ITP: Dimensional Analysis

There are several very good reasons to regard engineering problems, not in terms of dimensional quantities, but instead as dimensionless equations/relations:

DEFINITION:

Dimensional analysis is the process by one predicts and identifies the number and form of all relevant dimensionless parameters for a given problem.

As stated above, dimensional analysis is a formal method of predicting which and how many dimensionless groups are important (and independent!) in a given problem.

The procedure for doing so is rather simple (and is called the Buckingham Pi method):

1. Make a list of relevant parameters:

Here you brainstorm on each of the material and process variables that may have some effect on your problem of interest.

2. Identify the base units represented in each of these variables:

In this step you take the variables from step 1, which usually have some form of compound units (like force) and break them into their base units/dimensions (like mass*length per time squared).

3. Use the Buckingham Pi Theorem to determine the number of independent dimensionless variables to make:

If you take the number of variables, V (from step 1), and subtract off the number of base dimensions, D, you get the number of independent dimensionless groups, G, possible: G = V - D.

4. Chose a subset of your variables, V, equal to the number of dimensions, D, that are recurring variables for building your dimensionless groups:

This is the trickiest step for two main reasons: a) you need to choose these variables well (meaning that they represent all of the base dimensions in an independent way, if possible -- avoid repeating the same type of variable!) or else you may have difficulty getting reasonable groups; b) the variables you choose will actually affect your answer (but you can always show how new dimensionless groups can be made via combinations of other dimensionless groups, so it is no problem).

5. With each of the remaining non-recurring variables form "dimensionless groups" that have undetermined powers:

Here you form V-D candidate groups by taking each of your recurring variables to a different unknown power (that is different for each candidate group!) and multiply it by one of your non-recurring groups (usually to the first power).

6. Solve for your unknown powers:

Step five will give you D algebraic equations for the powers in each of the G candidate dimensionless groups. You get these equations by simply recognizing that the sum of the powers for each base dimension must be equal to zero for a dimensionless group. Solve the handful of simultaneous algebraic equations and you get your real dimensionless group. repeat this for each candidate and you are done!

OUTCOME:

Perform simple dimensional analysis, using the Buckingham Pi method.