Mass Conservation

Mixture of Gases Continuity:

$${\frac{{\partial}}{{\partial t}}}$$$$\left(\vphantom{ \alpha_m\rho_m }\right.$$$$\alpha_{m}^{}$$$$\rho_{m}^{}$$$$\left.\vphantom{ \alpha_m\rho_m }\right)$$ + $${\frac{{\partial}}{{\partial z}}}$$$$\left(\vphantom{ \alpha_m\rho_m V_m }\right.$$$$\alpha_{m}^{}$$$$\rho_{m}^{}$$V_m$$\left.\vphantom{ \alpha_m\rho_m V_m }\right)$$ = $$\sum_{{x=l,s}}^{}$$$$\Gamma_{{xg}}^{}$$

Noncondensable Gas Continuity:

$${\frac{{\partial}}{{\partial t}}}$$$$\left(\vphantom{ \alpha_m\rho_m X_n }\right.$$$$\alpha_{m}^{}$$$$\rho_{m}^{}$$X_n$$\left.\vphantom{ \alpha_m\rho_m X_n }\right)$$ + $${\frac{{\partial}}{{\partial z}}}$$$$\left(\vphantom{ \alpha_m\rho_m X_n V_m }\right.$$$$\alpha_{m}^{}$$$$\rho_{m}^{}$$X_nV_m$$\left.\vphantom{ \alpha_m\rho_m X_n V_m }\right)$$ = $${\frac{{\partial}}{{\partial z}}}$$$$\left(\vphantom{ \alpha_m D_n^X \frac{\partial X_n}{\partial z} }\right.$$$$\alpha_{m}^{}$$D_n^X$${\frac{{\partial X_n}}{{\partial z}}}$$$$\left.\vphantom{ \alpha_m D_n^X \frac{\partial X_n}{\partial z} }\right)$$ + $${\frac{{\partial}}{{\partial z}}}$$$$\left(\vphantom{ \alpha_m D_n^\rho \frac{\partial \rho_m}{\partial z} }\right.$$$$\alpha_{m}^{}$$D_n^$\rho$$${\frac{{\partial \rho_m}}{{\partial z}}}$$$$\left.\vphantom{ \alpha_m D_n^\rho \frac{\partial \rho_m}{\partial z} }\right)$$

Liquid Continuity:

$$\epsilon_{v}^{}$$$${\frac{{\partial}}{{\partial t}}}$$$$\left(\vphantom{ \alpha_l\rho_l }\right.$$$$\alpha_{l}^{}$$$$\rho_{l}^{}$$$$\left.\vphantom{ \alpha_l\rho_l }\right)$$ + $$\epsilon_{v}^{}$$$${\frac{{\partial}}{{\partial z}}}$$$$\left(\vphantom{ \alpha_l\rho_l V_l }\right.$$$$\alpha_{l}^{}$$$$\rho_{l}^{}$$V_l$$\left.\vphantom{ \alpha_l\rho_l V_l }\right)$$ = $$\sum_{{x=g,s}}^{}$$$$\Gamma_{{xl}}^{}$$

Solid Continuity:

$$\epsilon_{v}^{}$$$${\frac{{\partial}}{{\partial t}}}$$$$\left(\vphantom{ \alpha_s\rho_s }\right.$$$$\alpha_{s}^{}$$$$\rho_{s}^{}$$$$\left.\vphantom{ \alpha_s\rho_s }\right)$$ = $$\sum_{{x=g,l}}^{}$$$$\Gamma_{{xs}}^{}$$