  We consider the design of dimensional analysis experiments when there is more
than a single response. We first give a brief overview of dimensional analysis
experiments and the dimensional analysis (DA) procedure. The validity of the DA
method for univariate responses was established by the Buckingham $\Pi$-Theorem
in the early 20th century. We extend the theorem to the multivariate case,
develop basic criteria for multivariate design of DA and give guidelines for
design construction. Finally, we illustrate the construction of designs for DA
experiments for an example involving the design of a heat exchanger.
