Approximate models of microsegregation with coarsening
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Approximate Models of Microsegregation with Coarsening V. R. VOLLER and C. BECKERMANN Microsegregation refers to the processes of solute rejection and redistribution at the scale of the dendrite arm spaces in the mushy region of a solidifying alloy. A representative geometry for a microsegregation analysis is the half-arm spacing in a “platelike” morphology (Figure 1). Models of microsegregation are based on a solute balance within this domain. A recent review by Kraft and Chen[1] covers the range of available models. Common assumptions used in modeling include a binary eutectic alloy; a fixed average composition, C0; equilibrium at the solid-liquid interface; a constant partition coefficient k ,1; and a straight liquidus line in the phase diagram. Two key features of the solute balance that need to be included in a comprehensive model are the following. (1) The mass diffusion of the solute. Typically, the solute diffusion in the liquid is rapid, and, at each instant in time, a uniform distribution of solute, Cl(t), can be assumed. In the solid, however, diffusion is much slower and the solute balance needs to account for the so-called “back-diffusion” of solute into the solid. (2) Changes in morphology. As solidification proceeds, the arm spacing will coarsen. If the overall solute balance is maintained in the half-arm domain, this feature will dilute the solute in the liquid. One class of microsegregation models involves expressions that contain integrals. When coarsening is not accounted for and the solid growth is parabolic, Wang and Beckermann[2] obtain an integral expression that approximates the segregation ratio (C1/C0). At the opposite extreme, accounting for coarsening but neglecting back-diffusion, analytical expressions can also be obtained. In the case of a constant cooling rate, Mortensen[3] presents an analytical integral expression for the solid fraction, f, and Voller and Beckermann[4] present an analytical integral expression for the segregation ratio when solid growth is parabolic. In recent work, Voller and Beckermann[4] show, analytically, that coarsening can be included in a microsegregation model by using the enhanced diffusion parameter
V.R. VOLLER, Professor, is with the Saint Anthony Falls Laboratory, Department of Civil Engineering, University of Minnesota, MN 554550116. C. BECKERMANN, Professor, is with the Department of Mechanical Engineering, University of Iowa, Iowa City, IA 52243. Manuscript submitted April 5, 1999. 3016—VOLUME 30A, NOVEMBER 1999
a+ 5
X 2F 1 dX0 f 2 2 a 1 X0 X0 dt m 1 1
[1]
DtF X 2F
[2]
In Eq. [1],
a5
is the regular back-diffusion Fourier number, D is the diffusivity (m/s2) in the solid, XF is the length of the microsegregation domain at the conclusion of solidification—usually taken to be equal to half the final secondary arm spacing, and tF is the local solidification time. The other terms in Eq. [1] are X0(t), the time-dependent length of the half arm space; f, the solid fraction in the arm space; t 5 t/tF , the normalized time; and m, the order of the po
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