Dimensional Analysis (I)

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:

• a general discussion of dimensional analysis (already started)
• application of dimensional analysis to the equations of change
• discussion of the common dimensionless groups and their significance
• the Buckingham Pi Theorem
• methods (inspection, Rayleigh, Buckingham) for dimensional analysis of a list of variables
• the principles of similitude and how they can be used for scale-up from model behavior to real systems.

Using Dimensional Analysis

"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:

• showing that your problem is equivalent to one already solved
• giving you the requirements of an experimental test bed

Dimensional analysis relies on the ideas of dimensional consistency:

• in any equation, all terms must have the same dimensions or they cannot be added.
• both sides of an equation must have the same dimensions
• arguments of many functions (log, sin, exp, etc.) and all exponents must be dimensionless.

Based on these ideas, any approach to dimensional analysis will

1. establish a set of variables needed to describe a system
   • from a full mathematical model
   • by experience from known relationships and knowledge of the system
2. determine the dimensionless groups which describe the system
   • by transforming the model to dimensionless form
   • by algebraic combination of variables
3. verify the results by experiment

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.

References:

1. Brodkey, R.S. and H.C. Hershey, Transport Phenomena: A Unified Approach, McGraw-Hill, 1988, pp. 335-339.
2. Churchill, Stuart W., "A New Approach to Teaching Dimensional Analysis", Chemical Engineering Education, 30: 158-165, Summer 1997.
3. McCabe, W.L., J.C. Smith, P. Harriott, Unit Operations of Chemical Engineering, 5th Edition, McGraw-Hill, 1993, pp. 16-18.
4. Roberson, J.A. and C.T. Crowe, Engineering Fluid Mechanics, John Wiley, 1997, pp. 265-273.

R.M. Price
Original: 3/97; 5/21/99
Revised: 9/27/2000

Copyright 2000 by R.M. Price -- All Rights Reserved