## 3 Results and Discussions  
### Wall Effect on Packing Structures  
Fixed walls can significantly impact the packing characteristic of a pebble bed. The average packing factor of these pebble beds with different wall boundary conditions are shown in Figure 8. The results show that the smallest packing factor was observed in the pebble bed with four fixed walls. Utilizing four periodic boundaries in the pebble bed resulted in a higher packing factor and lower porosity. Therefore, as the quantities of fixed wall decreases, the packing factor increases and the porosity decreases. Compared to the pebble bed with four fixed walls, the average packing factor of the pebble beds with four periodic boundaries increased by 18.58% to 0.6178. This is mainly due to the lower packing factor of the pebble bed in the region close to the fixed wall. An increase in the number of fixed walls will increase the fixed wall's affected regions, which leads to a lower packing factor of the pebble bed. To investigate this phenomenon, the distribution of localized packing factors of the pebble bed in the x-y plane is calculated for different wall and boundary conditions, respectively, as presented in Figure 9. The figures show that the fixed walls have a notable influence on the local packing factor distribution. In the pebble bed with four walls, several areas near the wall exhibit high localized packing factor, resembling \"hot spots\". The corners feature greater localized packing factors as an effect of the double walls, whereas the inner regions of the pebble bed showcase more uniformly distributed packing factors. In the pebble bed with two fixed walls and two periodic boundaries, a high localized packing factor is found only near the parallel fixed walls. Meanwhile, the distribution of localized packing factor is more uniform in the inner region of the pebble bed and near the periodic boundary. As for the pebble bed with four periodic boundaries, the distribution of local packing factor is more uniform throughout the entire pebble bed. The observed distributions stem from the fact that pebble packing adjusts to the structure of a fixed wall, leading to a partially regular or layered distribution near the fixed wall. This is demonstrated in Figure 9a/c/, which exhibit similar distribution characteristics in the centers of the pebbles, while in the middle region of the pebble bed and the regions close to the periodic boundary, the influence of the fixed wall on the packing structure gradually decreases and the packing structures gradually become random and homogeneous. In addition, the area-averaged axial packing factors along different axial directions are calculated for the initial bed obtained by DEM simulation and the bed after a reduction in  
Figure 8: Packing factor of pebble beds with different boundary walls. pebble diameter to 0.98d; the results are shown in Figure 10. Similar evolutionary patterns are observed in the pebble bed before and after reductions in pebble diameter. The effect of pebble diameter reduction on the packing factor distribution of the pebble bed is minimal. To validate the physical model of pebble beds established by using DEM simulations, the axial packing factor distribution near the wall was analyzed in comparison with the calculated results of the empirical model proposed by [PERSON] based on the experimental results [39]. [PERSON]'s empirical model is shown as follows:  
\[\varepsilon(x)=\left\{ \begin{array}{l}2.14x^{2}-2.53x+1\qquad\quad(\text{if }x \leq 0.637)\\ \varepsilon_{b}+0.29\cos(2.3\pi(x-0.16))\times e^{-0.6x}+0.15e^{-0.9x}\;( \text{if }x>0.637)\end{array} \right. \tag{9}\]  
where \(\varepsilon(x)\) is the local porosity; packing factor \(\gamma(x)=1-\varepsilon(x)\). \(x\) is the non-dimensional distance to the fixed wall. \(\varepsilon_{b}\) is average porosity in the inner bulk region of the pebble bed.