Dimensional_Analysis

Dimensional_Analysis - Dimensional Analysis Author: John M....

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Dimensional Analysis Author: John M. Cimbala, Penn State University Latest revision, 29 August 2007 Primary Dimensions (Review) There are seven primary dimensions . All other dimensions can be formed by combinations of these. The seven primary dimensions are, along with their symbols: mass length time temperature current amount of light amount of matter {m} {L} {t} {T} {I} {C} {N} The primary dimensions of variables in an experiment or analysis can be used to our advantage – to reduce the required amount of effort . Dimensional Homogeneity We state the law of dimensional homogeneity as “ Every additive term in an equation must have the same dimensions. Example: The total energy ( E ) of a system is composed of internal energy ( U ), kinetic energy (KE), and potential energy (PE), i.e., . KE PE EU =+ + Let’s look at the primary dimensions of each term in this equation: { } { } { } energy force length E == { } { } 22 mL /t E = {} { } {} energy mass energy mass Um u ⎧⎫ = ⎨⎬ ⎩⎭ { } { } mL /t U = 2 2 2 1 length mass 2t KE mV i m e { } { } mL /t KE = {}{ } 2 length mass length time PE mgz { } { } mL /t PE = The law of dimensional homogeneity is the basis for the useful technique of dimensional analysis, which we discuss next. Dimensional Analysis and the Method of Repeating Variables Dimensional analysis is a simple, powerful tool that is useful in all disciplines (but unfortunately, is usually taught only in fluid mechanics). The goal of dimensional analysis is to reduce the number of independent variables in a problem. We accomplish this by converting all dimensional variables into nondimensional parameters , called “Pi’s”, and given the symbol Π (upper case Greek letter pi).
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Dimensional_Analysis - Dimensional Analysis Author: John M....

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