# Long-Term Thermo-Hydraulic Numerical Assessment of Thermo-Active Piles--A Case of Tropical Soils
## 2 Numerical, Modeling of the Long-Term Performance of Thermo-Active Piles
### Heat Transfer Mechanisms
#### 2.1.1 Heat Transfer between Thermo-Active Geostructures and Soil Masses  
In soils, conduction is usually the dominant process, but if the groundwater flow is present, then advection can also be important [26]. In thermo-active piles, conduction takes place in the concrete and the pipe wall, while convection happens within the pipes owing to the internal flow of the heat transfer fluid. This study assumes that the heat transfer in thermo-active pile systems can be simplified into two phenomena: the pile-ground conduction and the groundwater flow carried advection. To estimate the total thermal exchange between these two media, most existing studies use the following equation:  
\[Q_{total}=m_{f}c_{f}(T_{in}-T_{out}) \tag{1}\]  
where \(Q_{total}\) is the heat exchange (W), \(m_{f}\) is the mass flow rate of the heat carrier fluid (kg/s), \(c_{f}\)is the specific heat of the flowing fluid (J/kg/K), \(T_{in}\) and \(T_{out}\) (K) are respectively the fluid temperatures at inlet and outlet. Considering that the difference between inlet and outlet temperatures is usually lower than 5 degrees, it approximates the total thermal transfer efficiency as a first approach. However, this approach can only assess the total heat exchange. Also, an average temperature is applied to the finite line heat sources (see Section 2.2.2). Therefore, a strategy developed by [PERSON] (2017) and [PERSON] et al. (2018) to evaluate the conductive and advective heat exchange by thermo-active geostatructures and the surrounding soil is adopted [27; 28]. The energy balance equation can generally be expressed in its differential form, which represents the contribution of conduction and advection as:  
\[C_{eff}\frac{\partial T}{\partial t}+div\left(\overset{\rightarrow}{J}_{ cond}\right)+div\left(\overset{\rightarrow}{J}_{adv}\right)-j_{int}=0 \tag{2}\]  
where \(C_{eff}\) is the effective volumetric specific heat (J/m\({}^{3}\)/K), \(T\) is the temperature (K), \(\overset{\rightarrow}{J}_{cond}\) is the conductive heat flux (W/m\({}^{2}\)), \(\overset{\rightarrow}{J}_{adv}\) is the advective heat flux and \(j_{int}\) is the production of internal volumetric heat (W/m\({}^{3}\)) which is normally ignored in terms of the limited depth of the structures and the geology. \[C_{eff}=\rho_{g}c_{g}+n_{g}S\rho_{w}c_{w} \tag{3}\]  
where \(S\) is the degree of saturation, \(n_{g}\) is the porosity, \(\rho_{g}c_{g}\) and \(\rho_{w}c_{w}\) are the volumetric specific heat of soil and water (J/kg/K), respectively. This differential form of energy balance equation makes it possible to reveal the divergence of fluxes by conduction and advection in thermal transfers (depending on the presence or absence of groundwater flow). The conductive or advective divergence can be calculated at each instant for each solid element. Thus, the divergence of the advective flux is defined by the following equation:  
\[div\left(\overset{\rightarrow}{J}_{adv}\right)=\overset{\rightarrow}{\
abla \cdot}\overset{\rightarrow}{J}_{adv}=\overset{\rightarrow}{\
abla\cdot} \left(\rho_{w}c_{w}\overset{\rightarrow}{v}_{D}T_{(x,y,z)}\right)=\rho_{w}c_ {w}(\overset{\rightarrow}{v}_{D}\cdot\overset{\rightarrow}{\
abla}T_{(x,y, z)}+T_{(x,y,z)}\overset{\rightarrow}{\
abla\cdot}\overset{\rightarrow}{v}_{D}) \tag{4}\]  
where \(\overset{\rightarrow}{J}_{adv}\) is the vector of the advective term for each zone \(i\) and at each time \(t\), \(\rho_{w}\) is the density of water (kg/m\({}^{3}\)), \(c_{w}\) is the specific heat of the water (J/kg/K), \(\overset{\rightarrow}{v}_{D}\) is the fluid specific discharge (Darcy velocity in m/s), and \(T_{(x,y,z)}\) is the zone temperature.