Effective viscosity models for gas stirred ladles
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Y. SAHAI and R. I. L. GUTHRIE I.
INTRODUCTION
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In recent years, a variety of recirculatory flows encountered in metallurgical processing operations have been modeled mathematically with reasonable degrees of certainty and success (e .g. ,L2). So far, the general approach adopted by authors has been to predict one phase flow fields through solutions of the partial differential equations of continuity, motion, and turbulence over the flow domain of interest, using appropriate sets of boundary conditions. As many of these flows are inertial force dominated, effective viscosity often plays only a secondary role in affecting the flow patterns generated. Consequently, computational expenses and difficulties can be circumvented somewhat if the two equation turbulence models typically employed (i .e., k-W 3 or k-e 4 models) can be replaced with simpler one equation models. These bulk effective viscosity models, being time and space independent, can be particularly useful for incorporation into three dimensional or transient flow programs, provided they can be validated. Recently the present authors were required to develop a generalized mathematical model for describing recirculatory flows and related hydrodynamic phenomena taking place during the central injection of gas into the bottom of a 150 tonne cylindrical ladle of steel (see Figure 1). The purpose of the present work is to present an effective viscosity model for gas stirred liquid metal systems developed during the course of that research.
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Fig. 1 - Schematic of ladle gas injection system, illustrating hydrodynamic phenomena taking place, and the grid system used for their mathematical representation.
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FUEL
Fig. 2--Combustion chamber depicting typical fuel and air flows considered by Pun and Spalding (Reference 5) in developing an effective viscosity model.
viscosity model to carry out these computations. They proposed that
PREVIOUS WORK
In recent computational schemes for predicting flow in gas stirred liquid metal systems, a bulk effective viscosity formula presented by Pun and Spalding5 has been proposed and adopted.6'7 In reviewing previous work, it is useful to consider its formulation and applicability so as to focus attention on the need for a more appropriate formula* for gas stirred systems. Thus, in 1967 Pun and Spalding considered the case of an axisymmetrical combustion chamber in which fuel and air enter at one end and combustion product exit at the other (see Figure 2). While the main thrust of their paper aimed at demonstrating their very significant computational achievement in solving the governing differential equations for swirling chemically reacting flows in a duct, they needed to introduce a simple but plausible effective
P.e = K 0 2/3 L -1/3 p2/3 (rh U02)1/3.
Eq. [1] can be rewritten in dimensionless form as /~" = K
Y. SAHAI, Senior Research Associate, and R. I. L. GUTHRIE, Professor, are both with the Department of Mining and Metallurgical Engineering at
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