## 2 Mathematical Model of the Freezing Temperature Field and Related Physical Parameters
### Mechanical Model of the Freezing Temperature Field  
Vertical shaft freezing temperature field calculations are unstable heat conduction problems with phase transitions, moving boundaries, internal heat sources, and complex boundary conditions. Because the size of the frozen soil wall in the vertical direction is much larger than that in the horizontal direction and the heat conduction of the soil layer in the vertical direction is much weaker, the freezing temperature field of the vertical shaft can be simplified as a plane heat conduction problem. In addition, the differential equation of thermal conductivity in column coordinates is obtained by simplifying the freezing temperature field equation of a single tube. The development of the temperature field in different angles is the same under the axis of the freezing tube as the Z-axis for the freezing temperature field of a single tube; therefore, this model can be considered a spatial axisymmetric structure, then the equation is simplified to a one-dimensional parameter model, and its heat conduction equation is:  
\[\frac{\partial T_{n}}{\partial t}=a_{n}\bigg{(}\frac{\partial^{2}T_{n}}{ \partial r^{2}}+\frac{1}{r}\frac{\partial T_{n}}{\partial r}\bigg{)}(t>0,\;0< r<\infty) \tag{1}\]  
where \(T_{n}\) is the distribution of temperature (\({}^{\circ}\)C), n is the state of soil (\(n=1\) represents the soil is the unfrozen soil, \(n=2\) represents the soil is the frozen soil), \(t\) is the freezing time (s), \(r\) is the circular cylindrical coordinates (m), \(a_{n}\) is the coefficient of temperature conductivity (m\({}^{2}\).s\({}^{-1}\)), \(a_{n}=\lambda_{n}/c_{n}\) and \(\lambda_{n}\) are the thermal conductivity (J\(\cdot\)s\({}^{-1}\).m\({}^{-1}\).\({}^{\circ}\)C\({}^{-1}\)), and \(c_{n}\) is the volume specific heat (J\(\cdot\)m\({}^{-3}\).\({}^{\circ}\)C\({}^{-1}\)).  
Figure 2: Schematic diagram of cross-loop of frozen soil walls.  
The temperature of the soil is constant before the freezing process:  
\[T(r,0)=T_{0} \tag{2}\]  
The temperature field at infinity is not affected by freezing, and its temperature is:  
\[T(\infty,t)=T_{0} \tag{3}\]  
The temperature on the frozen front is always the freezing temperature \(T_{d}\):  
\[T(\xi_{N},t)=T_{d} \tag{4}\]  
The heat balance equation on either side of the frozen front is [41]:  
\[\lambda_{2}\frac{\partial T_{2}}{\partial r}\bigg{|}_{r=\xi_{N}}-\lambda_{1} \frac{\partial T_{1}}{\partial r}\bigg{|}_{r=\xi_{N}}=\psi\frac{d\xi_{N}}{dt} \tag{5}\]  
The temperature of the freezing tube arrangement ring is:  
\[T(R_{0},t)=T_{c} \tag{6}\]  
where \(T_{0}\) is the initial temperature (\({}^{\circ}\)C), \(T_{\rm d}\) is the freezing temperature of the soil (\({}^{\circ}\)C), \(T_{c}\) is the temperature of the brine (\({}^{\circ}\)C), \(R_{0}\) is the distribution radius of the freezing tube (m), \(\xi_{N}\) is the coordinates of the frozen front in the \(N\) region (m), \(N=1\) represents the internal distribution region of the freezing tube, \(N=2\) represents the external distribution region of the freezing tube, and \(\psi\) is the amount of latent heat per volume of soil during freezing (kJ\(\cdot\)kg\({}^{-1}\)).