Revision history [back]

If a large portion of the mechanical power of the impeller would be converted to rotational energy, the the fan would constantly accelerate. In reality what happens, if you feed a fan's motor with constant power, then the fan starts to accelerate, and after a while it rotates with a constant speed. As soon as the fan is at constant speed, the power fed in to increase the rotational energy of the fan is equal to the power, that the fan feeds into the mechanical energy of the air. These two powers (motor-> fan & fan-> air) are in balance, that is why the fan rotates at constant speed, and why all this energy goes to heat.
To me the equation looks correct. The minor things, that you may argue with, may be:

• Where exactly the mechanical energy of the air turns into heat. Now this calculation assumes, that this happens at the fan or in the duct, however some of the dissipation of mechanical energy of air to heat happens in the room, where the air stops moving due to friction. I think this is a very reasonable simplification.
• Other thing to nit-pick: With the extract fan, you throw away some of the mechanical energy of the air that is exiting the building at the air exhaust. So the system is not completely closed, a bit of energy is lost this way, which now is assumed to be heating the exhaust air.

If a large portion of the mechanical power of the impeller would be converted to rotational energy, the the fan would constantly accelerate. In reality what happens, if you feed a fan's motor with constant power, then the fan starts to accelerate, and after a while it rotates with a constant speed. As soon as the fan is at constant speed, the power fed in to increase the rotational energy of the fan is equal to the power, that the fan feeds into the mechanical energy of the air. These two powers (motor-> fan & fan-> air) are in balance, that is why the fan rotates at constant speed, and why all this energy goes to heat. heat (the air stops moving after a while and its energy dissipates to heat).
To me the equation looks correct. The minor things, that you may argue with, may be:

• Where exactly the mechanical energy of the air turns into heat. Now this calculation assumes, that this happens at the fan or in the duct, however some of the dissipation of mechanical energy of air to heat happens in the room, where the air stops moving due to friction. I think this is a very reasonable simplification.
• Other thing to nit-pick: With the extract fan, you throw away some of the mechanical energy of the air that is exiting the building at the air exhaust. So the system is not completely closed, a bit of energy is lost this way, which now is assumed to be heating the exhaust air.

If a large portion of the mechanical power of the impeller would be converted to rotational energy, the the fan would constantly accelerate. In reality what happens, if you feed a fan's motor with constant power, then the fan starts to accelerate, and after a while it rotates with a constant speed. As soon as the fan is at constant speed, the power fed in to increase the rotational energy of the fan is equal to the power, that the fan feeds into the mechanical kinetic energy of the air. These two powers (motor-> fan & fan-> air) are in balance, that is why the fan rotates at constant speed, and why all this energy goes to heat (the air stops moving after a while and its energy dissipates to heat).
To me the equation looks correct. The minor things, that you may argue with, may be:

• Where exactly the mechanical kinetic energy of the air turns into heat. Now this calculation assumes, that this happens at the fan or in the duct, however some of the dissipation of mechanical kinetic energy of air to heat happens in the room, where the air stops moving due to friction. I think this is a very reasonable simplification.
• Other thing to nit-pick: With the extract fan, you throw away some of the mechanical kinetic energy of the air that is exiting the building at the air exhaust. So the system is not completely closed, a bit of energy is lost this way, which now is assumed to be heating the exhaust air.