## 2 Establishment of the Calculation Model for FSPR Steady State Temperature Field  
### Model Simplifications and Assumptions  
In Figure 2, 18 concrete pipes and 18 hollow pipes were alternately arranged to form a super-large section pipe roof during the freezing construction of the pipe curtain of Gongbei Tunnel. The arrangement axis of the 18 hollow top pipes is a 5-segment circular arc with left and right symmetry, while the arc lengths and radii are \(Arc_{1}=5.45\) m and \(R_{1}=9.86\), \(Arc_{2}=4.87\) m and \(R_{2}=6.96\) m, \(Arc_{3}=11.75\) m and \(R_{3}=20.96\) m, \(Arc_{4}=3.86\) m and \(R_{4}=3.86\) m, \(Arc_{5}=6.17\) m and \(R_{5}=18.86\) m, respectively. The axes of the 18 concrete pipes are offset by 30 cm inward, thus, there is a slight dislocation of the circular freezing tubes in concrete pipes, and of the profiled freezing tubes in hollow pipes. The soil between the pipes is frozen through the cryogenic refrigerant circulation method in these two types of freezing tubes. The limiting tube is arranged on the outside of the axis of the concrete pipe; excess cold can be removed by circulating hot brine in the tube, while the thickness of the outside of the frozen soil curtain can be controlled to reduce frost heave of the stratum. The structural form and function of the FSPR method are extremely complex, therefore, appropriate assumptions and model simplification are required to ensure the feasibility of the analytical solution of the steady-state temperature field, including:  
1. The entire length of the underground excavation section of the Gongbei Tunnel is 255 m long and is curved. The actual tube curtain freezing is a three-dimensional heat conduction problem. The temperature deviation of longitudinal freezing is ignored, and it can be simplified to a two-dimensional plane problem. 2. Ignoring the irregular shape of the pipe curtain section and the slight offset between the hollow and concrete pipe axes, all 36 pipes are considered to be arranged on the same circumferential line, that is, the pipe curtain section is simplified to a circle. 3. In the actual project, due to the arrangement of the pipes, the outline of the frozen soil curtain is irregularly wavy. Considering the steady-state temperature field, the end of the freezing process is studied. For mathematical derivation convenience, it is assumed that the contour line of the frozen soil curtain is approximately a circle, and its rationality can be evaluated by verifying the analytical solution. 4. The profiled freezing tube in the hollow pipe contains a non-circular section, and its size is smaller compared with that of the jacking pipe. It is estimated to have the same section and size as the circular freezing tube in the concrete pipe. Flow and temperature differences of the low-temperature refrigerant in the two types of freezing pipes in the freezing process are ignored, and only the two types of freezing tubes with the same tube wall temperature are considered during derivation of the analytical solution. The effects of hollow and concrete pipes on the freezing temperature field are also ignored, and only the effects of freezing tubes are considered. Based on the above assumptions, the model in Figure 2 is simplified, and freezing tubes as well as frozen soil curtains are selected as the main research objects [32]. Two types of freezing tubes, A and B, with radii \(R_{0}\) are obtained, which are periodically arranged on the circumference line (\(R_{2}\)), and the spacing is set as the dislocation angle \(\beta\). The inner and outer boundaries (\(R_{1}\) and \(R_{3}\)) of the frozen soil curtain are circular (Figure 3):  
Figure 3: Simplified model of freezing temperature field of FSPR. Figure 2: Layout of pipe roof and freezing tube in Gongbei Tunnel. In Figure 3, the appropriate cartesian coordinate system is selected so that the center of a type A freezing tube is just on the positive semi-axis of the x-axis.