## 2 Models and Methods  
### Governing Equations  
#### 2.1.1 DEM Model  
To construct the actual geometry of a random packed bed, discrete element method (DEM) was employed to generate random packing recently. The DEM and CFD methods were developed and show better performance in many applications, especially in fixed bed reactor simulations. [PERSON] et al. introduced the DEM-CFD meshing method for fixed beds consisting of monodisperse spherical particles [24]. [PERSON] et al. combined particle packing structure with detailed reaction mechanisms of catalytic dry reforming of methane over rhodium and studied the interaction between chemical kinetics and transport of momentum, heat, and mass [25]. [PERSON] et al. studied the flow and heat transfer in fixed beds considering the packing structures of cylindrical particles and low tube to particle diameter ratios [26]. In this research, the DEM method was used to generate actual geometry of random packings with different particle shapes. The distinct characteristic of DEM is that inter-particle contact forces are included in the equations of motion. The size, shape, and orientation of each particle are taken into account. The governing equation describing particle motion is:  
\[m_{p}\frac{dv_{p}}{dt}=\mathbf{F_{s}}+\mathbf{F_{b}} \tag{4}\]  
where \(m_{p}\) mass of particle, \(v_{p}\) is particle velocity, \(t\) is time, \(\mathbf{F_{s}}\) and \(\mathbf{F_{b}}\) is surface and volume force respectively. \(\mathbf{F_{b}}\) could be calculated by:  
\[\mathbf{F_{b}}=\mathbf{F_{g}}+\mathbf{F_{c}} \tag{5}\]  
where \(\mathbf{F_{g}}\) is gravity, \(\mathbf{F_{c}}\) is the interaction force between particles and between particles and wall:  
\[\mathbf{F_{g}}=m_{p}\mathbf{g} \tag{6}\]  
\[\mathbf{F_{c}}=\sum_{\text{contacts}}\mathbf{F_{cm}} \tag{7}\]\(\mathbf{F_{cm}}\) is contact force which could be calculated as:  
\[\mathbf{F_{cm}}=\mathbf{F_{n}}+\mathbf{F_{t}} \tag{8}\]  
where \(\mathbf{F_{n}}\) is the normal and \(\mathbf{F_{t}}\) is the tangential force component:  
\[\mathbf{F_{n}}=-K_{n}d_{n}-N_{n}v_{n} \tag{9}\]  
If \(|K_{t}d_{t}|<|K_{n}d_{n}|C_{fs}\), \(\mathbf{F_{t}}\) is defined as:  
\[\mathbf{F_{t}}=-K_{t}d_{t}-N_{t}v_{t} \tag{10}\]  
Otherwise:  
\[\mathbf{F_{t}}=\frac{|K_{n}d_{n}|C_{fs}d_{t}}{|d_{t}|} \tag{11}\]  
where \(K_{n}\) and \(K_{t}\) are the normal and tangential spring constant, \(N_{n}\) and \(N_{t}\) are the normal and tangential damping, \(d_{n}\) and \(d_{t}\) are overlaps in the normal and tangential directions at the contact point respectively, \(C_{fs}\) is the static friction coefficient. \(N_{n}\) and \(N_{t}\) could be calculated as follows:  
\[N_{n}=2N_{ndamp}\sqrt{K_{n}M_{eq}} \tag{12}\]  
\[N_{t}=2N_{t}d_{amp}\sqrt{K_{t}M_{eq}} \tag{13}\]  
\(N_{n}\)_damp_ and \(N_{t}\)_damp_ are normal and tangential damping coefficient respectively, \(M_{eq}\) is the equivalent particle mass. These two damping coefficient could be defined as follows:  
\[\mathbf{F_{b}}=\mathbf{F_{g}}+\mathbf{F_{c}} \tag{5}\]0  
\[\mathbf{F_{b}}=\mathbf{F_{g}}+\mathbf{F_{c}} \tag{5}\]1  
where \(C_{n\,rest}\) and \(C_{t\,rest}\) are the normal and tangential coefficients of restitution respectively. The rolling of particles (especially non-sphere particles) has great impact on the motion of particles. The rotational momentum equation is given as:  
\[\mathbf{F_{b}}=\mathbf{F_{g}}+\mathbf{F_{c}} \tag{5}\]2  
where \(\mathbf{I_{p}}\) is the particle moment of inertia, \(\omega_{p}\) is particle angular velocity, \(\mathbf{M_{b}}\) is the moment that acts on the particle (rotational drag), \(\mathbf{M_{c}}\) is the moment that acts on an individual particle due to contact force, which in turn acts on the particle at a point other than the particle center of gravity:  
\[\mathbf{F_{b}}=\mathbf{F_{g}}+\mathbf{F_{c}} \tag{5}\]3  
where \(\mathbf{r_{c}}\) is the vector from the particle center of gravity to the contact point and \(\mathbf{M_{cm}}\) is the moment that act on the particle from contact models (rolling resistance).