Macrosegregation Model Research Articles

Modeling of interdendritic strain and macrosegregation for dendritic solidification processes: Part I. Theory and experiments

In the first part of this two-part article, mathematical models have been developed to characterize temperature, interdendritic stain, and segregation distributions during dendritic solidification. This aims to predict the effect of interdendric strain associated with sudden changes in the cooling conditions on the macrosegregation distributions, i.e., the combined effect of interdendric strain and macrosegregation on the dendritic structure. These theoretical models were verified on a laboratory scale. Four laboratory ingots of 0.53 and 0.9 wt pct C steels were cast horizontally and unidirectionally in a static mold under cooling conditions designed to approximate those in the continuous-casting process. Thermocouples recorded temperatures in the ingot at different locations from copper chill. The ingots were examined for macro-microstructure, and the extent of carbon macrosegregation was determined by wet chemical analysis. The experimental results indicate that static mold with sudden changes in the cooling conditions on the copper chill provides an approximately similar structure and macrosegregation profiles to those in a continuous-casting process. It is concluded that these cooling conditions have a significant effect on the fluctuated macrosegregation phenomenon. The sudden drop in the heat flux on the chill causes a positive segregation, whereas a sudden increase in heat flux results in a negative segregation. Also, the metallographic examination shows that there is high inelastic deformation of the dendrites due to the sudden drop in heat flux on the chill.

Modeling of interdendritic strain and macrosegregation for dendritic solidification processes: Part II. Computation of interdendritic strain and segregation fields in steel ingots

In Part II of this article, the models are applied to analyze the effects of various cooling conditions on the interdendritic-strain history and macrosegregation patterns. The temperature measurements are used to evaluate the heat-transfer coefficient between the ingot surface and the cooling fluid. The model predictions are compared to temperature measurements and carbon-concentration profiles in ingots, where good agreements are obtained. In examining the fluctuated cooling conditions on the interdendritic strain, the results show that the sudden increase in the heat flux is found to increase the compressive strain in the interdendritic-strain region. The drop in the heat flux shows completely opposite behavior, where the tensile strain generates through the same region. The model demonstrates also that interdendritic strain associated with the fluctuated cooling conditions is a major factor in determining macrosegregation quantitatively and qualitatively. The results show that the tensile interdendritic strain results in a positive segregation, whereas the compressive strain yields a negative one. The segregation level depends mainly on the interdendritic-strain history and its rate value. The mechanism of fluctuated-segregation phenomenon as a function of the interdendritic strain is extensively discussed.

Boundary element model of coupled heat and mass transfer in solidifying castings

The solidification and cooling processes taking place in the casting domain can be described by the system of partial differential equations and adequate boundary, geometrical, physical and initial conditions. From the mathematical point of view, the so-called moving boundary problem should be considered and the task can be effectively solved using numerical methods. In this paper, the alloy's solidification is considered. In differential equations describing the solidification process, the parameter called a substitute thermal capacity is introduced; in other words, the one domain approach is applied. The heat transfer model is coupled with the macrosegregation model, because the temperatures limiting the mushy zone sub-domain are a function of the alloy component's concentration (a binary alloy is taken into account). Two macrosegregation models are constructed on the basis of the lever-arm law and Scheil's equation, respectively. In terms of numerical computation, the boundary element method (heat transfer model) and the control volume method (macrosegregation) are applied. The typical numerical solutions presented in the literature do not take into account the relationships between heat transfer and macrosegregation processes or the models proposed are very complex and not effective. The solution presented in this paper seems to be sufficiently exact and very simple for numerical realisation.

INFLUENCE OF GRAVITY ACCELERATION ON MACROSEGREGATION AND MACROSTRUCTURE DURING THE UNIDIRECTIONAL SOLIDIFICATION OF CAST BINARY ALLOYS: A NUMERICAL INVESTIGATION

A comprehensive numerical approach was developed for modeling of macrosegregation during the solidification of cast binary alloy. The model accounts for the competition between dendritic (columnar and equiaxed) and eutectic structures through the use of the solidification-kinetics modeling for fraction of solid evolution. Microsegregation analytical calculations are performed assuming molecular diffusion in both solid and liquid. The coupling between the macroscopic and microscopic calculations is accomplished with the micro-latent heat method \\[1,2]. A numerical analysis was performed of the thermosolutal convection effects on macrosegregation and macrostructure (grain structure) during the solidification of a liquid Pb-10 wt. % Sn alloy cooled from below in terrestrial and low-gravity (g) environments. Below 0.01 g , the macrosegregation tendency is insignificant. Freckles, fingers (which are caused by plumes), and channels enriched in Sn as well as isolated pockets poor in Sn develop for 1 g and sufficiently low thermosolutal Rayleigh number. For the simulation conditions used in this work (e.g., high thermal gradients), small effects of the thermosolutal convection on the solidification macrostructure were observed.

Numerical model of macro-segregation during directional crystallization process

Abstract In the paper the mathematical model of macro-segregation proceeding during the directional crystallization process is presented. The boundary-initial problem considered is discussed. Next the numerical approximation constructed on the basis of the boundary element method supplemented by a procedure called the artificial heat source method is described. The boundary condition on the solidification front resulting from the alloy component balance is introduced, while in finally the practical aspects of computations concerning the course of the process are discussed.

The effect of macroscopic solute diffusion in the liquid upon surface macrosegregation

An existing one-dimensional mathematical model that predicts the macrosegregation formation due to solidification shrinkage has been modified to also account for the macroscopic diffusion of solute in the liquid. It is shown both numerically and analytically that such solute diffusion has an almost negligible influence on the predicted solute profile, except in a very thin layer near the chill surface, where severe solute depletion develops. This layer is related to a discontinuity in the diffusive solute flux at the surface, and for a moderately cooled Al-4.5 pct Cu alloy, the layer thickness is of the order 100 µm. When the lever rule is imposed, the solute concentration at the surface becomes equal to the partition coefficient multiplied by the nominal alloy concentration, and the boundary layer solidifies completely once the temperature drops below the liquidus temperature of the (initial) melt. This indicates that assuming local thermodynamical equilibrium at a domain boundary when simultaneously accounting for the macroscale solute diffusion should be reconsidered in macrosegregation modeling.

Modeling of macrosegregation due to thermosolutal convection and contraction-driven flow in direct chill continuous casting of an Al-Cu round ingot

MACROSEGREGATION continues to be one of the concerns in direct chill (DC) continuous casting of aluminum ingots. The variation in composition over the ingot cross section results in the need to adjust alloy composition in order to meet mechanical property specifications in finished products. The extent of macrosegregation is greatly affected by the casting conditions, such as casting speed, superheat, cooling rate, mold size, alloy composition, and grain-refining practices. Thus, it is of interest to develop a mathematical model to simulate macrosegregation in DC casting, enabling the selection of conditions that minimize macrosegregation while maintaining a high productivity. Various mechanisms for macrosegregation in aluminum DC casting have been discussed by Chu and Jacoby [1] and Yu and Granger. [2] In general, macrosegregation is caused by the relative movement of liquid and solid in the mushy zone, because the alloy constituents have different solubilities in the two phases. One cause of interdendritic liquid flow through the rigid solid structure in the mushy zone is the contraction of the liquid and the density difference between the solid and liquid phases during solidification. The commonly observed inverse segregation pattern near the outer ingot surface is known to be the result of such contraction driven backflow of solute-rich liquid. Positive segregation at the ingot surface can also be caused by exudation of interdendritic liquid through channels in the partially solidified shell into the contraction gap between the shell and the mold. This flow is driven by the metallostatic head. [3‐6] Less clear are the transport phenomena leading to either positive or negative segregation in the cen

Modeling of the peritectic reaction and macro-segregation in casting of low carbon steel

Macro-microscopic models have been developed to describe the macrosegregation behavior associated with the peritectic reaction of low carbon steel. The macrosegregation model has been established on the basis of previously published work and experimental data. A microscopic model of a three-phase reaction L+δ→γ has been modeled by using Fredriksson’s approach. Four horizontal and unidirectional solidified experimental groups simulating continuous casting have been performed with a low carbon steel containing 0.13 wt pct carbon. The extent of macrosegregation of carbon was determined by wet chemical analysis of millings. It is confirmed, by comparing calculated results with experimental results, that this model successfully predicts the occurrence of macrosegregation. The results indicate that a peritectic reaction which is associated with a high cooling rate generates high thermal contraction and a high tensile strain rate at the peritectic temperature. Therefore, the macrosegregation, particularly at the ingot surface, is very sensitive to the cooling rate, where extremely high positive segregation was observed in the case of a high cooling rate. However, in the case of slow cooling rate, negative segregation was noted. The mechanism of macrosegregation with peritectic reaction is discussed in detail.

A model for macrosegregation and its application to Al-Cu castings

A macrosegregation model has been developed to evaluate solute redistribution during solidification of casting alloys. The continuum formulations were used to describe the macroscopic transports of mass, energy, and momentum, associated with the microscopic transport phenomena, for two-phase systems. It was assumed that liquid flow is driven by thermal and solutal buoyancy, as well as by solidification contraction. The movement of free surface was also considered to ensure volume con-servation. In numerical calculations, the solidification event was divided into two stages. In the first stage, the liquid containing freely moving equiaxed grains was described through the relative vis-cosity concept. In the second stage, when a fixed dendritic network formed after dendrite coherency, the mushy zone was treated as a porous medium. After validation of the proposed model for the case of segregation in a bottom-chilled unidirectionally solidified casting of Al-Cu alloys, the nu-merical model was applied to the study of three different castings with simple geometry. It was found that solutal convection tends to decrease the macrosegregation generated by thermal convec-tion. When shrinkage-driven convection was also considered, segregation was again increased, with highly segregated areas forming away from the riser and next to the mold wall. It was demonstrated that solidification contraction has a stronger effect on the liquid flow in the mushy region than buoyancy. The model also was applied to assess the probability of pore formation based on the pressure drop concept. While in the absence of experimental data for the critical pressure drop it was not possible to uniquely predict the formation of porosity, it was possible to indicate the regions where porosity may form preferentially.

Mathematical modeling of macrosegregation of iron carbon binary alloy: Role of double diffusive convection

During alloy solidification, macrosegregation results from long range transport of solute under the influence of convective flow and leads to nonuniform quality of a solidified material. The present study is an attempt to understand the role of double diffusive convection resulting from the solutal rejection in the evolution of macrosegregation in an iron carbon system. The solifification process of an alloy is governed by conservation of heat, mass, momentum, and species and is accompanied by the evolution of latent heat and the rejection or incorporation of solute at the solid liquid interface. Using a continuum formulation, the goverming equations were solved using the finite volume method. The numerical model was validated by simulating experiments on an ammonium chloride water system reported in the literature. The model was further used to study the role of double diffusive convection in the evolution of macrosegregation during solidification of Fe 1 wt pct C alloy in a rectangular cavity. Simulation of this transient process was carried out until complete solidification, and the results, depicting the influence of flow field on thermal and solutal field andvice versa, are shown at various stages of solidification. Under the given set of parameters, it was found that the thermal buoyancy affects the macrosegregation field globally, whereas the solutal buoyancy has a localized effect.

An internal variable description of solidification suitable for macrosegregation modeling

A mathematical description of solidification suitable for macrosegregation modeling is presented. The concept incorporates the effect of finite solid diffusion locally in the dendrites, and remelting can be modeled without any need to trace the solute concentrations in the dendritic structure during solidification. This is accomplished by interpreting the mean solute concentrations in the solid as internal variables on which the solid fraction depends and by accounting for the rate of change of these variables due to solidification, remelting, and solid diffusion by additional evolution equations. The material coefficients needed in the model are estimated for a ternary AlFeSi alloy of commercial purity, and the internal variable equations are incorporated in a simple model problem for interdendritic melt flow leading to macrosegregation. The results are compared to similar studies based on the lever rule and the Scheil approach, and it is shown that the choice of solidification description has a pronounced effect on the predicted amount of macrosegregation.

Macrosegregation during solidification resulting from density differences in the liquid

Macrosegregation has been observed in directionally solidified Pb-20 pct Sn alloys, over a range of freezing rates and temperature gradients. The macrosegregation was shown to result from the upward flow of less dense, tin rich, interdendritic liquid during solidification, using radioactive tracer techniques. For comparison, it was shown that macrosegregation occurred in the opposite direction in a Sn-4 pct Pb alloy, where the interdendritic liquid was lead rich, and consequently more dense. Shrinkage trails and pipes were observed in some of the Pb-20 pct Sn ingots, similar to “freckles” observed in directionally cast superalloys. A mathematical model for macrosegregation in vertically solidified ingots is presented, the driving force being the density differences in the interdendritic liquid during solidification. Liquid flow through the dendritic array is estimated by considering the partially solidified alloy as a porous medium of variable porosity. For simplicity, the model neglects backflow due to volume shrinkage (inverse segregation). The experimental results are compared to the model predictions.