On the approximation of the partial areas method in the calculation of the fraction of solid
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Communications On the Approximation of the Partial Areas Method in the Calculation of the Fraction of Solid ANTONIOS ZAVALIANGOS and EVANGELOS TZIMAS
TL 5 TM 1 mL C0 and TS 5 TM 1 mL C0/k The determination of fraction of solid* in alloys in the *Most of the time in the literature, no distinction is made between volume, weight and mole fraction although substantial differences may occurs in some cases.
semisolid state is important for a number of applications (e.g., continuous casting and semisolid processing), and the corresponding analysis (e.g., heat transfer in solidification and rheology of semisolid alloys). It is commonly assumed that the enthalpy, associated with melting or solidification of an alloy, evolves linearly with the fraction of solid. For example, during melting of an alloy, the fraction solid is approximated by
e H˙ dt Y
the relation between the mole fraction of solid, XS , and any temperature TS , T , TL is given by T 5 TL 2
NS 5 XSN 5 NSA 1 NSB where NSA 5 XS(1 2 CS)N, NSB 5 XSCSN
L
tS
[1]
eH˙ dt 5 1 2 e HT˙ dT Ye HT˙ dT tL
T
˙L
TL
˙L
L
tS
TS
TS
where HL is the total latent heat of the alloy; HL(T ) is the latent heat released due to partial melting to a temperature T; tS , t, and tL are the times at which the solidus temperature TS , the temperature T, and the liquidus temperature TL are reached, respectively; T˙ is the rate of change of temperature; ˙ is the rate of change of enthalpy in the system due and H to melting. This is a common approximation in both numerical simulations and the experimental determination of the fraction of solid as a function of temperature.[1–5] In the latter case, this method is termed the “method of partial areas,” because the integrals in Eq. [1] correspond to the areas under a heat flux curve vs time or temperature corrected by a baseline for the heat content change of each phase due their heat capacity (e.g., in a differential scanning calorimetry experiment[2]). In this article, we examine the validity of this assumption. First, a simplified analysis of the evolution of latent heat in binary alloys during partial melting in the semisolid region is presented, in order to identify analytically possible sources of nonlinearity. Then, specific alloys are examined using a thermodynamics software[6] (ThermoCalc). In this article, it
ANTONIOS ZAVALIANGOS, Professor, is with the Department of Materials Engineering, Drexel University, Philadelphia, PA 19104. EVANGELOS TZIMAS, Ph.D., formerly with the Department of Materials Engineering, Drexel University, is with EC, JRC, Institute of Advanced Materials, Petten 175SZG, The Netherlands. Manuscript submitted July 12, 1999. METALLURGICAL AND MATERIALS TRANSACTIONS B
kXS (TL 2 TS) 1 2 XS 1 kXS
[3]
where TM is the melting temperature of pure A, k is the partition coefficient, and mL is the slope of the liquidus with respect to composition (defined so that mL(k 2 1) . 0). All parameters discussed subsequently are based on molar (atomic) values or fractions. The number of moles of A and B in the solid pha
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