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11.3.2: Speed of Sound in Ideal and Perfect Gases

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
\[
    P= constant\times \rho^{k}
    \label{gd:sd:eq:iserhoToP} \tag{11}
\]
and hence
\[
    \label{gd:sd:eq:idealGas1}
    c = \sqrt{\dfrac{\partial P}{\partial \rho}}  
      = k \times constant \times \rho^{k-1}  =k \times  
        \dfrac{\overbrace{ constant \times  \rho^k}^{P} }{ \rho}  
             =  k \times \dfrac{P }{\rho } \tag{12}
\]
Remember that \(P / \rho \) is defined for an ideal gas as \(RT\), and equation (12) can be written as

Ideal Gas Speed Sound

\[ 
\label{gd:sd:eq:sound} 
    c = \sqrt{ k\, R\, T}  \tag{13}
\]

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   
\[ 
    c \sim \sqrt{ \dfrac{\Delta P}{ \Delta \rho} }_{s=constant}       \tag{14}
\] 
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 
\[ 
    c = \sqrt{ 200000 \over 0.32823} = 780.5 \left[ m \over  
    sec \right]   \tag{15}
\] 
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

  • 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.