Analytical Fractal Model for Effective Thermal Conductivity of Open-Cell Metallic Foams With Coated Hollow Ligaments

Coating the hollow ligaments of open-cell (fluid-through) metallic foams (MFs) fabricated via the sintering route with a thin layer of graphene can improve the effective thermal conductivity (ETC) of the foam without significantly increasing its flow resistance, potentially important for thermal storage applications. However, the Euclidean geometry cannot accurately depict the random distribution of pores within MFs. To this end, the present study aims to analyze how such thin coatings affect the ETC of MF by employing the fractal theory to depict the random distribution of its open pores. Subsequently, a cubic representative structure is chosen for self-similar pores in the fractal to establish a correlation between the geometric parameters of MF and its fractal dimension. Upon determining the thermal resistance provided a representative structure of the foam having coated hollow ligaments, its ETC is derived as a function of fractal dimension, dimensionless parameters of pore size, porosity, and thermal conductivity of relevant materials (e.g., ligaments, coatings, and filling medium). For validation, existing experimental data are used to compare with analytical predictions, with good agreement achieved. It is demonstrated that the ligament hollowness weakens the thermal conduction of MFs. In addition, when the coating has a thermal conductivity greater than that of ligament, the coating enhances the ability of the foam to conduct heat. Although the ligament hollowness and coating thickness are imperative factors affecting the ETC, the material makes of ligament and coating plays a decisive role in the ETC.