# Preliminary Study on Reconstruction of Building Thermal Field Based on Iterative Algorithm Acoustic CT

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## Abstract

**:**

## 1. Introduction

## 2. Principle of 3D Temperature Field Reconstruction for Acoustic CT

_{j}is the Sound-slowness, whereby the acoustic wave ray passes through the reciprocal of the average velocity of each grid, in (m/s)

^{−1}.

## 3. Acquisition of Building Fire Foundation Temperature Field Function

- The geometric model was established by FDS, and the simulation parameters such as fire power and grid size were set.
- The data acquisition sensors were arranged in each grid to calculate the discrete data of the fire base temperature.
- The discrete data was fitted by the BP neural network algorithm, and the distribution function (the form of neural network) of the base temperature data at ahigh fit was obtained.

- (1)
- The geometric model and the setting of key parameters was established

- (2)
- Acquisition of the base temperature field function and the fitting of the function

## 4. Reconstruction of the Fire Temperature Field Based on Acoustic CT

#### 4.1. Arrangement of Acoustic Transceiver and Division of Grid

#### 4.2. Solution Method of the Acoustic Matrix Equation

- (1)
- ART algorithm

- (2)
- SART algorithm

## 5. Analysis of the Reconstruction Results

#### 5.1. Comparative Analysis of Three-Dimensional Overall Reconstruction Temperature Field Cloud Map

- (1)
- The reconstruction results under the 5 × 5 grid scheme (Figure 6(b1)) exhibit a high-temperature distribution pattern that is similar to the baseline temperature data (Figure 6a). The reconstruction results under the 6 × 6 grid scheme (Figure 6(b2)) also show significant high-temperature regions above and around the fire source, but with a distinctly different distribution pattern compared to the baseline temperature data (Figure 6a). It is evident that the reconstruction results under the 7 × 7 grid scheme (Figure 6(b3)) and the 8 × 8 grid scheme (Figure 6(b4)) do not show any high-temperature areas near the fire source, indicating that the reconstruction results are unsuccessful.
- (2)
- Under the 5 × 5 (Figure 6(b1)) and 6 × 6 (Figure 6(b2)) grid schemes, the maximum reconstructed temperatures are 600 K and 550 K, respectively. Both reconstructed temperatures are significantly lower than the baseline temperature data (Figure 6a), but the former demonstrates better reconstruction performance than the latter.

- (1)
- The reconstructions under the four grid schemes (Figure 7b) all exhibit significant high-temperature regions, with their locations consistent with the baseline temperature data (Figure 7a). However, the reconstruction performance, especially the distribution of high-temperature regions, varies significantly among the different grid schemes. Compared to the other two schemes, the high-temperature region morphology under the 5 × 5 (Figure 7(b1)) and 7 × 7 (Figure 7(b3)) grid schemes is more similar to the baseline temperature data.
- (2)
- The highest values of the reconstructed temperature field under the grid scheme of 5 × 5 (Figure 7(b1)), 6 × 6 (Figure 7(b2)) and 8 × 8 (Figure 7(b4)) were 600 K, 600 K and 550 K, respectively, which were lower than the maximum values of the baseline temperature data (Figure 7a). The morphology and maximum value of the high temperature zone under the 7 × 7 (Figure 7(b4)) meshing scheme are close to the baseline temperature (Figure 7a) data.

#### 5.2. Two-Dimensional Section Reconstruction Temperature Field Cloud Comparison Analysis

- (1)
- Among the four grid schemes, the reconstruction results of the 5 × 5 (Figure 8(b1)) scheme and the 6 × 6 (Figure 8(b2)) scheme showed a unimodal high temperature region similar to the baseline temperature data (Figure 8a), but the maximum reconstruction temperature was significantly lower than the baseline temperature data (Figure 8a), and had a large error. The reconstruction results of the 7 × 7 (Figure 8(b3)) scheme and the 8 × 8 (Figure 8(b4)) scheme showed a multimodal phenomenon and an abnormally high temperature, so the reconstruction result failed.
- (2)

- (1)
- In the four meshing schemes, a single-peak high temperature region similar to the baseline temperature data appeared, the location of the high temperature zone was basically consistent with the baseline temperature data (Figure 9a), and the reconstruction effect was acceptable. The data fluctuations of the 6 × 6 (Figure 9(b2)) scheme and the 8 × 8 (Figure 9(b3)) scheme in the low temperature region were more obvious, and the temperature distribution pattern of the 5 × 5 (Figure 9(b1)) scheme and the 7 × 7 (Figure 9(b3)) scheme in the low temperature region was closer to the baseline temperature data, but the 5 × 5 (Figure 9(b1)) scheme may be the result of the overall flattening caused by fewer data pixels.
- (2)
- In terms of the maximum reconstruction temperature, the reconstruction result (700 K) under the 7 × 7 (Figure 9(b3)) scheme was basically consistent with the baseline temperature data (Figure 9a), and the maximum reconstruction temperature value of the other three schemes was significantly low. Therefore, the reconstruction effect under the 7 × 7 (Figure 9(b3)) grid scheme under this algorithm is the best.

#### 5.3. Quantitative Error Analysis of the Reconstruction Effect

- In the process of solving the sparse matrix equation under the reconstruction scheme, neither the ART algorithm nor the SART algorithm can converge independently under the 6 × 6 grid, and the number of iterations should be limited. Under other mesh division schemes, both methods can achieve autonomous convergence.
- For solving the sparse matrix equation under the complex distributed temperature field of a building fire, the ART algorithm takes less time than the SART algorithm in terms of single iteration time and overall convergence time, but the latter shows better robustness in overall reconstruction accuracy.
- Under the SART algorithm, the 7 × 7 meshing scheme achieves the best reconstruction effect. The correlation error index is within 6%, the correlation coefficient is 0.97, and the single-layer calculation time is within 0.5 s. This shows that the appropriate meshing scheme can realize the real-time and accurate reconstruction of the complex temperature field of a building fire under the SART algorithm.
- The meshing scheme has significant control over the reconstruction results. When the number of grids is small, the stability and convergence of the matrix equation solution will be better, but the reconstruction accuracy is poor due to the small number of reconstructed pixels. The reconstruction accuracy can be improved when there are many reconstruction pixels; however, higher requirements are proposed for solving the matrix equations. Therefore, the reconstruction effect of the fire temperature field is the result of the sparse matrix solution and the effective pixels. An optimal matching ratio between pixel meshing and effective acoustic path data exists.

## 6. Conclusions

- With the early fire stage and the relatively simple temperature field distribution constructed in this paper as the reconstruction object, the acoustic CT temperature measurement technology under the preferred reconstruction scheme was selected to achieve a good reconstruction effect.
- Under the reconstruction plan proposed in this paper, the SART algorithm has better stability compared with the ART algorithm. The SART algorithm has better reconstruction quality and is expected to be used for the 3D temperature field reconstruction of building fires in the practical application to a fire site.
- The number of effective pixel grids and the accuracy of matrix equations jointly determine the reconstruction quality of the three-dimensional temperature field of building fires. Therefore, there is an optimal matching ratio between the pixel grid division and the effective acoustic path data for a specific fire geometry space. It can be selected according to a 7 × 7 × 8 grid division under the condition of setting parameters in this paper.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${\tau}_{\mathrm{TOF},\mathrm{i}}$ | the time-of-flight (TOF) of each acoustic wave traveling |

${\omega}_{\mathrm{ij}}$ | the length of the i th path through the j th grid |

${f}_{j}$ | the Sound-slowness (m/s)^{−1} |

x | the N-dimensional temperature field vector |

b | the M-dimensional data vector of time-of-flight (TOF) of acoustic wave traveling |

C | the velocity of sound propagation in the gas medium m/s |

γ | the adiabatic index of the gas |

R | ideal gas universal constant 8.3143 J/(mol·K) |

Z | the flue gas mixture 20.045 |

T | the absolute temperature of the gas medium K |

m | average molar mass of gas kg/mol |

k | the number of iterations |

λ | the relaxation factor |

${f}_{j}^{(k)}$ | the j th pixel values in the k th sub-iterations respectively |

${f}_{j}^{(k+1)}$ | the j th pixel values in the k + 1 sub-iterations respectively |

${I}_{\theta}$ | the set of all rays at the θ irradiation angle |

M | the total number of grids divided by the measured region |

${T}_{c(i)}$ | the reconstruction temperature of the i th grid center point |

${T}_{M(i)}$ | the base temperature of the i th grid center point |

${T}_{Mave}$ | the average temperature of foundation temperature field |

${T}_{Cave}$ | the average temperature of the reconstructed temperature field |

${T}_{M\mathrm{max}}$ | the highest temperature value of the foundation temperature field |

${T}_{C\mathrm{max}}$ | the highest temperature value of the reconstructed temperature field |

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**Figure 6.**Comparison between basic temperature field data and reconstructed temperature field data cloud maps based on the ART algorithm.

**Figure 7.**Comparison between basic temperature field data and reconstructed temperature field data cloud maps based on the SART algorithm.

**Figure 8.**Comparison between basic temperature field data and reconstructed temperature field data cloud maps based on the ART algorithm (z = −5).

**Figure 9.**Comparison between basic temperature field data and reconstructed temperature field data cloud maps based on the SART algorithm (z = −5).

Parameter Category | Overall Size (m^{3}) | Door Opening Size (m^{3}) | Grid | Combustion Source | ||
---|---|---|---|---|---|---|

Division Form | Mesh Size (m^{3}) | Fire Source Location Coordinates(m) | Fire Source Size (m^{2}) | |||

Parameter value | 10 × 10 × 10 | 3 × 0.2 × 4 | Evenness | 0.5 × 0.5 × 0.5 | (0,0,−5) | 1 × 1 |

Parameter Category | Initial Temperature (K) | Air Humidity (%RH) | Burning Materials | Wall Material | Wall Thermal Conductivity (W/(m·k)) | Ignition Source Power (MW/m^{2}) |
---|---|---|---|---|---|---|

Parameter value | 293.15 | 60% | Polyurethane_GM27 | Concrete | 1.8 | 10 |

Item | Content |
---|---|

Rebuild objects | Complex temperature field of building thermal field |

Sound path | 12 transceivers, 42 valid paths |

Mesh subdivision | 5 × 5, 6 × 6, 7 × 7, 8 × 8 |

Calculation method | ART, SART |

**Table 4.**Reconstruction error of the three-dimensional temperature field of a building fire and the average calculation time of a single layer.

Algorithm | Grid Numbers | ${\mathit{E}}_{\mathbf{mean}}$ (%) | ${\mathit{E}}_{\mathbf{max}}$ (%) | $\mathit{E}$ (%) | Correlation Coefficient (R) | Single Layer Average Calculation Time (s) | Stopping Criterion for Iteration |
---|---|---|---|---|---|---|---|

ART algorithm | 5 × 5 × 8 | 5.52 | 41.25 | 10.97 | 0.90 | 0.03 | Normal convergence |

6 × 6 × 8 | 9.40 | 21.84 | 16.22 | 0.81 | 26.9 | Iteration times | |

7 × 7 × 8 | — | — | — | — | — | Normal convergence | |

8 × 8 × 8 | — | — | — | — | — | Normal convergence | |

SART algorithm | 5 × 5 × 8 | 5.51 | 40.82 | 10.91 | 0.90 | 0.12 | Normal convergence |

6 × 6 × 8 | 25.81 | 36.78 | 30.64 | 0.59 | 32.5 | Iteration times | |

7 × 7 × 8 | 4.11 | 3.48 | 5.65 | 0.97 | 0.35 | Normal convergence | |

8 × 8 × 8 | 6.12 | 28.39 | 9.48 | 0.92 | 0.53 | Normal convergence |

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**MDPI and ACS Style**

Qin, H.; Wen, J.; Gao, Z.; Chai, L.; Yao, H.
Preliminary Study on Reconstruction of Building Thermal Field Based on Iterative Algorithm Acoustic CT. *Fire* **2023**, *6*, 199.
https://doi.org/10.3390/fire6050199

**AMA Style**

Qin H, Wen J, Gao Z, Chai L, Yao H.
Preliminary Study on Reconstruction of Building Thermal Field Based on Iterative Algorithm Acoustic CT. *Fire*. 2023; 6(5):199.
https://doi.org/10.3390/fire6050199

**Chicago/Turabian Style**

Qin, Hengjie, Jiangqi Wen, Zihe Gao, Lingling Chai, and Haowei Yao.
2023. "Preliminary Study on Reconstruction of Building Thermal Field Based on Iterative Algorithm Acoustic CT" *Fire* 6, no. 5: 199.
https://doi.org/10.3390/fire6050199