## 3 Numerical Simulation
### Governing Equations  
The coupled heat transfer between the water and soil was achieved by utilizing a non-isothermal pipe flow and a solid heat transfer module. The soil was added to the solid heat transfer module as a porous medium. In addition, the phase change interface was introduced to simulate the phase change process of the PCM, and the apparent heat capacity method was used to describe the latent heat. The control equation is as follows:  
(1) Non-isothermal pipe flow  
\[\frac{\partial A\rho_{\rm w}}{\partial t}+\
abla\cdot(A\rho_{\rm w}u)=0 \tag{2}\]  
\[\rho_{\rm w}\frac{\partial u}{\partial t}=-\
abla p-\frac{1}{2}f_{\rm D}\frac{ \rho_{\rm w}}{d_{\rm h}}|u|u+F \tag{3}\]  
\[\rho_{\rm w}A\C_{\rm P}\frac{\partial T}{\partial t}+\rho_{\rm w}A\C_{\rm P}u \cdot\
abla T=\
abla\cdot(Ak\
abla T)+\frac{1}{2}f_{\rm D}\frac{\rho_{\rm w}}{ d_{\rm h}}|u|u^{2}+Q_{\rm wall} \tag{4}\]  
where \(A\) is the cross-sectional area of the pipe (m\({}^{2}\)), \(\rho_{\rm w}\) is the density of water, \(\rho\) is the density of the ground soil (kg/m\({}^{3}\)), \(u\) is the flow velocity of the circulating medium (m/s), \(p\) is the pressure (N/m\({}^{2}\)), \(\C_{\rm P}\) is the constant pressure heat capacity (J/kg\(\cdot\)K), \(T\) is the temperature (K), \(F\) is the volume force considering the density change, \(k\) is the thermal conductivity (W/(m\(\cdot\)K)), and \(Q_{\rm wall}\) is the heat transfer between the fluid and the wall (W/m). In the equation, the volume force considering the density change was assumed, assuming that the circulating fluid was an incompressible Newtonian fluid, with \(F\) set to zero [40]. (2) Heat transfer in solids  
\[\rho\C_{\rm P}u\cdot\
abla T=\
abla\cdot(k\
abla T)-Q_{\rm wall} \tag{5}\]  
Equation (4) describes the heat exchange between the fluid and the pipe wall. Assuming that the soil was a porous medium material, the model did not consider the heat exchange phenomenon caused by groundwater flow and convective heat transfer. (3) Phase change interface  
The material parameters of PCM change during the phase transition process. The changes in its key parameters of volume density and thermal conductivity were calculated by the following formula [36]:  
\[\rho=\theta(T)\rho_{\rm phase1}+(1-\theta(T))\rho_{\rm phase2} \tag{6}\]  
\[k=\theta(T)k_{\rm phase1}+(1-\theta(T))k_{\rm phase2} \tag{7}\]  
where \(\theta(T)\) is the phase mass fraction, phase 1 represents the solid phase of PCM, and phase 2 represents the liquid phase of PCM. The equivalent heat capacity method was used to simulate the phase change process through the interface of PCM in a solid heat transfer module. During the test, a hollow steel ball with a wall thickness of 0.3 mm and a diameter of 19 mm was used to enclose the phase change paraffin. Due to the substantial quantity of steel balls in the FRPC pile, equivalent material parameters for the FRPC pile were experimentally determined and used as input parameters for the finite element model. The specific heat capacity \(C_{\rm p}\) of the phase change material was related to latent heat and calculated as follows:  
\[C_{\rm p}=\frac{1}{\rho}\Big{(}\theta(T)\rho_{\rm phase1}C_{\rm p,phase1}+(1- \theta(T))\rho_{\rm phase2}C_{\rm p,phase2}\Big{)}+L\frac{\partial\alpha_{m}(T)}{ \partial T} \tag{8}\]  
where \(\alpha_{\rm m}\) is a smooth transition function to prevent the numerical instability caused by the rapid change in the specific heat capacity. The equation for calculation is as follows:  
\[\alpha_{\rm m}(T)=\frac{1}{2}\frac{\Big{(}(1-\theta(T))\rho_{\rm phase2}- \theta(T)\rho_{\rm phase1}\Big{)}}{\theta(T)\rho_{\rm phase1}+(1-\theta(T)) \rho_{\rm phase2}} \tag{9}\]  
This parameter only works during the time interval of the phase transition temperature. In this study, the phase transition interval was 2 \({}^{\circ}\)C; a smaller interval would result in a sharper transition edge. This study aimed to investigate the effect of steel fibers and PCM on the performance of energy piles.