## 3 The Simple Model's Extension to the Moving System  
The case studied in the previous section may be extended to the case of a closed thermodynamic system interacting with its surroundings through work and heat. This interaction with the surroundings will lead to changes in mean entropy and pressure. In this article, we study such systems as having internal structure, which can be analyzed by internal exergy. To define such a system's internal exergy, we will derive an evolutionary differential equation using the assumption that irreversible entropy generation is fully defined by dissipative function and by heat exergy definition based on non-isothermal heat transfer. The initial conditions for this equation will be defined according to the results of the previous section.  
In this section, we will derive an equation for a second-order internal exergy. In this definition, it is assumed that the volume of averaging is fixed. The defining equation for higher-order exergy can be derived using the same method.  
We will use the following identity:  
\[\frac{\mathrm{d}}{\mathrm{dt}}\bigg{\langle}\frac{(\varphi-\langle\varphi \rangle)^{2}}{2}\bigg{\rangle}=\bigg{\langle}(\varphi-\langle\varphi\rangle) \frac{\mathrm{d}\varphi}{\mathrm{dt}}\bigg{\rangle}, \tag{18}\]  
with the following second-order expression of internal exergy:  
\[\mathrm{Ex}_{2}=\frac{\langle\rho\rangle\bigg{\langle}\bigg{|}\overrightarrow{ v^{\prime}}\bigg{|}^{2}\bigg{\rangle}}{2}+\frac{\big{\langle}(\mathrm{p}^{ \prime})^{2}\big{\rangle}}{2\mathrm{k}\langle\mathrm{p}\rangle}+\frac{\langle \mathrm{p}\rangle\big{\langle}(s^{\prime})^{2}\big{\rangle}}{2\mathrm{k} \mathrm{R}\mathrm{C}_{\mathrm{P}}}, \tag{19}\]  
and equations of gas dynamics for ideal gas in form:  
\[\rho\frac{\mathrm{d}u}{\mathrm{dt}}+\frac{\partial\mathrm{p}}{\partial \mathrm{x}}=\
abla(\sigma_{x}), \tag{20}\]  
\[\rho\frac{\mathrm{d}v}{\mathrm{dt}}+\frac{\partial\mathrm{p}}{\partial \mathrm{y}}=\
abla(\sigma_{y}), \tag{21}\]  
\[\rho\frac{\mathrm{d}w}{\mathrm{dt}}+\frac{\partial\mathrm{p}}{\partial \mathrm{z}}=\
abla(\sigma_{z}), \tag{22}\]  
\[\frac{\mathrm{d}\mathrm{p}}{\mathrm{dt}}+\mathrm{kp}\Big{(}\
abla\cdot \overrightarrow{v}\Big{)}=(\mathrm{k}-1)(\mathrm{Q}+\Phi), \tag{23}\]  
\[\frac{\mathrm{p}}{\mathrm{R}}\frac{\mathrm{d}s}{\mathrm{dt}}=(\mathrm{Q}+\Phi). \tag{24}\]  
After representing all quadratic parts in Equation (19) by Equation (18) and inserting Equations (20)-(24) into Equation (19), we will get the following result:  
\[\begin{array}{l}\frac{\mathrm{d}\mathrm{E}\mathrm{x}\mathrm{z}}{\mathrm{dt }}\mathrm{V}=-\int\limits_{\Gamma}\overrightarrow{v^{\prime}}\cdot\mathrm{d} \Gamma+\\ \\ +\int\limits_{\Gamma}\bigg{(}\Big{(}\Big{(}1-\frac{\mathrm{T}_{0}}{\mathrm{T}_ {0}+\Gamma^{\prime}}\Big{)}-1\Big{)}\big{(}\mathrm{Q}^{\prime}+\Phi^{\prime} \big{)}-\Phi^{\prime}\bigg{)}\mathrm{d}\mathrm{V}+\\ \\ +\int\limits_{\Gamma}\Bigg{(}\frac{\rho\big{|}\overrightarrow{v^{\prime}}\big{|} ^{2}}{2}+\frac{(\mathrm{p}^{\prime})^{2}}{2\mathrm{k}\langle\mathrm{p} \rangle}+\frac{(\mathrm{s}^{\prime})^{2}\langle\mathrm{p}\rangle}{2\mathrm{k} \mathrm{R}\mathrm{C}_{\mathrm{P}}}+\frac{(\mathrm{k}-1)\big{\langle}(\mathrm{p }^{\prime})^{2}\big{\rangle}}{2\mathrm{k}\langle\mathrm{p}\rangle}\Bigg{)} \overrightarrow{v}\cdot\mathrm{d}\Gamma+\\ \\ +\Bigg{(}\frac{\langle\mathrm{p}\rangle\big{\langle}(s^{\prime})^{2}\big{\rangle} }{2\mathrm{k}\mathrm{R}\mathrm{C}_{\mathrm{P}}}-\frac{\big{\langle}(\mathrm{p }^{\prime})^{2}\big{\rangle}}{2\mathrm{k}\langle\mathrm{p}\rangle}\Bigg{)} \frac{(\mathrm{k}-1)}{\langle\mathrm{p}\rangle}\langle\mathrm{Q}\rangle \mathrm{V},\end{array} \tag{25}\]  
where the first term is the power at which work is transferred to the studied system, the second term is the exergy of heat transferred to the system and loss of exergy by dissipation, the third term is additional exergy transfer, which is usually neglected due to the small velocity in the studied system (and this is the effect of flowing), and the last term is Brayton cycle correction.