## 3 Model Test Results
### Temperature Field Evolution Parallel to Freezing Pipes  
The temperature field parallels to the freezing pipe at the tunnel vertical center line after positive freezing for 60 min is shown in Figure 13.  
Figure 13 shows the morphology of the frozen wall, which is thicker at the ends and thinner in the middle; it is intuitive and obvious that the frozen wall is non-uniform. The average thickness of the frozen wall at three sections after positive freezing for 44 and 60 min is shown in Figure 14.  
Figure 14 shows that the average thickness of the frozen wall at three sections is not all equal after freezing for 44 min and 60 min. After freezing for 44 min, the average thickness of  
Figure 14: The average thickness of frozen wall at three sections.  
Figure 13: The temperature field parallels to the freezing pipe at tunnel vertical center line.  
the frozen wall at Sections 1-1, 2-2 and 3-3 is 35.9 mm, 30.7 mm and 37.6 mm, respectively; after freezing for 60 min, the average thickness of the frozen wall at Sections 1-1, 2-2 and 3-3 is 50.6 mm, 40.7 mm and 75.1 mm, respectively. The order of the average thickness of the frozen wall at three sections is Section 3-3 \(>\) Section 1-1 \(>\) Section 2-2, which presents the non-uniformity of a frozen wall along the length direction of the freezing pipe.  
The pipe wall temperature fitting curves along the length direction of the freezing pipe are shown in Figure 15 after freezing for 4 min, 28 min, 44 min and 60 min.  
Figure 15 shows that the pipe wall temperature first increases and then decreases along the length direction of the freezing pipe, where it reaches the maximum at Section 2-2. After freezing for 4 min, 28 min, 44 min and 60 min, the temperature fitting functions are as follows:  
\[\left\{\begin{array}{ll}T=-0.00015625z^{2}+0.06138z+14.92187&t=\quad 4\text{ min}\\ T=-0.00005z^{2}+0.0215z-11.55&t=28\text{min}\\ T=-0.0000425z^{2}+0.01725z-12.25625&t=44\text{min}\\ T=-0.000045z^{2}+0.0205z-13.5125&t=60\text{min}\end{array}\right. \tag{10}\]  
The conclusion obtained from the model test is inconsistent with the common sense that the alcohol temperature in the outer pipe gradually rises from the distal end to the proximal end of the freezing pipe; the main reason lies in the structure of the freezing pipe. As shown in Figure 16, the freezing pipe used in this model test is composed of an outer pipe, an inner pipe and a three-pipe joint; it is also commonly used in practical engineering. The low-temperature alcohol enters from the inner pipe, flows through the inner pipe to the end of the freezing pipe, and then flows back to the three-pipe joint through the outer pipe. When the alcohol flows in the outer pipe, it indirectly cools the soil by absorbing the heat transferred by the soil to the pipe wall, so as to achieve the purpose of freezing soil. Usually, the temperature of the pipe wall should gradually rise along the flow direction of the alcohol in the outer pipe, and the temperature of the pipe wall on each section should be shown as Section 3-3 \(<\) Section 2-2 \(<\) Section 1-1. However, the alcohol flowing in the outer pipe not only has heat exchange with the outer pipe wall, but also has heat exchange with the inner pipe wall, and there is also heat exchange between the inner pipe wall and the alcohol flowing in the inner pipe. The complex heat exchange process eventually leads to the non-uniform distribution of the temperature of the outer pipe wall of the freezing pipe, and the specific thermodynamic theory needs further study.  
Figure 15: Pipe wall temperature fitting curves along the length direction of the freezing pipe.