# Dimensional analysis

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Dimensional analysis is a basic math skill used in thermodynamics whereby a measure of something is related in terms of units of something else. It is a method of converting the units of one term into the units of another, knowing that multiplying or dividing the term by a conversion factor does not change the actual value of the term itself.

For instance, 1 kg = 2.20462 lb. Such conversions are done between different systems of measurement or within a single system of measurement (such as that 1 foot = 12 inches). Most commonly dimensional analysis is done to compare values between the English system of measurement and the International System of Units (SI).

## Unit Types

English units: This is a system that is used in the United States and a few other countries around the world. While the majority of the world, science, and industry uses SI units, "some in U.S. industry continues to avoid the use of SI units" (Potter v). As such, English units are important to understand when beginning to understand thermodynamics.

English units include: feet (ft, length), pound mass (lbm, mass), pound force (lbf, force), foot pound force (ft-lbf, work), and so on.

SI Units: Le Système international d'unités, or the International System of Units, is the internationally chosen metric of choice for measurement. This is the metric system and is used by most countries, in science, and in business.

SI units include meters (m, length), kilograms (kg, mass), newtons (N, force), joules (J, work), and so on.

## How to do Dimensional Analysis

When doing dimensional analysis, notice what units are present in the first term you are attempting to convert. For the first conversion factor, look for a unit that will cancel a unit from the first term. For example to convert 10 ft-lbf to Joules, multiply by the conversion value of 1.356. This would look like:

```                                       10 ft-lbf *  1.356 J
---------    -------- = 13.56 Joules.   This relatively simple dimensional analysis of converting between
1       1 ft-lbf                   ft-lbf to Joules demonstrates the basic process to be used.
```

When doing more complicated dimensional analysis, ensure that all of the terms that you want to cancel have cancelled. Check to see if the final units that you have remaining at the end of your work are the units that you expect.

## Common Conversions

1 lbf = 32.2 lbm-ft/sec2
1 N = 1 kg-m/s2

## References

Potter, Merle C., and Craig W. Somerton. Schaum's Outline of Thermodynamics for Engineers. New York: McGraw-Hill, 2009. Print.