## 2 Numerical Methodologies
### Numerical Setup  
A practical reformer is simulated and compared with available experimental data to validate the employed numerical methods. Figure 1 shows the steam methane reformer investigated. Due to symmetry, merely half the reformer is simulated, as shown in Figure 1b. The reformer has a total of 276 catalyst tubes and 432 burners, i.e., each side of the reformer has 138 catalyst tubes and 216 burners. Each burner has a diameter 197 mm, as shown in Figure 1f. Each catalyst tube has an outside diameter 136 mm and a thickness 13.4 mm. Figure 1: Cont. Figure 1: Steam methane reformer investigated: (**a**) Reformer configuration; (**b**) Reformer numerical model; (**c**) Simplified model containing one group of catalyst tubes and burners; (**d**) Reformer dimensions and burner positions; (**e**) Catalyst tube positions; (**f**) Burner configuration. The real operating conditions used by a petrochemical corporation are adopted as the inlet conditions of burners and catalyst tubes. The boundary conditions for numerical simulation in this study are described below. 1. At the Symmetry planes, the symmetric boundary conditions are used. 2. At walls, standard wall function is applied. 3. At the burner inlets, i.e., fuel and flue gas inlets, the real operating conditions used by a petrochemical corporation are adopted, including \(Q\) (Flowrate in the radial direction): 139,710 m\({}^{3}\)/hr \(T\) (Temperature): 673.15 K \(P_{gauge}\) (gauge Pressure): 1.04544 \(\times\) 10\({}^{4}\) N/m\({}^{2}\) Species mole fraction, including H\({}_{2}\) (Hydrogen): 0.0816 CH\({}_{4}\) (Methane): 0.0474 N\({}_{2}\) (Nitrogen): 0.49057 O\({}_{2}\) (Oxygen): 0.12818 CO\({}_{2}\) (Carbon dioxide): 0.25225
4. At the burner exits, i.e., fuel and flue gas exits, the diffusion flux in the outflow direction is zero for all flow variables, and the conservation of mass should be satisfied. 5. At the inlets of the catalyst tubes, the real operating conditions used by a petrochemical corporation are adopted, including \(Q\) (Flowrate in the axial direction): 24,740 m\({}^{3}\)/hr \(T\) (Temperature): 912.75 K \(P_{gauge}\) (gauge Pressure): 2.1658 \(\times\) 10\({}^{6}\) N/m\({}^{2}\) Species mole fraction, including CH\({}_{4}\) (Methane): 0.2029 H\({}_{2}\)O (Steam): 0.6 H\({}_{2}\) (Hydrogen): 0.12855 CO\({}_{2}\) (Carbon dioxide): 0.06565 CO (Carbon Monoxide): 0.00145 N\({}_{2}\) (Nitrogen): 0.00145
6. At the exits of the catalyst tubes, the diffusion flux in the outflow direction is zero for all flow variables, and the conservation of mass should be satisfied. 7. At the inlets of the burners or the catalyst tubes, the turbulence kinetic energy is assumed to be 10% of the inlet mean flow kinetic energy. The turbulence dissipation rate is calculated from: \[\varepsilon=C_{\mu}^{3/4}\frac{k^{3/2}}{l}\] (13) where \(l\) is connected with the hydraulic diameter, \(L\), by \(l\) = 0.07 \(L\). From Figure 1b,d, we see that the catalyst tubes and burners can be divided into six groups. In an attempt to save the computational time, we also calculate and compare a simplified model which contains only one group of catalyst tubes and burners. Figure 1c shows the simplified model, which contains 23 catalyst tubes and 36 burners. The prototype differs from the simplified model in the left- and right-hand-side boundaries: the prototype has solid walls, while the simplified model has periodic boundaries. In case the simulation results using the simplified model are close to those using the prototype, the periodic model can be used instead of the prototype to save simulation time in view of engineering applications. On the other hand, if obvious difference exists between the simulation results using the prototype and the periodic model, the prototype should be adopted for subsequent simulation and discussion. In this study, we use the software GAMBIT to generate grid mesh, which includes nearly four million cells for the prototype and one million cells for the periodic model, respectively.