<p>This article develops a numerical model to analyze freezing in a rectangular cold storage unit, integrating the Galerkin method with adaptive mesh refinement. To expedite the freezing and improve thermal performance, two strategies were introduced: (1) the dispersion of high-conductivity ternary nanoparticles (Al₂O₃–TiO₂–Ag) into water, and (2) the utilizing metal foam within the storage domain. These methods are specifically chosen to enhance the conduction-dominated heat transfer behavior typically observed during the solidification process. The governing equations were simplified by neglecting momentum terms, as the fluid motion is minimal during phase change, leading to a coupled model of energy conservation and solid fraction evolution. Simulation results reveal that the inclusion of metal foam decreases the freezing time by 76%. Besides, the dispersion of ternary nanoparticles boosts the freezing rate by 12.55%. When both enhancement methods are combined, the total freezing time is decreased by nearly 79%, demonstrating the synergistic effect of the dual approach.</p>