Further discussions on the solute redistribution during dendritic solidification of binary alloys
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I.
INTRODUCTION
S O L U T E redistribution during dendritic solidification is an important topic of the researches and the applications of the solidification process. In the early work of Flemings and Nereo, u] the so-called "local solute redistribution equation" (Eq. [1]) was presented as follows: 0f~ = 0Cl
V~VT) f~
(l-r)(1 ( 1 k)
[1]
Ct
where ft is the local liquid fraction in the mushy zone, C~ is the liquid composition, /3 is the solidification shrinkage, k is the equilibrium redistribution coefficient, e is the local cooling rate, V is the liquid flowing velocity of interdendritic liquid, and T is the temperature. This fundamental equation has been widely used for the evaluation of macrosegregation t2-51 and microsegregation tvl of binary alloys. It is seen that the diffusion in solid is ignored in the equation. Further work on the solute redistribution has been done by Poirier et al.,[8] where the solute conservation equation including the effect of the diffusion in the dendritic solid was established. The researches on solute redistribution during dendritic solidification means to identify the solute distributions in solid and liquid and their relationship with local solid fraction during solidification process. Since the local equilibrium at liquid/solid interface will be kept and the diffusion in the interdendritic liquid is usually sufficient, our object will only be to establish the relationship between the solid fraction and the constitution in the interdendritic liquid. The first equation for this relationship is the equilibrium one, which is written to be f~= 1 -k
1-
[2]
JIE WANQI, Professor, is with the Department of Materials Science and Engineering, Northwestern Polytechnical University, Xi'an 710072, People's Republic of China. Manuscript submitted May 13, 1993. METALLURGICAL AND MATERIALS TRANSACTIONS B
where Co and C~ are the average composition of the master alloy and that of interdendritic liquid, respectively. In the famous Scheil equation, 9 f, = 1
[3]
\Co/
both the solidification shrinkage and the diffusion in the dendritic solid were ignored. Equations [2] and [3] are based on two limited situations (sufficient diffusion in solid and no diffusion in solid). By ignoring the liquid flow (let V = O) in Eq. [1], the following equation was obtained: ul
,4, where the diffusion in solid is still not taken into account. With the hypothesis of platelike dendritic solidification with linear and parabolic solidification rates, Brody and Flemings tTl deduced the following two equations: f , = (1 +
ak)
1-
I51
and
f,
1
1 - 2ak
1 -
[6]
\-Coo/
J
where a is the diffusion parameter defined by a = D,~ where Ds is the diffusion coefficient in solid, rI is the local solidification time, and L is the dendritic spacings. Equation [5] is for the solidification with linear solidification rate and Eq. [6] is for that with parabolic solidification rate, Clyne and Kurz tg~ have discussed the diffusion in solid by an approximate method. Yeum VOLUME 25B, OCTOBER 1994--731
et al. u~ have dev
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