ArcelorMittal produces light, medium and heavy beams by hot rolling of beam blanks issued from continuous casting (CC). The present study focuses on the ...
Proc. 8th ECCC, European Continuous Casting Conference, Graz (Austria), 23-26 June 2014, ISBN-978-3-200-03664-2, The Austrian Society for Metallurgy and Materials (2014) 1220-1229
MICHEL BELLET 1, CRISTIAN FELIPE PEREZ-BROKATE 1, PIERRE HUBSCH 2
3D FINITE ELEMENT THERMOMECHANICAL MODELLING OF THE PRIMARY COOLING FOR BEAM-BLANKS CONTINUOUS CASTING Abstract The paper presents a thermomechanical numerical simulation of solidification in the mould and during the first part of secondary cooling in the case of continuous casting of heavy beam blanks, such as the ones cast by ArcelorMittal in Differdange. The simulation has been achieved using THERCAST®, a commercial software developed by CEMEF (MINES ParisTech) and Transvalor. The simulation is run in 3D, using the global non steady state approach. The simulation has been applied to three different geometries of the mould in view of evaluating the impact in terms of air gap formation and stress affecting the solidifying shell.
Keywords Beam blanks, thermomechanical modelling, 3D finite element, air gap, stress, strain.
1. Introduction ArcelorMittal produces light, medium and heavy beams by hot rolling of beam blanks issued from continuous casting (CC). The present study focuses on the continuous casting line for heavy beam blanks in Differdange. The objective is to develop a thermo-mechanical model of the casting process, in order to better understand and control the mechanical state of the solid shell. Such a model appears as a key tool to increase productivity while maintaining high quality standards. The thermomechanical model is focused on the primary cooling zone, which was also found critical in several studies from the literature [1] [2] [3]. The context of beam blanks continuous casting is quite specific and differs significantly from the continuous casting of billets of square or round section. In particular, the mould is made of several components of complex geometry (taper, curvature, design of water circuit). The commercial package THERCAST® used in this study is based on a 3D finite element thermo-mechanical solver jointly developed by CEMEF and TRANSVALOR [4] [5] [6] [7]. The main features of the solver are summarized in Section 2, as well as its application to the context of beam blanks CC. Material data and boundary conditions are presented in Section 3. In Section 4, the results are presented and discussed. A preliminary study consisted in a calibration of heat transfer coefficients, so that the mean heat flux extracted along the different mould plates is found in agreement with the temperature increase of the water in the mould cooling system. Then, after checking that the GNS approach converges to a kind of steady-state regime, the simulation is applied to three different geometries of the mould: - The “nominal” geometry of the mould (as it is used at present, configuration MLD1); 1 2
CEMEF, MINES ParisTech, UMR CNRS 7635, Sophia Antipolis, France ArcelorMittal, Differdange, Luxemburg
-
A geometry featuring the nominal mould taper but with a modified cooling system characterized by cooling channels closer to the interface (MLD2); A third geometry which differs by the taper definition but retails the nominal cooling system (MLD3).
2. Thermo-mechanical model In the thermomechanical solver of THERCAST®, the behaviour of solidifying alloys is modelled using a hybrid constitutive model. In the liquid (respectively, mushy) state, the metal is considered as a Newtonian (respectively, non-Newtonian) fluid. In the solid state, the metal is assumed to obey an elastic-viscoplastic model with strain hardening. Accordingly, an Arbitrary Lagrangian Eulerian (ALE) formulation is used. Solid regions are treated in a Lagrangian way. The position of the nodes of the mesh is updated by time integration of the material velocity: they are embedded in the material. By contrast, the position of the nodes in the mushy and liquid regions is defined in order to preserve a good aspect ratio of the finite elements [4] [5]. In the present continuous casting context, the Global Non Steady-state (GNS) approach, initially proposed by Bellet and Heinrich [6], is used. It permits a global non steady state thermomechanical modelling of the cast product. The submerged entry nozzle is not modelled. Instead, the meniscus is defined as an injection surface: surface nodes are considered Eulerian and the metal input flow rate is supposed to be uniformly distributed along this surface. This is an acceptable assumption when the effect of fluid flow over the thermomechanical phenomena in the solid shell is supposed negligible. At the other end of the computational mesh, that is along the lower surface, the nominal casting speed is prescribed. As a consequence of this GNS approach, it is necessary to use a remeshing method, since the computational domain is continuously expanding during the simulation. Remeshing only affects the extreme upper zone of the domain, near the meniscus, and is performed using a local extraction/remeshing strategy in order to minimise remeshing computation time [6] [7]. Contact with mould surface and with supporting rolls is taken into account using a specific penalty method [6]. However, in the present work, support rolls and mould components are supposed non deformable. A great advantage of the GNS approach is to overcome the limitations of the classical "slice method", which consists in conveying through the whole casting line a thin slice or a part of the material having a certain length in the casting direction. In this slice approach, the only possible mechanical boundary conditions (plane stress, plane strain, or generalized plane strain) are not relevant and make impossible any analysis of bulging between support rolls. It is also impossible to take into account axial gradients (in the casting direction) such as those occurring at mould exit for instance. Application to beam blanks continuous casting Half of the configuration is simulated, taking advantage of a longitudinal symmetry plane throughout the machine. Starting from the CAD surface definition of each of the three components constituting the mould, the automatic meshing procedure of THERCAST® can be used to create a surface mesh and transform it into a volumetric mesh [8]. The three mould components are treated separately. As can be seen in Figure 1, a specific remeshing procedure is used in order to refine the mesh around the 17 cylindrical water cooling channels in each of the two half main parts of the mould. A similar refinement is made in the lateral narrow face in which a manifold of 22 channels of rectangular cross section are machined. From the CAD
definition of the mould, the CAD definition of the initial configuration of the cast product can be automatically generated. It has an arbitrary initial height, about one quarter of the mould height (in blue in Figure 1). The product has a minimum mesh size of 5 mm and the mould plates have a minimum mesh size of 10 mm. Calculations were run on 32 cores of a Linux cluster AMD Opteron with capability of 1500 cores, 3 Tb RAM memory. The calculation time for the configurations presented is around 150 h.
Domain Product (init.) Intrados plate Extrados plate Narrow plate
Mesh data Nodes 22734 120669 124503 47653
Elements 106900 556762 579553 214204
Figure 1: Meshes of the different components. The blue domain represents the cast product and is shown in its initial configuration.
3. Material data and boundary conditions Constitutive equations In the present study, up to the solidus temperature TS, an elastic-viscoplastic constitutive model is used. Parameters are taken from Kozlowski et al. [9] who identified behaviour models for plain carbon steels, for strain rates ranging from 10-6 to 10-3 s-1, and temperatures in the austenite range, between 950 and 1400°C. The law II of Kozlowski et al. can be reformulated as follows:
σ = K (T )ε n (T )ε& m (T )
(1)
where σ is the von Mises flow stress, ε the generalized plastic strain, ε& the generalized plastic strain rate, T the temperature, K the viscoplastic consistency, n the strain hardening coefficient and m the strain rate sensitivity coefficient. For liquid steel, over the liquidus temperature TL, a Newtonian behaviour is considered. Denoting µ l the dynamic viscosity of liquid steel, we have:
σ = K l (T )ε& = 3µ l ε&
(2)
Between solidus and liquidus temperatures, the model derives from a mixture rule applied to the flow stress. As the values of flow stress vary by several orders of magnitude, a logarithmic linear variation is used, which leads to:
[
σ ( g l , ε& ) = K (TS )ε& m (T
S
] [K (T )ε& ]
) 1− g l
gl
l
L
= K (TS )1− g l K l (TL ) gl ε& (1− g l ) m (TS ) + gl
(3)
It can be seen that in the solidification interval, the viscoplastic consistency is a logarithmic linear interpolation between K (TS ) and K l (TL ) , while the strain-rate sensitivity is a linear interpolation between m(TS ) and ml (TL ) = 1 .
Thermal boundary conditions At the mould/product interface, THERCAST® allows to take into account the presence of a flux (made of molten powder for lubrication and thermal insulation) and the formation of an air gap. However, in the present study, because the properties and the thickness of the flux remain unknown or without sufficient accuracy, the boundary conditions are simplified as follows, considering only an air gap effect due to shrinkage, with a bounded heat exchange coefficient when this air gap tends to zero (effective contact). The extracted heat flux is then expressed as follows: − k∇T ⋅ n = heq (T − Tmld )
with
[
heq = Min hcond _ gap + hrad _ gap ; h0
]
(4)
where the different contributions to the heat contact resistance are: hcond _ gap =
λair δ gap
1 1 2 hrad _ gap = σε eq (T 2 + Tmld )(T + Tmld ) with ε eq = + − 1 ε ε mld
−1
(5)
The coefficient h0 is intended to limit the heat exchange coefficient when the thickness of the gap tends to a zero value. It is supposed to be close to the ratio λ flux / δ flux . In the above expressions, δgap and δflux denote respectively the air gap and the mould flux thickness, λair and λflux the air and the mould flux heat conductivity, T and Tmld the product and the inner mould surface temperature, ε and εmld the emissivity of steel and copper, and σ the StefanBoltzmann constant. At the interface between the cooling water and the mould, a convection-type boundary condition is used: φ = hmld −w (Tmld − Tw )
(6)
while a similar condition is used at mould/air interface: φ = hmld −air (Tmld − Tair )
(7)
The secondary cooling system is composed by a series of zones with water sprays and support rolls. Since the main objective of this study is the analysis of the primary cooling zone, each secondary cooling zone i s simplified using an equivalent heat transfer coefficient. This coefficient is obtained by an appropriate surface weighting of the heat fluxes associated with sprays (heat exchange coefficient obtained by Wendelstorf's expression [10]), with rolls (for which the roll contact area is estimated through Hertz contact theory) and with convection-radiation with surrounding air. A global averaged local heat transfer coefficient is used, s i m i l a r l y t o Eq. (4a ) to define heat extraction b e l o w m o u l d e x i t : φ = heq _ sc (T − Tair )
(8)
Figure 2 summarizes the different thermal boundary conditions applied to the different interfaces of the model. As mentioned above, the contact between the cast product and the mould is represented with an air gap dependent heat transfer coefficient. A convection heat transfer coefficient governs the heat exchange between mould components and water cooling channels. Along the top surface of the product the nominal casting temperature is prescribed. Along the bottom surface, an adiabatic (zero flux) condition is imposed.
Eq. (6)
Timp = 1576 °C Null pressure
Eq. (5)
Eq. (4) Sliding contact
Eq. (7)
Eq. (5) Eq. (8) Adiabatic v = vcast Figure 2: Schematics of the thermal boundary conditions.
4. Results
Before discussing results provided by the GNS thermomechanical approach, it is important to check that this non steady state approach converges appropriately with time, in order that only converged results be considered for further discussion. This was checked in a preliminary study [11] which is not reported here. From the thermal point of view, a permanent regime of the mould is found after about 100 s, corresponding to a displacement of the lower face of the mesh of about 0.8 m, that is 200 mm below mould exit. Hence the thermal convergence is quite fast. This can be easily understood by the fact that the axial temperature gradients are negligible and that heat diffusion is rapid in the copper mould. The time to obtain mechanical convergence (that is stabilization of equivalent stress in the solidified shell near the mould exit) is comparable. However, contrary to the thermal convergence, the mechanical convergence is less clear: the stress level is found fluctuant (by about 15 to 20% around an average value) and this is due to the progressive contact of the product with the supporting rolls below the mould exit. Calibration by reference to the experimental mean heat flux A main reference for the heat exchange in the primary cooling zone is the average heat flux extracted from the cast product. The local heat flux at the product/mould interface is defined as the product of the temperature gradient by the thermal conductivity of the material. From the simulation results obtained with THERCAST®, it is possible to calculate the average of the extracted heat flux on each of the three interfaces between the cast product and the three mould components. In order to validate the model by comparison with measurements, the real heat flux is calculated by means of the following expression, φ = Qw ρ wc p ,w
∆Tw S
(9)
in which Qw is the water flow rate, ρw the density of water, cp,w the specific heat of water, ∆Tw the temperature increase in the water cooling circuit (temperature difference between outlet and inlet) and S the steel/mould interface area. The following values are obtained: φNF = 1.40 MW/m2 and φMP = 1.23 MW/m2 for the narrow face and the main lateral plates respectively.
The local heat flux is defined by Eq. (4) in which the value of h0 can be calibrated by comparison with the average heat flux estimations above, based on experimental data. The result of this calibration is shown in Figure 3 where it can be seen that a unique coefficient h0 = 1400 W/m2/K provides a good agreement with experimental measurements. The figure also shows that the local heat flux tends to decrease with the distance to the meniscus, except in the top region of the mould.
Figure 3: Calibration of heat transfer conditions. On the left: narrow face; on the right: main lateral plates.
Figure 4: Left: temperature distribution along intrados side. Right: distribution of air gap.
Temperature distribution Figure 4 shows the temperature distribution at the mould interface and through the thickness of the product in the longitudinal symmetry plane. The temperature is found continuously decreasing with the distance to the meniscus. However, some clear differences appear in the flange region where the temperature decrease is found much lower than in the web region. This is clearly connected with the distribution of air gap, which can be seen on the right part of the figure. Air gap thickness is quite significant in the flange region, which reduces heat exchange and cooling of this region of the product. Air gap effect can also be seen in the fillet region and near the mould exit in the web region.
Figure 5 shows the temperature profiles along the product surface, at three different distances to the meniscus, at intrados and extrados. The variations with the horizontal position (web, fillet, flange region) are found greater than the differences between intrados and extrados, although the latter cannot be neglected, especially at mid-height of the mould.
Figure 5: Horizontal temperature profiles at three different heights in the mould region. MLD2
MLD
MLD1
MLD2
Figure 6: On the left, temperature and air gap profiles along mould surface at intrados, 0.3 m below meniscus, for MLD1 and MLD2 geometries. On the right: temperature distribution in a transverse section 0.16 m below meniscus for the same two geometries.
The temperature distribution in the mould is shown in Figure 6, with a comparison between the two mould geometries: MLD1 and MLD2. It can be seen that for MLD2, the shorter distance between cooling tubes and the surface has a marked influence in the web region, where the temperature profile takes higher values and shows oscillations with a wavelength equal to the distance between cooling channels. The distance channel/surface is too low and heat diffusion cannot smooth any more the temperature profile: this can also be seen on the transverse temperature distribution in the right part of the figure. This effect is less evident in the fillet and the flange regions because of air gap formation. Note also that the
peak of air gap at the corner between fillet and flange is reflected by the reduction of the temperature in the mould. This is because of a reduced heat transfer. Regarding quantitative comparison with measurements, a comparison with the thickness of the solid shell was made, thanks to the exploitation of a solid shell extracted from the machine after a break-out incident near the mould exit. The comparison between calculated and measured thickness profiles is given in Figure 7. Apart from the two last pairs of experimental measures near the mould exit, the agreement is excellent. MLD
MLD2
MLD1
Figure 7: Calculated solid shell thickness vs distance to the meniscus for the three mould geometries. Comparison with experimental measurements done when using the nominal mould geometry. Note that the circled experimental points should not be considered as they are affected by a break-out incident near the mould exit.
Stress analysis The stress in the horizontal transverse direction with respect to the casting direction (here the component σyy) is particularly considered. Figure 8 shows its distribution along the surface of the cast product. Because the stress component σyy cannot be analysed as a tangential stress component in the fillet region, a change of reference frame is operated in order to plot the tangential component, denoted σy'y'. Figure 9 shows those profiles along the three horizontal red lines that can be seen in Figure 8. Positive stress with maximum values of about 5 MPa are found in the web and fillet region in the upper part of the mould. It can be noted that at mould exit, due to the reheating of the product surface, the σy'y' component takes negative values at the product surface, expressing a compressive state in surface, while a positive stress remains in subsurface near solidification front. This is a classical stress state in the shell of CC products. The configuration MLD2 leads to systematically higher stresses than MLD1 in the upper part of the mould and at mid-height. The configuration MLD3, with a modified mould taper, seems efficient in reducing the level of tensile stresses in this upper half part of the mould.
MLD1
MLD2
MLD3
Figure 8: Stress component σyy along the surface of the product (y: horizontal transverse direction wrt casting direction). MLD3
MLD2
MLD1
Figure 9: Horizontal tangential stress component σy'y' along the surface of the product at three height levels in the mould region.
5. Conclusion
This study has shown the relevance of 3D thermal mechanical finite element modeling, using the GNS approach of THERCAST®, to investigate different features during beam blanks continuous casting: - The formation of air gap is different along the intrados and extrados side of the product and is of course strongly linked to taper definition. - Air gap formation has a significant impact on surface temperatures at the mould/product interface, which also strongly depends on the specific form of the lateral plates. - The three geometries tested present different stress levels and distribution in the solidifying shell.
References
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