a full 3d simulation of the beginning of the slab casting … · 2014-06-26 · a full 3d...

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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

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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)




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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




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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

[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) 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,

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) 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.


a full 3d simulation of the beginning of the slab casting … · 2014-06-26 · a full 3d...

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