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.

$\begin{figure}\centerline{\includegraphics {cont/sound/CompressiblityChart}} \end{figure}$

**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

$\displaystyle P = z\rho R T$

(3.19)

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

$\displaystyle Tds = dh - {dP \over \rho}$

(3.20)

The definition of pressure specific heat for a pure substance is

$\displaystyle C_p = \left( \partial h \over \partial T \right)_P = T \left( \partial s \over \partial T \right)_P$

(3.21)

The definition of volumetric specific heat for a pure substance is

$\displaystyle C_v = \left( \partial u \over \partial T \right)_{\rho} = T \left( \partial s \over \partial T \right)_{\rho}$

(3.22)

From thermodynamics, it can be shown ^3.4

$\displaystyle dh = C_p dT + \left[ v -T \left( \partial v \over \partial T \right)_P \right]$

(3.23)

The specific volumetric is the inverse of the density as

and thus

$\displaystyle \left( \partial v \over \partial T \right)_P = \left( \partial \l... ..._P + {zR \over P} \cancelto {1} { \left( \partial T \over \partial T \right)_P}$

(3.24)

Substituting the equation (3.24) into equation (3.23) results

$\displaystyle dh = C_p dT + \left[ v - T \left( \overbrace{RT \over P}^{v \over... ...er \partial T \right)_P + \overbrace{zR \over P}^{v \over T} \right) \right] dP$

(3.25)

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)

$\displaystyle Tds = \overbrace{C_p dT - {T \over z} \left(\partial z \over \par... ...{zRT} \left[ {T \over z}\left(\partial z \over \partial T \right)_P + 1 \right]$

(3.27)

Letting

for isentropic process results in

$\displaystyle {dT \over T} = {dP \over P } {R \over C_p} \left[ z + {T }\left(\partial z \over \partial T \right)_P \right]$

(3.28)

Equation (3.28) can be integrated by parts. However, it is more convenient to express

in terms of

and $d\rho / \rho$ as follows

$\displaystyle {dT \over T} = {d\rho \over \rho} {R \over C_v} \left[ z + { T }\left( \partial z \over \partial T \right)_{\rho} \right]$

(3.29)

Equating the right hand side of equations (3.28) and (3.29) results in

$\displaystyle {d\rho \over \rho} {R \over C_v} \left[ z + { T }\left( \partial ... ...R \over C_p} \left[ z + {T }\left(\partial z \over \partial T \right)_P \right]$

(3.30)

Rearranging equation (3.30) yields

$\displaystyle {d\rho \over \rho} = {dP \over P } {C_v \over C_p } \left[ z + {T... ...ght)_P \over z + { T }\left( \partial z \over \partial T \right)_{\rho} \right]$

(3.31)

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,

is defined as

$\displaystyle n = \overbrace{C_p \over C_v}^{k} \left( z + T \left( \partial z ... ...ight)_{\rho} \over z + T \left( \partial z \over \partial T \right)_{P} \right)$

(3.32)

Note that

approaches

when $z\rightarrow1$ and when

is constant. The integration of equation (3.31) yields

$\displaystyle \left(\rho_1 \over \rho_2 \right)^n = {P_1 \over P_2}$

(3.33)

Equation (3.33) is similar to equation (3.11). What is different in these derivations is that a relationship between coefficient n and

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.

$\displaystyle {dP \over d\rho} = nz R T$

(3.34)

$\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

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

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,