Finite Volume Method Analysis of Heat Transfer in Multi-Block Grid During Solidiﬁcation
Eliseu Monteiro1 , Regina Almeida2 and Abel Rouboa3
- Engineering Department of University of Tr´ s-os-Montes e Alto Douro, Vila Real a 2 CIDMA/UA - Mathematical Department of University of Tr´ s-os-Montes e Alto Douro, Vila Real a 3 CITAB/UTAD - Department of Mechanical Engineering and AppliedMechanics of University of Pennsylvania, Philadelphia, PA 1,2 Portugal 3 USA
Solidiﬁcation of an alloy has many industrial applications, such as foundry technology, crystal growth, coating and puriﬁcation of materials, welding process, etc. Unlike the classical Stefan problem for pure metals, alloy solidiﬁcation involves complex heat and mass transport phenomena. Formost metal alloys, there could be three regions, namely, solid region, mushy zone (dendrite arms and interdendritic liquid) and liquid region in solidiﬁcation process. Solidiﬁcation of binary mixtures does not exhibit a distinct front separating solid and liquid phases. Instead, the solid is formed as a permeable, ﬂuid saturated, crystal-line-like matrix. The structure and extent of this mushyregion, depends on numerous factors, such as the speciﬁc boundary and initial conditions. During solidiﬁcation, latent energy is released at the interfaces which separate the phases within the mushy region. The distribution of this energy therefore depends on the speciﬁc structure of the multiphase region. Latent energy released during solidiﬁcation is transferred by conduction in the solid phase, aswell as by the combined effects of conduction and convection in the liquid phase. To investigate the heat and mass transfer during the solidiﬁcation process of an alloy, a few models have been proposed. They can be roughly classiﬁed into the continuum model and the volume-averaged model. Based on principles of classical mixture theory, Bennon & Incropera (1987) developed a continuum model formomentum, heat and species transport in the solidiﬁcation process of a binary alloy. Voller et al. (1989) and Rappaz & Voller (1990) modiﬁed the continuum model by considering the solute distribution on microstructure, the so-called Scheil approach. Beckermann & Viskanta (1988) reported an experimental study on dendritic solidiﬁcation of an ammonium chloride-water solution. A numerical simulationfor the same physical conﬁguration was also performed using a volumetric averaging technique. Subsequently, the volumetric averaging technique was systematically derived by Ganesan & Poirier (1990) and Ni & Beckermann (1991). Detailed discussions on microstructure formation and mathematical modelling of transport phenomenon during solidiﬁcation of binary systems can be found in
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the reviews of Rappaz (1989) and Viskanta (1990). In the last few decades intensive studies have been made to model various problems, for example: to solve radiative transfer problem in triangular meshes, Feldheim & Lybaert (2004) used discrete transfer method (DTM can be see in the work of Lockwood &Shah (1981)), Galerkin ﬁnite element method was used by Wiwatanapataphee et al. (2004) and Tryggvason et al. (2005) to study the turbulent ﬂuid ﬂow and heat transfer problems in a domain with moving phase-change boundary and Dimova et al. (1998) also used Galerkin ﬁnite element method to solve nonlinear phenomena. Finite volume method for the calculation of solute transport in directionalsolidiﬁcation has been studied and validated by Lan & Chen (1996). Finite element method to model the ﬁlling and solidiﬁcation inside a permanent mold is performed by Shepel & Paolucci (2002). Three dimensional parallel simulation tool using a unstructured ﬁnite volume method with Jacobian-free Newton-Krylov solver, has been done by Knoll et al. (2001) for solidifying ﬂow applications. Also arbitrary...