The development of thermal energy storage systems is a possible solution in the search for reductions in the difference between the global energy supply and demand. In this context, the ability of some materials, the so-called phase change materials (PCMs), to absorb and release large amounts of energy under specific periods and operating conditions has been verified. The applications of these materials are limited due to their low thermal conductivity, and thus, it is necessary to associate them with high-conductivity materials, such as metals, to make the control of energy absorption and release times possible. Bearing this in mind, this paper presents a numerical analysis of the melting process of a PCM into a triplex tube heat exchanger (TTHX) with finned copper tubes, which allowed for the heat transfer between a heating fluid (water) and the phase change material to power a liquid-desiccant air conditioning system. Through the analysis of the temperature fields, liquid fractions, and velocities, as well as the phase transition, it was possible to describe the material charging process; then, the results were compared with experimental data, which are available in the specialized literature, and presented mean errors of less than 10%. The total required time to completely melt the PCM was about 105.5 min with the water being injected into the TTHX at a flow rate of 8.3 L/min and a temperature of 90 °C. It was observed that the latent energy that accumulated during the melting process was 1330 kJ, while the accumulated sensitive energy was 835 kJ. The average heat flux at the internal surface of the inner tube was about 3 times higher than the average heat flux at the outer surface of the TTHX intermediate tube due to the velocity gradients that developed in the internal part of the heat exchanger, and was about 10 times more intense than those observed in the external region of the equipment.