Exact Analytical Solution of Quasi-Equilibrium Mushy Zone Equations: Permeability and Dendrite Arm Spacing

Abstract—Exact analytical solutions to nonlinear heat and mass transfer equations describing the quasi-stationary solidification of binary melts or solutions with a mushy zone are constructed. The analytical solutions are obtained in parametric form, where the fraction of the solid phase in the mushy layer is a parameter. Specifically, the temperature and impurity concentration distributions in the solid, liquid, and mushy zones of the solidifying system; the rate of the solidification; and the spatial coordinate in the mushy zone as a function of the fraction of the solid phase in it are found. An algebraic equation is derived to determine the fraction of the solid phase at the solid–mushy zone interface. The impurity concentration in the mushy zone is shown to increase with the fraction of the solid phase. Here, the fraction of the solid phase at the crystal–mushy zone interface and its length increase with the temperature gradient in the solid material and the rate of the directional solidification. It is also shown that the fraction of the solid phase in the mushy layer decreases with an increase in the spatial coordinate directed from the solid phase to the liquid. Based on the analytical solutions to mushy zone equations, we have found its permeability and the characteristic dendrite arm spacing (DAS). The relative permeability in the mushy layer is found to increase from a certain known value at the boundary with a solid material to unity at the boundary with a liquid. The DAS in the mushy zone is shown to decrease when the temperature gradient in the solid phase increases, i.e., when the solidification rate increases. An increase in the impurity concentration in the melt leads to a decrease in the DAS in the mushy zone.