Modeling flows and mixing in steelmaking ladles designed for single- and dual-plug bubbling operations
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I.
INTRODUCTION
F O R high-quality steelmaking, gas bubbling in ladles is used to obtain chemical and thermal homogenization, as well as to accelerate the absorption of harmful nonmetallic inclusions into an overlaying slag. The main point in gas stirring operations is to identify procedures and equipment needed for achieving minimum mixing times and maximum recoveries of alloy additions at optimum gas flow rates. In order to reasonably predict these phenomena, detailed information on flow patterns, fluid velocities, and turbulent properties is needed. These have been the subject of study via ongoing physical and mathematical models over the last decade. Szekely et al. m were the first to attempt modeling the hydrodynamic behavior of liquid metal in an argon-stirred ladle. Velocity and turbulence energy fields were predicted through the solution of the turbulent Navier-Stokes equations in conjunction with the k-W two-equation model of turbulence. However, the boundary conditions (e.g., velocity and shear stress), adopted for an "interface" between the bulk fluid and the plume, proved unrealistic. DebRoy and Majumdar [2] and Grevet et al.[3] recognized the role of buoyancy in the gas/liquid mixture and proposed that the gas/liquid mixtures could be represented by a pseudo one-phase fluid of variable density. Sahai and Guthrie ta,Sj went on to develop mathematical and algebraic models to describe the interaction of a plume with its surrounding liquid, enabling plume dimensions, voidage, and centerline velocities to be specified and the whole flow field analyzed. Their results matched pilotscale water model results. These mathematical models were developed for axisymmetric gas stirring. In such systems, flows can be described via the two-dimensional continuity and momentum equations, expressed in cylindrical polar coordinates. While many flows within ladles can be idealized by assuming axisymmetric conditions, a major feature of industrial operations is their three-dimensional char-
acter (e.g., off-centered gas bubbling, multiplug gas bubbling, off-centered alloy/tracer additions, the Ruhrstahl-Heraeus (RH) degassing process, etc.). However, few studies on three-dimensional turbulent flows in gas-stirred ladles have been reported to d a t e . 16'7'8] The concept of mixing time, Zm, has commonly been used to represent the state of agitation in chemical and metallurgical processing vessels. Since Nakanishi et al. t91 fLrst correlated mixing times to stimng energy input, many empirical relationships of the type % = ke-", have been reported, t9-141 assuming that mixing times are independent of the experimental conditions. However, in general, the values of k and n vary with respect to the experimental situations studied by investigators. The fact that there are various empirical values for them reveals that the measured mixing times could be dependent on experimental conditions, such as vessel geometry, tracer injection point, monitoring point, gas bubbling location, gas injection rate, and the existence of a slag lay
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