A transient 2-D numerical study is carried out on the melting (energy storage) of an impure phase change material (paraffin wax) embedded in the annular gap formed by an outer equilateral pipe and an inner tube carrying the heat transfer fluid. A non-orthogonal curvilinear coordinate method is used to account correctly for the irregular boundaries of the annular domain. A set of parametric studies is performed by varying the bulk temperature and mass flow rate of the heat transfer fluid (HTF). Besides, numerical simulations are also made to ascertain the effects of the various cross-sectional shapes and positions of the inner tube as well as the initial state of the solid PCM on the melting process. The temporal velocity and temperature profiles are shown in addition to the cumulatively stored energy and total melt fraction as a function of the melting time. The results indicate that the bulk temperature of the HTF imposes a significant effect on the melting process compared to its mass flow rate. The lower eccentric position of the inner circular tube provides the maximum energy storage capacity compared to its concentric position. The concentrically placed circular inner tube provides the maximum energy storage in the annulus relative to the other two cross-sectional inner tubes namely prolate- and oblate-shaped elliptical tubes.