## 2 Materials and Methods
### Dynamic Modeling of Fixed-Bed Column  
The modeling of the continuous adsorption process was carried out to predict the dynamics of the process and obtain the corresponding kinetic parameters. From the \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Dye** & **Bed Height** & \(\mathbf{t_{b}}\) & \(\mathbf{V_{df}}\) & \(\mathbf{t_{oper}}\) & \(\mathbf{m}\) & \(\mathbf{q_{bed}}\) & \%_Removal_ & _MTZ_ \\ \hline Blue & 4 & 42 & 4029 & 523 & 1.0 & 1283.1 & 65.0 & 3.7 \\  & 2 & 20 & 3370 & 438 & 0.5 & 1165.6 & 34.7 & 1.9 \\ Red & 4 & 6 & 3098 & 446 & 1.0 & 705.3 & 42.2 & 3.9 \\  & 2 & 5 & 3445 & 447 & 0.5 & 1028.4 & 30.0 & 2.0 \\ Black & 4 & 36 & 3395 & 441 & 1.0 & 153.4 & 47.3 & 3.5 \\  & 2 & 6 & 1347 & 175 & 0.5 & 109.3 & 43.0 & 1.9 \\ \hline \end{tabular}
\end{table} 
Table 4: Estimated parameters obtained from BTCs corresponding to the adsorption of blue, red, and black dyes onto the PSAC in function of bed height (500 mg L\({}^{-1}\) for blue and red and 100 mg L\({}^{-1}\) for black dye; flow rate: 7.7 mL min\({}^{-1}\); temperature: 20 \({}^{\circ}\)C). Figure 7: A comparison of fixed-bed column three-dye adsorption onto the PSAC in function of time (initial dye concentration: 100 mg L\({}^{-1}\); flow rate: 7 mL min\({}^{-1}\); bed height: 2 cm; temperature: 20 \({}^{\circ}\)C). In general, for the cases in which it was not possible to obtain an adjustment, it is probably because, apart from not reaching saturation (\(C/C_{0}=1\)), the ideal \"S\" shape profile produced in ideal adsorption systems was not achieved once the dye began to be released at very short times. On the other hand, in cases where an adjustment is only obtained for the first part, the BTC becomes elongated in the second region without reaching saturation, which may be due to mass-transfer limitations or non-uniform flow [40]. Moreover, an \"S\" shape profile is often associated with the adsorption of the smallest and simplest dye molecules, which could be obtained in some cases for the blue dye, which is the dye with a less complex chemical structure [13; 41]. The results presented in Table 5 reveal good correlations (\(R^{2}>0.9\)) for both the Thomas and Yoon-Nelson models for the three dyes with a low \(SSE\). The Thomas model allows for us to assume irrelevant radial and axial dispersion and insignificant changes on temperature and pressure inside the column. Apart from that, and in agreement with batch experiments in which the intraparticle diffusion model showed multilinearity for blue and red dye adsorption with a good fit for the first stage and a good linear fit for the black dye without multilinearity, it suggests that external and intraparticle diffusion do not represent the rate-limiting stages, but both can have influence on the adsorption process [39]. The Yoon-Nelson model also represents a good fit, and many times is similar to the results of the [PERSON] model. [PERSON] and [PERSON] 2022 [39], [PERSON] et al. 2020 [1], and [PERSON] et al. 2021 [42] also reported the same observations. In addition, the worst fit to the Bohart-Adams model suggests that intraparticle diffusion forces and external mass-transfer resistances are not negligible [25]. Regarding the BTCs for the blue dye, in which it was possible to obtain a complete adjustment, the experimental (\(q_{exp}\)) and predicted (\(q_{t}\)) adsorption capacities did not differ significantly. Regarding the [PERSON] model, for the blue dye, \(K_{Th}\) decreased with an increasing initial dye concentration, while for the black dye, \(K_{Th}\) increased; in both cases, \(q_{t}\) decreased. Also, for the blue dye, \(K_{Th}\) increased and \(q_{t}\) decreased with an increasing flow rate. These changes can be related to the dye concentration gradient between the solid and liquid phases, which behaved as a driving force for the mass-transfer process [25; 35]. On the other hand, \(K_{Th}\) decreased and \(q_{t}\) increased with increasing bed height since the driving force became stronger due to the greater availability of active sites [43].