Phase-change materials (PCMs) can store large amounts of heat without significant change of their temperature during the phase-change process. This effect may be utilized in thermal energy storage, especially for solar-thermal power plants. In order to enhance the rate of heat transfer into PCMs, one of the most common methods is the use of fins which increase the heat transfer area that is in contact with the PCM. The present work deals with a latent heat thermal storage device that uses a finned tube with an array of radial fins. A heat transfer fluid (HTF) flows through the tube and heat is conducted from the tube to the radial fins that are in contact with the bulk of the PCM inside a cylindrical shell. The thermal storage charging/discharging process is driven by a hot/cold HTF inside the tube that causes the PCM to melt/solidify. The main objective of the present work is to demonstrate that close-contact melting (CCM) can affect the storage unit performance. Accordingly, two different types of experiments are conducted: with the shell exposed to ambient air and with the shell submerged into a heated water bath. The latter is done to separate the PCM from the shell by a thin molten layer, thus enabling the solid bulk to sink. The effect of the solid sinking and close-contact melting on the fins is explored. It is found that close-contact melting shortens the melting time drastically. Accordingly, two types of models are used to predict the melting rate: numerical CFD model and analytical/numerical close-contact melting model. The CFD model takes into account convection in the melt and the PCM property dependence on temperature and phase. The analytical/numerical CCM model is developed under several simplifying assumptions. Good agreement is found between the predictions and corresponding experimental results.