Lumped parameter model for one-dimensional melting in a slab with volumetric heat generation

In this study, a lumped parameter model was developed for one-dimensional heat conduction with melting of a phase change material (PCM) slab with volumetric heat generation. Two types of boundary conditions were considered: (a) adiabatic condition at the left side and isothermal condition at the right side and (b) isothermal condition at the left side and convective condition at the right side. The lumped model was obtained through two-point Hermite approximations for integrals. The two-side corrected trapezoidal rule (H1,1 approximation) was employed in the averaged temperature integrals for both the liquid and solid phases during melting process of the slab, and the plain trapezoidal rule (H0,0 approximation) was used to estimate the heat fluxes. For the melting problem with the boundary conditions (a), the lumped model results were verified by the analytical solution of melting in half-plane without internal heat source. Case study was performed to investigate the effect of the volumetric energy generation and the Stefan number on the instantaneous interface position. For the melting problem with the boundary conditions (b), the lumped model results were verified by the enthalpy method solution. Transient heat condition of the PCM slab was analyzed to illustrate the applicability of the proposed lumped model, with respect to different values of the volumetric energy generation, the Stefan number, the Biot number, the liquid-to-solid thermal conductivity ratio and the boundary temperature at the left side. Excellent agreement with available analytical or numerical solutions was achieved.