Turbulent Flow in Filling Ladles
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MATHEMATICAL MODEL Neglecting the fact that the jet may penetrate the bath at a slight angle, we may assume cylindrical symmetry, and consider the case of a vertical axisymmetric jet shown in Fig. 2. The problem may be stated mathematically by writing the equation of continuity and the equations of motion together with relevant boundary conditions and expressions for the production and dissipation of turbulence kinetic energy. For the latter purposes, use was made of the k - E model used by Launder et al. 3 Alternative two equation models include the early and now classical PrandtlKolmogorov model and the two equation models of Spalding. 4 In all cases, these models have proven more adequate in describing turbulent recirculatory flows than the simpler one equation models. Spalding's k- W model has been successfully applied to a number of metallurgical systems by Szekely et al and Evans et al and is well documented, in their papers,(e.g. Refs. 5 to 7). In general, these two equation models (which are somewhat empirical in nature) account for the transport of two characteristic magnitudes of turbulence, both of which are normally spatially dependent. The first 'property' is 'k' - the amount of turbulence kinetic energy per unit mass generated by the turbulent flow. The second is 'l' the length scale of turbulence, and characteristic of the way in which the kinetic energy 100
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MARTHA SALCUDEAN, is Assistant Professor, Department of Mechanical Engineering, University of Ottawa, and RODERICK I. L. GUTHRIE is Associate Professor, Department of Mining and Metallurgical Engineering, McGill University, Montreal, Canada. Manuscript submitted November 17, 1977. METALLURGICAL TRANSACTIONS B
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Fig. l-Recovery of vanadium and vanadium carbide as a function of addition timing during a B.O.F. tapping operation.
ISSN 0360-2141/78/1211-0673$00.75/0 © 1978 AMERICAN SOCIETY FOR METALS AND THE METALLURGICAL SOCIETY OF AIME
VOLUME 9B, DECEMBER 1978-673
Table I. Physical Parameters Used in Fullscale and Experimental Models of ladle Filling Operations
One Fourth
One Tenth (Cylindrical)
25
0.35
0.025
2.5
0.63
0.25
2.74 to 3.66
0.61 to 0.91
0.37 (diam)
0.12 to 0.20 0.05 to 0.1 240 to 480
0.029 0.0017 205
0.011 0.00031 108
Full Scale Volumetric capacity (rn") Ladle height (m) (as filled) Ladle dimensions (Width and length) Jet diameter (m) on impact (estimate) Flowrate (nr' .-1) Filling time (s)
Fig. 2-Schematic diagram of idealized axisymmetric turbulent flow system for simulating furnace tapping operations into teeming ladles.
of turbulence is dissipated. Small 'I' values typically reflect high rates of energy dissipation, while high length scales reflect the opposite. Since models based on a transport equation containing I explicitly have not proved successful, it is customary to express the dissipation of turbulence in terms of alternative turbulence quantities containing I implicitly. Thus, in the k - t: model presently used, I appears
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