## Mikro-makroskopowy model krzepnięia roztworu dwuskładnikowego

### Mirosław Seredyński

#### Abstract

This Ph.D. Thesis embarks upon difficulties related to efficient modelling of multiscale and multi-phase transport processes of mass, momentum, energy and species concentration accompanying the liquid-solid phase transition in a binary metal alloy. To account for all underlying phenomena, which directly influence a grain structure of a final casting product, one should develop a numerical model and perform calculations at least at a microscopic scale, i.e., at a size of individual crystals (or even at a finer one). On this micro-scale level, a main challenge involves precise modelling of very complex shapes of dendrite boundaries, on which balances of the quantities transferred between the liquid and solid phases must be directly included. Although several sophisticated mathematical methods have been successfully developed for this purpose over the last two decades (chapter 3.2.1), they are computationally very intensive, and therefore their applications are restricted to modelling the solidification of a single crystal or a small set of grains. To circumvent this problem, the micro-macroscopic approach has been developed (chapter 3.2.2), where key calculations are performed on the level of macroscopic conservation equations of mass, momentum, energy and species (chapter 5.3), obtained on the basis of the volume or stochastic averaging of the relevant microscopic equations (chapters 3.2.2, 5.1, 5.2, 5.3). And, special algebraic and/or experimental models are used to include information on a microscopic structure developing in the mushy zone, and on the phenomena occurring at this scale, into the macroscopic model - through both closing constitutive relations and effective material properties of the two-phase mixture (chapter 5.5). One of the main difficulties, that one encounters in the micro-macroscopic approach to modelling of binary alloy solidification involves proper numerical simulation of transport phenomena in the two phase region (mushy zone), developing between the solid and the liquid phases. This region may consist of two different grain structures, i.e. the matrix of motionless columnar dendrites immersed in the inter-dendritic liquid and the zone of equiaxed grains moving in the under-cooled melt (chapter 2). And, due to different mechanisms of momentum transfer occurring in these two dendritic structures, the recognition of the zones occupied by each of them is indispensable at any moment of the solidification process development, in order to properly model the convective diffusive transfer of momentum. Unfortunately, commonly used, nowadays, methods of the identification of columnar dendrites and equiaxed grain zones, on the macroscopic level (chapter 3.2.2.4), provide imprecise models, where specific mechanisms of heat, momentum and species transport in the equiaxed zone are not included (classical enthalpy –porosity method), or they are based on arbitrary assumed constant values of basic parameters of the model, such as the coherency point and the coefficient of switching function in the hybrid method (chapter 7.1). Therefore, the main objective of this thesis was to develop a new micro-macroscopic simulation model for binary alloy solidification, with a more precise, free from the above mentioned drawbacks, numerical technique for the recognition of regions of the two different dendritic structures developing within the mushy zone. The model developed is based on the theory of mixtures and the classical enthalpy formulation of the energy conservation equation, coupled with a special technique of tracking the interface between different dendrite structures on a macroscopic fixed grid of control volumes. The model is referred to as EP-FT (Enthalpy-Porosity & Front Tracking method). The border separating different grain structures is defined as a curve (or a surface in 3D) joining all columnar dendrite tips, which move in the under-cooled liquid according to the known, from theory or experiments, law of a crystal growth. Further assuming that this dendrite tip kinetics can be used to determine velocities of mass-less marker particles, used to represent temporal and local shapes and positions of the interface, one can distinguish the columnar dendrite zone from the equiaxed grain region developing within the mushy zone. In each of such identified parts of the two-phase region different representation of the momentum transfer is used, i.e.: the model of an anisotropic porous medium is applied in the region where the matrix of stationary columnar grains is formed, and the model of slurry is used in the zone of under-cooled liquid with floating equiaxed grains. The computer simulation model, corresponding to the above described mathematical model, has been obtained on the basis of the control-volume finite difference method. And, an original algorithm of a new, grid independent, front tracking technique has been developed, where the front is treated as an object consisting of repeatable line segments or triangles, respectively, in 2D and 3D geometry, and dynamic data structures along with the recurrence algorithm are used for the data access and the front shape modification (chapter 5.5.10 and Appendix). In order to perform calculations, home-made computer codes have been written for both: the EP-FT method and the hybrid method. Hierarchical verification procedure of both: the developed computer code correctness and the accuracy of the proposed simulation model has been performed, through comparisons between the EP-FT predictions and other available numerical solutions for the selected pertinent benchmark problems, and through the grid refinement study (chapter 6). They have proved a satisfactory accuracy of the EP-FT model and the correctness of its computer implementation. The next conducted validation analysis, based on comparisons of the predicted species concentration with the corresponding experimental findings, available in the literature, for the Pb-48 % mass Sn solidification in a rectangular mould, has confirmed an acceptable match between the calculated and experimental results. Further comparisons between the EP-FT results and the hybrid method predictions, for the selected problem of Pb-48 % mass Sn solidification in a rectangular mould, have been carried out in order to get the answer to the question of the influence of precision in the recognition of different dendrite structures on the obtained flow pattern and macrosegregation pictures (chapter 7.2). They have confirmed that the assumption of a constant coherency point value at any time of solidification - an underlying postulation of the hybrid method - is over-simplified because the solid volume fraction changes along the front of dendrite tips as well as in time when diverse crystal structures are developing. Moreover, results obtained from the hybrid method for arbitrary assumed different values of the coherency point have shown that the front separating the columnar and equiaxed dendrite zones (calculated by EP-FT model) always runs between two close lines of constant solid volume fractions, although values of the solid volume fraction change on these two isolines when the solidification process proceeds. This can be viewed as a further positive verification of the EP-FT model results. The proposed micro-macroscopic model for computer simulation of binary alloy solidification has been used for detailed analysis of the influence of various parameters of the closing microscopic models on numerically predicted macroscopic fields of velocity, temperature, species concentration and volumetric solid fraction in the two-phase mixture of a solidifying alloy (chapter 8). In particular, the problem is addressed of the impact of the permeability and anisotropy of the columnar dendrite mush (chapter 8.1) and of the chosen law of crystal growth (chapter 8.2) on the predicted fluid flow and concentration pictures. It has been shown that the degree of species segregation increases along with the growth of permeability of the columnar dendritic zone, and the anisotropy of this mush can cause evolution of the A-segregation channels – the phenomenon also observed in experiments (chapter 3.1). Moreover, the performed EP-FT calculations have revealed that the chosen model of dendrite tip kinetics has a small impact on the developing macroscopic fields of solid volume fraction and species concentration. But, visible differences have been observed in the cooling rates, in the equilibrium liquidus temperatures and in the volumetric solid fraction changes along the front, for different models of crystal growth (chapter 8.2). They can play an important role in precise modelling the event of blocking the growth of columnar dendrites by the equiaxed grains – the phenomenon known as the Columnar-to-Equiaxed Transition (CET, chapter 2). Having in mind the future applications of the proposed EP-FT model and its computer code in calculations of solidification of real casting products, having complex geometrical shapes, the method and its computer implementation have been extended to 3D geometry. In particular, the effective algorithms for tracking and approximation of curvilinear surfaces separating different dendritic structures have been elaborated (Appendix). The correctness of the 3D version of the EP-FT method has been verified by solving two selected test problems of solidification of Pb-48 % mass Sn in different 3D mould geometries and for different cooling conditions. The dissertation is summarized with a short discussion on future research concerning the further development and improvement of the micro-macroscopic modelling of binary alloy solidification (chapter 10). In particular, the need is emphasized for: (1) – more precise description of the slurry zone of equiaxed crystals by taking into account different phase densities and the resulting phenomenon of grains sedimentation; (2) - the elaboration of more enhanced models of porous medium imitating the columnar mush by accounting for time and spatial changes of the microstructure parameters; (3) – the replacing of classical algebraic models of micro-diffusion by more general description of species diffusion in each of the two phases and on their common interface; (4) – the development of a new dendrite growth kinetics involving the role of concurrent thermal and solutal convection. Further work is also needed to extend the front tracking technique on more general 2D triangular and 3D tetrahedral grids of control volumes or finite elements, in order to more precisely represent complex curvilinear geometry of real casting products. Finally, a more detailed validation analysis is desirable for the closing microscopic models used in the EP-FT method. A new dynamically developing laboratory visualization technique, i.e. the synchrotron x-ray video microscopy, whose time resolution allows observation of phenomena on the level of individual grains, seems to provide the promising tool for that purpose.Diploma type | Doctor of Philosophy | ||||

Author |
Mirosław Seredyński (FPAE / IHE)
Mirosław Seredyński
| ||||

Title in Polish | Mikro-makroskopowy model krzepnięia roztworu dwuskładnikowego | ||||

Language | pl polski | ||||

Certifying Unit | Faculty of Power and Aeronautical Engineering (FPAE) | ||||

Discipline | mechanics / (technology domain) / (technological sciences) | ||||

Start date | 04-07-2007 | ||||

Defense Date | 22-03-2010 | ||||

End date | 13-04-2010 | ||||

Supervisor |
Jerzy Banaszek (FPAE / IHE)
Jerzy Banaszek
| ||||

Pages | 201 | ||||

Keywords in English | x | ||||

Abstract in English | This Ph.D. Thesis embarks upon difficulties related to efficient modelling of multiscale and multi-phase transport processes of mass, momentum, energy and species concentration accompanying the liquid-solid phase transition in a binary metal alloy. To account for all underlying phenomena, which directly influence a grain structure of a final casting product, one should develop a numerical model and perform calculations at least at a microscopic scale, i.e., at a size of individual crystals (or even at a finer one). On this micro-scale level, a main challenge involves precise modelling of very complex shapes of dendrite boundaries, on which balances of the quantities transferred between the liquid and solid phases must be directly included. Although several sophisticated mathematical methods have been successfully developed for this purpose over the last two decades (chapter 3.2.1), they are computationally very intensive, and therefore their applications are restricted to modelling the solidification of a single crystal or a small set of grains. To circumvent this problem, the micro-macroscopic approach has been developed (chapter 3.2.2), where key calculations are performed on the level of macroscopic conservation equations of mass, momentum, energy and species (chapter 5.3), obtained on the basis of the volume or stochastic averaging of the relevant microscopic equations (chapters 3.2.2, 5.1, 5.2, 5.3). And, special algebraic and/or experimental models are used to include information on a microscopic structure developing in the mushy zone, and on the phenomena occurring at this scale, into the macroscopic model - through both closing constitutive relations and effective material properties of the two-phase mixture (chapter 5.5). One of the main difficulties, that one encounters in the micro-macroscopic approach to modelling of binary alloy solidification involves proper numerical simulation of transport phenomena in the two phase region (mushy zone), developing between the solid and the liquid phases. This region may consist of two different grain structures, i.e. the matrix of motionless columnar dendrites immersed in the inter-dendritic liquid and the zone of equiaxed grains moving in the under-cooled melt (chapter 2). And, due to different mechanisms of momentum transfer occurring in these two dendritic structures, the recognition of the zones occupied by each of them is indispensable at any moment of the solidification process development, in order to properly model the convective diffusive transfer of momentum. Unfortunately, commonly used, nowadays, methods of the identification of columnar dendrites and equiaxed grain zones, on the macroscopic level (chapter 3.2.2.4), provide imprecise models, where specific mechanisms of heat, momentum and species transport in the equiaxed zone are not included (classical enthalpy –porosity method), or they are based on arbitrary assumed constant values of basic parameters of the model, such as the coherency point and the coefficient of switching function in the hybrid method (chapter 7.1). Therefore, the main objective of this thesis was to develop a new micro-macroscopic simulation model for binary alloy solidification, with a more precise, free from the above mentioned drawbacks, numerical technique for the recognition of regions of the two different dendritic structures developing within the mushy zone. The model developed is based on the theory of mixtures and the classical enthalpy formulation of the energy conservation equation, coupled with a special technique of tracking the interface between different dendrite structures on a macroscopic fixed grid of control volumes. The model is referred to as EP-FT (Enthalpy-Porosity & Front Tracking method). The border separating different grain structures is defined as a curve (or a surface in 3D) joining all columnar dendrite tips, which move in the under-cooled liquid according to the known, from theory or experiments, law of a crystal growth. Further assuming that this dendrite tip kinetics can be used to determine velocities of mass-less marker particles, used to represent temporal and local shapes and positions of the interface, one can distinguish the columnar dendrite zone from the equiaxed grain region developing within the mushy zone. In each of such identified parts of the two-phase region different representation of the momentum transfer is used, i.e.: the model of an anisotropic porous medium is applied in the region where the matrix of stationary columnar grains is formed, and the model of slurry is used in the zone of under-cooled liquid with floating equiaxed grains. The computer simulation model, corresponding to the above described mathematical model, has been obtained on the basis of the control-volume finite difference method. And, an original algorithm of a new, grid independent, front tracking technique has been developed, where the front is treated as an object consisting of repeatable line segments or triangles, respectively, in 2D and 3D geometry, and dynamic data structures along with the recurrence algorithm are used for the data access and the front shape modification (chapter 5.5.10 and Appendix). In order to perform calculations, home-made computer codes have been written for both: the EP-FT method and the hybrid method. Hierarchical verification procedure of both: the developed computer code correctness and the accuracy of the proposed simulation model has been performed, through comparisons between the EP-FT predictions and other available numerical solutions for the selected pertinent benchmark problems, and through the grid refinement study (chapter 6). They have proved a satisfactory accuracy of the EP-FT model and the correctness of its computer implementation. The next conducted validation analysis, based on comparisons of the predicted species concentration with the corresponding experimental findings, available in the literature, for the Pb-48 % mass Sn solidification in a rectangular mould, has confirmed an acceptable match between the calculated and experimental results. Further comparisons between the EP-FT results and the hybrid method predictions, for the selected problem of Pb-48 % mass Sn solidification in a rectangular mould, have been carried out in order to get the answer to the question of the influence of precision in the recognition of different dendrite structures on the obtained flow pattern and macrosegregation pictures (chapter 7.2). They have confirmed that the assumption of a constant coherency point value at any time of solidification - an underlying postulation of the hybrid method - is over-simplified because the solid volume fraction changes along the front of dendrite tips as well as in time when diverse crystal structures are developing. Moreover, results obtained from the hybrid method for arbitrary assumed different values of the coherency point have shown that the front separating the columnar and equiaxed dendrite zones (calculated by EP-FT model) always runs between two close lines of constant solid volume fractions, although values of the solid volume fraction change on these two isolines when the solidification process proceeds. This can be viewed as a further positive verification of the EP-FT model results. The proposed micro-macroscopic model for computer simulation of binary alloy solidification has been used for detailed analysis of the influence of various parameters of the closing microscopic models on numerically predicted macroscopic fields of velocity, temperature, species concentration and volumetric solid fraction in the two-phase mixture of a solidifying alloy (chapter 8). In particular, the problem is addressed of the impact of the permeability and anisotropy of the columnar dendrite mush (chapter 8.1) and of the chosen law of crystal growth (chapter 8.2) on the predicted fluid flow and concentration pictures. It has been shown that the degree of species segregation increases along with the growth of permeability of the columnar dendritic zone, and the anisotropy of this mush can cause evolution of the A-segregation channels – the phenomenon also observed in experiments (chapter 3.1). Moreover, the performed EP-FT calculations have revealed that the chosen model of dendrite tip kinetics has a small impact on the developing macroscopic fields of solid volume fraction and species concentration. But, visible differences have been observed in the cooling rates, in the equilibrium liquidus temperatures and in the volumetric solid fraction changes along the front, for different models of crystal growth (chapter 8.2). They can play an important role in precise modelling the event of blocking the growth of columnar dendrites by the equiaxed grains – the phenomenon known as the Columnar-to-Equiaxed Transition (CET, chapter 2). Having in mind the future applications of the proposed EP-FT model and its computer code in calculations of solidification of real casting products, having complex geometrical shapes, the method and its computer implementation have been extended to 3D geometry. In particular, the effective algorithms for tracking and approximation of curvilinear surfaces separating different dendritic structures have been elaborated (Appendix). The correctness of the 3D version of the EP-FT method has been verified by solving two selected test problems of solidification of Pb-48 % mass Sn in different 3D mould geometries and for different cooling conditions. The dissertation is summarized with a short discussion on future research concerning the further development and improvement of the micro-macroscopic modelling of binary alloy solidification (chapter 10). In particular, the need is emphasized for: (1) – more precise description of the slurry zone of equiaxed crystals by taking into account different phase densities and the resulting phenomenon of grains sedimentation; (2) - the elaboration of more enhanced models of porous medium imitating the columnar mush by accounting for time and spatial changes of the microstructure parameters; (3) – the replacing of classical algebraic models of micro-diffusion by more general description of species diffusion in each of the two phases and on their common interface; (4) – the development of a new dendrite growth kinetics involving the role of concurrent thermal and solutal convection. Further work is also needed to extend the front tracking technique on more general 2D triangular and 3D tetrahedral grids of control volumes or finite elements, in order to more precisely represent complex curvilinear geometry of real casting products. Finally, a more detailed validation analysis is desirable for the closing microscopic models used in the EP-FT method. A new dynamically developing laboratory visualization technique, i.e. the synchrotron x-ray video microscopy, whose time resolution allows observation of phenomena on the level of individual grains, seems to provide the promising tool for that purpose. | ||||

Thesis file |
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Citation count* | 5 (2020-09-18) |

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