Porosity distribution optimization of insulation materials by the variational method

Heat transfer performance of thermal insulation materials is determined by many independent parameters, it is hard to optimize the internal structures of such materials through constructing a mathematical relation between the independent parameters and the optimization objective. In this paper, we theoretically analyze such factors as temperature, pressure and porosity influencing the heat transfer performance of porous materials, establish a mathematical model by using the entransy theory with such constraints as fixed mass and thickness of the porous material, and finally apply the variational principle to derive the governing equations for optimizing the porosity/solid fraction distribution in porous materials. Meanwhile, a 1-D and a 2-D physical model are taken as examples to show the applications. When the surface temperatures as well as the total mass and thickness of the high-porous structure are given, we get the optimal porosity distribution through solving the newly derived governing equations. The results show that both heat flux and the effective thermal conductivity of the optimized structure is the minimal. That is, the thermal insulation performance is optimal.