## 3 Results and Discussion
### Fixed-Bed Column Adsorption Evaluation
Therefore, a shallower bed depth has proven more advantageous for achieving optimal pharmaceutical adsorption capacity. The initial concentration of pharmaceuticals has a proportional effect on the mass present in the process. Lower concentrations of pharmaceuticals at a shallower bed height (Figure 5a,d) were sufficient to yield favorable process responses without the need for higher concentrations. A similar trend was noted for the flow rate (Figure 5c,f). This was expected, as an increase in feed flow rate reduces the contact time between the adsorbate and the adsorbent, which in turn decreases bed capacity [67; 68]. [PERSON] et al. [69] observed that while increasing the initial pollutant concentration generally enhances adsorption capacity, it can also reduce process efficiency (bed utilization). This finding aligns with the present study, where higher concentrations of pharmaceuticals led to decreased process efficiency. An ANOVA analysis and a Pareto diagram were conducted for the two response factors to assess the significance of the variables applied in the process (see Table S2 and Figure S3 in the Supplementary Material). The validity of the polynomial model was evaluated by calculating the coefficient of determination (R\({}^{2}\)). By examining the Pareto chart, one can determine the magnitude and importance of the effects within the process. This analysis revealed that only the qmaxPAR response factor was influenced by the evaluated variables in the process, specifically the initial concentration (C\({}_{0}\)) and the interaction between the solid mass (W) and flow rate (Q). The R\({}^{2}\) and adjusted R\({}^{2}\) values for this factor were 0.5297 and 0.0, respectively. In contrast, for the qmaxMTZ response factor, although no significant variables were identified, it yielded R\({}^{2}\) and adjusted R\({}^{2}\) values of 0.811 and 0.526, respectively. These values raise questions about the reliability of the relationship between predicted and experimental data, and in some cases, they may be considered statistically unsatisfactory. Furthermore, when investigating each parameter and their interactions individually (Figure S3), it was noted that parameters with negative coefficients corresponded to undesirable or antagonistic effects, while those with positive coefficients indicated desirable and/or synergistic effects on the efficiency of the adsorption process. To confirm the previous statements and approximate the mathematical relationship of the experimental matrix, as in batch adsorption, two second-order polynomial equations were obtained. The first equation refers to qmaxPAR (Equation (14)), and the second equation to q\({}_{\text{maxMTZ}}\) (Equation (15)). These equations are valid only for the conditions applied in the present study, and only the significant factors were considered. \[\text{q}_{\text{maxPAR}(\frac{\text{mg}}{\text{g}})}=112.827-1.8604\text{x}_{1}-5.5251\text{x}_{3}+1.3102\text{x}_{2}\text{x}_{3} \tag{14}\]  
\[\text{q}_{\text{maxMTZ}(\frac{\text{mg}}{\text{g}})}=133.046 \tag{15}\]  
where q\({}_{\text{maxPAR}}\) and q\({}_{\text{maxMTZ}}\) (mg/g) represent the maximum adsorption capacity for paracetamol and metronidazole, respectively; x\({}_{1}\) is the initial concentration of the pharmaceuticals; x\({}_{2}\) represents the mass (\(W\); g); and x\({}_{3}\) is the feed flow rate (Q; mL/min). Considering these equations and the optimal experimental conditions obtained through the experimental procedure, it can be observed that the values of q\({}_{\text{maxPAR}}\) = 32.665 mg/g and q\({}_{\text{maxMTZ}}\) = 133.046 mg/g are derived from the model, with the latter showing that the variables involved do not significantly influence the adsorption of the compound. When analyzing these results in conjunction with the R\({}^{2}\) value obtained in this study, it becomes evident that the experimental design and the applied model do not statistically support the proposed experiment.