Mathematical model of heat flow and austenite-pearlite transformation in eutectoid carbon steel rods for wire
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THE prediction of phase transformations in metal heat-treatment processes, such as the controlled cooling of wire rod, is made difficult by the complex nature of the coupled heat transfer and phase-transformation kinetics. Heat transfer at the surface of the metal, for example, depends on local cooling conditions which change with temperature, fluid properties and relative fluid velocity. Likewise within the metal, the thermophysical properties vary with temperature; and heat due to the transformation is evolved. Also the transformation kinetics depend on cooling rate, steel composition and austenite grain size, and have been inadequately characterized, from the processing standpoint, in terms of isothermal or constant cooling conditions, which do not obtain in real processes. For these reasons the mathematical modelling of heat-treatment processes has not developed to the same extent as in other processes like continuous casting. The need to develop such models, however, is clear since then design of the processes can be based on first principles with less guesswork involved, and the solution to problems such as those arising from macrosegregation achieved rapidly and rationally. Thus in the present work, a mathematical model has been formulated which incorporates the important physical phenomena in the heat treatment of eutectoid carbon steel rods. In the present paper, the validity of the model is tested against limited experimental measurements reported in the literature; and the influence of variables, viz rod diameter, cooling conditions, finishing temperature and steel composition, on the temperature field, PRAKASH K. AGARWAL, formerly Graduate Student in the Department of Metallurgical Engineering, University of British Columbia, is now with IVACO, L'Orignal, Ontario. J. K. BRIMACOMBE is Stelco Professor of Process Metallurgy, Department of Metallurgical Engineering, University of British Columbia, Vancouver, B. C. V6T IW5, Canada. Manuscript submitted March 17, 1980.
and austenite-pearlite transformation is predicted. In subsequent papers a rigorous evaluation of the model, employing temperature and transformation data measured in our laboratory, will be presented together with examples of the use of the model in the analysis, improvement and design of controlled cooling processes such as Stelmor. 1. MATHEMATICAL MODEL The model has been formulated for the case of an infinitely long steel rod moving at high speed through a cooling system or alternatively, for a rod subjected to uniform cooling over its length. In both cases axial heat conduction can be ignored, in the former because it is negligible compared to the bulk flow of heat, and in the latter because axial temperature gradients are small. The following assumptions have also been made in the formulation: i) uniform initial temperature ii) radial symmetry iii) temperature independent of angular displacement iv) uniform circular cross-section Under these conditions, heat flow within the rod is governed by the following equation: -Or -
k
+ -r
Ofr
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