A generalized formulation for evaluation of latent heat functions in enthalpy-based macroscopic models for convection-di
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MAN CHAKROBORTY, Research Scholar, Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India. On leave from Department of Power Plant Engineering, Jadavpur University, Calcutta 700 091, India. PRADIP DUTTA, Assistant Professor, is with the Department of Mechanical Engineering, Indian Institute of Science. Manuscript submitted September 18, 2000. 562—VOLUME 32B, JUNE 2001
As described in Brent et al.,[2] a general form of the enthalpy updating expression can be written as [⌬HP]n⫹1 ⫽ [⌬HP]n ⫹
␣P [{hP}n ␣0P
[1]
⫺ cF ⫺1{⌬HP}n] where ⌬HP is the latent heat content of the computational cell surrounding the grid point P, h is the sensible enthalpy, c is the specific heat, is a relaxation factor, n is the iteration level, ␣P and ␣0P are the coefficients of finite volume discretization equation,[7] and F ⫺1 is the inverse of the latent heat function. The physical interpretation of the terms in Eq. [1] is described in detail in Brent et al.[2] As a staring point, we consider the metallurgical phase diagram in a general functional form of T ⫽ T(TL , Tm , C0 / Cl), where TL , Tm , C0, and Cl are the liquidus temperature, melting temperature (of a pure component), nominal alloy concentration, and liquid composition, respectively. As a specific example, this function may take the form
冢
冣
Cl Cl T ⫽ TL ⫺ Tm ⫺1 C0 C0
[2]
for the case of a linearized phase diagram, which is a common assumption in many of the macroscopic models quoted in the literature.[4] The next step is to substitute the proper metallurgical relation for C0/Cl as a function of liquid fraction (depending upon the metallurgical model under consideration), appropriately representing the microscopic solute balance. For the case of a nonequilibrium solidification situation,[4] the preceding may be described by Scheil’s model[8] as (Cl ⫺ Cs)dfS ⫽ (1 ⫺ fS)dCl
[3]
where fs is the mass fraction of the solid and Cs is the solid phase composition. On integrating Eq. [3], we obtain
再
fL ⫽ exp ⫺
兰
cl
c0
dCl Cl(1 ⫺ kp)
冎
[4]
where fL is the mass fraction of the liquid and kP is the partition coefficient. Equation [4], in principle, can be integrated when the variation of kP with Cl is known. For the specific case of a constant partition coefficient (or, a partition coefficient independent of composition), integration of Eq. [4] gives Cl ⫽ C0 f kLp⫺1
[5]
It can be noted that kP in Eq. [5] can be corrected on account of solutal undercooling, in which case it can be expressed as a function of solutal diffusion boundary layer thickness, interface speed, and species diffusion coefficient in the liquid.[5] Thus, more general cases of nonequilibrium solidification may effectively be addressed. Now, substituting Eq. [5] in Eq. [2], and using fL ⫽ ⌬H/L, we obtain h ⫺ cTL ⌬H ⫽1⫺ h ⫺ cTm L
(1⫺ kp)
冢 冣
[6]
From Eq. [6], we get F ⫺1(⌬H ) ⫽ Tm ⫺ (Tm ⫺ TL)
(kp⫺1)
冢 冣 ⌬H L
[7]
METALLURGICAL AND MATERIALS TRANSACTIONS B
Fig. 1—Variation of inverse of latent heat function with liquid fraction, corresponding to various models (fo
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