# Study on the Effect of Insulation Materials on the Temperature Field of Piles in Ice-Rich Areas
## 3 Modeling of Pile-Soil Heat Transfer  
### Numerical Calculation Method  
The infiltration and flow of soil moisture during the thawing and freezing of permafrost were not considered in any of the finite element calculations. The mathematical control equations of the entire computational model were described by the heat transfer equation in transient three-dimensional Cartesian coordinates with a phase transition [16; 17] heat balance control differential equations for the problem of phase change temperature field are as follows:  
Figure 4: Comparison of the thickness of different insulation materials with identical thermal conductivity.  
Figure 3: The construction site situation.  
The differential equation for the thermal conductivity of the transient temperature field in the solid phase region \(\Omega_{f}\) was:  
\[\frac{\partial T_{f}}{\partial t}=\frac{\lambda_{f}}{\rho C_{f}}\Bigg{(}\frac{ \partial^{2}T_{f}}{\partial x^{2}}+\frac{\partial^{2}T_{f}}{\partial y^{2}}+ \frac{\partial^{2}T_{f}}{\partial z^{2}}\Bigg{)}+\frac{q_{v}}{\rho C_{f}} \tag{1}\]  
The differential equation for the thermal conductivity of the transient temperature field in the liquid phase region \(\Omega_{u}\) was:  
\[\frac{\partial T_{u}}{\partial t}=\frac{\lambda_{u}}{\rho C_{u}}\Bigg{(}\frac{ \partial^{2}T_{u}}{\partial x^{2}}+\frac{\partial^{2}T_{u}}{\partial xy^{2}}+ \frac{\partial^{2}T_{u}}{\partial z^{2}}\Bigg{)}+\frac{q_{v}}{\rho C_{u}} \tag{2}\]  
where \(T\) is the transient temperature of the object (\({}^{\circ}\)C); \(t\) is the time for a process to occur (s); \(\lambda_{f},~{}\lambda_{u}\) are the thermal conductivity of frozen soil and thawed soil in W/(m\({}^{\circ}\)C), respectively; \(\rho\) is the density of the material; \(C_{f},~{}C_{u}\) are the heat capacities of frozen soil and thawed soil, respectively kJ/(kg\({}^{\circ}\)C), and \(q_{v}\) is the internal heat source intensity (W/m\({}^{3}\)). The two temperature fields at the phase transition interface \(\xi(t)\) need to satisfy both the temperature continuity condition and the energy conservation condition:  
\[T_{f}(\xi(t),t)=T_{u}(\xi(t),t)=T_{m} \tag{3}\]  
\[\lambda_{f}\frac{\partial T_{f}}{\partial n}-\lambda_{u}\frac{\partial T_{u} }{\partial n}=L\rho\frac{d\xi(t)}{dt} \tag{4}\]  
\(T_{m}\) is the freezing surface temperature (\({}^{\circ}\)C); \(n\) is the direction normal to the freezing surface; \(L\) is the latent heat of phase change of the substance, and the latent heat of phase change of water is 334.56 kJ/kg.  
The calculation process uses the sensible heat capacity method to simulate the phase change calculation of the material model. That is, the phase change of water in permafrost does not occur at a fixed temperature but in a small range. We assume that the phase change occurs within a temperature range (\(T_{m}\)\(\pm\)\(\Delta\)T) around \(T_{m}\) so that we obtain the Expressions (5) and (6) for the heat capacity and thermal conductivity:  
\[C=\begin{cases}\begin{array}{c}C_{f}~{}T\leq T_{m}-\Delta\text{T}\\ \frac{L}{2\Delta\text{T}}+\frac{C_{f}+C_{u}}{2}~{}T_{m}-\Delta\text{T}\leq\text {T}_{m}+\Delta\text{T}\\ C_{u}~{}T\geq T_{m}-\Delta\text{T}\end{array}\\ \lambda=\begin{cases}\begin{array}{c}\lambda_{f}~{}T\leq T_{m}-\Delta\text{T} \\ \lambda_{f}+\frac{\lambda_{u}-\lambda_{f}}{2\Delta\text{T}}[T-(T_{m}-\Delta \text{T})]~{}T_{m}-\Delta\text{T}\leq\text{T}\leq T_{m}+\Delta\text{T}\\ \lambda_{u}~{}T\geq T_{m}-\Delta\text{T}\end{array}\\ \end{cases}\end{cases} \tag{5}\]