Microsegregation in cellular microstructure
- PDF / 532,730 Bytes
- 5 Pages / 597 x 774 pts Page_size
- 26 Downloads / 251 Views
I.
INTRODUCTION
E X T E N S I V E work has been done to predict the microsegregation in cellular microstructures (dendritic segregation can be simplified as a cellular one). In current models, t~-4~ the cellular segregation is taken as the result of the cell thickening process, i.e., as the consequence of the "directional solidification" occurring between and perpendicular to the cells. The segregation profile is then described by various modified Scheil I5~ expressions as a function of the local solid fraction fs. The influence of the morphological feature is normally neglected. By directional solidification of metallic systems, cells generally assume a flat, rounded shape, as apparent in Figure 1, which shows the cellular structure of the superalloy IN* 939 (48 Ni, 22.6 Cr, 19.0 Co, 2.0 W, 3.79 Ti, *IN is a trademark Huntington, WV.
of
Inco
Alloys
International,
Inc.,
1.9 A1, 1.3 Ta, 1.1 Nb, and 0.15 C, in weight percent). Therefore, an overemphasized solute flux, rejected from the cell interface, has to be directed toward the melt, as against the intercellular volume. This may, compared to the existing models, result in a significant reduction of the solute enrichment in the intercellular liquid and, therefore, a reduction of the segregation profile across the cells. Although Bower et al. t2} include in their model the longitudinal diffusion in the intercellular liquid and the solute buildup at the tip such that it predicts some reduced cell segregation, they do not consider geometric effects. In this work, a modified model is being proposed which takes into account the effect of cell geometry on the solute rejection in both longitudinal and lateral directions and, thus, on the final segregation profile. II.
PROPOSED
MODEL
As an approximation for the directionally solidified cells from Figure 1, a semielliptical shape can be assumed (Figure 2). As the problem is axisymmetric, it may be treated two-dimensionally. The shape of the interface is described by Z 2
where a and b are the major and minor half-axes of the ellipse corresponding to length and radius ( i . e . , halfspacing) of the cells, respectively. The cell is assumed to grow continuously in the outward normal direction of the interface. The local growth velocity v, can be resolved into two parts, namely, into the axial velocity vz (in solidification direction) and the lateral velocity Vr (perpendicular to the solidification direction). The geometrical relationship leads to 2
2
2
Vz q'- Vr = Vn
[2]
and
a2r
Vr
vz
-
[3]
bZz
Now the cell growth can be analyzed independently in the z and r directions. The growth velocity in the z direction is constant along the entire interface and equal to the isotherm velocity for directional solidification, i.e., v~ = v. The solute buildup at the interface caused by vz is Acz = c, - Co, with c, being the liquid concentration at the solidification front. The mass balance at the interface with stationary solidification leads to
Ocl
vAcz = - D - -
[4]
Oz
or as given by Bower et al.12} c, = (1 + a
Data Loading...
