Internationally, energy-storage technologies have facilitated the large-scale utilization of renewable energy, reducing reliance on conventional power generation and enhancing energy efficiency. In the pursuit of strengthening the efficiency of phase-change energy-storage systems, the focus lies on further enhancing the efficiency of vertical shell-and-tube energy-storage systems. This study investigates the influence of two different heat-transfer fluid (HTF) injection directions on the melting of phase-change materials (PCM) in a vertical shell-and-tube latent heat storage (LHS) system. The melting behavior of PCM is analyzed under both pure conduction and natural convection conditions. The research findings reveal that during the initial melting stage, both HTF injection methods primarily rely on thermal conduction, resulting in no significant changes in PCM melting. However, in the later stages of natural convection, bottom HTF injection exhibits superior heat-transfer efficiency compared to top injection. Under a constant volume of phase-change material, both pipe length and pipe thickness affect the PCM melting process. As the pipe length increases within the range of 1.6 m to 0.2 m, the PCM melting time also increases. The results show that the melting time of the PCM is reduced by almost 15,000 s when the tube length H exceeds 800 mm, regardless of whether the heat-transfer fluid is injected at the top or bottom. In this paper, we also obtained results that the three composites containing 10% expanded graphite save 5.3%, 10.2%, and 14.3% of melting time, respectively, compared to pure paraffin when H = 200 mm and top injection are considered. For bottom injection, the three composites saved 7.7%, 12.5%, and 17.2% of melting time, respectively. This further emphasizes the more significant effect of priming in improving melting time.