## 4 Conclusions  
The purpose of this study was to develop a PSM in a structurally arranged porous medium made of discrete alumina spheres to produce a quantitatively accurate porous micro-combustor. A quantification study was undertaken to examine and systematically compare the heat recirculation, including preheating and heat loss, in the porous micro-combustor using the VAM and PSM. A pore-scale numerical model was developed in a structurally arranged porous medium made of discrete alumina spheres to better understand the coupling between the porous media and combustor wall. Experimental and numerical studies showed that the porous media allowed more effective heat recirculation through the solid matrix and combustor walls in the micro-/meso-scale. In addition, the combustion characteristics and flame stability limits of H\({}_{2}\)/air premixed combustion were examined. A parametric study is subsequently carried out to examine the effects of solid matrix thermal conductivity (\(k_{s}\)) on the PSM and VAM, and then the results were analyzed briefly. Under the conditions (\(U_{0}=2\) m/s, \(\Phi=0.65\), \(\varepsilon=0.48\), \(d_{s}=1\) mm, \(k_{\text{up}}=20\) W/m-K and \(k_{s}=1\)-\(100\) W/m-K) considered in the present study, the following conclusions can be made from the simulation results:  
1. The VAM predicts a parabolic flame front and the highest temperature region at the centerline, but the PSM predicts a scattered flame zone in the pore areas;
2. Under the same flow conditions and properties of porous medium, the PSM gives a larger flame stability range, a lower flame temperature and peak solid matrix temperature and a higher peak wall temperature than a VAM counterpart;  
Figure 14: Effects of \(k_{s}\) on the ratio of preheating to heat loss obtained by the PSM and VAM.  
3. In the PSM combustor, solid-to-wall heat exchange (\(q_{s-w}\)) between the solid matrix and wall decreases with the decrease in \(\Phi\);
4. Ratio of preheating to heat loss (\(R_{p-hl}\)) of the PSM and VAM both decrease with the increase in \(\Phi\), and the PSM combustor provides a larger \(R_{p-hl}\);
5. Under the same flow conditions and properties of porous media, the difference of temperature and \(R_{p-hl}\) between the PSM and VAM decreases with the increase in \(\Phi\), and the decrease in \(k_{s}\).  
Therefore, the PSM porous micro-combustor is more suitable for simplifying to a VAM with a larger \(\Phi\) and a smaller \(k_{s}\), which is only for specific configurations of a porous micro-combustor considered in the present study, and therefore it should be noted that the results in present paper are of limited applicability. However, the calculation and analysis methods in principle can be applied to other porous structure in micro-combustors as well, which is necessary to provide some qualitative guidance to the application of porous micro-combustors.  
Conceptualization, Q.L. and J.L.; methodology, Q.L. and J.L.; software, Q.L.; validation, Q.L. and J.S.; formal analysis, Q.L.; investigation, Q.L.; resources, J.L.; data curation, Q.L.; writing--original draft preparation, Q.L.; writing--review and editing, Q.L. and J.L.; visualization, Q.L.; supervision, J.W. and J.L.; project administration, J.L.; funding acquisition, J.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 51776136).  
Not applicable.  
The authors gratefully acknowledge the financial support provided by the National Natural Science Foundation of China (Grant No. 51776136).  
The authors declare no conflict of interest.