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A unit is defined as a measure of a physical extent, while a dimension is a description of the physical extent. Units, unlike physical laws, can be considered as either derived or basic. There is a certain latitude in choosing the basic units, and, unfortunately, this free choice has resulted in the aforementioned mild form of confusion. As described earlier, two systems of units have arisen: metric, the cgs, or centimeters-gram-second system and the English, the fps, or foot–pound–second system of engineering.

There are hundreds of conversion constants employed by engineers and scientists. Some of the more common “conversion constants” are provided in Tables 3.2 to Table 3.4 .

Conversion of units can be accomplished by the multiplication of the quantity to be converted by appropriate unit ratios, i.e. the conversion constants. For example, suppose an energy of 50 Btu must be converted to units of ft-lbf . From the energy section in Table 3.2, one notes that to convert from Btu to ft-lbf , on simply multiplies by 778, Therefore:

(3.2)

The conversion constant, or unit ratio, is:

(3.3)

The 50 Btu may then be multiplied by the earlier conversion constant without changing its value. Therefore:

(3.4)

with the Btu units cancelling, just like numbers.

Problems are frequently encountered in environmental studies and other engineering work that involve several variables. Engineers and scientists are generally interested in developing functional relationships (equations) between these variables. When these variables can be grouped together in such a manner that they can be used to predict the performance of similar pieces of equipment, independent of the scale or size of the operation, something very valuable has been accomplished. In addition, one of the properties of equations that has a rational basis and is deduced from general relations is that they must be dimensionally homogeneous, or consistent. Dimensional analysis is a relatively “compact” technique for reducing the number and the complexity of the variables affecting a given phenomenon, process, or calculation. It can help obtain not only the most out of experimental data but also scale-up data from a model to a prototype. To do this, one must achieve similarity between the prototype and the model. This similarity may be achieved through dimensional analysis by determining the important dimensionless numbers, and then designing the model and prototype such that the important dimensionless numbers are the same in both.