# Numerical Simulation of Natural Gas/Hydrogen Combustion in a Novel Laboratory Combestor
## 2 Materials and Methods
### Mathematical and Physical Models  
#### 2.2.1 Governing Equations  
The mathematical model is based on the numerical solution of the Favre-averaged conservation equations for mass, momentum, energy, and mass fractions of chemical species, which may be written as follows for steady state conditions [38]:  
\[\frac{\partial\left(\overline{\rho}\,\widetilde{u}_{j}\right)}{\partial\,x_{j} }=0 \tag{1}\]  
\[\frac{\partial}{\partial\,x_{j}}\left(\overline{\rho}\,\widetilde{u}_{j}\, \widetilde{u}_{i}\right)=-\frac{\partial\,\overline{\rho}}{\partial\,x_{i}}+ \frac{\partial\,}{\partial\,x_{j}}\left[\mu\left(\frac{\partial\,\widetilde{u} _{i}}{\partial\,x_{j}}+\frac{\partial\,\widetilde{u}_{i}}{\partial\,x_{i}} \right)-\frac{2}{3}\,\mu\,\frac{\partial\,\widetilde{u}_{k}}{\partial\,x_{k}} \,\delta_{ij}-\overline{\rho}\,\overbrace{u^{\prime\prime}\,\,u^{\prime \prime}\,\,i^{\prime\prime}}\,\right]+\overline{\rho}\,g_{i} \tag{2}\]  
\[\frac{\partial\left(\overline{\rho}\,\widetilde{u}_{j}\,\widetilde{u}_{k} \right)}{\partial\,x_{j}}=\widetilde{u}_{j}\frac{\partial\,\overline{\rho}}{ \partial\,x_{j}}+\overline{u^{\prime\prime}\,\,\beta\,\overline{\rho}}{ \partial\,x_{j}}+\overline{\tau}_{ij}\frac{\partial\,\widetilde{u}_{i}}{ \partial\,x_{j}}+\overline{\tau}_{ij}\frac{\partial\,\overline{u^{\prime \prime}}}{\partial\,x_{j}}+ \tag{3}\]  
\[\frac{\partial\left(\overline{\rho}\,\widetilde{u}_{i}\,\widetilde{y}_{i} \right)}{\partial\,x_{j}}=\frac{\partial\,}{\partial\,x_{j}}\left(\overline{ \rho\,\,D_{i}^{M}\frac{\partial\,y_{i}}{\partial\,x_{j}}}-\overline{\rho} \,\overbrace{u^{\prime\prime}\,\,j^{\prime\prime}\,\,i}\right)+\overline{ \omega_{i}} \tag{4}\]  
where the overbar denotes a time-averaged value and the tilde denotes a density-weighted average value, while the double prime identifies a fluctuation about the density-weighted averaged value. In these equations, \(u_{j}\) and \(x_{j}\) are the velocity component and the coordinate in the \(j\)th direction, respectively, \(h\) is the specific enthalpy (sum of sensible and chemical enthalpy), \(y_{i}\) the mass fraction of species \(i\), \(p\) the pressure, \(\rho\) the density, \(\mu\) the dynamic viscosity, \(\lambda\) the thermal conductivity, \(c_{p}\) the specific heat capacity, Le\({}_{i}\) the Lewis number of species \(i\), \(g_{i}\) the component of the gravity vector in the \(i\)th direction, \(\widetilde{u}_{p}^{\prime\prime}\) the radiative \begin{table}
\begin{tabular}{c c c} \hline  & **Hydrogen** & **Methane** \\ \hline Molar mass (kg/kmol) [34] & 2.016 & 16.043 \\ Density (kg/m\({}^{3}\)) [34] & 0.0818 & 0.6528 \\ Dynamic viscosity (kg/m\(\cdot\)s) [34] & 8.938 \(\times\) 10\({}^{-6}\) & 1.114 \(\times\) 10\({}^{-5}\) \\ Thermal conductivity (W/m\(\cdot\)K) [34] & 0.1867 & 0.0342 \\ Specific heat capacity (J/kg\(\cdot\)K) [34] & 14,313 & 2235.9 \\ Adiabatic flame temperature (K) [35] & 2483 & 2236 \\ Lower heating value (MJ/kg) [35] & 120 & 50 \\ \hline \end{tabular}
\end{table} 
Table 1: Physical properties of hydrogen and methane at 300 K and 1 atm. heat source, and \(\dot{\omega}_{i}\) the volumetric reaction rate of species \(i\). These equations are based on the assumptions that the fluid is Newtonian, mass diffusion is governed by [PERSON]'s law, the Dufour effect is negligible, and the mixture behaves as an ideal gas. The second term on the right side of Equation (3) is usually neglected. Similarly, the third and fourth terms on the right side of Equation (3), which account for viscous dissipation, are negligible, except in high Mach number flows. Furthermore, the second term into the square brackets in Equation (3) is also often neglected, assuming that the Lewis number is equal to unity for all species.