One Dimensionless Group Problem

The simplest problem concerns only one pi term. The complexity of the analysis increases rapidly with increasing number of pi terms because then the choice of the experimental plan related to the proposed relationship for the dimensionless pi groups cannot be solved by an automatic procedure. 

When a one-dimensional analysis problem shows that the difference m - n (m = number of process variables, n= number of basic dimensions associated with the process variables) is unitary, then only one pi term is required to describe the process. The functional relationship that can be used for one pi term is  

The case of Stokes settling velocity is considered as an illustrative example. If we assume that the stationary settling velocity, Wg, of a small particle flowing into a liquid or gaseous medium is a function of its diameter, d, specific weight, gAp, and the viscosity of the gaseous or liquid medium, it follows that  

We observe that four variables and three basic dimensions (M, L, T) are required to describe the variables. For this problem, one pi term (group) can be produced according to the pi theorem. This pi group can easily be expressed as  

Since there is only one pi group, it follows that  

---