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Happy's Essential Skills: Metrics and Dimensional Analysis

Sometimes, the interactions are straightforward, and the relationships are easy to derive.

Other times, the interactions are more complex and time-consuming testing must be done to determine the relevant equations. 

A more scientific approach, however, is available to determine how various design factors interact. Called dimensional analysis, the method involves deriving dimensionless combinations of variables, which then can be plotted easily. Dimensional analysis can be used to: 

• Quickly and easily discover certain errors in analytically derived relationships
• Reduce the amount of testing needed to establish empirical relationships among variables
• Simplify the presentation of theoretically or experimentally derived relationships
• Serve as a basis for all model laws, which are important devices in reducing the cost and complexity of design and scientific problems 

Example of a Pseudo Independent Variable—The Reynolds Number

If you are not familiar with dimensionless parameters, let me pick one that you might have heard of—the Reynolds number. Laminar and turbulent flow of liquids in pipe was first described quantitatively by Osborne Reynolds in 1883 from work done by Sir George Stokes. Reynolds found that that fluid velocity, fluid density, fluid viscosity and pipe diameter determined the energy loss/nature of fluid flow in pipes. The dimensional analysis of these four variables combine to form a single dimensionless parameter we call the Reynolds number. Figure 1 shows the Moody diagram using the Darcy-Weisbach friction factor plotted against Reynolds number (Re) used as the independent variable for a graph for various relative roughness of pipes in fluid flow. 

From the equation below, you can see how the dimensional terms cancel out.

Figure 1: Moody diagram using the Darcy-Weisbach friction factor plotted against Reynolds number (Re). (Source: Wikimedia, Moody Diagram)  

In engineering, there are 154 dimensionless parameters used to facilitate the analysis of physical topics. The Reynolds number is one of those 154. The others are shown in Figures 2A and 2B.

References

1. Holden, H., “The Complexity Index,” IPC Technical Bulletin, May 1978.
2. Holden, H. et al, The HDI Handbook, chapter 3, pp 115–117, published by I-Connect007, 2008.
3. Holden, H., “Calculating your Fabrication Capability Coefficients,” CircuiTree Magazine, February 2006.
4. Discussion on “What_is_the_purpose_of_dimensionless_equations?” by A.H. Rodriquez, ResearchGate. 

Figure 2A: 77 of the dimensionless parameters in engineering.

Figure 2B: Additional 77 of the dimensionless parameters.

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