Eigenfunction-based solution for solid-liquid phase change heat transfer problems with time-dependent boundary conditions

Abstract

Solid-liquid phase change heat transfer problems appear in a broad variety of engineering systems, such as thermal management and latent energy storage. Boundary conditions in such systems often vary with time, for example, due to sinusoidal heating, or step changes in externally applied temperature or heat flux. Unfortunately, phase change problems with time-dependent boundary conditions do not generally admit an exact solution, and therefore, approximate analytical solutions are of much interest. This paper presents an eigenfunction expansion based technique for solving one-dimensional phase change heat transfer problems with time-dependent temperature or heat flux boundary conditions. The temperature field is expressed as a series solution of appropriate eigenfunctions, and with time-dependent coefficients, which are determined by deriving and solving an ordinary differential equation that accounts for the time-dependent nature of the boundary condition. Solutions for Cartesian, cylindrical and spherical problems are derived. Results are found to be in excellent agreement with past work and numerical simulations. The effect of Stefan problem on the nature of phase change propagation is studied. Two practical problems involving sinusoidal and step function boundary conditions are also solved and analyzed in detail. In addition to improving the fundamental understanding of phase change heat transfer, this work may also contribute towards design and optimization of practical engineering problems involving phase change heat transfer.