The enhancement of heat transfer in a thermal storage system consisting of a cylindrical vertical tube with internal longitudinal fin arrangement is discussed in this paper. This configuration that forms a V-shaped enclosure for the phase change material(PCM) gives maximum benefit to the fin arrangement. A theoretical model that also accounts for the circumferential heat how through the tube wall is developed using enthalpy formulation and is employed in conjunction with the fully implicit finite difference method to solve the solidification in the convectively cooled vertical tube. A generalised enthalpy-temperature (H-T) relationship, developed by Date, is suitable for constant phase change temperature and this is modified in the present work to accommodate materials having either constant or a range of phase change temperatures. The numerical model is validated with experimental data. The instantaneous surface heat flux and the solidified fraction for different number of fins and tube radii, that are of interest for the design of latent heat thermal storage system consisting of an array of vertical cylindrical tubes containing PCM inside it, are also discussed in this article.