Comparison of numerical modeling techniques for complex, two-dimensional, transient heat-conduction problems
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I.
INTRODUCTION
M A N Y numerical methods have been employed to solve for the temperature distribution in transient heat-conduction problems with or without change of phase. Traditionally, finite-difference techniques have been applied with considerable success; but as interest has grown in complex shapes and combined heat flow/stress problems, an example of which is the solidification of steel ingots with corrugations, attention has turned to finite-element methods developed originally for stress analysis of structures. As a result, the number of numerical methods and versions of each, available for use in tackling a given heat-flow problem, has increased rapidly; however, the comparative advantages of the different techniques with respect to accuracy, stability, and cost remain unclear. Thus, in the present paper, this question has been examined by comparing the temperature predictions of several different formulations of the standard finite-element method, ~the matrix method of Ohnaka, 2 and the alternating-direction, implicit finite-difference method 3 against analytical solutions for two problems. Because this study is the first part of a larger project on heat flow and stress generation in steel ingots, the two problems have been chosen to approximate different stages in ingot processing: reheating in a soaking pit and solidification in the mold. These problems also test the ability of the numerical methods to handle temperature-dependent boundary conditions and the latent heat of solidification, respectively. Two-dimensional heat flow in the transverse mid-plane of the ingot has been considered. II.
(0.762 m • 1.524 m) "convectively" heated in a soaking pit, as shown in Figure 1. Heat was assumed to transfer uniformly to all four sides of the ingot, giving rise to a temperature distribution with two-fold symmetry; thus only one-quarter of the ingot section need be considered. The initial temperature of the ingot at charging, To, convective heat transfer coefficient, h, surrounding pit temperature, T~, and thermophysical properties, k, p, and Cp, were all assumed to be constant. Heat conduction within the ingot then is described mathematically by the well-known partial differential equation* *All symbols are defined in a Nomenclature section at the end of this paper.
y m)
~762
q=O
///////~///////(~,, Steel ingot
Soaking
pit
atmosphere
TcO:llO0*C
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