## 2 Method and Materials
### Establishment of the Numerical Simulation Models  
#### 2.2.1 Mathematical Models  
Due to the high-velocity inlet flow and the complex structure of the bag filter, turbulence is present in both the inlet and internal flow field. The Reynolds number (\(Re\)) at the hopper inlet reaches 201,900 and approximately 306,935 in the porous medium, which indicates turbulent flow. For this reason, the standard \(k\)-\(\epsilon\) turbulence model is employed to simulate the turbulent flow. The continuous phase of the flue gas is treated as isothermal, incompressible, and steady. Considering the objective of this study is to analyze the airflow distribution within the bag filter, to simplify the process, the discrete phase model (DPM) uncoupled method in Fluent 2022 R2 software was used for simulation, where the interaction between the discrete and continuous phases is neglected. The particle sizes of 2.5 \(\mu\)m and 5 \(\mu\)m were modeled as discrete sources.  
The key governing equations of continuous phase simulation are outlined below.  
Continuity equation:  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{ \partial z}=0\, \tag{1}\]  
Momentum conservation equation:  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial(\rho u_{i}u_{i})}{ \partial x_{j}}=-\frac{\partial p}{\partial x_{i}}+\mu\frac{\partial^{2}u_{i} }{\partial x_{j}^{2}}+\rho f_{i}, \tag{2}\]  
where \(\rho\) is the density of the fluid (kg/m\({}^{3}\)), \(u_{i}\) is the velocity component in the \(i\)-direction (m/s), \(p\) is the static pressure (Pa), \(\mu\) is the dynamic viscosity (kg/m\(\cdot\)s), \(f_{i}\) is the external force in the \(i\)-direction (N), \(x_{j}\) is the spatial coordinate in the \(j\)-direction (m).  
To solve for the turbulent kinetic energy (\(k\)) and dissipation rate (\(\epsilon\)), the corresponding transport equations are as follows:  
Turbulent kinetic energy (\(k\)) equation:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_{ i}}=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}} \right)\frac{\partial k}{\partial x_{j}}\right]+G_{k}-\rho\epsilon+S_{k}, \tag{3}\]  
Figure 1: 3D modeling and numbering scheme of the bag filter. (**a**) 3D Models of the single-chamber and multi-chamber bag filter; (**b**) numbering scheme for cross-sections and filter bags.  
Turbulence dissipation rate (\(\epsilon\)) equation:  
\[\frac{\partial(\rho\epsilon)}{\partial t}+\frac{\partial(\rho\epsilon u_{i})}{ \partial x_{i}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{t }}{\sigma_{\epsilon}}\bigg{)}\frac{\partial\epsilon}{\partial x_{j}}\Bigg{]}+ C_{1\epsilon}\frac{\epsilon}{\xi}G_{k}-C_{2\epsilon}\rho\frac{\epsilon^{2}}{\xi}+S_{ \epsilon}, \tag{4}\]  
where \(G_{k}\) is the generation term of the turbulent kinetic energy caused by the average velocity gradient, \(\sigma_{k}\) and \(\sigma_{\epsilon}\) are Prandtl numbers, \(C_{1\epsilon}\) and \(C_{2\epsilon}\) are empirical constants, and \(S_{k}\) and \(S_{\epsilon}\) are user-defined source items. The values used are \(\sigma_{k}\) = 1.1, \(\sigma_{\epsilon}\) = 1.3, \(C_{1\epsilon}\) = 1.44, \(C_{2\epsilon}\) = 1.92, and \(S_{k}\) = \(S_{\epsilon}\) = 0.  
The turbulent viscous coefficient \(\mu_{t}\) is related to \(k\) and \(\epsilon\), and its expression is shown in Equation (5):  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\epsilon}. \tag{5}\]  
Additionally, the hydraulic diameter (\(\phi\)) and turbulence intensity (\(I\)) were calculated based on various models and operating conditions [36]:  
\[\phi=\frac{Re\mu}{\rho v}, \tag{6}\]  
\[I=0.16(Re)^{-1/8}, \tag{7}\]  
where \(v\) is the airflow velocity (m/s) and \(\mu\) is the kinematic viscosity of the fluid (m\({}^{2}\)/s).