In this paper, a model of the directional solidification process of binary melts with a two-phase zone is developed in which the fraction of the liquid phase is described by the spatio-temporal scaling dependence. Self-similar variables with the growth laws of interphase boundaries, which are inversely proportional to the square root of time, are used. The mathematical model of the process is reformulated using the self-similar variables. Exact self-similar solutions to the heat and mass transfer equations are determined in the presence of two moving boundaries of the phase transition: solid phase–two-phase zone and two-phase zone–liquid phase. The distributions of temperature and impurity concentration in the solid phase, two-phase region, and melt are found in the form of integral expressions. A decrease in the dimensionless temperature of the cooled boundary leads to an increase in the crystallization rate and an increase in the liquid phase fraction. The crystallization rate, parabolic growth constants, and the liquid phase fraction at the solid-phase–two-phase zone are determined depending on the scaling parameter and the thermophysical constants of the solidified melt. The positions of the phase transition boundaries between the solid phase and the two-phase region, as well as the two-phase region and the binary melt, are found. The dependences for the solidification rate (inversely proportional to the square root of time) are analyzed. It was shown that the scaling parameter significantly affects the rate of the solidification process and the liquid phase fraction in the phase transformation region. The developed model and the method of its solution can be generalized to the case of directional solidification of multicomponent melts in the presence of several regions of phase transformation (for example, the main and cotectic two-phase zones during crystallization of three-component melts).
|Translated title of the contribution||ON THE THEORY OF DIRECTIONAL SOLIDIFICATIONWITH A PHASE TRANSFORMATION DOMAIN|
|Number of pages||11|
|Publication status||Published - 2020|