The formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel is simulated by simultaneously solving macroscopic mass, momentum, energy, and species conservation equations with full coupling of the temperature and concentrations through thermodynamic equilibrium at the solid/liquid interface. The flow field, solid fraction evolution, and macrosegregation patterns for four cases are presented. The results show both the formation of channel segregates and the formation of islands of mush surrounded by bulk melt. In examining the solidification of a ten-element steel, the global extent of macrosegregation of an element is found to be linearly dependent on its partition coefficient (more severe segregation for small partition coefficient), although such scaling is not possible locally. Results for the solidification of a binary Fe-C alloy (with the same carbon content as the ten-element alloy) are similar to those for the ten-element alloy due solely to the large contribution of carbon to buoyancy driven flow in the ten-element steel chosen for study. While including only those elements that make significant contributions to buoyancy driven flow reproduces the global extent of macrosegregation seen in the ten-element alloy, local differences in the predictions are visible. Finally, comparison of results for the solidification of the same ten-element steel using two different sets of data to describe the partition coefficients and change in liquidus temperature with concentration of the elements shows completely opposite behavior,i.e., upward flow through the mushy zone for one case and downward flow for the other. Thus, the need to have accurate phase-equilibrium data when modeling multicomponent macrosegregation is illustrated. Together, the results give an indication of what areas require more careful examination if accurate modeling of multicomponent solidification is to be accomplished.

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Formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel

Mathematical modeling of mass, momentum, heat, and species transport phenomena occurring during solidification of metal alloys is reviewed. Emphasis is placed on the incorporation of the effects of the solid structure and the interactions between the solid and liquid phases on a microscopic scale into a (macroscopic) model of the transport phenomena occurring at the system scale. Both columnar and equiaxed growth structures, as well as laminar convection of liquid and solid crystals are considered. The macroscopic conservation equations are introduced via a volume averaging approach and commonly made simplifications are examined. Basic constitutive relations for the phase interactions occurring in alloy solidification are presented. Recent progress in including nucleation, microsegregation, undercooling and other microscopic phenomena in the macroscopic equations is reviewed. The specific areas where future theoretical and experimental research is needed are identified.

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Mathematical Modeling of Transport Phenomena During Alloy Solidification

Micro-macrosegregation calculations have been performed for a rectangular cavity containing either a Pb-48 wt pct Sn alloy or a Sn-5 wt pct Pb alloy. The numerical results calculated with a finite volume method (FVM) and a finite element method (FEM) are compared with experimental results previously obtained by Hebditch and Hunt. The two methods are based on the same average conservation equations governing heat and mass transfer and the same assumptions: lever rule, equal and constant density of the solid and liquid phases (except in the buoyancy term), permeability of the mushy zone given by the Carman-Kozeny relation, and no transport of the solid phase. Although the same parameters are used in both calculations, small differences are observed as a result of the different formulations. In particular, the instabilities appearing in the mushy zone (channels) of the Sn-5 wt pct Pb alloy are more pronounced with the FVM formulation as compared with FEM, whereas the opposite trend is observed for the Pb-48 wt pct Sn alloy. Nevertheless, the final segregation maps at the end of solidification compare fairly well with the experimental findings.

Numerical simulation of macrosegregation: a comparison between finite volume method and finite element method predictions and a confrontation with experiments

A numerical study of the combined effects of microsegregation, mushy zone permeability and fllow, caused by volume contraction and thermosolutal convection, on macrosegregation and eutectic formation in binary alloy solidification

Extension of the continuum model for transport phenomena occurring during metal alloy solidification—I. The conservation equations

Numerical simulations of a binary metal alloy (Pb-Sn) undergoing solidification phase change are performed using a continuum model for conservation of total mass, momentum, energy, and species. The system is contained in an axisymmetric, annular mold which is cooled along its outer vertical wall. Results show that thermosolutal convection in the melt and mushy zones is strongly coupled and that macrosegregation is reduced with increased cooling rate. For low cooling rates, solutally induced convection in the mushy zone favors the development of channels, which subsequently spawn macrosegregation in the form of A-segregates. With increasing solidification rate, however, thermosolutal interactions in the melt contribute to reducing the formation of channels and A-segregates.

P. J. Prescott

Papers

Numerical simulation of a solidifying Pb-Sn alloy: The effects of cooling rate on thermosolutal convection and macrosegregation