Uniqueness of Pi Terms

A review of the method of an incomplete group of repeating variables used for identifying pi terms reveals that the specific pi terms obtained depend on the somewhat arbitrary selection of this incomplete group. For example, in the problem of studying the heat transfer from a wall to a fluid flowing in the pipe, we have selected d, p, p, and X as repeating variables. This has led to the formulation of the problem in terms of pi terms  

What will the result be if we select d, p, p, and Cp as repeating variables A quick check will reveal that the pi term involving the heat transfer coefficient (a) becomes  

It must be emphasized, however, that the required number of pi terms cannot be reduced by this manipulation only their form is altered. Thanks to this technique. 

One may ask which form is the best for the pi groups Usually, we recommend keeping pi terms as simple as possible. In addition, it is easier to use the pi terms that could be improved by the experimental methodology used. The final choice remains arbitrary and generally depends on the researcher s background and experience. 

The previously presented method of the incomplete group of repeating variables, provides a systematic procedure which, when properly executed, provides a correct, complete and unique set of pi terms. In other words, this method offers an excellent algorithm for the calculus. In this case, only the list of variables has to be determined by the researcher. Since the only restrictions for the pi terms are to be (a) correct in number, (b) dimensionless, and (c) independent, it is possible to produce other identifying procedures. One of them is the production of pi terms by inspection, without resorting to a more formal methodology. 

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