In this study, we develop a unified variational approach for the mathematical modeling of non-isothermal two phase flows with liquid-vapour phase transition. The framework is a generalization to the Onsager's variational principle widely used for over-damping system. The dynamic equations and boundary conditions for non-isothermal one component two-phase system is systematically derived based on the conservation of mass, momentum, energy, and the proposed unified variational approach. To develop an efficient numerical method for the dynamic equations, the mesoscopic lattice Boltzmann method with double distribution function framework is proposed, one distribution function is developed for the Navier-Stokes-Korteweg equations, and the other is designed for the total energy balance equation. A series numerical experiments, including the droplet evaporation, bubble nucleation and departure and Leidenfrost droplet impact on a flat plate, are carried out to validate the capability and performance of the present model. The numerical results of the proposed model are found to be in excellent agreement with the results of theoretical and the experimental data. The present model provides an effective predictive tool for simulating non-isothermal two phase flows with liquid-vapour phase transition.

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