## 3 Results and Discussion
### Differentiating the Thermal Conductivity between Inner and Outer Walls
#### 3.2.2 Optimal Combination of Materials  
The above-presented results have shown that heat recirculation creates a distinction of the roles played by the inner and outer walls in determining stability. The outer wall controls blowout. It also plays a role in affecting extinction. However, this is mainly ruled by the inner wall.  
Figure 10 shows the inlet gas velocity at blowout (V\({}_{\text{in}}\) @ blowout) as a function of the inlet gas velocity at extinction (V\({}_{\text{in}}\) @ extinction) for the four cases All Cordierite, Cordierite + SiC, SiC + Cordierite, and All SiC, and the additional case As SiC + Cordierite. In this latter case, the thermal conductivity of both outer and inner walls was set equal to the same value of volume-averaged solid thermal conductivity as the case SiC + Cordierite (23.3 W/m K--see Table 3).  
These results clearly contradict the intuitive idea that the best combination of materials is that of a low-conductive outer wall, which minimizes the heat losses toward the external environment, and a highly conductive inner wall, which facilitates pre-heating of the cold feed to get easier ignition. Indeed, the case Cordierite + SiC was the worst case, as it exhibited lower resistance to extinction than the case All Cordierite and, at the same time, lower resistance to blowout than the case All SiC. Conversely, the case SiC + Cordierite exhibited much better resistance to extinction than the case All SiC and, at the same time,  
Figure 10: V\({}_{\text{in}}\) @ blowout as a function of V\({}_{\text{in}}\) @ extinction for different cases: All Cordierite; Cordierite + SiC; SiC + Cordierite; All SiC; As SiC + Cordierite.  
much better resistance to blowout than the case All Cordierite. This can be attributed only to a lesser extent to the fact that the volume-averaged solid thermal conductivity for the case SiC + Cordierite was lower than the thermal conductivity of SiC and higher than the thermal conductivity of cordierite. Indeed, better resistance to extinction was mainly imparted by the low thermal conductivity of the inner wall, whereas better resistance to blowout was mainly imparted by the high thermal conductivity of the outer wall.  
Global assessment of Figure 10, thus, demonstrates that the distinction of the roles played by the inner and outer walls offers the opportunity of differentiating the thermal conductivities to enlarge the stable operating window of the U-bend catalytic micro-combustor compared to the use of the same material for both walls. From this perspective, the configuration SiC + Cordierite, with a low thermal conductivity for the inner wall and a high thermal conductivity for the outer wall, represents the optimal combination of materials. This combination is suitable for both low- and high-power systems.  
For the optimal configuration with an insulating inner wall and conductive outer wall, simulations were also run by varying the thermal conductivity of the outer wall, while keeping the thermal conductivity of the inner wall constant and equal to that of cordierite. The results of these simulations are shown in Figure 11, where V\({}_{\text{in}}\) @ extinction and V\({}_{\text{in}}\) @ blowout are plotted as a function of the volume-averaged solid thermal conductivity. In the same figure, the results obtained for the configuration with both walls made of the same material are also shown.  
At a fixed solid thermal conductivity, as a result of higher resistance to both extinction and blowout, the optimal configuration exhibited a wider stable operating window than the configuration with both walls made of the same material. Moreover, the benefit in terms of widening of the stable operating window increased with increasing solid thermal conductivity.  
Figure 11: V\({}_{\text{in}}\) @ extinction and V\({}_{\text{in}}\) @ blowout versus the solid thermal conductivity: Optimal configuration and configuration with both walls made of the same material. (These are all cases of Table 3).