Generalized enthalpy method for multicomponent phase change
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I.
INTRODUCTION
MOVING boundary problems [~-SJ are characterized by the existence of an interphase front that moves through a given body. The task of numerical modeling is to solve transport equations and, at the same time, compute shape and position of the phase change regions. A prominent example is the propagation of a solid/liquid interface during solidification. Solidification is usually assumed to be driven by diffusion of heat. Other phase transformations are controlled by solute diffusion. Numerical schemes are available for phase change processes governed by a single diffusion equation, e.g., for heat. Among these, the enthalpy approach is attractive because it eliminates the need to track phase fronts, especially when their geometric shapes are complex. Interface positions are recovered after the diffusion problem is solved. Since multicomponent systems are the rule in practice, there is a need to extend such schemes to situations where heat and several solute species are transported simultaneously. It is the aim of the current article to develop a generalization of the enthalpy method such that nonisothermal multicomponent systems can be handled in a similar style. In previous literature, numerical modeling of alloy solidification with coupled heat and mass transport has a certain tradition. Several studies use liquidus, solidus, or other curves from equilibrium TX-phase diagrams, in many cases by assuming the curves to be linear. An early example is the article by Fix.t6~ Crowley and OckendonF1 noted that activity and concentration are formally analogous to temperature and enthalpy and use a diffusion equation in this sense. Linear curves in the phase diagram are assumed; two-phase zones are ignored. This line has been further
A. MACKENBROCK, formerly with the Max-Planck-Institut flu" Eisenforschung GmbH, Diisseldorf, is with the GFTA Institut far Trendanalysen GmbH, D-40699 Erkrath, Germany. K.-H. TACKE, is with the Department of Metallurgy, Max-Planck-lnstitut flit Eisenforschtmg GrnbH, D-40237 Diisseldorf, Germany. Manuscript submitted July 17, 1995. METALLURGICAL AND MATERIALS TRANSACTIONS B
developed in Reference 8, which introduces kinetic undercooling, and in the models of Wilson and co-workers,tg-tn where mushy zones are included. Sundarraj and VollerU3I also use an activity-driven diffusion equation for a eutectic system. From a more mathematical viewpoint, Donnelly[t4j presents enthalpy and concentration diffusion balances in terms of temperature and chemical potential differences. This approach was extended toward nonequilibrium thermodynamics in the article by Luckhaus and Visintin, [~51 which has some aspects in common with the approach of this article. Metallurgical phenomena of multicomponent phase transformations have been modeled by several authors. A model for peritectic solidification in the iron-carbon system, starting with equilibrium diagram relations between temperatures and concentrations, has been proposed by Chuang et al. II6l Dendritic iron-carbon peritectic solidification
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