Parametric study and approximation of the exact analytical solution of the Stefan problem in a finite PCM layer in a steady periodic regime

In this paper a parametric study and an approximation of the exact analytical solution of the Stefan problem in steady periodic regime conditions (Mazzeo et al., 2015) is presented. The physical model describes the thermal behaviour of a PCM (phase change material) layer subject to phase transition, and the considered thermal regime ensures continuous cycles of phase changes under the action of the periodic boundary conditions. The exact analytical solution determines, through implicit transcendental equation with complex thermal parameters and unknowns, the bi-phase interface position, the thermal field and the sensible and latent stored energy in the layer. The dimensionless solution is a function of the Fourier and Stefan numbers calculated in the two phases, and of the dimensionless bi-phase interface position corresponding to the steady regime. For the parametric study, the thermophysical properties of the most commonly used PCMs and the variation range of attenuation and time lag between the temperature loadings operating on the internal and external surfaces were considered. This study has allowed for the identification of the thermal parameters that mainly influence the dynamic thermal behaviour of the PCM layer and the mathematical structure of the frequency response of the layer. Since an analytical expression in an explicit form of the position of the bi-phase interface in the layer is not available, an approximate analytical solution was obtained, which makes the bi-phase interface position depends on the product between the Fourier number and the Stefan number calculated in the two phases. The limits of validity of such a solution were determined evaluating the relative error, which is committed in the determination of the amplitude and of the argument of the oscillating component of the bi-phase interface position. Finally, the fields of variation of the thermal parameters that ensure a relative error value lower than 3% and the corresponding values of the maximum errors of the amplitude or the argument are determined. In PCM layer thermal analysis it is useful to have an expression in an explicit form of the oscillating component of the bi-phase interface position to obtain the mathematical expression of the temperature and heat flux field as a function of only dimensionless thermal parameters and boundary loadings.