Fractal-Based Thermal Conductivity Prediction Modeling for Closed Mesoporous Polymer Gels

The closed mesoporous polymer gels have garnered significant attention as advanced thermal insulation materials due to their superior lightweight characteristics and excellent thermal management capabilities. To accurately predict their thermal performance, this study develops a novel mathematical model that integrates fractal geometry theory, Kirchhoff's thermal conduction principles, comprehensive Rosseland diffusion approximation, and Mie scattering theory. The conductive thermal conductivity component was formulated based on a diagonal cross fractal structure, while the radiative component was derived considering microscale radiative effects. Model predictions exhibit strong agreement with experimental results from various mesoporous polymer gels, achieving a prediction error of less than 11.2%. Furthermore, a detailed parametric analysis was conducted, elucidating the influences of porosity, cell size, temperature, refractive index, and extinction coefficient. The findings identify a critical cell size range (1-100 µm) and porosity range (0.74-0.97) where minimum thermal conductivity occurs. This proposed modeling approach offers a robust and efficient theoretical tool for designing and optimizing the thermal insulation characteristics of closed mesoporous polymer gels, thereby advancing their application in diverse energy conversion and management systems.