# Fan Speed Formula

Is there any formula for computing fan speed by using air mass,air flux, air density or specific heat? I have computed air mass and air flux, and found the values for air density and specific heat, but now I am stuck at finding a correlation between these and fan speed.

• Can you reproduce your computing here much appreciated – user6760 May 18 '15 at 12:20
• I start from 2 equations for computing the power consumption of a data center: P = mCp*(Tout-Tin) and P=pf*Cp(Tout-Tin), where m=air mass, Cp= specific heat, p=air density (1.205kg/m^3 at 20degreesC),f= air flow rate = m/p, Tin = the inlet temperature (temperature supplied by a CRAC unit), Tout=exhaust temperature. I also know that the fan power Pfan=(FanSpeed^3) (equation 3). What I need, is to find a correlation between the first 2 equation and the 3rd one – Ade May 19 '15 at 10:05
• The fan laws state that the air flow is proportional with the fan speed. So, basically, I need to convert airflow (m^3/s) to speed – Ade May 19 '15 at 13:54

## 2 Answers

From an engineering perspective, there are many different fan designs including axial and centrifugal configurations, along with various blade designs including forward curved, backward curved, and radial. There is no formula to calculate the required fan speed, but if a specific fan configuration is known then fan similarity laws can be used to calculate performance based on known performance of a similar fan. For example, for a given fan design the flow varies linearly with speed, static pressure varies with the square of the speed, and power consumption varies with the cube of the speed. Similarly, at the same speed flow varies with the cube of the impeller diameter, and static pressure varies as the square of the diameter.

These formulas are described here for example.

Using a dimensional analysis indeed can help you find dimensionless numbers as The Dark Side suggested Floris might help with, but beyond that there is no direct analytical method, no closed form solutions that can relate fan speed, shaft torque, flow rate and delta pressure. The issue is that at best the flow behavior is two dimensional, but more likely three dimensional in its behavior. There are two approach's to find the relationships:

(1) Do the dimensionless analysis, come up with dimensionless numbers, then do experiments with your fan to find the relationship of speed to the other parameters. Then you can fit the relationships to polynomials.

(2) Model the fan in CAD, then do a dimensional CFD simulation to relate your parameters of interest.

The difficulty in trying to obtain an analytical solution directly from the fundamental energy, momentum and continuity principles comes from the nonlinear nature of the Navier Stokes equations and the intractability of the equations in particular 2 dimensional and 3 dimensional problems. If fluid rotation can occur you are stuck doing 3 dimensional analysis. The other factor is the complex description of the fan's surface. Not all fans are alike.