N1 n2 n3 n nn

Figure 9.1 Pressure gauges at two points along a pipeline of diameter d, with a mean velocity of the pipe of V.

model, we will begin with a relatively small set of variables and when the model does not suitably represent the prototype we may find out that we need to add variables. For this example problem, we will assume that the pressure gradient Ap/L is related to average velocity in the pipe, V, the diameter of the pipe, d, the viscosity of the fluid flowing in the pipe, m, and the density of the fluid, p.

The third step in the process is to write the dimensions of each variable. For example, the variable AP/L has the dimensions of (force/length3) as shown next.

The other variables in the example have dimensions as shown in Eqs. (9.5) through (9.7):

Now that all of the dimensions have been described, we can use Buckingham's Pi theorem to determine the number of required Pi terms to describe this model.

n = k — r = 5 dimensions — 3 basic dimensions = 2 Pi terms (9.8)

The result of this analysis is that we have confirmed the following fact. To conduct an experiment to determine the relationship between AP/L

and the other important variables, we only need to measure two dimen-sionless Pi terms and not five different variables.

Pi terms are dimensionless terms and in this case, it would be possible to choose the following two Pi terms to describe the system.

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