Modeling of the formation of under-riser macrosegregation during solidification of binary alloys

The formation of macrosegregation in a rectangular ingot with reduced cross section from the riser to the casting, chilled from the bottom, has been studied numerically. In addition to positive inverse segregation occurring near the chilled surface, very severe negative segregation around the under-riser region and moderate positive segregation near the top corners of the casting were found. Although large circulating vortexes are created by natural convection in the under-riser region during the early stage of solidification, the fluid flow in the mushy zone is dominated by solidification shrinkage. As a result, the final solute distribution in the casting is determined by the flow of solute-rich liquid in the mushy zone owing to the combined effects of solidification shrinkage and change of cross section from casting to riser. Detailed explanations regarding the effect of different flow phenomena on the formation of the segregations are provided. The effects of riser size and cooling condition at the bottom of the ingot on the formation of macrosegregation also were studied. The predicted negative and positive macrosegregations in the casting compared very well with the available experimental data.     

Influence of Forced/Natural Convection on Segregation During the Directional Solidification of Al-Based Binary Alloys

We analyzed the columnar solidification of a binary alloy under the influence of an electromagnetic forced convection of various types and investigated the influence of a rotating magnetic field on segregation during directional solidification of Al-Si alloy as well as the influence of a travelling magnetic field on segregation during solidification of Al-Ni alloy through directional solidification experiments and numerical modeling of macrosegregation. The numerical model is capable of predicting fluid flow, heat transfer, solute concentration field, and columnar solidification and takes into account the existence of a mushy zone. Fluid flows are created by both natural convection as well as electromagnetic body forces. Both the experiments and the numerical modeling, which were achieved in axisymmetric geometry, show that the forced-flow configuration changes the segregation pattern. The change is a result of the coupling between the liquid flow and the top of the mushy zone via the pressure distribution along the solidification front. In a forced flow, the pressure difference along the front drives a mush flow that transports the solute within the mushy region. The channel forms at the junction of two meridional vortices in the liquid zone where the fluid leaves the front. The latter phenomenon is observed for both the rotating magnetic field (RMF) and traveling magnetic field (TMF) cases. The liquid enrichment in the segregated channel is strong enough that the local solute concentration may reach the eutectic composition.     

Macrosegregation Formation in an Al–Si Casting Sample with Cross-sectional Change During Directional Solidification

A unidirectional solidification experiment of hypoeutectic Al-7.0 wt% Si alloy against gravity direction in a cylindrical mold with cross-sectional change was made, and the macrosegregation in different parts of the as-solidified sample was investigated (Ghods et al. in J Cryst Growth 441:107–116, 2016; J Cryst Growth 449:134–147, 2016). The current study is to use a two-phase columnar solidification model to analyze the segregation mechanisms as used in this experiment. Following flow phenomena and their contributions to the formation of macrosegregation are simulated and compared: (1) solidification shrinkage-induced feeding flow; (2) thermo-solutal convection; and (3) combined thermo-solutal convection and shrinkage-induced feeding flow. The shrinkage-induced feeding flow leads to an inverse (positive) segregation in the bottom part, and a severe negative segregation in the part below cross-sectional change. Thermo-solutal buoyancy leads to a so-called steepling convection in the main part of the sample (away from the bottom and cross-sectional change), and this kind of flow leads to a positive macrosegregation near the sample surface. The calculations have successfully explained the experimental result of macrosegregation.     

Three‐dimensional Simulation of Thermosolutal Convection and Macrosegregation in Steel Ingots

Thermosolutal convection and macrosegregation formation during the solidification of steel ingots are numerically simulated in three dimensions. The simulation is based on a fully coupled model for mass, momentum, energy, and species conservation equations. The interdendritic flow in the mushy zone is governed by Darcy's law, and the permeability term is discretized using an interpolated liquid fraction method. The numerical results for a benchmark test of macrosegregation in a Pb‐Sn alloy are compared with experimental data taken from the literature. The present model is applied to simulate the solidification of industrial steel ingots. Preliminary predictions are obtained, including the positive segregation in the hot top, and the conically shaped negative segregation zone at the bottom of the ingot. The predicted variation of the segregation ratio in carbon along the vertical centreline of an ingot is compared with measurements, and generally good agreement is observed. Future attention should be paid to the precision of prediction by considering complex solidification issues, such as the sedimentation of free equiaxed grains and the formation of shrinkage cavity.     

Modeling of macrosegregation caused by volumetric deformation in a coherent mushy zone

A two-phase volume-averaged continuum model is presented that quantifies macrosegregation formation during solidification of metallic alloys caused by deformation of the dendritic network and associated melt flow in the coherent part of the mushy zone. Also, the macrosegregation formation associated with the solidification shrinkage (inverse segregation) is taken into account. Based on experimental evidence established elsewhere, volumetric viscoplastic deformation (densification/dilatation) of the coherent dendritic network is included in the model. While the thermomechanical model previously outlined (M. M’Hamdi, A. Mo, and C.L. Martin: Metall. Mater. Trans. A, 2002, vol. 33A, pp. 2081–93) has been used to calculate the temperature and velocity fields associated with the thermally induced deformations and shrinkage driven melt flow, the solute conservation equation including both the liquid and a solid volume-averaged velocity is solved in the present study. In modeling examples, the macrosegregation formation caused by mechanically imposed as well as by thermally induced deformations has been calculated. The modeling results for an Al-4 wt pct Cu alloy indicate that even quite small volumetric strains (≈2 pct), which can be associated with thermally induced deformations, can lead to a macroscopic composition variation in the final casting comparable to that resulting from the solidification shrinkage induced melt flow. These results can be explained by the relatively large volumetric viscoplastic deformation in the coherent mush resulting from the applied constitutive model, as well as the relatively large difference in composition for the studied Al-Cu alloy in the solid and liquid phases at high solid fractions at which the deformation takes place.     

Modeling of inverse segregation and porosity formation in directionally solidified aluminum alloys

A model has been developed which simulates inverse segregation and microporosity formation in directionally solidified alloys. Based upon a finite difference scheme, the model takes into account volume changes associated with density variations during solidification. The continuity equations for the mass, the solute, and the energy together with the Darcy equation describing the flow in the mushy zone are solved in a mixed Lagrangian-Eulerian representation. All nodal points within the liquid phase move with the fluid velocity, whereas nodes are fixed in space as soon as they are reached by dendrite tips. When the dendrite tips arrive at the end of the ingot, the remaining interdendritic liquid partially compensates for the solidification shrinkage occurring deeper within the volume. Since the size of the ingot remains fixed from that point on (absence of a purely liquid region), air (macroporosity) is introduced at the mesh points to satisfy the mass balance, starting from the top of the mushy zone. The formation of microporosity is also accounted for in the model through a calculation of local hydrogen segregation. Using this model, it is shown that inverse segregation decreases with increasing hydrogen content (or volume fraction of microporosity). The results of the simulation are compared with experimental results obtained on an Al-Cu alloy solidified under well-controlled directional conditions.     

Mathematical simulation on coupled flow, heat, and solute transport in slab continuous casting process

A three-dimensional comprehensively coupled model has been developed to describe the transport phenomena, including fluid flow, heat transfer, solidification, and solute redistribution in the continuous casting process. The continuous casting process is considered as a solidification process in a multicomponent solid-liquid phase system. The porous media theory is used to model the blockage of fluid flow by columnar dendrites in the mushy zone. The relation between flow pattern and the shape of the solid shell is demonstrated. Double diffusive convection caused by thermal and concentration gradients is considered. The change in the liquidus temperature with liquid concentration is also considered. The formation mechanism of macrosegregation is investigated. Calculated solid shell thickness and temperature distribution in liquid core are compared with the measured quantities for validating the model.     

Chimney Formation in Solidifying Ga-25wt pct In Alloys Under the Influence of Thermosolutal Melt Convection

The directional solidification of Ga-25wt pct In alloys within a Hele-Shaw cell under the influence of thermosolutal convection was observed by means of X-ray radioscopy. The unstable density stratification at the solidification front causes the formation of rising plumes containing solute-rich liquids. The development of the chimneys and the probability of their surviving depend sensitively on the spatial and temporal properties of the flow field. Variations of the vertical temperature gradient along the solidification cell lead to the observation of different mechanisms for chimney formation. Perturbations of the dendritic structure are the origin of development of segregation freckles in case of low temperature gradients. The long-term stabilities of these segregation channels are strongly influenced by the transient nature of the melt convection. The situation at higher temperature gradients is characterized by two dominating convection rolls in the liquid phase which are driven by a lateral temperature gradient and the convex shape of the solidification front. The penetration of this flow pattern into the mushy zone results in continuous accumulation of solute in the central part of the mushy zone followed by a remelting of the solid fraction and the occurrence of a stable chimney.     

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.     

Formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel

The formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel is simulated by simultaneously solving macroscopic mass, momentum, energy, and species conservation equations with full coupling of the temperature and concentrations through thermodynamic equilibrium at the solid/liquid interface. The flow field, solid fraction evolution, and macrosegregation patterns for four cases are presented. The results show both the formation of channel segregates and the formation of islands of mush surrounded by bulk melt. In examining the solidification of a ten-element steel, the global extent of macrosegregation of an element is found to be linearly dependent on its partition coefficient (more severe segregation for small partition coefficient), although such scaling is not possible locally. Results for the solidification of a binary Fe-C alloy (with the same carbon content as the ten-element alloy) are similar to those for the ten-element alloy due solely to the large contribution of carbon to buoyancy driven flow in the ten-element steel chosen for study. While including only those elements that make significant contributions to buoyancy driven flow reproduces the global extent of macrosegregation seen in the ten-element alloy, local differences in the predictions are visible. Finally, comparison of results for the solidification of the same ten-element steel using two different sets of data to describe the partition coefficients and change in liquidus temperature with concentration of the elements shows completely opposite behavior,i.e., upward flow through the mushy zone for one case and downward flow for the other. Thus, the need to have accurate phase-equilibrium data when modeling multicomponent macrosegregation is illustrated. Together, the results give an indication of what areas require more careful examination if accurate modeling of multicomponent solidification is to be accomplished.     

Modeling freckle formation in three dimensions during solidification of multicomponent alloys

The formation of macrosegregation defects known as “freckles” was simulated using a three-dimensional finite element model that calculates the thermosolutal convection and macrosegregation during the dendritic solidification of multicomponent alloys. A recently introduced algorithm was used to calculate the complicated solidification path of alloys of many components, which can accommodate liquidus temperatures that are general functions of liquid concentrations. The calculations are started from an all-liquid state, and the growth of the mushy zone is followed in time. Simulations of a Ni-Al-Ta-W alloy were performed on a rectangular cylinder until complete solidification. The results reveal details of the formation of freckles not previously observed in two-dimensional simulations. Liquid plumes in the form of chimney convection emanate from channels within the mushy zone, with similar qualitative features previously observed in transparent systems. Associated with the formation of channels, there is a complex three-dimensional flow produced by the interaction of the different solutal buoyancies of the alloy solutes. Regions of enhanced solid growth develop around the channel mouths, which are visualized as volcanoes on top of the mushy zone. The prediction of volcanoes differs from our previous calculations with multicomponent alloys in two dimensions, in which the volcanoes were not nearly as apparent. These and other features of freckle formation phenomena are illustrated.     

Modeling Freckle Segregation with Mesh Adaptation

Modeling the formation of macroscopic segregation channels during directional solidification processes has important applications in the casting industry. Computations that consider thermosolutal convection involve different length scales ranging from the small solute boundary layer at the dendrite tips to the characteristic size of the casting. In general, numerical models of solidification in the presence of a developing mushy zone are computationally inefficient because of nonlinear transport in an anisotropic porous medium. In the current work, mesh adaptation with triangular finite elements is used in conjunction with an efficient fractional-step solver of the momentum equations to predict the occurrence of channel-type segregation defects or freckles. The triangulations are created dynamically using an unstructured grid generator and a refinement criterion that tracks the position of the channel segregates. The efficiency of mesh adaptation is illustrated with simulations showing channel formation and macrosegregation in directional solidification of a Pb–Sn alloy.     

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.     

Simplified computation of macrosegregation in multicomponent aluminum alloys

An approximate method for calculating the macrosegregation in a multicomponent aluminum alloy is proposed. This method is based on the use of a predefined solidification path (i.e., relation between the solute concentration in the liquid phase and the solid fraction) instead of addressing the fully coupled micro-macrosegregation problem. In determining the solidification path, it is assumed that the total solute concentration is constant, and that the solidification history is the same everywhere in the casting. In this manner it becomes quite easy to take into account how the macrosegregation development is affected by the solute diffusion in the dendrites and the precipitation of secondary phases, provided that such effects are accounted for in the model used for determining the solidification path. In order to demonstrate the approximate method, the inverse segregation formation at a chill surface of an Al-4 pct Mg-0.2 pct Fe-0.15 pct Si-0.3 pct Mn (AA5182) alloy is calculated. In this case study, the solidification path is determined prior to the macrosegregation computation by a microsegregation model discussed elsewhere, and the solid and liquid densities are related to the concentrations of the different alloying elements by a simple mixture law without distinguishing between the different solid phases that are formed. The accuracy of the approximate method is discussed by considering a binary alloy. It turns out that the macrosegregation formation at a chill surface of an Al-4 pct Mg alloy is fairly close to that resulting from a modeling in which the variation of the total solute concentration is taken into account. Furthermore, the mixture law is compared to a more elaborate treatment of the densities involving both primary and eutectic solid phases. This comparison is carried out for an Al-4.5 pct Cu alloy for which literature data exist. The mixture law is found to give a reasonable accuracy in the calculated macrosegregation.     

The energy and solute conservation equations for dendritic solidification

The energy equation for solidifying dendritic alloys that includes the effects of heat of mixing in both the dendritic solid and the interdendritic liquid is derived. Calculations for Pb-Sn alloys show that this form of the energy equation should be used when the solidification rate is relatively high and/or the thermal gradients in the solidifying alloy are relatively low. Accurate predictions of transport phenomena in solidifying dendritic alloys also depend on the form of the solute conservation equation. Therefore, this conservation equation is derived with particular consideration to an accounting of the diffusion of solute in the dendritic solid. Calculations for Pb-Sn alloy show that the distribution of the volume fraction of interdendritic liquid (g _L) in the mushy zone is sensitive to the extent of the diffusion in the solid. Good predictions ofg _L are necessary, especially when convection in the mushy zone is calculated. Formerly Research Associate, The University of Arizona. Formerly Graduate Student, The University of Arizona.     

Convection in the two-phase zone of solidifying alloys

The analysis is applicable to alloy solidification which proceeds horizontally to the center of a mold. The model follows the growth of the solid-liquid zone adjacent to the chill face (the initial transient), the movement of the zone across the mold, and the region of final solidification adjacent to the centerline (the final transient). During solidification the density of the liquid varies across the twophase zone. Consequently, there is natural convection which is treated as flow through a porous medium. The equations for convection are coupled with the equation of solute redistribution between the phases in order to calculate macrosegregation after solidification is complete. Results were computed for alloys which show: (1) “inverse segregation≓ at a cooled-surface; (2) macrosegregation resulting from solidification with the initial transient, a period with a complete two-phase zone, and a final transient; and (3) macrosegregation when the width of the two-phase zone exceeds the semi-width of the mold.     

Investigation of Macrosegregation Formation in Aluminium DC Casting for Different Alloy Systems

Direct chill (DC) casting of aluminum involves alloys employing different solute elements. In this article, a qualitative analysis and comparison of macrosegregation formation is presented for three different alloy systems: Al-Mg, Al-Zn and Al-Cu. For this purpose, a multiphase, multiscale solidification model based on a volume-averaging method accounting for shrinkage-induced flow, thermal-solutal convection and grain motion is used and applied to an industrial-scale DC-cast ingot. The primary difference between these alloys is the thermal-solutal convection with Al-Mg having a competing thermal and solutal convection, whereas the other two systems have a cooperating thermal and solutal convection. In the study, the combined effect of the macrosegregation mechanisms is analyzed for each alloy to assess the role of the alloy system on the final macrosegregation.

     

A mathematical model for surface segregation in aluminum direct chill casting

A two-dimensional mathematical model for the development of macrosegregation at and close to the ingot surface during direct chill (DC) casting of aluminum rolling sheet ingots is presented. The model accounts for macrosegregation caused by exudation of interdendritic melt and macrosegregation associated with solidification shrinkage. Equations for the conservation of energy, solute, momentum, and mass during the stationary phase of the process are solved numerically by a finite-element method. The solution domain corresponds to a vertical cross section at the middle of the longest side of the slab. The main simplifications in the modeling concept are to assume that the solid in the mushy zone moves with the casting speed, and that the alloy is binary and solidifies according to the lever rule. The thickness and solute concentration of the surface layer and the macrosegregation close to the surface are calculated, and modeling results are compared to measurements on full-scale castings.     

Numerical Modeling of Macrosegregation in Binary Alloys Solidifying in the Presence of Electromagnetic Stirring

A model for predicting solidification and solute segregation of binary alloys undergoing electromagnetic stirring has been developed. A dual-zone formulation was employed to describe the velocity fields in the mushy region. The key feature of this model lies in its accounting for flow damping in the suspended particle region via turbulent interactions the crystallite surfaces. The damping force is given in terms of the turbulent kinetic energy, fraction solid, and the crystallite sphericity. The computed macrosegregation results for Al-4.5 pctCu alloy were validated against, and were found to agree with, experimental measurements. The effect of final grain size and frequency on segregation was also determined. This validated model represents a rigorous mathematical framework for describing the flow behavior and solute segregation in electromagnetically stirred melts.