The present numerical study is conducted to analyze melting process within a rectangular enclosure filled by phase change material (PCM) vertically heated from one side. The right hot wall and the left cold wall are maintained at temperatures, Th=38.3 °C and Tc =28.3 °C, respectively, and it was filled by solid PCM Gallium initially at temperature Tc. The horizontal walls are insulated. A transient numerical model is developed to study the heat transfer and melting behaviours, and the natural convection is accounted. To enhance the heat transfer and the melting process of the PCM, fins with a rectangular and triangular shape are proposed. Moreover, the effects of both thermophysical properties and fins integration on the flow structure and heat transfer characteristics are investigated in detail. The melt fraction contours with the natural convection driven flow are performed and compared, as well as the temperature distributions for a Rayleigh number of around Ra= 106. It is found that the rate of the melting increases with the elevation in the values of specific heat capacity Cp as well as the thermal conductivity λ of the PCM Gallium. The results show also that the rectangular fin accelerates the PCM melting faster than the triangular fin’s shape thanks to the increased exchange area.