Dynamic similarity considerations in gas-stirred ladle systems
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Communication Dynamic Similarity Considerations in Gas-Stirred Ladle Systems DIPAK MAZUMDAR In recent years, submerged gas injection into melts contained in ladies or similar vessels has been the subject of considerable interest. Since gas injection plays key roles in determining the efficiency of numerous transport processes (viz., dispersion, mixing, inclusion float out, etc.) carded out in these vessels, the fundamental and applied aspects of gas injection phenomena have consequently been the subject of numerous investigations. [1-8] In general, until relatively recently, industrial ladle stirring operations have been generally studied using aqueous analogues of different sizes. More recently, in conjunction with low-temperature physical models, mathematical models have also been applied. These have accordingly lead to improved understanding of the related process dynamics, providing better insight into the submerged gas injection operations. So far, although many theoretical and experimental investigations have been carded out on various aspects of argon-stirred ladles, the physical modeling principles, particularly the dynamic similarity criterion between model and full-scale systems, have not been adequately considered. Therefore, appropriate scale-up (or scale-down) equations for gas flow rates, etc., which are essentially derived through such similarity considerations, are lacking. However, considering laboratory/pilot-scale simulation of the industrial gas injection operations and subsequent scale-up of results, the need for appropriate scaling equations vis-a-vis modeling criteria become readily apparent. Since hydrodynamic studies on ladle flows are often not concerned with thermal and chemical similarity effects, the equivalence between a model ladle and a full-scale system can be adequately described via the geometric and dynamic similarities. Geometric similarity provides the necessary means for scaling the characteristic physical dimensions of the system, while dynamic similarity entails the correspondence among various forces (e.g., inertial, viscous, body forces, etc.) acting on the system. It is through consideration of these forces that the required criterion between the model and the full-scale systems can be established. In any flow system, the balance between various forces acting on a fluid element can be described via the NavierStokes equation. [91 For a multidimensional flow situation, under steady-state conditions, the force balance in tensorial form can be expressed as
OXj (flUjUi) --
0( % OXj/
Op + - -
OXj
OXj
tx
+ F,
[11
D I P A K M A Z U M D A R , Assistant Professor, is with the Department of Metallurgical Engineering, Indian Institute of Technology, Kanpur 208016, India. Manuscript submitted August 11, 1989. METALLURGICAL TRANSACTIONS B
1 = O(NR~, NFr)
[2]
NEu
Equation [2] states that the ratio of the pressure force to the kinetic energy contained in the fluid (e.g., NEu) in a flow system is a function of the inertial, viscous, and body (viz., the buoyancy, in the presen
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