A Generalized Enthalpy Update Scheme for Solidification of a Binary Alloy with Solid Phase Movement
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cillations in the iterative procedure and thus provides stability to achieve a converged solution.[1,3] However, the preceding methods do not take care of multiphase nature of the flow, in which the solid phase is free to move within the liquid zone. However, several existing numerical models involve multiphase flow,[4–8] but these multiphase flow schemes require intricate microscopic morphological parameters. Also, the existing enthalpy based models for multicomponent problems require different functional relationships to handle various microsegregation models such as lever rule, Scheil’s model, and so on. The current work proposes a more generalized numerical scheme that accounts for the multiphase nature of the flow, while reducing the functional relationships for various segregation models to a single function. TO start with, we consider the generalized energy Eq. [1] that takes into account the phase change process along with multiphase advection. @ @ qcp T þ r qcpl uT ¼ r:ðkrTÞ ðqDHÞ @t @t r ðqul DHÞ þ r: qfs us cpl cps T
½1
where The numerical prediction of macroscopic phase change processes under the influence of convection and diffusion requires special attention to model the phase change mechanism. The system being multicomponent in nature complicates the numerical prediction of the phase change mechanism. The problem becomes even more difficult to deal with if the convection field is multiphase in nature (i.e., if the solid phase is also allowed to move within the liquid zone). A standard approach for numerical modeling to deal with the phase-change processes in a convection diffusion problem is the so-called ‘‘fixed-grid enthalpy-based method.’’[1,2] Brent et al.[2] demonstrated the robustness of enthalpy updating scheme by capturing the phase change phenomena of pure metal successfully without encountering solution instability. The incorporation of microscopic issues in macromodels of multicomponent systems have been realized effectively through appropriate devising of generalized enthalpy-based scheme.[3] According to the enthalpy update model proposed by Chakraborty and Dutta,[3] the latent heat content of each computational cell is updated according to the temperature- and volume-averaged species concentration values predicted by the macroscopic conservation equations. The updating of latent enthalpy has to be done during each iteration within a time step. In a physical sense, such an updating scheme activates the source term in the energy equation to take into account the enthalpy associated with the phase change process. This method PRODYUT R. CHAKRABORTY, Doctoral Student, and PRADIP DUTTA, Professor, are with the Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India. Contact e-mail: [email protected] Manuscript submitted May 18, 2011. Article published online October 4, 2011. METALLURGICAL AND MATERIALS TRANSACTIONS B
u ¼ fl ul þ fs us ;
cp ¼ fl cpl þ fs cps ;
k ¼ fl k l þ fs k s ;
fl þ fs ¼ 1
½2
and DH ¼ fðT; Cl Þ ¼ fl ððDHÞliq
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