# Thermomechanical Behavior of Parallel Multi-U-Shaped Energy Piles Under the Summer Condition
## 3 Numerical Simulation of the Thermal Response of the Parallel Multi-U Energy Pile
### Physical Equation of Numerical Simulation  
Based on the above assumptions, the main physical field equations utilized in the model are as follows:  
Figure 2: Full- scale parallel 5U energy pile test flowchart. Under quasi-static conditions, the equilibrium equations are shown below by neglecting the inertia effect during pile installation:  
\[\
abla\cdot\sigma+\rho g=0 \tag{1}\]  
In Equation (1), \(\
abla\cdot\sigma\) is the divergence of the total stress tensor; \(\sigma\) is the total stress tensor, positive in tension and negative in compression; \(\rho\) is the density of the material; and \(g\) is the gravitational acceleration vector. The total stress tensor \(\sigma\)[40] can be written as  
\[\sigma=C(\varepsilon+\alpha_{c}I\Delta T) \tag{2}\]  
In Equation (2), \(C\) represents the stiffness tensor, which includes the material's Poisson's ratio and Young's modulus; \(\varepsilon\) denotes the total strain tensor, whereas \(\alpha_{c}\) represents the material's coefficient of thermal expansion; \(I\) represents the unit matrix; and \(\Delta T=T-T_{0}\) represents the temperature difference. The heat exchange of the fluid in the heat exchanger tube takes place by convection, and the fluid motion can be described by the Navier-Stokes equations. \[\frac{\partial}{\partial t}\rho_{w}-\
abla\cdot(\rho_{w}u)=0 \tag{3}\]  
\[\rho_{w}\bigg{(}\frac{\partial}{\partial t}u+u\cdot\
abla u\bigg{)}=-\
abla p +\
abla\cdot\bigg{\{}\mu\Big{[}\
abla u+(\
abla u)^{T}\Big{]}-\frac{2}{3}\mu (\
abla\cdot u)I\bigg{\}}+F_{w} \tag{4}\]  
In Equation (4), \(u\) is the fluid velocity; \(p\) is the fluid pressure; \(\rho_{w}\) is the densities of fluid; \(\mu\) is the hydrodynamic viscosity; and \(F_{w}\) is the external forces acting on the fluid. The equation of conservation of energy [8] when only thermal effects are considered is  
\[\rho c_{p}\frac{\partial T}{\partial t}-\
abla\cdot(\Lambda\
abla T)=0 \tag{5}\]  
In Equation (5), \(c_{p}\) is the constant pressure heat capacity of the material; \(t\) is the time; \(\
abla\cdot\) is the divergence; and \(\lambda\) is the thermal conductivity of the material. When heat transmission in a porous medium is considered in the pile-soil heat transfer process, the energy conservation equation becomes [38; 41]  
\[[\theta_{s}\rho_{s}c_{s}+(1-\theta_{s})\rho_{w}c_{w}]\frac{\partial T}{ \partial t}-\
abla\cdot\{[\theta_{s}\lambda_{s}+(1-\theta_{s})\lambda_{w}] \
abla T\}=0 \tag{6}\]  
In Equation (6), \(\theta_{s}\) is the volume fraction of the solid; \(\rho_{s}\) is the densities of solid particles in the pores; \(c_{s}\) and \(c_{w}\) are the constant pressure heat capacities of the two materials; and \(\lambda_{s}\) and \(\lambda_{w}\) are the thermal conductivity of the two materials. In this study, the circulating medium in the heat exchanger tube of the energy pile is water, the cross-section of the heat exchanger tube is set as a circle in the model, and the energy conservation equation of its circulating liquid is [42; 43]  
\[\rho_{f}A_{p}c_{f}\frac{\partial T}{\partial t}+\rho_{f}A_{p}c_{f}u_{f}\cdot \
abla T_{f}=\
abla\cdot\Big{(}A_{p}\lambda_{f}\
abla T_{f}\Big{)}+Q_{wall} \tag{7}\]  
In Equation (7), \(\rho_{f}\) is the density of the water; \(\rho_{p}\) is the density of the pipe; \(c_{f}\) is the constant pressure heat capacity of the water; \(u_{f}\) is the flow rate of the water; \(T_{f}\) is the temperature of the water; \(\lambda_{f}\) is the thermal conductivity of the water; and \(Q_{wall}\) expresses the heat flux generated by the pipe, which is given by [38; 44; 45]  
\[Q_{wall}=h_{eff}\Big{(}T_{ext}-T_{f}\Big{)} \tag{8}\]In Equation (8), \(h_{eff}\) is the effective value of the heat transfer coefficient of the pipe and \(T_{ext}\) is the external temperature of the heat exchange pipe.