Magnitude_Estimates_and_Characteristic_Times

Magnitude_Estimates_and_Characteristic_Times - I.C .8 M...

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I.C.8 M agnitudes: Length Scales and Characteristic Times The objects of concern in environmental engineering span an enonnous range of scales, as illustrated in Figure 1.C.6. The lengths of the objects depicted span almost 17 orders of magnitude, from roughly 10-10 m for the spacing between molecules in fluids to 107 m for the diameter of the earth. O n a mass scale, because mass varies in 0 ~ - ~- '2 "" 0- t.S)' fJ ~ >" ~ ~ I - Q Q ~ l' " ..oS ~ ,.., t' ~ 10-10 10-8 Spacing between molecules in liquid water Distance between air molecules (T = 293 K, P = 1 atm) Virus W avelength of visible light Bacterium O ayparticle O oud droplet 10-6 Sand grain Rain drop Human height Residential building (horizontal) W idth of O hio River e '-" g .~ ~ ~ ~ 0) :E (J .~ .e u ; .-= U 104 1~ Figure 1.C.6 Length scales of some objects of interest in environmental engineering. Height of M t. Everest above sea level Urban air basin G reat Lakes Continent Diameter of earth - 1~ <- . ~ <$' ~ ~ "2 .. u \. 3' 0 V) proportion to volume (i.e., as the cube of the linear dimension), the objects of interest vary over m~re than 50 orders of magnitude, from 3 X 10-24 g for a hydrogen mole- cule to 6 x 1027 g for the mass of the earth! O ften. in a preliminary analysis, we seek to determine only the magnitude of the answer. It may take very little effort to get the correct ariswer to within a factor of 10 compared with the work required to get the correct answer to within, say, 10-30 per- cent. To decide whether a process or parameter is important. we may need to deter- mine only the magnitude of the result. W e will make extensive use of magnitude estimates in this book. In making these estimates, we use the symbol - to mean "has the same scale as'' or "is of the same or- der of magnitude as.'' Although the idea of two parameters having the same order of magnitude defies precise definition, it is an enormously valuable concept. Here is an example that provides an approximate defmition of the concept. Consider two param- eters, a and b. If the ratio a/b lies within the range 0.33-3, we would conclude that a and b are of the same order of magnitude. If the ratio a/b is less than 0.03 or greater than 30, we would conclude that the parameters do not have the same magnitude. Be- tween these bounds, that is, if a and b differ by a factor in the range 3-30, the decision of whether or not the parameters are of the same scale varies with circumstances.
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This note was uploaded on 05/04/2010 for the course CEE 320 taught by Professor Mcmahon during the Spring '10 term at Wisconsin.

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Magnitude_Estimates_and_Characteristic_Times - I.C .8 M...

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