13.1. Perfect Ideal Gas (PIG)
todo Add details, and link to Thermodynamics to deal with thermodynamic potentials, coefficients and derivatives in the most general way
Equation of state.
\[p = \rho R T \ ,\]
with \(R = \frac{R_u}{M_m}\), being \(M_m\) the molar mass of the gas and \(R_u\) the universal gas constant
\[R_u = 8134.46 \frac{\text{J}}{\text{kmol} \, \text{K}} \ .\]
Example 13.1 (Molar mass of dry air - mixture of gas)
Let the composition of dry air: \(\text{N}_2: \, 78.08\%\), \(\text{O}_2: \, 20.95 \%\), \(\text{Ar}: \, 0.93 \%\), \(\text{CO}_2: \, 0.04\%\)
…
\[M_{m, air} = 28.97 \, \frac{\text{kg}}{\text{kmol}} \ .\]
Thus the value of gas constant for dry air is
\[R_{air} = \frac{R_u}{M_{m,air}} = \frac{8314.46 \frac{\text{J}}{\text{kmol} \,\text{K}}}{ 28.97 \frac{\text{kg}}{\text{kmol}}} = 287.00 \frac{\text{J}}{\text{kg} \, \text{K}} \ .\]
Internal energy and enthalpy. In perfect ideal gas, energy and enthalpy are function of temperature only, with constant heat coefficients \(c_v\), \(c_p\)
\[\begin{split}\begin{aligned}
e & = c_v T \\
h & = c_p T
\end{aligned}\end{split}\]
Heat coefficient ratio.
\[\gamma = \frac{c_p}{c_v}\]
Meyer relation.
\[c_p - c_v = R\]
Thus,
\[\begin{split}\begin{aligned}
\frac{c_v}{R} & = \frac{ 1}{\gamma-1} \\
\frac{c_p}{R} & = \frac{\gamma}{\gamma-1}
\end{aligned}\end{split}\]
Entropy. Starting from the first principle of thermodynamics
\[de = T ds + \frac{p}{\rho^2} d \rho \ ,\]
with \(d e = c_v dT\), it’s possible to write the differential of entropy as
\[\begin{split}\begin{aligned}
d s
& = c_v \frac{dT}{T} - R \frac{d \rho}{\rho} = \\
& = c_v \frac{dp}{p} - c_p \frac{d \rho}{\rho} = \\
& = c_p \frac{dT}{T} - R \frac{d p}{p} \ ,
\end{aligned}\end{split}\]
having used the equation of state, Meyer relation and the relation between differentials of thermodynamic quantities, like
\[\frac{d T}{T} = \frac{ d\left(\frac{p}{\rho R} \right)}{ \frac{p}{\rho R} } = \frac{\rho}{p} \left( \frac{dp}{\rho} - \frac{p}{\rho^2} d \rho \right) = \frac{dp}{p} - \frac{d \rho}{\rho} \ .\]
The finite difference between two thermodynamic states immediately follows from integration (being \(R\), \(c_v\), \(c_p\) constant for a PIG),
\[\begin{split}\begin{aligned}
s - s_0
& = c_v \ln \left( \frac{T}{T_0} \right) - R \ln \left( \frac{\rho}{\rho_0} \right) = \\
& = c_v \ln \left( \frac{p}{p_0} \right) - c_p \ln \left( \frac{\rho}{\rho_0} \right) = \\
& = c_p \ln \left( \frac{T}{T_0} \right) - R \ln \left( \frac{ p}{ p_0} \right) \ ,
\end{aligned}\end{split}\]
Speed of sound. The speed of sound in a medium is defined as
\[a^2 = \left( \dfrac{\partial p}{\partial \rho} \right)_s \ .\]
For a PIG, relation (todo Add ref) links pressure, density and entropy. This relation can be recast as
\[\frac{p}{p_0}(\rho, s; \rho_0, s_0) = \frac{\rho^\gamma}{\rho_0^\gamma} e^{ (\gamma-1) \frac{s-s_0}{R} } \ .\]
Direct evaluation of the partial derivative at constant \(s\) gives the expression of speed of sound for a PIG
\[a^2 = \gamma R T = \gamma \frac{p}{\rho} \]
\[\begin{split}\begin{aligned}
a^2
& = \left( \dfrac{\partial p}{\partial \rho} \right)_s = \\
& = \left( \dfrac{\partial}{\partial \rho} \right)_s \frac{\rho^{\gamma}}{\rho_0^{\gamma}} e^{(\gamma-1)\frac{s-s_0}{R}} = \\
& = \gamma \rho^{-1} \frac{\rho^{\gamma}}{\rho_0^{\gamma}} e^{(\gamma-1)\frac{s-s_0}{R}} = \\
& = \gamma \frac{p}{\rho} = \\
& = \gamma R T \ .
\end{aligned}\end{split}\]
