Dimensional Analysis gleans information from the fundamental dimensions that describe a process or problem and use that information to gain insight into the problem. It is based on dimensional consistency or dimensional homogeneity -- the principle that all additive or equated terms in an equation must have the same net dimensions.
There are two broad ways of applying dimensional analysis.
The method provides the most insight when a complete mathematical model is available. Dimensional analysis of a model can indicate the important dimensionless groups of variables, reduce the model to a minimal form, and make it easy to assess asymptotic behavior of the system. In a way, dimensional analysis of a describing equation is the source of the dimensionless groups we look at to understand a problem (for example, the Reynolds number, which results from dimensional analysis of the equation of motion).
In the second approach, the starting point is nothing more than a list of variables that are believed to be important to understanding the problem. Applied this way, the analysis is speculative -- all choices and results are tentative and should be confirmed by experiment. The list of variables you start with makes a big difference. Selection of variables for the list depend on experience, and so application of this approach is tricky when dealing with unfamiliar systems. On the other hand, if you can do a reasonable job of selecting variables, a lot of useful information may be gained.
In this course, we'll look at both approaches, and you will be expected to work problems applying the second approach.
Our study of dimensional analysis will thus follow the following program:
"Real" transport problems are often too complicated to solve analytically, so solutions rely on empirical data and correlations.
For common situations (flow in a pipe, etc.) data and correlations are readily available in handbooks. But what if you're confronted with a new or unique situation? Dimensional analysis can help by:
Dimensional analysis relies on the ideas of dimensional consistency:
Based on these ideas, any approach to dimensional analysis will
If you start with a full model, the dimensionless form can be used to study limiting behavior of the system -- what happens when one group becomes much larger or much smaller than other -- and the dimensionless groups can be interpreted in terms of fundamental behaviors.
If you start from a list of variables, trying to correlate their relationship, algebraic combination of the variables into dimensionless groups reduces the number of parameters. Then, when you try to fit the data, you need fewer experiments. Be aware, though, that dimensional analysis does not specify the form of the correlating equation.
Original: 3/97; 5/21/99
Copyright 2000 by R.M. Price -- All Rights Reserved