Simplified computation of macrosegregation in multicomponent aluminum alloys
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I.
INTRODUCTION
MATHEMATICAL modeling of macrosegregation formation is based upon volume-averaged conservation principles for the global balance of mass, solute, energy, and momentum. In addition, the local solidification path has to be determined by a microsegregation model in which the volume fraction of solid, temperature, and solute concentrations in the solid and liquid are related to the specific enthalpy and solute concentration of the two-phase volume elements. This reflects a coupling between the micro and macro scale: while the solute diffusion at the micro scale influences the development of the local solidification path and, thereby, the global transport of solute mass leading to the macrosegregation, a change in total solute concentration in a two-phase volume element (macrosegregation development) influences the final microsegregation and, thus, the amount of different solid phases that appear. While some models address the microsegregation formation in industrial multicomponent alloys,[1–6] most macrosegregation models are restricted to binary systems, even though the transport equations at the macroscopic level can in many cases, be easily extended to a multicomponent situation. For example, in modeling the inverse segregation formation close to a chill surface,[7] as well as surface segregation formed by exudation,[8] it can be assumed that the solid phase be fixed. Then, the only difference between ˚ VARD J. THEVIK, ASBJØRN MO, Research Director, and HA Research Scientist, are with SINTEF Materials Technology, N-0314 Oslo, Norway. Manuscript submitted October 20, 1997. METALLURGICAL AND MATERIALS TRANSACTIONS A
binary and multicomponent alloys is the number of solute conservation equations: one equation per alloying element is needed. The problem in extending such macrosegregation models to a multicomponent situation lies, however, in the coupling to the microscopic scale. Many microsegregation models are not necessarily applicable for determining the solidification path in macrosegregation computations, since they are restricted to systems in which the total solute concentration is constant. Furthermore, the dendritic growth kinetics are often simplified in a manner which does not apply to an industrial casting situation, which can involve remelting as well as solidification.[9] For a binary alloy, coupled micro-macro segregation models accounting for backdiffusion of solute in the dendrites can be found in References 10 and 11, and the solidification-path model discussed by Combeau et al.,[9] accounting also for remelting, was applied in Reference 12. Two main problems arise if such coupled micromacro modeling concepts are to be extended to a multicomponent situation. One of these is the handling of the eutectic reaction at the end of the solidification. This becomes quite complicated even for a binary alloy, because the total concentration within the volume element is nonconstant during macrosegregation formation.[13] The other problem is to access the thermodynamic equilibrium phase diagram d
