## 3 Results and Discussion
### Modeling Study
#### 3.2.3 The \(k\)-C\({}^{*}\) Model  
For the \(k\)-C\({}^{*}\) model, the results in Figure 5A,B indicate a correlation between \(ln(C/\!C_{0})\) and \(H/\!u\) only during the initial 10-20 min of operation, and the obtained \(k_{C}\) values were negative. Moreover, no significant correlation was observed for the period of 30\(-\)300 min, regardless of filtration velocity. These results can be anticipated by the examination of Figure 2B above, which indicates that the SS removal is independent of both the operation time and the filtration velocity. Similarly, the results in Figure 5C,D indicate a correlation between \(ln(L/\!L_{0})\) and \(H/\!u\) only during the initial 10\(-\)60 min, with negative \(k_{L}\) values. The negative filtration rate constants indicate that the \(k\)-C\({}^{*}\) model, which is based on pseudo first-order SS removal, cannot accurately describe the system used in the present study. Some previous studies have reported that the \(k\)-C\({}^{*}\) model cannot be applied to the removal of SS from stormwater by filtration over the lifetime of the system because the parameters, i.e., \(k_{C}\), \(k_{L}\), \(A_{C}\), or \(A_{L}\), fluctuate with SS concentration and hydraulics [45].  
Nevertheless, the results in Figure 5E indicate a good correlation between \(ln(\sigma)\) and \(H/\!u\) throughout the period of operation. Moreover, the results in Figure 5F indicate a gradual increase in the calculated \(k_{\sigma}\) and \(A_{\sigma}\) values during operation, thus indicating that the SS removal is enhanced with increasing \(\sigma\) and decreasing porosity. The increase in \(k_{\sigma}\) is consistent with the limited variation in \(C/\!C_{0}\) with filtration velocity and increasing operation period (Figure 2B). Moreover, the increase in \(A_{\sigma}\) is in accord with the decrease in \(\sigma_{m}\) with increasing filtration velocity (Figure 3). Hence, although no relationship between the parameters and the filtration conditions (e.g., SS concentration, SS characteristics, and hydraulic condition) is considered in the \(k\)-C\({}^{*}\) model [49], a modified version of this model that considers a specific deposit might successfully describe the performance of the system used in the present study. Meanwhile, the values of \(k_{\sigma}\) were significantly higher than the \(k_{\sigma}\) obtained in a wetland in Texas, USA, i.e., 0.228-6.849 h\({}^{-1}\)[44], indicating a faster SS removal by the media in his study.  
On the other hand, it should be noted that the good correlation between the filtration velocity (\(L/\!u\)) and \(ln(X_{\sigma})\) does not mean that this model describes the time variation of the influent SS concentration. The concentration of the pollutants of the influent of a stormwater treatment system generally varies with time, i.e., first, middle, and final flush, where the load of the pollutants is concentrated during the prophase, metaphase, and anaphase, respectively [50]. Therefore, a more advanced model must be developed to a better description of the SS removal in a stormwater filtration device.  
Figure 4: The deep bed filtration model: (**A**) the time evolution of \(\lambda\); (**B**) the correlations between \(\lambda\) and \(\sigma\).