## 2 Simple Model
\begin{array}{l}\frac{\mathrm{Ex}}{\mathrm{V}\mathrm{\varepsilon^{2}}}=\frac {\mathrm{E_{1}}-\mathrm{E_{2}}}{\mathrm{V}\mathrm{\varepsilon^{2}}}=\frac{ \mathrm{\rho_{0}}\bigg{\langle}\bigg{|}\bigg{|}^{2}\bigg{\rangle}}{2}+\frac{ \left\langle\mathrm{(p^{\prime})^{2}}\right\rangle}{2\mathrm{k}\mathrm{P_{0} }}+\frac{\mathrm{p_{0}}\big{\langle}\left\langle s^{\prime}\right\rangle^{2} \big{\rangle}}{2\mathrm{k}\mathrm{RC_{p}}}+\\ -\frac{\epsilon\left(2\mathrm{k}-1\right)\left\langle\left(p^{\prime}\right)^ {3}\right\rangle}{6\mathrm{k}^{2}\left(\mathrm{P_{0}}\right)^{2}}+\frac{ \left\langle\mathrm{p^{\prime}}\left\langle s^{\prime}\right\rangle^{2} \right\rangle}{2\mathrm{k}\mathrm{RC_{p}}}-\frac{\epsilon\mathrm{p_{0}}\big{ \langle}\left\langle s^{\prime}\right\rangle^{3}\big{\rangle}}{3\mathrm{k} \left(\mathrm{C_{p}}\right)^{2}}+\\ +\frac{\epsilon\mathrm{\rho_{0}}\bigg{\langle}\mathrm{p^{\prime}}\bigg{|}^{2} \bigg{\rangle}}{2\mathrm{k}\mathrm{P_{0}}}-\frac{\epsilon\mathrm{\rho_{0}} \big{\langle}s^{\prime}\bigg{|}^{2}\bigg{\rangle}}{2\mathrm{C_{p}}}+O\big{(} \mathrm{\mathrm{\epsilon^{2}}}\big{)}.\end{array}  \tag{12}\]  
If we take only the quadratic part of this expression, it is the same as in works [PERSON] (1958); Lifshitz and Lifshitz (1958); [PERSON] (1958). From [PERSON] and [PERSON]'s work [PERSON] (1958), it is:  
\[\mathrm{E_{III}}=\frac{\mathrm{\rho(v^{\prime})}^{2}}{2}+\frac{\left(\mathrm{ p^{\prime}}\right)^{2}}{2\mathrm{pc}^{2}}+\frac{\mathrm{\rho T(s^{\prime})}^{2}}{2 \mathrm{C_{P}}}. \tag{13}\]From [PERSON] and [PERSON]'s work [8], it is:  
\[e_{tot}=\rho_{0}\frac{\overset{\rightarrow}{(\mathrm{v^{\prime}})^{2}}}{2}+ \frac{1}{\rho\mathrm{c}^{2}}\frac{\left(\mathrm{p^{\prime}}\right)^{2}}{2}+ \frac{\mathrm{P}_{0}\left(\mathrm{s^{\prime}}\right)^{2}}{2\mathrm{RC}_{\mathrm{ P}}}. \tag{14}\]  
The third power term part of Expression (12) can be found only in [PERSON]'s work [27]:  
\[ \begin{array}{l}\mathrm{E}_{3}=\frac{1}{2}\rho_{0}(\overset{\rightarrow}{ \mathrm{v^{\prime}}})^{2}+\frac{1}{\rho_{0}\mathrm{c}^{2}}\frac{\left(\mathrm{ p^{\prime}}\right)^{2}}{2}+\\ +\frac{\rho_{0}\mathrm{T}_{0}\left(\mathrm{s^{\prime}}\right)^{2}}{2\left( \mathrm{C}_{\mathrm{P}}\right)^{2}}+\\ +\frac{\left(1-2\mathrm{k}\right)\left(\mathrm{p^{\prime}}\right)^{3}}{6\left( \rho_{0}\right)^{2}\mathrm{c}^{4}}+\frac{\mathrm{p^{\prime}}\left(\mathrm{s^ {\prime}}\right)^{2}}{2\mathrm{KR}_{\mathrm{P}}}-\\ -\frac{\rho_{0}\mathrm{T}_{0}\left(\mathrm{s^{\prime}}\right)^{3}}{3\left( \mathrm{C}_{\mathrm{P}}\right)^{3}},\end{array}  \tag{15}\]  
but it does not have all the terms in Equation (12) that can be found in this work, mainly because the author of that work think they are negligible. It is valuable to note that this equation for internal exergy does not contain the mean temperature of volume. This peculiarity can be explained as follows: no heat is added or removed from the system, and the entropy is kept the same during this thermodynamic process, so the thermodynamic state is fully described by pressure and entropy. We can also study such situations as when fluctuations of parameters diminish due to the dissipation process while mass in volume and internal energy are kept constant. This will lead to an increase in system's entropy. This state will be named as \"state 3\". Using asymptotic expansion with a small parameter \(\varepsilon\) we will get the following result:  
\[\frac{\mathrm{P}_{0}}{\mathrm{R}}\Delta\mathrm{s}_{1\to 3}=\mathrm{Ex}. \tag{16}\]  
The mean thermodynamic temperature may be determined according to the Gouy-Stodola theorem [28]:  
\[\mathrm{T}_{0}=\frac{\mathrm{Ex}}{\langle\rho\rangle\Delta\mathrm{s}_{1 \to 3}}=\frac{\mathrm{P}_{0}}{\mathrm{R}\langle\rho\rangle}, \tag{17}\]  
or the mean temperature in this case is calculated from the mean pressure and density. This method to find the internal exergy of a closed volume may be extended to a reacting mixture according to the internal exergy definition.