## 3 Results and Discussion
### Factors That Impact the Filtration Process  
#### 3.2.1 Aspect Ratio  
To determine the effect of the aspect ratio of elliptical fibers, particle sizes in the range of the Greenfield gap were chosen. In the case of small particles (\(d_{\mathrm{P}}\) = 0.4 \(\upmu\)m), Cases 5 (\(\epsilon\) = 1), 2 (\(\epsilon\) = 2), and 6 (\(\epsilon\) = 3) are considered. As shown in Figure 7, the particle dendrites of these three cases possess the same mass load per unit area of woven fibers, consisting of 36,508, 53,504, and 70,712 particles on the surfaces, respectively. Cylindrical fibers have an aspect ratio of exactly 1. With regard to such fibers, submicron particles can be deposited on any side, irrespective of the front or back arrangement of two fibers. As the aspect ratio increases, an increasing number of particles are liable to be deposited on the fiber's  
Figure 6: Dendrite structures with different particle parameters: (**a**) Case 1, \(d_{\mathrm{p}}\) = 0.2 \(\upmu\)m; (**b**) Case 2, \(d_{\mathrm{p}}\) = 0.4 \(\upmu\)m; (**c**) Case 3, \(d_{\mathrm{p}}\) = 1.0 \(\upmu\)m; (**d**) Case 4, \(d_{\mathrm{p}}\) = 2.0 \(\upmu\)m.  
lateral and front sides (as indicated by Figure 7b,c), and the branched dendrites grow larger, indicating that the interception and impaction become more important as the cross-sectional area of the fiber media increases. Both the pressure drop and capture efficiency are known to be significant in the filtration process, and these two parameters vary dynamically during non-steady-state filtration. [PERSON] et al. [19] proposed exponential and linear functions of the deposited mass to describe the normalized pressure drop (\(\Delta P/\Delta P_{0}\)) and normalized capture efficiency (\(\eta/\eta_{0}\)) of non-woven fibers, respectively. As depicted in Figure 8, similar patterns can be observed in the woven fibers. As shown in Figure 8a, there is an exponential relationship between the normalized pressure drop and the total deposited mass; this can be written as \(\ln(\Delta P/\Delta P_{0})=kM+b\). The fitting parameters listed in Table 3 produce coefficients of determination (R\({}^{2}\)) greater than 0.997. The normalized pressure drop \(\Delta P/\Delta P_{0}\) can be modeled as \(\ln(\Delta P/\Delta P_{0})=0.88M-0.32\) (Case 5), \(\ln(\Delta P/\Delta P_{0})=3.36M-1.02\) (Case 2), and \(\ln(\Delta P/\Delta P_{0})=4.94M-2.31\) (Case 6). The cylindrical fibers give the lowest rate of increase in \(\Delta P/\Delta P_{0}\), and the rate of increase in \(\Delta P/\Delta P_{0}\) is positively correlated with the fiber shape (aspect ratio). This is due to the significant growth of branched dendrites on the fiber's lateral sides (as indicated by the front view in Figure 7). Namely, the slope \(k\) is linearly dependent on the perimeter of the fiber cross-sectional area, and the linear relation has the form \(k=0.0384\)_perimeter_\(-1.7885\) (\(71.7<\)_perimeter_\(<177\)), where R\({}^{2}>0.995\), which mainly comes from the remarkable reduction in fluid cross-sectional area. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & **Packing Density** & **Aspect Ratio** & \(\mathbf{k}\) & \(\mathbf{b}\) & \(\mathbf{\lambda}\) & \(\mathbf{\gamma}\) \\ \hline
5 & 5\% & 1 & 0.88 & \(-0.32\) & 0.19 & 1.00 \\
2 & 5\% & 2 & 3.36 & \(-1.02\) & 0.53 & 1.00 \\
6 & 5\% & 3 & 4.94 & \(-0.82\) & 1.02 & 1.01 \\
7 & 2.5\% & 2 & 1.93 & \(-0.52\) & 0.36 & 0.99 \\
8 & 7.5\% & 2 & 4.53 & \(-1.43\) & 0.57 & 1.00 \\
9 & 10\% & 2 & 6.78 & \(-2.31\) & 0.58 & 1.01 \\
10 & 5\% & 2 & 3.37 & \(-1.01\) & 0.53 & 1.00 \\
11 & 5\% & 2 & 3.33 & \(-1.04\) & 0.53 & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fitting parameters of the simulation cases. As shown in Figure 8b, during the initial dust-loading process, the rate of increase in the normalized capture efficiency (\(\eta/\eta_{0}\)) decreases as the mass load (\(M\)) increases and then reaches a constant value (a linear rate of increase), which is similar to the conclusions of previous studies [19].