Dimensional analysis and modeling : dimensions and units

in this topic, we first review the concepts of dimensions and units. We then review the fundamental principle of dimensional homogeneity, and show how it is applied to equations in order to nondimensionalize them and to identify dimensionless groups. We discuss the concept of similarity between a model and a prototype. We also describe a powerful tool for engineers and scientists called dimensional analysis, in which the combination of dimensional variables, nondimensional variables, and dimensional constants into nondimensional parameters reduces the number of necessary independent parameters in a problem. We present a step-by-step method for obtaining these nondimensional parameters, called the method of repeating variables, which is based solely on the dimensions of the variables and constants. Finally, we apply this technique to several practical problems to illustrate both its utility and its limitations.

DIMENSIONS AND UNITS
A dimension is a measure of a physical quantity (without numerical values), while a unit is a way to assign a number to that dimension. For example, length is a dimension that is measured in units such as microns ( m), feet (ft), centimeters (cm), meters (m), kilometers (km), etc. There are seven primary dimensions (also called fundamental or basic dimensions)—mass, length, time, temperature, electric current, amount of light, and amount of matter.
All nonprimary dimensions can be formed by some combination of the seven primary dimensions.
For example, force has the same dimensions as mass times acceleration (by
Newton’s second law). Thus, in terms of primary dimensions,

Dimensional analysis and modeling : dimensions and units

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