A two-dimensional (2D) numerical study is carried out to investigate the thermal performance of an impure phase-change material (PCM) in an equilateral triangular-shaped double pipe heat exchanger. To tackle the irregular boundaries, a nonorthogonal body-fitted coordinate (BFC) transformation technique is employed. The nondimensional transformed curvilinear conservation equations for mass, momentum, and energy are written in terms of physical variables and they are solved using a control-volume based finite difference method on a staggered grid arrangement. The developed model is then used to study the effects of the inner tube wall temperature, the initial temperature of the solid PCM, and the shape, as well as the position of the inner tube in the annulus on the melting characteristics, and cumulatively stored energy. Various quantities such as average Nusselt numbers over the inner tube surface, the total and complete melt fractions, and the latent and total stored energies all as a function of the melting time are reported. A correlation for the average Nusselt number on the inner tube wall is also provided. The numerical results show that the shape and the placement of the inner tube are crucial for the efficient design of a latent heat thermal energy storage (LHTES) system. The storage of energy is greatly influenced by the change of the inner tube wall temperature compared to the change of initial solid PCM temperature.