On an Anomaly in the Modeling of Electromagnetic Stirring in Continuous Casting
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I.
INTRODUCTION
ELECTROMAGNETIC stirring (EMS) has been used in the continuous casting of steel[1] since as early as the 1970s as a way to control solidification structures, thereby increasing yield and productivity. In tandem, mathematical modeling has played an important role in the implementation of EMS, as regards providing understanding of exactly what effect stirring has. A cornerstone of the modeling literature in this area is a sequence of papers by Schwerdtfeger and co-workers[2–7] which explore, both experimentally and theoretically, the effect of stirring in the round billet, rectangular bloom, and slab geometries that are characteristic for the continuous casting of steel. The models in question consist of the Navier–Stokes equations for the velocity field of the molten metal and Maxwell’s equations for the induced magnetic flux density; in principle, these are two-way coupled, since the alternating magnetic field gives rise to a Lorentz force which drives the velocity field, which can in turn affect the magnetic field. Moreover, the frequency of the magnetic field is typically great enough to allow the use of the time average of the Lorentz force as input to the Navier– Stokes equations. In calculating the induced magnetic field, an assumption is necessary as regards the applied oscillating field surrounding the domain of interest, typically the steel strand. To obtain adequate data for this, it may in practice mean using a Hall probe magnetometer to make measurements of the magnetic field at a point or points
M. VYNNYCKY is with the Department of Materials Science and Engineering, KTH Royal Institute of Technology, Brinellva¨gen 23, 100 44 Stockholm, Sweden. Contact e-mail: [email protected] Manuscript submitted April 23, 2017.
METALLURGICAL AND MATERIALS TRANSACTIONS B
in the space between the outer surface of the steel strand and the periodic winding of the inductor on an iron comb,[2,3,7] or a Gauss meter[8,9]; indeed, measurements acquired in the former way were used as the basis for prescribing the normal component of the magnetic flux density at the surface of the strand. However, some time later, and in a mathematically related problem, McKee et al.[10] prescribed the tangential component in their model for particle tracking within a turbulent cylindrical electromagnetically driven steel melt. Consequently, there appears to be some uncertainty as to what should be the correct boundary condition in this situation: indeed, McKee et al.[10] followed Moffatt[11] in initially assuming that both the normal and tangential components of the magnetic flux density are required as boundary conditions, only to ultimately just use the latter. Moreover, the fact that the expressions for the components of the Lorentz force for round billets[2,7] and for rectangular strands[3] have been cited and used on numerous occasions since, even up to the present day,[9,12–19] suggests that a resolution of the issue is timely. Here, we focus on the analytically simpler case of the round billet. In particular, we demonstrate that presc
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