Macrotransport-solidification kinetics modeling of equiaxed dendritic growth: Part I. Model development and discussion
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INTRODUCTION
SIMULATION of casting solidification has lately received increased attention because of the potentially significant savings in the time required for prototyping and in the cost associated with defective castings. The most recent breakthrough was the incorporation of solidification kinetics models in macrotransport codes. A second generation of computer models for solidification, the macrotransportsolidification kinetics (MT-SK) codes, has been launched and applied by a few progressive foundries. This resulted in increased accuracy of model prediction, as well as in an increased number of predictors, such as amount and spacing of phases, grain size, microstructural transitions, etc. While MT-SK modeling of solidification of eutectic alloys has found an early satisfactory solution,[1,2] equiaxed dendritic solidification has proven more difficult to tackle. This is not surprising, considering the complicated geometry of dendrites. Maxwell and Hellawell[3] have presented a model for the nucleation and the initial stages of solidification of spherical grains. Dustin and Kurz[4] developed a model in which heterogeneous nucleation, solute, and thermal undercooling were considered. Although their model predicts interesting features such as grain number and recalescence, it ignores the overall solute balance and assumes that the internal volume fraction of solid is constant during growth. L. NASTAC, formerly Graduate Research Assistant, Department of Metallurgical and Materials Engineering, The University of Alabama, is Senior Staff Engineer, Concurrent Technologies Corporation, Johnstown, PA 15904. D.M. STEFANESCU, University Research Professor and Director of the Solidification Laboratory, is with the Department of Metallurgical and Materials Engineering, The University of Alabama, Tuscaloosa, AL 35487. Manuscript submitted March 11, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS A
Rappaz and Thevoz[5] have proposed a model based on solute diffusion in the liquid, heat, and solute balances, and diffusion-controlled growth kinetics. They assumed no back diffusion, complete mixing of solute within the interdendritic liquid, and ignored the role of thermal undercooling in growth kinetics. The main goal of this research effort was to develop a model for equiaxed dendritic solidification that capitalizes on previous models, but relaxes some of their most restrictive assumptions. Indeed, in the case of relatively high cooling rates and large grain size, the thermal undercooling might become an important factor. In addition, even for substitutional elements, where the solute-solid diffusivity is very small, back diffusion has to be included to satisfy the overall mass balance equation. This is particularly important in the last stage of solidification, when the solute-liquid diffusivity is less significant. For interstitially dissolved elements, e.g., carbon in Fe-C alloys, kinetics models without diffusion in the solid phase would considerably overestimate the growth of dendrites. An important parameter in equiaxed
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