p0282

# p0282 - 2.82. CHAPTER 2, PROBLEM 82 277 2.82 Chapter 2,...

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2.82. CHAPTER 2, PROBLEM 82 277 2.82 Chapter 2, Problem 82 Problem: The time τ required to drain all of the fluid through a hole of diameter d in the bottom of a tank is a function of initial fluid depth, h o , initial fluid volume, V o , gravitational acceleration, g , and fluid kinematic viscosity, ν . How many dimensionless groupings are there? Determine the dimensionless groupings. Solution: The dimensional quantities and their dimensions are [ τ ]= T, [ d L, [ h o L, [ V o L 3 , [ g L T 2 , [ ν L 2 T There are 6 dimensional quantities and 2 independent dimensions ( L, T ) ,sot h a tth e number of dimensionless groupings is 4. The appropriate dimensional equation is [ τ ]=[ d ] a 1 [ h o ] a 2 [ V o ] a 3 [ g ] a 4 [ ν ] a 5 Substituting the dimensions for each quantity yields T = L a 1 L a 2 L 3 a 3 L a 4 T 2 a 4 L 2 a 5 T a 5 = L a 1 + a 2 +3 a 3 + a 4 +2 a 5 T 2 a 4 a 5 Thus, equating exponents, we arrive at the following two equations. 0= a 1 + a 2 +3 a 3 + a 4 +2 a 5 1= 2 a 4 a 5 We can solve immediately for a 4 from the second equation, viz., a 4 = 1 2 1 2 a 5 Using this value for a 4 in the first equation shows that a 1 + a 2 a 3 1 2 1 2 a 5 a 5 = a 1 = 1 2 a 2

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## This note was uploaded on 02/09/2012 for the course AME 309 taught by Professor Phares during the Spring '06 term at USC.

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p0282 - 2.82. CHAPTER 2, PROBLEM 82 277 2.82 Chapter 2,...

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