# Analysis of Pressure Effect on Three-Dimensional Flame Surface Density Estimation
## 5 Results and Discussion
### Global Fuel Consumption Rate Tests  
Once three-dimensional flame surface density \(\Sigma\) on the measurement plane is estimated, the global fuel consumption rate of the burner, \(W\), can be obtained by the integration of the \(\Sigma\) over the flame brush [3],  
\[W=\int_{0}^{+\infty}\int_{0}^{+\infty}\Sigma\rho_{0}S_{L}\tau xdxdy \tag{18}\]  
where \(\rho_{0}\) is the density of the unburned mixture and \(S_{L}\) represents the unstretched laminar flame speed. \(S_{L}\) and \(\rho_{0}\) are obtained by CHEMKIN II [33] with the GRI 3.0 [34] mechanism in the current study. The properties and the parameters of the tested mixtures for different pressures are summarized in Table 1. We can see that the laminar flame speed decreases with pressure, with its value halved under 1.0 MPa compared to 0.1 MPa. Moreover, the unburned gas density increases linearly with pressure, since we consider the unburned mixture as an ideal gas. The global fuel consumption rate \(W\) integrated from Equation (18) represents the fuel consumption speed of the system. Therefore, \(W\) should be identical to the fuel supply from the burner outlet \(\rho_{0}US_{0}\), with \(U\) being the bulk outlet velocity from the burner exit and \(S_{0}\) measuring the area of the burner exit. Thus, the ratio of global fuel consumption rate to the amount of unburned mixture supplied to burner \(R\) can be viewed as a parameter that can be used to test the 3D flame surface density correction. \[R=\frac{\int_{0}^{+\infty}\int_{0}^{+\infty}\Sigma\rho_{0}S_{L}\tau xdxdy}{ \rho_{0}US_{0}} \tag{19}\]  
In current study, the area of the burner exit is 3.14 \(\times\) 10\({}^{-4}\) m\({}^{2}\) and the outlet velocities are listed in Table 1. Table 2 lists the ratios obtained from models described in earlier sections at different pressures. We can see that the \(R\) values obtained by M1 and M2 are larger than those of M3 and M4 for atmospheric-pressure flames, indicating that M3 and M4 represent more reasonable methods to evaluate flame surface density and the fuel consumption rate. This would be due to the fact that the 3D FSD obtained by M1 and M2 only considers the difference between 3D and 2D through a common proportional constant within the flame brush. In other words, these two models neglect the difference between the flame wrinkles at different positions within the turbulent flame zone. However, flames often acquire more wrinkled structures at the flame's tip, while having less wrinkled structures at the flame's base; this could be the main cause of the overestimation. Moreover, the models from the normal vector analysis can obtain more satisfactory results for the flame at atmospheric pressure. For M3 and M4, the 3D FSD is overestimated only by about 10% on average, which can be considered as a good result. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Mixture** & \(\mathbf{\phi}\) & **P (MPa)** & \(S_{L}\) (cm/s) & \(\rho\) (kg/m\({}^{3}\)) & \(\mathbf{U}\) (m/s) & \(\mathbf{U}\tau\mathbf{r}^{2}\) (m\({}^{3}\)/s) \\ \hline CH\({}_{4}\)/air & 0.9 & 0.1 & 30.97 & 1.12 & 2.92 & 0.000917 \\ CH\({}_{4}\)/air & 1.0 & 0.5 & 18.78 & 5.57 & 3.0 & 0.000917 \\ CH\({}_{4}\)/air & 1.0 & 1.0 & 14.49 & 11.14 & 3.0 & 0.000917 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the CH\({}_{4}\)/air mixtures at different pressures. \begin{table}
\begin{tabular}{c c c c c c} \hline
**P (MPa)** & **2D** & **M1** & **M2** & **M3** & **M4** \\ \hline
0.1 & 0.96 & 1.40 & 1.23 & 1.19 & 1.09 \\
0.5 & 0.52 & 0.75 & 0.66 & 0.57 & 0.56 \\
1.0 & 0.45 & 0.65 & 0.58 & 0.51 & 0.49 \\ \hline \end{tabular}
\end{table} 
Table 2: The value of \(R\) obtained by different models at pressure of 0.1, 0.5 and 1.0 MPa. In comparison, when the flame is operated at 0.5 MPa, \(R\) shows a clear decrease for all models.