## 3 Parametric Study
### Numerical Models  
The analytical model for the pile foundation and surrounding soils was developed using the general-purpose finite element software ABAQUS, as shown in Figure 6. The 2D plane strain model was constructed to represent a quarter of a whole pile section because of symmetry. The 2D model conservatively ignores the small heat transfer through the bottom of the pile foundation. Symmetry boundary conditions without heat flux passing through were applied. The surrounding soil was modeled for a thickness of 15 m, which is far enough to be not affected by the temperature change inside the pile foundation based on preliminary analysis results. The outer face of the soil was assigned a constant temperature at 12 \({}^{\circ}\)C, which falls in the range of nature temperature for the depth of the pile foundation (10 to 15 \({}^{\circ}\)C according to [36]). A relatively fine mesh (-10 to 20 mm) was applied for the concrete section. On the other hand, the mesh size in the surrounding soil gradually increases along the radius (\(r\)) direction when moving far from the pile foundation in order to save computer memory and storage. The mesh size was selected by balancing the time integration increment and computer running time to achieve the required numerical stability.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Thermal Conductivity (\(k\))** & **Specific Heat Capacity (\(c\))** & **Inner Diameter (\(d_{i}\))** \\
**[W/m-\({}^{\circ}\)C]** & **[J/kg.\({}^{\circ}\)C]** & **[mm]** \\ \hline
1.0 & 830 & 200 \\
1.5 & 890 & 300 \\
2.0 & 940 & 400 \\
2.4 & 1000 & 600 \\
2.8 & - & 800 \\
3.2 & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Study parameters.  
Figure 5: Maximum temperature in the stored compressed air: (**a**) \(5\times 5\) m column spacing; (**b**) \(6\times 6\) m column spacing; (**c**) \(7\times 7\) m column spacing.  
The thermal properties (\(k\) and \(c\)) presented in Table 1 were assigned to the concrete. For the soil, thermal conductivity and specific heat capacity for typical sand with a moisture content of 5% were used as \(k\) = 1.0 W/m-\({}^{\circ}\)C and \(c\) = 1220 J/kg-\({}^{\circ}\)C. An interface element was defined with a thermal conductance of 25 W/m\({}^{2.\circ}\)C, considering that complete contact between the concrete and soil is impossible. The density of the concrete and soil was adopted as 2450 kg/m\({}^{3}\) and 1600 kg/m\({}^{3}\), respectively.  
The non-steady state heat transfer analysis was conducted by applying the temperature change with time at the inner surface of the pile foundation. Time integration adopted the backward Euler method with unconditional stability. The Courant-Friedrichs-Lewy is controlled to be less than 1.0 by automatically adjusting time integration incrementation during the simulation. Nonlinear thermal properties and the concrete cracking effect were not considered in the simulation due to the fact that the proposed new energy storage foundation requires limited cracks or even crack-free by using fiber reinforced concrete and high strength [8; 9]. The temperature change follows the 24 h cycle, as defined in Figure 3a. The energy storage pile foundation is supposed to be subjected to many temperature loading cycles for its entire life. Therefore, for each simulation, 150 cycles were applied to ensure the variation of the temperature inside the concrete section at the same hour between adjacent cycles to be less than 0.05 \({}^{\circ}\)C. The initial temperature of the pile foundation and surrounding soils was assumed at 12 \({}^{\circ}\)C, which falls in the range of nature underground temperature for the depth of the pile foundation (10 to 15 \({}^{\circ}\)C according to [36]).