# Kinetic Modeling and Analysis of Fuel Element Pneumatic Lifting System in Pebble Bed Reactors
## 3 Analysis of the Motion Characteristics of Fuel Elements  
Based on the above fuel-element pneumatic-lifting-system kinetic model, the numerical-simulation method is used to analyze the motion characteristics in the fuel-element pipeline. The values of the basic parameters of the model are shown in Table 2. The diameter ratio of the spheres/pipe, the inclination angle of the pipe, the collision recovery coefficient, and the bending radius of the pipe are important parameters to describe the pipe structure [28], and the range of parameters is based on the basic working conditions of the pneumatic lifting system of the fuel elements of the pebble-bed reactor. The pneumatic forces \(F_{\text{d}}\), \(F_{\text{L}}\), and \(M_{\text{o}}\) on the fuel elements in the straight pipe, typical, and atypical bends, respectively are calculated according to Equations (18)-(20), respectively [24].  
\[F_{\text{d}}=\frac{1}{2}C_{\text{d}}A_{\text{c}}\rho_{\text{g}}{v_{\text{d}}}^ {2}=\frac{1}{2}f_{\text{d}}(k,w^{\prime},e/e_{\text{max}})A_{\text{c}}\rho_{ \text{g}}{v_{\text{d}}}^{2} \tag{18}\]  
\[F_{\text{L}}=\frac{1}{2}C_{\text{L}}A_{\text{c}}\rho_{\text{g}}{v_{\text{d}}}^ {2}=\frac{1}{2}f_{\text{L}}(k,w^{\prime},e/e_{\text{max}})A_{\text{c}}\rho_{ \text{g}}{v_{\text{d}}}^{2} \tag{19}\]  
\[M_{\text{o}}=\frac{D_{\text{s}}}{4}C_{\text{m}}A_{\text{c}}\rho_{\text{g}}{v_{ \text{d}}}^{2}=\frac{D_{\text{s}}}{4}f_{\text{o}}(k,w^{\prime},e/e_{\text{max} })A_{\text{c}}\rho_{\text{g}}{v_{\text{d}}}^{2} \tag{20}\]  
where \(C_{\text{d}}\) is the traction coefficient, \(C_{\text{g}}\) is the lateral-force coefficient, \(C_{\text{m}}\) is the torque coefficient, \(A_{\text{c}}\) is the windward area of the fuel element, \(d\) is the diameter of the fuel element, \(\rho_{\text{g}}\) is the gas density, and \(v_{\text{g}}\) is the incoming-gas velocity.  
The fuel element runs along the conveying pipeline under the action of gas power, which will show three kinds of motion states. These are the acceleration process, balance-lifting process, and bend-motion process, as shown in Figure 6. Under each motion state, the pipeline-structure parameters have a significant influence on the motion characteristics of the fuel element.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Symbol** & **Unit** & **Value** & **Symbol** & **Unit** & **Value** \\ \hline \(D_{\text{s}}\) & mm & 60 & \(k\) & mm & 0.92–0.97 \\ \(m\) & kg & 0.196 & \(\mu_{1}\) & & 0.126 \\ \(A_{\text{C}}\) & mm\({}^{2}\) & 2826 & \(E_{1}\) & GPa & 10.16 \\ \(\theta\) & \(\circ\) & –30–90 & \(\mu_{2}\) & & 0.3 \\ \(\rho_{\text{g}}\) & kg/m\({}^{3}\) & 10.26 & \(E_{2}\) & GPa & 200 \\ \(R\) & mm & 400–800 & \(f\) & & 0.41 \\ \(v_{g}\) & m/s & 3–10 & \(\lambda\) & & 0.5–0.8 \\ \hline \hline \end{tabular}  
where fuel-element diameter \(D_{\text{s}}\), fuel-element mass \(m\), and windward area \(A_{\text{C}}\) are the standard parameters of the fuel element of a pebble-bed reactor; pipe inclination angle \(\theta\), pipe bending radius \(R\), gas density \(\rho_{\text{g}}\), and vertical lifting speed \(v_{\text{g}}\) are the design parameters in actual engineering; the spherical/ pipe diameter ratio \(k\) of 0.92–0.97 corresponds to a pipe diameter of 62–65 mm, respectively; and \(\mu_{1}\), \(\mu_{2}\), \(E_{1}\), \(E_{2}\), \(f\), and \(\lambda\) are the [PERSON]’s parameters, moduli of elasticity, friction factor, and recovery coefficient for the graphite shell of the fuel element and steel channel, respectively.  
\end{table}
Table 2: Basic parameters of fuel-element motion.