Modeling of Convection and Macrosegregation ... - ACS Publications

Ind. Eng. Chem. Res. 2009, 48, 8789–8804

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Modeling of Convection and Macrosegregation through Appropriate Consideration of Multiphase/Multiscale Phenomena during Alloy Solidification R. Pardeshi,† P. Dutta,‡ and A. K. Singh*,† TCS InnoVation Labs-Tata Research DeVelopment and Design Centre, 54B, Hadapsar Industrial Estate, Pune, India 411013, and Department of Mechanical Engineering, Indian Institute of Science, Bangalore, India 560012

Solidification processes are complex in nature, involving multiple phases and several length scales. The properties of solidified products are dictated by the microstructure, the macrostructure, and various defects present in the casting. These, in turn, are governed by the multiphase transport phenomena occurring at different length scales. In order to control and improve the quality of cast products, it is important to have a thorough understanding of various physical and physicochemical phenomena occurring at various length scales, preferably through predictive models and controlled experiments. In this context, the modeling of transport phenomena during alloy solidification has evolved over the last few decades due to the complex multiscale nature of the problem. Despite this, a model accounting for all the important length scales directly is computationally prohibitive. Thus, in the past, single-phase continuum models have often been employed with respect to a single length scale to model solidification processing. However, continuous development in understanding the physics of solidification at various length scales on one hand and the phenomenal growth of computational power on the other have allowed researchers to use increasingly complex multiphase/multiscale models in recent times. These models have allowed greater understanding of the coupled micro/macro nature of the process and have made it possible to predict solute segregation and microstructure evolution at different length scales. In this paper, a brief overview of the current status of modeling of convection and macrosegregation in alloy solidification processing is presented. Introduction Solidification is an integral part of many important manufacturing processes such as casting, welding, laser processing, crystal growth, and so on. Many of these manufacturing processes provide a cost-effective route of forming a component. However, the properties of final products of these processes are not easy to control. A large number of interacting phenomena such as transport of energy and solute, convection, nucleation, and grain growth govern these processes and, in turn, the quality of the finished product, characterized in terms of segregation, microstructure, porosity, and mechanical properties.1,2 In order to control and improve the quality of cast products, it is important to have a thorough understanding of various mechanisms operating at the micro- and macro-level and the interactions between the physical and physicochemical phenomena operating at various length scales. In most of the alloy solidification processes, solute is rejected at the solid/liquid interface due to the difference in solubility of the solute in solid and liquid phases. The rejected solute can get redistributed at the microscopic length scale by diffusion (resultant inhomogenity at the microscale known as microsegregation). The solute-rich liquid can also get transported over longer distances by convection resulting in a defect called macrosegregation. Convection is usually present during the progress of solidification and can result from the buoyancy forces (natural convection) or induced by external forces (forced convection). Some of the common causes of natural convection are (1) thermal buoyancy, (2) solutal buoyancy, and (3) shrinkage. Examples of forced convection in the melt include * To whom correspondence should be addressed. E-mail: [email protected]. † TCS Innovation Labs-Tata Research Development and Design Centre. ‡ Indian Institute of Science.

(1) electromagnetic stirring, (2) mechanical stirring, and (3) rotation of cavity. Apart from being the driving force behind the evolution of macrosegregation, the presence of flow during solidification also affects the morphology of the solid/liquid interface, microstructure, macrostructure, microsegregation, freckles, and porosity. Flow during solidification affects the properties of the solidified products significantly. Some of these effects are listed below: (1) Morphology of solid/liquid interface: Melt flow plays an important role in the transition from planar-to-cellular and cellular-to-dendrite interfaces. (2) Microstructure: Melt flow affects the evolving microstructure through changes in cooling rate and solute concentration. (3) Macrostructure: Flow has a major effect on morphology and morphological transition (columnar-to-equiaxed transition, for example). (4) Segregation: Macrosegregation is widely attributed to convection in the mushy and superheated regions. It is also influenced by the morphological transition, which, in turn, is influenced by flow. (5) Microsegregation: It is caused primarily by diffusion. However, as both morphology and solute concentration are affected by flow, microsegregation is influenced by it as well. (6) Freckles: This defect is attributed to the plumes of soluterich liquid emanating from channels in the mushy region. A detailed discussion on all of the above-mentioned topics is outside the scope of this article; the present paper focuses on modeling of convection and the resultant constituent inhomogeneities, i.e., macrosegregation. Macrosegregation refers to variations in composition that occur in alloy castings or ingots and range in scale from several millimeters to centimeters or even meters (as shown in Figure 13). It is present in virtually all types of casting processes, including continuous casting, shape casting of steel and aluminum alloys, iron casting, casting of single crystal super-

10.1021/ie900164f CCC: $40.75  2009 American Chemical Society Published on Web 09/10/2009

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Ind. Eng. Chem. Res., Vol. 48, No. 19, 2009

Figure 1. Macrosegregation in ingot alloy solidification (reproduced with permission from ref 3).

alloys, semisolid casting, and even in semiconductor crystals. Macrosegregation is detrimental to the product quality as it affects the properties of cast materials and its subsequent processing behavior. Macrosegregation evolves during the process of solidification. Thus, in order to understand the casting process and correlate defects to processing parameters, it is important to understand the evolution of the solidification process. Solidification of binary mixtures does not exhibit a distinct front separating solid and liquid phases. Instead, the solid is formed as a permeable, fluid saturated, crystalline-like matrix. The structure and extent of this multiphase region, known as the mushy region, depends on numerous factors, such as the specific boundary and initial conditions. During solidification, latent energy is released at the interfaces which separate the phases within the mushy region. The distribution of this energy therefore depends on the specific structure of the multiphase region. Latent energy released during solidification is dissipated by conduction in the solid phase, as well as by the combined effects of conduction and advection in the liquid phase. The same is true for the solute rejected at the solid/liquid interface. The length scales involved in these phenomena range from a few angstroms to a few micrometers. More details about the multiple length scales associated with various phenomena are described in subsequent sections. Length Scales in Solidification. Solidification processes are multiscale in nature and have influence of phenomena occurring at different length and time scales, as shown in Figure 2.4 The influence of electronic potential on the kinetics of atomic rearrangement is a process happening at the atomic length scale. The atoms self-assemble to form clusters, and the surface energy of the clusters balances volumetric contributions to control nucleation and growth. The formation of microstructures, growth of dendrites/eutectics, and microsegregation during dendrite growth are phenomena happening at microscale. At the macroscopic scale, transport processes of heat, solute, and solid grains occur at the actual geometric length scale. The range of time and length scales is clearly too large to include all of them in a single computational model. Accordingly, the changes in solid-liquid interface during solidification can be studied at three different length scales, namely the macro-, micro-, and nanoscales. However, models often analyze solidification events at an intermediate scale, called the mesoscale. A brief description of the different scales and the significant parameters in each scale are presented here.5

Figure 2. Multiple scales in solidification (reproduced with permission from ref 4).

Macroscale. This scale is of the order of 10-3 m. Macroscale features include shrinkage cavity, macrosegregation, cracks, surface roughness, and casting dimensions. These macrostructural features may sometimes influence casting properties. Boundary conditions for a casting process are imposed at the macroscale. Mesoscale. Microstructure features at grain level, without resolving the grain boundary, are described at this scale. Generally, it can be assumed that the mesoscale is of the order of 10-4 m. There is no clear demarcation between solid and liquid in the mesoscale also. The interface is treated as a mushy zone. Microscale. This scale is of the order of 10-6-10-5 m. The morphology of the solidification grain is resolved at the microscale. The grain morphology includes the mushy solid size and type (columnar or equiaxed). In a typical casting, mechanical properties depend on the solidification structure at the microscale level. Details such as the length scale of the microstructure (e.g., interdendritic arm spacing), the chemical microsegregation profile, the amount and size distribution of microshrinkage, porosity, and inclusions are described at this scale. Nanoscale (Atomic Scale). This scale is of the order of 10-9 m (nanometers) and describes the atomic arrangement at the solid-liquid interface. At this scale, solidification is described in terms of transfer of individual atoms from the liquid to the solid, leading to nucleation and growth kinetics. Currently, there is no computational model and database linking the nanostructure with the properties of castings. Among the different length scales described in this section, the system scale (macro) and the dendritic morphology can contribute to the inhomogeneous scale (micro) and are of major engineering interest. For predicting and controlling the property values of as-cast material through boundary and initial conditions, system scale study is important. But the dendritic scale phenomena are directly influenced by the system scale and may result in both microscopic and macroscopic parameters. Convection and Macrosegregation during Solidification. There are various causes of fluid flow and solid movement during solidification, and these are highlighted below. The movement of fluid during solidification results from the buoyancy forces caused by the density gradients in the liquid.


The foremost cause of density gradient is the cooling process, which causes a thermal gradient in the liquid and leads to thermal buoyancy. The resultant flow is called thermal conVection. In addition to this, alloy solidification is accompanied by solute rejection at the solid/liquid interface, setting up a concentration gradient, and thus density gradient, in the liquid. This results in solutal buoyancy, and the resultant convective flow is called solutal conVection. In most situations, both forms of natural convection are significant, and the resulting convection is termed thermo-solutal conVection or double-diffusiVe conVection.6 During dendritic solidification, the density of the interdendritic liquid varies spatially as well as temporally due to constitutional and thermal effects, which leads to thermosolutal convection in the interdendritic region. The thermo-solutal convection affects the transport of energy and solute in the mushy phase and leads to macrosegregation due to long-range solute transport from the mush to the bulk melt, as well as sedimentation and flotation of equiaxed grains. Convection can also arise from suction caused by solidification shrinkage and contraction of solid and liquid during cooling. Besides these, surface shear stresses at free surfaces, which result from surface tension gradients (Marangoni effect), can also create significant fluid flow. Apart from the natural convection generated flow, external forces such as electromagnetic stirring or centrifugal acceleration may also be imposed to control the nature of flow. Forced convection can also be caused by the motion of gas bubbles, stirring, rotation, vibration, etc. in many casting situations. Most alloy elements have a lower solubility in the solid than in the liquid phase. Such differences lead to the selective rejection of constituents at microscopic phase interfaces. During freezing, the solutes are therefore rejected into the liquid phase, leading to a continual enrichment of the liquid and lower solute concentrations in the primary solid. This segregation occurs on the scale of the microstructure that is forming, which often consists of dendrites having arm spacings on the order of 10-250 µm. It is therefore termed microsegregation and results in a nonuniform cored solute distribution in the dendrite arms. Consider now a small volume element that contains several dendrite arms and the liquid between them, i.e., an element inside the liquid-solid (mushy) zone. The flow of solute-rich liquid or the movement of solute-poor solid in or out of the volume element will change the average composition of the volume element away from the nominal composition. Since solute can be advected over large distances, macrosegregation results. Positive (or negative) segregation refers to compositions above (or below) the nominal alloy composition, and the overall macrosegregation averages out to zero over the entire casting. Modeling of the Solidification Process As solidification is an important phenomenon in many industrial applications, it has been a subject of extensive research for the past several decades. Several research groups have carried out experimental study, analytical solution, scaling analysis, and numerical modeling of solidification processes. This has led to better understanding of solidification phenomena. Mathematical modeling of solidification processes has evolved over the years and has been playing an increasingly important role in understanding the phenomena. The present day understanding of modeling of solidification phenomena, in general, and macrosegregation, in particular, is the result of extensive research of the last four decades, and these works are summarized in various review articles.1,7-12 Models of macrosegregation have generally aimed at understanding the basic

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mechanisms involved, quantitatively predicting the occurrence and severity of macrosegregation, and for obtaining the effect of various process parameters on macrosegregation. As may be noted from these studies, mathematical models have provided good insight into the formation of macrosegregation during the progress of solidification. In the following section, some of the important methods employed in modeling of convection and macrosegregation are summarized. Single-Phase Models. Early macroscopic models that included convection effects were highly simplified and neglected the coupling that exists between flows in the mushy and liquid zones. Among the earliest models that link macrosegregation to interdendritic fluid flow are those of Fleming and coworkers.13-15 They developed a differential solute redistribution equation for application in the mushy phase. The flow in the bulk liquid region was not treated explicitly. Interdendritic flow was assumed to be induced by solidification shrinkage, and solute was transported by fluid advection. However, the analysis required prescription of the temperature and fluid velocity fields. Although the model had limited utility, it did demonstrate how interdendritic flow is responsible for macrosegregation. Later, Mehrabian et al.16 extended the solute redistribution model by incorporating an equation for buoyancy driven flow in the mushy zone. Using Darcy’s Law, the mushy zone was treated as a porous medium, and its permeability was a prescribed function of the liquid volume fraction. However, a shortcoming of the solute redistribution model is that it does not account for coupling, which exists between the mushy and fully melted zones. The first model to incorporate coupling between mushy and bulk liquid zones was reported by Szekely and Jassal,17 who predicted thermal convection both in the melt and in the mushy regions. Transport equations were developed for a solid-liquid mixture in the mushy zone, and traditional single-phase equations were used for the melt. However, the effects of solutal buoyancy were ignored, and macrosegregation was not predicted. These models17 and other similar models have been classified as multidomain models, because the mathematical solution domain is distinctly divided according to the physical domains (i.e., solid, mushy, and melted zones). Because of practical considerations related to their numerical solution algorithms, multidomain models are not well-suited for predicting irregular interface shapes. Single domain models, which overcame many of the limitations of the multidomain models, emerged in the 1980s. Bennon and Incropera18 presented a set of equations for momentum, energy, and species transport in binary, solid-liquid phase change systems, which was concurrently applied in all regions (solid, mushy, and liquid). It required only a single, fixed numerical grid system and a single set of boundary conditions to effect a solution.19 The mushy zone in these models was viewed as a solid-liquid mixture with averaged macroscopic properties. Equations for solid and liquid phases, which were valid in the respective single-phase regions, were averaged to form a set of conservation equations for solid-liquid mixtures in the mushy zone. Limiting assumptions were invoked (a nondeforming solid phase, Ts ) Tl, and no diffusion through the solid phase) to reduce the number of dependent variables. The solidus and liquidus interfaces, as well as individual phase variables, were implicitly determined by solving the mixture equations. Prakash and Voller20 investigated a number of alternative approaches to account for the nature of the mushy region in a model of a solidification system. The results presented in this paper indicated that the mushy region modeling approach used

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in a solidification analysis can have a dramatic effect on the predictions obtained. Voller et al.21 presented a general twophase description based on the two-fluid model. Three limiting cases were identified, which result in one-phase models of binary systems. The governing equations for the three different cases based on dendritic morphology (columnar/equiaxed) have been derived. Illustrative examples of double diffusive convection in a square cavity with the three different cases were shown, and differences between flow and macrosegregation predictions from these models were examined. Amberg22 reported a numerical study on Fe-C alloy using a continuum formulation based model. Sahani et al.23 presented experimental as well as modeling studies on Pb-Sn systems. The model showed a reasonably good agreement with experimental data on macrosegregation. Ni and Incropera24,25 extended a previously developed continuum model and proposed a new model which retains the operational convenience of the continuum model, while allowing for the inclusion of important features of the two-phase model. With the proposed model, it was possible to account for the effects of solutal undercooling, solidification shrinkage, and solid movement. Prescott and Incropera26,27 carried out numerical and experimental studies on macrosegregation in binary Pb-Sn alloys. More experimental validation of the continuum mixture model was reported by Krane and Incropera.28 Singh and Basu29 carried out simulations to study the role of thermo-solutal convection on macrosegregation during solidification of binary Fe-1 wt % C alloy. The effect of thermo-solutal convection on the extent of segregation and segregation profile were critically examined. To account for the larger dendrite arm spacing, the West30 correlation was modified. This resulted in prediction of the extent of macrosegregation normally encountered in steel casting. Krane et al.31 extended the continuum mixture model for ternary alloy solidification, and experimental validation of the model was performed with a Pb-Sb-Sn alloy.32 Rady et al.33 studied thermo-solutal convection and macrosegregation during solidification of hypereutectic and hypoeutectic binary alloys in statically cast trapezoidal ingots. Singh and Basu34 investigated the importance of thermo-solutal coupling on the evolution of macrosegregation in Fe-C alloy. It was shown that, in the evolution of macrosegregation, the role of the solutal field in defining the mush profile through thermo-solutal coupling was more important compared to its effect through buoyancy forces. Later, Singh and Basu35 carried out a numerical investigation to study the role of the cooling rate on the evolution of macrosegregation in binary Fe-1 wt % C alloy. Vreeman et al.36 employed a binary mixture model that accounted for the redistribution of alloying elements through the transport of free-floating dendrites and fluid flow in the melt and mushy zones, of a solidifying ingot. The model was used to study the evolution of macrosegregation during the direct chill casting of Al-Cu and Al-Mg aluminum alloys.37 Later, the model was used to simulate industrial scale billet casting.38 The disagreement between model predicted macrosegregation and experimental data was attributed to the columnar to equiaxed transition which was not accounted for in the model. Krane39 extended the previously developed model of Vreeman et al.37 to account for solid particle motion and transport phenomena occurring during the solidification of ternary alloy systems. Further, Mat and Illegbusi40 developed a hybrid mushy zone model to simulate flow in the mushy zone, in an attempt to predict the final macrosegregation pattern. Kumar and Dutta41 developed a macroscopic model for semisolid billet casting

which uses a separate solid fraction transport equation to simulate the transport of fragmented dendrites in the slurry. Later, Chowdhury et al.42 extended the solid fraction transport model to incorporate interactions among solid phase particles. With respect to solid movement, Ganguly and Chakraborty43 developed a comprehensive integrated model of stochastic convective transport in a solidifying binary melt where detailed transport phenomena in the particle and bulk phases are coupled together through a stochastic Eulerian-Lagrangian formalism, capturing the physical mechanisms and consequences of particle agglomeration, deagglomeration, and the underlying hydrodynamic interactions. Some of the important aspects in modeling of the solidification process are to properly account for turbulent flow and use of mesh adaptation for macrosegregation predictions. Chakraborty et al.44 analyzed the effects of turbulent transport during a binary alloy solidification process where turbulence effects were introduced through standard k-ε equations. The coefficients were appropriately modified to account for phase-change. Microscopically consistent estimates were made regarding temperature-solute coupling in a nonequilibrium solidification situation. Liu45 developed a finite element method based macrosegregation model in which mesh refinement was carried out to capture the fluid flow in the mushy zone close to the liquidus and in the liquid region just ahead of the liquidus front. It was shown that predictions of freckles can be improved using adaptive remeshing techniques. There are two basic formulations to arrive at a single-phase model: (a) the mixture method18-21 and (b) volume averaging method.6,46 While the mixture method is simple to work with, the volume averaging method provides more insight into the physical basis of averaging. Both approaches have evolved over a period of time. For example, Dasgupta et al.47 has used homogenization theory for effective property prediction in multiscale solidification modeling in the context of the mixture model. In this article, the single-phase modeling description is mostly restricted to the mixture method. In order to limit the size of the article, the volume averaging and other averaging techniques are not described in more detail, but they are available in the open literature.1,6,8,44 Mathematical Formulation in Mixtures Models. A typical single domain mixture model, derived by Bennon and Incropera18 for alloy solidification, is presented below for the purpose of illustration. The field variables {V(u,V), T, C} in these equations are mixture average values, where the mixture contains solid and liquid phases, each of which consists of two constituents.29 In the development of the momentum conservation equation, several terms were deemed negligible in all regions. The governing equations for a continuum mixture model for columnar solidification with a stationary solid phase are as follows: Continuity equation ∂ (F) + ∇ · (FV) ) 0 (1) ∂t Momentum equation in the x-direction ∂ F µ F ∂p (Fu) + ∇ · (FVu) ) ∇ · µ ∇u + (u - us) ∂t Fl Kx Fl ∂x FBx (2)

(

)

Momentum equation in the y-direction ∂ F µ F ∂p (FV) + ∇ · (FVV) ) ∇ · µ ∇V + (V - Vs) ∂t Fl Ky Fl ∂y FBy (3)

( )


Energy equation ∂ k k (Fh) + ∇ · (FVh) ) ∇ · ∇h + ∇ · ∇(hs - h) ∂t cs cs

( )

(

)

(4)

Solute equation ∂ (FC) + ∇ · (FVCl) ) ∇ · (D∇C) (5) ∂t The terms remaining in the equation are those that are significant in at least one region of the entire domain during solidification. For example, the Darcy damping term in the momentum equation renders the viscous term negligible within the mushy zone. The continuum model equations are mutually coupled and must be used with a closure model to determine local temperature, local composition, and local fraction solid. It must be added that there have been development in mixture model since the pioneering work of Bennon and Incropera.18,19 In other continuum models for equiaxed solidification, the solid phase motion is prescribed. Many researchers have used mixture models in the study of macrosegregation. It has been demonstrated that the numerical predictions are dependent on the choice of auxiliary relationships. The major auxiliary relationships used in the mixture model are the submodels to represent flow in the mush (eq 6 for example) model for average transport properties, Bousinessq approximation, and thermo-solutal coupling relationships. The following section describes some of the auxiliary relationships used in mixture models. Auxiliary Relationships for Macroscopic Models. Treatment of the Mushy Region. Darcy’s law is used in momentum equations to account for momentum exchange between interdendritic liquid and solid dendrites. The model assumes that the rate of momentum exchange between phases is proportional to the difference in their respective velocities and inversely proportional to permeability. The permeability of a dendritic array depends on several factors, including the local volume fraction of the solid phase and its structure. For example, the Kozeny-Carman equation for permeability has been extensively employed in the modeling of a mushy region which is given by the following expression: K ) Ko

(1 - fs)3 fs2

(6)

where the permeability constant Ko is an empirical constant that depends on dendrite arm spacing. Transport Properties. Transport properties, such as thermal conductivity, viscosity, and mass diffusivity, must be prescribed or modeled in order to solve the governing equations for either a mixture or a two-phase model. The most common modeling technique is to take constant values of properties for solid and liquid phases and obtain the value for the mushy region with the help of a mixture law. In general, the macroscopic diffusion transport coefficients for a phase within a multiphase system can be expected to be different from their microscopic versions. However, sufficient data are not yet available to construct a detailed model that accounts for all complexities. Various properties in the mushy region are accounted for through the mixture law as follows: µ ) µlfl + µsfs K ) Klfl + Ksfs C ) Clfl + Csfs D ) Dlfl + Dsfs

The above equation is the most simplified approximation for a mixture law, but there are several other ways mixture laws can be formulated. For example, Voller et al.21 describes three different ways to write above equations based on dendritic morphology. Boussinesq Approximation. Boussinesq approximation is applied to account for the thermo-solutal buoyancy. The correlation is described as follows: Feff ) F[βT(T - Tref) + βC(C - Cref)

(8)

Coupling of the Thermal and Solutal Field in the Mush. There are various ways to represent temperature-solute coupling. The Lever rule based correlation gives following expression for liquid fraction which is used in the macroscopic model: fl(T, C) )

C - kp(T - Tfus)/Pd (1 - kp)(T - Tfus)/Pd

(9)

The Scheil equation, which assumes zero backdiffusion in the solid, gives the following relationship: fl(T, C) ) [(T - Tfus)/(Tl - Tfus)]1/(kp-1)

(10)

The above two expressions for thermo-solutal coupling are quite popular, while there are other approaches to incorporate microsegregation phenomenon in a macroscopic framework, which is described subsequently in this paper. Chakraborty and Dutta51 have developed a more general formulation for evaluation of latent heat functions in macroscopic models for phase change processes. Later, Ganguly and Chakraborty52 have extended this generalized formulation for multicomponent alloy solidification. Improvement in Submodel for Mushy Region. The accuracy of macrosegregation prediction with mixture models is dependent on the submodels used. In order to highlight the importance of submodels, an example of improvement in the submodel for the mushy region representation is described. Although the Kozeny-Carman expression (eq 6) does not have some of the limitations of previous models (like Darcy’s law), it is not intended for use with a solid fraction less than 0.5. West30 proposed a piecewise continuous permeability model that differentiates regions in the mushy zone according to their proximity to the liquid interface. West’s model assumes capillary behavior in the regions far from the liquidus front, and it provides a transition to dispersed particle behavior in regions with a large liquid volume fraction. The West permeability model constants are chosen to fit the experimental data.48 However, as pointed out by Krane and Incropera,28 the lack of generality of the West model30 restricts its uses. Singh et al.49 have used the basic framework of the West model but generalized the two-part model by incorporating a microstructural parameter, i.e., dendrite arms spacing, thus enabling its usage in the study of different alloy systems. The combined model can be stated as follows: K ) Ko[(fl)3/(1 - fl)2], where Ko ) d2 /(180 × 1.75), for fl < 0.5 K ) (d2 /72)fs2/3[3 + 4/fs - 3(8/fs - 3)1/2],

(7)

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(11)

for fl > 0.5 (12)

Singh et al.47 studied the effect of various mush models50,28 on convection, and the resulting macrosegregation was examined

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Figure 3. Macrosegreation prediction by different mush models with experimental data at (a) Y ) 15 mm and (b) Y ) 39 mm (reproduced with permission from ref 49).

Figure 4. Macrosegreation prediction by different mush models and comparison with experimental data at (a) Y ) 69 mm and (b) X ) 19 mm (reproduced with permission from ref 49).

using a single domain model. The predicted macrosegregation profiles were compared with published experimental and numerical data as shown in Figures 3 and 4, which shows a comparison of macrosegregation prediction from different mush models. It was demonstrated that improvement in predictive power of the mathematical model is possible (in some parts) through selection of better mush models. It was also observed that the anisotropic mush model shows better agreement with the experimental data. Use of this generalized permeability model showed better agreement with the experimental data on the macrosegregation reported in the literature, which is demonstrated in Figures 4-6. Case Study: Effect of Cooling Rate on Macrosegregation. The final macrosegregation is dependent on several process parameters such as the geometry of the casting, thermophysical properties, cooling conditions, and so on. In this section, the effect of cooling rate on macrosegreation is studied using a single-phase model. Singh and Basu34 carried out a numerical investigation to study the role of cooling rate on the evolution of macrosegregation in binary Fe-1 wt % C alloy. A mathematical model was employed to simulate the effect of heat flux on thermo-solutal convection and, in turn, on the macrosegregation. The results of this study are briefly discussed. For the purpose of this study, solidification was considered in a rectangular cavity of length L (0.1 m) and height H (0.1 m). The material chosen for the study was Fe-1 wt % C, which

has a liquidus of temperature of 1463 °C. The initial temperature of the melt was 1463 °C. Simulations were carried out with heat fluxes varying in the range of 5 to 6000 kW/m2. At higher heat fluxes, the strength of melt flow is greater. At the same time, the mushy zone is thinner and the time required for solidification is shorter. Macrosegregation patterns at the end of complete solidification for three different heat fluxes are shown in Figures 7a-c. It may be noted that the overall nature of macrosegregation patterns undergoes drastic changes with heat flux. The severity of segregation (Cmax - Cmin) at q ) 60 kW/m2 is higher than those at 10 and 360 kW/m2. For the quantitative comparison of macrosegregation, a parameter called global extent of segregation (GES), which is defined as root-mean-square of deviation from nominal composition of all the nodal points, is employed sometimes. GES as a function of heat flux is shown in Figure 8. It may be noted from the graph that there is a drop in GES with an increase in heat flux in the beginning. However, GES starts to rise at around q ) 10 kW/m2 and goes through a peak at q ) 60 kW/m2. Beyond this point, there is a steady fall in GES with an increase in heat flux. Thus, it may be noted that there are three regimes in the GES curve. The first regime corresponds to q < 10 kW/m2, where GES falls monotonically with an increase in heat flux. The decrease in the rate of solidification with increase in heat flux leads to a decrease in macrosegre-


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Figure 5. Macrosegreation prediction by generalized mush models and comparison with experimental data at (a) Y ) 15 mm and (b) Y ) 39 mm (reproduced with permission from ref 49).

Figure 6. Macrosegreation prediction by generalized mush models and comparison with experimental data at (a) Y ) 69 mm and (b) X ) 19 mm (reproduced with permission from ref 49).

gation. Between q ) 10 and 100 kW/m2, GES first rises and then starts decreasing. This peculiar behavior may be attributed to the opposing nature of thermal and solutal convection. In this regime, thermal buoyancy completely overcomes solutal buoyancy and causes a rise in macrosegregation. However, this rise is arrested beyond q ) 60 kW/m2, as the higher heat fluxes also result in higher rates of solidification, which reduces the time for the evolution of macrosegregation. The fall in macrosegregation level at very high heat fluxes is mainly due to reduction of solidification time and thinning of the mushy region, thus allowing very little solute transport to the pure liquid region. Thus, due to the opposing nature of thermal and solutal convection, the GES curve shows a hump between two monotonically decreasing segments. The above results reveal a complex variation of GES with heat flux. To summarize, it is noted that GES goes down monotonically up to q ) 10 kW/ m2. Solutal buoyancy plays a crucial role in this regime. The lower the rate of solidification, the higher the GES. With heat flux between 10 and 100 kW/m2, the GES goes through a maximum point. This is due to the opposing nature of thermo-solutal convection. In this regime, thermal buoyancy plays an important role in the evolution of macrosegregation. For a higher heat flux (>100 kW/m2), the GES curve again goes down monotonically with increase in heat flux. Multiphase Models. Unlike single-phase models, multiphase models solve separate sets of equations for different phases. Prakash,50,51 in a two-part communication, had outlined a complete two-phase flow methodology for analyzing solid-liquid phase change in binary systems. The second part of the paper

was on an illustrative application of the model. This work was a good beginning of multiphase models in solidification, and further work with regards to formulation of interfacial exchange terms was identified. It was also concluded that the development of interfacial exchange expression needs calibration of the terms via controlled experiments. Ni and Beckermann55 presented a volume-averaged two-phase model for transport phenomena in multicomponent mixtures, in which basic forms of constitutive relations were developed. These constitutive relations link the macroscopic transport phenomena to microscopic processes. Hence, the model can simultaneously predict the structure and composition in a solidified material on both micro- and macroscopic length scales. Even though the basic forms of various constitutive relations were outlined, considerable additional work is needed to obtain suitable descriptions of the microscopic phenomena. Wang and Beckermann56 presented a multiphase approach of solute diffusion during denderitic alloy solidification which links microscopic phenomena to macroscopic models. Illustrative calculations have shown that the proposed model is successful in addressing a variety of phenomena, such as recalescence in equiaxed growth, dendrite tip undercooling during rapid columnar solidification, and columnar-to-equiaxed transition (CET). Beckermann and Schneider57 simulated solidification of low-alloy steel using a multiphase model assuming a stationary solid phase. The model was enhanced to incorporate finite-rate solute diffusion in solid and a micromodel of the peritectic transformation into a macroscopic framework. The model was used to investigate the solidification of multicom-

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Figure 7. Macrosegregation pattern upon complete solidification (reproduced with permission from ref 35).

Figure 8. Variation of global extent of segregation with heat flux (reproduced with permission from ref 35).

ponent steel alloys. The treatment for thermodynamics of multicomponent system was done using a pseudobinary system. A representative simulation of the solidification of a ten-element austenitic steel in a rectangular cavity, cooled from the side, showed the formation of macrosegregation, channel segregates, and “islands” of mush surrounded by the bulk melt. Wang and Beckermann58-60 presented a new multiphase model for alloy solidification to predict composition and structure evolution in an alloy solidifying with an equiaxed morphology. The model formulation was described in part I of the paper. In part II of the paper, the model was used to simulate solidification of Al-4 wt % Cu alloy in a side-cooled rectangular cavity. Part III described experimental validation of the model with data from experiments on NH4Cl-70 wt % H2O, performed in a square cavity cooled from all sides. The multiphase model has reproduced some key features of the experiments such as grain growth with grain movement, recalescence, and the rate of formation of the crystal sediment bed. The agreement with experiments is reasonably good, and it was concluded that a much better agreement is likely if dendrite fragmentation is properly accounted for. Subsequently, Ni and Beckermann61 used a two-phase model to simulate the

sedimentation process in globulitic alloy solidification. It was identified by simulations that the nucleation rate has a profound effect on grain transport and the final macrosegregation pattern. Reddy and Beckermann62 developed a modified version of the two-phase model of Ni and Beckermann,61 which was used to simulate the DC continuous casting of an Al-Cu round ingot. Gu and Beckermann63 used a multiphase model to study melt convection and macrosegregation in the casting of a large steel ingot. The final macrosegregation shows a reasonably good agreement with the experimental data, but it does not predict the negative segregation cone at the bottom of the ingot, as the model ignored the sedimentation of free equiaxed crystals. Martorano et al.64 applied the multiphase/multiscale model of Wang and Beckermann58 to study columnar-to-equiaxed transition (CET) in alloy solidification. Wu et al.65 used a twophase volume averaging model58 to study convection and grain movement, and their influence on the globular equiaxed solidification. Several simulation studies, with different modeling assumptions were carried out with/without grain movement, with slip/no-slip boundary conditions for the solid phase, and so on. These studies led to a better understanding of the grain evolution and macrosegregation formation in globular equiaxed solidification. Sun and Beckermann66 presented a diffuse interface model for two-phase flows with phase-change; surface tension is derived using an ensemble averaging approach. Wu and Ludwig67 presented a three-phase model for mixed columnarequiaxed solidification, with the parent melt as the primary phase, along with the solidifying columnar dendrites and globular equiaxed grains as two different secondary phases. The model was evaluated by comparing it to classical analytical models based on limited one-dimensional (1D) cases and showed satisfactory results. But, in order to apply this model for industrial castings, further improvements are still necessary. In their two-part paper, Ciobanas and Fautrelle68 presented a new multiphase Eulerian model for columnar and equiaxed dendritic solidification, where the mean conservation equations were derived by means of a statistical phase averaging technique, and the mathematical formulation of the model can be used for both columnar and equiaxed solidification. In part I, the equations were rigorously derived in the purely diffusive case, while in part II, one-dimensional simulations of Sn-Pb and Al-Cu directional solidification experiments involving CET phenomena was performed.


Figure 9. Geometry of steel solidification (reproduced with permission from ref 63).

Modeling of ingot solidification in steel was performed by Wu et al.69 using a multiphase model for predicting macrosegregation. Recently, Combeau et al.70 presented a multiphase solidification model tackling the motion and growth of equiaxed

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grains. The importance of the proper modeling of the grain morphology, while considering the motion of free-floating grains in steel ingots, was highlighted. Case Study: Macrosegregation in Large Steel Ingot Casting. In order to describe a typical case study in a multiphase modeling framework, the work done by Gu and Beckermann63 on the application of a multiphase model for simulating multicomponent steel solidification in a large ingot is presented here. It involved the solution of fully coupled two-phase conservation equations for the transport phenomena in the liquid, mush, and solid regions. Conjugate heat transfer in the mold and insulation materials, as well as the formation of a shrinkage cavity at the top, were taken into account. The effects of various elements in multicomponent steel on the liquidus temperature, partition coefficients, and liquidus slopes, backdiffusion and solid fraction evolution, liquid density, and buoyancy forces were accounted for. The ingot simulated was 1.016 m wide, 2.083 m deep, and 2.819 m high and was made up of AISI grade 4142 steel. The ladle mold had a taper of 11 pct, such that the ingot was 1.016 m wide at the top and 0.705 m wide at the bottom. The thickness of the cast iron mold varied from 0.337 m at the top to 0.276 m at the bottom (Figure 9). The simulation revealed that there were strong counterclockwise rotating convection cells in the initial stages of solidification, primarily because of the thermal buoyancy forces. In the thermal boundary layer along the mushy zone, and at the mush-liquid interface, the downward flowing melt reaches velocities of several centimeters per second. At subsequent times, as the mushy zone grows in size, the melt velocities decrease considerably. At t ) 15 000 s, the singlephase liquid region disappears completely and all the remaining flow is through the mush. Figure 10 shows the predicted variation of macrosegregation of carbon and sulfur along the vertical centerline, and its comparison with measurements from an industrial steel ingot. Although a good agreement is obtained in general, the neglect of sedimentation of free equiaxed grains prevents the prediction of the zone of negative macrosegregation, observed in the lower part of the ingot. It is also shown that the inclusion of the shrinkage cavity at the top and the variation

Figure 10. Macrosegregation of carbon and sulfur (reproduced with permission from ref 63).

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of the final solidification temperature due to macrosegregation were important in obtaining good agreement between the predictions and measurements. Macro/Micro Models. As discussed earlier, solidification is a multiscale process. However, inclusion of phenomena at various length scales in a single model framework is computationally prohibitive. Hence, many researchers focused on single-scale models (e.g., microscale or macroscale) by invoking the effects of other relevant scales through simplified or constitutive models. Models of macroscopic transport during solidification have been described in the previous section. Some prominent microscopic solidification models, addressing different aspects of microscale phenomena, namely, nucleation and growth, are discussed below. Microscopic Models. Nucleation and Growth. Microscopic modeling of solidification kinetics is based on nucleation and growth mechanisms. Invariably, the mechanism of nucleation considered in microscopic modeling is by bulk heterogeneous nucleation at foreign sites which are either already present within or intentionally added to the melt. Here, assuming a fixed number of initial nucleation sites, the nucleation rate is represented as a function of bulk undercooling by the following correlation:71 dn/dt ) K1[no - n(t)] exp(-K2 /T(∆T)2)

(13)

where K1 is proportional to the collision frequency of the atoms of the melt with the nucleation sites of the heterogeneous particles and K2 is related to the interfacial energy balance between the nucleus, the liquid, and the foreign substrate on which nucleation occurs. Although various researchers have used this approach to model equiaxed solidification, it fails to predict grain size accurately. The reason for this failure is the rapid site saturation depicted by the model. Accordingly, the final grain density is almost independent of solidification conditions, a fact which is not found experimentally. For the reasons cited above, the modeling of nucleation has been reconsidered recently using the pragmatic approach originally developed by Oldfield72 but with additional theoretical background of heterogeneous nucleation. Taking a statistical approach, Rappaz73 and Thevoz et.al.74 assumed that the quasi-instantaneous behavior occurs at a family of nucleation sites which is characterized by continuous rather than a discrete distribution. For a given undercooling, grain density is given simply by the integral of the distribution. n(∆T) )

∫

∆T

0

dn d(∆T') d(∆T')

(14)

If the extinction of nucleation site by the growing grain is taken into account, then the above equation transforms to n(∆T) )

∫

∆T

0

dn [1 - fs] d(∆T') d(∆T')

(15)

where fs is the volume fraction of solid already formed. Oldfield72 had used a simple polynomial law with two parameters in order to describe the nucleation in cast iron. However, other researchers have used a Gaussian function with three parameters. The characteristics of these distributions (mean undercooling, standard deviation, and total density of the grains) can be deduced from a few experimental measurements (e.g., differential thermal analysis types). Once solid nuclei are formed, they will grow. In contrast to nucleation, which is approached from an atomistic point of view, grain growth at microscopic scale is based on the conservation

of energy and species. This is similar to macroscopic conservation equations but additional phenomena such as capillary effects, local equilibrium of the various solid phases, and possible kinetic effects at high solidification rates are taken into account. This is presented in detail in a review work by Rappaz.73 Eutectic and Dendritic Growth. Eutectic alloys involve coupled growth of two solid phases from the liquid. The theory for eutectic columnar growth, proposed by Jackson and Hunt,75 permits calculation of ∆TE,c of the eutectic front as well as the lamellar spacing, λ, as a function of the growth rate. λ2Vs ) Kr/Kc

(16)

(∆TE,c)2Vs ) 1/(4KcKr)

(17)

and

where the constants Kc and Kr can be calculated from the properties of the alloy and arise from the solute diffusion and capillarity calculations, respectively. In the equaixed growth of eutectic grains, the relationships of eqs 16 and 17 can still be used if one neglects the thermal undercooling. For high cooling rates, however, one needs to consider the thermal undercooling as the temperature of the eutectic interface can be substantially higher than that of the undercooled liquid. The total undercooling then becomes: ∆TE,e ) ∆TE,c + ∆Tt

(18)

where the first undercooling is still given by eq 17 and ∆Tt is the thermal undercooling (i.e., the temperature difference between the solid-liquid interface and the temperature of the liquid). Using a quasi-stationary approximation, one gets ∆Tt ) g(R/Ro)RoVs

(19)

where the function g depends on the ratio of the grain size R, at time t, to the final grain size Ro. Putting the expressions for undercoolings in eq 19, one gets ∆TE,e ) 2(KcKrVs)1/2 + g(R/Ro)RoVs

(20)

Knowing the velocity Vs at a given total undercooling, ∆TE,e from eq 18, the λ spacing can then be computed using eq 16. Under normal solidification conditions, however, the value of thermal undercooling can be neglected. The formation of primary dendritic phases in binary alloys of noneutectic composition still lacks a complete theoretical solution. In constrained dendritic growth (columnar growth), the tip behavior is controlled primarily by solute diffusion and, at high rates by capillarity and kinetic effects.76 For equiaxed growth, Dustin and Kurz77 developed a model in which nucleation, solute diffusion, and thermal undercooling are considered in order to describe the solidification of a small volume whose temperature is otherwise uniform. Although their model is able to predict interesting features such as the number of grains and recalescence temperature, it ignores the overall solute balance. In order to predict final microstructural features, one needs to know the models for primary and secondary dendrite arm spacings. Also, to predict cooling curves, one needs to know how the solid fraction behind the tip changes with temperature. Macroscopic Transport-Transformation Kinetics Model. Most of the earlier work based on the macroscopic transport of heat have ignored transformation kinetics. The coupled model


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78

was provided by Stefanescu. Subsequently, the initial work of Oldfield71 on microstructure evolution was extended by Stefanescu79 to explain inoculation in gray iron. Thevoz et al.80 presented a general micro/macroscopic model of solidification that takes into account nucleation of new grains within the undercooled melt, the kinetics of the dendrite tips or of the eutectic front, and a solute balance at the scale of the grain in the case of dendritic alloys. The works by Su et al.,81 Thevoz et al.,80 and Stefanescu and Kanetkar82 were with no solid motion where grains nucleated do not move, ignoring grain coalescence and dissolution. Wang and Beckermann58 developed a model which had solid movement with liquid. In each macro-element, the empirical nucleation laws were used for nucleation and solid grains can transport for a macrovolume element. In order to improve accuracy and reduce computational time, there were coupling techniques developed: the latent heat method by Stefanescu and Kanetkar82 and Kanetkar and Stefanescu83 and the microenthalpy method by Rappaz and Thevoz,84 for example. One important aspect in this modeling approach is the comprehensive treatment of nucleation. There are several nucleation models proposed by different researchers,71,79,80,85,86 which requires parameters from experimental data. There is a debate as to which nucleation models works best. The macroscopic transport-transformation kinetics model has been used in several studies. Zou and Tseng87 simulated peritectic solidification using a macroscopic transport-transformation kinetics model for a binary system. After solidification is completed, the solid state transformation of carbon steel is also considered, but there is very little validation of the work. Nastac and Stefanscu88 proposed a deterministic model to account for the nucleation and growth of both white and gray eutectics, along with the effect of microsegregation of the third element. Nastac and Stefanscu88 used a macroscopic transporttransformation kinetics model for simulating solidification of Inconel 718 and experimentally validated it for variable cooling rates. It was demonstrated that the cooling rate affects the evolution of the solid fraction significantly, influencing the amount of Laves phases obtained at the end of solidification. Desbiolles et al.89 developed a 3D cellular automaton-finite element (CAFE´) model for simulating grain structure evolution in investment cast parts which was later used for investigating freckle formation.90 Case Study: Solidification of Al-Si Alloy in a Hollow Cylinder. In order to describe a typical case study using the macroscopic transport-transformation kinetics modeling framework, the work reported by Kanetkar and Stefanescu83 on the application of the latent heat method for a macro/micro model of eutectic solidification is presented. To account for the heat evolution term in the conductivity equation, an original latent heat approach was introduced, consisting of the calculation of the latent heat evolved by the fraction of solid formed as a function of time. In turn, the fraction of solid was calculated on the basis of nucleation and growth kinetics. The heat transfer coefficient, used for simulation, was determined by using an analytical approach for hollow cylinders. A 160 mm long air set (pep set type) sand mold with an outer diameter of 110 mm was used for casting 30 and 50 mm diameter cylindrical bars. Temperatures in the melt and the mold during solidification were measured using thermocouples. Two sets of three thermocouples were placed in the mold at a 90° angle to measure the heat transfer coefficient at the metal mold interface while the remaining two were used to record the cooling curves during solidification at two different locations.

Figure 11. Experimental and macromodel predicted thermal history for different size castings (reproduced with permission from ref 83).

Two eutectic alloys used in this investigation were Al-12 wt % Si and a 3.7 wt % C-2 wt % Si cast iron. A 2 kg charge of Al-Si of eutectic composition was melted in a resistance heating furnace. The melt was poured into the mold at 700 °C. Cast iron was melted in a high frequency induction furnace and poured at 1300 °C. The temperature data was recorded after every 2 s for a period of about 25 min to allow for complete solidification. The macroscopic transport-transformation kinetics solidification model developed by Kanetkar and Stefanescu83 was used to simulate these experiments. Figure 11 shows a plot of the cooling curves for Al-Si eutectic alloys at the center of experimental castings. The cooling curves exhibit undercooling of the melt at the beginning of solidification and subsequent recalescence. In addition, the degree of undercooling and the arrest temperature depend on the solidification time. It is seen that the prediction from the macroscopic transport-transformation kinetics eutectic model shows good agreement with the experimental data. Microsegregation in Macroscopic Models. One of the important aspects of interface morphology is microsegregation which requires tracking the evolution of the solutal field. The solute redistribution at the microscale can be considered by creating a microvolume element extending from the middle of the dendrite to the middle of the interdendritic liquid region and evaluating microscopic solute redistribution. There are several analytical microsegregation models derived using simplifying assumptions on geometry, solid diffusion, liquid diffusion, partition coefficient, and dendrite coarsening. Typical models based on analytical solution are the Lever rule and those of Scheil, Brody-Flemings, Clyne-Kurz, Ohanka, SarrealAbbasaschian, Kobayashi, and Nastac-Stefanescu.78 As far as metals growing under normal solidification conditions are concerned, local thermodynamic equilibrium is assumed to exist at the solid-liquid interface. At the interface, the solid and the liquid concentrations are linked by an equilibrium partition coefficient.91,92 Cs ) kpCl

(21)

This compositional difference always leads to concentration variations in the solidified alloy, which are known as segregation. Because solute can be transported by diffusion or by convection, the segregation pattern is quite different depending on the process involved. Convection can lead to the transport of mass over very large distances, compared to those involved

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in diffusional processes and may result in macrosegregation, i.e., compositional differences over distances comparable to the size of a large casting. In the equilibrium condition, the liquid and solid composition can be calculated by the Lever rule: Cl ) Co/[1 - (1 - kp)fs]

(22)

In actuality, however, the equilibrium solidification is difficult to attain due to low diffusivity of solute in solid. In nonequilibrium solidification, the local equilibrium between solid and liquid can, at best, be assumed to hold good at the interface. With respect to nonequilibrium situations, there are a host of possibilities. If the solute redistribution in a volume element between platelike dendrites is considered and if negligible undercooling at the solid-liquid interface with no net flow of solute through the volume element is assumed, the liquid composition can be calculated as described below (more details in the work of Kurz and Fisher93). First, in the case of Ds ) 0 and Dl ) ∞ (i.e., no diffusion in the solid and complete mixing in the liquid) Scheil’s91 equation can be used to calculate liquid concentration. Cl ) Co[1 - fs]p(k-1)

(23)

Second, in the case of Ds * 0 (finite diffusion in the solid) and Dl ) ∞, the following equation can be used: Cl ) Co[1 - ψfs]p(k-1)ψ

(24)

ψ ) 1 - 2Akp /(1 + 2A)

(25)

A ) 4Dstf /λ2

(26)

where,

and

In eq 26, Ds is the diffusion coefficient in the solid phase, tf is the local solidification time, and λ is the dendrite arm spacing. Equation 26 is not only valid for platelike dendrites but also for columnar dendrites if A in eq 25 is doubled. Third, there is a solid-state diffusion and solute buildup ahead of the solid-liquid interface. An analytical solution for this actual case is yet to be obtained. However, in the case where Ds ) 0 and a solute boundary layer controls the solute transfer in the liquid, an effective partition coefficient, keff, has been derived for semi-infinite, volume-element, and steady-state conditions. keff )

Cs ) Co

kp

( )

kp + (1 - kp) exp

-Rδc Dl

(27)

where, Cs is the solid composition at the interface, R is the growth rate, and δc is the solute boundary layer ahead of the interface. The coefficient keff can be used instead of k in eq 21 for evaluating Cl. For a generic case, a numerical solution is the only practical option available, and the same is described in detail in this paper in the section on mathematical modeling. In practice, microsegregation is usually evaluated by the microsegregation ratio, which is the ratio of the maximum solute composition to the minimum solute composition after solidification, and by the amount of nonequilibrium second phase in the case of alloys that form eutectic compounds. Equation 23 is often used in the case of low back diffusion parameter A (for

example, in the case of aluminum castings), and eq 24 is used in the case of higher A (for example, in the case of steel castings). However, the estimation of real microsegregation is difficult because the real phenomena are more complex and the solid composition after solidification cannot be calculated by eq 24 if solid diffusion is not negligible. The following points should be considered: Solidification Mode and Structure. Microsegregation varies considerably with the history of the growth of solid. For example, microsegregation often increases with cooling rate in the case of equiaxed dendritic solidification, but it decreases in the case of unidirectional dendritic solidification. Also, in equiaxed globular grain structure, the information on its formation mechanism and the history of the grain (i.e., dendrite melt-off and settling in the liquid) is important. Morphology of the Dendrite and Diffusion Path. In the case where solid-state diffusion is not negligible, the diffusion path or the morphology of the solid is very important. Although, eq 24 can be applied to the volume element in a primary or secondary dendrite, the real diffusion occurs three dimensionally in both dendrites. Therefore, careful attention is required to determine the dendrite spacing λ. Phase Transformation. If a phase transformation occurs during solidification, the microsegregation can change considerably because the equilibrium partition coefficient varies with phase. Movement of the Liquid Phase. In many cases, the interdendritic liquid does not remain stationary but moves by solidification contraction or by thermo-solutal convection, resulting in varying degrees of microsegregation. Other Effects. Other important aspects from a microsegregation point of view are the effects of the third solute element, dendrite coarsening, temperature and concentration dependence of the diffusion coefficient, undercooling, Soret effect, etc. Multiscale Approach. Quantitative evaluation of microsegregation can be done using analytical models but with limited predictability due to simplifying assumptions. For example, if the Scheil equation is used for microsegregation prediction, then diffusion of the solute into solid is neglected. However, in a real casting, the diffusion in the solid phase can be significant, especially during the final stages of solidification. Dutta5 had highlighted the issue of an unimaginably high computational power requirement for a purely microscopic approach based simulation, and it would not be a practical approach for simulation of large-scale solidification structures. In order to bridge these two scales, a combination of adaptive meshing and “homogenization” based “upscaling” strategy was proposed.46 Recently, Tan and Zabaras94 proposed a two-scale model based on a database approach to investigate alloy solidification. A microscale model was used to generate the database by solving the selected sample problems. A macroscale model was used to efficiently compute solidification with inputs from the database. The proposed multiscale framework for solidification was demonstrated for the two cases with the same irregular domain but with different boundary conditions. The predicted microstructure type (CET location), microstucture size, and volume fraction using the multiscale method compares well with the microscale model results. But the test cases were without considering convection. In the context of micromacro modeling, the lattice Boltzmann method (LBM) has certain merits compared to a continuum based formulation. The LBM is fundamentally based on a microscopic particle model and mesoscopic kinetics equations; hence, micro and meso can be bonded together. A recent work


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is reported by Chatterjee and Chakraborty on hybrid lattice Boltzmann methodology, and it is developed for simulating convection-diffusion transport processes pertinent to solidification problems. It was showed by Brooks et al.96 that little solid-state diffusion occurs during solidification and cooling of primary austenite solidified welds of Fe-Ni-Cr ternary alloys, whereas structures that solidify as ferrite may become almost completely homogenized as a result of diffusion. The experiments conducted by Hillert et al.97 on Fe-Cr-C alloys demonstrated that C and Cr have infinite diffusivity in the liquid, and C can be assumed as having infinite diffusivity in the fcc solid, the back diffusion of chromium cannot be ignored. Also, the back-diffusion during cooling after the end of solidification cannot be neglected. Many numerical microsegregation models have been proposed but use of this with macroscopic equations is rare, mainly due to issues of significant increase in computational time. Because of this, most of the work on numerical schemes for energy and mass transport use analytical models of microsegregation. Voller et al.98 and Voller99 developed a general numerical approach for coupling the temperature and concentration fields where an explicit time stepping scheme is used to solve the thermal and concentration conservation equations. The disadvantage of this approach is a small time step for stability, but it is offset by a straightforward numerical scheme that can readily incorporate multiscale behavior. The incorporation into the solution algorithm of local scale microsegregation models that can accurately account for both back diffusion and coarsening effects was also demonstrated in a one-dimensional problem involving the solidification of a binary alloy from below. The approach was different from some other approaches reported in the literature pertaining to incorporation of microscale issues in a macroscopic framework. For example, in the work of Chakraborty and Dutta,100 a modified partition coefficient has been used to account solutal undercooling near the solid-liquid interface. Case study: Solidification of Pb-15 wt % Sn in a Square Cavity. A numerical scheme with hybrid explicit and implicit time-stepping for macrosegregation which accurately considers microsegregation was developed recently by Pardeshi et al.101 An explicit time-stepping scheme was used for solving coupled temperature and concentration fields, while an implicit scheme was 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 and dendrite coarsening in solid are occurring at microscale (Voller99). This method offers distinct advantages of simplicity and flexibility in incorporation of microscale models, as demonstrated by the use of a back-diffusion parameter in the microsegregation model. The model was used to simulate solidification of Pb-15 wt % Sn alloy in a square cavity having dimensions 0.1 m × 0.1 m. For this test case, experimental data is reported in the literature23 along with simulation results assuming the Schiel model for microsegregation. The variations of boundary temperatures with time were in accordance with the experimental data reported in ref 23. The mushy region was modeled using West permeability relations as reported by Sahani et al.23 During solidification, Sn is rejected into the liquid, which is lighter than the solvent (Pb) and hence tends to move up. Hence, thermal and solutal buoyancy effects oppose each other for the nominal composition 15 wt % Sn in the alloy. Figures 12 and 13 show the final macrosgregation profiles at different vertical sections of the cavity. The results obtained by the present model (using

Figure 12. Final macrosegregation for Pb-15% wt Sn alloy at x ) 9.47 cm (reproduced with permission from ref 101).

Figure 13. Final macrosegregation for Pb-15% wt Sn alloy at x ) 8 cm (reproduced with permission from ref 101).

constitutive model for back diffusion) are compared with the experimental observations23 and with the corresponding numerical results obtained using Schiel model for microsegration. From these figures, it is evident that there is a positive macrosegregation at the top of the cavity, and negative macrosegregation in the rest of the cavity. Figures 12 and 13 depict concentration variations along vertical lines located at x ) 9.47 cm and x ) 8.0 cm, respectively, corresponding to the region close to the right wall. Near the right wall, solidification commences from the bottom region and progresses toward the top, because of the nature of melt convection. As a result, it can be expected that the bottom region will have negative segregation and the top region will experience positive segregation. The negative segregation in the bottom region will be diminished by back-diffusion effects, as observed in Figures 12 and 13. The positive segregation in the upper region of the cavity is enhanced by the melt convection, which carries the excess solute in a clockwise direction to the top by thermal buoyancy. At later stages of solidification, however, the thermal buoyancy is opposed by solutal buoyancy. With back diffusion, the strength of solutal buoyancy is reduced, as there is less solute build-up at the interface. Hence, positive segregation at the top region of the cavity is under-predicted by Schiel’s microsegregation model, as observed in Figures 12 and 13. It is evident

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from the figures that the results obtained from the present numerical simulation incorporating a constitutive model for back diffusion in solid are closer to the experimental results, as compared to the previous numerical simulation.23 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. Summary Solidification processes are multiscale in nature and are influenced by phenomena occurring at different length and time scales. In this review, several modeling approaches to solve the solidification problems for macrosegregation predictions were highlighted. For industrial size problems, it is not possible with the currently available computational power to resolve all length and time scales using a single scale model, and considerable time is required before such a level of computer power becomes available.102 In this scenario, the best approach going forward would be a micro/macro approach where it is important to enhance this modeling approach to capture more microscale physics with supporting sophisticated experiments for rigorous model validation. Although there has been considerable work done in the area of a macro/micro approach, more research efforts are required to incorporate more microscale physics and processes. Acknowledgment R.P. and A.S. would like to thank the management of TCS Innovation Labs-Tata Research Development and Design Centre for permission to publish this article and Dr. Pradip for encouragement and support. Additionally, P.D. gratefully acknowledges the continuous financial support for work in this field provided by the Ministry of Mines and Department of Science and Technology, Government of India. List of Symbols c ) specific heat B ) buoyancy term C ) composition D ) mass diffusion coefficient g ) gravitational fl ) liquid fraction fs ) solid fraction h ) enthalpy k ) thermal conductivity kp ) equilibrium partition coefficient K ) permeability Ko ) permeability coefficient Kc, Kr ) constants in thermal undercooling equation n ) nuclei density p ) pressure Pd ) phase diagram coefficient R ) grain size Ro ) final grain size Rg ) growth rate S ) source term t ) time Tf ) fusion temperature u, V ) velocity component in x and y directions V ) velocity vector x, y ) orthogonal axes µ ) dynamic viscosity F ) density

Φ ) general dependent variable. βT ) thermal expansion coefficient βC ) solutal expansion coefficient ∆T ) temperature undercooling λ ) dendrite arm spacing δc ) solutal boundary layer Subscripts c ) cold boundary e ) eutectic l ) liquid phase liq ) liquidus m ) mixture o ) initial condition s ) solid phase sol ) solidus fus ) fusion

Literature Cited (1) Beckermann, C.; Viskanta, R. Mathematical Modeling of Transport Phenomena During Alloy Solidification. Appl. Mech. ReV. 1993, 46, 1. (2) Ghosh, A. Segregation in cast products. Sadhana 2001, 26, 5. (3) Beckermann, C. Encyclopedia of Materials: Science and Technology; Pergamon: Elmsford, NY, 2002; 4733-4739. (4) Dantzig, J. Continuum Scale Simulation of Engineering Materials. Solidification Processes From Dendrites to Design; Raabe, D., Roters, F., Barlat, F., Chen, L., Eds.; Wiley-VCH Verlag: Weinheim, Germany, 2004; Chapter 34. (5) Dutta, P. Enthalpy based macro-scale and micro-scale models for solidification. 7th ISHMT-ASME Heat and Mass Transfer Conference and 18th National Heat and Mass Transfer Conference, Guwahati, Jan 4-6, 2006. (6) Beckermann, C.; Viskanta, R. Double-Diffusive Convection during Dendritic Solidification of a Binary Mixture. Phys. Chem. Hydrodyn. 1988, 10, 195. (7) Basu, B.; Date, A. W. Numerical modelling of melting and solidification problems - a review. Sadhana 1988, 13, 169. (8) Beckermann, C.; Wang, C. Y. Multi-Phase/-Scale Modeling of Transport Phenomena in Alloy Solidification. Annual ReView of Heat Transfer VI; Tien, C. L., Ed.; Begell House: New York, 1995; Vol. 6, pp 115-198. (9) Flemings, M. C. Our Understanding of Macrosegregation: Past and Present. ISIJ Int. 2000, 40, 833. (10) Rappaz, M. Modelling of microstructure formation in solidification processes. Int. Mater. ReV. 1989, 34, 93. (11) Beckermann, C. Modeling of macrosegregation: application and future needs. Int. Mater. ReV. 2002, 47, 243. (12) Lesoult, G. Macrosegregation in steel strands and ingots: characterization, formation and consequences. Mater. Sci. Eng. A. 2005, 413, 19. (13) Flemings, M. C.; Nereo, G. E. Macrosegregation: Part I. Trans. Metall. Soc., AIME 1967, 239, 1449. (14) Flemings, M. C.; Mehrabian, R.; Nereo, G. E. Macrosegregation: Part II. Trans. AIME 1968, 242, 41. (15) Flemings, M. C.; Nereo, G. E. Macrosegregation: Part III. Trans. AIME 1968, 242, 50. (16) Mehrabian, R.; Keane, M.; Flemings, M. C. Interdendritic fluid flow and macrosegregation; influence of gravity. Metall. Mater. Trans. B 1970, 1, 1209. (17) Szekely, J.; Jassal, A. S. An experimental and analytical study of the solidification of a binary dendritic system. Metall. Mater. Trans. B 1978, 9, 389. (18) Bennon, W. D.; Incropera, F. P. A continuum model for momentum, heat and species transport in binary solid-liquid phase change system - I. Model formulation. Int. J. Heat Mass Transfer 1987, 30, 2161. (19) Bennon, W. D.; Incropera, F. P. A continuum model for momentum, heat and species transport in binary solid-liquid phase change system - I. Model formulation. Int. J. Heat Mass Transfer 1987, 30, 2161. (20) Prakash, C.; Voller, V. R. On the numerical solution of continuum mixture model equations describing binary solid-liquid phase change. Numer. Heat Transfer B 1989, 15, 171. (21) Voller, V. R.; Brent, A. D.; Prakash, C. Modelling the mushy region in a binary alloy. Appl. Math. Modell. 1990, 14, 320. (22) Amberg, G. Computation of macrosegregation in an iron-carbon cast. Int. J. Heat Mass Transfer 1991, 34, 217.

Ind. Eng. Chem. Res., Vol. 48, No. 19, 2009 (23) Sahani, H.; Amberg, G.; Fredriksson, H. On the formation of macrosegregation in unidirectionally solidified Sn-Pb and Pb-Sn alloys. Met. Trans. A 1992, 23A, 2301. (24) Ni, J.; Incropera, F. P. Extension of the continuum model for transport phenomena occurring during metal alloy solidification 1. The conservation equations. Int. J. Heat Mass Transfer 1995, 38, 1271. (25) Ni, J.; Incropera, F. P. Extension of the continuum model for transport phenomena occurring during metal alloy solidification 2. microscopic Considerations. Int. J. Heat Mass Transfer 1995, 38, 1285. (26) Prescott, P. J.; Incropera, F. P. Convective - Transport phenomena and macrosegregation during solidification of a binary metal alloy 1. numerical Predictions. Trans. ASME J. Heat Transfer 1994, 116, 735. (27) Prescott, P. J.; Incropera, F. P.; Gaskell, D. R. Convective transport phenomena and macrosegregation during solidification of a binary metal alloy: II experiments and comparisons with numerical predictions. Trans. ASME J. Heat Transfer 1994, 116, 742. (28) Krane, M. J. M.; Incropera, F. P. Experimental validation of continuum mixture model for binary alloy solidification. Trans. ASME J. Heat Transfer 1997, 11, 783. (29) Singh, A. K.; Basu, B. Mathematical modeling of macrosegregation of iron carbon binary alloy: Role of double diffusive convection. Met. Trans. B 1995, 26B, 1069. (30) West, R. On the permeability of the two-phase zone during solidification of alloys. Met. Trans. A 1985, 16A, 693. (31) Krane, M. J. M.; Incropera, F. P.; Gaskell, D. R. Solidification of ternary metal alloys 0.1. Model development. Int. J. Heat Mass Transfer 1997, 40, 3827. (32) Krane, M. J. M.; Incropera, F. P.; Gaskell, D. R. Solidification of a ternary metal alloy: A comparison of experimental measurements and model predictions in a Pb-Sb-Sn system. Metall. Mater. Trans. A 1998, 29, 843. (33) Rady, M.; Satyamurty, V.; Mohanty, A. Thermosolutal convection and macrosegregation during solidification of hypereutectic and hypoeutectic binary alloys in statically cast trapezoidal ingots. Metall. Mater. Trans. B 1997, 28, 943. (34) Singh, A. K.; Basu, B. On convection in mushy phase and its effect on macrosegregation. Metall. Mater. Trans. A 2000, 31A, 1687. (35) Singh, A. K.; Basu, B. Numerical study of effect of cooling rate on double-diffusive convection and macrosegregation in iron-carbon System. ISIJ Int. 2001, 41, 1481. (36) Vreeman, C. J.; Krane, M. J. M.; Incropera, F. P. The effect of free-floating dendrites and convection on macrosegregation in direct chill cast aluminum alloys Part I: model development. Int. J. Heat Mass Transfer 2000, 43, 677. (37) Vreeman, C. J.; Incropera, F. P. The effect of free-floating dendrites and convection on macrosegregation in direct chill cast aluminum alloys Part II: predictions for Al-Cu and Al-Mg alloys. Int. J. Heat Mass Transfer 2000, 43, 687. (38) Vreeman, C. J.; Scholz, J. D.; Krane, M. J. M. Direct chill casting of aluminum alloys: Modeling and experiments on industrial scale ingots. Trans. ASME-J. Heat Transfer 2002, 124, 947. (39) Krane, M. J. M. Macrosegregation development during solidification of a multicomponent alloy with free-floating solid particles. Appl. Math. Modell. 2004, 28, 95. (40) Mat, M. D.; Ilegbusi, O. J. Application of a hybrid model of mushy zone to macrosegregation in alloy solidification. Int. J. Heat Mass Transfer 2002, 45, 279. (41) Kumar, A.; Dutta, P. Modeling of transport phenomena in continuous casting of non-dendritic billets. Int. J. Heat Mass Transfer 2005, 48, 3674. (42) Chowdhury, J.; Ganguly, S.; Chakraborty, S. Implication of solid phase interaction mechanisms on momentum, heat and solute transport in semi-solid material processing. Int. J. Heat Mass Transfer 2007, 50, 2692. (43) Ganguly, S.; Chakraborty, S. Stochastic convective transport in presence of fragmented dendrites in a solidifying binary melt. Eur. Phys. J. Appl. Phys. 2007, 40, 221. (44) Chakraborty, S.; Chakraborty, N.; Kumar, P.; Dutta, P. Studies on turbulent momentum, heat and species transport during binary alloy solidification in a top-cooled rectangular cavity. Int. J. Heat Mass Transfer 2003, 46, 1115. (45) Liu, W. Finite element modelling of macrosegregation and thermomechanical phenomena in solidification processes. Ph.D. thesis, ENSMP, France, 2005. (46) Ganesan, S.; Poirier, D. R. Conservation of mass and momentum for the flow of interdendritic liquid during solidification. Metall. Trans. B 1990, 21B, 173. (47) DasGupta, D.; Basu, S.; Chakraborty, S. Effective property predictions in multi-scale solidification modeling using homogenization theory. Phys. Lett. A 2006, 348, 386.

8803

(48) Piwonka, T. S.; Flemings, M. C. Pore formation in solidification. Trans. AIME 1966, 236, 1157. (49) Singh, A. K.; Basu, B.; Ghosh, A. Role of appropriate permeability model on numerical prediction of macrosegregation. Met. Trans. B. 2006, 37, 799. (50) Singh, A. K.; Pardeshi, R.; Basu, B. Modelling of convection during solidification of metal and alloys. Sadhana 2001, 26, 139. (51) Chakraborty, S.; Dutta, P. A Generalized formulation for evaluation of latent heat functions in enthalpy-based macroscopic models for convection-diffusion phase change process. Met. Trans. B 2001, 32B, 562. (52) Ganguly, S; Chakraborty, S. A Generalized formulation of latent heat functions in enthalpy-based mathematical models for multi-component alloy solidification systems. Met. Trans. B 2006, 37B, 143. (53) Prakash, C. Two-phase model for binary solid-liquid phase change, Part I: Governing equations. Numer. Heat Transfer B 1990, 18, 131. (54) Prakash, C. Two-phase model for binary solid-liquid phase change, Part II: some illustrative example. Numer. Heat Transfer B 1990, 18, 131. (55) Ni, J.; Beckermann, C. A volume-averaged two-phase mode1 for transport phenomena during solidification. Metall. Mater. Trans. 1991, 22B, 349. (56) Wang, C. Y.; Beckermann, C. A unified solute diffusion model for columnar and equiaxed dendritic alloy solidification. Mater. Sci. Eng., A 1993, 171A, 199. (57) Beckermann, C.; Schneider, M. C. Simulation of Micro-/macrosegregation during the solidification of a low-alloy steel. ISIJ Int. 1995, 35, 1300. (58) Wang, C. Y.; Beckermann, C. Equiaxed dendritic solidification with convection: part I. multiscale/multiphase modeling. Met. Trans. A 1995, 27A, 2754. (59) Wang, C. Y.; Beckermann, C. Equiaxed dendritic solidification with convection: part II. Numerical simulation for an Al-4 wt. pct. Cu alloy. Met. Trans. A 1995, 27A, 2765. (60) Wang, C. Y.; Beckermann, C. Equiaxed dendritic solidification with convection: part III. comparison with NH4Cl-H2O experiments. Met. Trans. A 1995, 27A, 2784. (61) Beckermann, C.; Ni, J. Simulations of sedimentation in equiaxed alloy solidification. Int. Commun. Heat Mass Transfer 1996, 23, 315. (62) Reddy, A. V.; Beckermann, C. Modeling of macrosegregation due to thermosolutal convection and contraction driven flow in direct chill continuous casting of an Al-Cu round ingot. Metall. Mater. Trans. B 1997, 28B, 479. (63) Gu, J. P.; Beckermann, C. Simulation of convection and macrosegregation in a large steel ingot. Metall. Mater. Trans. A 1999, 30A, 1357. (64) Martorano, M. A.; Beckermann, C.; Gandin, Ch.-A. A Solutal interaction mechanism for the columnar-to-equiaxed transition in alloy solidification. Metall. Mater. Trans. A 2003, 34A, 1657. (65) Wu, M.; Ludwig, A.; Bu¨hrig-Polaczek, A.; Fehlbier, M.; Sahm, P. R. Influence of convection and grain movement on globular equiaxed solidification. Int. J. Heat Mass Transfer 2003, 46, 2819. (66) Sun, Y.; Beckermann, C. Diffuse interface modeling of two-phase flow based on averaging: mass and momentum equations. Phys. D 2004, 198, 281. (67) Wu, M.; Ludwig, A. A three-phase model for mixed columnarequiaxed solidification. Metall. Mater. Trans. A 2006, 37, 1613. (68) Ciobanas, A. I.; Fautrelle, Y. Ensemble averaged multiphase Eulerian model for columnar/equiaxed solidification of a binary alloy: I. The mathematical model. J. Phys. D: Appl. Phys. 2007, 40, 3733. (69) Wu, M.; Ko¨no¨zsy, L.; Ludwig, A.; Schu¨tzenho¨fer, W.; Tanzer, R. On the formation of macrosegregations in steel ingot castings. Steel Res. Int. 2008, 79, 637. (70) Combeau, H.; Zaloznik, M.; Hans, S.; Richy, P. Prediction of macrosegregation in steel ingots: influence of the motion and the morphology of equiaxed grains. Metall. Mat. Trans. B 2009, 40, 289. (71) Kurz W.; Fisher, D. J. Fundamentals of Solidification; Trans Tech Publications: Switzerland, 1992. (72) Oldfield, W. Quantitative approach to casting solidification: freezing of casting iron. ASM Trans. 1966, 59, 945. (73) Rappaz, M. Modeling of microstructural evolution. ASM Int. Met. Handbook 1988, 15, 883. (74) Thevoz, Ph.; Desbiolles, J. L.; Rappaz, M. Modelling of equiaxed microstructure formation in casting. Met. Trans. A 1989, 20A, 311. (75) Jackson, K. A.; Hunt, J. D. Lamellar and rod eutectic growth. Trans. AIME 1966, 236, 1129. (76) Mullins, W. W.; Sekerka, R. F. Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys. 1964, 35, 444. (77) Dustin, I.; Kurz, W. Modeling of cooling curves and microstructure during equiaxed dendritic solidification. Z. Metall. 1986, 77, 265. (78) Stefanescu, D. M. Science and Engineering of Casting Solidification, 1st ed.; Kluwer Academic/Plenum Publishers: New York, 2002.

8804


(79) Stefanescu, D. M.; Trufinescu, S. Z. Metallkde 1974, 65, 610. (80) Thevoz, P.; Desbiolles, J. L.; Rappaz, M. Modeling of equiaxed microstructure formation in casting. Met. Trans. A 1989, 20A, 311. (81) Su, K. C.; Ohanka, I.; Yanauchi, I.; Fukusako, T. In The physical metallurgy of cast iron; Fredriksson, H., Hillert, M., Eds.; North Holland: New York, 1984. (82) Stefanescu, D. M.; Kanetkar, C. In State of the Art of Computer Simulation of Casting and Solidification Process; Fredriksson, H., Ed.; Les Edition de physique, Les Ulis, France, 1986. (83) Kanetkar, C.; Stefanescu. D. M. In Modeling of Casting and Welding Processes IV; Giamei, A., Abbaschian, G. J., Eds.; TMS: Warrendale, Pa, 1988. (84) Rappaz, M.; Thevoz, P. Solute diffusion-model for equiaxed dendritic growth - Analytical solution. Acta Met. 1987, 35, 2929. (85) Maxwell, I.; Hellawell, A. Simple model for grain refinement during solidification. Acta Met. 1975, 23, 229. (86) Goettsch, D. D.; Jonathan, A.; Dantzig, J. A. Modeling microstructure development in gray cast irons. Metall. Mater. Trans. A 1994, 25, 1063. (87) Zou, J.; Tseng, A. A. Microscopic modeling of fundamental phase transformations in continuous castings of steel. Metall. Trans. A 1992, 23, 457. (88) Nastac, L.; Stefanescu, D. M. Macrotransport-solidification kinetics modeling of equiaxed dendritic growth: Part II. Computation problems and validation on INCONEL 718 superalloy castings. Metall. Trans. A 1996, 27A, 4061. (89) Desbiolles, J. L.; Gandin, C. A.; Joyeux, J. F.; Rappaz, M.; Thevoz, P. Modeling of dendritic grain structures. In Modeling of Casting, welding and adVanced Solidification Processes VIII; Thomas, B. G., Beckermann, C., Eds.; TMS: Warrendale, PA, 1998. (90) Desbiolles, J. L.; Thevoz, P.; Rappaz, M. Micro-/Macrosegregation modeling in casting: A fully coupled 3D model. In Modeling of Casting, welding and adVanced Solidification Processes X; Stefanescu, D. M., Warren, J. A., Jolly, M. R., Krane, M. J. M., Eds.; TMS: Warrendale, PA, 2003. (91) Chalmers, B. Principles of Solidication; John Wiely & Sons Inc.: New York, 1964.

(92) Flemings, M. C. Solidification Processing; McGraw-Hill: New York, 1974. (93) Kurz, W.; Fisher, D. J. Fundamentals of Solidification; Trans Tech Publications: Switzerland, 1984. (94) Tan, L.; Zabaaras, N. Multiscale modeling of alloy solidification using a database approach. J. Comput. Phys. 2007, 227, 728. (95) Chatterjee, D.; Chakraborty, S. A hybrid lattice Boltzmann model for solid-liquid phase transition in presence of fluid flow. Phys. Lett. A 2006, 351, 359. (96) Brooks, J. A.; Baskes, M. I.; Greulich, F. A. Solidification modeling and solid-state transformations in high-energy density stainless steel welds. Met. Trans. A 1991, 22, 915. (97) Hillert, M.; Hoglund, L.; Schalin, M. Role of back-diffusion studied by computer simulation. Metall. Mater. Trans. 1999, 30, 1635. (98) Voller, V. R.; Mouchmov, A.; Cross, M. An explicit scheme for coupling temperature and concentration fields in solidification models. Appl. Math. Model. 2004, 28, 79. (99) Voller, V. R. Numerical methods for phase-change problems. In Handbook of Numerical Heat Transfer, second ed.; Minkowycz, W. J., Sparrow, E. M.; Murthy, J. Y., Eds.; John Wiley & Sons: Hoboken, NJ, 2006, Chapter 19. (100) Chakraborty, S.; Dutta, P. The effect of solutal undercooling on double-diffusive convection and macro-segregation during binary alloy solidification: a numerical investigation. Int. J. Numer. Meth. Fluids 2002, 38, 895. (101) Pardeshi, R.; Voller, V. R.; Singh, A. K.; Dutta, P. An explicitimplicit time stepping scheme for solidification models. Int. J. Heat Mass Transfer 2008, 51, 3399. (102) Voller, V. R.; Porte-Agel, F. Moore’s Law and Numerical Modeling. J. Comput. Phys. 2002, 179, 698.

ReceiVed for reView January 30, 2009 ReVised manuscript receiVed July 24, 2009 Accepted July 28, 2009 IE900164F

Modeling of Convection and Macrosegregation ... - ACS Publications

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