Convection in a Mushy Zone Forced by Sidewall Heat Losses

Abstract

We calculate the convective state due to weak sidewall heat losses in a mushy zone during the steady directional solidification of a binary alloy. The configuration consists of a warm liquid region and a cold solid region separated by a mixed phase region, the mushy zone, which is modeled as a reactive porous matrix. The structure of the convection that arises from horizontal temperature gradients, induced by the heat losses at the sidewalls, is characterized by a set of nondimensional parameters that describe the effects of latent heat, composition, permeability, and thermal and solutal buoyancy. We observe a wide range of behaviors and show that, as the critical Rayleigh number for convection in a horizontally uniform mush is reached, we begin to see the precursors of chimneys near the cooled boundaries. The strength of the cooling plays an important role in determining the strength and degree of localization of the convection near the boundary. We find, in common with other authors, that upflow, in this case caused by lighter fluid being released at the cooled sidewalls, leads to regions of dissolution, which are precursors to chimney formation. Although a treatment of the stability of the steady convective states presented is not considered, we identify the effects of the different physical parameters on the steady states.

Finite-sample-size effects on convection in mushy layers

“…Recent theoretical calculations show that chimney formation can be influenced by lateral heat transfer due to heat losses at the sidewalls. Sidewall heat transfer can have both direct thermal effects, and indirect effects by forming weak convective flows in the overlying fluid (Roper et al 2007(Roper et al , 2011. The cooling effect of sidewall heat losses can promote the formation of chimneys near to the walls (Roper et al 2007).…”

“…Sidewall heat transfer can have both direct thermal effects, and indirect effects by forming weak convective flows in the overlying fluid (Roper et al 2007(Roper et al , 2011. The cooling effect of sidewall heat losses can promote the formation of chimneys near to the walls (Roper et al 2007). Whilst localization of convection resulting from flow in the overlying fluid encourages chimney formation at the walls in three-dimensional mushylayer cells, in two-dimensional flow the localisation promotes chimney formation away from the cell walls.…”

Fluid Flow and Defect Formation in the Three-Dimensional Dendritic Structure of Nickel-Based Single Crystals

“…However, the analysis based on the Rayleigh number offered in this article indicates that full descriptions of vertical flow at values less than 0.4 fraction solid may not be entirely necessary as critical Ra h values occur in the vicinity of 0.1-0.2 fraction solid because of cross permeabilities. Furthermore, recent empirical models have suggested cross-flow permeabilities as the limiting flow type [38,45] in the lower ranges of fraction solid as well.…”

Convective instabilities during the solidification of an ideal ternary alloy in a mushy layer

“…Consequently, while there are a host of interesting, novel and industrially relevant buoyant convective phenomena that occur within binary alloy mushy layers (e.g. Worster 1992a,b;Amberg & Homsy 1993;Chen, Lu & Yang 1994;Anderson & Worster 1995, 1996Schulze & Worster 1998, 2001Chung & Chen 2000;Chung & Worster 2002;Guba & Worster 2006a,b;Roper, Davis & Voorhees 2007Katz & Worster 2008) double-diffusive convection driven from within the mushy layer itself is not one of them. Studies such as those by Nandapurka et al (1989) and Singh & Basu (1995) that specifically refer to double-diffusive, or thermosolutal, convection in binary mushy layer systems still impose the condition of thermodynamic equilibrium within the mushy layer and therefore implicitly refer to double-diffusive convection driven from within completely liquid regions rather than from within a binary mushy layer.…”