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To overcome the weak thermal conduction of the phase change materials (PCM), This investigation aims to study the effect of circular Y-shaped fins added to a two-pipe latent heat-saving unit compared with conventional circular fins. The system is placed in a straight-up orientation, and the PCM is located in the annulus, whereas hot water is moved inside the internal tube to charge the phase change material (PCM) with a phase change point of 35°C. Different independent geometric factors of the fins involving the length of the stem and angle of the tributaries to the horizontal line and the number of Y-shaped fins are analysed. The impact of the working fluid’s Reynolds number and temperature (as the input parameters) are evaluated as a sensitivity analysis to control the output (melting time and rate). The results show that increasing the height and number of the fins and reducing the angles of the tributaries results in higher performance of the system. For the optimal case, increasing the Reynolds number of the working fluid from 500 to 2000 results in 31% reductions in the melting time. Moreover, raising the working fluid’s temperature from 45° to 55° reduces the melting time by ∼44%.