Energy Conservation

$\bgroup\color{blue}$\displaystyle {\frac{{\partial}}{{\partial t}}}$\egroup$ $\bgroup\color{blue}$\displaystyle \left(\vphantom{ \alpha_m \rho_m U_m }\right.$\egroup$ $\bgroup\color{blue}$\displaystyle \alpha_{m}^{}$\egroup$ $\bgroup\color{blue}$\displaystyle \rho_{m}^{}$\egroup$ U_m $\bgroup\color{blue}$\displaystyle \left.\vphantom{ \alpha_m \rho_m U_m }\right)$\egroup$ + $\bgroup\color{red}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup$ $\bgroup\color{red}$\displaystyle \left(\vphantom{ \alpha_m \rho_m U_m V_m }\right.$\egroup$ $\bgroup\color{red}$\displaystyle \alpha_{m}^{}$\egroup$ $\bgroup\color{red}$\displaystyle \rho_{m}^{}$\egroup$ U_mV_m $\bgroup\color{red}$\displaystyle \left.\vphantom{ \alpha_m \rho_m U_m V_m }\right)$\egroup$ = - P_m $\bgroup\color[rgb]{.9,.8,0}$\displaystyle \left(\vphantom{ \frac{\partial}{\part... ... \textcolor[rgb]{.9,.8,0}{\frac{\partial \alpha_m}{\partial t}} }\right.$\egroup$ $\bgroup\color[rgb]{.9,.8,0}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup$ $\bgroup\color[rgb]{.9,.8,0}$\displaystyle \left(\vphantom{ \alpha_m V_m }\right.$\egroup$ $\bgroup\color[rgb]{.9,.8,0}$\displaystyle \alpha_{m}^{}$\egroup$ V_m $\bgroup\color[rgb]{.9,.8,0}$\displaystyle \left.\vphantom{ \alpha_m V_m }\right)$\egroup$ + $\bgroup\color[rgb]{.9,.8,0}$\displaystyle {\frac{{\partial \alpha_m}}{{\partial t}}}$\egroup$ $\bgroup\color[rgb]{.9,.8,0}$\displaystyle \left.\vphantom{ \frac{\partial}{\part... ... \textcolor[rgb]{.9,.8,0}{\frac{\partial \alpha_m}{\partial t}} }\right)$\egroup$ + $\bgroup\color{cyan}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup$ $\bgroup\color{cyan}$\displaystyle \left(\vphantom{ \alpha_m k_m \frac{\partial T_m}{\partial z} }\right.$\egroup$ $\bgroup\color{cyan}$\displaystyle \alpha_{m}^{}$\egroup$ k_m $\bgroup\color{cyan}$\displaystyle {\frac{{\partial T_m}}{{\partial z}}}$\egroup$ $\bgroup\color{cyan}$\displaystyle \left.\vphantom{ \alpha_m k_m \frac{\partial T_m}{\partial z} }\right)$\egroup$
		+ $\bgroup\color{green}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup$ $\bgroup\color{green}$\displaystyle \left(\vphantom{ \alpha_m D_n^X \left( h_n-h_g \right) \frac{\partial X_n}{\partial z} }\right.$\egroup$ $\bgroup\color{green}$\displaystyle \alpha_{m}^{}$\egroup$ D_n^X $\bgroup\color{green}$\displaystyle \left(\vphantom{ h_n-h_g }\right.$\egroup$ h_n - h_g $\bgroup\color{green}$\displaystyle \left.\vphantom{ h_n-h_g }\right)$\egroup$ $\bgroup\color{green}$\displaystyle {\frac{{\partial X_n}}{{\partial z}}}$\egroup$ $\bgroup\color{green}$\displaystyle \left.\vphantom{ \alpha_m D_n^X \left( h_n-h_g \right) \frac{\partial X_n}{\partial z} }\right)$\egroup$ + $\bgroup\color{green}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup$ $\bgroup\color{green}$\displaystyle \left(\vphantom{ \alpha_m D_n^\rho \left( h_n-h_g \right) \frac{\partial \rho_m}{\partial z} }\right.$\egroup$ $\bgroup\color{green}$\displaystyle \alpha_{m}^{}$\egroup$ D_n $\bgroup\color{green}$\displaystyle \left(\vphantom{ h_n-h_g }\right.$\egroup$ h_n - h_g $\bgroup\color{green}$\displaystyle \left.\vphantom{ h_n-h_g }\right)$\egroup$ $\bgroup\color{green}$\displaystyle {\frac{{\partial \rho_m}}{{\partial z}}}$\egroup$ $\bgroup\color{green}$\displaystyle \left.\vphantom{ \alpha_m D_n^\rho \left( h_n-h_g \right) \frac{\partial \rho_m}{\partial z} }\right)$\egroup$ + $\bgroup\color{magenta}$\displaystyle \sum_{{x=l,s}}^{}$\egroup$ $\bgroup\color{magenta}$\displaystyle \left(\vphantom{ Q_{xm} + Q^\Gamma_{xg} }\right.$\egroup$ Q_xm + Q_xg $\bgroup\color{magenta}$\displaystyle \left.\vphantom{ Q_{xm} + Q^\Gamma_{xg} }\right)$\egroup$

$\bgroup\color{blue}$\displaystyle \epsilon_{v}^{}$\egroup$ $\bgroup\color{blue}$\displaystyle {\frac{{\partial}}{{\partial t}}}$\egroup$ $\bgroup\color{blue}$\displaystyle \left(\vphantom{ \alpha_l\rho_l U_l }\right.$\egroup$ $\bgroup\color{blue}$\displaystyle \alpha_{l}^{}$\egroup$ $\bgroup\color{blue}$\displaystyle \rho_{l}^{}$\egroup$ U_l $\bgroup\color{blue}$\displaystyle \left.\vphantom{ \alpha_l\rho_l U_l }\right)$\egroup$ + $\bgroup\color{red}$\displaystyle \epsilon_{v}^{}$\egroup$ $\bgroup\color{red}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup$ $\bgroup\color{red}$\displaystyle \left(\vphantom{ \alpha_l\rho_l U_l V_l }\right.$\egroup$ $\bgroup\color{red}$\displaystyle \alpha_{l}^{}$\egroup$ $\bgroup\color{red}$\displaystyle \rho_{l}^{}$\egroup$ U_lV_l $\bgroup\color{red}$\displaystyle \left.\vphantom{ \alpha_l\rho_l U_l V_l }\right)$\egroup$	=	- $\bgroup\color[rgb]{.9,.8,0}$\displaystyle \epsilon_{v}^{}$\egroup$ P_l $\bgroup\color[rgb]{.9,.8,0}$\displaystyle \left(\vphantom{ \frac{\partial \alpha... ... \textcolor[rgb]{.9,.8,0}{\frac{\partial \alpha_l}{\partial t}} }\right.$\egroup$ $\bgroup\color[rgb]{.9,.8,0}$\displaystyle {\frac{{\partial \alpha_l V_l}}{{\partial z}}}$\egroup$ + $\bgroup\color[rgb]{.9,.8,0}$\displaystyle {\frac{{\partial \alpha_l}}{{\partial t}}}$\egroup$ $\bgroup\color[rgb]{.9,.8,0}$\displaystyle \left.\vphantom{ \frac{\partial \alpha... ... \textcolor[rgb]{.9,.8,0}{\frac{\partial \alpha_l}{\partial t}} }\right)$\egroup$ + $\bgroup\color{cyan}$\displaystyle \epsilon_{v}^{}$\egroup$ $\bgroup\color{cyan}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup$ $\bgroup\color{cyan}$\displaystyle \left(\vphantom{ \alpha_l k_l \frac{\partial T_l}{\partial z} }\right.$\egroup$ $\bgroup\color{cyan}$\displaystyle \alpha_{l}^{}$\egroup$ k_l $\bgroup\color{cyan}$\displaystyle {\frac{{\partial T_l}}{{\partial z}}}$\egroup$ $\bgroup\color{cyan}$\displaystyle \left.\vphantom{ \alpha_l k_l \frac{\partial T_l}{\partial z} }\right)$\egroup$
		+ $\bgroup\color{magenta}$\displaystyle \sum_{{x=m,s,w}}^{}$\egroup$ Q_xl + $\bgroup\color{magenta}$\displaystyle \sum_{{x=g,s}}^{}$\egroup$ Q_xl

$\bgroup\color{blue}$\displaystyle \epsilon_{v}^{}$\egroup$ $\bgroup\color{blue}$\displaystyle {\frac{{\partial}}{{\partial t}}}$\egroup$ $\bgroup\color{blue}$\displaystyle \left(\vphantom{ \alpha_s\rho_s U_s }\right.$\egroup$ $\bgroup\color{blue}$\displaystyle \alpha_{s}^{}$\egroup$ $\bgroup\color{blue}$\displaystyle \rho_{s}^{}$\egroup$ U_s $\bgroup\color{blue}$\displaystyle \left.\vphantom{ \alpha_s\rho_s U_s }\right)$\egroup$ = $\bgroup\color{cyan}$\displaystyle \epsilon_{v}^{}$\egroup$ $\bgroup\color{cyan}$\displaystyle {\frac{{\partial}}{{\partial z}}}$\egroup$ $\bgroup\color{cyan}$\displaystyle \left(\vphantom{ \alpha_s k_s \frac{\partial T_s}{\partial z} }\right.$\egroup$ $\bgroup\color{cyan}$\displaystyle \alpha_{s}^{}$\egroup$ k_s $\bgroup\color{cyan}$\displaystyle {\frac{{\partial T_s}}{{\partial z}}}$\egroup$ $\bgroup\color{cyan}$\displaystyle \left.\vphantom{ \alpha_s k_s \frac{\partial T_s}{\partial z} }\right)$\egroup$ + $\bgroup\color{magenta}$\displaystyle \sum_{{x=m,l,w}}^{}$\egroup$ Q_xs + $\bgroup\color{magenta}$\displaystyle \sum_{{x=g,l}}^{}$\egroup$ Q_xs