Modeling of metal-slag equilibrium processes using neural nets
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INTRODUCTION
A number of fundamental models exist for describing metal/slag equilibrium processes in pyrometallurgy, for example, (a) the regular-associated-solution model, tlj (b) Temkin's rule I21 used to estimate the activities in fusedsalt mixtures, (c) Schuhmann's approach 131to determine the activity coefficients, for example, in ternary systems by the application of the Gibbs-Duhem equation, (d) Masson's polymer theory I41 to determine the activities of basic oxide components in liquid silicate systems, and (e) the model of Flood and Grjotheim I51 for predicting the distribution of sulfur and partitioning of phosphorous between metal and slag. Turkdogan 161comments as follows on the slag models: "Although these theoretical models are helpful in improving our understanding of the structure of polymers, their application to slag-metal reaction equilibria is not particularly rewarding when compared with much simpler methods of representing the slag-metal equilibria presented here in generally accepted terms." The methods proposed by Turkdogan for defining equilibria are based on a hybrid of thermodynamic equilibria and empirical regression analysis. A set of linear regression equations is usually applied to model the effect of basicity, for example, which immediately excludes nonlinear effects and limits the expressive power of these equations. Numerous theoretical approaches exist for the modeling of equilibrium and other processes in pyrometallurgy. The equations proposed for the activities in liquid solutions as a function of variations in composition have been either empirical or semiempirical in nature. 17j In most cases, the parameters of these models have to be estimated by least-squares methods, rendering M.A. REUTER (Author for Correspondence), Research Metallurgist, is with the Institute for Non-Ferrous Process of Metallurgy, Aachen University of Technology, 5100 Aachen, Germany. T.J. VAN DER WALT, Doctoral Student, and J.S.J. VAN DEVENTER, Professor, are with the Department of Metallurgical Engineering, University of Stellenbosch, Stellenbosch 7600, Republic of South Africa. Manuscript submitted December 30, 1991. METALLURGICAL TRANSACTIONS B
the approaches semiempirical and, to a certain extent, blackbox. The objective of this article is to present an approach to modeling equilibrium processes in metallurgy which is even simpler than that presented by Turkdogan, for example. This proposed approach is based on neural nets (NNs) and permits a generalized nonlinear modeling of processes without predefining regression equations. Examples over a wide range of application will be considered in order to demonstrate the generality of NN modeling, namely, (a) the empirical modeling of activities in binary metal solutions, (b) the empirical modeling of activities in slags as a function of slag composition, temperature, and atmosphere, (c) the empirical modeling of activity coefficients in slags as a function of slag and metal phase compositions, (d) distributions between slags and metal phases as a func
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