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Next: Speed of Sound in Up: Speed of Sound Previous: Speed of sound in   Index


Speed of Sound in Real Gas

The ideal gas model can be improved by introducing the compressibility factor. The compressibility factor represents the deviation from the ideal gas.
Figure: The Compressibility Chart
\begin{figure}\centerline{\includegraphics
{cont/sound/CompressiblityChart}}
\end{figure}
Thus, a real gas equation can be expressed in many cases as
The speed of sound of any gas is provided by equation (3.7). To obtain the expression for a gas that obeys the law expressed by (3.19) some mathematical expressions are needed. Recalling from thermodynamics, the Gibbs function (3.20) is used to obtain
The definition of pressure specific heat for a pure substance is
The definition of volumetric specific heat for a pure substance is
From thermodynamics, it can be shown 3.4
The specific volumetric is the inverse of the density as $ v = zRT/P$ and thus
Substituting the equation (3.24) into equation (3.23) results
Simplifying equation (3.25) to became
$\displaystyle dh = C_p dT - \left[ {T v \over z} \left( \partial z \over \parti...
...ver z}\left(\partial z \over \partial T \right)_P {dP \over \rho} %\label{eq:}
$ (3.26)

Utilizing Gibbs equation (3.20)
Letting $ ds =0$ for isentropic process results in
Equation (3.28) can be integrated by parts. However, it is more convenient to express $ dT / T$ in terms of $ C_v$ and $ d\rho / \rho$ as follows
Equating the right hand side of equations (3.28) and (3.29) results in
Rearranging equation (3.30) yields
If the terms in the braces are constant in the range under interest in this study, equation (3.31) can be integrated. For short hand writing convenience, $ n$ is defined as
Note that $ n$ approaches $ k$ when $ z\rightarrow1$ and when $ z$ is constant. The integration of equation (3.31) yields
Equation (3.33) is similar to equation (3.11). What is different in these derivations is that a relationship between coefficient n and $ k$ was established. This relationship (3.33) isn't new, and in-fact any thermodynamics book shows this relationship. But the definition of n in equation (3.32) provides a tool to estimate n . Now, the speed of sound for a real gas can be obtained in the same manner as for an ideal gas.


\begin{examl}
Calculate the speed of sound of air
at $30\celsius$ and atmospher...
...as model
(compressibility factor).
Assume that $R = 287 [j /kg /K]$.
\end{examl}

SOLUTION $ \;$
According to the ideal gas model the speed of sound should be

$\displaystyle c = \sqrt{kRT} = \sqrt{1.407 \times 287 \times 300} \sim 348.1[m/sec]$    

For the real gas first coefficient $ n = 1.403 $ has
$\displaystyle c = \sqrt{znRT} = \sqrt{1.403 \times 0.995 times 287 \times 300} = 346.7 [m/sec]$    

Solution

According to the ideal gas model the speed of sound should be
$\displaystyle c = \sqrt{kRT} = \sqrt{1.407 \times 287 \times 300} \sim 348.1[m/sec]$    

For the real gas first coefficient $ n = 1.403 $ has
$\displaystyle c = \sqrt{znRT} = \sqrt{1.403 \times 0.995 times 287 \times 300} = 346.7 [m/sec]$    


The correction factor for air under normal conditions (atmospheric conditions or even increased pressure) is minimal on the speed of sound. However, a change in temperature can have a dramatical change in the speed of sound. For example, at relative moderate pressure but low temperature common in atmosphere, the compressibility factor, $ z=0.3$ and $ n\sim 1$ which means that speed of sound is only $ \sqrt{0.3 \over 1.4}$ about factor of (0.5) to calculated by ideal gas model.


next up previous index
Next: Speed of Sound in Up: Speed of Sound Previous: Speed of sound in   Index
Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21