## 2 Method and Materials
### Evaluation Indices  
To quantitatively assess the airflow uniformity within bag filters, the following four evaluation indices were selected.  
1. Flow distribution coefficient  
The flow distribution coefficient (\(K_{qi}\)) for each filter bag can be calculated using the following formula:  
\[K_{qi}=\frac{Q_{i}}{\overline{Q}}, \tag{10}\]  
where \(Q_{i}\) represents the actual processed air volume of the \(i\)-th filter bag, and \(\overline{Q}\) is the average processed air volume across all filter bags, m\({}^{3}\)/s. Additionally, \(\Delta K_{qi}\) represents the maximum flow rate uneven amplitude, calculated as the difference between the maximum and minimum flow coefficients:  
\[\Delta K_{qi}=K_{qi\max}-K_{qi\min}. \tag{11}\]  
Under ideal conditions, where airflow distribution is perfectly uniform, \(K_{qi}=1.0\), and \(\Delta K_{qi}=0\). In cases of non-uniform airflow distribution, \(K_{qi}\) will fluctuate between 0 and 1, and \(\Delta K_{qi}\
eq 0\). A larger value of \(\Delta K_{qi}\) indicates poorer airflow uniformity, while a smaller value suggests a more uniform distribution. Typically, if the relative deviation of the processed air volume does not exceed 15%, the airflow distribution is considered relatively uniform.  
2. Integrated flow uneven amplitude  
The integrated flow uneven amplitude \(\Delta\overline{K}_{\xi}\) is the average of the absolute differences between the flow distribution coefficients of all filter bags and the ideal uniformity coefficient of 1.0, as expressed by the following formula:  
\[\Delta\overline{K}_{\xi}=\sum\bigl{(}\left|K_{qi}-1.0\right|\bigr{)}/N, \tag{12}\]  
where \(N\) is the total number of filter bags in the model. This parameter provides a comprehensive assessment of flow deviation across all filter bags, offering a more accurate evaluation of the overall airflow uniformity.  
3. Relative root mean square values (RMS)  
Figure 3: Mesh validation and mesh division for the single-chamber model: (**a**) Outlet velocity of bag X\({}_{8}\)Z\({}_{7}\) with different grid numbers; (**b**) the mesh division of the single-chamber model.  
The RMS value (\(\sigma\)) of the filtration air volume of the bags is used to assess the airflow uniformity across the entire bag filter, calculated as follows:  
\[\sigma=\sqrt{\frac{\sum_{i=1}^{n}\left(\frac{Q_{i}-\overline{Q}}{\overline{Q}} \right)^{2}}{N}}. \tag{13}\]  
From a statistical perspective, the RMS value reflects the degree of deviation between the airflow velocity at each filter bag and the average flow velocity. A larger RMS value indicates poorer uniformity in the airflow distribution. The following criteria are used to classify the uniformity: \(\sigma\leq 0.25\) (acceptable), \(\sigma\leq 0.15\) (satisfactory), and \(\sigma\leq 0.10\) (excellent). These criteria also allow the analysis of uniformity in the updraft along the filter bags.  
4. Relative standard deviation  
The relative standard deviation (\(C_{v}\)) is calculated using the exit center-axis velocity of each filter bag, where \(s\) represents the standard deviation. A smaller \(C_{v}\) indicates a more uniform velocity distribution. Generally, the flow distribution is considered uniform when the relative standard deviation of the processed air volume does not exceed 15%:  
\[C_{\sigma}=\frac{s}{\overline{v}}\times 100\%, \tag{14}\]  
\[s=\left(\sum(v_{i}-\overline{v})^{2}/N-1\right)\frac{1}{2}. \tag{15}\]