A FULL 3D SIMULATION OF THE BEGINNING OF THE SLAB CASTING PROCESS USING A NEW COUPLED FLUID/STRUCTURE MODEL

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1 JAOUEN OLIVIE 1, COSTES FÉDÉIC 1, LASNE PATICE 1, FOUMENT CHISTIANE 1 A FULL 3D SIMULATION OF THE BEGINNING OF THE SLAB CASTING POCESS USING A NEW COUPLED FLUID/STUCTUE MODEL Abstract It is well known now that the slab defects like hot tears or cracks are rooted at the first beginning of the solid shell birth. Damages result from the competition between hydrostatic pressure within the turbulent flow of the liquid zone and the solidifying skin under tensile stresses and strains state. In addition, the thermal energy extracted from the cast product by the mould has huge effect of the thickness of the shell. Among other parameters, it depends on the gap growth issued from the shrinkage of the solidifying metal together with the deformation of the copper plates. Numerically speaking, the method able at taking all that phenomena into account through an accurate way is a fluid/structure model. Indeed, a standard CFD method does not represent the solid behaviour, so that the stresses, strains, gap evolution due to the shrinkage of the shell are not reachable. In that paper, a new 3D fluid/structure model involving the turbulent fluid flow and the solid constitutive equation is described. An application on a slab casting process taken into account the coupling with the deformation of the mould is presented. Keywords 3D finite elements, continuous casting, fluid mechanics, hot tearing, thermo-mechanical coupling, heat transfers 1. Introduction In the process of continuous casting, all the different phases of the steel, from the liquid to the complete solidified zones are coexisting at the same time all over the process, from the meniscus to the end of the casting length. For sure, behaviour of the different metal phases is fully coupled during the process. This is described for example in the illustration coming from B. Thomas [1], where the competition between solidifying skin and liquid metal is mentioned via the ferrostatic pressure (Fig.1). It appears that defects like porosities, cracks or hot tears take their roots from the strains, stresses and distortions occurring at the first instants of solidification in the brittle temperature range (BT) of the alloy. Depending on the tonnage, solidified areas at the end of the pouring of ingots can represent up to 30% to 40% of the total mass. Hence, it is easy to imagine that in such amount of transformed alloy, defects have 1 / 11

2 already occurred in the shell. In case of continuous casting, immediately after the meniscus, near the mould corners, the solidified shell is stressed and deformed thermally and mechanically, creating internal cracks and macro-porosities. Within this framework, thermomechanical modelling is of interest for steel makers. It can be helpful in the adjustment of the different process parameters in order to improve casting productivity while maintaining a satisfying product quality. Here, parameters are casting speed, secondary cooling piloting, bulging control, mould and machine bending, EMS, taper, etc. for the continuous caster concerned. However, optimization of the parameters requires a quite complex model that delivers very precise responses. From this point of view, the use of a CFD model sequenced with a structure model to simulate respectively the liquid and the solid phases, and to forecast the defects, is not well suited. Indeed, it is necessary to take into account at the same time liquid, mushy and solid areas in a coupled model. In addition, at each instant and locally, the gap growth should be taken into account for its influence on the heat transfers between metal shell and moulds that dramatically change throughout the solidification. In this paper, thermo-mechanical models developed in THECAST, casting software dedicated to the simulation of metal solidification are presented. The way of taking into account the coupling between metal and moulds during solidification is shown. A model of determination of the liquid and mushy zones constituted equation parameters is developed. Industrial applications in slab continuous casting are proposed. Turbulent Fluid Solid Skin Strong coupling Fig. 1: Schematic of phenomena in the mould region in continuous casting process [1]. 2. Thermo-mechanical model A 3D finite element thermo-mechanical solver based on an Arbitrary Lagrangian Eulerian (ALE) formulation within the cast product is used. egarding the components of the cooling system, copper plates, steel frames, running system, slag, etc. the formulation is Lagrangian. One of the special features of the casting software is that a specific contact analysis is applied in order to define the face to face correspondence between the different 2 / 11

3 meshes interfaces. Indeed, each body is independent from others with its own mesh. Only the geometric matching at interface has to be ensured initially. This will also be a boarded later. At any time, the mechanical equilibrium is governed by the momentum equation:. σ g γ 0 (1) where σ is the Cauchy stress tensor, g is the gravity vector, and γ is the acceleration vector. The very different behaviours of liquid and solid metal is considered by a clear distinction between constitutive equations associated to the liquid, the mushy and the solid states respectively. In order to fit the complex behaviour of solidifying alloys, a hybrid constitutive model is accounted. In the one-phase modelling, the liquid (respectively, mushy) metal is considered as a thermo-newtonian (respectively thermo-viscoplastic, VP) fluid. In the solid state, the metal is assumed to be thermo-elastic-viscoplastic (EVP) (Fig. 2). In particular, moulds are treated through an EVP model that can derive to elastic-plastic (EP) behaviour depending on yield stress values, when considered as deformable. Solid regions are treated in a Lagrangian formulation, while liquid regions are treated using ALE [2]. More precisely, a so called, transient temperature, or coherency temperature, is used to distinguish the two different behaviours. It is typically defined between liquidus and solidus, and usually set close to solidus. For more information, the interested reader can refer to [3] to [5]. Fig. 2: Schematic representation of the rheological behavior for the different phases of the metal in solidification conditions Following Fig. 2 scheme, the Cauchy stress tensor σ is that way, respectively locally expressed as the corresponding behavior to the cooling metal state. However, this scheme has a drawback. Indeed, the resolution is carried out in one shot, taking account the total range of temperature. But, what is possible in thermal point of view is not mechanically speaking. Due to the limits of actual algorithms and hardware precision, it cannot consider the corresponding total range of concerned viscosity. That is why, in order to override that limits and to account the very large range of data, namely the viscosity, from liquid to solid metal, a two steps scheme is applied for the global resolution of mechanical equations (1). So that, one step is dedicated to the liquid and mushy zones, the liquid solver, and the second one is dedicated to the mushy and solid ones, the solid solver, typically, the one above (Fig. 2). Under that context, two cases are possible. On the one hand, option 1, with the transient temperature that bounds the two steps. The full coupling liquid/solid is ensured by the control of liquid 3 / 11

4 velocities and pressure with the solid corresponding ones at transient temperature volume interface [6]. On the other hand, option 2, an overlap within the mushy zone is also available. So that, both liquid solver and solid solver are applied on the total or partial range between liquidus and solidus (Fig. 3). Another advantage of such a scheme is that any model can be associated to each solver. In particular, turbulent fluid flow within the liquid zone of the metal is managed by the Navier- Stokes equations completed by terms coming from LES method [7]. But, any other model can be called. Fig. 3: Schematic representation of the option 2 for the 2 steps algorithm. The high level of temperature for the step 1 (Tstep1) is within the mushy zone range. Same, the low level of temperature for the step 2 (Tstep2) is within the mushy zone, such that Tstep 1> Tstep2. In case of Tstep1 = Tstep2, it is similar to the option 1. The choice of the two temperatures Tstep1 and Tstep2 is depending on the structure of the alloy within the mushy zone, typically, it can depend on the viscosity and/or the solid fraction and the composition. The thermal problem treatment is based on the resolution of the heat transfer equation, which is the general energy conservation equation: dh( T) dt.( ( T) T ) (2) where T is the temperature, (W/m/ C) denotes the thermal conductivity and H (J) the specific enthalpy which can be defined as: T H( T) ( ) C ( ) d g ( T) L( T ) (3) T0 p l s 4 / 11

5 T 0 ( C) is an arbitrary reference temperature, (kg/m 3 ) the density, T s ( C)the solidus temperature, C p (J/kg/ C) the specific heat, g l the volume fraction of liquid, and L (J/kg) the specific latent heat of fusion. In the one-phase modeling, g l (T ) can be previously calculated using the micro-segregation model PTIMEC_CEQCSI [8] or results from micro-segregation model that can be used [9]. The boundary conditions applied on free surface of the metal could be of classical different types: average convection: T n h( T T ) where h (W/m²/ C) is the heat transfer. ext coefficient, and T ext is the external temperature 4 4 radiation: T.n stef ( T T ext ), where is the steel emissivity, stef is the Stephan Boltzmann constant. external imposed heat flux: T. n n denotes the outward normal unit vector. At part/molds interface, heat transfers are taken into account with a Fourier type equation: imp 1 T. n ( T Tmold ) (4) eq where T mold is the interface temperature of the mold and eq (W/m²/ C) 1, the heat transfer resistance that can depend on the gap and/or the local normal stress, as presented below: eq eq 1 s min(, ) 0 rad 1 s if e if e 0 (5) where e and e s s with s e and es respectively the gap and an eventual other body (typically slag) thickness and and s the and the eventual other body thermal conductivity. 0 is a nominal heat resistance depending on the surface roughness, rad stef ( T T mold 2 mold 1 )( T T mold with mold the emissivity of the mold, ) A a heat 1 resistance taking into account the normal stress n, A and m being the parameters of the law. As mentioned above, the transfers, thermal and mechanical, between components of the cooling system are carried out via connections established through a dedicated contact analysis. Yield, at each Gauss point of the surface mesh and at each time step, the distance to the in front faces is computed. Hence, the gap growth is permanently updated, so the heat m n 5 / 11

6 transfers following (5). That way, the full thermo-mechanic coupling between cast product and molds is ensured. 3. Slab casting application As an example of the power of such a model, a slab casting application is proposed. Fig. 4 presents the configuration of the initial setting. The case is representing the very first beginning of the slab casting. The dummy bar is firstly fixed until the cavity is fulfilled. Once the volume full at 100%, the dummy bar is moving down at casting speed. The water channel cooling is ensured by a convection boundary condition at external face of the copper plates. The slag is taken into account by a dedicated meshed body on top of the cavity. In that scheme, the step 2 of the algorithm is not only dealing with the liquid metal, but also with the within the cavity till this one is full. Fig. 5 shows how the flow front of the liquid metal is evolving during the pouring. This is possible thanks to use of a level-set method. Level-set represents the distance ( to the interface ( between fluids at time t, here liquid metal and ; its expression is: ( x, dist( x, ( ) in ( ( x, 0 on ( ( x, dist( x, ( ) in / ( (6) where is the cavity space, ( is the space occupied by the liquid metal into the cavity, and / ( is the remaining space, meaning the space. Considering definition (6), flow front of the metal is the 0 value iso-surface of ( [6], [7]. efractory nozzle Slag Initial cavity mesh Dummy bar Copper plates Fig. 4: Illustration of the initial setting of the slab casting. The second wide copper plate has been hidden. 6 / 11

7 On Fig. 5, it can be seen that the copper plates are warmed at internal side by the hot metal filling the cavity. At the same time, the cooling channels are maintaining the external side quite cold, ensuring the control of the heat extraction from the cast product. The same phenomenon is continuing all over the process as shown Fig. 6, where the action of the dummy bar is on. As explained above, the 2 steps algorithm allows the full thermo-mechanical coupling between liquid and solid, so that, the competition presented Fig. 1 is perfectly caught by the solver as illustrated Fig.7. It shows on the one hand, the ferrostatic pressure within the liquid zone, on the other hand, the strain and stresses into the solid skin. From these distributions, it is easy to determine the probability to get some cracks or hot tearing localized at end of solidification into the brittle range of temperature of the alloy. Indeed, Yamanaka criterion is based on strain under tensile stress state and is dedicated to the prediction of cracks occurrence [10]. Associated to thermo-mechanic computation, the shrink of the solidifying alloy is rendered. Fig 8 shows how the solidified skin shrinks at corner creating an gap. This gap has immediate influence on heat transfers that is taken account through (5). The thickness of solid skin changes depending on heat extraction. Hence it is thinner at corner due to the gap that modifies heat transfers acting like a local insulator. This is the phenomena at origin of under solidification at corner and can lead to a break out at mould exit. 150 C 20 C a) b) 150 C 20 C c) d) 7 / 11

8 Fig. 5: Illustration of the pouring of the cavity, before activating the dummy bar. 0 value isosurface represents the liquid metal flow front. Iso-values are the corresponding temperature distribution within the cooling system. a) 10%, b) 25% c) 50% and d) 75% of filling. With such a tool, it is so possible to control the cooling of the moulds with water channels and prevent from troubles by ensuring solid skin thickness big enough to avoid break out. Coupled with the cast product history, the copper plates are also taken into account as deformable bodies. Hence, the stresses and strains are so calculated within the moulds while the metal is cast. Fig. 9 shows how the copper plates are thermally and mechanically loading following of the passing of the hot metal. Also, the tensile stresses into the copper are illustrated. 150 C 150 C a) 20 C b) 20 C c) d) Fig. 6: Illustration of the pouring of the cavity, once the dummy bar activated. The temperature within the cooling system is shown, together with the mesh adaptation that applies fine mesh size in order to catch gradients of fields, like heat gradient or cooling rate. 8 / 11

9 75000 Pa 5 MPa 0 Pa a) b) -5 MPa c) Fig. 7: Ferrostatic pressure within the liquid metal (a). Tensile stresses distribution into solidifying alloy (b). Strain distribution at solid skin (c). Illustration on a cutting plan at middle of the mould. White lines show the mushy zone. 9 / 11

10 Fig. 8: Illustration of the gap effect on solid shell thickness on a cutting plane at mould exit. The solid skin is thinner at gap location, while it is broader at contact with copper plates. Depending on its thickness at moulds exit, the solid wall can break under the hot molten alloy pressure if it s not strong enough to resist to, yielding a catastrophic breaking out. Note the mesh adaptation, in that case around the mushy zone. 50 Mpa 10 Mpa Fig. 9: Illustration of hot metal passing in front of the copper plates (two cutting planes have been applied on the wide and small plates). The tensile stresses are shown on the plates; the expansion is exaggerated 100 times. Note the shape of the stress distribution at corner. It results from the gap effect, changing the local heat transfers and so impacting the resulting stresses. 4. Conclusion THECAST is industrially used. Thanks to the original model aimed at coupling liquid behaviour together with solid deformation, considering the whole range of data 10 / 11

11 variations, it allows determining the thermo-mechanical comportment of the solidifying metal in continuous casting processes. In addition, associated to specific boundary conditions, it leads to forecast accurately the defects of slabs or billets. The power of this model is such that it gives access at the same time, to phenomena occurring at different scales. Indeed, fluid flow with turbulent behaviour in the liquid zone of the alloy is shown, together with stresses and strains within the solid metal. Hence, it allows to better understand the impact of phases on each other through a two steps full coupling algorithm. In particular, the root of defects occurring at the very beginning of solidification is now much easier to catch. With such a tool, steel makers are able to foresee and so, to control and optimize their process. The presented example illustrates how nowadays numerical models could be used in the steel industry to improve the quality of production and the productivity [11], [12]. eferences [1] C. Li, B.G. Thomas, Maximum casting speed for continuous cast steel billets based on submold bulging computation, 85th Steelmaking Conf. Proc., ISS, Warrendale, PA (2002) [2] M. Bellet, V.D. Fachinotti, ALE method for solidification modelling, Comput. Methods Appl. Mech. and Engrg. 193 (2004) [3] O. Jaouen, Ph D. thesis, Ecole des Mines de Paris, [4] F. Costes, Ph D. thesis, Ecole des Mines de Paris, [5] M. Bellet et al, Proc. Int. Conf. On Cutting Edge of Computer Simulation of Solidification and Casting, Osaka, The Iron and Steel Institute of Japan, pp , [6] M. Bellet, O. Boughanmi, G. Fidel, A partitioned resolution for concurrent fluid flow and stress analysis during solidification: application to ingot casting, Proc. MCWASP XIII, 13th Int. Conf. on Modelling of Casting, Welding and Advanced Solidification Processes, Schladming (Austria), June 17-22, 2012, A. Ludwig, M. Wu, A. Kharicha (eds.), IOP Conference Series 33 (2012) , 6 pages [7] G. François, Ph D. thesis, Ecole des Mines de Paris, [8] N. Triolet et al, The thermo-mechanical modeling of the steel slab continuous casting: a useful tool to adapt process actuators, ECCC [9] A. Kumar, M. Zaloznik, H. Combeau,International Journal of Thermal Sciences, vol. 54, (2012) [10] O. Cerri, Y. Chastel, M. Bellet, Hot tearing in steels during solidification Experimentalcharacterization and thermomechanical modeling, ASME J. Eng. Mat. Tech. 130 (2008) 1-7. [11]. Forestier et al, Finite element thermomechanical simulation of steel continuous casting, MCWASP XII, TMS, 2009 [12] J. Demurget et al., Increase the productivity of the vertical continuous machine of Hagondange plant. FFA JSI 2012 proceedings, (2012) / 11