## 2 Modeling
### Pore-Scale Model
Energy equation in the solid phase:  
\[\frac{\partial}{\partial x}\left(k_{s}\frac{\partial T_{s}}{\partial x}\right)+ \frac{\partial}{\partial y}\left(k_{s}\frac{\partial T_{s}}{\partial y}\right)+ \frac{\partial}{\partial z}\left(k_{s}\frac{\partial T_{s}}{\partial z}\right)=0 \tag{6}\]  
Mass conservation for the \(i\)th species:  
\[\rho_{g}u_{g}\frac{\partial Y_{i}}{\partial x}+\rho_{g}y_{g}\frac{\partial Y_{ i}}{\partial y}+\rho_{g}w_{g}\frac{\partial Y_{i}}{\partial z}=\frac{\partial}{ \partial x}\left[D_{i}\frac{\partial\big{(}\rho_{g}Y_{i}\big{)}}{\partial x} \right]+\frac{\partial}{\partial z}\left[D_{i}\frac{\partial\big{(}\rho_{g}Y_ {i}\big{)}}{\partial y}\right]+\frac{\partial}{\partial z}\left[D_{i}\frac{ \partial\big{(}\rho_{g}Y_{i}\big{)}}{\partial z}\right]+\omega_{i}W_{i} \tag{7}\]  
Energy equation in the combustor wall:  
\[\frac{\partial}{\partial x}\left(k_{w}\frac{\partial T_{w}}{\partial x}\right) +\frac{\partial}{\partial y}\left(k_{w}\frac{\partial T_{w}}{\partial y} \right)+\frac{\partial}{\partial z}\left(k_{w}\frac{\partial T_{w}}{\partial z }\right)=0 \tag{8}\]  
where the wall thermal conductivity \(k_{w}\) is 20 W/m-K [39]. The boundary conditions applied in the PSM are shown in Figure 1b. Premixed H\({}_{2}\)/air is fed into the inlet of the micro-scale combustor, and has a given velocity, equivalence ratio and mixture temperature (\(T_{0}\) = 300 K). The far-field pressure outlet is set to 1 atm at the exit. A simplified reaction mechanism comprising 6 species and 7 elementary reactions is taken from Ref. [38] for the combustion chemistry. The solid radiative heat transfer between the inner surfaces is analyzed using the discrete ordinate model [40]. Heat loss from the non-insulated walls is composed of two parts, convection and radiation, and is governed by:  
\[q_{w}=h\left(T_{w0}-T_{a}\right)+\varepsilon_{r}\sigma\left(T_{w0}{}^{4}-T_{a }{}^{4}\right) \tag{9}\]  
where the convective heat loss coefficient \(h\) is 20 W/m\({}^{2}\)-K [39], the ambient temperature \(T_{a}\) is 300 K [33; 36; 39], the emissivity of non-insulated wall is 0.8 [36] and the Stephan-Boltzmann constant \(\sigma\) is 5.67 \(\times\) 10\({}^{-8}\) W/m\({}^{2}\)-K4 [2; 33; 36; 39; 41; 42]. The governing equations are discretized and solved numerically using the CFD software FLUENT(r) [43]. The non-conformal grids with tetrahedral elements are generated by Pointwise(r). A reasonable grid density is determined using a mesh independence study, as seen in Figure 2a. The results demonstrate that a grid with 357,355 nodes is sufficiently fine to capture both the centerline and wall temperature profiles, and a higher number of nodes (434,459) does not yield significant improvement. As such, non-uniform meshes with 357,355 nodes are used in the following tests, and the local mesh gradation is shown in Figure 2b. In order to accurately capture the temperature and velocity gradients, the highest grid density is applied near the gas-solid and gas-wall interfaces with four layers of T-Rex grids for a layer thickness of \(\triangle\)s= 1 \(\upmu\)m (Region I in Figure 2b). \[\rho_{g}u_{g}\frac{\partial u_{g}}{\partial x}+\rho_{g}v_{g}\frac{\partial u _{g}}{\partial y}+\rho_{g}w_{g}\frac{\partial u_{g}}{\partial z}=-\frac{ \partial p}{\partial x}+\mu\left(\frac{\partial^{2}u_{g}}{\partial x^{2}}+ \frac{\partial^{2}u_{g}}{\partial y^{2}}+\frac{\partial^{2}u_{g}}{\partial z ^{2}}\right) \tag{2}\]0 Mesh generation. (**a**) Result of grid independence test obtained by the PSM, \(U_{0}\) = 2 m/s, \(\Phi\) = 0.65, \(d_{s}\) = 1 mm, \(\varepsilon\) = 0.48, \(k_{s}\) = 5 W/m-K, and \(k_{w}\) = 20 W/m-K; (**b**) local representation of the mesh with 357,355 nodes (\(d_{s}\) = 1 mm). A moderate density is used in the fluid (Region II) and solid (Region III) regions, while the lowest density is in the wall (Region IV).