Abstract
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.
Highlights
The study of phase change processes through the analysis of moving boundary problems has a wide variety of applications
These applications rely on the use of phase change materials (PCMs) in concentrating solar power (CSP) plants for thermoelectric generation as described by Gil and Mathur [1-2]
The solutions estimated through the proposed method, which is based on a complex Fourier series expansion of the temperature profiles, are consistent with those obtained with the previously established heat balance integral method (HBIM)
Summary
The study of phase change processes through the analysis of moving boundary problems has a wide variety of applications. Et al [8] have studied salt hydrates as PCMs for thermal energy storage and thermal shielding applications due to the high latent heat values and low to mild liquid-solid saturation temperatures at standard pressure. Savovic [17] and Mazzeo [18-19] have used front tracking methods to determine the thermal performance of hydrate salts and paraffins during isobaric phase change processes in wallboards subjected to periodic boundary conditions. The base real functions are given by ffff1324nnnn((((xxxx,,,,tttt))))====eeee−−γγγγ000δδ0nnδδnnxxxxcscsoioinsns((((γγγγ00δ00δnδnδnnxxxx++−−2222γγγγ00220022nnnn2222tttt)))) These functions can be applied to layers subjected to time-dependent boundary conditions. Using the results previously discussed, the equation of motion for the interface takes the following form dξ ρjLf dt k2π L−ξ nDsn(−1)ne−δ(s2Lπ−2ξn)22t n=1 This equation is solved numerically with the RK4 method; the proposed method provides semi-analytical solutions to the dynamics of the phase transition.
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