Convection in a Mushy Zone Forced by Sidewall Heat Losses
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DURING the directional solidification of a multicomponent alloy, a morphological instability[1] of the liquid-solid interface occurs when the liquid ahead of the solidification front is constitutionally supercooled. As the instability progresses, a region forms that is composed of solid dendrites and interstitial fluid—the mushy zone. Interactions between this region and the liquid region through convective fluid motion can lead to the formation of chimneys, which are narrow channels carrying a flux of lower-melting-point fluid. The solidification of these chimneys causes regions of lowermelting-point solid to be present in the final solidified material and the flow in the chimneys can also cause the nucleation of new crystal grains; both phenomena are often called ‘‘freckles.’’ This macrosegregation is highly undesirable as it can lead to mechanical failure of manufactured components. The process of chimney formation has enjoyed a wide variety of experimental, theoretical, and numerical study primarily aimed at understanding how convective motions arise and lead to channel formation. The mushy layer is a reactive medium because it can change its resistance to fluid flow as the local solid fraction changes, which, in turn, evolves as a result of changing thermal and compositional conditions brought about by fluid motion. Fluid motion can arise from unstable density gradients produced by gradients in temperature S.M. ROPER, Postdoctor, and S.H. DAVIS, Professor, Engineering Sciences and Applied Math, and P.W. VOORHEES, Professor, Materials Science and Engineering, are with Northwestern University, Evanston, IL 60208-3125. Contact e-mail: [email protected] Manuscript submitted June 5, 2006. Article published online May 12, 2007. METALLURGICAL AND MATERIALS TRANSACTIONS A
and composition. These interactions naturally give rise to a positive feedback mechanism, whereby fluid motion causes a reduction in solid fraction, which promotes fluid motion in that part of the mush, which itself transports heat and solute and further promotes dissolution. This process culminates in the formation of channels of fluid within the mushy layer. Various formulations of the governing equations have been considered.[2–5] All of them involve the assumption that the mush solidifies under global equilibrium conditions, so that given the bulk composition, the liquid and solid compositions in the mush can be read from the phase diagram. In some of the derivations, volume averages are taken to reduce the detailed structure of the mush to a continuum model of a reactive porous matrix.[5] A single set of governing equations, valid throughout the liquid and mushy regions, is derived. Simple mixture models can also be posited for the mushy layer.[2,3,4] In this case, the mushy layer is coupled to the completely fluid region through boundary conditions at the mush-liquid interface. For the present study, we use a two-layer formulation.[4] It has been the goal of some experimental studies[6] to develop a predictive tool to determine when a directionall
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