# Heat-Flow Coupling Law for Freezing a Pipe Reinforcement with Varying Curvatures
## 6 Effect of Working Conditions
### Analysis of Results  
The relevant results are shown in Figure 12. The primary focus of the analysis here is on the waterfront side, with the key distinction being the difference in freezing pipe types, while all other parameters and conditions remain the same. A layer of frozen soil curtain was wrapped around the tunnel after freezing reinforcement, with frozen soil within the brown area (\(-1\)\({}^{\circ}\)C contour) and stable frozen soil curtain within the blue area (\(-10\)\({}^{\circ}\)C contour). Since the curtain range exceeds the boundary of the highly permeable water soil layer, the final result shows that there is almost no water infiltration into the tunnel, and a waterproofing effect is achieved. After comparative analysis, it was observed that in the case of the straight pipe freezing, the \(-1\)\({}^{\circ}\)C isotherm of the freezing curtain began to close the intersection circle on the 7 th day and continued to expand outward, completing its basic closure on the 10 th day. Figure 11: Geometric schematic diagram of shallowly buried concealed tunnel model. Figure 12: Equivalent surface diagram for different tubes frozen at 58 d.  
Conversely, the \(-10\)\({}^{\circ}\)C isotherm in this configuration started intersecting with the circle on the 10 th day and thickened over time. For the arc-shaped pipe freezing, the \(-1\)\({}^{\circ}\)C isotherm initiated its closure of the intersection circle as early as the 6 th day and continued to expand outward, reaching its basic closure on the 10 th day. The \(-10\)\({}^{\circ}\)C isotherm, in this case, began closing the intersection circle on the 11 th day and also exhibited a gradual thickening trend. This indicates that when using freezing methods for curved and straight tubes, the contour of the curtain exhibits a consistent pattern. This consistency can be attributed to the alteration of the freezing tube's curvature while maintaining constant freezing tube spacing and cross-section dimensions. Under these conditions, the cold generated by different freezing tubes diffuses at an equal rate in the direction of the tunnel excavation, aligning with the hydrostatic conditions outlined in Section 4.2. Consequently, various tube curvatures result in the same thickness for the freezing wall, and the time for achieving \(-1\)\({}^{\circ}\)C and \(-10\)\({}^{\circ}\)C closures follows a similar pattern. For the curtain on the water-facing side, the excavated face of the section with the most unfavorable reinforcement zone and the largest freezing distance is taken as the object of study, the thickness of the curtain is analyzed, and the change in the thickness of the stabilized freezing curtain of the section with the freezing process is plotted, along with the changes at the temperature-measurement apertures C2 and C3 on the water-facing side. The thickness of the stable freezing curtain of the section with the freezing process is analyzed (Figure 13). As can be seen from the temperature plot, the temperature changes in three distinct steps--the sudden change freezing section, continuous freezing section, and stable freezing section. The freezing process occurs in distinct stages. Initially, at the onset of freezing until the phase-change stage, the stratum temperature undergoes a rapid decline, with intense heat exchange, leading to a steep curve drop, known as the sudden change in the freezing section. As freezing progresses, temperatures continue to decrease, marking the transition into the continuous freezing section. During this phase, the temperature difference between the freezing tube and the stratum diminishes, resulting in a reduction in the rate of heat exchange and a slower decline in temperature compared to the continuous freezing section, signifying the stabilization of the freezing section. When analyzing temperature measurements from specific holes, it becomes evident that the temperature drop is more pronounced and results in a lower final freezing temperature in the No. 2 hole for both freezing tubes. However, the temperature in the No.