Enthalpies of a binary alloy during solidification
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INTRODUCTION
THEmathematical modeling of the solidification of alloys involves mass, energy, and momentum balances. In order to numerically simulate solidification, one has to know the physical properties accurately; these include diffusivities and conductivities, densities, specific heats, and heats of fusion. In this paper we compute enthalpies and heats of fusion during solidification for a particular binary system, but the methodology can be readily applied to other alloys, as well. When treating the enthalpy of the alloy, we indicate a realistic method of representing enthalpies of the interdendritic liquid and of the dendritic solid and the heat of fusion that are consistent with the redistribution of solute and the relative amounts of the phases during solidification. The Pb-Sn system is selected as the example because it exhibits a significant heat of mixing, and because alloys of this system are often studied by solidification researchers. Our major objectives are to present relationships for the enthalpies of a binary alloy as a function of temperature and composition and then apply these enthalpies to the twophase mixture that exists during dendritic solidification. In metallurgical research, Pb-Sn alloys have been used for solidification studies because of the convenience of working with relatively low temperatures and the availability of physical and thermodynamic data. However, even for this relatively simple system, the data are not given in a form which can be directly or conveniently used in quantitative analyses of solidification processes. To this end of providing quantitative properties for solidification models, the densities and viscosities of aluminum-rich A1-Cu alloys l~'2j and the densities of Pb-Sn alloys c31were reviewed and extrapolated to the respective solidification temperature ranges of the alloys. This paper represents a continuation of this overall effort.
D.R. POIRIER, Professor, and P. NANDAPURKAR, Research Associate, are with the Department of Materials Science and Engineering, The Umversity of Arizona, Tucson, AZ 85721. Manuscript submitted November 20, 1987.
METALLURGICALTRANSACTIONS A
ENTHALPIES OF TIN AND LEAD
The enthalpies of the pure components, H i for Sn and H i for Pb, are from the correlations given by Hultgren et al. {4] The enthalpies, in J 9 (g atom) -~, can be written in the form: H ~ - H~
= ao + a l T + a2 T2 + b T -I
[1]
where the subscript i is 1 (Sn) or 2 (Pb), the superscript denotes the standard state ( v i z . , pure Sn and pure Pb in their equilibrium states), TR is the reference temperature (298.16 K), T is temperature in K, and the coefficients a0, a~, a 2, and b are given in Table I. The correlations given by Eq. [1] and Table I are from the melting points to approximately 1100 K for the liquids and from 298 K to melting points for the solids. However, our primary interest is in the range from 600.6 K (the melting point of Pb) to 456 K (the eutectic temperature). Hence Eq. [1] and Table I are extrapolated in this work to estimate the en
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