# Heat-Flow Coupling Law for Freezing a Pipe Reinforcement with Varying Curvatures
## 5 Modeling and Analysis of Multiple Pipes
### Discussion  
In Section 4, the different two-dimensional types of single-pipe model-calculation results were compared. Then, the multi-pipe model was mainly used in the thickness direction of the temperature field and seepage field to carry out the study. Two solid frozen single-pipe intermediate weak zones were selected for analysis, where the straight pipe and the length of the arc ratio of the s = 7 arc pipe model were selected with a Y = 0.25 m cross-section.  
As can be seen from Figure 8, after 58 days of freezing, the weakest points in the two multi-pipe models had completely frozen in a cross-circle pattern. The overall formation indicates an average temperature consistently lower than \(-10\)\({}^{\circ}\)C within the stable freezing area. As shown by the temperature field in the cloud diagram, particularly within the red line boundary defined by the freezing wall, the main freezing area of the straight pipe's weakest cross-section exhibits an average thickness of approximately 1.12 m. As one moves away from the main surface, at a distance of \(-2000\) mm from it, the thickness of the freezing wall measures around 1.19 m. The average thickness of the freezing wall in the main freezing area of the s = 7 curved pipe is about 1.19 m, and the thickness of the freezing wall at the distance of \(-2000\) mm from the main surface is up to 1.31 m. In the arc-shaped pipe, the average thickness of the freezing wall is 70 mm, and at a position \(-2000\) mm away from the main surface, the difference in the freezing wall's thickness is 120 mm, as it is for the single pipe in Section 3.2. This verifies that the arc-shaped pipe is able to maintain a relatively thick freezing wall in the positive freezing period under the action of seepage flow. Secondly, the inter-circle time of the freezing wall can also reflect the freezing effect of different tubes. The intersection of the \(-10\)\({}^{\circ}\)C isotherm and \(-1\)\({}^{\circ}\)C isotherm is different: the minimum freezing zone of the curved pipe starts to appear on the 4 th d, and the maximum freezing zone starts to appear on the 5 th d, while the minimum straight freezing zone starts to appear on the 5 th d, and the maximum freezing zone starts to appear on the 6 th d. To summarize, the arc-shaped pipe freezing is nearly 1 d earlier than the intersection time of the straight pipe freezing.  
In addition, in order to further study the freezing effect of different tubes, Figure 9 shows the isovelocity line variation cloud map of the X = 2.7 m cross-section, and Table 4 makes a comparison of different ranges of isovelocity line envelopes.  
In Section 4.3, two-dimensional calculations yielded smaller seepage velocities at the same location on the water surface of the curved pipe model, which provides a better blockage of the water flow. The same velocities are also chosen here for comparison. Observation of the cloud diagram reveals that the magnitude of the velocity change is large in the middle part, which is because it is in the upstream region, i.e., in the lower end of the freezing tube, and it gradually decreases higher up as the seepage velocity increases progressively away from the freezing tube. When the seepage velocity is the same, the isovelocity line envelope range is larger for the s = 7 curved pipe, while the straight pipe's envelope range is slightly smaller and the inner velocity of the envelope is smaller than the outer velocity. This suggests that as the coverage increases, the velocity within the region becomes broader and slower, leading to a more significant waterproofing  
Figure 8: Cloud view of the freezing temperature field of multiple tubes with Y = 0.25 m cross-section.  
effect on the waterfront side. Conversely, when the coverage is smaller, this effect is less pronounced. This observation further confirms the conclusion that the curved pipe demonstrates superior water-blocking capabilities.