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Energy storage systems have become increasingly important to reduce environmental impact and to solve the mismatch between temporal and methodical energy demanded and the energy produced. The purpose of this paper is to describe the results obtained from the numerical simulation of the Latent Heat Thermal Energy Storage System (LHTESS) with a geometry of a vertical shell and tube, made of aluminum metal foam filled with a paraffin wax phase change material (PCM). In the numerical simulations the inner surface of the hollow cylinder is assumed to be at a constant temperature above the PCM melting temperature, the external surface to lose heat toward the outside external ambient, and the top and bottom surfaces are considered adiabatic. The phase change process is modeled with the enthalpy-porosity theory, while the Darcy-Forchheimer model and the Local Thermal Non-Equilibrium (LTNE) assumption are adopted to analyze the aluminum foam-filled by the paraffin. The results of numerical simulations, concerning LHTESS charging phase, are reported as a function of time and are compared in terms of melting time, average temperature, and energy storage rate. The presence of the metal foam is known to significantly improve heat transfer in the LHTESS, and the obtained results show that it is necessary to consider systems with an external heat loss to simulate real operating conditions and understand how this different heat transfer coefficient affects system storage.