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• Topology optimization for diffusion–reaction process in porous media is conducted. • The lattice Boltzmann method and the level-set method are coupled. • Delicately designed hierarchical porous structure are obtained. • Total reaction rate and material utilization ratio are evaluated. • Topology optimized structures overwhelm random hierarchical porous structure. Hierarchical porous media possess both high reactive sites and low transport resistance, yet the limit of the total reaction rate is not clear. In the present study, topology optimization (TO) of three-dimensional porous structures where diffusion–reaction processes occur is conducted by the lattice Boltzmann method and the level-set method to optimize the material distributions and to obtain the highest total reaction rate. Hierarchical porous structures are generated, and more mesoscopic pores are incorporated as the process gradually becomes diffusion-limited. The total reaction rate and utilization ratio of the microscopic materials are significantly improved in the TO structures. Compared with spherical structures, the TO results of spherical shell structures can significantly improve the material utilization ratio at the cost of slightly reduced total reaction rate. Finally, the TO structures are compared with random hierarchical structure, and remarkable increase of the total reaction rate is achieved. [ABSTRACT FROM AUTHOR]

• 3D multi-scale topology optimization of natural convection is studied. • Different microstructures of truss type and TPMS type are considered. • Optimized porous heat sinks better than conventional fins are obtained. • Heat dissipation performance of optimized designs is verified by FEM. • Effects of cell geometry, cell porosity and volume fraction are studied. This work presents a novel multi-scale topology optimization (TO) method for designing the three-dimensional porous heat sink to enhance the natural convection heat transfer. The pore-scale lattice Boltzmann model is applied to predict the effective transport properties while the level-set-based TO is performed at the representative elementary volume (REV)-scale. Six predetermined unit cells of distinct two types are considered, and the pore-scale study shows that the truss-type cells have higher permeability while lower effective thermal conductivity compared with the TMPS-type cells. The REV-scale TO generates the optimized designs when Grashof number (Gr) ranges from 2.4 × 102 to 1.2 × 105, which can enhance the heat transfer by 37.7 %∼49.1 % compared with the reference designs, and their performance is validated a posteriori with the relative error lower than 3 %. Parametric study finds that the optimized porous heat sinks made up of Primitive cell perform the best at all Gr values due to the higher effective thermal conductivity. Either decreasing the underlying porosity or increasing the filling volume of the porous material brings a better heat transfer performance, while the former approach is more effective. This work provides useful guidance to the choice of the underlying microstructure and to the topological design of porous heat sink. [ABSTRACT FROM AUTHOR]