A conduction model is developed to describe the phase change between the plates of a thermal storage system. The diffusion equation and the associated boundary, initial, and interface conditions are approximated numerically by finite differences and implicit approach with variable time-step. The developed computational code is validated against data and good agreement was found. It is found that the reduction of the surface temperature of the cold plate increases the interface advance rate and reduces the full solidification time. Opposite effects are found due to the increase of the spacing between plates. Further, fractions of Al2O3 nanoparticles are mixed with the phase change material (PCM) to enhance the thermal conductivity of the PCM. For 7% volumetric fraction of Al2O3, the full solidification time and latent heat values decreased by 25.5% and 4.5%, respectively.