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Semi-analytical solutions to the classical two phase Stefan problem are proposed. Time dependent solutions to the one-dimensional liquid-solid phase transition in a PCM wallboard subjected to isothermal and periodic Dirichlet boundary conditions are obtained. Transient and steady state solutions are found in finite size systems, and the semi-analytical solutions are validated through the asymptotic time limit behaviour of the phase transition. In this work, complex Fourier methods are proposed to find the solutions in the transient and steady state periodic regimes. Semi-analytical solutions based on the heat balance integral method (HBIM) are used to verify the consistency of the proposed method. The Fourier method can be pictured as a generalization of the phasors based method recently introduced by other authors. The proposed method incorporates a complete set of complex functions, which allows finding the transient and steady state response of the system. Finally, solutions for the time dependent interface position, liquid and solid temperature distributions and the thermal energy penetrating through the PCM wallboard, are shown. The solutions from the proposed method are found to be consistent when compared to the semi-analytical solutions estimated through the HBIM.

Complex Fourier series expansion for the liquid-solid phase transition in PCM layers: transient and steady state periodic regimes