Analysis of self-similar solutions for the ice and hydrophilic liquid interaction problems

Abstract

One-dimensional self-similar solutions allows us to establish the fundamental analytical properties for processes. Unlike the well-known Stefan problem concerning the freezing of pure water in contact with ice, the phase transition temperature is not constant and depends on the concentration of the hydrophilic liquid, which is described by a diffusion equation. We use a linear approximation to relate the equilibrium temperature and concentration at the phase boundary. The temperature in the ice and in the liquid is described by the heat conduction equations. Heat and mass transfer occur at the phase boundary when water from the hydrophilic liquid freezes onto the ice surface, increasing its concentration, or conversely, ice melting at the boundary occurs, decreasing the concentration of the hydrophilic liquid. This is determined by the input parameters of the problem. The paper analyzes such solutions for the case of ice - hydrophilic liquids interactions, for example, seawater and an ethanol solution that take place in problems of the interaction of ice shelves with the ocean and in the thermal drilling of boreholes in glaciers. The study also investigates peculiarities of choice of the thermal diffusivity of the liquid in applied problems.