## 2 Establishment of the Calculation Model for FSPR Steady State Temperature Field
### Conformal Mapping and Calculation Model Transformation  
Considering that it is difficult to directly solve Equation group 1, conformal mapping should be considered during the conversion of circular boundary conditions of the model into the corresponding linear boundary conditions, thus, the logarithmic transformation function is introduced [33; 34]:  
\[\left\{ \begin{array}{l}\zeta=i\ln\left(\frac{Z}{R_{2}}\right)\\ Z=re^{i\theta};\text{Object plane}\\ \zeta=x+iy;\text{Image plane}\end{array} \right. \tag{2}\]  
where by \(Z\) represents the object plane (i.e., the original plane in Figure 3), \(r\) and \(\theta\) represent a point in the object plane, \(\zeta\) represent the image plane, \(x\) and \(y\) represent a point in the image plane, we can obtain:  
\[x+iy=-\theta+i\ln\left(\frac{r}{R_{2}}\right) \tag{3}\]  
From Equations (2) and (3), we can convert the computational model in Figure 3 into the non-equidistant single-row tube with asymmetric development of the frozen curtain in the image plane, as shown in Figure 4:  
Figure 4: Freezing temperature field model in image plane of non-equidistant single-row tube with asymmetric development of frozen curtain. This model can be calculated using Equation groups (4) and (5):  
\[\left\{ \begin{array}{l}x=-\theta,y=\ln\frac{r}{R_{2}}\\ \xi_{1}=\ln\frac{R_{2}}{R_{1}},\xi_{2}=\ln\frac{R_{3}}{R_{2}}\\ l=\frac{2\pi}{n},d=\beta,r_{0}=\frac{R_{0}}{R_{2}}\end{array} \right. \tag{4}\]  
\[\left\{ \begin{array}{l}\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{ 2}T}{\partial y^{2}}=0\\ T(nl,r_{0})=T_{f};\text{Boundary condition for A'-type freezing tubes}\\ T(d+nl,r_{0})=T_{f};\text{Boundary condition for B'-type freezing tubes}\\ T(x,\xi_{2})=T_{0};\text{Frozen soil boundary II, }T_{0}\
eq 0\\ T(x,-\xi_{1})=T_{0};\text{Frozen soil boundary I, }T_{0}\
eq 0\end{array} \right. \tag{5}\]  
where by, Equation group (4) denotes the conformal mapping function relationship between the models in Figures 3 and 4. Equation group (5) denotes the expressions of the model in Figure 4.  
If the two types of freezing tubes A\({}^{\prime}\) and B\({}^{\prime}\) are separated, two linear single-row tubes' (equidistantly spaced) models of asymmetric frozen soil curtains can be obtained. Therefore, based on the separation variable solution method [35; 36; 37], the model in Figure 4 is regarded as a superposition of the two types of linear single-row tube equidistant arrangement models A\({}^{\prime}\) and B\({}^{\prime}\) (Figure 5):  
Mathematical expressions of A\({}^{\prime}\)-type and B\({}^{\prime}\)-type linear single-row tube models are shown in Equation groups (6) and (7), respectively:  
\[\left\{ \begin{array}{l}\frac{\partial^{2}T_{1}}{\partial x^{2}}+\frac{ \partial^{2}T_{1}}{\partial y^{2}}=0\\ T_{1}(nl,r_{0})=T_{f}-a;\text{Boundary condition for A'-type freezing tubes}\\ T_{1}(d+nl,r_{0})=b;\text{Boundary condition for B'-type freezing tubes}\\ T_{1}(nl,\xi_{2})=T_{0};\text{Frozen soil boundary II, }T_{0}\
eq 0\\ T_{1}(nl,-\xi_{1})=T_{0};\text{Frozen soil boundary I, }T_{0}\
eq 0\end{array} \right. \tag{6}\]  
Figure 5: Freezing temperature field model in image plane of two kinds of linear single-row tubes: (**a**) A\({}^{\prime}\)-type linear single-row tube; (**b**) B\({}^{\prime}\)-type linear single-row tube. \[\left\{ \begin{array}{l}\frac{\partial^{2}T_{2}}{\partial x^{2}}+\frac{\partial^{2 }T_{2}}{\partial y^{2}}=0\\ T_{2}(nl,r_{0})=a;\text{Boundary condition for A'-type freezing tubes}\\ T_{2}(d+nl,r_{0})=T_{f}-b;\text{Boundary condition for B'-type freezing tubes}\\ T_{2}(nl,\xi_{2})=T_{0};\text{Frozen soil boundary II},\ T_{0}\
eq 0\\ T_{2}(nl,-\xi_{1})=T_{0};\text{Frozen soil boundary I},\ T_{0}\
eq 0\end{array} \right. \tag{7}\]  
where the model temperature field \(T\) (Equation group (5)) is the superposition of temperature fields \(T_{1}\) and \(T_{2}\) of two different frozen soil boundary conditions, namely: \(T=T_{1}+T_{2}\). \(a\) and \(b\) are the coefficients to be solved.