Effect of computational domain size on the mathematical modeling of transport processes and segregation during direction
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I. INTRODUCTION
THE present study is concerned with obtaining an improved understanding of the transport phenomena in the documented experiments of Tewari and Shah,[1] who directionally solidified Pb-Sn alloys under rather steep thermal gradients of approximately 70 to 100 K cm21. In our previous work and in that of others, the simulation of the directional solidification (DS) process with uniform gradients has been performed. Directional solidification, however, is usually done by moving the specimen (or casting) relative to the coordinates of a furnace with a hot zone and cold zone (i.e., a Bridgeman furnace); hence, the liquid above the advancing solidification front has a rather steep thermal gradient near the front, which drops off away from the front and approaches zero in the hot zone, where the temperature is almost uniform. Therefore, the overlying liquid approximately comprises a layer with a uniform thermal gradient under a layer with an almost isospatial temperature. An approximation to simulating the temperature field in DS was the work of Schneider et al.[2] For thermal-boundary conditions, they imposed adiabatic sidewalls “while the top and bottom surfaces were maintained at time-varying hot and cold temperatures.” Since the transport phenomena in DS can be dominated by thermosolutal convection, this series of simulations was undertaken to see whether the entire length of the overlying liquid should be included in the computational domain of computer simulations, if one wishes to capture the interaction between the convecting liquid and the advancing solid. Tewari and Shah[1] reported macrosegregation along the lengths of small-diameter (7 mm) rods of Pb-Sn alloys (10 to 58 wt pct Sn) when they were directionally solidified in strong positive thermal gradients (melt on top, solid below, and gravity pointing downward). The results presented here are numerical simulations of the convection and segregation in C. FRUEH, Graduate Professional Intern, is with the Mechanical Reliability and Melting Department, Sandia National Laboratories, Albuquerque, NM 87185. D.R. POIRIER, Professor, is with the Department of Materials Science and Engineering, The University of Arizona, Tucson, AZ 85721. S.D. FELICELLI, Research Scientist, is with Centro Atomico Bariloche, 8400 S.C. de Bariloche, Argentina. Manuscript submitted March 21, 2000. METALLURGICAL AND MATERIALS TRANSACTIONS A
a directionally solidified Pb-23.2 wt pct Sn alloy, using the thermal conditions specified by Tewari and Shah. With an “in-house” finite-element simulator, the solidification process is modeled by solving the fully coupled equations of momentum, energy, and solute transport, along with the constraint of assuming local equilibrium on the complex solid-liquid interface in the mushy zone. The simulations are based on a mathematical model[3–6] of dendritic solidification, in which the mushy zone is treated as an anisotropic porous medium of variable porosity. At each node, the simulator solves for temperature, the fraction of liquid, the vel
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