Now for a non-isothermal flow, we have the moisture transfer, the liquid transfer plus vapour transfer. We have heat transfer as well, because of temperatural dependence. We will restrict ourselves on only two variables, that are the volumetric liquid content and temperature.
As for volumetric vapour content, we will transfer volumetric vapour content to volumetric liquid content by using the equilibrium condition. And then again we will apply conservation of mass to liquid transfer, to vapour transfer and substitute corresponding expression for the equilibrium velocity and vapour velocity.