Augmenting meting and solidification rates of latent heat storage unit (LHSU) is very much essential for its efficient operation. By the effective utilization of natural convection, rate of heat transfer can be enhanced and the conical shell is beneficent in this regard. Using fins further improves the charging and discharging rates. Hence, the current study is focused on analyzing melting and solidification characteristics of a conical shell and tube LHSU along with the effect of fin parameters viz., fin diameter and number of fins. Numerical analysis is chosen for this purpose and the performance is compared via melting/solidification times, energy stored, energy/exergy efficiencies. Initially, the performance of unfinned conical shell is compared with the cylindrical shell without fins and then the effect of fin parameters is presented. For melting process conical shell is found to be superior to cylindrical shell. 34.46% reduction in melting time is noted by using conical shell and rate of energy stored is also higher for conical shell. Increase in fin diameter caused an increase in melting time when 20 number of fins are used, whereas melting time got decreased with the increase in fin diameter when five number of fins are used. Hence, when a greater number of fins are used, lesser diameter is preferred for melting. For discharging process, conical shell took 60% more time than cylindrical shell. Even after using fins, solidification time is not drastically reduced in comparison with cylindrical shell.