Multigrain and Multiphase Mathematical Model for Equiaxed Solidification
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INTRODUCTION
CASTINGS with fine equiaxed grains are frequently pursued by the casting industry. Mathematical models have been proposed to elucidate the phenomena underlying the solidification of equiaxed grains. Oldfield[1] presented one of the first deterministic models of equiaxed solidification, especially developed for the solidification of eutectic cells that were assumed as solid spheres. Rappaz et al.[2] extended Oldfield’s[1] model considering that the critical undercooling for heterogeneous nucleation on substrate particles within the melt follows a normal distribution, predicting the final average density or size of eutectic cells. Maxwell and Hellawell[3] and later Greer et al.[4] proposed models in which grains of the primary phase were solid spheres, i.e., globulitic, subdivided into classes of different grain sizes, nucleated at different undercoolings. These models are valid before globulitic grains change into dendritic and before interactions occur between neighboring MARCELO AQUINO MARTORANO, Associate Professor, is with the Department of Metallurgical and Materials Engineering, University of Sa˜o Paulo, Av. Prof. Mello Moraes, 2463, Sa˜o Paulo, SP 05508-900, Brazil. Contact e-mail: [email protected] DAVI TEVES AGUIAR, Materials Engineer, is with the Materials Department, Petro´leo Brasileiro S.A., Av. Henrique Valadares, 28, Rio de Janeiro, RJ 20231-030 Brazil. JUAN MARCELO ROJAS ARANGO, Professor, formerly with the Department of Metallurgical and Materials Engineering, University of Sa˜o Paulo, is now with the Department of Metallurgical and Materials Engineering, University of Antioquia, Medelli´n, Colombia. Manuscript submitted February 6, 2014. Article published online October 25, 2014 METALLURGICAL AND MATERIALS TRANSACTIONS A
grains. These conditions usually exist until the end of the nucleation period, when the final average grain size can be predicted. Rappaz and The´voz[5,6] developed the first deterministic model for the solidification of equiaxed dendritic grains. They assumed that around each grain an imaginary spherical envelope existed, including internal solid and liquid, rather than only solid as in previous models, predicting grain sizes in agreement with experiments.[7,8] The undercooling for grain nucleation was defined by a normal distribution, and all grains were locally assumed to have the same size, rather than being subdivided into classes of sizes. The composition of the external liquid surrounding grain envelopes was assumed constant and unaffected by solute rejection from grain envelopes, disregarding an important source of interaction between grains. Wang and Beckermann[9,10] used the volume averaging technique to calculate the average solute concentration in the external liquid. They also adopted an effective diffusion length to obtain the solute fluxes into the solid and external liquid. The final grain size could not be predicted, however, because an instantaneous nucleation model was adopted. Gandin et al.[11] presented a deterministic mathematical model to simulate the equiaxed dend
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