Abstract: The present study explores numerically the process of melting of a phase-change material (PCM) in a heat storage unit with internal fins open to air at its top. Heat is transferred to the unit through its horizontal base, to which vertical fins made of aluminum are attached. The phase-change material is stored between the fins. Its properties used in the simulations, including the melting temperature of 23–25 °C, latent and sensible specific heat, thermal conductivity and density in solid and liquid states, are based on a commercially available paraffin wax. A detailed parametric investigation is performed for melting in a relatively small system, 5–10 mm high, where the fin thickness varies from 0.15 mm to 1.2 mm, and the thickness of the PCM layers between the fins varies from 0.5 mm to 4 mm. The ratio of the PCM layer to fin thickness is held constant. The temperature of the base varies from 6 °C to 24 °C above the mean melting temperature of the PCM. Transient three- and two-dimensional simulations are performed using the Fluent 6.0 software, yielding temperature evolution in the fins and the PCM. The computational results show how the transient phase-change process, expressed in terms of the volume melt fraction of the PCM, depends on the thermal and geometrical parameters of the system, which relate to the temperature difference between the base and the mean melting temperature, and to the thickness and height of the fins. In search for generalization, dimensional analysis of the results is performed and presented as the Nusselt numbers and melt fractions vs. the Fourier and Stefan numbers and fin parameters. In some cases, the effect of Rayleigh number is significant and demonstrated.