Happy's Essential Skills: Metrics and Dimensional Analysis
After reading 20 of my columns, by now you probably realize that I am an analytical person. Thus, I dedicate this column to the topic of metrics—the method of measuring something. I mentioned the four levels of metrics in my column entitled Producibility and Other Figures of Merit back in June of this year. Additionally, I introduced the five stages of metrics in the second part of the following column, Design for Manufacturing and Assembly, Part 2 later that same month. This column completes the discussion of metrics with the subject of dimensionless quantities.
The use of a dimensionless quantity is a common way to study a wide variety of physical phenomena or engineering problems. A dimensionless equation, algebraic or differential, involves variables without physical dimensions. For example, most people prefer to know the Mach number rather than the speed because it's easier to compare performance. These numbers (coefficients or factors) are part of the physical world and exist in many applications. More than 1200 dimensionless quantities are used in science, economics, medicine, geophysics and ecology. Avogadro’s number is one used in chemistry. This column will talk only about those used in engineering for thermodynamics, thermomechanics, fluid mechanics, solid mechanics and electromagnetics.
These numbers or factors are essential in modern industry and science. These numbers are the Figure of Merit that I describe in Column #9. They allow us to quantify those physical or mental activities that otherwise would fall into the realm of opinion or conjecture. One that I have used for the last 40 years for printed circuit fabrication is the complexity index. This index allows me to model (from design information) the first pass yield in PCB fabrication with only a 3−7% error.[1–3]. Using this process, many other factors can be created that will simplify working with complex systems in design, fabrication or assembly. Use the process any place where you need to measure something that right now is just an opinion.
Every numerical characteristic used in engineering comprises three elements:
There are two types of dimensions: fundamental and derived. Fundamental dimensions cannot be expressed in terms of other dimensions, and they serve as building blocks for more complex, derived dimensions.
Sometimes, a group of fundamental dimensions has a dedicated name, which is then called a unit. A special class of units are those with a dimension of one, which result when all the fundamental dimensions of the unit cancel. Although not entirely accurate, such units are called dimensionless parameters. These units are frequently called number, ratio, index, factor, effect, or coefficient. Examples include Mach number, Slenderness ratio, Toms effect, Reynolds number, and friction coefficient. The advantage of these absolute numbers is that their numerical values are independent of the dimensional system used.
All dimensional systems in use today can be classified according to whether they use the metric or American/British system and whether they use force or mass as a fundamental dimension. Force-based systems include the metric MKS force and American/British engineering systems. Mass-based systems include the SI and British mass systems.
- MKS: Fundamental dimensions are meter (m), kilogram (kg), and second (s), where kg means force. Mass has a derived dimension of kg-s2/m. Until recently, this system was widely used in engineering in continental Europe and is still favored by many there.
- American/British engineering: Fundamental dimensions are foot (or inch), pound (lb), and second, where lb means force. Accordingly, the dimensions of mass are lb-s2/ft (also called the slug) or lb-s2/in.
- SI: Fundamental dimensions are kilogram (for mass), meter, and second. Force is a derived dimension, with the unit Newton. SI actually has seven fundamental dimensions, and also includes ampere, degrees Kelvin, mole, and candella.
- British Mass: Here, pound is the dimension of mass and is legally equivalent to 0.45359237 kg in SI. Force is a derived dimension, and its unit is poundal, which is numerically equivalent to the force necessary to impart l-ft./s2 acceleration to a 1-lb mass. One poundal is equivalent to 0.138254954 kg-m/s2 in SI, which is nearly 0.5 oz. in the American/British engineering system.
Mass-based systems have the advantage that this dimension is independent of the numerical value of gravitational acceleration g, which even at sea level can vary as much as 0.5% with latitude. Therefore, the use of a mass-based system is uniform anywhere in the universe.
On the other hand, having mass as a fundamental dimension is impractical in everyday life because mass cannot be directly perceived or measured. Rather, mass must be determined indirectly by subjecting it to a known inertial or gravitational acceleration and measuring the force generated.
In engineering, dimensional analysis is the analysis of the relationships between different physical factors/quantities by using their fundamental dimensions (such as mass, length, time, electrical charge, or temperature, etc.) and unit of measure (such as Fahrenheit vs. Centigrade, miles vs. kilometers, or pounds vs. kilograms) and comparing these dimensions as calculations. Converting from one-dimensional unit to another is often somewhat complex. Dimensional analysis, or more specifically the factor-label method, also known as the unit-factor method, is a widely used technique for such conversions using the rules of algebra.
From a discussion on ResearchGate, the Professor Issam Sinjab commented, “But on a more abstract level, dimensional analysis allows you to abstract over a wide variety of phenomena, and gives insight to regimes where some parameters vanish and can be ignored. The Reynolds number in hydrodynamics is a good example of this. At very low Reynolds number, many terms in the equations vanish, and thus systems can be approximated by very simple equations (laminar flow) even though flow at all Reynolds numbers is fundamentally governed by the Navier-Stokes equation.”
For another interesting example of this, the mathematician G. I. Taylor used dimensionless forms of certain equations to figure out highly classified and secret information about the atomic bomb in the 1940s.
The purposes of dimensionless equations are:
- To simplify the equation(s) by reducing the number of variables used
- To analyze system behavior regardless of the unit used to measure variables
- To rescale parameters and variables so that all computed quantities are of same order (relatively similar magnitudes)
- To scale-up physical systems from experiment to production
The concept of physical dimension was introduced by Joseph Fourier in 1822. Physical quantities that have the same common measure have the same dimension; if they have different dimensions, they are incommensurable. For example, it is pointless to ask whether a gram is less, the same, or more than a minute. A major job in engineering design is to determine how the physical factors influencing a part or system interact. This interaction is expressed as a system of equations, which can be illustrated as a graph or series of graphs.
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.
- Holden, H., “The Complexity Index,” IPC Technical Bulletin, May 1978.
- Holden, H. et al, The HDI Handbook, chapter 3, pp 115–117, published by I-Connect007, 2008.
- Holden, H., “Calculating your Fabrication Capability Coefficients,” CircuiTree Magazine, February 2006.
- 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.