For my previous post, I neglected gear/belt efficiency.
Was that the 80% you were referring to?
If it is, I used 100% - not realistic.
I am attaching a revised spreadsheet - I believe I have resolved the error.
I have included the Calculations you provided.
At 100% efficiency, the motor mechanical power and the fan mechanical power are identical.
Along the way, I added an efficiency factor for the gear/belt system.
80% => N = ~4.678, Mechanical motor power is ~25 watts, Electrical motor power is ~30 Watts.
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I’m assuming that you’re going to give the motor 12V now…
As the power efficiency is constant, and the power required to run the fan is a monotonically increasing function of fan speed, the maximum speed you can spin the fan is such that the motor is running at maximum power.
Max power out of the motor at 12V is 348.1W*, which is 13.92 times as much as in the original problem (20W/80%=25W). As the fan power requirement is proportional to the cube of the speed**, the fan is running 2.406 times as fast as the original problem, or 6,014 RPM. At max power output, the motor is running at 6087 RPM*. So the required gearing/belting is a 1.012:1 reduction. In practice, 1:1.
These numbers from the previous post.
** From Kevin, but I’ll buy it because energy is proportional to mass times speed squared and mass per unit time is proportional to speed, making power proportional to speed cubed.