## 3 Results and Discussion
### Fundamental Combustion Characteristics for the VAM and PSM  
Upon validating the PSM numerical model, a comparative study was carried out to examine the temperature distribution differences between the VAM and PSM. In order to ensure comparability, the same key parameters (\(U_{0}\), \(\Phi\), \(k_{s}\), \(k_{w}\), \(\varepsilon\), and \(d_{s}\)) were used for both calculation models. Figure 4 compares the temperature contours and OH concentration of the planar porous micro-combustor. It can be found in Figure 4a that the VAM predicts a parabolic flame front and the highest temperature region at the centerline, while the flame profile obtained by the PSM is irregular in three dimensions as the flame front is distributed along the surface of alumina spheres. It should be noted that the PSM predicts a more scattered high temperature distribution region and a lower peak flame temperature. To further understand the temperature distribution and flame characteristics of the PSM, three planes (\(x\) = 0, 0.25, and 0.5 mm) were selected for this test component, as shown in Figure 4b,c. The flame fronts are distributed along the surface of the alumina spheres for the PSM, forming tiny flames in the pore areas. A higher OH concentration and flame temperature are found with a more concentrated flame distribution at \(x\) = 0, as shown in Figure 4c. The flame zone becomes scattered when x increases from 0 to 0.25 mm. At the same time, the OH concentration and flame temperature greatly reduce. Furthermore, a minimum flame temperature and minimum temperature difference between the fluid and porous media is predicted at \(x\) = 0.5 mm. Figure 5 shows the flame temperatures, peak wall temperature and peak solid matrix temperature obtained by the VAM and PSM variation with \(\Phi\) in the porous micro-combustor partially filled with alumina spheres with \(d_{s}\) of 1 mm, \(\varepsilon\) of 0.48 and \(k_{s}\) of 5 W/m-K. It is shown that under the same inlet flow conditions (\(U_{0}\) and \(\Phi\)), the porous combustor using PSM gives a much lower flame temperature and peak solid matrix temperature and a higher peak wall temperature than the VAM counterpart. There are basically three  
Figure 4: Temperature distribution and OH concentration obtained by the VAM and PSM, with \(U_{0}\) = 2 m/s, \(\Phi\) = 0.65, and \(d_{s}\) = 1.0 mm. (**a**) Local temperature distribution obtained by the VAM and PSM; (**b**) temperature distribution obtained by the PSM; (**c**) OH concentration obtained by the PSM. causes for this. First, in the VAM combustor, the heat recirculation through solid-wall heat exchange is ignored, resulting in the heat flux error. The solid-to-wall heat exchange (\(q_{s-w}\)) between the solid matrix and wall in the PSM combustor leads to a higher wall temperature and a lower porous temperature. Second, different from the flame central region of the VAM combustor located in along the centerline, the PSM combustor forecasts a primary flame zone at the pore position, which is closer to the wall, and consequently poses a positive effect on the wall temperature. Third, the volumetric heat transfer between the gas mixture and porous media exhibits differences because of the different temperature distribution and surface heat transfer areas. In other words, based on the verification of the PSM, there are errors in the volumetric heat transfer coefficient between the gas mixture and alumina spheres calculated by the empirical correlation in the VAM. Another obvious pattern shown in Figure 5 is that the difference between two calculation models in terms of flame temperatures, peak wall temperature and peak solid matrix temperature decreases with the increase in \(\Phi\). In Figure 5, an equivalence ratio range is obtained in which the flame can be stabilized within the porous medium. According to a previous study [39], critical flame stability limits, \(\Phi_{3}\) and \(\Phi_{4}\) marked by the circle and square in Figure 5, are key features of planar micro-combustor designs.