## 2 Numerical Methodologies  
### Numerical Methods and Physical Models  
In this research, we use ANSYS FLUENT V.17 [23] to analyze the flow development in a reformer. The governing equations are solved by the SIMPLE algorithm [24]. The convection terms are discretized by the second order upwind scheme, and the diffusion terms are discretized by the central difference scheme. Concerning the choice of physical models, FLUENT provides Spalart-Allmaras one-equation model, \(k\)-\(\varepsilon\) two-equation model, \(k\)-\(\omega\) two-equation model, transition k-kl-\(\omega\) three-equation model, transition SST four-equation model, Reynolds stress seven-equation model, scale-adaptive simulation (SAS), detached eddy simulation (DES), and large eddy simulation (LES) for simulation of turbulence. For simulation of radiation, FLUENT provides Rosseland radiation model, P-1 radiation model, discrete transfer radiation model (DTRM), surface-to-surface (S2S) radiation model, and discrete ordinates (DO) radiation model for choice. With respect to chemical reaction simulation, FLUENT provides species transport, non-premixed reaction, premixed reaction, partially premixed reaction, and composition PDF transport for choice. By referring to the SMR research by CFD mentioned in Section 1 and from the viewpoint of model accuracy, stability, and engineering application, turbulence is simulated by the standard \(k\)-\(\varepsilon\) model [25], radiation is simulated by the DO (discrete ordinate) radiation model [26], and chemical reaction is simulated by the FRED (finite rate/eddy dissipation) model [27]. The near-wall flow variables, including the velocity, temperature, and turbulence quantities, are solved by means of the standard wall functions [28]. The catalyst is modeled by the porous zone model, which is available in ANSYS FLUENT and has also been employed by [PERSON] et al. [15; 16; 17; 18; 19] and [PERSON] et al. [20] in their studies of steam methane reformer. This research considers the 3D steady flow with chemical reaction. The governing equations include the continuity equation, momentum equation, turbulence model equation (\(k\)-\(\varepsilon\) model), energy equation, radiation model equation (DO radiation model), and chemical reaction model equation (FRED model). Among these models, only the FRED chemical reaction model is described below while the others and the convergence criterion have been stated in detail in one of the author's previous studies [29]. The \(r\)th chemical reaction can be expressed as follows:  
\[\sum_{i=1}^{N}v^{\prime}_{i,r}\mu_{i} \begin{array}{l}k_{f,r}\\ k_{b,r}\end{array} \sum_{i=1}^{N}v^{\prime\prime}_{i,r}\mu_{i} \tag{1}\]  
where  
\(N\) = number of chemical species in the system  
\(v^{\prime}_{i,r}\) = stoichiometric coefficient for reactant \(i\) in reaction \(r\)  
\(v^{\prime\prime}_{i,r}\) = stoichiometric coefficient for product \(i\) in reaction \(r\)  
\(\mu_{i}\) = species \(i\)  
\(k_{f,r}\) = forward rate constant for reaction \(r\)  
\(k_{b,r}\) = backward rate constant for reaction \(r\)  
Equation (1) applies for both reversible and non-reversible reactions. For non-reversible reactions, the backward rate constant, \(k_{b,r}\), in Equation (1) is simply omitted. The species transport equation for a chemical reaction system can be express as  
\[\
abla\cdot\left(\rho\overset{\rightarrow}{v}Y_{i}\right) =\
abla\cdot\left(\rho D_{i,m}+\frac{\mu_{t}}{Sc_{t}}\right)\
abla Y _{i}+R_{i}+S_{i} \tag{2}\]  
where \(Y_{i}\), \(D_{i,m}\), \(Sc_{t}\), \(R_{i}\), and \(S_{i}\) are the mass fraction, diffusion coefficient, turbulent Schmidt number, net generation rate, and extra source term of species \(i\), respectively. The net sourceof chemical species \(i\) due to reaction is computed as the sum of the Arrhenius reaction sources over the \(N_{R}\) reactions that the species participate in.