Mobilized thermal energy storage units have a vital role in reducing energy consumption in buildings by enabling industrial waste heat to be used in buildings. High conductive fins can enhance the heat transfer performance of mobilized thermal energy storage tanks which suffer significantly from the low thermal conductivity of phase change materials. On the other hand, investment costs of the mobilized thermal energy storage tanks need to be decreased to compete with fossil fuel-driven systems in buildings. The present study numerically investigates the effect of innovative fin structures on the melting performance for fixed fin material volume to disable cost increase. Two-dimensional models with phase change were simulated for shell-and-tube heat exchangers. The shell geometry was designed sufficiently large to observe the melting growth of phase change material independent from shell walls within the given charging time. Straight and Branched type fin structures with the fin numbers of Nfin=2, 4, and 6 were simulated to uncover the effect of shape and length scale of fins on natural convection-driven melting. It was found that Straight fin type is more suited than Branched fins as they do not show significant melting enhancement with increased complexity and cost. The fin structures in all cases performed better when located at the top of the heat transfer fluid tube, even though the literature considers that top-located fins inhibit natural convection circulations. Varying the number of fins from (2-fin) to (4-fin) causes 15.8% increase in melting ratio, but further increase in the fin number (6-fin) reduces melting ratio below the (4-fin) case. Within (4-fin) structures located at the top, using distinct fin lengths yields melting ratio to increase 28.1%. Overall, the results show that heat transfer could be improved by varying the fin structure without increasing total fin volume and cost. The melting region growth shape with optimized fin structure forms the basis for the multitube arrangement of mobilized thermal energy storage units to enhance heat transfer performance with low cost.