## 3 Results and Discussions
### Wall Effect on Pressure Distribution  
To investigate the affecting mechanisms of the wall effect on the pressure distribution in randomly packed pebble beds, the helium gas flow behaviors in randomly packed pebble beds were numerically investigated. The detailed data of the helium flow in the pebble bed are obtained by the steady state simulation. Figure 11 illustrates the pressure distribution inside the pebble bed with three different wall boundaries. Figure 11a-c display the pressure distribution inside the pebble bed and on the surface of the pebbles, while Figure 11a'-c' reveal the pressure distribution in the mid-plane of the pebble bed. The findings demonstrate that the pressure distribution is uniform with minimal variation before helium enters the pebble bed. As helium passes through the bed, the helium pressure progressively decreases. Although there is a slight difference in local pressures due to the void distribution within the pebble bed, the pressures exhibit a gradual decline in the flow direction. Following the helium gas discharge from the pebble bed, the pressure promptly reverts to a uniform distribution, with its value gradually approaching the outlet pressure. In order to quantitatively analyze the along-travel pressure drop along the flow direction within three types of pebble beds, the averaged pressures of the cross-sectional plane were calculated at different positions along the flow direction. The resulting data are  
Figure 10: Axial packing factor variation in pebble beds with (**a–c**) 4 fixed walls, (**d–f**) 2 fixed walls and 2 periodic boundaries, and (**g–i**) 4 periodic boundaries. presented in Figure 12a, indicating minimal variance in helium pressure prior to entering the pebble bed. After the fluid passes through the pebble bed, the pressure decreases linearly. As it exits the pebble bed, the pressure rapidly stabilizes and reaches equilibrium with the relative pressure at the outlet position. This is mainly due to the fact that the pebble packed bed is a porous medium. As the helium passes through the pebble bed, it encounters resistance, resulting in pressure loss, which is closely related to the characteristics of the pebble bed porous media. Figure 11: Pressure distribution in pebble beds with different boundaries: (**a**) 3D view and (**a\({}^{\prime}\)**) midplane distribution with 4 fixed walls, (**b**) 3D view and (**b\({}^{\prime}\)**) midplane distribution with 2 fixed walls and 2 periodic boundaries, and (**c**) 3D view and (**c\({}^{\prime}\)**) midplane distribution with 4 periodic boundaries. Figure 12: Pressure distribution in pebble beds with different boundaries: (**a**) along the flow direction, (**b**) pressure drop per unit length. Additionally, the pressure drop gradient can be obtained by linearly fitting the cross-sectional averaged pressure loss over the pebble bed position after entering the pebble bed regions, as follows:  
\[ \begin{array}{l}y=-0.71238x+7.0389,@\,4\text{ walls,}\\ y=-0.82072x+8.2351,@\,2\text{ walls and }2\text{ periodics,}\\ y=-0.99546x+9.8626,@\,4\text{ periodics.}\end{array}  \tag{10}\]  
where \(x\) is in mm, \(y\) is in Pa. Finally, the pressure drop gradient of helium in pebble beds are 712.83 Pa/m, 820.72 Pa/mm, and 995.46 Pa/m, respectively, with three different wall boundaries. Figure 12b displays the helium pressure drop gradient in pebble beds with different wall boundaries. It is evident that the pressure drop gradient is the least for the pebble bed with four fixed walls, while the pebble bed with the four periodic boundaries experiences the greatest pressure drop gradient. The pressure drop gradient in the pebble bed, featuring two fixed walls and two periodic boundaries, falls between the pressure drop gradient in pebble beds comprising four fixed walls or four periodic boundaries. Compared to the pebble bed with four fixed walls, the pressure drop gradient of helium in the pebble beds with four periodic boundaries increased by 39.73% to 995.46 Pa/m.