## 3 Numerical Calculation Analysis
### The Effect of the Freezing Tube Diameter  
Figure 14a shows the temperature evolution curves of the interlayer (\(I_{1}\)) and outside (\(O_{1}\)) wall of the shaft wall with different freezing tube diameters (\(D\)) and freezing time (\(t\)). The temperatures in the interlayer and outside wall position show a decreasing trend when the freezing tube diameter increases, while the variation magnitude is smaller than that induced by other factors, such as the freezing tube pitch and layout radius. A larger amount of freezing medium will provide more cooling capacity per time unit as the freezing tube diameter increases, resulting in a higher heat exchange efficiency and greatly contributing to the reduction in the shaft temperature. Since the diameter design of the freezing tube is in accordance with certain standards, the temperature difference brought by the limited freezing tube diameter change is not as obvious; this value is approximately 8.3 \({}^{\circ}\)C for the 140 mm freezing tube diameter and approximately 8.4 \({}^{\circ}\)C for the 159 mm freezing tube diameter (within 180 d).  
Figure 12: Variance in temperature and thickness under different freezing tube pitches and freezing time: (**a**) temperature in interlayer and outside wall of shaft, (**b**) temperature and thickness in the inside and outside of frozen soil wall and its average or sum value with freezing tube pitches of 1.3 m and 2.5 m.  
Figure 13: Temperature contours within 30 d, 90 d, and 180 d: (**a**) freezing tube pitch of 1.3 m, (**b**) freezing tube pitch of 2.5 m.  
The temperature (\(T\)) and thickness (\(\delta\)) evolution curves inside (\(l_{2}\)) and outside (\(O_{2}\)) the frozen soil wall and average (\(\overline{T}\)) or sum value (\(\delta_{0}\)) with different freezing tube diameters of 140 mm and 159 mm are presented in Figure 14b. As expected, the increase in the freezing tube diameter will decrease the frozen soil wall temperature and increase its thickness, while this effect will gradually weaken with the increase in freezing time. For example, the sum thickness (\(\delta_{0}\)) of the frozen soil wall increases by 0.18 m within 30 d when the freezing tube diameter increases from 140 mm to 159 mm, but the increase range is only 0.08 m when the freezing time reaches 180 days. This result indicates that increasing the freezing tube diameter has relatively less effect on the cross-loop of the frozen soil wall,  
Figure 14: Variance in temperature and thickness under different freezing tube diameters and freezing time: (**a**) temperature in interlayer and outside wall of shaft, (**b**) temperature and thickness in the inside and outside of frozen soil wall and its average or sum value with freezing tube diameters of 140 mm and 159 mm.  
frozen soil wall thickness, and average temperature during long-term freezing. As shown in Figure 15a,b, the temperature of the freezing interface decreases from 0.5 \({}^{\circ}\)C to \(-\)13 \({}^{\circ}\)C (below 0 \({}^{\circ}\)C) as the freezing time increases, with freezing tube diameters of 140 mm and 159 mm, indicating that the cross-loop is completed in 180 d under ventilation conditions. The freezing temperature region with freezing tube diameters of 140 mm and 159 mm gradually increases from a small semicircle to a large ellipse throughout the model. Thus, the variance in the freezing tube diameter produced based on the standard may not be the main factor controlling the formation of the frozen soil wall under ventilation conditions, while the increase in freezing tube diameter can promote the frozen soil wall to a certain extent.  
Figure 15: Temperature contours within 30 d, 90 d, and 180 d: (**a**) freezing tube diameter of 140 mm, (**b**) freezing tube diameter of 159 mm.