a full 3d simulation of the beginning of the slab casting … · 2014-06-26 · a full 3d...
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TRANSVALOR S.A. – Parc de Haute Technologie – 694, av. du Dr. Maurice Donat – 06255 Mougins cedex – France
Phone: +33 (0)4 9292 4200 – Fax: +33 (0)4 9292 4201 – http://www.transvalor.com
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JAOUEN OLIVIER1 , COSTES FRÉDÉRIC
1, LASNE PATRICE
1, FOURMENT
CHRISTIANE1
A FULL 3D SIMULATION OF THE BEGINNING OF THE SLAB
CASTING PROCESS USING A NEW COUPLED FLUID/STRUCTURE
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
air 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, air 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 (BTR) 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
TRANSVALOR S.A. – Parc de Haute Technologie – 694, av. du Dr. Maurice Donat – 06255 Mougins cedex – France
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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, thermo-
mechanical 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
air 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 THERCAST®, 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.
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. Regarding 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
Turbulent
Fluid
Solid Skin
Strong
coupling
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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:
0. γgσ (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
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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:
))(.()(
TTdt
TdH (2)
where T is the temperature, (W/m/°C) denotes the thermal conductivity and H (J) the
specific enthalpy which can be defined as:
)()()()()(
0
s
T
T
lp TLTgdCTH (3)
TRANSVALOR S.A. – Parc de Haute Technologie – 694, av. du Dr. Maurice Donat – 06255 Mougins cedex – France
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0T (°C) is an arbitrary reference temperature, (kg/m3) the density, sT (°C)the solidus
temperature, pC (J/kg/°C) the specific heat, lg the volume fraction of liquid, and L (J/kg) the
specific latent heat of fusion. In the one-phase modeling, )(Tgl 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: )(n. extTThT where h (W/m²/°C) is the heat transfer
coefficient, and extT is the external temperature
radiation: )(n.44
extstef TTT , where is the steel emissivity, stef is the
Stephan – Boltzmann constant.
external imposed heat flux: impT n. n denotes the outward normal unit vector.
At part/molds interface, heat transfers are taken into account with a Fourier type
equation:
)(1
n. mold
eq
TTR
T (4)
where moldT is the interface temperature of the mold and eqR (W/m²/°C) 1 , the heat transfer
resistance that can depend on the air gap and/or the local normal stress, as presented below:
011
1
0
)11
,1
min(
1
0
0
airseq
airs
radair
eq
eifR
RR
R
eifR
RRR
R
(5)
where air
air
air
eR
and
s
s
s
eR
with aire and se respectively the air gap and an eventual other
body (typically slag) thickness and air and s the air and the eventual other body thermal
conductivity. 0R is a nominal heat resistance depending on the surface roughness,
))((
111
22
moldmoldstef
mold
radTTTT
R
with mold the emissivity of the mold, m
nAR 1 a heat
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 air gap growth is permanently updated, so the heat
TRANSVALOR S.A. – Parc de Haute Technologie – 694, av. du Dr. Maurice Donat – 06255 Mougins cedex – France
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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 air 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 )(t to the interface )(t
between fluids at time t ,
here liquid metal and air; its expression is:
)(/))(,(),(
)(0),(
)())(,(),(
tintxdisttx
tontx
tintxdisttx
(6)
where is the cavity space, )(t is the space occupied by the liquid metal into the cavity,
and )(/ t is the remaining space, meaning the air space. Considering definition (6), flow
front of the metal is the 0 value iso-surface of )(t [6], [7].
Fig. 4: Illustration of the initial setting of the slab casting. The second wide copper plate has
been hidden.
Refractory nozzle
Initial cavity mesh
Copper plates
Slag
Dummy bar
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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 air gap. This air
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 air
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.
a) b)
c) d)
150°C
20°C
150°C
20°C
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Fig. 5: Illustration of the pouring of the cavity, before activating the dummy bar. 0 value iso-
surface 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.
a) b)
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.
20°C
150°C
150°C
20°C
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a) b)
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.
0 Pa
0
75000 Pa 5 MPa
-5 MPa
0.05
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Fig. 8: Illustration of the air gap effect on solid shell thickness on a cutting plane at mould
exit. The solid skin is thinner at air 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.
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 air gap effect, changing the local heat transfers and so impacting the resulting
stresses.
4. Conclusion
THERCAST® is industrially used. Thanks to the original model aimed at coupling
liquid behaviour together with solid deformation, considering the whole range of data
50 Mpa
10 Mpa
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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].
References
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submold bulging computation, 85th Steelmaking Conf. Proc., ISS, Warrendale, PA
(2002) 109-130.
[2] M. Bellet, V.D. Fachinotti, ALE method for solidification modelling, Comput. Methods
Appl. Mech. and Engrg. 193 (2004) 4355-4381.
[3] O. Jaouen, Ph D. thesis, Ecole des Mines de Paris, 1998.
[4] F. Costes, Ph D. thesis, Ecole des Mines de Paris, 2004.
[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 173 – 190, 1999.
[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,
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Schladming (Austria), June 17-22, 2012, A. Ludwig, M. Wu, A. Kharicha (eds.), IOP
Conference Series 33 (2012) 012052, 6 pages
[7] G. François, Ph D. thesis, Ecole des Mines de Paris, 2011.
[8] N. Triolet et al, The thermo-mechanical modeling of the steel slab continuous casting: a
useful tool to adapt process actuators, ECCC 2005.
[9] A. Kumar, M. Zaloznik, H. Combeau,International Journal of Thermal Sciences, vol. 54,
33-47 (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] R. 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) 94-95.