## 1 Introduction
In other words, the VAM could not predict the overall pore-level simulation results, including temperature and species distribution. Additionally, solid radiative heat transfer was ignored because of the computational cost. [PERSON] et al. [26] extended the above study by including solid-solid radiative heat recirculation. They developed a 2D pore-scale model of two simple porous structures to study the effect of multidimensionality on flames, and determined the limitations of one-dimensional models. In a PSM study by [PERSON] et al. [27], pore-scale filtration combustion within discrete square cylinders was numerically simulated to investigate the effects of turbulence and multidimensionality on flame characteristics. They revealed that turbulence in the preheating zone could increase the diffusion before the flame front, but it was not enough to affect key flame characteristics such as temperature distribution and maximum flame velocity of the porous media. [PERSON] et al. [28] experimentally and numerically studied the effects of pressure and equivalence ratio on flame stability within the pores of porous inert media, employing a 1D VAM and 3D PSM on real geometries of sponge-like structures. They found that the PSM could be used to predict the average temperature distribution along the space coordinates of the combustors. The VAM demonstrated a steeper slope, indicating lower flame thickness than in the experiments. [PERSON] et al. [29] simplified the complex construction of a discrete-modeled porous medium to a regular periodic structure composed of three elements: solid particles, gas flow zone and gas stagnation zone. A numerical study on the fundamental combustion process of premixed methane-air combustion was carried out based on the simplified approach. It showed that the velocity of flame propagation increased considerably near the gas stagnation zone when elementary cells were larger than 3 mm. As mentioned above, notable advances have been achieved in filtration combustion at the pore scale. However, heat transfer through the combustor wall is ignored in recent PSM numerical simulations of porous combustors [21; 24; 25; 26; 27; 28; 29]. In micro-/meso-scale combustors, heat recirculation through the combustor wall is crucial to self-sustaining and stabilizing flames, confirmed by a series of free flame cases and VAM numerical simulations [30; 31; 32; 33; 34; 35; 36]. In [PERSON] and [PERSON] paper [30], computational fluid dynamics simulations of methane and propane combustion were performed to study the effects of key parameters on combustion characteristics and flame stability. They found that at the micro-/meso-scale, the walls were responsible for the majority of the external heat losses and upstream heat transfer. Therefore, the wall thermal conductivity was the primary parameter that determined the stabilization and self-sustainability of the flames. [PERSON] and [PERSON] [31] investigated the effects of a novel thermally orthotropic material on flame stability based on a full understanding of the significance of the combustor wall. [PERSON] and [PERSON] [32] affirmed that the heat recirculation via thermal conduction in the combustor wall was vital for determining the flame speed. [PERSON] et al. [33] considered the solid matrix and combustor wall to be the pathways for heat recirculation in the porous micro-combustor. The thermal conduction in the combustor wall was significant for preheating the unburned gas mixture. This study develops a PSM of a structurally arranged porous media made of discrete alumina spheres to produce a quantitatively accurate porous micro-combustor. To understand reaction and heat transfer in porous micro-combustors more accurately, this study considers the PSM model and the heat transfer between the gas mixture and combustor wall. The remainder of this paper discusses the following:  
1. The combustion of H\({}_{2}\)/air in a regularly packed bed of alumina spheres is numerically simulated using a 3D PSM.