This paper presents a unique solution to the problem of heat transfer intensification in shell-and-tube latent heat thermal energy storage units by means of high conducting fins. We developed a design approach using topology optimization and multi-phase computational fluid dynamics. No assumption is made about the fins layout, which freely evolves along the optimization process resulting in more efficient non-trivial geometries. At each optimization iteration, the fluid-dynamic response in the phase change material is computed by solving the transient Navier-Stokes equations augmented with a phase-change porosity term. Coupling large design freedom to detailed physics modeling allowed studying the effect of convective transport on both design and performance of latent heat thermal storage units. Results indicate that accounting for fluid flow in design optimization studies is crucial for performance. It is shown that melting and solidification can be enhanced remarkably through natural convection by using well engineered fins with specific design features, that could hardly be revealed with alternative design routes. These features make designs optimized for melting fundamentally different from those optimized for solidification.