Hydrodynamics of gas stirred melts: Part II. Axisymmetric flows
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INTRODUCTION
IN Part I, the hydrodynamics of gas stirred melts were considered and a simple macroscopic model for the representation of plumes and attendant recirculatory flow fields developed. In the present paper, it will be shown that the approach produces results which (1) are consistent with an advanced differential model of gas stirred systems, (2) are consistent with experimental flow fields, plume profiles, and so forth presently available in model and pilot scale ladle systems, and (3) provide the necessary framework to predict behavior in full scale injection systems. II.
PREVIOUS
WORK
As previously noted, early experimental studies by hydraulic engineers were largely confined to the case of water entrainment into 'plume curtains' rising through large bodies of water (section I). Little work seems to have been conducted outside the field of process metallurgy regarding stirring produced by gas injection into vessels of finite diameter or width. Szekely, Wang, and Kiser ~were the first to attempt hydrodynamic modeling using the example of an argon stirred ladle. Through solution of the Navier-Stokes equation (using the k-W two equation model of turbulence) z, they predicted flow patterns and turbulence energy fields in a water model of the system. In the analysis, they considered the upward movement of gas to be equivalent to the vertical motion of a centrally placed solid core of material. At the interface between this core and the liquid, the hypothesis required that radial components of velocity were zero. Consequently, only axial components of velocity were taken to be responsible for generating recirculation within the liquid. Boundary velocity values were obtained by hot wire anemometry measurements in the water model. Predicted velocity and turbulence energy fields were said to be in qualitative agreement with experiment. Szekely, Dilawari, and Metz 3 then studied the recirculatory flow pattern generated by a continuously moving vertical cylindrical belt running at a velocity of 5 m per second. In this case the authors found that predicted velocity and turbulence fields were in more satisfactory agreement with those measured experimentally. In a subsequent paper, Szekely, Lehner, and Chang4 employed the same vertical shear stress conditions to predict flow fields and eddy diffusivity values in seven ton and 60
ton argon stirred ladles. Boundary velocity values adjacent to the cylindrical core of gas were estimated from the'plume curtain' rise velocity relationship given by Bulson. 5 Mixing times based on predicted eddy diffusivities were calculated and claimed to be in good agreement with mixing times measured experimentally. More recent mathematical models include a paper by Deb Roy, Majumdar, and Spalding 6 and an equivalent by Szekely, E1-Kaddah, and Grevet. 7 In these, both sets of authors recognized the importance of buoyancy and proposed computational schemes wherein the gas-liquid mixture contained within the jet region was represented by fluid of variable density. In both schemes, the i
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