Equiaxed dendritic solidification with convection: Part I. Multiscale/multiphase modeling

Equiaxed dendritic solidification in the presence of melt convection and solid-phase transport is investigated in a series of three articles. In part I, a multiphase model is developed to predict com-position and structure evolution in an alloy solidifying with an equiaxed morphology. The model accounts for the transport phenomena occurring on the macroscopic (system) scale, as well as the grain nucleation and growth mechanisms taking place over various microscopic length scales. The present model generalizes a previous multiscale/multiphase model by including liquid melt convec-tion and solid-phase transport. The macroscopic transport equations for the solid and the interdendritic and extradendritic liquid phases are derived using the volume averaging technique and closed by supplementary relations to describe the interfacial transfer terms. In part II, a numerical application of the model to equiaxed dendritic solidification of an Al-Cu alloy in a rectangular cavity is dem-onstrated. Limited experimental validation of the model using a NH₄C1-H₂O transparent model alloy is provided in part III.     

Prediction of Columnar to Equiaxed Transition during Diffusion-Controlled Dendritic Alloy Solidification

A model is presented to predict the columnar to equiaxed transition (CET) in alloy castings. The model is based on a multiphase approach and accounts for heat and solute diffusion, as well as for grain nucleation, growth, and morphology. The model equations are applicable to both columnar and equiaxed dendritic solidification, thus offering an efficient single-domain formulation. A fixed grid, fully implicit finite-difference procedure is employed in the numerical solution, and a novel front tracking technique is incorporated that is also implicit in nature and readily applies to multidimensional situations. Calculations are performed for one-dimensional (1-D) and two-dimensional (2-D) castings of Al-Cu and Sn-Pb alloys. The calculated CET positions are compared with previous measurements in a (1-D) ingot cast under well-controlled conditions, and good agreement is found. The effects of various casting parameters on the CET are numerically explored.     

A three-phase model for mixed columnar-equiaxed solidification

A three-phase model for mixed columnar-equiaxed solidification is presented in this article. The three phases are the parent melt as the primary phase, as well as the solidifying columnar dendrites and globular equiaxed grains as two different secondary phases. With an Eulerian approach, the three phases are considered as spatially coupled and interpenetrating continua. The conservation equations of mass, momentum, species, and enthalpy are solved for all three phases. An additional conservation equation for the number density of the equiaxed grains is defined and solved. Nucleation of the equiaxed grains, diffusion-controlled growth of both columnar and equiaxed phases, interphase exchanges, and interactions such as mass transfer during solidification, drag force, solute partitioning at the liquid/solid interface, and release of latent heat are taken into account. Binary steel ingots (Fe-0.34 wt pct C) with two-dimensional (2-D) axis symmetrical and three-dimensional (3-D) geometries as a benchmark were simulated. It is demonstrated that the model can be used to simulate the mixed columnar-equiaxed solidification, including melt convection and grain sedimentation, macrosegregation, columnar-to-equiaxed-transition (CET), and macrostructure distribution. The model was evaluated by comparing it to classical analytical models based on limited one-dimensional (1-D) cases. Satisfactory results were obtained. It is also shown that in order to apply this model for industrial castings, further improvements are still necessary concerning some details.     

Macrotransport-solidification kinetics modeling of equiaxed dendritic growth: Part I. Model development and discussion

An analytical model that describes solidification of equiaxed dendrites has been developed for use in solidification kinetics-macrotransport modeling. It relaxes some of the assumptions made in previous models, such as the Dustin-Kurz, Rappaz-Thevoz, and Kanetkar-Stefanescu models. It is assumed that nuclei grow as unperturbed spheres until the radius of the sphere becomes larger than the minimum radius of instability. Then, growth of the dendrites is related to morphological instability and is calculated as a function of melt undercooling around the dendrite tips, which is controlled by the bulk temperature and the intrinsic volume average concentration of the liquid phase. When the general morphology of equiaxed dendrites is considered, the evolution of the fraction of solid is related to the interdendritic branching and dynamic coarsening (through the evolution of the specific interfacial areas) and to the topology and movement of the dendrite envelope (through the tip growth velocity and dendrite shape factor). The particular case of this model is the model for globulitic dendrite. The intrinsic volume average liquid concentration and bulk temperature are obtained from an overall solute and thermal balance around a growing equiaxed dendritic grain within a spherical closed system. Overall solute balance in the integral form is obtained by a complete analytical solution of the diffusion field in both liquid and solid phases. The bulk temperature is obtained from the solution of the macrotrasport-solidification kinetics problem.     

A solutal interaction mechanism for the columnar-to-equiaxed transition in alloy solidification

A multiphase/multiscale model is used to predict the columnar-to-equiaxed transition (CET) during solidification of binary alloys. The model consists of averaged energy and species conservation equations, coupled with nucleation and growth laws for dendritic structures. A new mechanism for the CET is proposed based on solutal interactions between the equiaxed grains and the advancing columnar front—as opposed to the commonly used mechanical blocking criterion. The resulting differences in the CET prediction are demonstrated for cases where a steady state can be assumed, and a revised isotherm velocity (V _T ) vs temperature gradient (G) map for the CET is presented. The model is validated by predicting the CET in previously performed unsteady, unidirectional solidification experiments involving Al-Si alloys of three different compositions. Good agreement is obtained between measured and predicted cooling curves. A parametric study is performed to investigate the dependence of the CET position on the nucleation undercooling and the density of nuclei in the equiaxed zone. Nucleation undercoolings are determined that provide the best agreement between measured and calculated CET positions. It is found that for all three alloy compositions, the nucleation undercoolings are very close to the maximum columnar dendrite tip undercoolings, indicating that the origin of the equiaxed grains may not be heterogeneous nucleation, but rather a breakdown or fragmentation of the columnar dendrites. An erratum to this article is available at .     

Advanced Solute Conservation Equations for Dendritic Solidification Processes Part I: Experiments and Theory

The macrosegregation formed in dendritic equiaxed structure during early stages of solidification of Al‐4.5%Cu alloy has been studied by experimental work and by metallurgical study of cast samples taken from the experimental work. An experimental work was conducted to study the coupled effect of natural convection streams, interdendritic strain and mushy permeability of Al‐4.5%Cu aluminum alloy solidified in horizontal rectangular parallelepiped cavity at different superheats. The metallurgical study includes macro‐microstructure evaluation, measurements of grain size of equiaxed crystals and macrosegregation analysis. This study shows that the level of surface segregation exhibiting as positive segregation varies with superheat whereas the rest of inner ingot areas show the light fluctuation in segregation values. In addition to experimental work, there is a mathematical study which contains a complete derivation of local solute redistribution equations based on Fleming's approach under different solute diffusion mechanisms in the dendritic solid. This derivation includes also the effects of interdendritic strain and mushy permeability on the local solute redistribution distribution. Owing to the length of the study, it is presented in two parts. The first part describes the experimental work and its results as well as a detail derivation of solute conservation equations. This part also involves comparison and discussion between existing and proposed solute conservation equations. The second part contains the mathematical analyses of a two dimensional mathematical model of fluid flow, heat flow, solidification, interdendritic strain and macrosegregation. Also, this part also contains the numerical simulations by using finite difference technique “FDT” to create convection patterns, heat transfer, interdendritic strain, and macrosegregation distributions. This part also includes comparisons between the available measurements and model predications as well as full discussion of different model simulations. The mechanism of interdendritic strain generation and macrosegregation formation during solidification of dendritic equiaxed structure under different diffusion mechanisms in dendritic solid has also been explained and discussed.     

Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method

A comprehensive model is developed for solving the heat and solute diffusion equations during solidification that avoids tracking the liquid—solid interface. The bulk liquid and solid phases are treated as regular solutions and an order parameter (the phase field) is introduced to describe the interfacial region between them. Two-dimensional computations are performed for ideal solutions and for dendritic growth into an isothermal and highly supersaturated liquid phase. The dependence upon various material and computational parameters, including the approach to conventional sharp interface theories, is investigated. Realistic growth patterns are obtained that include the development, coarsening, and coalescence of secondary and tertiary dendrite arms. Microsegregation patterns are examined and compared for different values of the solid diffusion coefficient.     

Solute diffusion model for equiaxed dendritic growth

A new approach to the modeling of the equiaxed solidification of dendritic alloys is proposed. It is assumed that, in metallic alloys, microstructure formation is primarily controlled by solute diffusion (i.e. there is complete “thermal mixing” at the scale of one grain), and that the dendrite interface is an iso-concentrate at all times. The evolution of one dendritic grain is therefore modelled as follows: (i) complete mixing of solute within the interdendritic liquid; (ii) no back-diffusion in the solid; (iii) spherical solute diffusion in the liquid around the grain envelope; (iv) overall solute balance; (v) overall thermal balance; (vi) growth velocity, υ_g, of dendrite tips governed by the kinetic equation derived for the isolated dendrite case. By using an explicit finite difference scheme to solve these coupled equations, the concentration profiles, cooling curve, fraction of solid and evolution of dendritic grain envelope can be calculated. The intitial conditions used to start the calculation are provided by two parameters related to nucleation: the initial undercooling and the density of gains. The effect of nucleation and thermal conditions on equiaxed growth are studied. The theoretical predictions of recalescence and of the distribution of an interdendritic eutectic phase are in good agreement with experimental observations.     

Modeling of globular equiaxed solidification with a two-phase approach

A two-phase volume averaging model for globular equiaxed solidification is presented. Treating both liquid and solid (disperse grains) as separated but highly coupled interpenetrating continua, we have solved the conservation equations for mass, momentum, species mass fraction, and enthalpy for both phases. We also consider the conservation of grain density. Exchange or source terms take into account interactions between the melt and the solid, such as mass transfer (solidification and melting), friction and drag, solute redistribution, release of latent heat, and nucleation. An ingot casting with a near globular equiaxed solidification alloy (Al-4 wt pct Cu) is simulated. Results including grain evolution, melt convection, sedimentation, solute transport, and macrosegregation formation are obtained. The mechanisms producing these results are discussed in detail.     

双辊连铸不锈钢薄带凝固组织特点

 通过金相观察分析了同径双辊薄带连铸机上生产的奥氏体不锈钢薄带的凝固组织,结果表明：铸带凝固组织包括2个柱状晶区和1个等轴晶区,其等轴晶呈近球形或蔷薇形。与传统连铸板坯相比,其柱状晶区一次及二次枝晶的间距较小,等轴晶粒内部为非枝晶结构,其尺寸大约是连铸坯等轴晶的1/10,凝固组织更致密。     

Using a Two-Phase Columnar Solidification Model to Study the Principle of Mechanical Soft Reduction in Slab Casting

A two-phase columnar solidification model is used to study the principle of mechanical soft reduction (MSR) for the reduction of centerline segregation in slab casting. The two phases treated in the model are the bulk/interdendritic melt and the columnar dendrite trunk. The morphology of the columnar dendrite trunk is simplified as stepwise growing cylinders, with growth kinetics governed by the solute diffusion in the interdendritic melt around the growing cylindrical columnar trunk. The solidifying strand shell moves with a predefined velocity and the shell deforms as a result of bulging and MSR. The motion and deformation of the columnar trunks in response to bulging and MSR is modeled following the work of Miyazawa and Schwerdtfeger from the 1980s. Melt flow, driven by feeding of solidification shrinkage and by deformation of the strand shell and columnar trunks, as well as the induced macrosegregation are solved in the Eulerian frame of reference. A benchmark slab casting (9-m long, 0.215-m thick) of plain carbon steel is simulated. The MSR parameters influencing the centerline segregation are studied to gain a better understanding of the MSR process. Two mechanisms in MSR modify the centerline segregation in a slab casting: one establishes a favorable interdendritic flow field, whereas the other creates a non-divergence-free deformation of the solid dendritic skeleton in the mushy region. The MSR efficiency depends not only on the reduction amount in the slab thickness direction but also strongly on the deformation behavior in the longitudinal (casting) direction. With enhanced computation power the current model can be applied for a parameter study on the MSR efficiency of realistic continuous casting processes.     

Prediction of the As-Cast Structure of Al-4.0 Wt Pct Cu Ingots

A two-stage simulation strategy is proposed to predict the as-cast structure. During the first stage, a 3-phase model is used to simulate the mold-filling process by considering the nucleation, the initial growth of globular equiaxed crystals and the transport of the crystals. The three considered phases are the melt, air and globular equiaxed crystals. In the second stage, a 5-phase mixed columnar-equiaxed solidification model is used to simulate the formation of the as-cast structure including the distinct columnar and equiaxed zones, columnar-to-equiaxed transition, grain size distribution, macrosegregation, etc. The five considered phases are the extradendritic melt, the solid dendrite, the interdendritic melt inside the equiaxed grains, the solid dendrite, and the interdendritic melt inside the columnar grains. The extra- and interdendritic melts are treated as separate phases. In order to validate the above strategy, laboratory ingots (Al-4.0 wt pct Cu) are poured and analyzed, and a good agreement with the numerical predictions is achieved. The origin of the equiaxed crystals by the “big-bang” theory is verified to play a key role in the formation of the as-cast structure, especially for the castings poured at a low pouring temperature. A single-stage approach that only uses the 5-phase mixed columnar-equiaxed solidification model and ignores the mold filling can predict satisfactory results for a casting poured at high temperature, but it delivers false results for the casting poured at low temperature.     

电渣重熔GH984G定向凝固组织CAFE法模拟

 电渣重熔凝固组织的控制直接关系到高温合金的品质与实际生产应用。针对电渣重熔GH984G的定向凝固过程,同时考虑传热和溶质扩散,基于CAFE法与C语言结合,建立了三维电渣重熔凝固过程组织演变的CAFE模型,并对凝固过程温度场和凝固组织演变进行模拟预测。结果表明,铸锭温度场和熔池深度都是首先为较浅平状态,然后不断加深至最后稳定;在电极熔化初始,金属熔池浅平,枝晶生长方向是竖直向上,之后金属熔池不断加深,底部竖直向上的柱状晶方向变为斜向上约26°。同时在铸锭的中心线上出现了等轴晶,等轴晶形核长大后与柱状晶镶嵌生长。此外,随着电极熔速变大,渣金界面上涨速度也随之变大,且熔池深度相应变宽变深。模拟结果与试验结果基本吻合,从而验证了模型与形核参数的适用性。     

Equiaxed dendritic solidification with convection: Part III. Comparisons with NH4Cl-H2O experiments

This third article on equiaxed dendritic solidification is intended to provide experimental validation of the multiphase model developed in part I. Numerical and experimental results are presented for the solidification of a NH₄C1-70 wt pct H₂O solution inside a square cavity cooled equally from all sidewalls. The numerical simulations were performed using the numerical procedures developed in part II. The experiments were conducted to measure the temperature historiesvia thermocouples and to record the images of the solidification process using a shadowgraph system. Preliminary validity of the multiphase model is demonstrated by the qualitative agreement between the measurements and predictions of cooling curves as well as of the evolution of the crystal sediment bed. In addition, several important features of equiaxed dendritic solidification are identified through this combined experimental and numerical study, including the grain generation and growth behaviors in the presence of liquid flow, the sedimentation of equiaxed crystals, the formation of a crystal sediment bed, and a bottom zone of negative segregation resulting from the countercurrent solid-liquid multiphase flow. Quantitative comparisons between the numerical simulation and experiment reveal several areas for future research.     

Natural Convection and Columnar-to-Equiaxed Transition Prediction in a Front-Tracking Model of Alloy Solidification

A meso-scale front-tracking model (FTM) of nonequilibrium binary alloy dendritic solidification has been extended to incorporate Kurz, Giovanola, and Trivedi (KGT) dendrite kinetics and a Scheil solidification path. Model validation via comparison with thermocouple measurements from a solidification experiment, in which natural convection is limited by design, is presented. Via solution of the flow field due to natural thermal buoyancy, it is shown that resultant liquid-phase convection creates conditions in which equiaxed solidification is favored. Comparison with simulations in which casting solidification is diffusion controlled show that natural convection has greatest effect at intermediate times, but that at early and late stages of columnar solidification, the differences are relatively small. It is, however, during the time of greatest divergence between the simulations that the authors’ predictive index for equiaxed zone formation is enhanced most by convection. Finally, the columnar-to-equiaxed transition is directly simulated, in directional solidification controlled by diffusion. This article is based on a presentation made in the symposium entitled “Solidification Modeling and Microstructure Formation: in Honor of Prof. John Hunt,” which occurred March 13–15, 2006 during the TMS Spring Meeting in San Antonio, Texas, under the auspices of the TMS Materials Processing and Manufacturing Division, Solidification Committee.     

Advanced Solute Conservation Equations for Dendritic Solidification Processes Part II: Numerical Simulations and Comparisons

The mathematical model of derived solute equations in part I for equiaxed dendritic solidification with melt convection streams and interdendritic thermo‐metallurgical strain is applied numerically to predict macrosegregation distributions with different diffusing mechanisms in dendritic solid. Numerical and experimental results are present for solidification of a Al–4.5% Cu alloy inside horizontal rectangular cavity at different superheats. The numerical simulations were performed by using simpler method developed by Patanker. The experiments were conducted to measure the cooling curves via thermocouples and the metallurgical examinations to measure the grain size and macrosegregation distributions in Part I. Preliminary validity of the model is demonstrated by the qualitative and quantitative agreements between the measurements and predications of cooling curves and predicted macrosegregation distributions including mushy permeability and interdendritic strain. In addition, several important features of macrosegregation in equiaxed dendritic solidification are identified through this combined experimental and numerical study. Also, quantitative agreements between the numerical simulations and experiments reveal several areas for future research work. The differences and errors between predicted macrosegregation results under different diffusing mechanisms have been discussed.     

An analytical model for solute redistribution during solidification of planar,columnar, or equiaxed morphology

Existing models for solute redistribution (microsegregation) during solidification were reviewed. There are no analytical models that take into account limited diffusion in both the liquid and the solid phases. A new analytical mathematical model for solute redistribution was developed. Diffusion in liquid and in solid was considered. This model does not require a prescribed movement of the interface. It can be used for one-dimensional (1-D) (plate), two-dimensional (cylinder), or three-dimensional (3-D) (sphere) calculations. Thus, it is possible to calculate microsegregation at the level of primary or secondary arm spacing for columnar dendrites or for equiaxed dendrites. The solution was compared with calculations based on existing models, as well as with some available experimental data for the segregation of base elements in as cast Al-4. 9 wt pct Cu, INCONEL 718, 625, and plain carbon (0. 13 wt pct C) steel.     

An integrated model for dendritic and planar interface growth and morphological transition in rapid solidification

Rapid solidification can be achieved by quenching a thin layer of molten metal on a cold substrate, such as in melt spinning and thermal spray deposition. An integrated model is developed to predict microstructure formation in rapidly solidified materials through melt substrate quenching. The model solves heat and mass diffusion equations together with a moving interface that may either be a real solid/liquid interface or an artificial dendrite tip/melt interface. For the latter case, a dendrite growth theory is introduced at the interface. The model can also predict the transition of solidification morphology, e.g., from dendritic to planar growth. Microstructure development of Al-Cu alloy splats quenched on a copper substrate is investigated using the model. Oscillatory planar solidification is predicted under a critical range of interfacial heat-transfer coefficient between the splat and the substrate. Such oscillatory planar solidification leads to a banded solute structure, which agrees with the linear stability analysis. Finally, a microstructure selection map is proposed for the melt quenching process based on the melt undercooling and thermal contact conditions between the splat and the substrate.     

Modeling of casting solidification stochastic or deterministic?

Solidification modeling has had a phenomenal impact on metalcasting in the last decade. Following its initial success in predicting the occurrence of porosity defects, it has grown to be an essential casting engineering tool. As more complex models have been developed (in response to the realization that simple heat flow models were not adequate to solve the shrinkage problem) , they are being used to design more efficient gating and risering systems, which minimize the amount of metal poured to produce a good casting. Models today include predictions of fluid flow during mold filling, casting distortion, mold–metal interface reactions and cast structure.Until a few years ago solidification simulation was based only on deterministic models. As prediction of microstructural evolution became a contemporary problem, the limitation of deterministic models in predicting such features as dendrite coherency, columnar-to-equiaxed transition, dendrite fragmentation and movement of dendrites by the liquid, became evident. Recently developed stochastic models for solidification are capable of simulating and displaying the growth of columnar and equiaxed grains. However, the physics of dendritic growth is rather approximate. The growth of dendrite arms and their branching are ignored, and only a bulk representation of the grain growth is provided.A micro-scale approach for more accurate dendritic growth simulation in casting processes is presented in this paper. The model couples stochastic modeling at a length scale of 10⁻⁶ m, with deterministic modeling at a length scale of 10⁻⁴ m. A deterministic tip-velocity model is used to calculate the advance of the dendrite tip. Arm thickening is also calculated with a deterministic law derived from the dendrite tip velocity law and crystallographic considerations in combination with a deterministic coarsening model. However, the overall growth of dendrite arms is derived from probabilistic calculations. Branching of dendrites arm is allowed to occur based on morphologic instability. Thus the dendrite morphology, rather than the gain structure can be simulated.A discussion on the advantages and limitations of contemporary deterministic and stochastic models is also included.     

Modeling of the Coupling of Microstructure and Macrosegregation in a Direct Chill Cast Al-Cu Billet

The macroscopic multiphase flow and the growth of the solidification microstructures in the mushy zone of a direct chill (DC) casting are closely coupled. These couplings are the key to the understanding of the formation of the macrosegregation and of the non-uniform microstructure of the casting. In the present paper we use a multiphase and multiscale model to provide a fully coupled picture of the links between macrosegregation and microstructure in a DC cast billet. The model describes nucleation from inoculant particles and growth of dendritic and globular equiaxed crystal grains, fully coupled with macroscopic transport phenomena: fluid flow induced by natural convection and solidification shrinkage, heat, mass, and solute mass transport, motion of free-floating equiaxed grains, and of grain refiner particles. We compare our simulations to experiments on grain-refined and non-grain-refined industrial size billets from literature. We show that a transition between dendritic and globular grain morphology triggered by the grain refinement is the key to the explanation of the differences between the macrosegregation patterns in the two billets. We further show that the grain size and morphology are strongly affected by the macroscopic transport of free-floating equiaxed grains and of grain refiner particles.