The variation of pressure with density in a thick gas

Problem 138P Chapter 3

Fluid Mechanics | 2nd Edition

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Fluid Mechanics | 2nd Edition

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Problem 138P

The variation of pressure with density in a thick gas layer is given by where C and n are constants. Noting that the pressure change across a differential fluid layer of thickness dz in the vertical z–direction is given as dP= ‒ρg dz, obtain a relation for pressure as a function of elevation z. Take the pressure and density at z = 0 to be P0 and ρ0, respectively.

Step-by-Step Solution:

Step 1:

To obtain the relation for the variation of pressure with altitude with respect to the density of the air. The change in pressure P with altitude is given by

$dP$ = $-\rho{}gdz$ ----(1)

Where $\rho{}$ is density of the air

g is the acceleration due to gravity (assuming constant)

z is the vertical height (altitude)

Step 2:

Variation of the pressure with density is termed as

P = C ${\rho{}}^{n}$ -----(2)

Where C and n are constants $\$

Step 3:

If the density of the air at sea level is ${\rho{}}_{0}$ and the pressure at sea level is P0, the equation (1) can be written as

P0 = C ${{\rho{}}_{0}}^{n}$ -----(3)

Dividing (2) and (3)

$\frac{P}{{P}_{0}}$ = $\frac{{\rho{}}^{n}}{{{\rho{}}_{0}}^{n}}$

$(\frac{P}{{P}_{0}}{)}^{1/n}$ = $\frac{\rho{}}{{\rho{}}_{0}}$

$\rho{}$ = ${\rho{}}_{0}(\frac{P}{{P}_{0}}{)}^{1/n}$ -----(4)

Step 4:

Substituting (4) in (1)

$dP$ = $-{\rho{}}_{0}(\frac{P}{{P}_{0}}{)}^{1/n}gdz$

separating variables, we get

${P}^{-1/n}dp$ = $-\frac{{g\rho{}}_{0}}{P{\ }_{0}^{1/n}}dz$ ----(5)

Step 5 of 7

Chapter 3, Problem 138P is Solved

Step 6 of 7

Textbook: Fluid Mechanics

Edition: 2

Author: Yunus A. Cengel, John M. Cimbala

ISBN: 9780071284219

Fluid Mechanics was written by and is associated to the ISBN: 9780071284219. This full solution covers the following key subjects: pressure, given, layer, Density, across. This expansive textbook survival guide covers 15 chapters, and 1547 solutions. Since the solution to 138P from 3 chapter was answered, more than 246 students have viewed the full step-by-step answer. The answer to “The variation of pressure with density in a thick gas layer is given by where C and n are constants. Noting that the pressure change across a differential fluid layer of thickness dz in the vertical z–direction is given as dP= ??g dz, obtain a relation for pressure as a function of elevation z. Take the pressure and density at z = 0 to be P0 and ?0, respectively.” is broken down into a number of easy to follow steps, and 69 words. This textbook survival guide was created for the textbook: Fluid Mechanics, edition: 2. The full step-by-step solution to problem: 138P from chapter: 3 was answered by , our top Engineering and Tech solution expert on 07/03/17, 04:51AM.