11.3.2: Speed of Sound in Ideal and Perfect Gases
- Page ID
- 792
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The speed of sound can be obtained easily for the equation of state for an ideal gas (also perfect gas as a sub set) because of a simple mathematical expression. The pressure for an ideal gas can be expressed as a simple function of density, \(\rho\), and a function "molecular structure'' or ratio of specific heats, \(k\) namely
\[
P= constant\times \rho^{k}
\label{gd:sd:eq:iserhoToP}
\]
\[
\label{gd:sd:eq:idealGas1}
c = \sqrt{\dfrac{\partial P}{\partial \rho}}
= k \times constant \times \rho^{k-1} =k \times
\dfrac{\overbrace{ constant \times \rho^k}^{P} }{ \rho}
= k \times \dfrac{P }{\rho }
\]
Remember that \(P / \rho \) is defined for an ideal gas as \(RT\), and equation (12) can be written as
Ideal Gas Speed Sound
\[\label{gd:sd:eq:sound}
c = \sqrt{ k\, R\, T}
\]
Example 11.2
Calculate the speed of sound in water vapor at \(20 [bar]\) and \(350^{\circ}C\), (a) utilizes the steam table, and (b) assuming ideal gas.
Solution 11.2
The solution can be estimated by using the data from steam table
\[
c \sim \sqrt{ \dfrac{\Delta P}{ \Delta \rho} }_{s=constant}
\]
At \(20[bar]\) and \(350^{\circ}C\): s = 6.9563 \(\left[ \dfrac{kJ }{ K\, kg}\right]\) \(\rho \) = 6.61376 \(\left[ \dfrac{kg }{ m^3} \right]\)
At \(18[bar]\) and \(350^{\circ}C\): s = 7.0100 \(\left[\dfrac{ kJ }{ K\, kg}\right]\) \(\rho \) = 6.46956 \(\left[ \dfrac{ kg }{ m^3}\right]\)
At \(18[bar]\) and \(300^{\circ}C \): s = 6.8226 \(\left[\dfrac{ kJ }{ K\, kg}\right]\) \(\rho \) = 7.13216 \(\left[ \dfrac{kg }{ m^3} \right] \)
After interpretation of the temperature:
At \(18[bar]\) and \(335.7^{\circ}C \): s \(\sim\) 6.9563 \(\left[\dfrac{ kJ }{ K\, kg} \right]\) \(\rho \sim\) 6.94199 \(\left[ \dfrac{kg }{ m^3} \right]\)
and substituting into the equation yields
\[
c = \sqrt{ 200000 \over 0.32823} = 780.5 \left[ m \over
sec \right]
\]
for ideal gas assumption (data taken from Van Wylen and Sontag, Classical Thermodynamics, table A 8.)
\[
c = \sqrt{k\,R\,T} \sim \sqrt{ 1.327 \times 461 \times
(350 + 273)} \sim
771.5 \left[ \dfrac{m }{ sec} \right]
\]
Note that a better approximation can be done with a steam table, and it \(\cdots\),
Contributors and Attributions
Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.