# Kinetic Modeling and Analysis of Fuel Element Pneumatic Lifting System in Pebble Bed Reactors
## 2 Kinetic Modeling of Fuel-Element Pneumatic Lifting System
### Modeling the Motion of a Fuel Element in an Atypical Bend  
In the piping design of the fuel-element pneumatic lifting system, there are certain atypical bends in the second half of the accelerating section of the piping, the force state of which is different from that of the fuel elements in the typical bends mentioned above. The pneumatic thrust \(F_{\mathrm{d}}\) on the fuel element in an atypical bend is in the plane where the bend axis is located, whereas \(F_{\mathrm{L}}\), \(M_{\mathrm{o}}\), and gravity \(F_{\mathrm{g}}\) can be decomposed into two components, namely, the parallel and vertical components of the bend axis plane, as shown in Figure 4. \(F\) in Figure 4a refers to one of \(F_{\mathrm{g}}\), \(F_{\mathrm{L}}\), and \(M_{\mathrm{o}}\). Plane B is the plane where the bend axis is located, plane C is the vertical plane of the bend axis plane, and \(F_{\mathrm{B}}\) and \(F_{\mathrm{C}}\) are the components of \(F\) decomposed in planes B and C, respectively, where the \(z\)-axis is tangential to the bend axis in plane B and points along the airflow, the \(x\)-axis is in plane B along the bend radius and the direction is away from the bend center, and the \(y\)-axis is in plane C and perpendicular to plane B in the opposite direction of \(F_{\mathrm{gC}}\). Figure 4b shows the C-plane components of each force. \(F_{\mathrm{LC}}\), \(F_{\mathrm{gC}}\), and \(M_{\mathrm{oC}}\) will result in translational and rotational motion of the fuel element in plane C.  
Figure 3: Force on the fuel element in a typical bend. Equations (4)-(8) are the kinetic model of the fuel element in an atypical bend. \[m\frac{\mathrm{d}v_{z}}{\mathrm{d}t}+m\frac{v_{x}v_{z}}{R}=F_{\mathrm{d}}-F_{ \mathrm{gB}}\sin\theta_{\mathrm{B}} \tag{4}\]  
\[m\frac{\mathrm{d}v_{x}}{\mathrm{d}t}-m\frac{v_{z}v_{z}}{R}=F_{\mathrm{LB}}+F_{ \mathrm{gB}}\cos\theta_{\mathrm{B}} \tag{5}\]  
\[m\frac{\mathrm{d}v_{y}}{\mathrm{d}t}=F_{\mathrm{LC}}-F_{\mathrm{gC}} \tag{6}\]  
\[(\frac{1}{10}mD_{\mathrm{s}}{}^{2})\frac{\mathrm{d}w_{\mathrm{B}}}{\mathrm{d} t}=M_{\mathrm{oB}} \tag{7}\]  
\[(\frac{1}{10}mD_{\mathrm{s}}{}^{2})\frac{\mathrm{d}w_{\mathrm{C}}}{\mathrm{d} t}=M_{\mathrm{oC}} \tag{8}\]  
where \(v_{x}\), \(v_{y}\), and \(v_{z}\) are the components of the \(x\)-axis, \(y\)-axis, and \(z\)-axis of the velocity of the fuel element, respectively. \(\theta_{\mathrm{B}}\) is the angle between \(F_{\mathrm{gB}}\) and \(F_{\mathrm{LB}}\). \(w_{\mathrm{B}}\) and \(w_{\mathrm{C}}\) are the components of \(w\) in the B and C planes, respectively. Equations (9)-(13) are the kinetic model of the collision process of the fuel element in an atypical bend according to the law of momentum. \[mv_{z}{}^{\prime}-mv_{z}=(F_{\mathrm{d}}-F_{\mathrm{gB}}\sin\theta_{\mathrm{B }}-m\frac{v_{x}v_{z}}{R})t_{\mathrm{P}}\pm fNt_{\mathrm{P}} \tag{9}\]  
\[mv_{x}{}^{\prime}-mv_{x}=(F_{\mathrm{LB}}+F_{\mathrm{gB}}\cos\theta_{\mathrm{B }}+m\frac{v_{z}v_{z}}{R})t_{\mathrm{P}}\pm N_{\mathrm{B}}t_{\mathrm{P}} \tag{10}\]  
\[mv_{y}{}^{\prime}-mv_{y}=(F_{\mathrm{LC}}-F_{\mathrm{gC}})t_{\mathrm{P}}\pm N _{\mathrm{C}}t_{\mathrm{P}} \tag{11}\]  
Figure 3:0  
Figure 3:1  
where \(v_{x}{}^{\prime}\), \(v_{y}{}^{\prime}\), and \(v_{z}{}^{\prime}\) are the components of the \(x\)-axis, \(y\)-axis, and \(z\)-axis of the velocity of the fuel element after collision, respectively; \(w_{\mathrm{B}}{}^{\prime\prime}\) and \(w_{\mathrm{C}}{}^{\prime\prime}\) are the components of the rotational speed \(w\) of the fuel element in surfaces B and C, respectively; \(N_{\mathrm{B}}\) and \(N_{\mathrm{C}}\) are the components of the average positive pressure of the collision process in surfaces B and C, respectively; and \((\frac{D_{\mathrm{s}}}{2})_{\mathrm{B}}\) and \((\frac{D_{\mathrm{s}}}{2})_{\mathrm{C}}\) are the components of the radius of the fuel element at the collision point in surfaces B and C, respectively. If the radius \(R\) is infinite, Equations (1)-(13) define the kinetic model of the basic motion of the fuel element in a straight pipe.