In fact, any two independent Pi terms would suffice. The inverse of either of the above Pi terms would also be valid Pi terms, for example, 1/Hi or 1/n2. The most important criteria of the two Pi terms is that they are independent terms. Another important factor in choosing the best Pi terms for the problem being studied, is that it is important to choose one Pi term that includes the dependent variable to be investigated and that it appears only in that term.
By using dimensional analysis, we have now learned that to run this experiment, we do not need to vary density, diameter, or viscosity. It is possible to vary only velocity and measure only pressure gradient. Not only have we reduced the variables from five to two, but we have also created dimensionless terms so that our results are independent of the set of units that we choose. Figure 9.2 shows a series of experiments that compare the pressure gradient, designated in these charts, to the four variables V, d, p, and m. These four comparisons require four separate graphs. This means that we will be able to generate the same amount of data from one experiment that we would have needed four experiments to generate, had we not used dimensional analysis.
Figure 9.3 shows a single dimensionless plot that contains the same information included in all four plots in Fig. 9.2. The plot shows the pressure gradient as a function of V, d, p, and m-
In some models, Pi terms are relatively easy to determine. It is useful here to discuss an algorithm for generating independent Pi terms from a list of variables. To generate a set of independent Pi terms for this example problem, begin by writing down all of the variables, each raised to the power of an unknown as shown below.
(AP/L)a (V )b (d)c (m)e (p)f Next, write the dimensions of those variables, in the same format. (F/L3)a (L/T )b (L)c (FT/L2)e (FT2/L4)f
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