An explicit–implicit time stepping scheme for solidification models

A numerical scheme with hybrid explicit and implicit time stepping in solidification problems is presented. An explicit time stepping scheme is used for solving coupled temperature and concentration fields, while an implicit scheme is used for solving equations of motion. The explicit approach results in a local point-by-point coupling scheme for the temperature and concentration fields that uses constitutive model for back diffusion in solid. The present method offers distinct advantages of simplicity and flexibility in incorporation of micro-scale models, as demonstrated by the use of a back diffusion parameter in the microsegregation model. Results from the present method are compared with those in the literature using a fully implicit method, and they show significant improvement in final macrosegregation prediction.     

A numerical study of the combined effects of microsegregation, mushy zone permeability and fllow, caused by volume contraction and thermosolutal convection, on macrosegregation and eutectic formation in binary alloy solidification

Numerical simulations of the columnar dendritic solidification of a Pb-20 wt% Sn alloy in a square cavity cooled from one side and fed by a rectangular riser are reported. Overall macrosegregation patterns predicted using Scheil and lever-rule type microsegregation models are found to be similar, although the predicted eutectic fraction is significantly higher with the Scheil-type model. The choice of mushy zone permeability function significantly affects the predicted number, length and orientation of segregated channels. The inclusion of shrinkage-driven flow leads to the prediction of the well-known inverse macrosegregation pattern. However, macrosegregation caused by thermosolutal convection readily masks the inverse segregation. The microsegregation models predict different solid concentrations and eutectic fractions, leading to different solid density distributions which, in turn, cause differences in the extent of contraction-driven flow.     

Numerical study of macrosegregation in Aluminum alloys solidifying on uneven surfaces

Solidification of Aluminum alloys is modeled on uneven surfaces characterized by sinusoidal curves. Wavelengths and amplitudes of these surfaces are varied to study the effect of changing surface topography on fluid flow, macrosegregation and inverse segregation in the solidifying alloy. Solidification is initiated by convective heat removal from the uneven surfaces and simulations are carried out in both vertical and horizontal configurations. Stabilized finite element methods, recently used for modeling solidification in the presence of shrinkage and buoyancy driven flows, are used to discretize and solve the governing transport equations derived by volume averaging. The effect of varying amplitudes and wavelengths is observed in heat transfer, fluid-flow, macrosegregation and inverse segregation processes. In vertical solidification, inverse segregation, that usually occurs at the bottom of the cavities, is studied for different sinusoidal topographies quantified by a particular wavelength and amplitude. The fluid flow here is driven by a combination of shrinkage and thermosolutal buoyancy. Shrinkage driven flow arises due to different densities of solid and liquid phases. During horizontal solidification of an Aluminum alloy from uneven surfaces, thermosolutal buoyancy plays a dominant role in fluid flow and the effect of shrinkage is neglected by assuming the individual phase densities to be equal. Convection in this case is much stronger than that in the vertical case and large scale redistribution of the solute element occurs. To measure variation in macrosegregation with changing surface topography, global extent of segregation and difference between maximum and minimum solute concentrations are calculated for different amplitudes and wavelengths. In both the cases, the main aim is to quantify changes in macrosegregation due to changing surface topography accomplished by varying amplitudes or wavelengths or both.     

Influence of convection and grain movement on globular equiaxed solidification

A two-phase volume averaging model was used to study convection and grain movement, and their influence on the globular equiaxed solidification. Both liquid and solid phases were treated as separate interpenetrating continua. The mass, momentum, species and enthalpy conservation equations for each phase and a grain transport equation were coupled. An ingot casting (Al-4 wt.% Cu) with near globular solidification morphology was simulated. Case studies with different modeling assumptions such as with and without grain movement, and with slip and non-slip boundary conditions for solid phase were presented and compared. Understanding of grain evolution and macrosegregation formation in globular equiaxed solidification was improved.     

Effects of Solutal Undercooling on Three-Dimensional Double-Diffusive Convection and Macrosegregation during Solidification of a Binary Alloy

ABSTRACT

A three-dimensional transient numerical model is developed for simulation of double-diffusive convection during binary alloy solidification processes, taking into account nonequilibrium effects due to solutal undercooling. Such an effect arising from microscopic convection near the diffusion boundary layer adjacent to the mushy region is captured by devising a macroscopic model based on a fixed-grid, enthalpy-based, control-volume approach. Microscopic features pertaining to solutal undercooling are incorporated through a modification of the partition coefficient by means of a number of macroscopically observable parameters. Numerical simulations are performed for solidification of a metallic alloy system kept in a side-cooled cubic enclosure. Typical curvatures of the streamlines and their nonequidistant characteristics, as projected on various cross-sectional planes, show an element of three-dimensionality in the double-diffusive convection (originating from the solidification process itself) and its interaction with the progressing solidification front. The three-dimensional transport leads to a global macrosegregation, with significant composition variations across the longitudinal planes, as dictated by the modified partition coefficient and thermosolutal convection mechanisms.     

Control of macrosegregation during the solidification of alloys using magnetic fields

Numerical modeling of convection damping and macrosegregation suppression during solidification of alloys with prominent mushy zones through the use of tailored magnetic fields is demonstrated here. Macrosegregation leads to commonly observed defects such as freckles, channels and segregates in cast alloys that severely affect the performance and suitability of the alloy for further applications. The current work demonstrates the successful use of magnetic fields in suppressing thermosolutal convection and eliminating some of these defects in solidifying metallic alloys. The computational model presented utilizes volume-averaged governing transport equations and stabilized finite element techniques to discretize these equations. A finite-dimensional optimization problem, based on the continuum sensitivity method is considered to design the time history of the imposed magnetic field required to effectively damp convection. The coefficients that determine this time variation are the main design parameters of this optimization problem. Continuum sensitivity equations are derived by design-differentiating the governing equations of the direct problem. The cost functional here is given by the square of the L₂ norm of an expression representing the deviation of the volume-averaged velocity corresponding to conditions of convection less growth. The cost functional minimization process is realized through a non-linear conjugate gradient algorithm that utilizes finite element solutions of the continuum direct and sensitivity problems. Design of the time history of the imposed magnetic field is highlighted through different examples with the main objective being the suppression of convection and macrosegregation during alloy solidification.     

合金凝固传热传质宏微观耦合数值研究及验证

建立了描述二元合金凝固的平面枝晶一维微观偏析数学模型,考虑溶质在固相中有限扩散,在液相中完全扩散。通过数值模拟,分析比较了A l-Cu和F e-C合金的微观偏析特性。同时,进一步将微观数值模型与宏观凝固实验的传热传质数学模型相耦合,实现了凝固宏微观复合尺度的全数值模拟。研究表明,数值计算结果与实验数据吻合良好,证明微观模型能较准确地反映微观质量传输并能可靠地与宏观相变传热传质模型相耦合。此外,从微观到宏观的计算结果都说明F e-C合金的凝固过程几乎接近平衡凝固。     

Thermosolutal transport and macrosegregation during freeze coating of a binary substance on a continuous moving object

The process of freeze coating of a binary substance on a continuous moving plate is investigated theoretically. A comprehensive model describing the momentum, heat, and mass transport in the freeze-coating system has been developed that accounts for the coupling between the macroscopic and microscopic aspects of the process. The problem is formulated using the single-domain approach and the governing equations are solved by the finite difference method. Effects of various controlling parameters on the freeze-coat thickness and the macrosegregation pattern have been determined. It is found that macrosegregation could be important in the freeze-coating process. As the distance from the surface of the plate is increased, the solid species concentration considerably decreases, reaching a minimum value and rising toward the ambient concentration. The macrosegregation pattern appears to be most sensitive to the equilibrium partition ratio. As the latter is increased, the difference between the solid and liquid species concentrations tends to decrease, leading to a substantial reduction of macrosegregation within the freeze coat.     

Effect of macro scale phenomena on microsegregation

A metal solidification system consists of solid, mushy and liquid regions. In many systems the two phase mushy region has a fine scaled columnar dendritic morphology. Microscopic models of metal solidification systems focus on the mass diffusion (microsegregation) and movement of the solid/liquid interface in the dendrite arm spaces. In this paper the effects of macroscopic variables on the microsegregation predictions are studied. In particular, the effect of cooling and macrosegregation histories on the solid solute profile, the eutectic fraction formed and solid/liquid interface movement in the arm spacing will be investigated.     

Three-dimensional double-diffusive convection and macrosegregation during non-equilibrium solidification of binary mixtures

A three-dimensional transient mathematical model (following a fixed-grid enthalpy-based continuum formulation) is used to study the interaction of double-diffusive natural convection and non-equilibrium solidification of a binary mixture in a cubic enclosure cooled from a side. Investigations are carried out for two separate test systems, one corresponding to a typical model “metal-alloy analogue” system and other corresponding to a real metal-alloy system. Due to stronger effects of solutal buoyancy in actual metal-alloy systems than in corresponding analogues, the convective transport mechanisms for the two cases are quite different. However, in both cases, similar elements of three-dimensionality are observed in the curvature and spacing of the projected streamlines. As a result of three-dimensional convective flow patterns, a significant solute macrosegregation is observed across the transverse sections of the cavity, which cannot be captured by two-dimensional simulations.     

Modeling of Solidification Process in a Rotary Electromagnetic Stirrer

A macroscopic model of the solidification process in a rotary electromagnetic stirrer is presented. The fluid flow, heat, and mass transfer inside a rotary stirrer are modeled using, 3-D swirl flow equations in which turbulent flow is simulated using a k ? ? model. A hybrid model is used to represent the mushy zone, which is considered to be divided into two regions: a coherent region and a noncoherent region. Each region is represented by a separate set of governing equations. An explicit time-stepping scheme is used for solving the coupled temperature and concentration fields, while an implicit scheme is used for solving equations of motion. The coupling relations also include eutectic solidification, which is an important feature in modeling solidification with electromagnetic stirring, especially in the context of the formation of semi-solid slurry. The results from the present numerical solution agree well with those corresponding to experiments reported in literature.     

A novel fixed-grid interface-tracking algorithm for rapid solidification of supercooled liquid metal

Abstract

In this study, we present a novel fixed-grid interface-tracking method using finite volume method to simulate multidimensional rapid solidification (RS) of under-cooled pure metal. The discretized advection equation for solid fraction function is solved using the THINC/WLIC method, which is a VOF method. The governing equations for fluid flow are solved numerically using pressure-velocity coupling SIMPLE algorithm in a 2-D model with incompressible Newtonian fluid. The energy equation is modeled using an enthalpy-based formulation. The nonequilibrium solidification kinetics, interface tracking, undercooling, nucleation, heat transfer, and movement of liquid are included in the presented RS model.     

The formation of negative- and positive-segregated bands during solidification of aluminum-copper alloys

The evolution of macrosegregation, appearing in the form of negative- and positive-segregated bands, in an aluminum-copper alloy unidirectionally solidified from the bottom has been analyzed numerically. It is found that a negative-segregated band is formed in the solidified casting if the casting is quenched during solidification. On the other hand, a positive-segregated band is created resulting from a sudden decrease of heat extraction rate at the bottom of the casting. Multiple negative- and positive-segregated bands could be formed in the solidified casting if the thermal boundary condition at the bottom of the casting fluctuates with time. The predicted banding segregation in the solidified alloy compares favorably with the available experimental data.     

A model for latent heat energy storage systems

In this study, a theoretical approach is proposed for the prediction of time and temperature during the heat charge and discharge in the latent heat storage of phase changed materials (PCM). By the use of the average values of the mean specific heat capacities for the phase‐changed materials, analytical solutions are obtained and compared with the available experimental data in the literature. It is shown that decreasing the entry temperature of the working fluid from ?4 to ?15°C has a very dominant and strong effect on the PCM solidification time. The effect of the working fluid flow rate and the material of PCM capsules on the time for complete solidification and total charging is also investigated. The agreement between the present theoretical model results and the experimental data related to the cooling using small spheres and the heat storage using rectangle containers is very good. The largest difference between the present results and the experimental data becomes about 10% when the fluid temperature approaches the phase change temperature at high temperatures. Copyright © 2006 John Wiley & Sons, Ltd.     

Effects of mould filling on evolution of the solid–liquid interface during solidification

This work presents a numerical analysis of simultaneous mould filling and phase change for solidification in a two-dimensional rectangular cavity. The role of residual flow strength and temperature gradients within the solidifying domain, caused by the filling process, on the evolution of solidification interface are investigated. An implicit volume of fluid (VOF)-based algorithm has been employed for simulating the free surface flows during the filling process, while the model for solidification is based on a fixed-grid enthalpy-based control volume approach. Solidification modeling is coupled with VOF through User Defined Functions developed in the commercial computational fluid dynamics (CFD) code FLUENT 6.3.26. Comparison between results of the conventional analysis without filling effect and those of the present analysis shows that the residual flow resulting from the filling process significantly influences the progress of the solidification interface. A parametric study is also performed with variables such as cooling rate, filling velocity and filling configuration, in order to investigate the coupled effects of the buoyancy-driven flow and the residual flow on the solidification behavior.     

Macrosegregation modeling during direct-chill casting of aluminum alloy 7050

ABSTRACT

A fully transient model of the direct-chill casting process is used to predict the macrosegregation development of aluminum alloy 7050. The ingot diameter, casting speed, superheat, secondary cooling, and thickness of pure Al at startup are varied. Predicted radial composition distributions are fit to Weibull probability density functions at each axial location, and the normalized standard deviation describes the macrosegregation level and the time when the process reaches steady state. The sump depth, steady-state height, and macrosegregation level were most affected by changes in casting speed and ingot diameter. The pure Al dilutes the alloy and delays compositional steady state.     

Solidification in a finite, initially overheated slab

An approximate theory of solidification in a finite, initially overheated slab is developed for small Stefan numbers. One wall of the slab is taken to be insulated and the other is subject to an instantaneous temperature drop below the freezing point. Our approach combines the heat-balance integral method and the time-dependent perturbation theory. The resulting solution is valid uniformly in time. It predicts quantitatively the deviations of the process considered from solidification with no overheating. Simple expressions for the solidification time are derived. The accuracy of the present model is examined by comparing it with various asymptotic solutions.     

Effects of cell size and macrosegregation on the corrosion behavior of a dilute Pb–Sb alloy

The aim of this study was to examine the effect of cooling rate on the cellular growth of a Pb–0.85 wt%Sb alloy and to evaluate the influences of cell size and of the corresponding macrosegregation profile on the resultant corrosion behavior. In order to obtain the as-cast samples a water-cooled unidirectional solidification system was used. Such experimental set-up has permitted the development of a clear cellular structural array even for relative high cooling rates and has allowed a wide range of solidification conditions to be analyzed. Macrostructural and microstructural aspects along the casting were characterized by optical microscopy and scanning electron microscope (SEM) techniques. The electrochemical impedance spectroscopy technique and potentiodynamic curves (Tafel extrapolation) were used to analyze the corrosion resistance of samples collected along the casting length and immersed in a 0.5 M H₂SO₄ solution at 25 °C. It was found that the corrosion rate decreases with increasing cell spacing and that the pre-programming of microstructure cell size can be used as an alternative way to produce as-cast components of Pb–Sb alloys, such as battery grids, with better corrosion resistance.