## 2 Mathematical Model  
Two-dimensional steady-state CFD simulations were carried out to investigate the stability of the U-bend catalytic heat-recirculating micro-combustor schematized in Figure 1.  
Figure 1: Two-dimensional computational domain (not to scale): U-bend catalytic heat-recirculating micro-combustor. The green lines denote the catalytic surfaces where heterogeneous combustion takes place. After entering the micro-combustor, the gas stream flows through the inlet channel, turns around the inner wall, flows through the outlet channel and then exits. Table 1 lists the values of the geometrical parameters for the configuration of Figure 1.  
The model was developed using the commercial CFD software package ANSYS Fluent (release 2020 R1). The processes simulated include conduction and convection in the gas phase, solid-phase conduction within the walls, external convective heat losses, and chemical reactions both in the gas phase and at the gas-solid interface (green lines in Figure 1). The model equations are the conservation equations for mass, momentum, energy, and chemical species in the fluid as well as the conservation equation for energy in the solid walls. Laminar flow was assumed. In the momentum equation, the stress tensor took the standard form for a Newtonian fluid. Simulations were run for lean pre-mixed propane/air. Among alkanes, propane is particularly suitable for catalytic micro-combustion [11; 37]. The reaction rate of homogeneous (i.e., gas-phase) propane combustion was calculated according to the single-step reaction rate by [PERSON] and Dryer [38]:  
\[\mathrm{r_{hom}}=-\ 4.836\ \times\ 10^{9}\ \mathrm{exp}\ \ (\frac{-\ 1.256\ \times\ 10^{8}}{ \mathrm{R\,T}})\,\mathrm{C_{C_{3}H_{8}}}^{0.1}\,\mathrm{C_{O_{2}}}^{1.65}\ \ \ \ \left[\mathrm{kmol}/\left(\mathrm{m^{3}s}\right)\right], \tag{1}\]  
where the activation energy is expressed in J/kmol and the species concentrations in kmol/m\({}^{3}\). A single-step reaction rate was also assumed for heterogeneous (i.e., catalytic) propane combustion (occurring at the gas-solid interface) [39]:  
\[\mathrm{r_{cat}}=-\ 2.4\ \times\ 10^{5}\ \mathrm{exp}\ \ (\frac{-\ 9.06\ \times\ 10^{7}}{ \mathrm{R\,T}})\,\mathrm{C_{C_{3}H_{8}}}\ \ \ \left[\mathrm{kmol}/\left(\mathrm{m^{2}s}\right)\right], \tag{2}\]  
where the activation energy is expressed in J/kmol and the concentration of propane in kmol/m\({}^{2}\). At the inlet boundary, uniform profiles were specified for (axial) velocity, temperature and species mole fractions. The inlet conditions are detailed in Table 2. Stability was assessed in terms of critical inlet gas velocities, i.e., the velocity below which extinction occurs, \(\mathrm{V_{in}}\) @ extinction, and the velocity beyond which blowout occurs, \(\mathrm{V_{in}}\) @ blowout. The (inlet) Reynolds number, Re, defined based on the inlet gas velocity, gas kinematic viscosity at ambient conditions, and channel gap size, is also reported in Table 2. At the highest value of \(\mathrm{V_{in}}\) explored (24.5 m/s), Re is equal to around 940, thus confirming the assumption of laminar flow. At the outlet boundary, a condition of fixed static pressure was assigned. At the gas-solid interface, no-slip boundary conditions were used for both velocity components (the \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \hline Channel gap size, d [mm] & 0.6 \\ Wall thickness, \(\delta\) [mm] & 0.2 \\ Length of the micro-combustor, L [mm] & 12.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the geometrical parameters for the U-bend configuration of Figure 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \hline Inlet gas velocity, \(\mathrm{V_{in}}\) [m/s] & 0.12–24.5 \\ Inlet gas temperature [K] & 300 \\ Inlet fuel equivalence ratio [-] & 0.5 \\ (Inlet) Reynolds number, Re [-] & 4.6–937.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Inlet conditions. fluid had zero velocity relative to the boundary).