This study deals with heat transfer from pin-fin aluminum heat sinks to a phase-change material (PCM) which fills the inter-fin space. The sinks have a horizontal base and accordingly their fins are vertical. The sink base dimensions are 100 mm by 100 mm, with fin height of 10 mm, 20 mm or 30 mm, and cross section of 4 mmx4 mm. The number of fins varies, e.g. 49, 64, 81, etc. The applied power is between 50 W to 250 W, corresponding to the heat fluxes of 5-25 kW/m2. The present paper reports mostly numerical results, but the numerical model is validated using the findings from an ongoing experimental investigation, in which a commercially available paraffin wax RT-35 is used as the PCM, with the melting temperature of about 35 °C. The simulations reflect the material properties, geometry, and other features of the experimental set-up, including heating with an electrical foil heater. Accordingly, the base temperature serves as the dependent parameter. Numerical simulations, performed using the Fluent 6.2 software, serve to obtain detailed melting patterns and explain the effect of fin size and number on sink performance.