Chapter02

Chapter02 - Dimensional Analysis Dimensional This technique...

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Dimensional Analysis Dimensional Analysis z This technique is useful in all branches of science and engineering z Basic premises are 1. Physical quantities have dimensions 2. All equations developed from basic laws of physics are dimensionally homogeneous z We can develop a useful theory from this information
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Dimensions Dimensions - - 1 1 z Consider viscosity, μ, which has SI units kg/(m•sec) and USCS units slug/(ft •sec) z We write z [ μ ] means “dimensions of μ ” while M , L and T denote dimensions of mass, length and time, respectively z If temperature is relevant, we denote its dimension by Θ z M , L , T and Θ are called independent dimensions
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Dimensions Dimensions - - 2 2 z All dimensions can be expressed in terms of the independent dimensions z For example, consider pressure, p , which has dimensions of force per unit area…appealing to Newton’s law, we have z Thus, for dimensional analysis purposes, pressure has SI units of kg/(m•sec 2 ) and USCS units of slug/(ft•sec 2 )
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Dimensional Homogeneity Dimensional Homogeneity - - 1 1 z Universal equations are valid regardless of the choice of units z Counter-example… theoretical hull speed for a sailboat (single hull, non-surfing) U hull = Sailboat speed in knots L w = Hull length at the waterline in feet
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Dimensional Homogeneity Dimensional Homogeneity - - 2 2 z Can’t use this empirical formula if L w is given in units other than feet z The problem is, the coefficient “1.34” has dimensions of L ½ /T z This empirical formula is not dimensionally homogeneous z We can fix the problem by simply observing that [ g ] = L/T 2 so that [ g ½ ] = L ½ /T …this tells us that U hull /( g L w ) ½ is dimensionless
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Dimensional Homogeneity Dimensional Homogeneity - - 3 3 z A sailboat with hull length at the waterline of 20 feet has a hull speed of 6 knots…a little arithmetic tells us z The coefficient “0.40” is dimensionless z So, this dimensionally homogeneous equivalent of our original equation holds in all units
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General Observations General Observations Dimensional analysis has at
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Chapter02 - Dimensional Analysis Dimensional This technique...

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