The usage of Phase change material (PCM) for a shell and tube heat exchanger can be considered as a prominent method for the effective storage of thermal energy. The study pertaining to the influence of various design and operating parameters on the melting characteristics of a PCM is necessary for the understanding of thermal hydraulics. This study presents the numerical investigations on melting characteristics of a PCM placed in the annulus of a vertical shell and tube thermal storage unit. Enthalpy-porosity approach is adapted to discretize the governing equations and Boussinesq approximation is used to include the effect of natural convection. The parametric analysis has been carried out in terms of non-dimensional parameters such as Stefan number (0.2-0.6), Rayleigh number (2.04 x 10(5)-2.32 x 10(6)), Reynolds number (500-2000), L/D ratio (1-10), thermal diffusivity ratio (45.29-1500), and tube thickness to diameter ratio (0.036-0.113). Stefan number and Rayleigh number are found to be the most influencing parameters in dictating the total melting time. A correlation is developed to determine the melt fraction evolution. Another correlation is developed to estimate the total melting time which will be useful in selecting various design and operating parameters while designing a shell and tube thermal storage unit.