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With the gradual increase in energy demand in global industrialization, the energy crisis has become an urgent problem. Due to high heat storage density, small volume change, and nearly constant transition temperature, phase change materials (PCMs) provide a promising method to store thermal energy. In this work, we designed and fabricated three kinds of porous metal structures with hexagonal, rectangular, and circular pores and explored the phase change process of PCMs within them. A two-dimensional numerical model was established to investigate the heat transfer process of PCMs within different shapes of porous metal structures and analyze the influence of heat source location on the thermal performance of the thermal storage units. Visualization experiments were also carried out to reveal the melting process of PCMs within different porous metal structures by a digital camera. The results show that paraffin in a porous metal structure with hexagonal pores has the fastest melting rate, while that in a porous metal structure with circular pores has the slowest melting rate. Under the bottom heating mode, the melting time of the paraffin in porous metal structures with hexagonal pores is shortened by 18.6% compared to that in porous metal structures with circular pores. Under the left heating mode, the corresponding melting time is shortened by 16.7%. These findings in this work will offer an effective method to design and optimize the structure of porous metal and improve the thermal properties of PCMs.