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  • Menghuai Wu*, Abdellah Kharicha and Andreas Ludwig

    Discussion on Modeling Capabilityfor Macrosegregation

    DOI 10.1515/htmp-2016-0133Received June 28, 2016; accepted December 23, 2016

    Abstract: Macrosegregation originates from the solute par-titioning at the liquid-solid interface and the relativemotion between liquid and solid phases during solidifica-tion of metal alloys. A suitable macrosegregation modelshould incorporate solidification thermodynamics, crystalgrowth kinetics, multiphase computational fluiddynamics, and even thermal-structural mechanics.No current model includes all those phenomena, henceleading to assumptions having to be made. This paperdiscusses some modeling assumptions regarding the treat-ment of (1) diffusion kinetics of crystal growth, (2) crystaldendritic morphology and (3) solidification shrinkage.Theoretical analyses based on test calculations weremade. We find that some previous models, which over-simplified some of the aspects mentioned above for thepurpose of computational efficiency, can only be appliedto study macrosegregation qualitatively. They lead to sig-nificant error estimations of macrosegregation. When thequantitative accuracy for macrosegregation is of primaryimportance, the multiphase-based models with an appro-priate treatment of these aspects, despite the sophisticatedcomputational requirements, are recommended.

    Keywords: macrosegregation, solidification, crystal growth,heat and mass transfer, porosity, diffusion kinetics


    Macrosegregation of alloy castings is caused by microse-gregation and relative motion between liquid and solidphases during solidification [14]. As categorized in

    Figure 1, solute partitioning at the solid-liquid interfaceinduces the compositional difference between solid andliquid phases, and diffusion-governed crystal growthkinetics plays an important role in the formation of micro-segregation. The liquid/solid relative motion can beinduced by the flow, the crystal sedimentation, or thedeformation of a dendritic network of crystals in theliquid-solid two-phase region. Causes for the flow can bedifferent, e. g. thermo-solutal buoyancy, shrinkage, etc. Asuitable macrosegregation model should consider theaforementioned phenomena by incorporating solidificationthermodynamics, crystal growth kinetics, multiphase fluidmechanics, and even thermal-structural mechanics. Nocurrent model can include all those phenomena; thereforeassumptions are a practical necessity.

    Different solidification models were proposed for cal-culating macrosegregation in alloy castings. Some ofthose models are simple and computationally efficient,incorporating only one mixture-continuum phase andsimplified growth kinetics [57]; others are relativelycomplex and costly, incorporating the nature of multi-phase/multiphysics [821]. Although plenty of experi-mental evaluation efforts have been made, it is stilldifficult to judge the validity of some model assumptions,merely based on the claimed agreements between calcu-lation and experiments. This is because there are toomany other uncertain variables, such as process para-meters and physical properties that may influence themodeling result. Comprehensive reviews of this topiccan be found elsewhere [34]. This article discussessome model assumptions, many of which stemmingfrom previous models, and their validity for modelingmacrosegregation quantitatively. Those assumptionsare: (1) diffusion kinetics of crystal growth; (2) crystaldendritic morphology; (3) solidification shrinkage.

    Diffusion-governed crystal growthkinetics

    The early continuum model for macrosegregation did notconsider the diffusion-governed crystal growth kineticsexplicitly [5, 6]. Furthermore it assumed that the alloy

    *Corresponding author: Menghuai Wu, Christian Doppler Laboratoryfor Advanced Process Simulation of Solidification and Melting,Montanuniversitaet Leoben, A-8700 Leoben, Austria; Department ofMetallurgy, Montanuniversitaet Leoben, A-8700 Leoben, Austria,E-mail: Kharicha, Christian Doppler Laboratory for AdvancedProcess Simulation of Solidification and Melting,Montanuniversitaet Leoben, A-8700 Leoben, Austria; Department ofMetallurgy, Montanuniversitaet Leoben, A-8700 Leoben, AustriaAndreas Ludwig, Department of Metallurgy, MontanuniversitaetLeoben, A-8700 Leoben, Austria

    High Temp. Mater. Proc. 2017; 36(5): 531539

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  • solidification follows a predefined solidification path, i. e.the evolution of solid phase and the concentrations in thesolid and liquid states are a function of temperature. In thesolidification mushy zone, the interdendritic melt isassumed to have completely mixed, and the solute in thesolid phase is either completely mixed due to strong backdiffusion (following the assumption of lever rule), or noback diffusion (following the assumption Gulliver-Scheil).It is known that complete mixing of interdendritic melt is agood assumption for the deep mushy region where thevolume fraction of solid is large, but it is an unrealisticassumption for the primary dendrite tip region where thevolume fraction of solid is small [22, 23], as schematicallyshown in Figure 2. A solute diffusion profile around thegrowing crystal initially forms for the columnar dendritetip region, and the volume-averaged concentration in theinterdendritic melt, hc,i, is significantly different from thethermodynamic equilibrium concentration, c*,, whichapplies only at the liquid-solid interface. For simplicity,hc,i will be written as c,. It is only in the tip region, wherethe relative motion by flow is significant because of thehigh permeability. Therefore, the macrosegregation modelbased on the assumption of the lever rule or the Gulliver-

    Scheil equation would potentially cause quantitativeerrors.

    In order to demonstrate the significance of the diffu-sion-governed growth kinetics in macrosegregation, a 2Dtest case was calculated: a square casting (50 50mm2) ofa ternary alloy (Fe-0.45 wt.%C-1.06 wt.%Mn). The liquidmelt with an initial temperature of 1,777 K is cooled fromall sides of the boundary with a constant heat transfercoefficient (300W m2 K1) and a constant ambienttemperature of 373 K. The crystal morphology during soli-dification is purely globular (spherical). The nucleationparameters are: maximum crystal number densitynmax = 2.0 10

    9m3, undercooling for maximum nucleationrate TN = 5 K, and Gaussian distribution width T = 2K.There is no solidification shrinkage. Flow is induced bysedimentation of equiaxed crystals and thermo-solutalbuoyancy of the melt. A volume-averaged two-phasemodel [8] was used. Both melt (liquid phase) andequiaxed crystals (solid phase) are treated as separatedand interpenetrating continua. The liquid and solidphases have different velocities, but they are coupledthrough the drag forces. The Boussinesq approach isemployed to consider the buoyancy force for the crystal


    Figure 1: Origin of macrosegregation.










    (a) (b)

    Figure 2: Schematic of the solute distribution field in the dendrite tip region.

    532 M. Wu et al.: Modeling Macrosegregation

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  • sedimentation. Other settings for the simulation and ther-mal physical properties are given in references [24, 25].A diffusion-governed solidification model [17, 21]was used to calculate the solidification rate. As shownin Figure 2(b), the solute partitioning during crystalgrowth causes pile-up of the solute in the liquid regionat the solid-liquid interface. In turn, the solute pile-upslows down the crystal growth rate. Actually, thefinal growth rate of the crystal can be determined bysolving a Stefan problem analytically [21]. The solutepartitioning at the solid-liquid interface is balanced bythe solute diffusions in both the liquid and solid phaseregions. The diffusion lengths, l, and ls, are related tothe radius of the spherical grain, R. The diffusion coeffi-cients of carbon in both liquid and solid are 2.0 108

    and 1.0 109m2 s1, respectively; the diffusion coeffi-cients of manganese in liquid and solid are 4. 0 109

    and 1.2 1013 m2 s1. Three calculations for thistest case were made: I) with the consideration of diffu-sion-governed growth kinetics, II) with the assumptionof Gulliver-Scheil, III) with the assumption of thelever rule. For the latter two cases, the diffusion coeffi-cients in liquid are increased to an enormous value(106m2 s1), to mimic the complete mixing; the

    diffusion coefficients for the solid are either very large(lever rule) or zero (Gulliver-Scheil).

    Final macrosegregation patterns of the casting withthe three calculations are compared in Figure 3. Thedistribution of cindexi are evaluated, where c

    indexi is the

    so-called segregation index which is calculated as(cmix, i c0, i)/c0, i. cmix, i, is the mixture concentration ofthe co-existing primary solid and liquid (or rest eutecticat the end of solidification) phases. The complexity of thecrystal sedimentation during solidification induces nega-tive segregation in the middle and lower bottom region,and the solute-enriched melt is pushed upwards to thetop surface and side-wall regions. Detailed analysis of theformation of such segregation patterns was performedelsewhere [24, 25].

    As the global flow/sedimentation patterns of all threecalculations are quite similar, their macrosegregation pat-terns exhibit certain qualitative similarities. However,large differences in segregation intensity were observed.