A general enthalpy method for modeling solidification processes
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I.
INTRODUCTION
A. Background A standard approach for the numerical modeling of metallurgical solidification processes are so-called "fixed" grid methods. I~-4l The essential feature of these methods is that the evolution of latent heat is accounted for by the definition of enthalpy. Consequently, the numerical solution can be carried out on a space grid that remains fixed throughout the calculation. The major advantage of fixed grid methods is that they permit modeling of solidification phase change through simple modifications of existing heat-transfer numerical methods and/or software. A "generic" solidification phase change system can be considered to be that of a binary alloy. This system contains a two-phase (solid/liquid) region, often referred to as the "mushy" region, over which the latent heat associated with the phase change is evolved. Appropriate numerical treatments for the latent heat evolution can best be illustrated by considering heat conduction-controlled phase change problems (i.e., convection effects due to density changes at the phase interface or density variations in the liquid phase are neglected). In such a case, an appropriate governing equation for the system is l~'31 OH --
=
V.(kVT)
[1]
Ot where k is a mixture conductivity given as k = (1 - g)ks + gk~
[2]
and H is a mixture enthalpy written as H = (1 - g)Hs + gH,
[3]
Note that g is the volume fraction of the liquid and the subscripts [~s and [~t represent solid and liquid phases, respectively. In general, the enthalpy could be a function of a number of variables, such as temperature, concentration, cooling rate, etc. In many solidification models, however, the enthalpy in the mushy region can be assumed to be a function of temperature alone. Four possible C.R. SWAMINATHAN, Graduate Student, Department of Mechanical Engineering, and V.R. VOLLER, Associate Professor, Department of Civil and Mineral Engineering, are with the University of Minnesota, Minneapolis, MN 55455. Manuscript submitted March 4, 1991. METALLURGICAL TRANSACTIONS B
enthalpy-temperature curves are shown in Figure 1. Curve A represents the isothermal solidification of a pure metal or an alloy of eutectic composition where the latent heat is evolved at a unique temperature. In most solidification systems, however, the phase change from liquid to solid and the accompanying evolution of latent heat will occur over a temperature range in which both liquid and solid coexist. Curve B represents the case where there is a linear evolution of the latent heat over the solidification range T~-Ts. Curve C depicts a nonlinear evolution of the latent heat over the solidification range T~-TE, e.g., the Scheil equation. ISJ The step discontinuity in curve C represents the isothermal transformation of the remaining liquid at the eutectic temperature TE. The curves A through C represent systems in which the specific heats are functions of temperature alone and the latent heat of fusion is constant. In such cases, the enthalpy can be written as
(
H = ( 1 - g) j r pc, dT + g re
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