## 3 Simulation Tests and Analysis
### Simulation Model  
In the test model, gas fuel (propane) and air entered a burner through different pipes, and a mixture of gas fuel and air burned in the furnace. The furnace dimensions were 0.8 m in length, 0.5 m in width, and 0.5 m in height. The radius of the air pipe and fuel pipe was 0. 016 m. The coordinates of the air pipe center and fuel pipe center were 0.25, 0, and 0.1 m and 0.25, 0, and 0.15 m, respectively. The tail flue transverse dimensions were 0.15 m in width, 0.15 m in height, and 0.2 m in length. Figure 1 shows the geometry of the simulation model. The simulation data from the plane at 0.08 m of the furnace height were used to test the proposed method and to present the reconstruction results. Tetrahedral grids were employed for the fuel pipe and the air pipe. The furnace was structured by hexahedral grids to improve mesh quality. Five different mesh grids with node numbers of 452,997, 609,363, 784,370, 1,019,816 and 1,210,028, were used to test the mesh independency. The grid convergence index (GCI) [38] was also used in this test:  
\[\text{GGI}=\text{F}_{\text{S}}\frac{|\varepsilon|}{\text{r}^{\text{P}}-1}, \tag{13}\]  
\[\varepsilon=\frac{\text{T}_{\text{k}}-\text{T}_{\text{k}+1}}{\text{T}_{\text{ k}}}, \tag{14}\]  
\[\text{r}_{\text{k},\text{k}+1}=\frac{\text{h}_{\text{k}}}{\text{h}_{\text{k}+1 }}=\sqrt[3]{\frac{\text{N}_{\text{k}+1}}{\text{N}_{\text{k}}}}, \tag{15}\]where \(\rm{F_{S}}\) is the safety factor and \(\rm{p}\) is the order of grid convergence. \(\rm{T_{k}}\) is the highest temperature in the simulation result using the k-grid, \(\rm{T_{k+1}}\) is the highest temperature in the simulation result using the k + 1 grid, \(\rm{N_{k}}\) is the grid nodes number in the k grid, and \(\rm{N_{k+1}}\) is the grid nodes number in the k + 1 grid. The convergence condition is defined as follows:  
\[\rm{c}=\frac{\rm{GCI_{k,k+1}}}{\rm{r_{k,k+1}^{p}}\rm{GCI_{k,k-1}}}, \tag{16}\]  
GCI converges when \(\rm{c}\approx 1\). The result is shown in Table 1.  
As shown in Table 1, \(\rm{c}\) in the 784,370 grid nodes is 0.91, and the 784,370, 1,019,816 and 1,210,028 grid nodes are in the asymptotic range of convergence. In consideration of the efficiency and accuracy, 784,370 grid nodes were used in the simulation model. The basic governing equations in the simulation process were the conservation equations of mass, momentum, energy, species, and ideal gas state equation [39], which are as follows:  
(1) Mass conservation equation:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho V)=0, \tag{17}\]  
(2) Momentum conservation equation:  
\[\frac{\partial(\rho V)}{\partial t}+\
abla\cdot(\rho VV)=-\
abla\rm{p}+\
abla \cdot(\tau)+\rho g+F, \tag{18}\]  
(3) Energy conservation equation:  
\[\frac{\partial(\rho E)}{\partial t}+\
abla\cdot[V(\rho E+\rm{p})]=\
abla\cdot[ k_{\rm{eff}}\
abla T-\sum_{i}h_{i}\rm{J_{i}}+(\tau_{\rm{eff}}\cdot V)]+S_{h}, \tag{19}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Nodes Number** & **GCI (\%)** & \(\rm{r^{p}}\) & \(\rm{c}\) \\ \hline
452,297 & 2.59 & 1.21 & – \\
609,363 & 2 & 1.18 & 0.65 \\
784,370 & 2.18 & 1.19 & 0.91 \\
1,019,816 & 2.33 & 1.12 & 0.95 \\
1,210,028 & – & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 1: GCI convergence test. Figure 1: Schematic diagram of the simulation model.