Project topics and materials
Get Complete Project Material File(s) Now! »
Macrosegregation in Alloy Solidification
Segregation at Microscopic Scale
A typical equilibrium phase diagram for a simple binary alloy A-B at constant pressure is shown in Fig.1. 4. Nucleation of solid crystals occurs at a temperature slightly below the liquidus temperature of an alloy at a given composition. Due to the partition of chemical species during the phase transformation, in most cases (with a solute partition coefficient less than unity), the solid phase is formed at a smaller composition than the average value. The rejection of solute from the solid to liquid leads to an increase of the concentration in the liquid phase. There exists a mushy zone consisting simultaneously of the solid and liquid phases over a temperature range.
Under equilibrium and well-mixed conditions that are often assumed for simplicity, the evolution of concentrations in the solid and liquid phases follows the solidus and liquidus lines, respectively, up to their maximum solubility at the eutectic temperature.
However, accounting for limited solute diffusion was demonstrated to be important in order to correctly model the formation and development of solidification structures. In fact, mass diffusivity being smaller than thermal diffusivity in metal alloys, solidification at the microscopic scale is principally governed by solute concentration gradients on each side of the solid-liquid interface. Due to a limited solute diffusion in the solid and liquid phases, rejected species from the solid phase are accumulated at the solid-liquid interface and then diffused into the liquid phase. So there exists gradients of composition in the solid and liquid phases as schematized in Fig.1. 5.
In turn, the morphology of microstructures has a crucial impact on the solute diffusion flux. Solidification microstructures can be divided into columnar (constrained-growth) and equiaxed (unconstrained-growth) structures. In columnar solidification, most of the solute is rejected principally in the lateral direction perpendicular to the heat flow. While in the case of equiaxed crystals that are surrounded by undercooling melt, rejection of solute takes place in all directions on their solid-liquid interface and eventually for an accumulation layer outside the grains. For each structural type, solid grains can grow with more or less dendritic shape, corresponding to different growth and solidification kinetics at the soli-liquid interface. The formed structures and the grain morphology depend on many factors such as phase change conditions, composition, thermodynamic properties of phase transformations, impacts of phase convection, etc. Various subjects related to the nucleation and growth of solidification structures can be consulted in [Kurz and Fisher, 1989] [Dantzig and Rappaz, 2009].
Advanced solidification models are able to account for the effect of convection on solute gradients in a small layer near the solid-liquid interface as well as for complex shapes of interfacial structures that are associated to instability conditions and curvatures of interfaces. Studies of these topics can be found in [Flemings, 1974] [Kurz and Fisher, 1989] [Langer, 1989] [Pines et al., 1990] [Wang and Beckermann, 1993][Martorano et al., 2003].
Due to the movement of the liquid and solid phases during the solidification process, the segregation of chemical components at the scale of solid-liquid interface is manifested on a system scale, the so-called macrosegregation.
Generally, melt flow can be generated by many sources such as external forces, surface tension gradients, residual flow due to filling of the mold, buoyancy forces due to temperature and compositional gradients, drag forces from solid motion, etc. In the present work, the liquid movement is governed by the two later listed mechanisms. The thermal and solutal buoyancy forces can oppose or corporate to each other, depending on heat exchange conditions and on the weight of chemical species rejected in comparison with that of the bulk liquid. Additionally, during solidification these effects can be enhanced or weakened by the solid motion, according to the orientation of these forces relative to the gravity direction.
In consideration of the solid phase, its movement can take place within a small distance, for instance deformation-induced displacements in continuous casting processes. The solid motion also manifests over large domains such as the settling and floating of equiaxed grains in ingot castings. Relating to the later mechanism, its influence on macrosegregation was experimentally observed for different solidified alloys, i.e. undercooled Pb-Sn eutectic alloys [de Groh III, 1994] and aluminum alloys (Al-1wt%Cu, Al-10wt%Cu with different amounts of grain refiner) [Rerko et al., 2003]. Settling of equiaxed crystals is also known as the principal factor causing the negative segregation cone at the bottom casting in Fe-C alloys solidification [Flemings, 1974]. The origin of the equiaxed crystals has not been entirely clear. Two principal formation mechanisms were proposed including: heterogenous nucleation and detachment and transport of dendrite arms initially formed in a columnar zone.
It can be seen that macrosegregation involves physically different multi-scale phenomena. Consequently, in order to predict macrosegregation, it is necessary to simultaneously model the microscopic growth processes and the macroscopic happenings such as heat transfer, mass transport, phases advection… From the numerical point of view, accounting for the solid motion requires dealing with a complex mathematical system because of its close interaction with other phenomena.
Objectives and Outline
In the aim of modeling macrosegregation with ultimate applications to industrial castings of large sizes or under complex geometries, developing FE solidification model would expand the capacity of numerical predictions, especially being promising for such industrial applications.
With interest in successes achieved from the FV solidification model developed by Založnik and Combeau [Založnik and Combeau, 2010a, 2010b] in simulating macrosegregation, in particular for heavy ingots as presented above, our study’s objective is to adapt and implement this model into the FE framework on which the 3D simulating program Thercast® is developed, in order that it is able to model the equiaxed transport during solidification.
Following this chapter in which main physical phenomena related to macrosegregation have been introduced,
• a review of solidification models will be presented in Chapter 2, consisting of the multi-scale modeling, volume-averaged method, governing equations used to describe phenomena during solidification and volume-averaged solidification models accounting for solid motion.
• Finite element implementation and adaptation will be detailed in Chapter 3, in which the resolutions for macro-micro equations are established by using the splitting scheme.
• Chapter 4 will present two-dimensional numerical results obtained from the current model that will be compared to references and analyzed for the purpose of verifying and validating our work, taking in first for purely growth process, then with melt convection, then with only transport phenomena and finally solidification cases using the complete growth-transport model.
• Three-dimensional simulations and industrial applications are the subject of Chapter 5.
• Conclusions and perspectives are presented at the end of the report.
Résumé en français
Afin de modéliser la macroségrégation pour des produits industriels de grandes tailles ou de géométries complexes, un modèle de solidification par éléments finis sera développé. Une modélisation par éléments finis peut permettre d’étendre de façon prometteuse la capacité de prédictions numériques du phénomène de macroségrégation dans les applications industrielles.
Au vue de l’intérêt du modèle de volumes finis développé par Založnik et Combeau [Založnik and Combeau, 2010a, 2010b] pour la simulation de la macroségrégation, en particulier pour de larges lingots d’acier, notre étude a pour but d’adapter et de mettre en œuvre ce modèle dans une formulation de type éléments finis utilisée dans le logiciel industriel Thercast®, afin qu’il soit capable de modéliser la solidification en présence du transport des grains solides.
Suite à ce chapitre où les principaux phénomènes physiques liés à la macroségrégation ont été introduits,
• une revue bibliographique des modèles de solidification sera présentée dans le chapitre 2, composée de quatre parties : Modélisation multi-échelle ; Méthode des prises de moyenne ; Ensemble d’équations gouvernantes ; Modèles de solidification utilisant la technique de prise de moyenne avec la présence du transport des cristaux.
• La mise en œuvre et l’adaptation numérique seront détaillées dans le chapitre 3, dans lequel la résolution des équations macro-micro est établie en utilisant le schéma de splitting.
• Le chapitre 4 est consacré à présenter et à analyser des résultats numériques obtenus par le modèle actuel dans une configuration bidimensionnelle, en les comparant aux solutions de référence.
• Des simulations tridimensionnelles et des applications industrielles sont menées au chapitre 5.
• Les conclusions et les perspectives seront portées à la fin du manuscrit.
Table of contents :
Chapter 1 – Introduction
1.1 Motivation
1.2 Macrosegregation in Alloy Solidification
1.2.1 Segregation at Microscopic Scale
1.2.2 Segregation at Macroscopic Scale
1.3 Objectives and Outline
1.4 Résumé en français
Chapter 2 – Literature Reviews
2.1 Multi-scale Modeling
2.2 Volume-Averaged Method
2.3 Conservation Equations
2.4 Volume-Averaged Solidification Models accounting for Solid Transport
2.4.1 Models of Beckermann and co-workers
2.4.2 Models of Wu and co-workers
2.4.3 Models of Combeau and co-workers (SOLID software (*))
2.4.4 Models at CEMEF
2.5 Summary
2.6 Résumé en français
Chapter 3 – Finite Element Implementation and Adaptation
3.1 Two-Phase Solidification Model
3.1.1 Modeling of Macroscopic Transport Phenomena with FEM
3.1.1.1 Pure Transport Equations and Artificial Diffusion
3.1.1.2 Energy Equation
3.1.1.3 Momentum Equations
3.1.2 Modeling of Microscopic Processes
3.1.3 Coupling between Microscopic and Macroscopic Scales
3.2 Extension to a Three-Phase Model
3.3 Summary
3.4 Résumé en francais
Chapter 4 – Numerical Simulation and Validation
4.1 Mono-dimensional Solidification
4.1.1 Test Case Description
4.1.2 Modeling of Well Mixed and Partially Mixed Solute Diffusions
4.2 Thermo-solutal Liquid Convection during Solidification
4.2.1 Test Cases Description
4.2.2 “Infinite” Solute Diffusion at the Microscopic Scale
4.2.3 Limited Solute Diffusion at the Microscopic Scale
4.3 Purely Convective Transport during Sedimentation
4.3.1 Test Case Description
4.3.2 Analyses and Coherency Verification
4.3.3 Effects of Artificial Diffusion
4.4 Complete Solidification Model
4.4.1 Simulation Results and Analyses
4.4.2 Effects of Artificial Diffusion
4.5 Dendritic Solidification Modeling
4.5.1 Purely Diffusive Solidification
4.5.2 Complete Growth-Transport Solidification
4.6 Summary
4.7 Résumé en français
Chapter 5 – Tests 3D et Applications Industrielles
5.1 Tests 3D (Benchmark de Hebditch-Hunt)
5.1.1 Etude sur une pièce mince de 1 mm d’épaisseur avec deux plans de symétri
5.1.2 Etude du cas réel en simulant la moitié de la cavité
5.2 Applications Industrielles
5.2.1 Configurations des simulations
5.2.2 Etude sur le lingot en configuration plane cartésienne 2D
5.2.3 Etude sur le lingot cylindrique 3D
5.3 Résumé
Chapter 6 – Conclusions and Perspectives
Bibliography
Models for the Solute Diffusion Lengths
Data and Simulation Parameters
