Unidirectional solidification of binary melts from a cooled boundary: analytical solutions of a nonlinear diffusion-limited problem

Abstract

A model is presented that describes nonstationary solidification of binary melts or solutions from a cooled boundary maintained at a time-dependent temperature. Heat and mass transfer processes are described on the basis of the principles of a mushy layer, which divides pure solid material and a liquid phase. Nonlinear equations characterizing the dynamics of the phase transition boundaries are deduced. Approximate analytical solutions of the model under consideration are constructed. A method for controlling the external temperature at a cooled wall in order to obtain a required solidification velocity is discussed.

Solidification dynamics under random external-temperature fluctuations

“…The random external temperature fluctuations caused by various natural noises lead to the formation of a mushy zone, and the rate of increase of the mushy zone thickness can reach several centimeters per day. The heat flow on the cooling boundary surface, which is inversely proportional to b(t) [14][15][16][17][18], weakens as the mushy zone thickness increases. Due to the gener ality of the thermodiffusion Stefan model, this effect should also be characteristic of magma solidification models [41,42] and the evaporation of metals with the formation of liquid films [43][44][45][46][47].…”

“…The authors of [14][15][16][17][18] obtained the law of motion of the boundary of the mushy zone-melt (solution) phase transition b(t), which actually represents the thickness of the region having formed by time t, (1) where Here, we introduced the following designations:…”

Self-Similar Solidification of Binary Alloys

“…Taking into account that relaxation times of the solute concentration field is several orders of magnitude higher than the thermal relaxation time, the temperature field in the mushy layer can be written as a linear function of coordinate z (see, among others [4][5][6][7]). Furthermore, taking into account that the diffusion field in the mushy layer is practically frozen and the mass transport is caused by the impurity displacement into the liquid phase we use a linear temperature approximation and the Scheil equation for the impurity distribution in the mushy layer [4][5][6][7].…”

“…Furthermore, taking into account that the diffusion field in the mushy layer is practically frozen and the mass transport is caused by the impurity displacement into the liquid phase we use a linear temperature approximation and the Scheil equation for the impurity distribution in the mushy layer [4][5][6][7]. Thus, instead of (16), we have…”

On the theory of dendritic growth: Soret and temperature-dependent diffusion effects

“…In the latter case, the system dynamics depends substantially on the time variation of the cooling boundary temperature. To describe the processes that take place under these con ditions, the authors of [53][54][55][56][57][58][59][60][61][62][63][64][65] developed methods for analytical solution of quasi equilibrium mushy zone equations.…”