An analysis of unidirectional solidification of pure metals cooled through an interface resistance
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resistance are analogous, and the conclusions would be the same. The method in question is based on an interesting idea: an assumption of "virtual adjunct of solid," where Newtonian resistance is considered as equivalent to a previously solidified layer of metal. No theoretical justification of this assumption has been presented. We will first show that this idea works nicely as long as the temperature distribution in the solidified layer is assumed to be linear. Consider a casting poured at the constant solidification temperature Ts with its surface in contact with a cooling medium of temperature To through a constant Newtonian resistance 1 / h , Assume a linear temperature distribution in the solidified layer (Figure l(a)). We try to replace the Newtonian resistance with a solid adjunct of equivalent resistance, S o ~ k , = l / h i . Its surface is kept at To (Figure l(b)). Virtual identity of the two cases is assured as the flux in the solid OT qs = k,-
= k s ( Tf - T o ) / S '
= k~(T,-
To)/So
Ot
is equal to the Newtonian flux at the surface qN = h i ( T i - To) = ( k f f S o ) (T, - To)
The real case (a) (without prime) and the virtual case (b) (with prime) are related by x' =x+S
0
S' =S+So
An Analysis of Unidirectional Solidification of Pure .Metals Cooled through an Interface Resistance EISUKE N I Y A M A and K O I C H I A N Z A I Most of the quantitative analyses conducted today on actual solidification problems are based on numerical methods, as distinct from analytical methods, which have only limited application. Nevertheless, analytical equations are still of high value, because unlike numerical methods, they are generally suited to give clear insight into the nature of the problems. When using one of them, however, it is important to understand its background, e . g . , whether it is an exact solution or an approximate one. The purpose of this communication is to show that an analytical method proposed by Garcia e t al. tlj for the problem of unidirectional solidification of a pure metal cooled through a constant interface resistance is an approximate one and not an exact one, as claimed by the original authors. To the present authors' knowledge, this point has not been questioned since the publication. Only the case of a constant ambient temperature will be taken up here for simplicity. The cases with mold EISUKE NIYAMA, Professor, and KOICHI ANZAI, Associate Professor, are the Department of Materials Processing, Faculty of Engineering, Tohoku University, Sendal 980, Japan. Manuscript submitted May 28, I992. METALLURGICAL TRANSACTIONS B
t' = t + t 0 where to is solidification time for thickness So. From the well-known Stefan's solution t2l for the virtual case S' = ~r s - To)/dsH]t' [1] a solution for the real case is obtained by simple algebra, S=S
' -So
= V'[2ks(Ty- To)/d~H]t + (kffh,) 2 - (ks~h,)
[2]
Tf Ti /
(b)
/
,"
J J
f
To-
/
so s' s Fig. 1--Schematic temperature distribution: (a) real case and (b) virtual case. VOLUME 23B, DECEMBER 1992--881
This gives solid thickness as
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