# Dimensionless Number

Dimensionless numbers are widely applied in the field of mechanical and chemical engineering. According to the Pi-Theorem, the functional dependence between a certain number of variables (e.g.: n) can be reduced by the number of independent dimensions occuring in those variables (e.g. k) to give a set of independent, dimensionless numbers (e.g. p=n-k). The dimensionless numbers can be derived by dimensional analysis.

Example: Stirrer

The power-consumption of a stirrer is a function of the density and the viscosity of the fluid to be stirred, the size and the particular geometry of the stirrer given by the diameter of the stirrer, and the speed of the stirrer. Therefore, we do have n=5 variables representing our example.

Those n=5 variables are build up by k=3 dimensions being:

• Length L [m]
• Time T [s]
• Mass M [kg]

According to the Pi-Theorem, the n=5 variables can be reduced by the k=3 dimensions to form p=n-k=5-3=2 independent dimensionless numbers which are in case of the stirrer

There are literally thousands (to be precise: infinite) dimensionless numbers including those being used most often: (in alphabetical order indicating the field of use)