Determine fan sound (dB) based on its rotational speed (RPM) I am trying to find the sound amplitude or level produced by a fan, knowing its RPM.
Is there a simple formula for it?
What other parameters does it depend on? Assume I have all the fan properties, such as input power, number of blades, blade area, blade length and so on. Also, assume the atmospheric pressure is known. Thanks!
 A: From the class notes from this Mechanical Engineering course about noise and its perception at Penn State, it looks like, all else being equal, there is a log-linear relationship between the rotational speed of a fan and the noise level (expressed in decibels):
$$
\Delta N ~\sim~ 50 \, \Delta \log_{10}{\left[\text{rpm}\right]}
$$
This won't give you the noise level of a given fan.
However, knowing its reference noise level (in dB) at a reference RPM_0 speed, you'll be able to infer its noise at a higher or lower RPM_1 speed:
$$
N_1 = N_0 + 50 \, \left(\log_{10} \left[\text{rpm}\right]_1 \, – \, \log_{10} \left[\text{rpm}\right]_0\right)
$$
Bear in mind this is an approximation. Hopefully it does help you reduce the fan noise in your system.
A: There isn't a simple formula for fan noise, but the physics can be worked out from the fundamental equations of fluid mechanics and acoustics. It isn't a simple problem however. The noise created by fans is complex and from several fundamental sources, and its amplitude depends on frequency.
Here is an example of a fan noise frequency spectrum for a cooling fan.  Credit to the Wikibooks site Acoustics/Noise From Cooling Fans. That site also goes into a bit more detail about noise sources than is possible here, and has some references for further research.

As can be seen in the diagram, there are several kinds of noise created by rotating fans. These include narrow band noise at the fundamental frequency of fan blade rotation, and noise created by harmonics of that fundamental blade frequency. This is fundamentally caused by the oscillating pressure field produced by the motion of the fan
There is also low and higher frequency noise created by boundary layer and turbulence effects at the fan's blades, and from the rotation of the fan shaft and motor.
As discussed at the Wikibooks site, they produced the above diagram experimentally, and that is probably the best approach with the specific fan you're trying to model - capture the power spectrum experimentally and fit some empirical curve to it. Solving the problem analytically or computationally is not simple. It can be done but computational fluid dynamics is an expensive problem both in modeling and computing effort.
