# Group Pile Effect on Temperature Distributions inside Energy Storage Pile Foundations
## 3 Description of Thermal Loading and Models  
### Determination of Storage Temperature  
Thermodynamic cycles of CAES can be used to calculate the temperature and pressure in the air stored inside the pile. The equations obtained from the energy balance principals and ideal gas equations can be expressed as follows:  
\[\frac{V_{s}}{C^{7/2}}\left(\frac{T-\eta_{2}T_{1}}{1-\eta_{2}}\right)^{5/2}= \frac{\dot{w}_{in}\eta_{1}t_{in}}{3.5R}\Big{/}\left(\frac{T-\eta_{2}T_{1}}{1- \eta_{2}}-T_{1}\right)+\frac{\rho_{i}V_{s}}{\mu} \tag{1}\]  
\[P=\frac{RT}{C^{7/2}}\left(\frac{T-\eta_{2}T_{1}}{1-\eta_{2}}\right)^{5/2} \tag{2}\]  
where the definitions and values of the parameters in Equations (1) and (2) are presented in Table 1. By solving the equations above with available storage energy (\(\dot{w}_{in}\)) and available volumes (\(V_{s}\)) for storage, the temperature and pressure demand for the pile can be determined. The derivation of Equations (1) and (2) was explained in detail in [6].  
Using Equations (1) and (2), the storage pressure and temperature were calculated for different building geometries and design parameters. The detailed calculations were presented in [6] for different cases including numbers of building stories, different column spacings, and different inner diameters of the pile. From the calculations, a critical case was chosen, which has a 7-m column spacing, two stories, and a 200-mm pile inner diameter. This case produces the highest storage pressure and temperature, as show in Figure 3. As the column spacing increases, the stored energy for one single column will increase due to the increase in the tributary area. On the other hand, the larger column spacing will result in a longer pile length due to the higher tributary loading area, which can provide a larger available volume for energy storage. Therefore, the storage temperature and pressure are not sensitive to the change in column spacing. For a given column spacing, different pile group designs can be developed which permit the study of different pile spacings using the same temperature loading for the 7-m column spacing case.  
\begin{table}
\begin{tabular}{c l c} \hline \hline
**Parameters** & **Definitions** & **Adopted values** \\ \hline R & Universal gas constant & 8.31 J/(mol-K) \\ T\({}_{1}\) & Ambient temperature & 20 \({}^{\circ}\)C \\ C & Thermal constant & 10.89 \\ t\({}_{in}\) & Compressing time & 3600 s \\ \(\rho_{i}\) & The initial density of air & 1.2 kg/m\({}^{3}\) \\ \(\mu\) & The molar mass of air & 0.029 \\ \(\eta_{1}\) & Efficiency of compression & 75\% \\ \(\eta\)\({}_{2}\) & The efficiency of heat exchange and storage & 90\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters in the thermodynamic equations.  
Figure 4 illustrates storage energy (Figure 4a), calculated storage temperature (Figure 4b), and storage pressure (Figure 4c) for a 24-hour cycle from 7:00 to 7:00 (next day) for the case of a 2-story building, \(7\times 7\) m column spacing, and 200-mm inner diameter (\(d_{\text{i}}\)). When the solar energy starts to be generated and the supply is higher than the demand, the available storage energy starts accumulating at 7:00 (see Figure 4a). When the solar energy becomes unavailable and the supply is less than the demand, the stored energy starts releasing at 19:00 (see Figure 4a). The temperature and pressure follow the energy storage and release cycle increase from 7:00, reaching maximum values at 19:00 (the middle of the cycle) when the storage energy starts to become unavailable. Then, temperature and pressure decrease to release the energy, returning to their original value at 7:00 of the next day. This temperature and pressure loading cycle might have lower values if the available storage energy reduces. However, for engineering design purposes, the temperature and pressure shown in Figure 4 is treated as the most critical case and is applied for the analyses presented in this paper.