A Numerical Method of Macrosegregation Using a Dendritic Solidification Model, and Its Applications to Directional Solid
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e are two major types of macrosegregation depending on alloy compositions and casting processes, and they differ in the mechanism of their formations. The first type (type I) consists of positive or negative segregation and of the channel segregation widely known as freckle, ‘‘A’’ or inverse ‘‘V’’ segregation. The second type (type II) is that of V segregation, which is represented in continuous casting of steels as well as in a steel ingot or casting. The mechanism of V segregation is not understood fully. However, the mechanisms of the formations of type I macrosegregation have been well established both theoretically and experimentally by a series of pioneering works by Flemings and his coworkers.[1–10] The macrosegregation theories of Flemings are concisely written in his book,[11] and some unsolved macrosegregation problems are discussed in his review.[12] YOSHIO EBISU, Doctor of Engineering, is with the EBIS Corporation, Kanagawa-ken 252-0325, Japan. Contact e-mail: yoshio@ ebiscorp.jp Manuscript submitted July 23, 2010. Article published online January 19, 2011. METALLURGICAL AND MATERIALS TRANSACTIONS B
More recently, after the numerical simulations by Fujii et al.[10] on the channel segregation of steel ingots, Schneider and Beckerman[13] developed a more comprehensive macroscopic model that incorporates mass, momentum, heat, and solute conservation equations, and they performed numerical simulations on some of the commercial steels with similar casting conditions as those of Fujii et al.[10] With respect to the directional solidification (DS), Neilson and Incropera[14] performed numerical simulations of freckles in an aqueous NH4 Cl solution. Subsequently, Schneider et al.[15] conducted numerical simulations of the macrosegregation and freckle formation of DS Ni-base superalloys based on the same macroscopic conservation equations as those of Reference 11. This research is followed by Felicelli et al.[16] who studied the freckles of DS Ni-base alloys based on a similar macroscopic model as abovementioned. This article presents a numerical method for analyzing macrosegregation with special emphasis paid on the type I macrosegregation. The physical variables for describing the solidification behavior are temperature, solute concentration in bulk liquid and mushy phases, and fluid flow velocity vectors and pressures of liquid in both these phases. The corresponding governing equations to VOLUME 42B, APRIL 2011—341
calculate these variables comprise the energy equation, the solute redistribution equations, and the momentum and pressure equations with Darcy’s law incorporated. In addition to these equations, the equation for describing the relationship between the temperature and the volume fraction solid has been introduced to obtain the liquidus temperature during the solidification of multicomponent alloys. Thus, the total number of the variables and the corresponding equations becomes n + 6 with n being the number of alloy elements. Furthermore, because the permeability within the mushy zone plays a major role i
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