Numerical simulation of deformation-induced segregation in continuous casting of steel
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I. INTRODUCTION
MACROSEGREGRATION, i.e., an inhomogeneity of solute concentration at the scale of the whole product, is a common defect encountered in most solidification processes. It is always due to a relative velocity of the liquid with respect to the solid phase, and, as such, it can be induced by thermosolutal convection,[1,2] forced convection,[3] solidification shrinkage,[4,5,6] transport and sedimentation of grains,[7] or deformation of the solid skeleton.[8,9] In continuous casting of steel, bulging of the cast slab between two successive rolls has been shown to play a major role in the centerline segregation, which occurs at a fairly late stage of solidification.[8,9] This phenomenon, which is schematically illustrated in Figure 1, can be explained as follows. Assuming that the solid and liquid have the same specific mass (i.e., no solidification shrinkage), let us consider the stationary situation between two successive rolls of a continuous casting machine. The casting direction points toward the right, and the gravity vector direction is arbitrary. The ferrostatic pressure associated with the deep liquid sump at this position is such that the solid crust that has formed has a tendency to creep between the rolls. The resulting bulging of the slab (enhanced in Figure 1 for the purpose of illustration) makes solid material move out and then in of a rectangular domain outlined by the inner roll position. The velocity at any point within the solid crust is simply the sum of the T. KAJITANI, formerly Research Fellow, Ecole Polytechnique Fe´de´rale de Lausanne, is Senior Researcher with the Kimitsu R&D Laboratory, Nippon Steel Corporation, Kimitsu, Kimitsu-city Chiba, Japan. J.-M. DREZET, Senior Researcher, is with Calcom SA, Lausanne, Switzerland, and Ecole Polytechnique Fe´de´rale de Lausanne. M. RAPPAZ, Professor, is with the Laboratoire de Me´tallurgie Physique, De´partement des Mate´riaux, Ecole Polytechnique Fe´de´rale de Lausanne, CH-1015 Lausanne, Switzerland. Manuscript submitted February 2, 2000. METALLURGICAL AND MATERIALS TRANSACTIONS A
horizontal casting speed, vc ⫽ vc ex , and the vertical bulging speed, vb ⫽ vc tan ␣ ey , where ␣ is the angle that the bulged surface makes locally with the horizontal (Figure 1(a2)). For the sake of clarity, let us now assume that the temperature field is such that the position of the liquidus at the beginning of this small section (TL,in) is at the center of the slab. If there was no heat extraction at the surface of the slab, the isotherms would follow the bulging profile (Figure 1(a1)). Accordingly, the position of the liquidus at the exit of the domain (TL,out) would also be at the center of the slab. The velocity field being always tangent to the liquidus surface (Figure 1(a2)),* the mass balance integrated over *From a mass-balance point of view, the velocity of the liquid within this pocket could be set equal to zero everywhere, except at the boundary where it is tangent. This will ensure that mass is conserved for any closed contour of this domain. How
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