Modeling of macrosegregation in direct-chill casting of aluminum alloys: Estimating the influence of casting parameters

Materials Science and Engineering A 413–414 (2005) 85–91

Modeling of macrosegregation in direct-chill casting of aluminum alloys: Estimating the influence of casting parameters b ˇ Miha Zaloˇznik a,∗ , Boˇzidar Sarler b

a Impol d.d., R&D Department, Partizanska 38, SI-2310 Slovenska Bistrica, Slovenia Nova Gorica Polytechnic, Laboratory for Multiphase Processes, Vipavska 13, SI-5000 Nova Gorica, Slovenia

Received in revised form 29 June 2005

Abstract The influences of the most important process parameters on macrosegregation in Al–5.25 wt.% Cu direct-chill cast billets of 218 and 282 mm diameter were studied by performing numerical simulations. Solidification and transport of heat, momentum, and species were modeled using the Bennon–Incropera one-phase continuum mixture model. A rigid, coalesced solid phase was assumed in the mushy zone, which was modeled as a porous medium. The model was solved numerically, using the finite volume method in axisymmetric geometry. Five cases, encompassing variations of billet diameter, casting velocity, casting temperature, and mold cooling type were simulated. A detailed analysis of the interplay of transport of species by flows induced by thermal and solutal natural convection as well as by solidification shrinkage and their effect on the final macrosegregation pattern is given. It was shown that the process parameters affect macrosegregation through their direct impact on the thermosolutal flow in the liquid pool and the mushy zone. The main factors were shown to be the depth of the liquid pool, and the temperature difference that drives the thermal natural convection in the liquid pool. © 2005 Elsevier B.V. All rights reserved. Keywords: Direct-chill casting; Macrosegregation; Process parameters; Melt flow; Numerical modeling

1. Introduction Macrosegregation is a common defect in the direct-chill (DC) casting of aluminum alloys. In vertical DC casting, the melt is poured from the furnace through a troughing system into a bottomless mold with the cross-sectional shape of the casting. A starter head mounted to a hydraulic ram forms a false bottom to the mold. As it continuously lowers at a controlled casting velocity, the metal inside the mold partly solidifies to form a solid shell strong enough to support the melt. Below the mold, water jets spray water onto the billet surface to cool the billet and complete the solidification. Macrosegregation is an inhomogeneous distribution of alloying elements at the scale of the solidified casting. This can lead to nonuniform mechanical properties that affect the behavior of the metal during subsequent processing and impair the quality of the final product. It is therefore desirable to be able to simulate the casting process in order to predict the influence of the casting parameters on the resulting

∗

Corresponding author. Tel.: +386 1 4701918; fax: +386 1 4701919. E-mail address: [email protected] (M. Zaloˇznik).

0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.09.056

macrosegregation pattern and improve the process in terms of mitigation of macrosegregation. Besides prediction, modeling is aimed at improved understanding of the basic mechanisms involved. Macrosegregation is caused by solute transport primarily due to the flow of segregated liquid in the mushy zone, which is a result of buoyancy forces due to thermal gradients (thermal natural convection), buoyancy forces due to concentration gradients (solutal natural convection), density differences between the two phases (solidification shrinkage), and inlet flow (bulk convection). Except for shrinkage-induced flow, the flow in the mushy zone is confined to the high-liquid-fraction part of the mush close to the liquid–mush interface and is driven by more severe natural and forced convection flow in the bulk liquid part of the casting. The most pronounced and commonly observed [1–3] solute distribution pattern in a DC cast billet is a radial concentration distribution. A solute-depleted region is present in the billet center, adjoined by a positive segregation zone spreading into the outward radial direction, an adjacent thin negative segregation zone and another positive segregation layer at the surface. The enriched subsurface layer is attributed to solidificationshrinkage induced flow, while the other patterns are generally

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a consequence of an interplay of thermal convection, solutal convection and shrinkage flows. The exact mechanism is not yet completely understood. Attempts at explanation of observed macrosegregation in industrial and small-size DC castings were made by Jalanti et al. [4,5] who simulated 5xxx-series alloys and concluded that shrinkage flow has a dominant effect in Al–Mg castings. However, the general picture seems to be more complex, as is suggested by the results presented in this paper. It is now also recognized, that the negative centerline segregation is at least partly a consequence of the deposition of low-concentration free-floating crystal grains, which has been shown by several researchers [1,6]; but could partly also be a consequence of modified flow conditions due to a changed mushy-zone permeability in grain refined castings [2,6]. First attempts at the modeling of these particular effects were made by Vreeman and coworkers [7,8] as well as by Reddy and Beckermann [9,10]. For a more detailed overview of macrosegregation modeling the reader is directed to the most recent extensive review [11], which is, among others, also dealing with the particular case of DC casting. Several experimental studies were also published, which should, with the support of simulation, help to understand the mechanisms of macrosegregation. In addition to the already mentioned [1,2,6], a recent paper of Eskin et al. [3] presents a systematic experimental investigation of the dependence of macrosegregation and structure on process parameters. The purpose of the present paper is to investigate the mechanisms through which the most important DC-casting parameters affect macrosegregation in an industrial-size Al–5.25 wt.% Cu billet. The study is performed by simulating the process at varying casting velocities, casting temperatures, and types of mold heat transfer. A one-phase mixture model, described in the next section, is used and it is solved with the finite volume method. Although the assumption of a rigid mushy zone precludes quantitatively accurate results, the following study is able to construct a conceptual picture of the interplay of thermally, solutally and solidification-shrinkage induced flows in the liquid and mushy parts of the billet and to explain the effect of casting parameters on macrosegregation through their influence on these flows. 2. Model description 2.1. Governing equations

Vk , Vm

fs + fl = 1,

fk =

mk ρk = gk , mm ρm

ρ m = gs ρ s + g l ρ l ,

vm = fs vs + fl vl ,

hm = fs hs + fl hl ,

Cm = fs Cs + fl Cl ,

(1) (2)

(3)

where v is the velocity, h the enthalpy, and C is the concentration. The governing equations are formulated in terms of the mixture quantities, for which they are solved. Nevertheless, it is inevitable that terms containing phase quantities are retained. They are expressed as correction “source” terms, appearing as last terms on the right-hand side in the equations given below. The continuity equation for the mixture retains the form valid for one phase: ∂ρm + ∇ · (ρm vm ) = 0. ∂t The mixture momentum conservation equation is

(4)

∂ρm vm + ∇ · (ρm vm vm ) ∂t
 µl ρm ρm = −∇p + ∇ · µl ∇vm − (vm − vs ) ρl K ρl − ρ0 g [βT (T − T0 ) + βC (Cl − C0 )] − ∇ · [ρm (fs vs vs + fl vl vl − vm vm )],

(5)

where p is the pressure, µl the liquid viscosity, K the permeability, g the gravity acceleration, βT and βC the thermal and solutal expansion coefficients, and ρ0 , T0 , and C0 are the reference density, temperature, and concentration, respectively. The velocity of the solid is defined to be equal to the casting velocity (vs = vcast ) everywhere, since all solid mush is assumed to be rigid, coalesced in a porous matrix and connected with the bulk solid. The permeability is modeled by the Kozeny–Carman relation K = K0 gl3 /(1 − gl )2 , where K0 is the permeability constant. The diffusive term in the energy conservation equation (Fourier law) is reformulated in terms of the mixture enthalpy using the following supplementary thermodynamic state equations:  T hs (T ) = href + cps dT, Tref

The macroscopic conservation equations are transport equations for heat, mass, momentum, and species, simultaneously valid in the liquid, mushy and solid regions. A rigid and connected solid phase is assumed, which forms a porous mush in the phase change zone. The one-phase continuum mixture model of Bennon and Incropera [12] is used, where the two-phase (solid– liquid) mixture is described using volume fractions (gk ) and mass fractions (fk ) of solid (k = s) and liquid (k = l) phases, defined as follows: gk =

where V is the volume, m the mass, ρ the density, and the subscript “m” denotes the solid–liquid mixture. The mixture quantities are then defined in terms of phase quantities:

hl (T ) = hs (T ) + Leut +



T

Teut

(cpl − cps ) dT,

(6)

where cps and cpl are the specific heats of solid and liquid, respectively, Leut the latent heat of solidification at the eutectic temperature Teut , and href is the enthalpy at the reference temperature Tref . The mixture energy conservation equation is thus written ∂(ρm hm ) + ∇ · (ρm vm hm ) ∂t

 k k =∇· ∇hm + ∇ · ∇(hs − hm ) cps cps −∇ · [ρm (fs vs hs + fl vl hl − vm hm )],

(7)

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where k is the thermal conductivity of the solid–liquid mixture. The mixture species conservation equation is

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boundary conditions were prescribed at the right boundary for the contact of the liquid and the mold. The heat transfer to the mold in the non-insulated part was modeled by a heat transfer coefficient dependent on the liquid fraction at the contact

∂(ρm Cm ) + ∇ · (ρm vm Cm ) ∂t = ∇ · (ρm Dm ∇Cm ) + ∇ · [ρm (fs Ds ∇Cs + fl Dl ∇Cl − Dm ∇Cm )] − ∇ · [ρm (fs vs Cs + fl vl Cl − vm Cm )], where D is the diffusivity of the respective species. The phase quantities (fs , fl , vs , vl , hs , hl , Cs , Cl ) and the temperature T, which still appear in the mixture transport equations (4), (5), (7), and (8) are modeled in the following way. According to the aforementioned assumption, vs = vcast and, consequently, Eqs. (2) and (3) yield vl = [vm − (1 − fl )vcast ]/fl . Further, the unknowns fs , hs , and hl can be eliminated using Eqs. (2) and (6). The solid concentration Cs is determined by the partitioning that follows from the inverse lever rule: Cs = kp Cl .

(9)

The remaining three unknowns fl , Cl , and T are determined by solving a system of three equations: the definition of mixture enthalpy (Eq. (10)), derived from Eqs. (3) and (6), the inverse lever rule (Eq. (11)), which can be derived from Eqs. (3) and (9), and an additional equation, describing the linearized liquidus line equation (Eq. (12)):    T  T hm (T, fl ) = cps dT + fl Leut + (cpl − cps ) dT , Tref

Teut

f

(8)

(1−f )

l surface: hmold = hhigh hlow l , where hhigh = 5000 W/m2 K and hlow = 400 W/m2 K. At the insulation ring the heat transfer coefficient was approximated to be zero. The secondary cooling was modeled using the Weckman–Niessen correlation [13] for the heat transfer coefficient at a vertical surface cooled by a subcooled-nucleate-boiling falling-water film. The water entry temperature was 300 K and a heat balance was used to calculate the increase of the water temperature along the billet surface. The bottom of the computational domain was assumed to be adiabatic. While this assumption is not exact due to the short domain length of 0.6 m it should not significantly affect neither flow nor macrosegregation. All boundaries, except the inflow boundary, are impermeable for species transport. The left boundary (billet centerline) is a symmetry boundary. The thermophysical properties of Al–5.25 wt.% Cu used in the simulations are given in Table 1 together with their sources. They were obtained from the materials properties modeling software JMatPro [14] and from Ref. [7]. All process parameters, which vary from case to case and are not explicitly given here, are summarized in Table 2.

(10) 3. Numerical solution procedure

Cm Cl = , kp + fl (1 − kp )

(11)

T (Cl ) = Tf + mL Cl ,

(12)

where Tf is the melting temperature of the pure solvent, mL the slope of the liquidus line in the binary phase diagram, and kp is the solid–liquid equilibrium partition ratio. By substitution of variables in Eqs. (10) and (11) a quadratic equation for fl in the form F (fl ) = 0 is obtained, which is solved analytically for fl . Then the temperature T is calculated from Eq. (10) and the liquid concentration Cl from Eq. (12). Further, the solid concentration Cs is calculated from Eq. (9), and the phase enthalpies hs and hl from their definitions in Eq. (6). 2.2. System specifications and boundary conditions Billets of two different sizes were simulated, with diameters of 282 and 218 mm. Axisymmetry was assumed. Two different molds were used for the smaller dimension. The first one was a hot-top mold, which is 4 cm high and incorporates an insulation ring on the upper 3 cm of height. The second one was a conventional water-cooled mold of 7 cm height. In both cases the metal entry is only through part of the mold opening; the exact inlet diameters are given in Table 2. The melt enters with the nominal composition of 5.25% Cu, the prescribed casting temperature (Tcast ) and a uniform velocity determined by the casting velocity (the withdrawal velocity of the false bottom) and the solidification shrinkage; i.e. the inflow mass flux must be the same as the outflow from the domain at the bottom. Slip

3.1. Numerical method The set of macroscopic transport equations (Eqs. (4), (5), (7) and (8)) was solved with the finite volume method (FVM) in axisymmetric geometry and on an orthogonal grid. A grid of 100 × 356 control volumes (CV) with additional gridpoints at Table 1 Thermophysical properties of Al–5.25 wt.% Cu Solid density, ρs (kg/m3 ) Liquid density, ρl (kg/m3 ) Solid specific heat, cps (J/kg K) Liquid specific heat, cpl (J/kg K) Solid thermal conductivity, ks (W/m K) Liquid thermal conductivity, kl (W/m K) Solid diffusivity, Ds (×10−12 m2 /s) Liquid diffusivity, Dl (×10−9 m2 /s) Latent heat at eutectic, Leut (×105 J/kg) Liquid viscosity, µ (×10−3 Pa s) Thermal expansion coefficient, βT (×10−4 K−1 ) Solutal expansion coefficient, βC Reference temperature, T0 (K) Reference concentration, C0 Reference density, ρ0 (kg/m3 ) Permeability constant, K0 (×10−11 m2 ) Pure Al melting temperature, Tf (K) Al–Cu eutectic temperature, Teut (K) Eutectic solidification interval, Teut (K) Eutectic concentration, Ceut (wt.% Cu) Cu solubility in Al at Teut , Cα,eut (wt.% Cu) Partition coefficient, kp

2750 [14] 2460 [14] 1030 [14] 1130 [14] 180 [14] 80 [14] 5 [7] 5 [14] 3.77 [14] 1.3 [14] 1.17 [7] −0.73 [7] 973.15 0.0525 2460 6.67 [7] 933.6 [14] 821.2 [14] 1 33.21 [14] 5.74 [14] 0.173

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Table 2 Process parameters for the simulated cases Case

Diameter (mm)

vcast (mm/s)

Tcast (K)

mold type

Qcool (×10−3 m3 /s)

Inlet (mm)

1 2 3 4 5

282 218 218 218 218

1.25 1.43 1.14 1.43 1.43

973.15 973.15 973.15 953.15 973.15

Hot-top Hot-top Hot-top Hot-top Water cooled

1.75 1.56 1.56 1.56 1.56

172 140 140 140 140

the domain boundaries (thus a total of 102 × 358 gridpoints) was used in all cases and it was strongly biased in the top part (the height of the liquid and mushy regions). The top part (0.15 m in the large billet and 0.12 m in the small billet) was discretized into 100 × 200 equal CV and the remaining bottom part into 100 × 256 CV with a grid expansion factor of 1.0175. The equations were solved implicitly and the SIMPLE algorithm was used for pressure–velocity coupling. For the discretization of advective fluxes the centered difference scheme was used in the energy equation and the first-order upwind scheme in the species and momentum equations. The upwind scheme was used to prevent stability problems with higher order schemes, appearing in the parameter range of advection-diffusion transport of species [15]. It was also used to cure severe problems with iterative convergence of the coupled equation system, which were encountered in the use of higher order schemes for momentum advection. One should note that this is only a preliminary computational framework that works efficiently. It is used before the full implementation of more accurate numerical schemes, which have been shown to be necessary [16], is completed. 3.2. Solution procedure The implicit iterative solution of the strongly coupled equations can be rather unstable at the beginning of the calculation. The development of a strong thermosolutal convective flow in the liquid pool and its interaction with the mushy zone can severely slow down convergence if the solution of the full problem is started right from the beginning with an unfavorable initial state. Therefore, a special procedure was used to speed up the computations. First, only an approximation of the temperature, liquid fraction and flow fields was obtained by solving the energy and momentum equations with both thermal and solutal expansion coefficients (βT , βC ) set to zero. Then the realistic expansion coefficients and the solute conservation equation were invoked, with species diffusion coefficients initially increased by 100 times. The latter was done to somewhat relax the gradients of the solutal buoyancy body force driving the flow, thus easing convergence during the initial redistribution of copper in the mushy and liquid zones. After a calmed down flow was obtained, the diffusion coefficients were set to their realistic values and the computations were continued until a quasi-steady radial macrosegregation profile was obtained. A complete steady state of the system was not found, since slight oscillations of the flow persisted even long after the quasi-steady segregation profile was reached. This is not entirely surprising, since physically oscillatory flow was reported before in computations of heat transfer and fluid flow in DC casting [16]. Contrarily, Vree-

man and Incropera [7], who reported unsteady flow in some of their macrosegregation computations, presumed a numerical problem. It is important to note that in the cases treated in this paper the flow oscillations had only a negligible effect on the macrosegregation pattern. 4. Results 4.1. Simulated cases Five different cases are presented in the present paper. They illustrate variations of billet size, casting velocity (vcast ), casting temperature (Tcast ), and mold type and the resulting macrosegregation profiles. These are believed to be the casting parameters with the strongest influence on macrosegregation. Other parameters with smaller effects (e.g. the water flow rate [3]) remain to be investigated. The specifications of the five cases are given in Table 2. Case 1 is observed in more detail to shed some light on the flow structures and species transport in the sump and on the way they result in the final macrosegregation pattern. The other cases are discussed afterwards, focusing on the mechanism of the effect of the different parameters on the macrosegregation profile. The main distinguishing features of the flow and species transport are explained. 4.2. Case 1: formation of macrosegregation To make a clear presentation of the origins of the thermosolutal buoyancy-driven flow in the liquid sump, Fig. 1 shows the flow field together with the liquid-density field (calculated according to the Boussinesq approximation used in the momentum equation (Eq. (5))). As can be seen in the plot, the liquid is relatively quiescent at the bottom and in the center of the liquid sump, which appears to be caused by the counteracting effect of thermal buoyancy and the layering tendency of the heavy solute-rich liquid. Next to the liquid–mush interface a complex flow structure, consisting of small vortices, forms. Five vertically arranged flow cells can be identified, delimited by sharp density gradients. Each cell contains one or two vortices of thermosolutal origin. Such a multicellular structure occurs in thermosolutal flows, when a stable solutal stratification is destabilized by a large enough lateral temperature gradient [17]. The coupling of the thermal and solutal driving forces and the advective transport of heat and solute also causes a vertical density bump inside each thermosolutal cell and raises the splitting of the flow into two vortices. In the solidification zone the liquid is enriched due to segregation of the solute (Cl > Cm ; Cs < Cm ). Since copper has larger

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age, as the liquid fills up the space left by the shrunken solidified metal. The liquid in the low-fl mushy region is highly segregated, thus even small relative velocities can result in considerable net solute transport. This can be clearly seen along the whole mushy zone. The direction of the shrinkage-induced flow is opposite to the liquid fraction gradient. As the enriched liquid flows towards the solid front it is replaced by the leaner liquid from regions of the mushy zone with higher fl . The enriched liquid accumulates at the front and the concentration rapidly increases in the direction opposite to the fl -gradient. Because of the characteristic sump shape and the direction of the shrinkage-induced solute transport, this flow tends to decrease the concentration in the center and increase the concentration in the outer part of the billet. This is also the mechanism causing typical positive subsurface segregation. 4.3. Simulation of parameter influence Fig. 1. Liquid density (ρl ) and velocity (vm ) fields in the billet for Case 1. The density field is shown only in the liquid and part of the mushy zone (fl > 0.1).

density than aluminum, a solutal downward flow of enriched liquid can be observed in the high-fl portion of the mushy zone (along the liquid–mush interface). This flow carries enriched liquid away from the top-outer part (top right in Fig. 2) of the mushy zone, replacing it with solute-lean liquid, which penetrates into the mush from the bulk liquid region. The heavy, enriched liquid flows down the slope of the mushy zone front, accumulating at the bottom of the sump. As the circulation, driven by thermal convection, carries some high concentration liquid out of the mushy zone into the bulk-liquid sump, smaller circulations, driven by strong solutal buoyancy can be observed. They are characterized by a rapid deflection back downward due to the strong effect of increased density. In the low-fl regions of the mushy zone, the permeability of the porous mush rapidly decreases; consequently, drag forces dominate over buoyancy and inertia. The flow is driven mostly by solidification shrink-

Fig. 2. Mixture Cu concentration (Cm ) field in the billet for Case 1.

The influence of the main process parameters on macrosegregation is estimated by calculations of the segregation patterns in the smaller (218 mm-diameter) billet for variations of casting parameters considered realistic for the process. Case 2 represents the basic case. The casting velocity is decreased by 20% in Case 3 and the casting temperature by 20 K in Case 4. In Case 5, a change of mold type is simulated causing a significant change of heat transfer in the mold. The parameters for all cases are shown in Table 2. The thermosolutal flow structure driving the species transport in the basic Case 2 is similar to the structure described for Case 1. In both cases there is positive segregation in the center and a smooth negative segregation minimum, which is shown in Fig. 3. As expected, the segregation is more pronounced in the larger billet. In both cases severe oscillations are present in the subsurface part of the segregation profile. Their origin is numerical and was analyzed before in detail [18]. A remedy is alignment of the grid with the liquid–mush interface or local grid refinement. In Case 3, where the casting velocity was decreased, the sump is much shallower, as shown in Fig. 4. This restricts the flow, resulting in a simpler flow structure with one dominant main circulation. In the absence of small circulations less solute is advected from the mushy zone into the bulk liquid and there-

Fig. 3. Comparison of macrosegregation profiles as a function of the normalized radius for different billet sizes.

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Fig. 4. Impact of process parameters on the sump shape. The (fl = 0) and (fl = 1) lines are shown (left: casting velocity, center: casting temperature, right: mold cooling).

fore the settling of enriched liquid at the sump bottom is less pronounced. Consequently, the overall segregation is weaker at lower casting velocity, as can be seen in Fig. 5. A decrease of casting temperature in Case 4 causes only a slight shift of the shape of the mushy zone as is shown in Fig. 4. The key to the explanation of the lower segregation is the flow in the liquid region. The primary effect of a smaller temperature difference in an otherwise preserved sump is a smaller driving force for thermal natural convection. The consequence is a less intensive main circulation resulting into less transport of enriched liquid from the mush into the bulk liquid and along the liquid–mush interface towards the centerline. Due to the preserved sump shape, on the other hand, shrinkage transport with its tendency of lowering the concentration at the centerline gains importance in relation to advective transport. The resulting decrease of positive segregation in the billet center and slight increase of concentration in the outer part of the billet, shown in Fig. 6, is fully consistent with this interpretation. Case 5, shown in Fig. 7, simulates a variation in mold cooling. The conventional water-cooled mold causes an earlier formation of a solid shell and a lower heat flux in the direct-chill zone. Consequently, the thickness of the mushy zone is significantly increased. This allows a deeper penetration of the downward flowing current into the mushy zone. The throughout concave shape of the liquid–mush interface on the outer half of the billet

Fig. 5. Prediction of the influence of casting velocity variation on macrosegregation.

radius also appears to favor less discharge of solute-rich liquid from the mushy zone into the liquid pool, which explains the steep transition from negative subsurface macrosegregation to positive macrosegregation at the mid-radius. The flow of the enriched liquid stays in the mushy zone and transports solute towards the center. In the central part this effect is less pronounced and it seems to approximately balance with solutetransport by shrinkage flow, thus causing the zone of relatively constant positive macrosegregation in the center.

Fig. 6. Prediction of the influence of casting temperature variation on macrosegregation.

Fig. 7. Prediction of the influence of mold cooling on macrosegregation.

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5. Conclusions and future research

Acknowledgments

Effects of casting velocity, casting temperature, and mold cooling on macrosegregation during DC casting of an Al– 5.25 wt.% Cu alloy were numerically simulated and analyzed. The results showed that these parameters influence the macrosegregation pattern through their direct impact on the thermosolutal natural convection flow in the liquid pool and on the shape and thickness of the mushy zone. The thermosolutal flow has the tendency to increase macrosegregation in the billet center, especially when instability causes the emergence of smaller flow cells, which make the transport of solute-rich liquid from the mushy zone even more effective. It is generally intensified by a deeper liquid pool and by a higher casting temperature. The opposite, or better to say, alleviating effect is that of the shrinkage flow, which transports solute-rich interdendritic liquid away from the centerline. Its magnitude depends primarily on the shape of the mushy zone, i.e. locally on its slope and globally on its depth. Higher casting velocity and higher casting temperature both promote macrosegregation. The first through an increase of the liquid pool depth and the latter by an increase of the temperature difference driving the thermal natural convection the liquid pool. The estimations of the effects of the casting parameters presented in the paper show, that although the model is not capable of properly predicting macrosegregation patterns, it can be used to predict trends caused by variations of the parameters. These trends, in the sense of changes in the degree of macrosegregation, also correspond to experimentally observed trends, e.g. in [3]. The next steps in the development towards a predictive model shall consist of extensions of the physical and the numerical model. The former concerns the effect of the transport of free-floating grains on macrosegregation, which significantly affects centerline segregation. The latter is an improvement of the numerical accuracy of the model by the use of more accurate higher order discretization schemes for advection of momentum and species. The importance of numerical accuracy in the simulation of DC casting was recently shown by the authors [16].

The work was supported by the Slovenian Ministry of Higher Education, Science and Technology (MVZT) and the Ministry of the Economy through the Young Researchers program, as well as by MVZT through grant L2-5387-1540-03. References [1] H. Yu, D.A. Granger, in: E.A. Starke, T.H. Saunders (Eds.), Aluminum Alloys: Their Physical and Mechanical Properties, EMAS, Sheffield, UK, 1986, pp. 17–29. [2] T.L. Finn, M.G. Chu, W.D. Bennon, in: C. Beckermann, L.A. Bertram, S.J. Pien, R.E. Smelser (Eds.), Micro/Macro Scale Phenomena in Solidification, ASME, New York, NY, USA, 1992, pp. 17–24. [3] D.G. Eskin, J. Zuidema Jr., V.I. Savran, L. Katgerman, Mater. Sci. Eng. A384 (2004) 232–244. [4] T. Jalanti, M. Swierkosz, M. Gremaud, M. Rappaz, in: K. Ehrke, W. Schneider (Eds.), Continuous Casting, DGM Publ./Wiley/VCH, Weinheim, Germany, 2001, pp. 191–198. ´ [5] T. Jalanti, Etude et mod´elisation de la macros´egr´egation dans la coul´ee semi-continue des alliages d’aluminium, PhD Thesis, EPFL, Lausanne, Switzerland, 2000. [6] A. Joly, G.U. Gr¨un, D. Daloz, H. Combeau, G. Lesoult, Mater. Sci. Forum 329/330 (2000) 111–120. [7] C.J. Vreeman, F.P. Incropera, Int. J. Heat Mass Transfer 43 (2000) 687–704. [8] C.J. Vreeman, J.D. Schloz, M.J.M. Krane, J. Heat Transfer 124 (2002) 947–953. [9] A.V. Reddy, C. Beckermann, in: V.R. Voller, S.P. Marsh, N. El-Kaddah (Eds.), Materials Processing in the Computer Age II, TMS, Warrendale, PA, USA, 1995, pp. 89–102. [10] A.V. Reddy, C. Beckermann, Metall. Mater. Trans. B28 (1997) 479–489. [11] C. Beckermann, Int. Mater. Rev. 47 (2002) 243–261. [12] W.D. Bennon, F.P. Incropera, Int. J. Heat Mass Transfer 30 (1987) 2161– 2170. [13] D.C. Weckman, P. Niessen, Metall. Mater. Trans. 13B (1982) 593–602. [14] N. Saunders, X. Li, A.P. Miodownik, J.-P. Schill´e, in: J.-C. Zhao, M. Fahrmann, T.M. Pollock (Eds.), Materials Design Approaches and Experiences, TMS, Warrendale, PA, USA, 2001, pp. 185–197. ˇ [15] M. Zaloˇznik, B. Sarler, D. Gobin, Mater. Technol. 38 (2004) 249–255. ˇ [16] M. Zaloˇznik, B. Sarler, in: H. Kvande (Eds.), Light Metals, TMS, Warrendale, PA, USA, 2005, pp. 1031–1036. [17] D. Gobin, R. Bennacer, Int. J. Heat Mass Transfer 39 (1996) 2683–2697. [18] B.C.H. Venneker, L. Katgerman, in: P.R. Sahm, P.N. Hansen, J.G. Conley (Eds.), Modelling of Casting, Welding and Advanced Solidification Processes IX, Shaker Verlag, Aachen, Germany, 2000, pp. 670–687.