Modeling of micro-macrosegregation in solidification processes
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cussed in terms of equilibrium conditions at the solidliquid interface during local solidification and remelting of the alloy. First, the continuity equations for heat, mass, and momentum are briefly summarized. Here, emphasis is placed on the derivation of the solute transport equation. It will be convenient to follow the recent work of Bennon and Incropera t5,61 and Voller and Prakash t91 in deriving average continuity equations which govern heat, mass, and fluid flow. For the sake of simplicity, it will be assumed here that: (1) the solid phase is stationary; (2) the material is incompressible and the densities of the solid and liquid phases are equal (with no shrinkage porosity formation[l~ and (3) the specific heat at a given temperature is the same in both the solid and the liquid phases. It is to be noted that although equiaxed grains can move, up to a certain volume fraction of solid, the assumption of a fixed solid is more appropriate for treating most solidification
I.
fs
Fig. 1 --Schematic representation of the one-domain method. VOLUME 21A, MARCH 1990--749
microstructures. Accordingly, an average velocity, v, within a small volume element can be obtained directly from the real velocity of the liquid phase, v,, at the same location: v =
(1 -f~)v,
[1]
where fs is the volume fraction of solid. If the same volume element is fully liquid or solid, then v = v~ or v = 0, respectively. Energy equation. The average local enthalpy per unit volume is defined to be H = f ~ H s + (1 - f s ) n ,
[2]
where Hs and H, are the enthalpies per unit volume of the solid and liquid phases, respectively: I-Is =
[4]
cp(O)dO + L
and cp(T) and L are the specific heat and latent heat per unit volume, respectively. It should be noted that local thermodynamic equilibrium has been assumed, i.e., that the temperature, T, of the solid at a given location is the same as that of the liquid. The average velocity and enthalpy now having been defined, the heat flow equation can be written as OH div (K grad T) = m + v. grad H + Sh Ot
[5]
where K is the thermal conductivity, averaged over the phase fractions,* *Due to the intricate shape of the solid-liquid interface in the m u s h y region, the effect of the growth direction is not taken into account when averaging the thermal conductivity.
K =f~K~ + (1
-f~)K,
[6]
The source term, Sh, has to be introduced in order to balance the latent heat term. Latent heat is included in the term v. g r a d H but is not transported when the solid remains fixed. Thus, Sh
Substituting Eq. simplification,
= - L v " grad (1 - f~) [7]
div (K grad
into
Eq.
0t
[5]
[7] yields,
after
+ Cp.
v- grad T
/x S~ = - - - v K
[11]
where K is the local permeability of the mushy region (i.e., a function of f~). With increasing volume fraction of solid, the permeability tends toward zero, and the source term will dominate all of the other terms apart from the pressure gradient and the body force. Hence, the NavierStokes equation will reduce to the Darcy law in the mushy region. [9] It should be
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