## 2 Modeling  
### Pore-Scale Model  
Figure 1a illustrates the design features of a porous micro-combustor. The length, width and height of the combustor are 20, 10 and 1 mm, respectively. The wall thickness \(t\) is 0.5 mm. The planar micro-combustor is partially filled with alumina spheres with \(d_{s}=1\) mm, producing a porosity \(\varepsilon\) of 0.48. The alumina spheres have a thermal conductivity of 5 W/m-K [37]. The combustor is made of SS 316 L and has a wall thermal conductivity of 20 W/m-K. The large aspect ratio (=10) leads to the same temperature distribution in the wall surface and channels along the x direction. In view of the symmetry, only half of a channel is considered to save computational time, as shown in Figure 1b. In numerical simulations, sphere-sphere and wall-sphere contact points often present difficulties in mesh quality. In order to eliminate the contact points, cylinders with a radius of 0.1 \(d_{s}\)[38] are artificially created between contacted spheres or a sphere and wall surface that are in contact, as shown in Figure 1b. To eliminate the non-essential factors in the mathematical formulation, the following assumptions are made: (1) ideal gases and incompressible flows [36]; (2) steady-state combustion [33]; (3) no Dufour and [PERSON] effects; (4) no gas radiation; (5) no work done by pressure and viscous forces; (6) inert porous medium [33]. The conservation equations are written as follows:  
Continuity:  
\[\frac{\partial\left(\epsilon\rho_{g}u_{g}\right)}{\partial x}+\frac{\partial \left(\epsilon\rho_{g}v_{g}\right)}{\partial y}+\frac{\partial\left(\epsilon \rho_{g}w_{g}\right)}{\partial z}=0 \tag{1}\]  
\(x\)-Momentum:  
\[\rho_{g}u_{g}\frac{\partial u_{g}}{\partial x}+\rho_{g}v_{g}\frac{\partial u _{g}}{\partial y}+\rho_{g}w_{g}\frac{\partial u_{g}}{\partial z}=-\frac{ \partial p}{\partial x}+\mu\left(\frac{\partial^{2}u_{g}}{\partial x^{2}}+ \frac{\partial^{2}u_{g}}{\partial y^{2}}+\frac{\partial^{2}u_{g}}{\partial z ^{2}}\right) \tag{2}\]  
\(y\)-Momentum:  
\[\rho_{g}u_{g}\frac{\partial v_{g}}{\partial x}+\rho_{g}v_{g}\frac{\partial v _{g}}{\partial y}+\rho_{g}w_{g}\frac{\partial v_{g}}{\partial z}=-\frac{ \partial p}{\partial y}+\mu\left(\frac{\partial^{2}v_{g}}{\partial x^{2}}+ \frac{\partial^{2}v_{g}}{\partial y^{2}}+\frac{\partial^{2}v_{g}}{\partial z ^{2}}\right) \tag{3}\]  
\(z\)-Momentum:  
\[\rho_{g}u_{g}\frac{\partial w_{g}}{\partial x}+\rho_{g}v_{g}\frac{\partial w _{g}}{\partial y}+\rho_{g}w_{g}\frac{\partial w_{g}}{\partial z}=-\frac{ \partial p}{\partial z}+\mu\left(\frac{\partial^{2}w_{g}}{\partial x^{2}}+ \frac{\partial^{2}w_{g}}{\partial y^{2}}+\frac{\partial^{2}w_{g}}{\partial z ^{2}}\right) \tag{4}\]  
Energy equation in the gas phase:  
\[\rho_{g}c_{g}u_{g}\frac{\partial T_{g}}{\partial x}+\rho_{g}c_{g}v_{g}\frac{ \partial T_{g}}{\partial y}+\rho_{g}c_{g}w_{g}\frac{\partial T_{g}}{\partial z }=\frac{\partial}{\partial x}\left(k_{g}\frac{\partial T_{g}}{\partial x} \right)+\frac{\partial}{\partial y}\left(k_{g}\frac{\partial T_{g}}{\partial y }\right)+\frac{\partial}{\partial z}\left(k_{g}\frac{\partial T_{g}}{ \partial z}\right)-\omega_{i}W_{i}Q+q_{r} \tag{5}\]  
Figure 1: Illustration of porous micro-combustor in the PSM. (**a**) Physical model; (**b**) channel along the spheres.