## 4 Results and Discussion  
Acoustic Thermometry Reconstruction of Two-Dimensional Temperature Field for Experimental Conditions  
Acoustic thermometry is a well-established and reliable temperature measurement technique for power plant boilers that operate continuously. Acoustic thermometry devices are typically installed above the top layer of the burner in practical power plants and boiler applications due to limitations imposed by the combustion chamber and boiler structure. This positioning allows them to obtain temperature distribution information in a 2D plane. In this study, the traditional acoustic thermometry algorithm is used to reconstruct the temperature field at a height of 1 m in the experimental conditions considered for the mentioned boiler model. According to the arrangement shown in Figure 3(a), eight acoustic measurement points are set up, and a receiver and transmitter are arranged at each measurement point. After receiving the sound waves, all receivers move clockwise to the next transmitter position; this process is repeated until all transmitters have emitted sound waves, completing one cycle. This configuration creates an 8-transmitter, 24-channel acoustic thermometry system. Using the relationship between the sound velocity and temperature, the TOF for sound transmission on each channel can be calculated, obtaining TOF data for 24 paths. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Experimental Condition**} & **Primary Air** & **Secondary** & **Primary Air** & **Secondary Air** & **Tertiary Air** & **Tertiary Air** \\  & **Velocity** & **Air Velocity** & **Temperature** & **Temperature** & **Velocity** & **Temperature** \\  & **(m/s)** & **(m/s)** & **(K)** & **(K)** & **(m/s)** & **(K)** \\ \hline
1 & 28.75 & 40 & 400 & 600 & 50 & 600 \\
2 & 29.75 & 40 & 400 & 600 & 50 & 600 \\
3 & 30.75 & 40 & 400 & 600 & 50 & 600 \\
4 & 36.75 & 40 & 400 & 600 & 50 & 600 \\
5 & 37 & 40 & 400 & 600 & 50 & 600 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Boundary conditions for the test datasets. Figure 4: Schematic diagram of the acoustic measurement point location path and coarse grid temperature. **(a)** Arrangement and path of acoustic measurement points; **(b)** coarse grid temperature distribution. On the one hand, based on the TOF data from the 24 paths, the average temperature across 16 discrete grid points at the height of 1 m can be reconstructed. On the other hand, the TOF data from these 24 paths can be compiled into a single-layer TOF vector, denoted as Tm in Equation (10), to provide a small amount of actual measurement data for the subsequent reconstruction algorithm. Considering condition 2, the average temperature can be solved at a height of z = 1 m within the 16 coarse grid points shown in Figure 3(b). The 16 coarse grid cell temperatures obtained from Figure 3(b) are treated as the geometric center temperatures of each coarse grid cell. Moreover, third-order polynomial interpolation is applied in the 2D plane. The result regarding the temperature distribution of the planar model is shown in Figure 4(b), presenting a reconstruction error rate of 16.11%. The reconstruction results are then compared with the reconstructed temperature field obtained using the new method. As depicted in Figure 4(a), the temperature field generated by simulating the combustion of highly volatile coal powder using the Fluent software maintains a skewed single-peaked pattern with a four-corner cut-circle shape, presenting a complex temperature distribution. Traditional acoustic thermometry is constrained by the limited number of measurement points and paths; thus, it only provides a small number of average temperatures for coarse grids, as shown in Figure 3(b). Subsequently, the conventional interpolation method is employed to derive finer grid temperatures; however, it also suffers from inherent limitations, such as large extrapolation errors and low smoothness.