Eutectic reaction and nonconstant material parameters in micro-macrosegregation modeling
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UCTION
MICROSEGREGATION and macrosegregation refer to the inhomogeneous distributions of alloying elements on the scale of the dendrites and of the whole casting, respectively. These two segregation phenomena are coupled. Microsegregation development influences the development of the local solidification path (i.e., the relation between enthalpy, solute concentration, solid fraction, temperature, and solute concentrations in the solid and liquid parts of the two-phase volume elements) and, thereby, the global transport of solute mass leading to the macrosegregation. On the other hand, a change in average solute concentration in the coexisting solid and liquid phases during solidification (macrosegregation development) influences the final microsegregation and, thus, the amount of the different solid phases that appear. Modeling of segregation at the two different scales has traditionally been based upon different assumptions and simplifications. Microsegregation modeling quantifies the solute profile within the dendrites and the amount of different phases appearing during the eutectic reaction. It is usually assumed that there is no transfer of solute into or out of the solution domain (closed system) and remelting is not considered.[2–13] In macrosegregation modeling where there is global transfer of solute (open system), attention is mainly paid to solving the equations for the global balance of mass, solute, energy, and momentum. However, quite simple models (lever rule or the Scheil approximation) are usually applied for calculating the local solidification path.[14–22] It should be noted also that the eutectic reaction
HERVE´ COMBEAU, Assistant Professor, is with the Laboratoire de Science et Ge´nie, des Materiaux Me´talliques, INPL, Ecole des Mines de Nancy, F-54042 Nancy, Cedex, France. ASBJØRN MO, Senior Scientist, is with SINTEF Materials Technology, N-0314 Oslo, Norway. Manuscript submitted January 3, 1997. METALLURGICAL AND MATERIALS TRANSACTIONS A
is taken into account in a quite simplified manner in these kinds of models. The implementation of a more elaborate microsegregation model into macrosegregation computations was addressed by Voller and Sundarraj,[23,24,25] who numerically solved the exact diffusion problem on the microscale simultaneously with the solution of the macroscopic conservation equations. Beckermann and co-workers made a thorough study on combining the two orders of magnitude in micro-macrosegregation modeling by volume-averaging techniques,[26] and treated the microsegregation by introducing an approximation for the diffusion length in the dendrites.[27,28,29] The method was also applied for a multicomponent alloy in which complete solidification was obtained without any eutectic reaction.[30] Remelting locally in the casting may occur during the process due to complex boundary conditions.[31–35] This complicates the solidification path in macrosegregation computations considerably,[1] as it can influence the possible formation (or dissolution) of eutectics. Mo[36] proposed a model
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