The effective thermal conductivity (ETC) of roughened porous media (RPM) is of interest in a number of applications of heat transfer. In this work, a fractal analytical model for the ETC of RPM with microscale effect is proposed. The proposed fractal model is expressed in terms of relative roughness, the molecular mean free path, porosity, fractal dimensions (pore area fractal dimension and tortuosity fractal dimension), maximum pore diameter, capillary straight length, and the thermal conductivity of the solid matrix and gas. It is observed that the dimensionless ETC of RPM decreases with increasing relative roughness, pore area fractal dimension and tortuosity fractal dimension. Besides, it is found that the dimensionless ETC of RPM increases with thermal conductivity ratio of gas phase over solid phase. In addition, it is found that the dimensionless ETC of RPM is slightly dependent on the relative roughness and tortuosity fractal dimension when kg/ks < 1. The determined dimensionless ETC of RPM is in good agreement with experimental data and existing models reported in the literature. With the proposed fractal model, the physical mechanisms of heat transport through RPM with microscale effect are better elucidated. Every parameter in the fractal analytical model has clear physical meaning, with no empirical constant.

中文翻译：

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具有微尺度效应的粗化多孔介质有效热导率的分形模型

粗糙多孔介质 (RPM) 的有效热导率 (ETC) 在传热的许多应用中都受到关注。在这项工作中，提出了具有微尺度效应的RPM ETC的分形分析模型。所提出的分形模型用相对粗糙度、分子平均自由程、孔隙率、分形维数（孔面积分形维数和曲折分形维数）、最大孔径、毛细管直线长度和固体基质的热导率和气体。可以观察到，RPM的无量纲ETC随着相对粗糙度、孔面积分形维数和曲折分形维数的增加而减小。此外，发现RPM的无量纲ETC随着气相与固相的热导率比的增加而增加。此外，ķG/ķs < 1. 确定的 RPM 的无量纲 ETC 与实验数据和文献中报道的现有模型非常吻合。使用所提出的分形模型，可以更好地阐明通过具有微尺度效应的 RPM 进行热传输的物理机制。分形分析模型中的每一个参数都有明确的物理意义，没有经验常数。