# Heat-Flow Coupling Law for Freezing a Pipe Reinforcement with Varying Curvatures
## 4 Discussion--Single-Pipe Model
### Analysis of Freezing Wall  
When assessing the freezing effect, one crucial criterion for evaluation is the thickness of the freezing curtain (or freezing wall) in the freezing project. This measurement plays a pivotal role in determining the overall effectiveness of the freezing process. Taking the main surface as the dividing line, the pipe is divided into three parts--the upstream section, the middle section, and the downstream section--and the thickness of the curtain is measured both when the water is stagnant and when it is flowing. Figure 4 illustrates the variation in the difference in curtain thickness away from the main surface after 58 days of freezing under both static water and dynamic water conditions in various pipes. The permafrost curtains formed by different tubes under hydrostatic conditions exhibit approximately equal thickness. This similarity arises from the fact that despite variations in tube shapes, they all share the same cross-sectional area and provide an equivalent cooling effect within the unit cross-section. Figure 4 shows that, except for the curved pipe for which s = 7, the difference in thickness is smaller with the increase in distance; i.e., the isothermal change before and after seepage is small. This also confirms that seepage has a larger impact upstream and a smaller impact downstream as per the above cloud diagram analysis; i.e., the cold volume in the upstream section is brought to the downstream section due to the seepage effect.  
Figure 3: Temperature field clouds for 58 d of freezing under hydrostatic (**left**) and dynamic (**right**) conditions for each pipe type.  
The difference between the cross-section at positions \(-\)500 mm and 500 mm from the main surface and at 1250 mm and 2000 mm is not much different, and the interpolated value at the middle section is slightly higher, which indicates that there is not much difference in the effect of seepage either. However, there is a small difference in the value of the change in the thickness of the freezing wall of the different tubes. When analyzing the upstream sections at positions \(-\)2000 mm and \(-\)1250 mm from the main surface, it is observed that the difference in the thickness of the freezing curtain diminishes as the long arc ratio s decreases (resulting in an increase in 'H'). This suggests that as the arc of the freezing pipe becomes higher, and there is less variation in the thickness of the freezing wall, indicating a greater \"blocking\" capacity of the fluid. The difference is smallest for s = 7, and the decrease in the difference is large compared to the other tubes, which indicates that a tube with this shape freezes better than the other tubes and produces a curtain to resist deformation. This is because the freezing pipe in the upstream section has a larger inclination with the direction of water flow, which leads to a larger angle between the freezing wall and the water flow, and consequently a better water-insulation effect. As the value of s decreases and the curvature increases, the freezing wall upstream of the freezing pipe gradually becomes perpendicular to the water flow, and the coverage of this \"vertical wall\" is wider; hence, the larger the curvature of the curved pipe structure is, the more that the freezing effect of the water barrier becomes the most significant in the upstream region, followed by the middle section, and with the smallest value in the downstream section. In summary, the curved freezing tube exhibits an improved freezing-wall structure when subjected to force in the context of water flow, as compared to the scenario where the freezing-wall thickness remains relatively unchanged with no seepage. This suggests that the curved freezing tube offers a certain \"offset\" effect compared to the freezing wall created by a straight tube. Furthermore, the \"counteracting\" effect becomes more pronounced with smaller s-values, emphasizing the heightened \"offset\" effect in such cases.