Numerical simulation of a solidifying Pb-Sn alloy: The effects of cooling rate on thermosolutal convection and macrosegregation

Numerical simulations of a binary metal alloy (Pb-Sn) undergoing solidification phase change are performed using a continuum model for conservation of total mass, momentum, energy, and species. The system is contained in an axisymmetric, annular mold which is cooled along its outer vertical wall. Results show that thermosolutal convection in the melt and mushy zones is strongly coupled and that macrosegregation is reduced with increased cooling rate. For low cooling rates, solutally induced convection in the mushy zone favors the development of channels, which subsequently spawn macrosegregation in the form of A-segregates. With increasing solidification rate, however, thermosolutal interactions in the melt contribute to reducing the formation of channels and A-segregates.     

Modeling of solute redistribution in the mushy zone during solidification of aluminum-copper alloys

A mathematical model has been established to predict the formation of macrosegregation for a unidirectional solidification of aluminum-copper alloys cooled from the bottom. The model, based on the continuum formulation, allows the calculation of transient distributions of temperature, velocity, and species in the solidifying alloy caused by thermosolutal convection and shrinkage-induced fluid flow. Positive segregation in the casting near the bottom (inverse segregation) is found, which is accompanied by a moving negative-segregated mushy zone. The effects of shrinkage-induced fluid flow and solute diffusion on the formation of macrosegregation are examined. It is found that the redistribution of solute in the solidifying alloy is caused by the flow of solute-rich liquid in the mushy zone due to solidification shrinkage. A higher heat-extraction rate at the bottom increases the solidification rate, decreasing the size of the mushy zone, reducing the flow of solute-rich liquid in the mushy zone and, as a result, lessening the severity of inverse segregation. Comparisons between the theoretical predictions from the present study and previous modeling results and available experimental data are made, and good agreements are obtained.     

Macrosegregation in a multicomponent low alloy steel

Macrosegregation theory is extended to predict the formation of channel-type segregation for multicomponent systems. Specifically, calculations are carried out for 0.7 pct C steel, by considering heat, mass and momentum transport in the mushy zone. In the model used for calculations the momentum transport equation and the energy equation were solved simultaneously. It is confirmed, by comparing calculated results with experimental results, that this model successfully predicts the occurrence of channel-type segregation. This analysis is also more rigorous than previous works on macrosegregation because previous analyses were done by solving for convection in the mushy zone with an “uncoupled” temperature field. Using the model, the effects of adjusting the compositions of silicon and molybdenum in steel were quantitatively evaluated in order to show how channel-type segregates can be avoided by adjusting alloy composition. A method of optimizing composition to minimize segregation is presented. It is recommended that this methodology be applied to alloy design so that ingots of alloys amenable to commercial practice can be obtained readily with a minimum amount of “trial-and-error” development work and expense. Formerly Research Affiliate, Department of Materials Science and Engineering, Massachusetts Institute of Technology AR, was Visiting Scientist, Department of Materials Science and Engineering, Massachusetts Institute of Technology     

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.     

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.     

Macrosegregation during directional solidification of Pb-Sb alloys

Macrosegregation of Sb was investigated during directional solidification of binary Pb-Sb alloys containing 2.2 and 5.8 wt% Sb over growth rates varying from 0.8 to 30 μm s^?1. The cellular to dendritic transition was observed at a growth rate of 3.0 μm s^?1 in Pb-2.2 Sb alloy in contrast to a growth rate of 1.5 μm s^?1 in Pb-5.8 Sb alloy. The chemical analysis data revealed considerable macrosegregation of Sb along the longitudinal section of alloys. The degree of macrosegregation increased with a decrease in the growth rate. This behavior is discussed in light of thermo-solutal convection in the mushy zone as well as that in the melt ahead of the solid-liquid interface.     

Macrosegregation during dendritic arrayed growth of hypoeutectic pb- sn alloys: Influence of primary arm spacing and mushy zone length

Thermosolutal convection in the dendritic mushy zone occurs during directional solidification of hypoeutectic lead tin alloys in a positive thermal gradient, with the melt on the top and the solid below. This results in macrosegregation along the length of the solidified samples. The extent of macrosegregation increases with increasing primary dendrite spacings for constant mushy zone length. For constant primary spacings, the macrosegregation increases with decreasing mushy zone length. Presence of convection reduces the primary dendrite spacings. However, convection in the interdendritic melt has significantly more influence on the spacings as compared with that in the overlying melt, which is caused by the solutal buildup at the dendrite tips. Formerly Graduate Student, Chemical Engineering Department, Cleveland State University     

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.     

Modeling of micro- and macrosegregation and freckle formation in single-crystal nickel-base superalloy directional solidification

The formation of macrosegregation and freckles by multicomponent thermosolutal convection during the directional solidification of single-crystal Ni-base superalloys is numerically simulated. The model links a previously developed thermodynamic phase equilibrium subroutine with an existing code for simultaneously solving the macroscopic mass, momentum, energy, and species conservation equations for solidification of a multicomponent alloy. Simulation results are presented for a variety of casting speeds and imposed thermal gradients and for two alloy compositions. It is found that for a given alloy composition, the onset of convection and freckle formation occurs at a critical primary dendrite arm spacing, which agrees well with previous experimental findings. The predicted number and shape of the freckle chains in the unstable cases also agree qualitatively with experimental observations. Finally, it is demonstrated how the onset and nature of convection and macrosegregation vary with alloy composition. It is concluded that the present model can provide a valuable tool in predicting freckle defects in directional solidification of Ni-base superalloys.     

A thermodynamic prediction for microporosity formation in aluminum-rich Al-Cu alloys

A computer model is used to predict the formation and the amount of microporosity in directionally solidified Al-4.5 wt pct Cu alloy. The model considers the interplay between so-called “solidification shrinkage” and “gas porosity” that are often thought to be two contributing and different causes of interdendritic porosity. There is an accounting of the alloy element, Cu, and of dissolved hydrogen in the solid- and liquid-phase during solidification. Consistent with thermodynamics, therefore, a prediction of forming the gas-phase in the interdendritic liquid is made. The local pressure within the interdendritic liquid is calculated by macrosegregation theory that considers the convection of the interdendritic liquid, which is driven by density variations within the mushy zone. Process variables that have been investigated include the effects of thermal gradients and solidification rate, and the effect of the concentration of hydrogen on the formation and the amount of interdendritic porosity. These calculations show that for an initial hydrogen content less than approximately 0.03 ppm, no interdendritic porosity results. For initial hydrogen contents in the range of 0.03 to 1 ppm, there is interdendritic porosity. The amount is sensitive to the thermal gradient and solidification rate; an increase in either or both of these variables decreases the amount of interdendritic porosity.     

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.     

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.     

Thermosolutal convection during dendritic solidification of alloys: Part II. Nonlinear convection

A mathematical model of thermosolutal convection in directionally solidified dendritic alloys has been developed that includes a mushy zone underlying an all-liquid region. The model assumes a nonconvective initial state with planar and horizontal isotherms and isoconcentrates that move upward at a constant solidification velocity. The initial state is perturbed, nonlinear calculations are performed to model convection of the liquid when the system is unstable, and the results are compared with the predictions of a linear stability analysis. The mushy zone is modeled as a porous medium of variable porosity consistent with the volume fraction of, interdendritic liquid that satisfies the conservation equations for energy and solute concentrations. Results are presented for systems involving lead-tin alloys (Pb-10 wt pct Sn and Pb-20 wt pct Sn) and show significant differences with results of plane-front solidification. The calculations show that convection in the mushy zone is mainly driven by convection in the all-liquid region, and convection of the interdendritic liquid is only significant in the upper 20 pct of the mushy zone if it is significant at all. The calculated results also show that the systems are stable at reduced gravity levels of the order of 10⁻⁴ g ₀ (g ₀=980 cm·s⁻¹) or when the lateral dimensions of the container are small enough, for stable temperature gradients between 2.5≤G _l≤100 K·cm⁻¹ at solidification velocities of 2 to 8 cm·h⁻¹.     

Convection and channel formation in solidifying Pb-Sn alloys

A suite of experiments on the dendritic solidification of Pb-Sn melts has been carried out. The first goal has been to quantify the longitudinal macrosegregation, and hence the convective vigor through the dendritic (“mushy”) zone during solidification, as a function of the mushy zone Rayleigh number. The mushy zone Rayleigh number Ra _m is a ratio of the driving compositional buoyancy force to the retarding Darcy frictional force. The second goal has been to characterize the formation of convection channels as a function of Ra _m . In a fixed furnace, the melts were program cooled and solidified from beneath, at various cooling rates. Two different temperature gradients were examined. Each pairing of cooling rate and temperature gradient results in a different Ra _m . As expected, the measured longitudinal macrosegregation increased with Ra _m . The vestiges of convection channels on a solidified ingot surface (which we call “freckle trails”) were observed for all conditions except for the most rapid cooling rate with the smaller temperature gradient (i.e., the smallest Ra _m ) and for the slowest cooling rate with the larger temperature gradient (i.e., the largest Ra _m ). Under the latter solidification conditions, the vestiges of convection channels in an ingot interior (which we call “chimneys”) were observed. Chimneys were not observed in other ingots. When present, the number of freckle trails decreased and the width of the trails increased with increasing Ra _m . The trails became more diffuse as well. It appears that Ra _m may control channel characteristics as well as convection and the resulting macrosegregation. There appear to be two critical values, a lower one for surface freckle trails and a higher one for interior chimneys. Conditions at the Earth’s inner-outer core boundary (ICB) may be those exhibiting high Ra _m convection, so that convection channels, if they exist, could be as large as several hundred meters in width.     

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.     

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.     

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.     

Convection and channel formation in solidifying Pb−Sn alloys

A suite of experiments on the dendritic solidification of Pb–Sn melts has been carried out. The first goal has been to quantify the longitudinal macrosegregation, and hence the convective vigor through the dendritic (“mushy”) zone during solidification, as a function of the mushy zone Rayleigh number. The mushy zone Rayleigh number Ra _m is a ratio of the driving compositional buoyancy force to the retarding Darcy frictional force. The second goal has been to characterize the formation of convection channels as a function of Ra _m . In a fixed furnace, the melts were program cooled and solidified from beneath, at various cooling rates. Two different temperature gradients were examined. Each pairing of cooling rate and temperature gradient results in a different Ra _m . As expected, the measured longitudinal macrose gregation increased with Ra _m . The vestiges of convection channels on a solidified ingot surface (which we call “freckle trails”) were observed for all conditions except for the most rapid cooling rate with the smaller temperature gradient (i.e., the smallest Ra _m ) and for the slowest cooling rate with the larger temperature gradient (i.e., the largest Ra _m ). Under the latter solidification conditions, the vestiges of convection channels in an ingot interior (which we call “chimneys”) were observed. Chimneys were not observed in other ingots. When present, the number of freckle trails decreased and the width of the trails increased with increasing Ra _m . The trails became more diffuse as well. It appears that Ra _m may control channel characteristics as well as convection and the resulting macrosegregation. There appear to be two critical values, a lower one for surface freckle trails and a higher one for interior chimneys. Conditions at the Earth's inner-outer core boundary (ICB) may be those exhibiting high Ra _m convection, so that convection channels, if they exist, could be as large as several hundred meters in width.     

Effect of magnetic field on the microstructure and macrosegregation in directionally solidified Pb-Sn alloys

An investigation into the influence of a transverse magnetic field (0.45 T) on the mushy zone morphology and macrosegregation in directionally solidified hypoeutectic Pb-Sn alloy shows that the field has no influence on the morphology of dendritic arrays. The field does, however, cause severe distortion in the cellular array morphology. Cellular arrayed growth with the magnetic field results in an extensive channel formation in the mushy zone, as opposed to the wellaligned and uniformly distributed cells formed in the absence of the field. The channels are produced due to the anisotropy in the thermosolutal convection caused by the magnetic field. Macrosegregation, however, along the length of the directionally solidified samples is not influenced by this magnetic field for either the cellular or dendritic arrays. Formerly Graduate Student, Chemical Engineering Department, Cleveland State University.     

Solutal convection induced macrosegregation and the dendrite to composite transition in off-eutectic alloys

The effect of solute gradient induced convection during vertical solidification on the macrosegregation of Pb-rich Pb-Sn off-eutectic alloys is determined experimentally as a function of composition and growth rate. In many cases macrosegregation is sufficient to prevent the plane front solidification of the alloy. The transition from dendritic to composite structure is found to occur when the composition of the solid is close enough to the eutectic composition to satisfy a stability criterion based onG _L /V (liquid temperature gradient/growth rate). A vertical or horizontal magnetic field of 0.1 T (1 kilogauss) does not reduce macrosegregation, but downward solidification (liquid below solid) virtually eliminates macrosegregation in small (∼3 mm) diameter samples.