## 3 Numerical Calculation Analysis
### The Effect of the Convective Heat Transfer Coefficient  
Figure 8a shows the temperature evolution curves of the interlayer (\(I_{1}\)) and outside (\(O_{1}\)) wall of the shaft wall with different convective heat transfer coefficients (\(\alpha\)) and freezing time (\(t\)). The results indicate that the temperatures in the interlayer and outside wall of the shaft all gradually decrease with an increase in freezing time, while the temperature difference between these two positions is no more than 9 \({}^{\circ}\)C approximately. Due to the huge decrease in the trend difference before and after 30 freezing days, a kink to represent the obvious temperature change in the first 30 freezing days is added in the temperature figures. The temperature of the interlayer dropped by approximately 6.32 \({}^{\circ}\)C after 180 freezing days, and the lowest temperature was approximately 13.18 \({}^{\circ}\)C with a convective heat transfer coefficient of 30 W\(\cdot\)m\({}^{-2.\circ}\)C\({}^{-1}\), but the lowest outside temperature dropped to 4.98 \({}^{\circ}\)C. In addition, the variance in the temperature of these two positions is relatively small when the convective heat transfer coefficient constantly increases, and the freezing time is invariable. For example, the greatest temperature difference in the interlayer is approximately 0.5 \({}^{\circ}\)C. Thus, the value of the convective heat transfer coefficient has little influence on the temperature of the interlayer and outside wall under ventilation conditions. Figure 8b shows 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 a convective heat transfer coefficient of 30 W\(\cdot\)m\({}^{-2.\circ}\)C\({}^{-1}\). The temperature drop is mainly concentrated in the initial freezing stage. The temperature inside and outside the frozen soil wall first rapidly decreases during the initial 60 freezing days, while the average value decreases from 19.5 \({}^{\circ}\)C to \(-\)7.5 \({}^{\circ}\)C, and then the reduction rate of temperature decreases while the average value decreases by approximately 1 \({}^{\circ}\)C in the next 120 freezing days. Meanwhile, the temperature in the outside position (\(T_{(O_{2}/\alpha=30)}\)) is constantly higher than that in the inside position (\(T_{(I_{2}/\alpha=30)}\)) under normal freezing construction, while the temperature difference between these two positions is basically no more than 0.7 \({}^{\circ}\)C. The thickness of the frozen soil wall starts to grow when the temperature passes 0 \({}^{\circ}\)C, and the growth rate also increases with continuous freezing, while the largest increase occurs during the 30-60 freezing days, and the sum thickness increases by approximately 1.2 m. With the continuation of freezing and the gradual thickening of the frozen soil wall, the growth rate gradually decreases while the thickness in the outside position exceeds the thickness inside the frozen soil wall. The relatively more concentrated cooling capacity transfer leads to a lower temperature in the inside position, while the larger external fields and the inside ventilation effect tend to make the thickness outside of the frozen soil wall continue to increase and exceed that in the inside position. To better show the development process of the frozen soil wall, the temperature contours of the calculation analysis within 30 d, 90 d, and 180 d are drawn and shown in Figure 9. The distribution of the freezing temperature gradually changes from a circle to an oval with a decrease in temperature, as the cooling capacity in the freezing region between the two freezing tubes is the most sufficient, while the temperature drops fastest at the mid-point of the center line of the two freezing tubes and then gradually diffuses outward, with this point as the center. Thus, a bulge exists in the contour when the shape of the freezing region changes.