A multi-phase model for plumes in powder injection refining processes
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INTRODUCTION
IN recent years the submerged injection of powders has gained importance as a means to carry out smelting and refining reactions because of the intimate contact between the fine particles and the metal. Familiar examples include the Mitsubishi Copper Smelting Process, the desulfurization of blast furnace iron with calcium carbide or mixtures of lime and magnesium, t h e desiliconization and dephosphorization of iron with iron oxide mixtures, and the calcium treatment of steel for desulfurization and sulfide shape control. In spite of the importance of these types of reaction, the details of the particle trajectories, velocities, heating rates, and reaction rates are unknown. Work by the present authors has elucidated the states of flow in the conveying lines J and the implications for the regimes of flow in the descending or jet portion of the injection process/ The present work is concerned with the ascending or plume region of the injection process. There have been several plume models proposed for ladles agitated by gas alone. Generally the aim of these models has been the prediction of the recirculating flow patterns or mixing time, and hence the treatment of the two-phase region has been simplified. They have all used mixture models which use volume-averaged velocities and densities, and thereby the two-phase region has been treated as a single phase. The constitutive relation between void fraction and velocity has either come from Wallis' Drift-Flux Model 3 for gas-water flow in a rigid vertical pipe, 4'5 an analysis of the slip between bubbles and liquid in a plume 6 or by solving the single phase flow equations allowing for diffusion of liquid. 7 The disadvantage of such mixture models is that the discreteness of each phase is ignored, and average velocity, void fraction, and temperature are assigned to a particular point. Therefore the transport of momentum, heat, and mass between phases cannot be addressed by such models. It is only by formulating equations of continuity, momentum, L. R. FARIAS, formerly with McMaster University, is Research Metallurgist with HyL Research and Development Center, Monterrey, N.L., Mexico. G.A. IRONS is Associate Professor, Department of Materials Science and Engineering, McMaster University, Hamilton, ON L8S 4L7, Canada. Manuscript submitted November 29, 1984.
METALLURGICALTRANSACTIONS B
and energy for each phase which include the interphase transport terms (for example, in the form of drag, heat transfer, and mass transfer coefficients) that one can calculate the changes in void fractions, velocities, temperatures, and compositions throughout the plume. In the present work, this approach is adopted for the calculation of phase velocity and volume fraction from the continuity and momentum equations. Among other aspects, this allows the investigation of the effects of changes in injection rates, liquid head, and bubble size. In addition the effects of powder addition can be assessed by simply including extra equations for this phase. The present model is
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