An Improved Model for the Flow in an Electromagnetically Stirred Melt during Solidification
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ETIC (EM) stirring has found extensive use in solidification processing as a means to produce a fine, equiaxed grain structure.[1–3] The role of the flow is to homogenize the solute and temperature profiles in the bulk liquid,[4,5] which affects segregation in the cast product. It also increases the nucleation potential in the melt through a combination of dendrite fragmentation,[6–8] a decrease of the temperature gradient at the solidification front,[9] and an increase the dissipation rate of the melt superheat.[10] Clearly, an improved understanding of the flow characteristics in the bulk and mushy zones during solidification is critical in order to control the structure of the final product. There are essentially two basic approaches for modeling the flow in the mushy zone, namely single-zone[11] and dual-zone models.[12] The single-zone model treats the entire two-phase region as a porous medium, with the flow being damped via Darcy’s law. Although this model provides an adequate representation of the flow in columnar solidification, it is not suitable for equiaxed solidification, as the crystallites have been shown to travel with the flow prior to coalescence.[13] The dualzone model addresses this problem by dividing the mushy zone into two domains: (i) the suspended particle region, and (ii) the fixed particle region, with the transition between regions occurring at the coherency point. Recently, efforts have been made to model EM solidification processes.[14–21] With no exception, all GREGORY M. POOLE, Graduate Student, and NAGY EL-KADDAH, Professor, are with the Department of Metallurgical and Materials Engineering, University of Alabama, Tuscaloosa, AL. Contact e-mail: [email protected] Manuscript submitted May 24, 2013. Article published online September 13, 2013. METALLURGICAL AND MATERIALS TRANSACTIONS B
models used an approximate analytical solution for describing the EM force field in the bulk liquid and two-phase regions. In modeling the flow in the mushy zone, all of these studies, except for the works of Pardeshi et al.[20] and Budenkova et al.,[21] employed a single-zone model for describing fluid flow in the mushy region. Several approaches have been used to model the turbulent field in the two-phase region. Prescott and Incropera[15,16] and Shyy et al.[17,18] determined the turbulent field using a low-Re k-e model, together with ad-hoc expressions for the decay of the turbulent viscosity in the two-phase region. Pardeshi et al.[20] and Budenkova et al.[21] employed high-Re k-e model to describe the turbulent characteristics in the bulk liquid and suspended particle regions. Their formulation does not account for the damping of turbulence in both the suspended and fixed particle regions. Furthermore, the interaction between the turbulent eddies and the solid crystallites, which is known to produce an additional damping force on the flow field,[22–24] was not addressed in these models. This paper presents a comprehensive model for the numerical calculation of the EM, velocity, and temperature fields in an axisymmetrical EM
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