# Kinetic Modeling and Analysis of Fuel Element Pneumatic Lifting System in Pebble Bed Reactors
## 3 Analysis of the Motion Characteristics of Fuel Elements
### Analysis of the Motion of the Vertical-Lifting Part of the Fuel Element  
Figure 8a-d shows the curves of axial velocity \(v_{z}\), transverse velocity \(v_{x}\), transverse displacement x, and rotational angular velocity \(w\) of the fuel element in a straight-pipe section perpendicular to the horizontal plane (i.e., \(\theta\) = 90\({}^{\circ}\)), respectively, as a function of time, and Figure 8e shows the effect of the sphere/pipe-diameter ratio and recovery factor on the motion of the fuel element. When the fuel element enters the vertical-lifting section from the bend-turning section at the beginning (i.e., within 0.1 s time), its transverse velocity \(v_{x}\) is larger and the collision between the fuel element and the inner-wall surface of the pipe is frequent and violent, which makes the axial velocity \(v_{z}\) drop rapidly. As shown in Figure 8a, \(v_{z}\) drops rapidly from 5.1 m/s to 3.8 m/s. In 0.1-0.6 s, the collision intensity decays gradually. As shown in Figure 8b,c, the transverse velocity \(v_{x}\) of the fuel element decreases significantly, the frequency of the collision decreases, and its axial velocity \(v_{z}\) differs from the gas-flow velocity \(v_{g}\) (shown as a dashed line in Figure 8a), and the fuel element completes the re-acceleration process rapidly due to the pneumatic thrust \(F_{\text{d}}\). After 0.6 s, the fuel element no longer collides with the inner wall of the pipe; its axial velocity \(v_{z}\) is basically stable, referring to the kinetic model of the fuel-element pneumatic lifting system. The difference between this and the lifting gas-flow velocity \(v_{g}\) is mainly caused by gravity, and the difference is the delayed velocity \(v_{gx}\) of the fuel element. The lateral velocity \(v_{x}\) and lateral displacement x of the fuel element fluctuate reciprocally around 0 and have a tendency to gradually decrease. With reference to the kinetic model of the fuel-element pneumatic lifting system, the lateral force \(F_{\text{L}}\) on the fuel element is always directed to the center of the pipe, which has the effect of maintaining its movement along the axis of the pipe and gradually weakening the lateral movement. The rotational angular velocity \(w\) of the fuel element is also basically stable during this period, and the variation in \(w\) under the action of the aerodynamic component \(M_{\text{o}}\) is very small. In summary, at this time, the fuel element enters the smooth lifting stage without collision with the pipe and has a high and stable axial velocity, which is the best state for the pneumatic lifting process of the pebble-bed-reactor fuel element. Figure 8: Vertical lifting\(-\)section parameters: (**a**) axial velocity, (**b**) lateral velocity, (**c**) lateral displacement, (**d**) angular velocity, (**e**) sphere/pipe\(-\)diameter ratio and recovery factor. The effect of the sphere/pipe-diameter ratio and recovery factor on the motion characteristics of fuel-element vertical-lifting process is shown in Figure 8e. Due to the change in the direction of motion at the beginning of the vertical-lifting process, the fuel element collides violently with the inner wall of the pipe, and the axial velocity of the fuel element drops significantly. Lowering the recovery factor \(\lambda\) could reduce the strength and frequency of collisions, which in turn reduces the magnitude of the velocity drop and the level of cyclic stress on the pipe (setting a bellows at the front of the vertical-lifting section and relaxing the pipe constraint can effectively reduce \(\lambda\)); then, the collision starts to decay, and the fuel element is lifted vertically with the characteristic axial velocity \(\overline{v_{z}}\) (the small fluctuation of \(\overline{v_{z}}\) is caused by the transverse motion \(v_{x}\) and rotational motion \(w\) of the fuel element). The difference between \(\overline{v_{z}}\) and the gas-flow velocity (6 m/s, for example) is the delay velocity \(v_{gx}\).