Chemical non-equilibrium modelling of columnar solidification

This paper deals with the macroscopic modeling and numerical simulation of columnar dendritic solidification of binary alloys. The macroscopic governing equations and associated effective transport properties were previously derived using a volume averaging technique with local closure. The macroscopic model takes into account the spatial variation of the pore-scale geometry within the mushy zone, which leads to additional terms involving porosity gradients. The second important feature concerns solute mass conservation, which is described by considering a macro-scale non-equilibrium accounting for chemical exchanges at the solid–liquid interface. A simplified version of the model is validated through a comparison of the numerical solution to three experiments available in the literature. Porosity extra terms are systematically estimated on the basis of these numerical simulations, and the influence on solidification of effective transport properties such as permeability and interfacial solute exchange coefficients is investigated.

[1]  Michel Quintard,et al.  Macroscopic modeling of columnar dendritic solidification , 2004 .

[2]  M. Velarde,et al.  Momentum transport at a fluid–porous interface , 2003 .

[3]  S. Whitaker.  Flow in porous media I: A theoretical derivation of Darcy's law , 1986 .

[4]  Michel Quintard,et al.  Two-phase flow in heterogeneous porous media: The method of large-scale averaging , 1988 .

[5]  M. Rappaz,et al.  Numerical simulation of macrosegregation: a comparison between finite volume method and finite element method predictions and a confrontation with experiments , 1998 .

[6]  S. Whitaker,et al.  One- and Two-Equation Models for Transient Diffusion Processes in Two-Phase Systems , 1993 .

[7]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[8]  Frank P. Incropera,et al.  A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems. II: Application to solidification in a rectangular cavity , 1987 .

[9]  J. Hunt,et al.  Observations of ingot macrosegregation on model systems , 1974, Metallurgical and Materials Transactions B.

[10]  Christoph Beckermann,et al.  Formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel , 1995 .

[11]  Christoph Beckermann,et al.  Equiaxed dendritic solidification with convection: Part I. Multiscale/multiphase modeling , 1996 .

[12]  J. Ni,et al.  A volume-averaged two-phase model for transport phenomena during solidification , 1991 .

[13]  Pierre M. Adler,et al.  Taylor dispersion in porous media: analysis by multiple scale expansions , 1995 .

[14]  Michel Quintard,et al.  Local thermal equilibrium for transient heat conduction: theory and comparison with numerical experiments , 1995 .

[15]  Stephen Whitaker,et al.  Momentum transfer at the boundary between a porous medium and a homogeneous fluid-I. Theoretical development , 1995 .

[16]  Frank P. Incropera,et al.  Numerical simulation of a solidifying Pb-Sn alloy: The effects of cooling rate on thermosolutal convection and macrosegregation , 1991 .

[17]  Michel Quintard,et al.  Average momentum equation for interdendritic flow in a solidifying columnar mushy zone , 2002 .

[18]  D. R. Poirier,et al.  Flow of interdendritic liquid and permeability in pb-20 Wt Pct Sn alloys , 1985 .

[19]  D. R. Poirier,et al.  Permeability for flow parallel to primary dendrite arms , 1992 .

[20]  W. Kurz,et al.  Fundamentals of Solidification , 1990 .

[21]  V. Voller,et al.  The modelling of heat, mass and solute transport in solidification systems , 1989 .

[22]  S. Whitaker.  The Forchheimer equation: A theoretical development , 1996 .

[23]  Christoph Beckermann,et al.  Prediction of Columnar to Equiaxed Transition during Diffusion-Controlled Dendritic Alloy Solidification , 1994 .

[24]  D. R. Poirier.  Permeability for flow of interdendritic liquid in columnar-dendritic alloys , 1987 .

[25]  David R Poirier,et al.  The energy and solute conservation equations for dendritic solidification , 1991 .

[26]  Stephen Whitaker,et al.  Heat transfer at the boundary between a porous medium and a homogeneous fluid , 1997 .

[27]  J. Desbiolles,et al.  3-MOS: a general FEM code for the prediction of microstructures in castings , 1990 .

[28]  Christoph Beckermann,et al.  Modeling of micro- and macrosegregation and freckle formation in single-crystal nickel-base superalloy directional solidification , 1997 .

[29]  S. Torquato,et al.  Random Heterogeneous Materials: Microstructure and Macroscopic Properties , 2005 .

[30]  C. Beckermann,et al.  MULTIPHASE/-SCALE MODELING OF ALLOY SOLIDIFICATION , 1995 .

[31]  F. Incropera,et al.  A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems—I. Model formulation , 1987 .

[32]  Frank P. Incropera,et al.  Convective Transport Phenomena and Macrosegregation During Solidification of a Binary Metal Alloy: II—Experiments and Comparisons With Numerical Predictions , 1994 .

[33]  M. Quintard,et al.  Numerical calculation of the permeability in a dendritic mushy zone , 1999 .

[34]  S. Whitaker,et al.  Transport in ordered and disordered porous media: volume-averaged equations, closure problems, and comparison with experiment , 1993 .

[35]  H. Combeau,et al.  Numerical model for prediction of the final segregation pattern of bearing steel ingots , 1993 .

[36]  Chaoyang Wang,et al.  A multiphase solute diffusion model for dendritic alloy solidification , 1993, Metallurgical and Materials Transactions A.

[37]  S. Whitaker.  The method of volume averaging , 1998 .

[38]  Michel Quintard,et al.  Transport in ordered and disordered porous media II: Generalized volume averaging , 1994 .

[39]  V. Voller,et al.  A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems , 1987 .

[40]  Michel Quintard,et al.  Two-medium treatment of heat transfer in porous media: numerical results for effective properties , 1997 .

[41]  Christoph Beckermann,et al.  Simulation of convection and macrosegregation in a large steel ingot , 1999 .

[42]  Michel Quintard,et al.  Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment , 1994 .

[43]  D. R. Poirier,et al.  Conservation of mass and momentum for the flow of interdendritic liquid during solidification , 1990 .

[44]  Michel Quintard,et al.  Averaged Momentum Equation for Flow Through a Nonhomogenenous Porous Structure , 1997 .

[45]  R. W. Wright,et al.  CORE MELT PROGRESSION: STATUS OF CURRENT UNDERSTANDING AND PRINCIPAL UNCERTAINTIES , 1995 .

[46]  M. Quintard,et al.  Averaged solute transport during solidification of a binary mixture: Active dispersion in dendritic structures , 2002 .

[47]  V. Voller,et al.  Modeling of micro-macrosegregation in solidification processes , 1990 .

[48]  M. Quintard,et al.  Passive dispersion in dendritic structures , 2002 .