Some fundamental aspects of mixing in metallurgical reaction systems
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INTRODUCTION
IT is an established fact that the overall kinetics of many metals processing operations is limited by the rate at which reactants may be supplied to the reaction zone; or in other words, mixing in the system is the rate-controlling factor. Typical examples include pneumatic steelmaking, desulfurizafion, and a whole range of ladle metallurgical operations, such as vacuum degassing, deoxidation, temperature homogenization, and the like. It has been realized that the rate of mixing in these systems, which in turn governs the overall kinetics, will depend on the rate of energy input or energy dissipation. The general nature of this behavior has been illustrated by a correlation first proposed by Nakanishi e t al. 1 and then generalized by Sano and Mori 2 which enabled one to represent a broad range of ladle metallurgy systems by plotting the 'mixing time' against the rate of energy density input into the melt. In this context mixing time is generally defined as the time required to attain 95 pct homogenization. These two plots, which are reproduced as Figure 1 and Figure 2, indicate that on a log-log plot the mixing time shows a straight line dependence on the energy density i n p u t - - that is, power input per unit mass, with a slope in the range of - 0 . 3 3 to - 0 . 4 . It is perhaps remarkable that one may represent the behavior of such a broad range of systems on a single plot. At the same time this is also extremely useful because it enables one to provide a quick assessment of a projected performance without doing extensive calculations or experimental work. Up to the present, the material presented in Figures 1 and 2 had a largely empirical basis without a real justification of these findings; Sano and Mori did present an analysis, a critical review of which will be given subsequently. The purpose of the present paper is to develop a fundamental justification for these plots and thus provide additional support for their validity in the assessment of a broad range of liquid metals processing operations. II.
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Fig. 1 - - Relationship of vessel mixing time with stirring power density in the melt.
in Figure 3, mixing takes place due to two mechanisms, namely bulk transport as caused by the macroscopic circulation in the melt and diffusion (both eddy and molecular). It should be stressed here that if the flow were laminar, a given tracer would follow the path of the streamlines and very little mixing would take place. For this reason we shall postulate that mixing rate will be controlled primarily by eddy diffusion and hence will seek to relate the mixing process to the turbulence characteristics of the system. This assumption is thought to be reasonable in light of the large Schmidt numbers encountered in melts of interest. To sustain
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