# Flame Evolution Characteristics for Hydrogen/LPG Co-Combustion in a Counter-Burmer
## 3 Numerical Modeling
### Flamelet Modeling  
For the purpose of developing computational models for flame front simulations, numerical approaches have been enhanced through past studies, which have analyzed the operations of reaction kinetics [28; 29]. Steady-state assumptions and partial equilibrium depend, in general, on these methods, assuming that all chemical reactions happen in shorter scale of time than the mixture flow time scale. All these presumptions, moreover, show low approximations in the 'unburnt' areas of a flame front, where the borderline-burned regions are also important to describe flame front limits. The Flamelet-Formed Manifold (FFM) method can be considered useful to simplify descriptions in flame calculations. The method has shown that the approach of a multi-dimensional flame may be considered a group of one-dimensional flames [30]. For the numerical simulation of premixed counter-flames, Steady Laminar Flamelet (SLF) was used. Also, in order to carry out the numerical simulation, continuity, momentum, and energy equations were solved for the counter-flame burner system. The K-\(\varepsilon\) viscous model was used in this simulation. The simulation was established with several assumptions: \begin{table}
\begin{tabular}{c c c c} \hline
**No.** & **Number of Nodes** & **Number of Elements** & **Flame Temperature (\({}^{\circ}\)C)** \\ \hline
1 & 145,620 & 95,788 & 1225 \\
2 & 280,630 & 115,678 & 1142 \\
3 & 415,630 & 278,944 & 1105 \\
4 & 623,451 & 467,782 & 972 \\
5 & 788,467 & 697,833 & 966 \\ \hline \end{tabular}
\end{table} 
Table 2: Grid independency test result. Figure 9: Variation in average time with total element number. * There was one-dimensional laminar flame in the outward area. * Air and fuel were assumed to be blended together consistently so that the ratio of air to fuel was of a unified order and phenomena of evaporation could be ignored. * There was no interaction with fuel and air prior to the combustion process. * Both gasses were regarded as two interaction continua with constant densities. * One of the main steps that was beneficial for calculations of premixed flamelet models was using the pressure-velocity coupling in ANSYS Fluent, which was handled using the [SIMPLE/SIMPLEC/PISO] algorithm. This choice was made to balance numerical stability and convergence efficiency, particularly in handling the flame characteristics observed in this study. * For gradient evaluation in the spatial discretization scheme, the Green-Gauss Cell-Based method was used. This selection was based on its robustness in capturing flow gradients with minimal numerical diffusion, ensuring the accurate interpolation of velocity and temperature fields. The development of chemical reaction flows is represented through a group of differential equations that appear as the conservation of mass and momentum, energy, and species mass fractions [30]. The equations are the continuity equation,  
\[\
abla.(\rho v)=0 \tag{1}\]  
where \(\rho\) is the mass density and \(v\) is the flow velocity; the momentum equation,  
\[\rho v.\
abla v=-\
abla p+\
abla.\tau \tag{2}\]  
where \(p\) denotes the pressure and \(\tau\) is the viscous stress tensor; the conservation equation for the \(B\) species mass fractions \(J_{i}\),  
\[\
abla.(\rho vJ_{i})-\
abla.\bigg{(}\frac{1}{Le_{i}}\frac{k}{cp}\
abla J_{i} \bigg{)}=\psi_{i}^{+}-\psi_{i}^{-},\ \ \ \ \ \ i=1,\dots,\ B \tag{3}\]  
and the balance equation for the enthalpy \(h\),  
\[\
abla.(\rho vh)-\
abla.\bigg{(}\frac{k}{cp}\
abla h\bigg{)}=\
abla.\bigg{(} \sum_{1}^{B}h_{i}\bigg{(}\frac{1}{Le_{i}}-1\bigg{)}\frac{k}{cp}\
abla J_{i} \bigg{)} \tag{4}\]  
where the Lewis numbers \(Lei=k/\rho Dicp\), where \(Di\) is the diffusion coefficient of the species \(i\), are introduced, the thermal conductivity is \(k\), the specific heat is \(cp\). The chemical source term is partitioned into a production component, \(\psi+\), and a consumption component, \(\psi-\), representing the positive and negative contributions, respectively. The pressure is determined by the ideal gas law.