## 3 Accuracy Verification of the Analytical Solution
### Comparative Analysis of Calculation Results  
Calculation results and the cloud map of the temperature field are compared as shown in Figures 8-13:  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Group** & \(\mathbf{R_{1}}\)**/**m** & \(\mathbf{R_{2}}\)**/**m** & \(\mathbf{R_{3}}\)**/m** & \(\mathbf{\zeta_{1}}\)**/\(\mathbf{\zeta_{2}}\)** & \(\mathbf{D}\)**/m** & \(\mathbf{R_{0}}\)**/m** & \(\mathbf{n}\) & \(\mathbf{\beta}\)**/(\({}^{\circ}\))** & \(\mathbf{T_{0}}\)**/\({}^{\circ}\)**C** \\ \hline
1 & 6.0 & 7.0 & 8.0 & 1 & 1.46 & 0.06 & 20 & 4 & 0 \\
2 & 6.0 & 7.0 & 8.0 & 1 & 1.02 & 0.06 & 25 & 4 & 0 \\
3 & 8.0 & 9.0 & 10.0 & 1 & 1.62 & 0.06 & 36 & 2 & \(-1.5\) \\
4 & 8.0 & 9.0 & 10.0 & 1 & 1.46 & 0.08 & 36 & 3 & \(-1.5\) \\
5 & 7.9 & 9.0 & 10.0 & 1.1 & 1.59 & 0.08 & 36 & 2.6 & \(-0.5\) \\
6 & 7.9 & 9.0 & 10.0 & 1.1 & 1.59 & 0.06 & 36 & 2.6 & \(-0.5\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Characteristic parameters for analytical solution.  
Figure 6: Computational periodic element model of the analytical solution.  
Figure 8: Comparison of the 1 st group of calculation results: (**a**) Comparison curve between analytical and numerical solutions of section; (**b**) Steady-state temperature field.  
Figure 10: Comparison of the 3 rd group of calculation results: (**a**) Comparison curve between analytical and numerical solutions of section; (**b**) Steady-state temperature field.  
Figure 9: Comparison of the 2 nd group of calculation results: (**a**) Comparison curve between analytical and numerical solutions of section; (**b**) Steady-state temperature field.  
From Figures 8-13, the analytical solution and numerical solution curve for each model section under six groups of different parameters coincide, and maximum temperature  
Figure 11: Comparison of the 4 th group of calculation results: (**a**) Comparison curve between analytical and numerical solutions of section; (**b**) Steady-state temperature field.  
Figure 12: Comparison of the 5 th group of calculation results: (**a**) Comparison curve between analytical and numerical solutions of section; (**b**) Steady-state temperature field.  
Figure 13: Comparison of the 6 th group of calculation results: (**a**) Comparison curve between analytical and numerical solutions of section; (**b**) Steady-state temperature field.  
differences at each point do not exceed 0.9 \({}^{\circ}\)C. The data indicate that the steady-state temperature field model of the complicated freezing tube layout in FSPR can be simplified, and conformal mapping and the separation-variable method can be employed to solve the problem. The derived analytical solution is highly accurate, and can accurately calculate the temperature value at any point in the model.  
In the vicinity of the inner and outer boundaries of the model's frozen soil, temperatures of the three sections in each group are relatively close, corresponding to symmetrical and uniform temperature distributions on both sides of steady-state temperature field cloud map. In the area closer to the freezing tube, temperature differences among the three sections are greater, and the maximum temperature difference is located at the axial surface \(R_{2}\) of the freezing tube. The temperature curve of Section 1 is located between Section 2 and the main section, and is closer to the latter.  
In actual engineering, Section 1 is the position of the midline between the adjacent concrete pipe and hollow pipe, and is an important area for \"freeze-sealing between pipes\" in FSPR. The above figures show that the temperature range of this area within the size range of the pipe is \(-10\)\({}^{\circ}\)C\(\cdots\)\(-28\)\({}^{\circ}\)C and the distribution is relatively uniform, implying that a reliable frozen soil curtain can be formed between the jacking pipes to ensure \"freeze-sealing\" effects and safety.