Thermodynamic coupling of temperature and composition fields in phase-change problems has been a challenge for decades. A compromise has been always desired between numerical efficiency and detailed physical consideration, toward a general scheme. In the present work, a macro-micro numerical method is proposed to link the conservation equations of energy and species with the thermodynamics of the solidification problems. Firstly, the basic structure of the method, simplified with a local equilibrium assumption, is presented. The method is then extended to a multi-phase model, demonstrating a three-phase approach to the solidification of a eutectic binary alloy. Relaxing the limitations imposed by the equilibrium assumption, non-equilibrium and microscale considerations was also included subsequently by a suggested modification to the macroscopic mathematical model. Advantages gained through the general algorithm proposed are concerned with two features of the method; (a) consistency with the energy and species equations. (b) No need of a predefined solidification path; that allows for the usage of raw phase diagram curves and offers simplicity and generality for extension through complex problems (i.e. microscopic, multi-phase or non-equilibrium). A benchmark problem was employed to test the performance of the proposed method in two cases of local equilibrium and Scheil-like solidification. The obtained results were validated in comparison with available semi-analytical solution. © 2011 Elsevier Inc.
Jafari, A., Seyedein, S. H., & Aboutalebi, M. R. (2011). Semi-implicit method for thermodynamically linked equations in phase change problems (SIMTLE). Applied Mathematical Modelling, 35(10), 4774–4789. https://doi.org/10.1016/j.apm.2011.03.051