Solidification Of Pure Metals Research Articles

Thermoelastic Stability Analysis of Solidification of Pure Metals on a Coated Planar Mold of Finite Thickness

A theoretical model for investigating the thermoelastic instability/mechanism during pure metal solidification on a coated mold of finite thickness is developed. This study extends the previous theoretical works on growth instability during solidification process by investigating the effects of an added coating layer. Mold coating is one of the most important factors controlling the heat transfer rate, and hence it has a very important role on the solidification rate and the development of microstructure. In this model, thermal and mechanical problems are coupled through the pressure-dependent contact resistances at mold/coating and coating/shell interfaces. The thermal diffusivities of solidified shell, coating, and mold materials are assumed to be zero. This assumption provides us to solve heat transfer problem analytically. A linear perturbation method is used to simplify complexity of the modeled solidification problem, and governing equations are solved numerically using a variable step variable order predictor–corrector algorithm. The effects of coating layer thickness and coupling rates at shell/coating and coating/mold interfaces are investigated in detail. The results show that coating thickness has destabilizing effect on the growth instability when the coupling rates are small. However, when these coupling rates are increased individually or together, the destabilizing effect of coating thickness turns to be stabilizing. On the other hand, coupling rates have generally destabilizing effects on the process, but an increase in the thickness of coating leads to diminishing coupling rates effect in some cases.

Multi-GPUs parallel computation of dendrite growth in forced convection using the phase-field-lattice Boltzmann model

Melt flow drastically changes dendrite morphology during the solidification of pure metals and alloys. Numerical simulation of dendrite growth in the presence of the melt flow is crucial for the accurate prediction and control of the solidification microstructure. However, accurate simulations are difficult because of the large computational costs required. In this study, we develop a parallel computational scheme using multiple graphics processing units (GPUs) for a very large-scale three-dimensional phase-field-lattice Boltzmann simulation. In the model, a quantitative phase field model, which can accurately simulate the dendrite growth of a dilute binary alloy, and a lattice Boltzmann model to simulate the melt flow are coupled to simulate the dendrite growth in the melt flow. By performing very large-scale simulations using the developed scheme, we demonstrate the applicability of multi-GPUs parallel computation to the systematical large-scale-simulations of dendrite growth with the melt flow.

Effect of coating material on the growth instability in solidification of pure metals on a coated planar mold of finite thickness

In this study, a theoretical basis of thermomechanical instability during the solidification of pure metals on a coated planar mold is presented. This study extends the previous works by taking into account the presence of a deformable coating layer on the inner mold surface. The thermal and mechanical problems are assumed to be coupled through the pressure dependent thermal contact resistances at the shell/coating and coating/mold interfaces. On the other hands, the thermal capacitance of solidified shell, coating layer and mold materials are neglected for the sake of simplicity. A linear perturbation method is used to reduce the spatial dimension of the problem. The model leads to two coupled differential equations for the shell thickness perturbation and residual stress which are solved numerically. The results document the variation of the perturbed solidification front as a function of ratio between the thermal conductivities of the shell and coating materials for combinations of other process parameters such as the combination of shell and mold materials, the values of coating thickness and coupling rates. In case of weak coupling, the maximum magnitude of this perturbation decreases regardless of the value of coating thickness and shell-mold material combinations when the thermal conductivity of the coating material is decreased. In case of strong couplings however, a critical thermal conductivity ratio between shell and coating materials which leads to a formation of greatest or lowest perturbation in the solidification front depending on the values of other process parameter has been found.

Phase-field modeling of microstructure evolution during solidification in presence of gas bubble

The microstructure evolution during solidification of pure Aluminum metal with gas bubble nucleation and growth is modeled and simulated by the phase-field method. Interface microstructure formation in the presence of gas bubble is simulated to investigate the interaction between gas bubble and microstructural evolution, and also a pressure correction equation is implemented to study the effect of pressure difference between the bubbles and melts. Results indicate that pressure difference between gas bubble and the liquid melt can significantly affect the gas atoms diffusion near the gas–liquid interface, which will lead to the bubble growth velocity change. These simulation results are in agreement with experimental observations.

What is the Process Window for Semi-solid Processing?

In semi-solid processing, the liquid fraction vs temperature is commonly used to define the process window. Conventionally, it is assumed that a freezing range is required for semi-solid processing but recently both high-purity aluminum and binary Al-Si eutectic alloy have been rheo-processed. Here, the kinetics during melting and solidification of pure metal are analyzed, and a comparison between the liquid fraction vs temperature and the liquid fraction vs time is presented. It is found that liquid fraction vs time is a significant criterion for semi-solid processing.

Numerical modeling of pure metal solidification using the one domain approach

Numerical modeling of pure metal solidification on the basis of the wellknown Stefan model is rather difficult. The knowledge of temporary solidification front position and the local values of the solidification rate in the normal direction for time t are neces� sary in order to determine the new position of moving boundary for time t + ∆t. The prob� lem is especially complicated for 2D and 3D tasks. The concept greatly simplifying the modeling of solidification process boils down to the introduction of the artificial region cor� responding to the mushy zone subdomain. For this region the su bstitute thermal capacity is defined and the mathematical model corresponds to the one domain approach. The artificial mushy zone appears owing to conventional enlargement of solidification point on a certain interval of temperature ∆T. The basic goal of the paper is the numerical analysis of the influence of the interval ∆T on the numerical solution simulating the thermal processes in the domain of the solidifying metal.

Solidification Modeling: Evolution, Benchmarks, Trends in Handling Turbulence, and Future Directions

A systematic evolution of the solidification modeling is presented in this article. An approach starting from the basic governing equations to the intricate modeling of the alloy solidification using different approaches has been reviewed. Important advantages and issues related to different formulations and the use of fixed/moving grids for the modeling of solidification have been discussed. This article outlines the important solidification modeling approaches used in the literature. The mathematical description of the most frequently employed methods for modeling of solidification has been presented providing adequate references for other solidification models. This article highlights an important subdomain of solidification modeling, namely, the modeling of solidification processes having significant turbulence (such as welding, casting, and Czochralski crystal growth). A review of the use of different turbulence models along with the state-of-the-art techniques in these areas is presented. The paper also describes the important benchmarking studies (both experimental and numerical modeling results) used for the validation of solidification of both pure metals and alloys. Finally, the physical and numerical complexities associated with the solidification modeling phenomena along with the important challenges and future directions are presented.

Approximate Analytic Solutions for the Two-Phase Stefan Problem Using the Adomian Decomposition Method

An Adomian decomposition method (ADM) is applied to solve a two-phase Stefan problem that describes the pure metal solidification process. In contrast to traditional analytical methods, ADM avoids complex mathematical derivations and does not require coordinate transformation for elimination of the unknown moving boundary. Based on polynomial approximations for some known and unknown boundary functions, approximate analytic solutions for the model with undetermined coefficients are obtained using ADM. Substitution of these expressions into other equations and boundary conditions of the model generates some function identities with the undetermined coefficients. By determining these coefficients, approximate analytic solutions for the model are obtained. A concrete example of the solution shows that this method can easily be implemented in MATLAB and has a fast convergence rate. This is an efficient method for finding approximate analytic solutions for the Stefan and the inverse Stefan problems.

Thermal Kinetics Phase Field Model for the Metals' Solidification - Mathematic Derivation of the Dendrite Growth Phase Field Variable Diffusion Model

Phase field equations for simulation of dendritic growth have a history of nearly twenty years. The existing phase field equations are directly derived through the Ginzburg-Landau equations. However, though widely used, the physical meaning of each variable in the equations is not clear. So the domestic and foreign researchers have made a lot of mistakes and interpretation. In this paper, with the solid fraction as a phase field variables in the field, based on thermodynamics and heat transfer theory, author gives a rigorous scientific phase variable diffusion model of the pure metal solidification and its derivation process.

A two-grid adaptive volume of fluid approach for dendritic solidification

A two-grid approach to simulating dendritic growth in the solidification of pure metals is presented. The approach is based on using a uniform Cartesian grid to solve the energy equation and an adaptive Cartesian grid of higher-resolution to solve the interface evolution, providing a more accurate representation of the interface. The energy equation is solved using a diffuse-interface method, and the advection equation of a discretized solid fraction function is solved using an unsplit VOF (volume of fluid) method, with PLIC (piecewise linear interface calculation) reconstruction of the interface. The thermal gradients at both sides of the interface, which are needed to obtain the front velocity, are calculated with the aid of the distance function to the reconstructed interface, which is obtained using an efficient method described in detail in this work. The influence of the grid resolution used to solve the advection equation on the accuracy of the method is analyzed. For the dendritic growth cases considered in this work, particularly, it was found that using a grid resolution for the advection equation two times higher than that used for the energy equation considerably improves the results, while keeping the CPU time consumed at an acceptable level. To underline the importance of an accurate estimation of the interface curvature, the results obtained using three techniques commonly used in VOF methods of different degrees of accuracy (a convolved VOF technique, a distance function technique and an improved height function technique, which provides second-order accuracy) are compared. The proposed methodology is assessed by comparing the numerical results with analytical solutions and with results obtained by different authors for the formation of complex dendritic structures in two and three dimensions.

Experimental verification of immune recruitment mechanism and clonal selection algorithm applied for solving the inverse problems of pure metal solidification

In the paper an application of two immune algorithms — Immune Recruitment Mechanism and Clonal Selection Algorithm — in procedures for solving the inverse problems of pure metals solidification is presented. Discussed problems consist in reconstruction of boundary conditions (forms of the heat transfer coefficient and the heat flux) on the basis of temperature measurements. For verifying the effectiveness of investigated algorithms the experimental data obtained in the solidification of aluminum have been used. An example of applying considered procedures for determining the cooling conditions of the continuously cast ingot is also presented.

Phase-field simulation of solidified microstructure evolution in the presence of lateral constraint

Lateral constrain in the presence of melting has a significant effect on microstructure evolution of crystal growth, and this effect is related to the size and property of lateral constrain, thus determining microstructure formation during solidification. In the paper, microstructure evolution in the presence of lateral constrain during the solidification of pure Ni metal is simulated using a non-isothermal phase-field model. Effects of size and properties of lateral constrain are simulated and studied, also microstructures formed at different initial dendritic arm distances are discussed. Results indicate that the presence of lateral constrain has a direct influence on pattern evolution which determines the microstructure formation. Microstructure changes significantly with lateral constrain distance turning small, initial constrain temperature low becoming low and initial dendrite arm distance growing. Different heights of lateral constrains have almost the same effects on microstructure change during solidification.

An implicit lattice Boltzmann model for heat conduction with phase change

In this work, a new variation of the lattice Boltzmann method (LBM) was developed to solve the heat conduction problem with phase change. In contrast to previous explicit algorithms, the latent heat source term was treated implicitly in the energy equation, avoiding iteration steps and improving the formulation stability and efficiency. The Bhatnagar–Gross–Krook (BGK) approximation with a D2Q9 lattice was applied and different boundary conditions including Dirichlet and Neumann boundary conditions were considered. The developed model was tested by solving a one-dimensional melting problem for a pure metal, and one and two-dimensional solidification problems for a binary alloy. The results of the LBM solution were compared with analytical and finite element solutions and a good consistency was observed. Considering the special capabilities that LBM offers, like local characteristic, and inherent parallel structure, the developed model is an interesting alternative to traditional continuum models.

Quasi-steady thermal analysis of solidification of cylindrical rod in Ohno continuous casting process

The Ohno continuous casting process, associated with unidirectional crystallisation of high purity metals, is of practical importance because of its increasing applications in electronic industries. This paper deals with analytical solution of axisymmetric heat transfer during solidification of pure metals in this casting process. The two-dimensional quasi-steady heat conduction equations for solid and liquid phases along with associated interface and boundary conditions are solved by the method of separation of variables. The expressions for temperature field in the melt and the solid regions are obtained as a function of various model parameters such as Biot number, Peclet number, dimensionless latent heat of solidification and liquid to solid thermal conductivity ratio. Besides, the casting rates of three different pure metals are compared for the known operating conditions.

Restoring boundary conditions in the solidification of pure metals

The scope of the paper is an algorithm reconstructing the boundary conditions (heat flux and heat transfer coefficient) in the solidification of pure metals on the grounds of temperature measurements. For the verification of the algorithm experimental data derived in the course of the solidification of aluminum were used. An example of the application of the algorithm for designating the cooling conditions in continuous casting is provided, when the values of temperature at selected points on the boundary of the casting are known.

A relook at the preferred growth direction of the solid–liquid interface during solidification of pure metals

A simple model for the prediction of the principal growth direction of a solid–liquid interface during solidification of pure metals of different crystal structures is developed. The existing interface controlled growth model for the growth of a solid in a liquid is suitably modified with the incorporation of density change during solidification. The idea of conjugate planes is introduced to construct virtually a unit cell of a solid and, finally, the growth of the conjugate planes is presented with respect to the plane normal. A very good match is shown between the calculated directions of highest growth rate and the experimentally found directions of the primary dendrites of a number of crystal structures. It is also shown that the differential growth of a solid in a liquid along different crystallographic directions is inherently associated with the density change and spatial distribution of atoms on the facing liquid and solid planes on solidification.

Effect of Mold Surface Topography on Uneven Shell Growth in Solidification

The unstable shell growth pattern in the early solidification stages results mainly from the unevenness of the heat flux drawn from the mold surface. This heat flux is determined either by the size of the air-gaps formed between the solidifying shell and mold surface or from the value of the contact pressure. Present work examines the role of the mold surface topography on gap nucleation and the contact pressure variation in pure metal solidification. The results of the present study can be used to tune mold surfaces for the control of cast surface morphologies.

Role of the Mold Properties on Gap Nucleation in Solidification

The role of the mold properties on gap nucleation in pure metal solidification is investigated. The mold is assumed to be finite and deformable, and has a sinusoidal surface micro-geometry. Unlike previous models, the model developed herein assumes that the mold material has a non-negligible thermal capacitance. Of particular interest are the roles played by the mold thickness and mold thermal capacitance on the existence of critical mold surface wavelength that corresponds to the situation where both contact pressure and its time derivative simultaneously fall to zero. The present work also assumes that the thermal and mechanical problems in the mold-shell interface are uncoupled. It is shown that the inclusion of the thermal capacitance of the mold material, together with thermal capacitance of the shell and the mold distortion, may be sufficient to predict a critical wavelength beyond which no gap nucleation occurs at the troughs. The role of the mold properties is examined through qualitative comparisons of the present and previous models. Gap nucleation times, associated mean shell thicknesses, and critical wavelengths are calculated for pure copper and pure iron molds under identical process conditions. It is found that a copper mold leads to faster gap nucleation compared to an iron mold. The associated critical wavelengths of iron molds are shown to be larger than those of copper. An optimum mean mold thickness corresponding to the longest gap nucleation time for a given set of process parameters is determined. The effect of the mean pressure on the optimum mold thickness is also investigated.

Combined Effects of Mold Deformation and Shell Thermal Capacity on Growth Instability During Unidirectional Solidification of Pure Metals

ABSTRACT Previous models of thermomechanically induced freezing front growth instability have neglected either the mold distortion or the thermal capacitance of the solidifying shell material. While these assumptions are useful in providing insight into solidification thermomechanics, they fail to account for the combined effects of the mold deformation and the thermal capacity of the shell material. In this paper, growth instability during solidification of pure metals is reexamined under the assumption that the shell material with nonnegligible thermal capacitance solidifies on a deformable mold with a mean finite thickness. The mold is assumed to have infinitely large thermal diffusivity, and a sinusoidal surface microgeometry for which the ratio of the amplitude to the wavelength in much less than one. This makes the aspect ratio a convenient perturbation parameter. The contact pressure in the troughs of the mold surface is calculated for different mold-shell material combinations that consist of pure copper, aluminum, or iron. The influence of mold surface wavelength and mean contact pressure on shell growth is examined for these cases. The effects of mold thickness on gap nucleation times for selected wavelengths are investigated. The role of the thermal capacitance of the solidifying shell material when combined with thermoelastic distortion of the mold is examined through qualitative comparisons between results from the present model and those from previous models. Conclusions are drawn about the relative significance of these factors on the unstable growth of the shell thickness. Communicated by Jim Barber on February 16, 2005.

Simulation of casting solidification and its grain structure prediction using FEM

Metal casting is one of the basic manufacturing processes. Numerical simulation of solidification is useful in getting high quality castings and minimising product cost and scrap. Finite element analysis in heat transfer can be used for this purpose. In this study, a program for finite element modelling of casting solidification was developed. The salient features of the program are: facility to incorporate latent heat through enthalpy method, incorporation of air gap by coincident node technique, ability to handle non-linear transient heat conduction through temperature dependent material properties, and object oriented programming. The code was written in C++ language and Gaussian elimination method was used to solve the matrices. This program can handle two-dimensional solidification of pure metals and alloys. Using this program, solidification of infinite slab of water was simulated and the results were compared with the reported literature and ANSYS and were found to be in good agreement. Also, solidification of Al–6wt.% Si alloy in sand and metal mould was simulated and compared with the experimental findings and simulation results reported in the literature and ANSYS. For a given geometric profile, the solidification time obtained from the present program was 1200s as against the literature of 1100s and ANSYS package of 900s. So it was found to be in good agreement with the research and experimental findings. The program was limited to 2D heat transfer analysis using linear quadrilateral elements. The program could be extended for predicting the grain structure. From the simulation results, the solid–liquid interface velocity (R) and solidification rate (dT/dt) could be calculated and using these values thermal gradient (G) could be found and there by the most probable grain structure could be predicted.