## 2 Materials and Methods
### Procedure and Experimental Design
#### 2.4.2 Fixed-Bed Column Adsorption  
Similar to batch adsorption, fixed-bed column adsorption was performed using a 2\({}^{3}\) design, with duplicates at the central point (*), and analyzed with STATISTICA 10.0(r) \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & **Variables** & \\ \hline
**Level** & **pH** & **Solid Mass (g)** & **Time (min)** \\ \hline (\(-\)1) & 3 & 0.50 & 10 \\
0 & 7 & 1.25 & 30 \\ (+1) & 11 & 2.00 & 60 \\ \hline \end{tabular}
\end{table} 
Table 1: Values of the variables evaluated for the simultaneous batch adsorption of paracetamol and metronidazoleazole. software. The effects of the variables and their interactions were examined through analysis of variance (ANOVA), while response surface methodology (RSM) was used to forecast the optimal conditions based on the F-value and \(p\)-value. The experimental design included three factors: the initial concentration of pharmaceuticals in the solution (C\({}_{0}\), x\({}_{1}\)), the bed mass (W, x\({}_{2}\)), and the volumetric feed flow rate (Q, x\({}_{3}\)), each evaluated at three levels, along with triplicates at the central point, resulting in a total of 11 experiments (see Table 2). The maximum adsorption capacity of the fixed-bed column was chosen as the response factor for this phase (q\({}_{\text{max}}\)--Equation (6)). \[\text{C}_{\text{sat}}=0.95\text{C}_{0} \tag{5}\]  
\[\text{q}_{\text{max}}=\frac{\text{C}_{0}\text{Q}}{1000\text{W}}\int_{0}^{t_{ \text{f}}}\left(1-\frac{\text{C}_{\text{sat}}}{\text{C}_{0}}\right)\text{dt} \tag{6}\]  
where C\({}_{\text{sat}}\) is the solute concentration at saturation time (mg/L); C\({}_{0}\) is the initial concentration of pharmaceuticals at the column inlet (mg/L); q\({}_{\text{max}}\) is the maximum adsorption capacity of the bed (mg/g); t\({}_{\text{f}}\) (min) is the final time; W (g) is the mass of adsorbent applied to the bed; and Q (mL/min) is the volumetric flow rate of the column. The experiments were carried out using a borosilicate glass column with an internal diameter of 12 mm and a total length of 20 cm. The pharmaceutical solution was pumped upward through the column at a controlled flow rate using an ISMATEC peristaltic pump (model MCP), in accordance with the system used by [PERSON] et al. [39]. The length of the bed, formed by adding granular activated carbon, varied according to the amount of adsorbent used. To prevent void spaces and preferential pathways, as well as the carryover of GAC particles by the treated solution, the bed was positioned between 6 g of glass beads and a filter paper with a grammage of 80 g/m\({}^{2}\) and a thickness of 205 um (JProlab, Sao Jose dos Pinhais, PR, Brazil). Distilled water was pumped as the influent solution for at least 20 min to equilibrate the column prior to each operation. All experiments were conducted at 25 \({}^{\circ}\)C, using distilled water to dilute the pharmaceuticals from the stock solution. Breakthrough curves were generated by continuously monitoring the pollutant concentration over time during the process. For simulating the breakthrough curve, mathematical models that simplify the mass balance, typically represented by a system of partial differential equations [40], were employed to facilitate potential process scaling and to identify the limiting step of the adsorption process while determining the kinetic parameters of the column. The modeling of multicomponent adsorption systems is complex, due to simultaneous effects and displacement of the curves, thus has been little studied in the literature. For modeling the multicomponent breakthrough curves the empirical models proposed and established by [40], these being the logistic (Equation (7)) and Gompertz (Equation (8)), were applied in this work. \[\frac{\text{C}}{\text{C}_{0}}=\frac{1}{1+\exp[\text{k}(\tau-\text{t})]}+\frac {\text{ck}^{*}\text{exp}[\text{k}^{*}(\tau^{*}-\text{t})]}{\left\{1+\exp\exp[ \text{k}^{*}(\tau^{*}-\text{t})]\right\}^{2}} \tag{7}\]