## 2 Mathematical Model of the Freezing Temperature Field and Related Physical Parameters
### Calculation Procedure
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Layout radius of freezing tube (\(R_{d}\))/m & 5.5 & Tube pitch (\(L_{d}\))/m & 1.45 \\ Thermal conductivity of unfrozen soil & 1.4 & Number of freezing tubes & 24 \\ The thickness of the wall (\(\delta\))/m & 1.68 & Moisture content (\(\omega\))/\% & 20 \\ Diameter of freezing tube (\(D\))/mm & 140 & Initial ground temperature /\({}^{\circ}\)C & 25 \\ Wet density of soil /kg\(\cdot\)m\({}^{-3}\) & 2000 & Annual average temperature of wind & 25 \\ Density of concrete/kg\(\cdot\)m\({}^{-3}\) & 2500 & Outside temperature of the freezing & \(-\)30 \\ Specific heat of unfrozen soil/kg\(\cdot\)m\({}^{-1.\circ}\)C\({}^{-1}\) & 0.8 & Inside temperature of the shaft wall/\({}^{\circ}\)C & 25 \\ Specific heat of unfrozen soil /kj\(\cdot\)kg\({}^{-1.\circ}\)C\({}^{-1}\) & 0.9 & Specific heat of concrete /kj\(\cdot\)kg\({}^{-1.\circ}\)C\({}^{-1}\) & 1.1 \\ \hline \end{tabular}
\end{table} 
Table 3: Typical parameter values in the single-factor calculation. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Layout radius of freezing tube (\(R_{d}\))/m & 5.5 & Tube pitch (\(L_{d}\))/m & 1.45 \\ Thermal conductivity of unfrozen soil & 1.4 & Number of freezing tubes & 24 \\ The thickness of the wall (\(\delta\))/m & 1.68 & Moisture content (\(\omega\))/\% & 20 \\ Diameter of freezing tube (\(D\))/mm & 140 & Initial ground temperature /\({}^{\circ}\)C & 25 \\ Wet density of soil /kg\(\cdot\)m\({}^{-3}\) & 2000 & Brine temperature /\({}^{\circ}\)C & \(-\)35 \\ Density of concrete/kg\(\cdot\)m\({}^{-3}\) & 2500 & Outside temperature of the freezing & \(-\)30 \\ Thermal conductivity of concrete /kg\(\cdot\)m\({}^{-3}\) & 2500 & Brine temperature /\({}^{\circ}\)C & \(-\)35 \\ Thermal conductivity of concrete /W\(\cdot\)m\({}^{-1.\circ}\)C\({}^{-1}\The plane quadrilateral element is used to divide the grid, the thinnest grid is applied to the frozen area, followed by the predicted frozen area, and the sparsest grid is set at the outer boundary. The schematic diagrams of grid division are shown in Figure 6a,b. In the temperature analysis of ANSYS software, the convergence criterion for temperature is 0.5 \({}^{\circ}\)C.  
To verify the reliability of this model, the field temperature of the frozen soil wall in the east ventilation shaft of the Chengjiao coal mine was obtained from the thermometer hole (with the temperature sensors of Ocean-1010), as shown in Figure 1, and is compared with the average temperature of the frozen soil wall in the theoretical calculation in this study. The fixed depths of the frozen soil wall (below ground level) in the east ventilation shaft are selected as \(-100\), \(-200\) and \(-305\) m, respectively, and the freezing time is approximately 180 d. In this freezing project, the criterion for the correspondence between experimental measurements and theoretical calculation temperature is the coincidence of these two curves and the temperature difference at 90 d and 120 d. It can be found that the decreasing law of these two kinds of temperatures is consistent, these two curves gradually match, and the temperature difference is below 1 \({}^{\circ}\)C at 120 d, as shown in Figure 7a,b, and these two kinds of values are also similar when the initial ground temperature gap caused by the temperature gradient of topsoil is ignored. In addition, the temperatures of the frozen soil wall (all factors were selected as the typical value) were simulated, and the calculated time delay ratio and relative error were selected as evaluation factors for grid sensitivity tests, as presented in Figure 7b. Although the increase in the grid refinement ratio could decrease the relative error and improve the accuracy of the model, the calculation time was found to rapidly increase with an increase in the grid refinement ratio. Thus, the mesh C7 (as shown by red pentagram) was selected for the calculation model.