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Chapter7 - 07-R3868 10:16 AM Page 201 CHAPTER D I M E N S I...

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DIMENSIONAL ANALYSIS AND MODELING I n this chapter, 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 engi- neers and scientists called dimensional analysis , in which the combination of dimensional variables, nondimensional variables, and dimensional con- stants 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 con- stants. Finally, we apply this technique to several practical problems to illus- trate both its utility and its limitations. 201 CHAPTER 7 OBJECTIVES When you finish reading this chapter, you should be able to Develop a better understanding of dimensions, units, and dimensional homogeneity of equations Understand the numerous benefits of dimensional analysis Know how to use the method of repeating variables to identify nondimensional parameters Understand the concept of dynamic similarity and how to apply it to experimental modeling A 1 : 46.6 scale model of an Arleigh Burke class U.S. Navy fleet destroyer being tested in the 100-m long towing tank at the University of Iowa. The model is 3.048 m long. In tests like this, the Froude number is the most important nondimensional parameter. Photograph courtesy of IIHR-Hydroscience & Engineering , University of Iowa . Used by permission .

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7–1 DIMENSIONS AND UNITS A dimension is a measure of a physical quantity (without numerical val- ues), while a unit is a way to assign a number to that dimension. For exam- ple, length is a dimension that is measured in units such as microns ( m), feet (ft), centimeters (cm), meters (m), kilometers (km), etc. (Fig. 7–1). 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, Dimensions of force : (7–1) where the brackets indicate “the dimensions of” and the abbreviations are taken from Table 7–1. You should be aware that some authors prefer force instead of mass as a primary dimension—we do not follow that practice. {Force} e Mass Length Time 2 f {mL/t 2 } 202 CHAPTER 7 3.2 cm 1 2 3 cm Length FIGURE 7–1 A dimension is a measure of a physical quantity without numerical values, while a unit is a way to assign a number to the dimension. For example, length is a dimension, but centimeter is a unit.
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