Convection in the two-phase zone of solidifying alloys
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I.
INTRODUCTION
THIS paper describes
a model of binary alloys which solidify horizontally. The majorpurpose o f the model is to calculate macrosegregation which results from the convection o f interdendritic liquid during solidification. This convection, which is driven by solidification contractions and by gravity, is modeled as flow through a porous medium. The model can be used to obtain (1)the pressure field, (2) the velocity field, and (3) the solute flux in the solid plus liquid zone during solidification. With these established, macrosegregation after complete solidification is calculated. Included are the macrosegregation and flow field predictions for tin-lead, aluminum-copper, and tin-bismuth alloys as well as comparisons with empirical data. The numerical model is the basis for a FORTRAN program which has been run on a Prime 400 computer system and on a VAX 11/780 computer system. The documentation o f the FORTRAN code and operating manuals are given in a series of reports.~-s Figure 1 shows a binary alloy undergoing horizontal solidification from chill surfaces located at opposite sides of an otherwise insulated mold with the gravity force in the - y direction. Initially the alloy is liquid with a uniform composition of Co; when heat is extracted the alloy solidifies from the opposing chills, and during solidification there must be a zone containing solid and liquid phases ( i . e . , the S / L zone). In Figure l(a) the S / L zone is not completely developed because at the chill surface the temperature o f the alloy is greater than the final solidification temperature and there is liquid. This is referred to as the initial transient. The S / L zone is complete in Figure l(b), which depicts the situation after the period o f the initial transient; in so-called unsteady-state solidification, the width of the S / L zone varies (typically increases) as it moves away from the chill. Finally, in Figure l(c) the S / L zone has reached the center of the mold, and now its width must decrease until the isotherm o f TE reaches the center and solidification is complete. This is the final transient. A.L. MAPLES is Systems Analyst, General Electric Company, Huntsville, AL 35807. D.R. POIRIER is Professor, Department of Metallurgical Engineering, The University of Arizona, Tucson, AZ 85721. Manuscript submitted December 6 , 1982.
METALLURGICAL
TRANSACTIONS B
The structure of the S / L zone is shown in Figure 2(a) in which the solid phase is shown as dendrites. Figure 2(a) does illustrate the manner in which volume fraction o f solid varies within the S / L zone, and this is plotted in Figure 2(b). Since there is redistribution of solute between the two phases during solidification, the temperature and composition o f the interdendritic liquid vary as depicted in Figures 2(c) and 2(d), respectively. The mathematical formulation is based upon that given by Mehrabian e t al.6 for unidirectional and horizontal solidification, but Mehrabian e t a l . did not analyze the transients of the S / L zone. In a later work,
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