## 4 Results and Discussion
### Hydrodynamics  
#### 4.3.1 Control Equations and Boundary Conditions  
In general, the pore Reynolds number (Re) can be used to assess the flow dynamics of fluids within the accumulation bed. When Re \(<\) 100, the fluid exhibits laminar flow. When 100 \(<\) Re \(<\) 300, the fluid undergoes a transition from laminar to turbulent flow. At Re \(>\) 300, the fluid displays high instability and irregularity, resulting in an overall turbulent flow state. In this study, the pore Re of the calculation model was determined using the corresponding equation. For fully developed turbulent flow on a solid wall, it is possible to categorize the flow into a wall zone and a core zone in relation to the perpendicular direction of the wall. The core zone is universally regarded as having fully turbulent flow, while the wall zone is partitioned into viscous, transition, and mathematical layers. As a result, the standard wall function was used for calculations. Additionally, for simplicity, we assumed that the gas was an incompressible ideal gas. Within this simulation, the three-dimensional control equation was resolved across the \(x\), \(y\), and \(z\) directions.  
The continuity equation is expressed as  
\[\frac{\partial\big{(}\rho_{g}\big{)}}{\partial t}+\
abla\cdot\big{(}\rho_{g} \overline{v}\big{)}=0, \tag{3}\]  
where \(\rho_{g}\) is the gas density (kg/m\({}^{3}\)), \(\overline{v}\) is the gas velocity vector (m/s), and \(t\) is the time (s).  
The momentum equation is expressed as  
\[\rho\Bigg{(}\frac{\partial\overline{u_{i}u_{i}}}{\partial x_{j}}\Bigg{)}=- \frac{\partial\overline{p}}{\partial x_{j}}+\frac{\partial}{\partial}\Bigg{[} \mu+\mu_{i}\Bigg{(}\frac{\partial\overline{u_{i}}}{\partial x_{j}}+\frac{ \partial\overline{u_{j}}}{\partial x_{j}}\Bigg{)}\Bigg{]}. \tag{4}\]  
The k\(-\epsilon\) equation is expressed as  
\[\rho\frac{\partial\overline{u_{j}k}}{\partial x_{j}}=\Bigg{[}\frac{\partial}{ \partial x_{j}}\bigg{(}\mu+\frac{\mu_{i}}{\sigma_{k}}\bigg{)}\frac{\partial k} {\partial x_{j}}\Bigg{]}+\mu_{i}\Bigg{(}\frac{\partial\overline{u_{i}}}{ \partial x_{j}}+\frac{\partial\overline{u_{j}}}{\partial x_{ji}}\Bigg{)}\frac {\partial\overline{u_{i}}}{\partial x_{j}}-\rho\epsilon, \tag{5}\]  
\[\rho\frac{\partial\overline{u_{j}\epsilon}}{\partial x_{j}}=\Bigg{[}\frac{ \partial}{x_{j}}\bigg{(}\mu+\frac{\mu_{i}}{\sigma_{\epsilon}}\bigg{)}\frac{ \partial\epsilon}{\partial x_{j}}\Bigg{]}+C_{1\epsilon}\frac{\epsilon}{k}\mu_ {i}\Bigg{(}\frac{\partial\overline{u_{i}}}{\partial x_{j}}+\frac{\partial \overline{u_{j}}}{\partial x_{i}}\Bigg{)}\frac{\partial\overline{u_{i}}}{ \partial x_{j}}-C_{2\epsilon}\rho\frac{\epsilon^{2}}{k}. \tag{6}\]  
Figure 5: Comparison of axial porosity under different particle size ratios: (**a**) \(D/d_{p}\) = 4.44; (**b**) \(D/d_{p}\) = 5.6; and (**c**) \(D/d_{p}\) = 6.25.  
According to Equations (5) and (6), we have the following values:  
\[\mu=c_{\mu}\rho\frac{k^{2}}{\epsilon},c_{\mu}=0.09,C_{1\epsilon}=1.44,C_{2 \epsilon}=1.92,\sigma_{k}=1.0,\text{ and }\sigma_{\epsilon}=1.3.\]  
The boundary equations for the entrance, exit, and walls are, respectively, expressed as  
\[z=z_{in},\overline{u}=\overline{v}=0,\overline{w}=w_{0}, \tag{7}\]  
\[z=z_{out},\frac{\partial u}{\partial z}=\frac{\partial v}{\partial z}=0, \tag{8}\]  
and  
\[\overline{u}=\overline{v}=\overline{w}=0. \tag{9}\]  
The fluid medium was air with a density of 1.205 kg/s and a viscosity of \(1.8189\times 10^{5}\) kg/(m/s).