = {1 \over 1.33}
{P_2 \over P_1} = {1.33}
So, raising the flow rate or speed by 10% raises the power by about one-third.
Motor Power Consumption
If the speed of a fan is doubled, the mechanical power goes up by a factor of 8 since the fan law says power is proportional to fan speed cubed.
What does this mean for the voltage and current of the motor?
On the mechanical side, the motor power is given by the speed times the torque. If the speed doubles, the torque must go up by a factor of 4 to get the overall factor of 8 increase.
P_1 = \omega_{1} \tau_1 P_2 = 2 \omega_{1} 4 \tau_{1} = 8 P_1
On the electrical side, if the speed doubles, the back-EMF must double. If the torque goes up by a factor of four, so must the current. The electrical power will be greater than the mechanical power since there will be more current flowing through the internal resistance being converted to heat.
Blower Doors
Blower doors use the pressure difference and flow rate created by a fan to determine how well sealed a building is.
Kinetic Energy of Air Flow
P = {1 \over 2} \rho A v^3
Exercises
1 Power Increase
You have a fan system delivering 20 CFM. If you increase the flow to 25 CFM, how much will the fan speed and mechanical power change?
2 Watt per CFM
About how much will the watts per CFM change if the same system is run faster and must deliver 20% more air?
3 Current and Voltage increase
About how much will the voltage and current increase if a system must deliver 20% more air? (You can ignore the internal resistance of the motor for this estimation.)
4 Title 24 W per CFM
Read Subchapter 7 Section 150.0.m.13 parts A-D of the Title 24 energy code.
What are the required values for power per CFM?
5 Fan Power Increase
You have a fan spinning at 1000 RPM while providing 50 watts of power. To increase your airflow, you increase the speed to 1250 RPM. What will the power delivery be after the increase?