The present study deals with melting in a horizontal double-pipe concentric storage unit with a longitudinally finned inner tube. This geometry, applicable for thermal energy storage, has been extensively studied in the past. However, close-contact melting (CCM), which may significantly increase the melting rate, has not been explored for this sort of systems. In the present experiments, a laboratory-scale three-fin unit is transparent and thus the processes inside it are observed and recorded. Close-contact melting is achieved by supplying heat to the outer shell of the unit: the solid phase is detached from the shell and moves in the liquid phase. In the upper part of the unit, the solid phase translates vertically and melts on the inclined fin surfaces. In the lower parts of the unit, the solid phase approaches the vertical fin by rotation, while its outward surface slides on the shell. It is shown that close-contact melting shortens the melting time by a factor of 2.5 in that specific laboratory-scale device. A novel theoretical model includes gravity-induced rotational motion of the solid, primary melting on a vertical fin with non-uniform temperature distribution, secondary melting at the shell, and frictional resistance at the latter. The model is validated using the experimental results, and then used for a parametric investigation and dimensional analysis. It is revealed that the melt fraction depends on the Fourier and Stefan numbers combined as FoSte(3/4), whereas the Nusselt number depends also on an additional group, Ste(1/4). Further generalization is achieved when different angles between the fins are considered. The present paper presents a further proof of the special role that close-contact melting (CCM) can play in thermal energy storage units and finned systems in general. It is demonstrated that the fins, when properly designed and oriented, induce CCM in their vicinity, leading to a very significant increase in the melting rate. (C) 2015 Elsevier Ltd. All rights reserved.