# 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 Acceleration-Process Motion of Fuel Elements  
Figure 7a-d shows the axial velocity, lateral velocity, lateral displacement, and rotational angular velocity versus time for the fuel element in the horizontal straight-pipe section, respectively, and Figure 7e shows the effect of the sphere/pipe-diameter ratio and pipe-inclination angle on the motion of the fuel element. Figure 7a shows that the axial velocity \(v_{z}\) of the fuel element under the action of pneumatic force increases rapidly from 0 to 5.1 m/s in 0.2 s, then \(v_{z}\) increases slowly until it is close to the lifting gas-flow velocity \(v_{g}\) (\(v_{g}\) is indicated by the dashed line in the figure, taking 6 m/s as an example). According to the kinetic model, the winding velocity of gas (the difference between \(v_{g}\) and \(v_{z}\)) is larger in 0-0.2 s, so the pneumatic thrust \(F_{\text{d}}\) is larger and the axial acceleration of the fuel element (the slope of \(v_{z}\)) is large. Its collision with the inner wall of the pipe leads to the instantaneous reduction of the axial velocity \(v_{z}\) (reflected as the zigzag step of the \(v_{z}\) curve), the collision-recovery coefficient \(\lambda\) (about 0.8) prompts the collision to gradually weaken, and finally \(v_{z}\) and \(v_{g}\) only have a small velocity difference caused by the rolling friction of the fuel element. Figure 7b,c displays the transverse velocity and transverse motion trajectory of the fuel element. When the transverse displacement reaches -1 mm and 1 mm, it means that the fuel element collides with the upper and lower wall surfaces of the pipe, respectively (the diameter of the fuel element \(D_{\text{s}}\) is 60 mm and the inner diameter of the pipe \(D_{\text{p}}\) is 62 mm). The collision time \(t_{p}\) is extremely short, and it is reflected as a point on the time axis t. The transverse velocity \(v_{x}\) of the fuel element gradually decreases from 1 m/s to nearly 0. The decay of the transverse motion is synchronized with the appearance of the jagged step on \(v_{z}\). At the end of each collision, the fuel element runs in the reverse direction  
Figure 6: Fuel-element pneumatic lifting system. with the velocity \(v_{x}\)\({}^{\prime}\) (the product of the velocity \(v_{x}\) and \(\lambda\) before this collision), and after 0.25 s it will roll against the lower wall of the pipe. Figure 7: Acceleration\(-\)section parameters: (**a**) axial velocity, (**b**) transverse velocity, (**c**) transverse velocity, (**d**) angular velocity, (**e**) sphere/pipe\(-\)diameter ratio and pipe\(-\)inclination angle. Figure 7d shows the change in the rotational angular velocity \(w\) of the fuel element with time. Within 0-0.25 s, when the fuel element collides with the pipe, its rotational angular velocity shows a positive or negative instantaneous step (the collision time is very short; the friction of the collision process produces the instantaneous torque acting on the fuel element). From the fuel-element pneumatic-lifting-system kinetic model, the instantaneous torque generated by the collision process is much larger than the torque \(M_{\text{o}}\) generated by the lifting airflow, so the change in \(w\) caused by the collision process will be much larger than the change in \(w\) caused by \(M_{\text{o}}\) (i.e., the \(w\) curve shows the step-like characteristics). When the collision ends after 0.25 s, the fuel element rolls against the lower wall of the pipe due to gravity. At this point, the sliding between the fuel element and the inner wall of the pipe can be ignored and only the pure rolling friction state is considered. The effect of the sphere/pipe-diameter ratio and pipe-inclination angle on the motion characteristics of the fuel-sphere acceleration process is shown in Figure 7e. the sphere/pipe-diameter ratio and pipe-inclination angle increase, the final velocity of the fuel element increases, and the time to reach the final velocity is shortened.