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.
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
Substituting the equation (3.24) into equation (3.23) results
Simplifying equation (3.25) to became
(3.26) |
Utilizing Gibbs equation (3.20)
Letting
Equation (3.28) can be integrated by parts. However, it is more convenient to express
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,
Note that
Equation (3.33) is similar to equation (3.11). What is different in these derivations is that a relationship between coefficient n and
SOLUTION
According to the ideal gas model the speed of sound should be
According to the ideal gas model the speed of sound should be
For the real gas first coefficient
Solution
According to the ideal gas model the speed of sound should be
For the real gas first coefficient |
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,
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