Mathematical modeling of mixing phenomena in a gas stirred liquid bath
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I.
INTRODUCTION
M I X I N G in a gas stirred liquid bath is a result of recirculatory flow caused by energy transfer from rising gas bubbles. Although experimentally measured mixing time is a useful parameter to characterize mixing in a bath, it alone is not sufficient to provide adequate understanding of the mixing process. The knowledge of parameters like liquid velocity and gas hold-up in the plume, recirculatory flow rate of liquid and liquid velocity outside the plume zone is desirable. Since the complex nature of the two phase plume and the existing turbulence in the bath often make direct measurements of these parameters difficult, prediction of these parameters using an appropriate mathematical model is of great practical significance. Mathematical models available in the literature to quantify mixing phenomena in gas stirred liquid baths can be broadly classified into three categories: (i) models based on momentum balance, (ii) models based on energy balance, and (iii) empirical models. Formulation of models in the first category basically involves setting up equations of motion and continuity along with appropriate boundary conditions, the solution of which gives velocity profiles. Integrating these profiles over the entire region outside the plume one can, in pri.'nciple at least, calculate average recirculatory flow rate (VL). The manner in which the effective turbulent viscosity and the boundary conditions are characterized leads to variations in models in this category. In spite of the fact that these models are conceptually sound, model predictions are not always satisfactory. This is often attributed to various assumptions which are frequently G.G. KRISHNA MURTHY, formerly a Graduate Student in the Department of Metallurgical Engineering at the Indian Institute of Technology, Kanpur, is Postdoctoral Fellow, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139. A. GHOSH and S. P. MEHROTRA, Professors, are with the Department of Metallurgical Engineering, Indian Institute of Technology, Kanpur 208016, India. Manuscript submitted August 19, 1987. METALLURGICAL TRANSACTIONS B
made to simplify the generalized model equations for easier solution, and characterization of a large number of empirical parameters contained in the model equations. Mathematical models based on the energy balance approach involve equating the steady state energy input rate to the .bath (E~) to the rate of energy losses due to dissipation (ED). Depending on how E~ and E o are characterized and quantified, variations in models result. One of the main drawbacks of almost all models in this category is that the gas kinetic energy contribution to input energy rate is assumed to be negligible. This assumption is perhaps based on experimental investigations by Lehrer, t~l who estimated that only 6 pct of the total gas kinetic energy contributed toward total energy input. However, this does not seem to be universally correct. In a later study, employing a water bath and an orifice
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