​It’s almost like a cheating – The ultimate hack for engineers

Or: Dimensional Analysis - The easiest way to use first principle

Ever since I came across it, I have been amazed by the utility and power of dimensional analysis. While many engineers vaguely recall if from their degrees as “some fluid mechanics thing”, I have used it to assess noise generated by spa pumps; optimise machine elements; augment the Design Of Experiments (DOE); review the potential to change a pressure sensor design; recall a formula in an exam that did not have a formula sheet, but did list key constants and their units; and assess the potential of a universe with different fundamental laws.
Because it can do so much more for engineers than would “some fluid mechanics thing”, it is a tool that any global engineer should master so that they can quickly adapt to new challenges. And that’s why dimensional analysis is the topic of this article.
The Universe Does Not Know What a Kilogram Is
The first thing to understand is that dimensional analysis is not really about units.
That might sound wrong because it is often taught as though it is about units. You check that metres are on both sides of an equation. You make sure seconds cancel out. You confirm that you haven’t added a force to a velocity and hoped nobody would notice.
That’s useful. But it is not the main power of the technique.
The real power is that dimensional analysis gets underneath the units.
The universe does not know what a metre is. It does not know what a mile is. It does not know what a second is or what a kilogram is. These are human inventions. They are labels we use so we can communicate consistently with one another.
But the universe does understand relationships.
It understands ratios.
If a pendulum behaves a certain way, it does not behave that way because we measured its length in metres instead of feet. If a structure vibrates at a certain frequency, it does not care if the mass was recorded in kilograms or pounds. If a fluid produces drag, it does not care if the engineer prefers SI units or imperial units.
The phenomenon is the phenomenon.
Our units are just our way of describing it.
Dimensional analysis works because it forces us to describe physical systems in a way that is independent of our arbitrary measuring sticks. It asks: what are the fundamental types of things involved here? Length. Time. Mass. Temperature. Charge. Maybe a few others depending on the problem.
Then it asks: what combinations of these things can actually matter?
That question is far more powerful than it first appears.
It is so powerful that even if the universe were to reform with different laws, dimensional analysis would still work. The laws might be different, but once there are quantities that interact, those quantities would still need to relate to each other in dimensionally consistent ways.
This is why it feels like cheating.
You can sometimes know something must be true before you know why it is true.
Rayleigh Knew This
This is not a new trick.
Lord Rayleigh used dimensional reasoning in the development of what is now often called Rayleigh’s method of dimensional analysis. He used it to explain why the sky is blue. His method is one of the classic approaches taught alongside the Buckingham Pi theorem, and Rayleigh’s method is commonly described as an early method of dimensional analysis.
In simple terms, Rayleigh’s method assumes that the dependent variable in a physical problem can be expressed as a product of the relevant independent variables, each raised to some power. You then solve for those powers by requiring the dimensions on both sides to match.
That sounds like a mathematical trick.
And in one sense it is.
But it is also more than that. It is a way of letting the structure of reality constrain your thinking before you do any testing. It tells you what forms of relationships are possible and what forms are impossible.
That is the part many engineers miss.
They think dimensional analysis is just something you use when you cannot remember the formula.
But it is really something you use when no formula exists.
Why People Think It Is Only a Fluid Mechanics Thing
The reason many engineers think of dimensional analysis as “some fluid mechanics thing” is understandable.
It is used heavily in fluid mechanics because fluid mechanics is hard. In fact, turbulence is so difficult that even with modern mathematics and computing power, we still cannot simply solve many turbulent flow problems from first principles in the nice clean way we might like to.
So fluid mechanics needed dimensional analysis.
Engineers needed Reynolds number, Mach number, Froude number, Nusselt number, Prandtl number and many others because we needed a way to understand and compare systems that were too complex to solve directly.
This is where dimensional analysis became famous.
But this also created a problem.
Because people saw the technique being used in fluid mechanics, they assumed the technique belonged to fluid mechanics.
That is like seeing a spanner used on a car and deciding spanners are only for cars.
The tool was useful there because the problems were hard there. That does not mean the tool is limited to that domain.
Dimensional analysis can apply anywhere physical variables interact.
Machines. Structures. Heat transfer. Acoustics. Electromagnetics. Manufacturing systems. Biological systems. Economic systems even, if you are careful with what you mean by dimensions.
And that should get your attention.
Because global engineers are constantly being thrown into systems they have not seen before.
Why Scientists Often Do Not Seem to Use It
There is something else interesting about dimensional analysis.
It gives insight without always explaining the mechanism.
That makes it extremely useful for engineers, but perhaps less satisfying for scientists.
A scientist often wants to know why.
Why does the system behave this way? What is the underlying mechanism? What is the causal structure? What is the theory beneath it?
And that is appropriate. That is science.
But engineers often need something slightly different.
An engineer needs to know what to do next.
Can I scale this? Can I reduce this? Can I make this quieter? Can I make this cheaper? Can I make this more reliable? Can I change this variable and get enough benefit to justify the cost?
Of course, the engineer would also like to know why. But if the bridge needs to stand up, the pump needs to be quieter, or the machine element needs to survive another duty cycle, then insight that supports a decision is already valuable.
This might be why dimensional analysis seems underused in scientific research.
I have only seen one paper where dimensional analysis was used in a way that really stood out to me, and I have only heard one physicist talk directly about using it as a research tool. That does not mean scientists do not use it. Clearly some do. But it does mean it does not seem to be emphasised as much as its power would suggest.
Engineers should not make the same mistake.
We do not need to wait until we can fully explain the phenomenon before using the insight.
How Engineers Can Use It
So how should an engineer actually use dimensional analysis?
The first use is to partially solve the relationship between variables.
Suppose you think a result depends on five or six variables. Without dimensional analysis, you might feel you need to test all combinations. That quickly becomes impossible. If each variable has several levels, your experiment count explodes.
But dimensional analysis can reduce the number of variables by combining them into dimensionless groups.
This does not usually solve the whole problem.
You may still need experiments. You may still need simulation. You may still need judgement.
But you need far fewer experiments than you would have needed otherwise.
This is where it works beautifully with Design of Experiments. Instead of designing experiments around raw variables, you can design experiments around dimensionless groups. You are then testing the structure of the phenomenon more directly.
It also works well with simulation.
If physical testing is expensive, slow, or difficult, then simulations can be used to explore the reduced relationship. You can use dimensional analysis to define the form of the problem and then use simulation to fill in the missing functional relationship.
That gives you something very close to a formula for optimisation.
Not a perfect formula necessarily.
But a useful one.
And useful is often what engineering needs.
Finding the Levers That Matter
Dimensional analysis can also tell you which dimensions you can change to get the effect you want.
This is one of the most valuable engineering uses.
Some variables give you much more bang for the buck than others.
If a quantity depends on one variable squared, another variable to the half power, and another variable inversely, then you now know something important. You know where the leverage is likely to be.
This helps prevent wasted effort.
Engineers can spend a lot of time changing things that are easy to change, rather than changing the things that matter. Dimensional analysis helps you see the structure of the problem before you fall into that trap.
It can tell you:

• This variable probably matters a lot.
• This variable probably matters, but weakly.
• This variable might not matter at all.
• This combination of variables is what you should really be thinking about.

That is powerful.
Especially when you are looking at a system you have never seen before.
The Global Engineer Advantage
This is why dimensional analysis belongs in the toolkit of the global engineer.
A global engineer cannot rely only on familiar systems.
You might find yourself working on a manufacturing problem in one country, a maintenance problem in another, a product issue somewhere else, and then a forensic investigation in a completely different industry.
You will not always have the right formula.
You will not always have the right standard.
You will not always have someone nearby who has seen the problem before.
So what do you do?
You start from first principles.
But first principles can be slow if you try to derive everything from scratch. Dimensional analysis gives you a shortcut. It lets you frame the problem quickly. It helps you identify the variables. It helps you reduce the complexity. It helps you see where  experimentation or simulation should be focused.
Dimensional analysis does not make you clever by itself. But it gives a clever engineer a way to move faster through unfamiliar territory.
And that is exactly what global engineering demands.
How to Master It
I cannot explain dimensional analysis fully here.
If I tried, this article would become a textbook chapter, and probably not a very good one.
But I do hope I have motivated you to take it seriously.
Not as something you vaguely remember from fluid mechanics.
Not as a unit-checking exercise.
Not as an academic trick.
But as one of the most powerful first-principles tools available to engineers.
If you do want to master it, then I strongly recommend having Applied Dimensional Analysis and Modeling by Thomas Szirtes in your book collection. The book provides mathematical background, procedures, and a wide range of engineering and applied science applications for dimensional analysis and dimensional modelling.
Dimensional analysis is one of those tools that can change how you see engineering problems. Once you get used to thinking dimensionally, you start seeing hidden structure everywhere.
And it might be one of the easiest ways to start using first principles properly.