# Numerical Simulation of Natural Gas/Hydrogen Combustion in a Novel Laboratory Combestor
## 2 Materials and Methods
### Mathematical and Physical Models
#### 2.2.3 Combustion Model  
The eddy dissipation concept (EDC) [43] was selected for combustion modelling. It requires the solution of the transport equations for the species mass fractions. The EDC model is based on the detailed description of dissipation of the turbulent eddies. The flow is divided into two regions, namely the region that contains the small scales of turbulence, called the region of fine structures of the turbulence, and the region surrounding it. The model assumes that chemical reactions occur only in the region of fine turbulence structures, where all the turbulent kinetic energy is dissipated into heat. The fine structures of turbulence are considered small chemical reactors, and it is assumed that the reaction rate is equal to the rate of mass transfer of species between the fine turbulence structures and their surroundings. The source term of the transport equation for the \(i\)th species, which represents the rate of mass production/consumption of that species per unit volume, \(\widetilde{\omega_{i}}\) [kg/m\({}^{3}\cdot\)s], is given by [44]:  
\[\widetilde{\omega_{i}}=\frac{\rho\ \gamma^{*}\ \chi\ \big{(}y_{i}^{*}-y_{i}^{ \rho}\big{)}}{\tau^{*}}=\frac{\rho\ \gamma^{*}\ \chi\ \big{(}y_{i}^{*}-\widetilde{y}_{i}\big{)}}{\tau^{*}(1- \gamma^{*}\,\chi)} \tag{12}\]  
where \(\gamma^{*}\) is the ratio of the mass in the fine structures to the total mass, \(\chi\) the reacting fraction of fine structures, \(y_{i}^{*}\) the fine-structure \(i\)th species mass fraction after reacting over the time scale \(\tau^{*}\), \(y_{i}^{o}\) the mass fraction of species \(i\) in the surroundings of the fine structures, and \(\widetilde{y}_{i}\) the mean mass fraction of species \(i\). The time scale is defined as follows [44]:  
\[\tau*=C_{\tau}\,\sqrt{\
u/\epsilon} \tag{13}\]  
where \(C_{\tau}\) is a constant of the model, taken as 0.4083, and \(\
u\) is the kinematic viscosity. Different definitions of \(\gamma^{*}\) and \(\chi\) have been proposed in the literature. In this work, a constant value of unity was used for \(\chi\), as proposed by [PERSON] and [PERSON] [45], and \(\gamma^{*}\) was defined as [44]:  
\[\gamma*=\gamma_{\lambda}^{3} \tag{14}\]  
where \(\gamma_{\gamma}\) is the mass of fine-structure regions divided by total mass, which is computed as  
\[\gamma_{\lambda}=C_{\gamma}\sqrt[4]{\
u\,\epsilon/k^{2}} \tag{15}\]  
The volume fraction constant, \(C_{\gamma}\), is equal to 2.1377. An operator splitting algorithm was used in every iteration to determine the species mass fractions. In the first step, the fine-structure species mass fractions were calculated by solving a system of ordinary differential equations for a constant pressure chemical reactor using the stiff CVODE ordinary differential equations solver [46]. A clustering technique was used to accelerate the chemistry calculations. This technique joins together control volumes with similar chemical compositions, integrates the reduced set of equations, and then interpolates the clusters back to the control volumes. The computed values of the species mass fractions in the fine structures at \(t=\tau^{*}\) are used in Equation (12) to determine the mean reaction rates, which are required to solve the transport equations for the mean species mass fractions in the second step of the operator splitting algorithm. The DRM-19 reaction mechanism [47] was used to describe the combustion of NG/H\({}_{2}\) mixtures. This is a reduced mechanism based on Gri-Mech 1.2, an earlier version of Gri-Mech 3.0 [48], comprising 21 species and 84 reactions. The DRM-19 mechanism was also used by other authors to model NG/H\({}_{2}\) flames, e.g., [49]. A test case was run using both DRM-19 and Gri-Mech 3.0. Since it was found that the results were quite close to each other and the DRM-19 mechanism requires fewer computational resources, it was selected for the present work.