Solute distribution during steady-state cellular growth
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I.
INTRODUCTION
M I C R O S E G R E G A T I O N in cast and wrought alloys has a profound influence on their mechanical and physical properties. Controlled unidirectional solidification experiments coupled with theoretical models for predicting the solute distribution under specific growth conditions can provide a better insight of the microsegregation characteristics in alloys. It is well known that parallel arrays of cells can form in controlled unidirectional solidification and that increasing pulling speed beyond a certain limit tends to generate parallel dendrites. Theoretical models tl-6] for the growth of dendrites are not readily applicable to cellular solidification for predicting solute distribution, because overlapping of the diffusion fields of adjacent cells has to be taken into account, whereas such impingement may not be significant tT] in the case of dendrites. In an attempt to visualize the initial breakdown of the planar solidification front, Ungar and Brown tS] and McFadden and Corriell tg] have computed the twodimensional steady-state cell shapes by means of finite element analysis. The approximate analysis by Kirkaldy I1~ based on the minimum entropy production rate criterion overlooks the lateral solute segregation near the cell tips. A lengthy numerical treatment [m of this problem on the same basis with a presumed cell shape has, however, yielded an optimum length and spacing of the cells. A more rigorous finite difference model, developed for steady-state cellular growth by McCartney and Hunt, [12j generates a stable cell shape prior to their solute distribution calculation. Predictions of the last model cannot be tested against the experimental data, unless a suitable growth criterion is presumed. The present work deals with the development of a solution for the solute distribution ahead of the steady-state cellular growth front in unidirectional solidification by using Galerkin's variational principles. [13,14,15]The influence of overlapping diffusion fields of the neighboring P.S. BASAK, formerly Research Scholar, Department of Metallurgical Engineering, Indian Institute of Technology-Kharagpur, is Assistant Manager with Shalimar Wire Industries Ltd., Uttarpara, West Bengal, India. A.S. GUPTA, Professor, Department of Mathematics, and S.K. PABI, Professor, Department of Metallurgical Engineering, are with the Indian Institute of Technology-Kharagpur, 721302 West Bengal, India. Manuscript submitted April 2, 1990. METALLURGICAL TRANSACTIONS A
cells and mass conservation at the entire interface are taken into account. The predictions of this model are compared with results of unidirectional solidification experiments on a Sn-0.9 wt pct Pb alloy. II.
ANALYSIS
A. Formulation of the Problem Imagine that the liquidus of a binary alloy is a straight line with slope m, and the equilibrium partition coefficient k is a constant. Uniformly spaced cells with smooth profiles are assumed to grow with a constant axial velocity, I7, into a liquid of bulk composition C=. Due to the interaction of the
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