Simulation Of Binary Alloy Solidification Research Articles

A 2D Numerical Simulation of Binary Alloy Solidification: Effect of Mesh Resolution on Formation of Channel Segregates

Mesh refinement is crucial for capturing the complex phenomena that governs the formation of channel segregates during binary alloy solidification. In this article, the influence of mesh size on the formation of channel segregates during the solidification of Sn-5wt%Pb alloy is numerically investigated. A solver is developed in OpenFOAM for solving the coupled transport equations of mass, momentum, energy and species. Subsequently, the simulations are performed for different mesh sizes to predict the flow field, temperature, species and solid fraction distribution including the morphology of channel segregates. From this study, it is observed that the mesh size significantly affects the morphology and the strength of channel segregates. For very fine mesh size, having sufficient number of grid point along their width, the formed channels are more continuous and the flow inside channels is resolved.

The two-domain model of solute transport in binary alloy

A mixed model for micro-macroscopic computer simulation of binary alloy solidification is proposed. It involves a two-domain approach to solute conservation equations in the liquid and solid phases, whereas transport of momentum and energy in the two-phase region is modelled using the phase mixture theory. To distinguish regions of columnar and equiaxed crystal structures evolving in a cast during solidification, the special front tracking technique on non-structural triangular grids is included in the model. In this two-domain approach, solute conservation equations are averaged across solid and liquid phases, and the solute transport at the phase interface is included. Additionally, the microstructure evolution is modelled to capture the development of various complex grain structures and more accurately describe the solute transport between the phases. The accuracy of the proposed model is first verified by a grid refinement analysis, and then the model is used to predict the solute concentration and macro-segregation in the example problem of Pb-48%wt Sn alloy solidification in a 2D mould. The results obtained are next compared with the relevant ones predicted by the fully single-domain model, earlier developed by authors. Thus, the role of finite diffusion in liquid and solid phases is identified and discussed.

Influence of crystal growth kinetics on prediction of macro segregation by micro-macroscopic simulation of binary alloy solidification

Using the micro-macroscopic computer simulation model of binary alloy solidification, based on a single domain control-volume calculations, and involving the tracking of columnar dendrite tips envelope for distinguishing different grain structures within the two-phase liquid-solid region, the paper reports the outcomes of the detailed analysis addressing the question of the influence of enhanced KGT crystal growth models on the predicted macro-segregation pattern in the Al-4wt.%Cu alloy cast. In particular, the analysis concerns the impact of a proper selection of the stability constant for the KGT model (based on a crystal-melt surface energy anisotropy strength and linear scaling law of the marginal stability theory) on the predicted dendrite tip under-cooling, volumetric solid fraction and actual solute concentration along the front of columnar dendrite tips. Differences in the resulting evolution of both: the under-cooled melt region within the mushy zone and the solute concentration fields calculated for the considered various kinetics of a grain growth are also reported and discussed.

Zone Partitioning Sequential Calculation Method for Three-Dimensional Phase-Field Material Simulation during Directional Solidification

Complex dendritic structures can be simulated directly by phase field method. However, phase field method needs a very fine mesh computing and memory requirements. A new calculation method named zone partitioning sequential calculation method is proposed to expand the simulation area of phase field method. The simulation area can be divided into several parts, and the parts can be calculated one by one in a certain order by the new method. This new method can reduce the computing and memory requirements of single calculation, because a part of the simulation area is less than the whole simulation area. Although this method could make error in the interface of different parts of the simulation area when the grains go through the interface, but the error has less effect on the grain growth. By using phase field method, coupled with zone partitioning sequential calculation method, the simulation of the directional solidification process of Al-Cu binary alloy is operated. The results show that the new method can be applied to phase field simulation of binary alloy solidification in a large area. The simulation results have certain accuracy and reliability.

Numerical Simulation of Solidification Microstructure based on Adaptive Octree Grids

Abstract The main work of this paper focuses on the simulation of binary alloy solidification using the phase field model and adaptive octree grids. Ni-Cu binary alloy is used as an example in this paper to do research on the numerical simulation of isothermal solidification of binary alloy. Firstly, the WBM model, numerical issues and adaptive octree grids have been explained. Secondary, the numerical simulation results of three dimensional morphology of the equiaxed grain and concentration variations are given, taking the efficiency advantage of the adaptive octree grids. The microsegregation of binary alloy has been analysed emphatically. Then, numerical simulation results of the influence of thermophysical parameters on the growth of the equiaxed grain are also given. At last, a simulation experiment of large scale and long-time has been carried out. It is found that increases of initial temperature and initial concentration will make grain grow along certain directions and adaptive octree grids can effectively be used in simulations of microstructure.

On the fully implicit solution of a phase-field model for binary alloy solidification in three dimensions

A fully implicit numerical method, based upon a combination of adaptively refined hierarchical meshes and geometric multigrid, is presented for the simulation of binary alloy solidification in three space dimensions. The computational techniques are presented for a particular mathematical model, based upon the phase-field approach, however their applicability is of greater generality than for the specific phase-field model used here. In particular, an implicit second order time discretization is combined with the use of second order spatial differences to yield a large nonlinear system of algebraic equations as each time step. It is demonstrated that these equations may be solved reliably and efficiently through the use of a nonlinear multigrid scheme for locally refined grids. In effect this paper presents an extension of earlier research in two space dimensions (J. Comput. Phys., 225 (2007), pp. 1271-1287) to fully three-dimensional problems. This extension is validated against earlier two-dimensional results and against some of the limited results available in three dimensions, obtained using an explicit scheme. The efficiency of the implicit approach and the multigrid solver are then demonstrated and some sample computational results for the simulation of the growth of dendrite structures are presented.

Three dimensional thermal-solute phase field simulation of binary alloy solidification

We employ adaptive mesh refinement, implicit time stepping, a nonlinear multigrid solver and parallel computation to solve a multi-scale, time dependent, three dimensional, nonlinear set of coupled partial differential equations for three scalar field variables. The mathematical model represents the non-isothermal solidification of a metal alloy into a melt substantially cooled below its freezing point at the microscale. Underlying physical molecular forces are captured at this scale by a specification of the energy field. The time rate of change of the temperature, alloy concentration and an order parameter to govern the state of the material (liquid or solid) are controlled by the diffusion parameters and variational derivatives of the energy functional. The physical problem is important to material scientists for the development of solid metal alloys and, hitherto, this fully coupled thermal problem has not been simulated in three dimensions, due to its computationally demanding nature. By bringing together state of the art numerical techniques this problem is now shown here to be tractable at appropriate resolution with relatively moderate computational resources.

On the Fully Implicit Solution of a Phase-Field Model for Binary Alloy Solidification in Three Dimensions

AbstractA fully implicit numerical method, based upon a combination of adaptively refined hierarchical meshes and geometric multigrid, is presented for the simulation of binary alloy solidification in three space dimensions. The computational techniques are presented for a particular mathematical model, based upon the phase-field approach, however their applicability is of greater generality than for the specific phase-field model used here. In particular, an implicit second order time discretization is combined with the use of second order spatial differences to yield a large nonlinear system of algebraic equations as each time step. It is demonstrated that these equations may be solved reliably and efficiently through the use of a nonlinear multigrid scheme for locally refined grids. In effect this paper presents an extension of earlier research in two space dimensions (J. Comput. Phys., 225 (2007), pp. 1271-1287) to fully three-dimensional problems. This extension is validated against earlier two-dimensional results and against some of the limited results available in three dimensions, obtained using an explicit scheme. The efficiency of the implicit approach and the multigrid solver are then demonstrated and some sample computational results for the simulation of the growth of dendrite structures are presented.

Comparison of Two Models for Simulation of Binary Alloy Solidification

The results of running a quasi two-dimensional experimental benchmark for horizontal directional solidification of binary Sn-3 wt.%Pb alloy in a rectangular region are presented. The computation was performed using the GIGAN software and two models to simulate the mushy zone were compared, namely, the one based on the equilibrium approach and the non-equilibrium one.

A fully implicit, fully adaptive time and space discretisation method for phase-field simulation of binary alloy solidification

A fully implicit numerical method based upon adaptively refined meshes for the simulation of binary alloy solidification in 2D is presented. In addition we combine a second-order fully implicit time discretisation scheme with variable step size control to obtain an adaptive time and space discretisation method. The superiority of this method, compared to widely used fully explicit methods, with respect to CPU time and accuracy, is shown. Due to the high nonlinearity of the governing equations a robust and fast solver for systems of nonlinear algebraic equations is needed to solve the intermediate approximations per time step. We use a nonlinear multigrid solver which shows almost h-independent convergence behaviour.

Solidification of a binary alloy: Finite-element, single-domain simulation and new benchmark solutions

A finite-element simulation of binary alloy solidification based on a single-domain formulation is presented and tested. Resolution of phase change is first checked by comparison with the analytical results of Worster [M.G. Worster, Solidification of an alloy from a cooled boundary, J. Fluid Mech. 167 (1986) 481–501] for purely diffusive solidification. Fluid dynamical processes without phase change are then tested by comparison with previous numerical studies of thermal convection in a pure fluid [G. de Vahl Davis, Natural convection of air in a square cavity: a bench mark numerical solution, Int. J. Numer. Meth. Fluids 3 (1983) 249–264; D.A. Mayne, A.S. Usmani, M. Crapper, h-adaptive finite element solution of high Rayleigh number thermally driven cavity problem, Int. J. Numer. Meth. Heat Fluid Flow 10 (2000) 598–615; D.C. Wan, B.S.V. Patnaik, G.W. Wei, A new benchmark quality solution for the buoyancy driven cavity by discrete singular convolution, Numer. Heat Transf. 40 (2001) 199–228], in a porous medium with a constant porosity [G. Lauriat, V. Prasad, Non-darcian effects on natural convection in a vertical porous enclosure, Int. J. Heat Mass Transf. 32 (1989) 2135–2148; P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Natural convective heat transfer in an enclosure filled with fluid saturated variable porosity medium, Int. J. Heat Mass Transf. 40 (1997) 3955–3967] and in a mixed liquid–porous medium with a spatially variable porosity [P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Natural convective heat transfer in an enclosure filled with fluid saturated variable porosity medium, Int. J. Heat Mass Transf. 40 (1997) 3955–3967; N. Zabaras, D. Samanta, A stabilized volume-averaging finite element method for flow in porous media and binary alloy solidification processes, Int. J. Numer. Meth. Eng. 60 (2004) 1103–1138]. Finally, new benchmark solutions for simultaneous flow through both fluid and porous domains and for convective solidification processes are presented, based on the similarity solutions in corner-flow geometries recently obtained by Le Bars and Worster [M. Le Bars, M.G. Worster, Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification, J. Fluid Mech. (in press)]. Good agreement is found for all tests, hence validating our physical and numerical methods. More generally, the computations presented here could now be considered as standard and reliable analytical benchmarks for numerical simulations, specifically and independently testing the different processes underlying binary alloy solidification.

A two-dimensional model of binary-alloy solidification with a Mushy-zone

The results of the experiments with solidification in space imply that the effect of gravity conditions should be included into the theoretical and computational models of solidification processes. The paper presents the results of computational simulation of binary-alloy solidification where the gravity acts as the driving force of chemical diffusion. A simplified version of a complex theoretical model constructed by the same authors was implemented on a computer and used to simulate the solidification of a two-dimensional sample of the pseudo-binary mercury-cadmiumtelluride alloy. The computational model was used to study the influence of gravity on the chemical diffusion in the sample. The results of our simulations have shown that the influence of gravity on the diffussion of the solute is a second-order effect only. Therefore, to describe the influence of the gravitational field on solidification processes adequately we would need a more complex model involving convective phenomena in the melt as well. In addition, a series of numerical experiments were performed to consider how the accuracy with which the material parameters had been determined affects the computed temperature and concentration fields. The results imply that the simulation of this kind could be seriously biased by insufficient accuracy in determining the primary material properties. Moreover, the results show that neglecting the thermal diffusion and the temperature and concentration dependence of the material parameters may lead to serious errors.