A Study on the mathematical modeling of turbulent recirculating flows in gas-stirred ladles
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I.
INTRODUCTION
IN recent years, there has been a considerable effort to develop a mathematical model of turbulent recirculating flows driven by an ascending gas bubble stream in steelmaking ladles. ~ Most of this effort has been focused on the turbulent viscosity models and the description of the two-phase region where gas bubbles and a liquid phase coexist. While using the k-e model for the turbulent viscosity is widely accepted, a quantitative description of the two-phase region has not been available. Szekely et al/3] assumed this region to be a solid core, having only an upward velocity component, and used the measured velocity as a boundary condition at the interface between the core and the liquid region. DebRoy et al.t4] considered the region to be liquid with a variable density determined by the gas volume fraction. The gas volume fraction, being the source of the buoyancy, was calculated as the ratio of gas flow rate to the overall gasliquid volume flow rate in the plume region, with the assumption of no slip between the liquid and the gas phase. Later, Grevet et al. ~6~applied the so-called drift flux model for the volume fraction, which allows partial slip between the two phases. More recently, Sahai and Guthrie ~sl modeled the region macroscopically. They calculated the volume fraction and the centerline velocity from their model and used these as the sources of the buoyancy. Although all these efforts have contributed a great deal to the prediction of the mean velocity field of the flow, the description of the turbulent characteristics of the system still remains unsatisfactory. However, knowledge of the turbulence (turbulent kinetic energy, turbulent viscosity, and energy dissipation) is thought to be one of the crucial aspects in determining the rates of various metallurgical processes which may occur in ladles. ~ The work described in this paper is aimed at improving the prediction of turbulence parameters in gas-stirred ladles using the experimental correlation for the gas vol-
ume fraction recently developed by Castillejos and Brimacombe. [11] The predictions were compared with the experimental results of Grevet tl~ obtained for the conditions given in Table I. II.
METALLURGICAL TRANSACTIONS B
MODEL
A. Governing Equations For the turbulent recirculating flow, the system was assumed to have axial symmetry. The turbulent viscosity was modeled using the k-e model, which introduces two additional transport equations for the turbulent kinetic energy, k, and its dissipation rate, e (a more detailed derivation of the equations is available in Reference 12). Thus, the governing equations used were of the following form in cylindrical coordinates: Equation of continuity: 10 0 r Or (prUr) + Oz (pUz) = 0
[1]
Equation of motion in axial direction: 0 0 - - (prUrU~) + - - (pU2:) Or Oz
OP l 0 [ OUz] O [ OUz] Oz + -r -or- Lrt*eff-~r J + 2 -Oz - /~eff Oz J
-
-~-
,o[
--
--
/~eff
r Or
r
1
Or J
+ pga
[2] Equation of motion in radial direction: 13 O - - - (prU 2) + - - (pUrU:) r Or 3z -
J.S. WOO
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