Fluid dynamics in bubble stirred ladles: Part II. Mathematical modeling
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I.
INTRODUCTION
free surface
IN the past decade
there has been considerable interest in the gas-stirring of metallurgical ladles, thanks to its low cost. This technique involves the introduction of gas into the system through one or a number of nozzles situated at the base of the ladle. The resulting column of bubbles rises due to buoyancy and leaves the reactor through the free surface at the top (see Figure 1). The momentum interaction between the gas and the melt induces a toroidal recirculation zone which facilitates metallurgical processes like degassing and desulfurization. Usually it is necessary to obtain a large contact area between the phases and to avoid dead water regions. Despite their popularity, however, the design of gasstirred ladles is currently based on trial and error methods and would be greatly enhanced by a better understanding of the underlying fluid dynamics. In the past a lot of work has been done to overcome this lack of understanding. Szekely et al. I~l were among the first to model bubble driven flows in ladles. They used a k-e turbulence model t61 and assumed that the bubbles were contained in a cylindrical region of given diameter. The boundary conditions at the bubble column-liquid interface were taken from measurements. Their work gave a qualitative, but not quantitative, agreement with measurements. A better handling of the bubbly phase was suggested by DebRoy et al. [21 They assumed the bubbles to be dispersed within a column with a given diameter. They related the volume fraction of gas to the gas flowrate through the nozzle. Later Sahal9 and Guth rie t3J and He Qinglin et al. I41 developed more elaborate models based on the k - e model. The former authors treated the two phase region with a phenomenological analysis. The latter authors employed measured void fraction profiles as an input to their calculations and obtained satisfactory quantitative agreement with the measurements. Recently Cross and Markatos ISj have attacked the problem of gas injection by solving transport equations for both liquid and gas phase in a Eulerian reference frame using a commercial computer code. I191This was, perhaps, a first S.T. JOHANSEN, Research Engineer, is with S1NTEF, Division of Metallurgy, N-7034 Trondheim, Norway. E BOYSAN, formerly with the Department of Chemical Engineering and Fuel Technology, Sheffield, United Kingdom, is with Flow Simulation Limited, 146 West Street, Sheffield SI 4ES, United Kingdom. Manuscript submitted July 30, 1986. METALLURGICAL TRANSACTIONS B
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plume
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porous plug
Fig. 1 - - A typical ladle with a porous plug at the center axis. The coordinate vectors (r, z) are shown.
attempt at a fundamentally based model, and therefore their predictions failed to reproduce the dispersion of the bubble plume. To the best of our knowledge a general model which can handle bubble driven flows accurately is conspicuously lacking. Previous models which are based on the concept of a fixed-radius column containing bubbles
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