## 2 Numerical Methodologies
### Numerical Methods and Physical Models
\[R_{i}=M_{w,j}\sum_{r=1}^{N_{R}}\hat{R}_{i,r} \tag{3}\]  
where \(M_{w,j}\) is the molecular weight of species \(i\) and \(\hat{R}_{i,r}\) is the Arrhenius molar rate of creation/destruction of species \(i\) in reaction \(r\). For a non-reversible reaction, the molar rate of creation/destruction of species \(i\) in reaction \(r\) is expressed as  
\[\hat{R}_{i,r}=\Gamma\left(v_{i,r}^{\prime\prime}-v_{i,r}^{\prime}\right) \left(k_{f,r}\prod_{j=1}^{N}[C_{j,r}]^{\left(\eta_{j,r}^{\prime}+\eta_{j,r}^{ \prime\prime}\right)}\right) \tag{4}\]  
For a reversible reaction,  
\[\hat{R}_{i,r}=\Gamma\left(v_{i,r}^{\prime\prime}-v_{i,r}^{\prime}\right) \left(k_{f,r}\prod_{j=1}^{N}[C_{j,r}]^{\eta_{j,r}^{\prime\prime}}-k_{b,r}\prod _{j=1}^{N}[C_{j,r}]^{\eta_{j,r}^{\prime\prime}}\right) \tag{5}\]  
where  
\(C_{j,r}=\) molar concentration of species \(j\) in reaction \(r\) (kgmol/m\({}^{3}\))  
\(\eta_{j,r}^{\prime}=\) rate exponent for reactant species \(j\) in reaction \(r\)  
\(\eta_{j,r}^{\prime\prime}=\) rate exponent for product species \(j\) in reaction \(r\)  
\(\Gamma=\) net effect of third bodies on the reaction rate  
The forward and backward rate constants for reaction \(r\), \(k_{f,r}\) and \(k_{b,r}\), are computed using the Arrhenius expression:  
\[k_{f,r}=A_{r}T^{\beta_{r}}e^{-Er/kT} \tag{6}\]  
\[k_{b,r}=\frac{k_{f,r}}{K_{r}} \tag{7}\]  
where  
\(A_{r}=\) pre-exponential factor (consistent units)  
\(\beta_{r}=\) temperature exponent (dimensionless)  
\(E_{r}=\) activation energy for the reaction (J/kgmol)  
\(R=\) universal gas constant (J/kgmol-K)  
\(K_{r}=\) the equilibrium constant for the \(r\)th reaction, is computed from  
\[K_{r}=\exp\left(\frac{\Delta S_{r}^{0}}{R}-\frac{\Delta H_{r}^{0}}{RT}\right) \left(\frac{p_{atm}}{RT}\right)_{i=1}^{N}\left(v_{i,r}^{\prime\prime}-v_{i,r} ^{\prime}\right) \tag{8}\]  
where \(p_{atm}\) denotes the atmospheric pressure (101,325 Pa). The term in the exponential function denotes the change in Gibbs free energy and its components are expressed as  
\[\frac{\Delta S_{r}^{0}}{R}=\sum_{i=1}^{N}\left(v_{i,r}^{\prime\prime}-v_{i,r} ^{\prime}\right)\frac{S_{i}^{0}}{R} \tag{9}\]  
\[\frac{\Delta H_{r}^{0}}{RT}=\sum_{i=1}^{N}\left(v_{i,r}^{\prime\prime}-v_{i,r} ^{\prime}\right)\frac{h_{i}^{0}}{RT} \tag{10}\]  
where \(S_{i}^{0}\) and \(h_{i}^{0}\) denote the standard-state entropy and standard-state enthalpy (heat of formation), respectively. In the past few decades, there have been many kinetics data reported for the SMR reaction. [PERSON], [PERSON], and [PERSON] [30] reviewed eight of them with their individual application range of temperature. In this paper, the kinetics constants proposed by [PERSON], [PERSON], and [PERSON] [31] are adopted because their application range of temperature matches the operating temperature of the reformer investigated in this study. In general, a reformer operates at high temperatures. For a non-premixed reaction, turbulence mixes the reactants and then advects the mixture to the reaction zone for quick reactions. For a premixed reaction, turbulence mixes the lower-temperature reactants and the higher-temperature products and then advects the mixture to the reaction zone for quick reactions. Therefore, the chemical reaction is generally mixing (diffusion) controlled. However, the flue gas, fuel gas, and air in a practical reformer are generally premixed before injecting into the reformer. Although the chemical reactions in most regions of a reformer are mixing controlled, in some regions, e.g., the neighborhood of the feed inlets, the chemical reactions are kinetically controlled. In existing chemical reaction models, the eddy dissipation model (EDM) [32] considers simultaneously the diffusion controlled and the kinetically controlled reaction rates.