## 3 Results and Discussion
### Heat Recirculation for VAM and PSM
#### 3.3.2 Heat Recirculation through Solid-To-Wall Heat Exchange  
As already mentioned, solid-to-wall heat exchange (\(q_{s-w}\)) between the solid matrix and wall has an important influence on the temperature distribution of the PSM combustor, as shown in Figure 5. Solid-to-wall heat exchange efficiency is represented by the following expression:  
\[\eta_{s-w}=\left|\int_{A_{p}}q_{s-w}dA\right|/Q_{c} \tag{10}\]  
where \(A_{p}\) refers to the contact heat transfer area between the solid matrix and wall. \(Q_{c}\) refers to the total heat release calculated from the lower heating value of H\({}_{2}\) (\(\sim\)120 MJ/kg). Figure 7 shows that in the PSM combustor, \(\Phi\) has an obvious effect on \(\eta_{s-w}\), that is, the larger \(\Phi\) is, the smaller is \(\eta_{s-w}\). Therefore, the influence of solid-to-wall heat exchange on the PSM calculation model decreases with the increase in \(\Phi\), which is consistent with the results shown in Figure 5, that is, the temperature difference between the PSM and VAM decreases with the increase in \(\Phi\). Figure 6: Delineation of solid-to-gas preheat zone and heat loss zone obtained by the PSM, with \(U_{0}=2\) m/s and \(\Phi=0.65\). 3.3.3. Ratio of Preheating to Heat Loss for VAM and PSM  
In a porous micro-combustor, there are basically two preheating pathways for heat recirculation: one is the wall-to-gas preheating, differing from the macro-scale counterpart, and the other is the solid-to-gas preheating, as illustrated in Figure 8. It indicates that the fraction of solid-to-gas preheating (\(\eta_{s-g}\)) of the PSM combustor is obviously higher than its VAM counterpart, because the scattered reaction zone of the PSM combustor gives a larger interfacial area of heat transfer between solid matrix and gas. At the same time, the higher wall temperature of the PSM combustor results in the greater fraction of wall-to-gas preheating (\(\eta_{w-g}\)). Combining the two preheating pathways, the total preheating efficiency can be obtained, which is found that the fraction of overall preheating of the PSM combustor is larger than its VAM counterpart. When the equivalence ratio changes from \(\Phi_{4}\) to \(\Phi_{3}\), the flame trends to be more upstream, thus the length (area) of the preheat zone decreases, which is believed to be the main reason for the reducing fraction of overall preheating for both PSM and VAM combustors. Figure 8: Fraction of preheating obtained by the PSM and VAM. Figure 7: Fraction of solid-to-wall heat exchange (\(\eta_{s-w}\)) obtained by the PSM. The fraction of heat loss with a change in \(\Phi\) is plotted in Figure 9 on the basis of the heat loss zone shown in Figure 6. A U-shaped curve was obtained for the VAM and PSM as the intention to increase the energy input rate would result in high heat loss. Figure 9 shows that the micro-combustors simulated using the PSM demonstrate a larger fraction of heat loss than the VAM counterpart. The multiple flame regions for the PSM indicate a larger flame area, and hence, more heat loss is understandable. Based on our earlier paper, it is found that a ratio of preheating to heat loss, \(R_{p-hl}\), is crucial to sustaining and stabilizing flames in a porous micro-combustor. The \(R_{p-hl}\) obtained by the PSM and VAM with \(U_{0}\) = 2 m/s are plotted in Figure 10, showing that a PSM combustor provides a larger \(R_{p-hl}\). Therefore, combining the preheating and heat loss, the preheating effect of the PSM combustor is much better than the VAM, which is consistent with the rule that the PSM combustor gives a larger stability range. Figure 10 shows that \(R_{p-hl}\) of the PSM and VAM combustor decreases with the increase in \(\Phi\). Figure 10: Variation of \(R_{p-hl}\) obtained by the PSM and VAM. Figure 9: Fraction of heat loss obtained by the PSM and VAM. Figure 13 quantifies the effects of \(k_{s}\) on two pathways of preheating; they are solid-to-gas preheating and wall-to-gas preheating.