Macroscopic Model for Predicting Columnar to Equiaxed Transitions using Columnar Front Tracking and Average Equiaxed Growth

A macroscopic model of Columnar-to-Equiaxed Transition (CET) formation is presented. The growth of a columnar zone and an equiaxed zone are treated separately and modeled on a fixed grid. The model uses a columnar Front Tracking (FT) formulation to compute the motion of the columnar front and the solidification of the dendritic columnar mushy zone. The model for the equiaxed zone calculates the average growth of equiaxed grains within the control volumes of undercooled liquid. The proposed model, which calculates the average equiaxed growth, is different from previous FT models which consider each equiaxed grain envelope separately. A lognormal size distribution model of seed particles is used for the equiaxed nucleation in the undercooled liquid zone. After nucleation, average equiaxed growth is computed by considering the equiaxed envelopes as spherical. The extended volume concept is used to deal with grain impingement. The Scheil equation is used to calculate the solid fraction and latent heat release. When the equiaxed fraction is great enough, advancement of the columnar front is halted and the CET position is determined. CET formation was simulated for directional solidification of an aluminium-silicon alloy. The results were compared with a previous FT-CET prediction model as well as with experimental data. Agreement was found in both cases. Introduction When columnar and equiaxed zones co-exist in a casting, a distinguishable transition is called the Columnar-to-Equiaxed Transition (CET). Over the years much research attention has been given to analytical, experimental and computer modelling of CET. Recently, Spittle [1] reviewed CET analysis. In other work, McFadden and Browne [2,3] presented a Front Tracking (FT) model to predict CET in alloy casting. CET models have been proposed at various scales [3-9]. Mechanical blocking [4] or solutal blocking [5] mechanisms are mostly used to halt the columnar growth and to define the CET position. The model presented here follows the equiaxed heterogeneous nucleation in the liquid ahead of growing solid/liquid interface. As an initial development, solutal interactions near impingement are not considered. In the present work, a FT model presented by Browne and Hunt [10, 11], which was later extended by McFadden and Browne [3], is used to compute the growth of the columnar zone. The proposed model does not track crystallographic or microscopic details for each dendrite, thus reducing the computational overhead considerably. A model, presented by Jacot et al.[8] has common features with this model in terms of growth and impingement of equiaxed grains. This volume-averaging approach to equiaxed solidification is being developed here because it will facilitate the integration of the effects of convection into the model, at low computational overhead, in comparison to the need to track flow between and motion of the individual equiaxed grains directly simulated in [3]. The aim of the current contribution is verification and validation of the new model of equiaxed nucleation, growth and impingement and CET criteria in diffusion-controlled solidification. The Model The FT model is used to simulate the advancement of the columnar zone and the development of undercooling in the liquid zone ahead of it. Four different zones can be identified (fig.1): solid, columnar mush, equiaxed mush in undercooled liquid, and superheated liquid. Fig.1 Solid, columnar mush, equiaxed mush in undercooled liquid and superheated liquid. The heat equation, equation (1) is explicitly solved in a fixed grid and a control volume (CV) approach is used for model calculations.