## 4 Numerical Results and Discussion
### Parametric Results  
This section discusses the effects of three parameters, including the inner diameter of the pile foundation (\(d_{i}\)), the thermal conductivity (\(k\)) of the concrete, and the specific heat capacity (\(c\)) of the concrete on the temperature changes and distributions inside the pile foundation. The results for different \(d_{i}\) are shown in this section with \(k\) = 2 W/m.\({}^{\circ}\)C and \(c\) = 940 J/kg.\({}^{\circ}\)C. The results for different \(k\) are presented for \(d_{i}\) = 200 mm and \(c\) = 940 J/kg.\({}^{\circ}\)C. The results for different \(c\) are presented for \(d_{i}\) = 200 mm and \(k\) = 2 W/m.\({}^{\circ}\)C.  
Figure 13 shows the temperature change in the first loading cycle for different \(d_{i}\) at: (a) the inner surface of the pile section; (b) the midpoint of the pile section; (c) the outer surface of the pile section. As seen in Figure 13a, the temperature change at the inner surface follows the input temperature loading (refer to Figure 4a) and decreases as the \(d_{i}\) increases. However, moving from the inner surface, the temperature change becomes more significant as the \(d_{i}\) increases (See Figure 13b,c). This observation indicates that the heat can be transferred to the outer surface much faster for the larger \(d_{i}\) cases with thinner section thickness, which in turn results in a more uniform temperature distribution in the concrete pile foundation, as shown in Figure 14.  
Figure 13: Temperature change in the first loading cycle for different \(d_{i}\) at: (**a**) the inner surface of the pile section; (**b**) the midpoint of the pile section; (**c**) the outer surface of the pile section. Figure 15 shows the maximum temperature in each loading cycle for different \(d_{i}\) at: (a) the midpoint of the pile section; (b) the outer surface of the pile section. For all cases, the maximum temperature increases with the loading cycle and is converged after a sufficient amount of cycles. The smaller \(d_{i}\) cases are converged at a higher temperature than the larger \(d_{i}\) cases because their input temperature loading is higher. On the other hand, the convergence speed increases with an increase of \(d_{i}\) since the heat can go through the concrete section faster for the larger \(d_{i}\) cases. Figure 16 shows temperature change in the first loading cycle for different \(k\) at: (a) the midpoint of the pile section; (b) the outer surface of the pile section. As seen in Figure 16, the temperature change becomes more significant as the \(k\) increases since the heat transfer becomes faster. As a result of the faster heat transfer, the temperature inside the pile foundation is consistently higher for the cases with larger \(k\) during the entire 24 h cycle. Also, it has been noticed that the temperature distribution in the concrete pile foundation along the radius direction tends to be slightly more uniform for the higher \(k\) cases, as shown in Figure 17.  
Figure 14: Temperature distribution in the first loading cycle for different \(d_{i}\) at: (a) 19:00; (b) 7:00 (end of the loading cycle). Figure 15: Maximum temperature in each loading cycle for different \(d_{i}\) at: (a) the midpoint of the pile section; (b) the outer surface of the pile section. Figure 18 shows the maximum temperature in each loading cycle for different \(k\) at: (a) the midpoint of the pile section; (b) the outer surface of the pile section. As expected, the maximum temperature increases with the loading cycle and is converged after a sufficient amount of cycles. The cases with higher \(k\) are converged at higher temperatures and show faster convergences compared to the lower \(k\) cases. Figure 19 shows temperature change in the first loading cycle for different \(c\) at: (a) the midpoint of the pile section; (b) the outer surface of the pile section.