# Design of experiments to determine proportionality

A common result of theoretical analysis in physics is some sort of relation derived from physical parameters and typically expressed in the form of a non-dimensional parameter. These scale relations are not equalities but proportional relationships. For instance, in turbulence you end with with

$$\frac{\eta}{l} \sim Re^{-3/4}$$

Assuming that some derivation results in:

$$f(\Pi_1,\Pi_2,\dots,\Pi_i) \sim C$$

where $C$ is the experimental measure, $f$ is a function of non-dimensional parameters, and $\Pi_i$ are the independent non-dimensional parameters, is there a minimum number of experiments to determine the constant of proportionality $a$ such that:

$$f(\Pi_1,\Pi_2,\dots,\Pi_i) = aC$$

sufficiently? I would expect that the number of experiments required is in some way related to the combination of all possible parameters. For instance, if $i = 2$ then I would expect a minimum of 4 experiments would be required. But is the minimum based on combinations sufficient or are more values needed?

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Part of the answer is going to depend on how well you want to characterize the exponents and to exclude non-power-law dependencies. The short short version is that a power law fit is a line fit on a log--log scale and as such takes at least 5 points to characterize both parameters and their errors (or 6 to get the correlation coefficient as well). –  dmckee Dec 12 '12 at 18:39