## 3 Numerical Calculation Analysis
### The Effect of the Layout Radius of the Freezing Tube  
The layout radius of the freezing tube has an important effect on the freezing effect and freezing time. Figure 10a shows the temperature evolution curves in the interlayer (\(I_{1}\)) and outside wall (\(O_{1}\)) of the shaft wall under different layout radii of freezing tube (\(R_{d}\)) and different freezing time. The temperatures in the interlayer and outside position gradually decrease with an increase in freezing time, while the variance in the temperature in these two positions is relatively clear, and the amplitude of variation gradually becomes greater when the layout radius of the freezing tube slowly decreases. For example, the temperature in the interlayer position between the layout radius of the freezing tube of 5.5 m and 5.7 m is 8.3 \({}^{\circ}\)C and 6.7 \({}^{\circ}\)C, respectively, and this value decreases to 6.3 \({}^{\circ}\)C when the layout radius of the freezing tube ranges from 6.1 m to 6.3 m (within 180 d). The amount of cooling per unit space becomes higher and the transfer rate of cooling capacity is greater when the layout radius of the freezing tube decrease. The incremental cooling effect would also increase due to the larger temperature gradients. Thus, the value of the layout radius of the freezing tube has a greater impact on the temperature distribution of the shaft wall under ventilation conditions. The temperature (\(T\)) and thickness (\(\delta\)) evolution inside (\(I_{2}\)) and outside (\(O_{2}\)) the frozen soil wall and its average (\(\overline{T}\)) or sum value (\(\delta_{0}\)) with the layout radius of the freezing tube of 5.5 m and 6.3 m are presented in Figure 10b. The decreasing trend of temperature and the increasing trend of thickness of the frozen soil wall are similar under different layout radii of freezing tubes, while the frozen soil wall thickness shows an obvious increase when the layout radius of the freezing tube increases. For example, the sum thicknesses of the frozen soil wall with the maximum and minimum layout radii of the freezing tube are 0.55 m and 0.44 m, respectively, when the freezing time reaches 30 days, and the sum thickness (\(\delta_{0}\)) of the frozen soil wall increases to 3.06 m and 2.65 m when the freezing time reaches 180 days, which means that the appropriate increase in the layout radius of the freezing tube can speed up the freezing efficiency, and the decrease in temperature and increase in thickness will make the frozen soil wall more stable and provide better shaft construction protection under ventilation conditions. Figure 9: Temperature contours within 30 d, 90 d, and 180 d.  
The temperature contours with the layout radius of the freezing tube of 5.5 m and 6.3 m are plotted to better demonstrate the freezing evolution laws and the effect of the layout radius of the freezing tube on this process, as shown in Figure 11a,b. The temperature of the freezing interface decreases from 0.5 \({}^{\circ}\)C to \(-\)8 \({}^{\circ}\)C (below 0 \({}^{\circ}\)C) as the freezing time increases when the layout radius of the freezing tube is 5.5 m, and this value decreases to \(-\)19 \({}^{\circ}\)C within 180 d when the layout radius of the freezing tube increases to 6.3 m, which means that the frozen soil wall completed the cross-loop in 1800 under ventilation with different freezing tube layout radii. The larger layout radius of the freezing tube shows a better cross-loop effect.