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INTRODUCTION

Process modeling for metal casting processes has been very successful for companies around the world. The early applications were two-dimensional (2-D), primarily axisymmetric. During the 1980s, process modeling propagated clearly from aerospace applications to a wide range of companies in transportation and energy. As a result of rapid improvement of algorithms and computing technologies, three-dimensional (3-D) process modeling became practical during the 1990s. Today, many small to mid-sized companies perform multiple simulations on a daily basis. Process modeling is no longer a luxury, but a necessity for survival in the casting industry.

Process simulation capabilities have been extended beyond thermal and flow modeling for shape casting. Today, coupled modules for the calculation of grain structure, porosity, hot tearing, and solid state transformation are commercially available. The goal of this article is to illustrate applications of numerical simulation to continuous casting processes, as well as to present some results of advanced models in investment casting. The examples shown hereafter were calculated with the software calcosoft^®, a finite element method software,¹ developed jointly by Calcom and the Swiss Federal Institute of Technology in Lausanne, Switzerland.

CONTINUOUS CASTING OF STEELS

Complex Cooling Pattern

Continuous casting of large steel slabs involves complex surface cooling patterns, due to the alternation of rolls and spray zones. Through user-defined functions, it is possible to define accurately such complex cooling profiles.

Figures 1a-b shows the stationary temperature field in the whole slab and near the mold, whereas Figure 1c presents a zoom on the fraction solidified in the middle longitudinal cross-section. In Figure 1d, the temperature evolution in five different locations in the thickness of the slab, from the surface to the center, is shown. One can well see the effect of the complex surface-cooling pattern. The surface temperature oscillates strongly when it passes a roll or a spray zone.

Due to the relatively low thermal conductivity of the steel, such surface temperature oscillations are not propagated very far inside the slab, as shown in Animation 1. In this animation, the evolution of the temperature profile across the slab thickness (from the surface to the center) is presented. The evolution of the profile is shown from 0.975 m. (exit of the mold) to 1.725 m. from the top surface (corresponding to 65-115 seconds).

Effect of Nozzle Design on Shell Thickness

In continuous casting of large steel slabs, controlling the onset of solidification is very important. At the exit of the mold, about one meter below the liquid level, the solidified shell is still very thin. In order to prevent any leak, one should guarantee that there is no weakness in the shell.

In this example, the effect of the fluid flow coming out of the nozzle on the thickness of the solid shell is studied. Two different nozzle geometries were tested: one with a large opening and the other with a small opening in order to reduce the flow toward the lateral faces of the slab.

Figure 1. (a-top left) The stationary temperature field in the whole slab and (b-top right) near the mold. (c-bottom left) The middle longitudinal cross-section, and (d-bottom right) the temperature evolution from the surface to the center of the slab.		Animation 1. The evolution of the temperature profile across the slab thickness (from the surface to the center).

Figure 2a shows the modeled geometry, with an enlargement of the two nozzles, which were used. In Figure 2b, the flow coming out of the large nozzle is presented, with a significant amount of flow going toward the lateral face. Figure 2c shows the solid fraction in various vertical cross-sections for the large nozzle case. The zoom shows that the shell is thinner close to the nozzle, due to the effect of the hot liquid jet coming out of the nozzle. In Figure 2d, a comparison between the large nozzle and the small nozzle shows that the shell thickness is much more regular in the small nozzle case.

Figure 2 (a-top left). The modeled geometry, with an enlargement of the two nozzles used in continuous casting. (b-top right) The large nozzle flow, and (c-bottom left) the solid fraction in various vertical cross-sections that resulted from that nozzle. (d-bottom right) A comparison of thickness resulting from the large nozzle (left) and the small nozzle (right).

Effect of Inoculation on Grain Structure

Controlling the grain structure is very important in continuous casting of steel. To avoid center-line segregation, it is favorable to promote equiaxed structures. In such structures, impurities are dispersed instead of being concentrated in the center of the slab, and are pushed ahead of the columnar zone. In the following example, the effect of nucleation is emphasized on the columnar-to-equiaxed transition. Figure 3a shows the thermal and the fraction of solid fields, and Figure 3b presents the grain structure in a transverse cross-section.^2-4 One can see that as nucleation in the bulk of the liquid is difficult, there are only columnar grains, up to the center line. However, if inoculation is promoted (by electromagnetic stirring, for example), equiaxed grains are favored (Figure 3c). Animation 2 and Animation 3 show the formation of these grain structures.

Figure 3 (a-left). The thermal and the fraction-of-solid fields in a continously cast steel slab. (b-top right) The grain structure in a steel slab transverse cross-section,2-4 with only columnar grains up to the center line. (c-bottom right). Equiaxed grains that result if inoculation is promoted (by electromagnetic stirring for example).

Solid-State Transformations During Continuous Casting

The solid-state transformations occurring upon solidification and cooling of steel can be modeled.⁵ Animation 4 shows the evolution of temperature (top) and fraction of solid (bottom) during the solidification and cooling of a continously cast steel slab. Animation 5 presents the appearance of the different phases, 200 and 7,000 mm below the liquid level. The fractions of delta-ferrite, austenite, alpha-ferrite, pearlite, and bainite are shown. The liquid is drawn in white.

Animation 2: The formation of grain structures during continuous casting of steel.		Animation 3: The formation of grain structures during the continuous casting of steel.
Animation 4: The evolution of temperature (top) and fraction of solid (bottom) during the solidification and cooling of a continuously cast steel slab.		Animation 5. The appearance of the different phases, 200 and 7,000 mm below the liquid level, during the solidification and cooling of a continously cast steel slab.

CONTINUOUS CASTING OF ALUMINUM ALLOYS

Twin Roll Strip Casting

The tuning of a strip-casting machine, which produces thin sheets of aluminum of a few millimeters in thickness, is a complex task. One should control carefully the size and location of the mushy zone as a function of the casting speed, the melt superheat, and the cooling parameters.

Figure 4 (a-left).The stationary thermal field in an aluminum alloy, and (b-right) in the wheel.				Animation 6. The solid fraction, in cross-sections from the side of the rolls to the center, of aluminum alloy in a strip casting machine.

Figures 4a and 4b show the stationary thermal field in the aluminum alloy as well as in the wheel. Animation 6 presents the solid fraction, in cross-sections from the side of the rolls to the center. One can well see that in such conditions, the fraction of solid repartition is not uniform at all, due to an uneven inlet velocity distribution.

Hot Tearing Sensitivity Calculations in DC Casting

Recently, Rappaz, Drezet, and Gremaud⁶ proposed a new model for hot cracking sensitivity (HCS). This model was applied to various cases of direct chill (DC) casting of aluminum billets.

Figure 5. (a-top left) The HCS (right) across the radius of the 200 mm diameter billet (left), for a 6063 aluminum alloy. The stationary temperature field is shown in red-yellow, while the fraction of solid field is shown in blue-white. (b-top right) Center-line cracking that results because hot cracking sensitivity is much higher in the center. (c-bottom left) A comparison of the HCS for different billet diameters (160, 200, and 240 mm.), and (d-bottom right) the influence of the secondary cooling conditions

Figure 5a shows the HCS across the radius of the 200 mm diameter billet for a 6063 aluminum alloy. The stationary temperature field is shown in red-yellow, whereas the fraction of solid field is shown in blue-white. One can see that the hot cracking sensitivity is much higher in the center, leading to center-line crack., as shown in Figure 5b.⁷

Figure 5c presents the comparison of the HCS for different billet diameters (160 mm, 200 mm, and 240 mm), whereas Figure 5d shows the influence of the secondary cooling conditions.

Primary Phase Formation in Globulitic Structures of Al-Si-Mg Alloys

In DC casting of aluminum, alloys are most often very highly inoculated in order to produce a very fine grain structure. Thus, the grains are globulitic, meeting their neighbors in the very early stage of dendritic growth. Since such alloys are most often heat treated, it is important to know the as-cast microstructure (i.e., primary phase and precipitates), which is the initial state of the alloy prior the homogenization and precipitation treatments.

Figure 6. (a-left) The Mg (left) and Si (right) concentration maps at the end of solidification in a globulitic sample of and Al-Mg-Si alloy. (b-right) The final concentration profile of Si across the sample (along the yellow horizontal line).				Animation 7. The evolution of the concentrations of Mg (left) and Si (right), in both the liquid and the solid phases, during the solidification of an Al-Mg-Si alloy.

Figure 6a shows the magnesium and silicon concentration maps at the end of solidification in a globulitic sample of and Al-Mg-Si alloy. Animation 7 presents the evolution of the concentrations of magnesium and silicon, in both the liquid and the solid phases, during the solidification. Figure 6b shows the final concentration profile of silicon across the sample (along the yellow horizontal line).

This calculation was performed using a "pseudo-front tracking technique"⁸ similar to phase field models.

Grain Movement in an Al-Si Casting

In large castings, when equiaxed solidification occurs, the equiaxed grains may move, due to fluid convection ahead of the mushy zone. A 2-D grain structure model 2 -4 was coupled with fluid flow to simulate the grain movement.^9,10

Figure 7. (a-left) A snapshot of the grain growth, together with the velocity vectors in the liquid and the streamlines, during solidification, in an Al-Si casting. (b-right)The final grain structure.				Animation 8. The evolution of the temperature and velocity fields during the solidification of an Al-Si casting.		Animation 9. The full grain growth in an Al-Si casting. As two grains touch, they form a cluster. When this cluster is linked to the mold wall, it no longer moves

Figure 7a shows a snapshot of the grain growth, together with the velocity vectors in the liquid and the streamlines, during solidification, in an Al-Si casting. In Animation 8, which presents the evolution of the temperature and velocity fields during the solidification, one can well see the vortex due to natural convection. Animation 9 shows the full grain growth, including the grain motion. One should notice that as soon as two grains touch, they form a cluster. When this cluster is linked to the mold wall, it does not move anymore. Figure 7b shows the final grain structure. In reality, one should consider the break-out of clusters, which could lead to avalanches.

CONTINUOUS CASTING OF MAGNESIUM ALLOYS

Horizontal Continuous Casting

To see the influence of liquid flow on the liquid pool and on the mushy zone during horizontal continuous casting of magnesium, two different ingate geometries were tested.

Figure 8. (a-top left) The surface temperature of a magnesium casting, including the chill and the ceramics (top), and a longitudinal cross-section of the fraction of solid in the middle vertical section (bottom). (b-top right) A non-symmetrical flow, due to a narrow and off-centered ingate. (c-bottom left) A comparison of the fraction of solid and velocity fields in the small ingate case (top), versus the larger ingate (bottom). (d-bottom right) Same as (c), with the velocities only.

Figure 8a shows the surface temperature of the casting, including the chill, the ceramics, and a longitudinal cross-section of the fraction of solid in the middle vertical section. Animation 10 presents the evolution of the temperature across the billet, in transverse cross-sections, whereas the evolution of the fraction of solid in longitudinal cross-sections is shown in Animation 11.

Animation 10. The evolution of the temperature across a magnesium casting, in transverse cross-sections.		Animation 11. The evolution of the fraction of solid of a magnesium casting in longitudinal cross-sections.		Animation 12. The skin of the casting after the exit of the die. A small region in the bottom of the billet is still mushy (oval spot in light blue or in yellow after the color change).

One can see in Animation 12 that the skin of the casting is not fully solid after the exit of the die. There is a small region in the bottom of the billet that is still mushy. This is the consequence of a non-symmetrical flow, due to a narrow and off-centered ingate (Figure 8b).

In order to prevent this hot spot, the ingate was enlarged. Figures 8c and 8d compare the fraction of solid and velocity fields in the small ingate case and the larger ingate, showing that the situation is improved in the second case.

CONTINUOUS CASTING OF COPPER ALLOYS

Start-up Phase in Semi-Continuous Casting of a Copper Alloy Billet

In the start-up phase of semi-continuous casting, one should control the rate of solidification, as a function of the withdrawal speed, in order to prevent leaks and to minimize cracking.

Figure 9. (a-left) The temperature (left) and the fraction of solid (right) fields in a copper alloy billet, when the steady state is reached. (b-center) The velocity field in the whole billet, and (c-right) the liquid pool.

Figure 9a shows the temperature and the fraction of solid fields in a copper alloy billet when the steady state is reached. Animation 13 shows an animation of the start-up phase (temperature), and Animation 14 presents the corresponding evolution of fraction of solid.

Animation 13. The start-up phase (temperature) of semi-continuous casting.		Animation 14. The evolution of fraction of solid fields in a copper alloy billet.		Animation 15. Grain growth (only the surface grains are shown). Many grains, which are nucleating at the bottom of the blade, against the copper chill, are selected and many are eliminated.

Figure 9b shows the velocity field in the whole billet, with a zoom in the liquid pool (Figure 9c).

INVESTMENT CASTING

Grain Evolution in DS and SX Turbine Blades

In directionally solidified (DS) or single crystal (SX) turbine blades, the knowledge of the formation of the grain structure is of primary importance. A 3-D microstructure model was applied, together with a thermal modeling in order to calculate the nucleation and growth of the grains,^2-4 in the case DS and SX blades.

Figure 10. (a-top left) The thermal field and the grain structure during solidification and at the end of the casting process. (b-top right) Stereographic projections (pole figures) show that few grains remain at the end. (c-bottom left) The calculated grain structure in a hollow SX blade, and (d-bottom right) The result of three calculations of the grain selector

Figure 10a shows the thermal field and the grain structure during solidification, as well as at the end of the casting process, of a DS blade. Animation 15 presents dynamically the grain growth (only the surface grains are shown). One can see how many grains, which are nucleating at the bottom of the blade, against the copper chill, are selected and many are eliminated. At the end (i.e., top of the blade), only a few grains remain, as shown in the stereographic projections (pole figures) presented in Figure 10b.

This 3-D model allows study of the stochastic nature of grain nucleation in SX blades. Figure 10c presents calculated grain structure in a hollow SX blade, and Figure 10d shows the result of three calculations of the grain selector shown in Figure 10c. The thermal history is exactly the same in all three cases, but the random number generator used in the stochastic nucleation was initialized differently. This allows statistics to be gathered on the grain orientations, which are obtained at the exit of a grain selector, and provides insight into the influence of process parameters on the distribution of SX orientations.

Modeling of Porosity Formation in an Equiaxed Turbine Blade

Most of the porosity models rely on criteria (e.g., Niyama criteria). Rappaz and Pequet¹¹ have proposed a new porosity modeling method, based upon the physics of pressure drop in the mushy zone, gas segregation, and cavitation, as well as nucleation and growth of pores.

Figure 11. (a-top left) The temperature (left) and the fraction of solid (right), in case an equiaxed turbine blade, case A, where the blade was cast in a thick shell with a wrap. (In the subsequent porosity results, case A is shown on the left and case B, where the blade was cast in a thin shell without a wrap, ion the right.) (b-top right) The calculated porosity on the surface of the blade, whereas (c-bottom left) and (d-bottom right) present the porosity in orthogonal cross-sections.

This model provides a description of both gas and shrinkage porosity, as well as macroporosity or holes, such as the piping observed at the top of the risers.
This model has been applied to two different cooling conditions of an equiaxed turbine blade. In calculation A, the blade was cast in a thick shell with a wrap, leading to a slow cooling rate, whereas in calculation B, a thin shell and no wrap was used, leading to a faster cooling. Figure 11a shows the temperature and the fraction of solid in case A. Animation 16 demonstrates the cooling of the blade (case A).

Animation 16. The cooling of an equiaxed turbine blade.		Animation 17. Porosity of turbine blade, shown in "x-ray" mode.

Figure 11b shows the calculated porosity on the surface of the blade, whereas Figure 11c and Figure 11d present the porosity in orthogonal cross-sections. Animation 17 shows the porosity in "x-ray" mode (all the metal with a porosity level lower than the threshold value is made transparent).

CONCLUSIONS

New developments are being reported on the simulation of almost all kinds of phenomena taking place during solidification. The challenge today is not only in the development of certain stand-alone models for some specific phenomena, but also in the integration of existing models into the whole process chain model. As a result, metal production, solidification, homogenization, extrusion, rolling, properties of the final products, and recycling will be studied and linked in the future mainly by computer modeling. In other words, the integrated process chain model will replace, to a large extent, the still-dominating approach of trial-and-error and will help establish computer modeling as the leading tool for R&D in the future by all major companies in the world.

ACKNOWLEDGEMENTS

The authors acknowledge the scientific contributions and results of Michel Rappaz, Jean-Luc Desbiolles, Alain Jacot, Jean-Marie Drezet, and Christel Pequet from the Swiss Federal Institute of Technology in Lausanne, Switzerland, as well as of Charles-André Gandin, from the Ecole des Mines of Nancy, France.

References

1. Calcosoft-3D manual, (Calcom SA, Lausanne, Switzerland, 2001)
2. Ch.-A. Gandin et al., "A Three-Dimensional Cellular Automaton-Finite Element Model for the Prediction of Solidification Grain Structures," Met. Mater. Trans., 30A (1999), pp. 3153-3165.
3. Ch.-A. Gandin and M. Rappaz, "A 3D Cellular Automaton Algorithm for the Prediction of Dendritic Grain Growth," Acta Mater., 45 (1997), pp. 2187-2195.
4. Ch.-A. Gandin and M. Rappaz, "A Coupled Finite Element-Cellular Automation Model for the Prediction of Dendritic Grain Structures in Solidification Processes," Acta Meltall. Mater., 42 (7) (1994), pp. 2233-2246.
5. A. Jacot et al., "Modelling of Electromagnetic Heating, Cooling and Phase Transformations during Surface Hardening of Steels," J. de Physique IV (C1) (1996), p. 203.
6. M. Rappaz, J.-M. Drezet, and M. Gremaud, "A New Hot Tearing Criterion," Met. Mater. Trans. A, 30A (1999), pp. 449-455.
7. I.. Farup, J.-M. Drezet, and M. Rappaz, "In-situ Observation of Hot Tearing Formation in Succinonitrile-Acetone," Acta Mater., 49 (2001), pp. 1261-1269.
8. M. Rappaz and A. Jacot, "Modelling of Solidification Microstructure Formation," Proc.of the Int. Conf. on the Sci. of Casting and Solidification, ed. D.M. Stefanescu et al. (Brasov, Romania: Editura Lux Libris, 28-31 May, 2001).
9. Ch.-A. Gandin, T. Jalanti, and M. Rappaz, "Modeling of Dendritic Grain Structures (Keynote)," Modeling of Casting, Welding and Advanced Solidification Processes VIII, ed. B.G. Thomas and C. Beckermann (Warrendale, PA: TMS, 1998), p. 363.
10. M. Rappaz et al., "Modelling of As-Cast Microstructures," Materials Science Forum, ed. H. Driver et al. (Uetikon-Zuerich, Switzerland: Transtec Publications Ltd., 1996), pp. 217-222.
11. C. Pequet and M. Rappaz, "Modeling of Porosity Formation during the Solidification of Al. Alloys Using a Mushy Zone Refinement Method," Modeling of Casting, Welding and Advanced Solidification Processes IX, ed. R. Sahm, N. Hansen, and G. Conley (Aachen, Germany: Shaker Verlag, 2000), pp. 71-79.

For more information, contact P. Thevoz, Calcom SA, Parc Scientifique-EPFL, CH-1015 Lausanne, Switzerland; e-mail mail@calcom.ch.