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

Thus, a real gas equation can be expressed in many cases as

$$P = z\rho R T$$ (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

$$Tds = dh - {dP \over \rho}$$ (3.20)

The definition of pressure specific heat for a pure substance is

$$C_p = \left( \partial h \over \partial T \right)_P = T \left( \partial s \over \partial T \right)_P$$ (3.21)

The definition of volumetric specific heat for a pure substance is

$$C_v = \left( \partial u \over \partial T \right)_{\rho} = T \left( \partial s \over \partial T \right)_{\rho}$$ (3.22)

From thermodynamics, it can be shown ^3.4

$$dh = C_p dT + \left[ v -T \left( \partial v \over \partial T \right)_P \right]$$ (3.23)

The specific volumetric is the inverse of the density as $v = zRT/P$ and thus

$$\left( \partial v \over \partial T \right)_P = \left( \partial \l... ..._P + {zR \over P} \cancelto {1} { \left( \partial T \over \partial T \right)_P}$$ (3.24)

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

$$dh = C_p dT + \left[ v - T \left( \overbrace{RT \over P}^{v \over... ...er \partial T \right)_P + \overbrace{zR \over P}^{v \over T} \right) \right] dP$$ (3.25)

Simplifying equation (3.25) to became

$$dh = C_p dT - \left[ {T v \over z} \left( \partial z \over \parti... ...ver z}\left(\partial z \over \partial T \right)_P {dP \over \rho} %\label{eq:}$$ (3.26)

Utilizing Gibbs equation (3.20)

$$Tds = \overbrace{C_p dT - {T \over z} \left(\partial z \over \par... ...{zRT} \left[ {T \over z}\left(\partial z \over \partial T \right)_P + 1 \right]$$ (3.27)

Letting $ds =0$ for isentropic process results in

$${dT \over T} = {dP \over P } {R \over C_p} \left[ z + {T }\left(\partial z \over \partial T \right)_P \right]$$ (3.28)

Equation (3.28) can be integrated by parts. However, it is more convenient to express $dT / T$ in terms of $C_v$ and $d\rho / \rho$ as follows

$${dT \over T} = {d\rho \over \rho} {R \over C_v} \left[ z + { T }\left( \partial z \over \partial T \right)_{\rho} \right]$$ (3.29)

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

$${d\rho \over \rho} {R \over C_v} \left[ z + { T }\left( \partial ... ...R \over C_p} \left[ z + {T }\left(\partial z \over \partial T \right)_P \right]$$ (3.30)

Rearranging equation (3.30) yields

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

$$n = \overbrace{C_p \over C_v}^{k} \left( z + T \left( \partial z ... ...ight)_{\rho} \over z + T \left( \partial z \over \partial T \right)_{P} \right)$$ (3.32)

Note that $n$ approaches $k$ when $z\rightarrow1$ and when $z$ is constant. The integration of equation (3.31) yields

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

$${dP \over d\rho} = nz R T$$ (3.34)

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

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

For the real gas first coefficient $n = 1.403$ has

$$c = \sqrt{znRT} = \sqrt{1.403 \times 0.995 times 287 \times 300} = 346.7 [m/sec]$$

Solution

According to the ideal gas model the speed of sound should be

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

For the real gas first coefficient $n = 1.403$ has

$$c = \sqrt{znRT} = \sqrt{1.403 \times 0.995 times 287 \times 300} = 346.7 [m/sec]$$

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.