We numerically study the thermohydrodynamics of boiling for a CO2 + ethanol mixture on lyophilic and lyophobic surfaces in both closed and open systems, based on a diffuse interface model for a two-component system. The corresponding wetting boundary conditions for an isothermal system are proposed and verified in this paper. New phenomena due to the addition of another component, mainly the preferential evaporation of the more volatile component, are observed. In the open system and the closed system, the physical process shows very different characteristics. In the open system, except for the movement of the contact line, the qualitative features are rather similar for lyophobic and lyophilic surfaces. In the closed system, the vortices that are observed on a lyophobic surface are not seen on a lyophilic surface. More sophisticated wetting boundary conditions for nonisothermal, two-component systems might need to be further developed, taking into account the variations of density, temperature, and surface tension near the wall, while numerical results show that the boundary conditions proposed here also work well even in boiling, where the temperature is nonuniform.