A stereological formulation for the source term in micromodels of equiaxed eutectic solidification
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composition. The time-dependent volume fraction solid, which is the sum of the contributions of all grains which have been nucleated at a time ~-(where ~- -< t) and grown between ~- and t, can be expressed as
REFERENCES 1. J.H. Yoon and F.M. Doyle: in Innovations in Materials Processing Using Aqueous, Colloid and Surface Chemistry, F.M. Doyle, S. Raghavan, P. Somasundaran, and G.W. Warren, eds., TMS, Warrendale, PA, 1989, pp. 195-211. 2. J.-C. Lee and F.M. Doyle: in Rare Earths: Resources, Science, Technology and Applications, R.G. Bautista and N. Jackson, eds., TMS, Warrendale, PA, 1992, pp. 139-50.
f,(t) =
fo'
ri(~-) 3 ~rR3(~" t) d~"
where ;i(r) is the nucleation rate at time r and R(~-, t) represents the radii of the grains which have been nucleated at that time and grown until time t. Differentiating Eq. [3] with respect to time and neglecting the contribution associated with new grains yields df,(t) _ dt
fo
ri(t) 4rrR2(~-, t) OR(r, t) d r Ot
Source Term in Micromodels of Equiaxed Eutectic Solidification
dfs(t) _ n(t) 47rR~u(t) dt
F.J. BRADLEY The development of micro-macroscopic models of solidification and their application to microstructure prediction is an active area of research. For the case of no convective transport, macroscopic modeling involves solution of the heat conduction equation subject to appropriate boundary conditions: [1]
where k is thermal conductivity, T is temperature, p is density, c is specific heat per unit mass, t is time, and g is the rate of heat generation per unit volume. Regardless of the approach to solving Eq. [1], predictions of microstructure evolution during solidification requires a realistic micromodel. In the case of solidification, the source term g in Eq. [1] can be expressed in terms of the rate of change of fraction solidified, fs, as g = Lv Ofs Ot
[2]
where Lv is latent heat of fusion per unit volume. Micromodeling essentially involves the application of nucleation and growth laws based on thermodynamic and kinetic considerations to the derivation of time-dependent fraction solidified relationships which can be utilized directly in F_x/. [2] to simulate the e v o l u t i o n of latent heat during solidification. Fundamental equations applicable to micromodeling of equiaxed eutectic solidification are briefly summarized. Rappaz Iu considered a volume element of uniform temperature containing a solidifying melt of eutectic F.J. BRADLEY, Assistant Professor, is with the Department of Materials Science and Engineering, University of Wisconsin-Madison, Madison, WI 53706. Manuscript submitted December 7, 1992.
[5]
where the number of grains per unit volume at time t, n(t), is given by n(t) =
f0
h(z)dz
[61
and the mean square grain radius, R e, is given by R 2 n(t) =
fl
h(~')R2(r, t)dr
[7]
In order to account for grain impingement, a weighting factor ~ is introduced to obtain the effective surface interfacial area between the solid equiaxed grains and the liquid phase so that Eq. [5] becomes dfs(t) - n(t)4~rR~v(t)qt(R) dt
[8]
which is applicable, in
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