# Group Pile Effect on Temperature Distributions inside Energy Storage Pile Foundations
## 3 Description of Thermal Loading and Models
### Analytical Model  
A quarter of the pile and surrounding soil with a thickness equal to \(s\)/2 was cut out because of the symmetric plan of the group pile foundation, as shown in Figure 5a. Then, it is modeled as 2D plane strain elements in the general-purpose finite element software Abaqus FEA (see Figure 5b), one of  
Figure 4: Daily energy storage cycle for 2-story, \(7\times 7\) m column spacing, \(d_{i}\) = 200 mm: (**a**) energy available for storage; (**b**) temperature; and (**c**) pressure. Figure 3: Maximum storage (**a**) temperature and (**b**) pressure vs. column spacing. typical 2D simplifications used in structural simulations [27]. Symmetry boundary conditions were applied as there is no heat flux passing through the boundaries. The inner diameter (\(d_{i}\)) was selected as 200 mm. The thickness of the surrounding soil was calculated based on the pile spacing values discussed in the section above. The concrete section was divided into a mesh size of 10 by 20 mm with equal dimensions, whereas the soil layer has a gradually increasing size for the mesh along the radial direction. The dynamic thermal transfer simulation was carried out by imposing cyclic thermal loading at the interior surface of the concrete section. The backward Euler method with unconditional stability was used in the time integration procedure, which uses a backward difference algorithm (Equation (3)) to solve the basic energy balance equation (Equation (4)). The detailed solution algorithm and equations can be found in [28]. \[\dot{U}_{t+\Delta t}=(U_{t+\Delta t}-U_{t})(1/\Delta t) \tag{3}\]  
\[\int_{V}\rho\dot{U}dV=\int_{V}qdS+\int_{V}rdV \tag{4}\]  
where \(V\) is the volume of the model; \(S\) is the surface area; \(\rho\) is the density of the material; \(\dot{U}\) is the material time rate of the internal energy; \(q\) is the heat flux per unit area; and \(r\) is the heat supplied internally per unit volume. The integration time increment was automatically adjusted during the simulation to control the Courant-Friedrichs-Lewy value to be no more than 1.0. The concrete cracking effect on the thermal properties was ignored because the proposed new energy storage foundation uses high-performance concrete limiting the development of cracks [5; 29]. The cyclic thermal loading follows the temperature change curve of a one-day cycle, as shown in Figure 4b. One thousand cycles of the thermal loading were conducted to ensure the convergence to a steady-state condition. The predefined initial temperature of the model was assigned as 12 \({}^{\circ}\)C, which resides within a range of normal underground temperatures (10 to 15 \({}^{\circ}\)C according to [30]). The thermal properties of concrete were selected based on the results of the parametric study discussed in [7]. Since the energy storage pile requires high-strength concrete according to [5], this paper selected values of the thermal conductivity (2 W/m-\({}^{\circ}\)C) and specific heat capacity (940 J/kg-\({}^{\circ}\)C), respectively, which are suitable for high strength concrete [31; 32]. Figure 5: Analytical model of (**a**) group piles and (**b**) 2D plane strain model. The thermal conductivity for the soil was selected for mild moist (9%) fine sand at 1.0 W/m-\({}^{\circ}\)C, according to [33]. This value will typically reduce with the decrease in moisture content in the soil. For the energy storage pile, the moisture content in the soil is expected to decrease due to heating. The thermal conductivity of the soil will reduce as the temperature increases, which makes using a temperature-dependent nonlinear thermal material model necessary. However, considering that the number of cycles required to a reach steady-state is quite large (approximately 1000 cycles), a linear thermal material model with a fixed thermal conductivity was used to save computational cost. This simplification is deemed reasonable and provides a conservative temperature distribution for the engineering design purpose by ignoring the possible reduction of thermal conductivity [7; 14; 19].