A three-phase model for mixed columnar-equiaxed solidification
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e two typical macrostructures in metal castings: columnar dendrites and equiaxed grains. To model mixed columnar-equiaxed solidification, it is necessary to consider the competitive growth of the columnar dendrites and the equiaxed grains in combination with melt convection and grain transport. An important feature in describing the mixed columnar-equiaxed solidification is the so-called columnar-to-equiaxed-transition (CET), to which great attention has been paid in the last decades.[1–5] A widespread opinion is that the CET is induced by the competitive growth of the columnar dendrites and equiaxed grains, and thus, most research work has been focused on this. The columnar tips are either blocked mechanically by the presence of equiaxed grains ahead of the columnar front, which is known as the ‘‘hard-blocking’’ mechanism,[1] or they are blocked by the disappearance of local constitutional undercooling, which is known as the ‘‘soft-blocking’’ mechanism.[4,5] Although the role of melt convection and grain transport in macrostructure formation has since long been known,[6] the study of grain transport and its influence on the CET is unfortunately very rare.[3] Obviously, mixed columnar-equiaxed solidification is a typical multiphase problem. This article reports about the extension of a two-phase model for describing the equiaxed globular solidification published earlier by the authors,[7,8] to a three-phase model by including an additional columnar phase. Both the columnar-equiaxed competitive growth and the grain rearrangement due to movement of the equiaxed MENGHUAI WU, Group Leader of Numerical Modeling and Simulation, and ANDREAS LUDWIG, Professor and Director, are with Simulation and Modeling of Metallurgical Processes, ChristianDoppler-Laboratory for Multiphase Modeling of Metallurgical Processes, Department of Metallurgy, University of Leoben, A8700 Leoben, Austria. Contact e-mail: [email protected] Manuscript submitted March 13, 2005. METALLURGICAL AND MATERIALS TRANSACTIONS
grains are taken into account. The origin of the equiaxed grains is thought to be heterogeneous nucleation. Dendrite fragmentation, although sometimes believed to play an important role in the mixed columnar-equiaxed solidification, is not included. Simulation results for a binary steel benchmark ingot (Fe-0.34 wt pct C) with a two dimensional (2-D) axis symmetrical and a three-dimensional (3-D) geometry are presented to demonstrate the potential of the model. Additionally, some one-dimensional (1-D) cases are simulated and evaluated by comparing them to classical analytical solutions. The limitations of the recent model and necessary further improvements are also discussed. II.
MODEL DESCRIPTION
A. General Assumptions (1) Three phases are defined: the primary liquid phase (l), the equiaxed phase as the first secondary phase (e), and the columnar as the second secondary phase (c). The corresponding phase fraction is given by fl, fe, and fc with fl 1 fe 1 fc 5 1. Both the primary and equiaxed phases are moving phases, for which
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