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Latent heat storage technology plays an important role in the effective utilization of clean energy such as solar energy in building heating, but the low thermal conductivity of heat storage medium (phase change material) affects its large-scale application. As a new heat storage enhancement technology, rotation mechanism has a good application prospect. In this paper, the solidification performance of a triplex-tube latent heat thermal energy storage unit at constant speed (0.5 rpm) is studied numerically. Different optimization design methods (Taguchi method and response surface method) are used for deep analysis. The influences of fin position, number, and material on solidification properties are explored by the Taguchi method. Then, the unit structure (fin angle, fin length, and fin width) is optimized by the response surface method. Compared with the original structure, the average heat release rate of 8 copper fins with all outer tubes is increased by 108.93%, and the solidification time is reduced by 52.06%. The optimal structure can further shorten the solidification time by 29.14% and increase the average heat release rate by 40.5%. Additionally, the study of wall temperature shows that increasing temperature difference makes solidification speed faster and heat energy release faster. This effect effectively eliminates the adverse effects of slow solidification during the later stages of the process on the system.