Self-similar solution of the problem on dissolution of ice by hydrophilic liquid

Abstract

The solution of the one-dimensional problem on interaction of ice and hydrophilic liquid is studied. Unlike the well-known Stefan problem of freezing of pure water upon contact with ice, the phase  transition temperature is not constant and depends on the concentration of hydrophilic liquid. The  concentration obeys the diffusion equation. We use a linear equation for the equilibrium temperature  and concentration at the contact boundary. The temperature in ice and liquid is described by the  heat conduction equations. Heat and mass transfer occurs at the phase boundary and water from the  hydrophilic liquid freezes onto the surface of the ice, its concentration increases, or, conversely, ice  dissolves at the boundary and the concentration of the hydrophilic liquid decreases. It is determined  by the input parameters of the problem. By introducing a self-similar variable, partial differential  equations are reduced to ordinary differential equations. Analytical solutions for the temperature in ice and liquid, and the concentration of the hydrophilic liquid are obtained. A transcendental equation  determines the self-similar parameter, and is solved numerically.