Finite difference heat-transfer modeling for continuous casting
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I.
INTRODUCTION
IN
this paper, we develop and compare a number of solution strategies for heat-flow models of continuous casting processes that in later studies are incorporated into optimization algorithms. These strategies are based on two-dimensional (2-D) slice models with nonlinear thermodynamic and transport properties. As a result of the comparison, a number of modifications were applied to enhance the accuracy of the models as well as the efficiency of the solution. Continuous casting is essentially a heat-extraction process. m Superheat and the heat of fusion are removed from liquid steel, and sensible heat is removed from solid steel. Heat transfer occurs by several mechanisms. In the liquid region of the strand, heat is transferred by conduction and convection, sometimes augmented by electromagnetic stirring. In the solid region, heat transfer is by conduction. Heat is removed from the strand in a variety of ways. Heat flow to the mold is governed by the nature of the gap that forms between the strand and the mold as a solid shell forms and shrinks. This shrinkage is partially offset by bulging of the shell caused by the hydrostatic pressure of the liquid inside, t21 After the mold, the strand is further cooled in most casters by water sprays or air/mist cooling. The purpose of the water spray cooling is to provide continued, controlled cooling of the strand and of the support rolls. After this, the strand is cooled by radiation and natural convection. In the continuous casting process, it is important to remove the heat in the strand in a controlled and predictable manner. Thus, mathematical modeling of the process plays an important role in producing quality strands.
B. LALLY and L. BIEGLER, Professor, are with the Department of Metallurgical Engineering and Materials Science and the Department of Chemical Engineering, respectively, Carnegie Mellon University, Pittsburgh, PA 15213. H. HENEIN, formerly with Carnegie Mellon University, is Professor with the Department of Mining, Metallurgical, and Petroleum Engineering, University of Alberta, Edmonton, AB T6G 2G6, Canada. Manuscript submitted February 22, 1989. METALLURGICAL TRANSACTIONS B
In order to develop accurate and efficient modeling tools for this process, a comparison of several numerical models is presented in this work. The next section describes the formulation of the mathematical model for heat flow in continuous casting. Following this, numerical methods for solving heat-flow equations as well as their refinements are discussed in Section III. Section IV presents the results of this model comparison and a comparison of computational efficiency. Finally, a summary and conclusions for the application of these models for optimization purposes are discussed in Section V. As a result of this model development, we found that with extensions to the alternating direction implicit (ADD technique, namely, application of the Kirchhoff transformation and iteration of temperature-dependent properties over time-steps, the heat flow in a 2-D slice of
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