## 4 Conclusions  
1. During the freezing process of FSPR, formation of the frozen curtain is largely dependent on two types of freezing tubes to freeze the soil between the jacking pipes, and achieve the purpose of sealing water. Taking this as the main research object of the freezing steady-state temperature field, the model is assumed and simplified in combination with actual situation of the Gongbei tunnel project. Using the conformal mapping function and the separation-variable solution method, the analytical solution expression for the steady-state temperature field of FSPR under different soil freezing points is deduced, which is a quick calculation method that can be used by engineers and technicians during the designing stage and to evaluate the effect of on-site freezing construction.
2. Different characteristic parameters and finite element software can be used to establish and solve the corresponding two-dimensional steady-state temperature field numerical calculation model. The correctness and accuracy of the analytical solution are verified by comparing the results. In this project, the calculation result is acceptable when the soil freezing point range is 0\(\sim\)--1.5 \({}^{\circ}\)C.
3. Combined with the contour map of the steady-state temperature field, it is shown that when the number of frozen tubes is large, that is, the spacing between the freezing tubes is small, the shapes of the inner and outer boundaries of the frozen soil curtain can be approximately regarded as circular rings in the steady state.
4. The temperature difference of the three sections is larger in the region closer to the freezing tube. 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 calculated results show that the temperature range of this area within the size range of the pipe is \(-10\)\({}^{\circ}\)C\(\sim\)\(-28\)\({}^{\circ}\)C, implying that a reliable frozen soil curtain can be formed between the jacking pipes to ensure the effect of \"freeze-sealing\" and safety.
5. How to adapt the analytical solution of temperature field to the operating state of various frozen tubes is a problem that requires further investigation. In addition, the actual tunnel section is closer to that of the ellipse, and considering these conditions, the analytical solution also deserves further exploration.  
Conceptualization, Y.D.; methodology, Y.D.; software, Y.D. and W.L.; validation, Y.D. and C.R.; writing--original draft preparation, Y.D.; writing--review and editing, Y.D. and X.H. All authors have read and agreed to the published version of the manuscript.  
Figure 14: Two-dimensional steady-state temperature field based on analytical solution.  
**Funding:** This research was funded by Scientific Research Startup Funding for Introducing Talents of Anhui University of Science and Technology, grant number 2022 jirc55; National Natural Science Foundation of China, grant number 51878005.  
**Data Availability Statement:** The datasets generated and analyzed during the current study are available from the corresponding author upon reasonable request.  
**Acknowledgments:** The authors would like to thank all the reviewers who participated in the review and MJEditor (www.mjeditor.com, accessed on 10 October 2022) for its linguistic assistance during the preparation of this manuscript.  
**Conflicts of Interest:** The authors declare no conflict of interest.