Dimensional analysis Buckingham method

There are two classical methods in dimensional analysis, Buckingham s pi theorem and the method of indices by Lord Rayleigh. Here we will briefly explain the more common of the two Buckingham s theorem. 

Dimensional Analysis is.a method by which the variables characterizing a phenomenon may be related. Accdg to Eschbach (Ref 2)> it is fundamentally identical with the analysis of physical equations, and in particular, with the analysis of physical differential equations. Methods of Lord Rayleigh and of E. Buckingham are used in ballistics, thermodynamics and fluid mechanics... 

Using the method of dimensional analysis described by Buckingham (4), these functions are related by ... 

The generation of such dimensionless groups in heat transfer (known generally as dimensional analysis) is basically done (1) by using differential equations and their boundary conditions (this method is sometimes called a differential similarity) and (2) by applying the dimensional analysis in the form of the Buckingham pi theorem. 

The dimensionless equation describing the transfer phenomena may be obtained either by direct reference to the ratios of the physical quantities or by recourse to the classical techniques of dimensional analysis, i.e., the Buckingham n Theorem or Rayleigh s method of indices. In addition, the basic differential equations governing the process may be reduced to dimensionless form and the coefficients identified. In general, the dimensionless equation for heat transfer through the combined film is... 

Buckingham is best known for his early work on thermodynamics and for his later study of dimensional theory. Attracted to problems that could not be solved by pure calculation but requiring experimentation as well, he demonstrated more clearly than anybody before him how the planning and interpretation of experiments can be facilitated by the method of dimensions, later referred to as dimensional analysis. He pointed out the advantages of dimensionless variables and how to generalize empirical equations. His frequently cited ir-theorem serves to reduce the number of independent variables and shows how to experiment on geometrically similar models so as to satisfy the most general requirements of physieal as well as dynamic similarity. 

Table 2.1 presents a non-exhaustive list of expressions of the characteristic times corresponding to the most commonly used phenomena involved in chemical reactors. The previous definitions unfortunately do not always enable one to build the expressions presented in Table 2.1. Various methods can be used such as a blind dimensional analysis, similar to the Buckingham method used for dimensionless numbers [7], which can be applied to the list of fundamental physical and chemical properties. Nevertheless, the most relevant method consists in extracting the expressions from a mass/heat/force balance.

An important way of obtaining these dimensionless groups is to use dimensional analysis of differential equations described in Section 3.11. Another useful method is the Buckingham method, in which the listing of the significant variables in the particular physical problem is done first. Then we determine the number of dimensionless parameters into which the variables may be combined. 

---