Explanation Given information: The speed of the sound is c , the pressure of the gas is P and the density of the gas is ρ . Expression for the speed of the sound, c = f ( P , ρ ) ...... (I) Expression for the number of the pi terms, N = n − j ...... (II) Here, the total number of the repeating variable is j and the total number of the variable is n . The total number of the variable are 3 ( c , P , ρ ) and the number of the repeating variables are 2 ( P , ρ ) . Substitute 3 for n and 2 for j in equation (II). N = 3 − 2 = 1 Expression for the first pi term, Π 1 = c P a ρ b ...... (III) Expression for the pressure using the equation of the ideal gas, P = ρ R g a s T ..... (IV) Here, the gas constant is R g a s and the absolute temperature of the gas is T . Dimensional expression for the speed of sound, c = [ L t − 1 ] ..... (V) Here, the length is L and the time is t . Dimensional expression for the pressure, P = [ m L − 1 t − 2 ] ..... (VI) Here, the mass is m . Dimensional expression for the density, ρ = [ m L − 3 ] ..... (VII) Calculation: Substitute [ m 0 L 0 t 0 ] for Π 1 , [ L t − 1 ] for c , [ m L − 1 t − 2 ] for P and [ m L − 3 ] for ρ in Equation (III) [ m 0 L 0 t 0 ] = [ L t − 1 ] [ m L − 1 t − 2 ] a [ m L − 3 ] b [ m 0 L 0 t 0 ] = [ L ] 1 − a − 3 b [ m ] a + b [ t ] − 1 − 2 a

Fluid Mechanics: Fundamentals and Applications

4th Edition

ISBN: 9781259696534

Author: Yunus A. Cengel Dr., John M. Cimbala

Publisher: McGraw-Hill Education

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Chapter 7, Problem 57P

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The functional relationship between given parameters.

The verification for the consistency of the result with the equation of the speed of the sound in an ideal gas.

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Chapter 7, Problem 56P

Chapter 7, Problem 58P

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Fluid Mechanics: Fundamentals and Applications

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Chapter 7 Solutions

The functional relationship between given parameters.

The verification for the consistency of the result with the equation of the speed of the sound in an ideal gas.