I'm about to conduct some experiments concerning a welding process. To prepare for this I wanted to do a dimensional analysis of the process. So I read a lot about the Π-Theorem and I was able to create 5 dimensionless quantities ($\Pi_1, \Pi_2, ..., \Pi_5$) out of 9 variables ($x_1, x_2, ..., x_9$).
e.g.: $$p_1 = \frac{\Pi_1}{\Pi_3 \cdot \Pi_5^2 \cdot \Pi_8}$$
My problem is that I am trying to find a relationship between the following:
$$y = f(\Pi_1, \Pi_2, ..., \Pi_5)$$
where
$$[y] = [f(\Pi_1, \Pi_2, ..., \Pi_5)] = 1$$
Is it possible to use the Π-Theorem if the function value y has no dimension?
Thanks for your help!