The melting heat transfer of nano-enhanced phase change materials was addressed in a thermal energy storage unit. A heated U-shape tube was placed in a cylindrical shell. The cross-section of the tube is a petal-shape, which can have different amplitudes and wave numbers. The shell is filled with capric acid with a fusion temperature of 32 °C. The copper (Cu)/graphene oxide (GO) type nanoparticles were added to capric acid to improve its heat transfer properties. The enthalpy-porosity approach was used to model the phase change heat transfer in the presence of natural convection heat transfer effects. A novel mesh adaptation method was used to track the phase change melting front and produce high-quality mesh at the phase change region. The impacts of the volume fraction of nanoparticles, the amplitude and number of petals, the distance between tubes, and the angle of tube placements were investigated on the thermal energy rate and melting-time in the thermal energy storage unit. An average charging power can be raised by up to 45% by using petal shape tubes compared to a plain tube. The nanoadditives could improve the heat transfer by 7% for Cu and 11% for GO nanoparticles compared to the pure phase change material.