Lattice Boltzmann Modeling for Melting/Solidification Processes

Abstract

The phenomena of melting and solidification are associated with many practical applications, such as metal processing, castings, environmental engineering, thermal energy storage system in space station and many more. In these processes, matter is subject to a change of phase and consequently, a boundary separating two different phases evolves and moves within the matter. Mathematical modeling of such 'moving boundary problems' are always a challenging task because of the dynamic evolution of the phase separating boundary, complex boundary conditions as well as varying thermophysical properties. Many macroscopic mathematical modeling strategies for the solidification/melting problems can be found in the contemporary literatures. An excellent review in this regard can be found in Hu & Argyropoulos (1996). Early efforts in melting/solidification modeling initiated with a moving/deforming grid approach (Rubinsky & Cravahlo, 1981; Voller & Cross, 1981; Voller & Cross, 1983; Weaver & Viskanta, 1986; Askar, 1987), in which independent conservation equations for each phase need to be initially formulated, and are to be subsequently coupled with appropriate boundary conditions at the inter-phase interfaces. However, such multiple region solutions require the existence of discrete interfaces between the respective phases. In fact, a major difficulty with regard to their implementation is associated with tracking of the phase interfaces (which are generally unknown functions of space and time). The need for moving numerical grids and/or coordinate mapping procedures complicates the application of this technique further, and generally, simplifying assumptions regarding the geometric regularity of the interfaces are made. Additionally, a serious limitation exists for modeling phase change behavior of multicomponent systems, since, unlike pure substances; such systems do not exhibit a sharp interface between solid and liquid phases, in a macroscopic sense. The phase-change behavior of such systems depends on many factors including the phase-change environment, composition, and thermodynamic descriptions of specific phase transformations. Moreover, solidification occurs over extended temperature ranges and solid formation often occurs as a permeable crystalline-like matrix which coexists with the liquid phase. In such cases, it would be virtually impossible to track a morphologically complex zone in a macroscopic framework, using any moving grid technique. In contrast, in fixed-grid mathematical models of phase change (Comini et al., 1974; Morgan et al., 1978; Roose & Storrer, 1984; Dalhuijsen & Segal, 1986; Pham, 1986; Dhatt et al., 1989; Comini et al., 1990; Voller et al., 1990), transport equations for individual phases are volume-averaged to