## 3 Results and Discussion
### Batch Adsorption Evaluation
[50] found that during the evaluation of batch adsorption for the pharmaceuticals paracetamol and atenol, both compounds exhibited rapid kinetics on activated carbon, reaching equilibrium at 60 min, with removal rates of 98% for PAR and 95% for ATL. The authors noted that prior to this time, the removal rates were approximately 50%, and after 60 min, the removal occurred at a much slower pace, with minimal changes in residual concentration over time. Similarly, [PERSON] et al. [41] observed that an increase in contact time resulted in higher pollutant removal rates and adsorption capacities, ultimately reaching process equilibrium. They attributed these results to the characteristics of the batch process, where initially, all active sites on the carbon activated surface are available facilitating rapid mass transfer. Over time, as these sites become occupied, their availability diminishes, leading to equilibrium. After evaluating the influence of various parameters on the adsorption process, the optimal conditions for simultaneous pharmaceutical degradation were established. Based on the proposed design, an ideal solution pH of 7, a solid mass of 2.0 g, and a contact time of 60 min were selected, resulting in a removal rate of over 98.1% and an adsorption capacity of 4.0 mg/g for the pharmaceuticals. These findings underscore the significance of careful planning and optimization of process variables to achieve high removal rates and adsorption capacities in water or effluent treatment applications. The results of the ANOVA analysis for the statistical model can be found in the Supplementary Material, specifically in Table S1. Significant variables and interactions were identified by examining the \(p\)-value at a 95% confidence level (\(p<0.05\)), and these  
Figure 5: Surface area and contour plots for simultaneous pharmaceuticals adsorption by fixed-bed column for \(\mathsf{q_{maxPAR}}\): (**a**) W and \(\mathsf{C_{0}}\), (**b**) Q and W, (**c**) \(\mathsf{C_{0}}\) and Q; \(\mathsf{q_{maxMT2}}\): (**d**) W and \(\mathsf{C_{0}}\), (**e**) Q and W, (**f**) \(\mathsf{C_{0}}\) and Q. Red areas indicate maximum response, green shows minimum, and yellow represents intermediate values. The response is maximized in red zones. are highlighted in red in the table. Additionally, the Pareto chart is presented in Figure S1 (Supplementary Material). The analysis indicates that the pH parameter (x\({}_{1}\)) does not show significant evidence for pharmaceutical removal or batch adsorption capacity, regardless of the conditions applied, which aligns with the observations in Figure 5. In contrast, solid mass (x\({}_{3}\)) and contact time (x\({}_{2}\)) demonstrate significant evidence for both response factors, particularly for Q\({}_{e}\). These findings are illustrated in the Pareto diagram, where the parameters that interfered with the system and exceeded the 95% confidence level were process time and solid mass (Figure S1, Supplementary Material). Under these conditions, the model effectively captured the relationship between the variables and the responses. Additionally, the values of the coefficient of determination (R\({}^{2}\)) and the adjusted coefficient of determination (R\({}^{2}\)Adj) were R\({}^{2}\) = 0.9754; R\({}^{2}\)Adj = 0.9569 for %R and R\({}^{2}\) = 0.8893; R\({}^{2}\)Adj = 0.8063 for Q\({}_{e}\) (mg/g), respectively, indicating strong agreement between actual and predicted values and further confirming the satisfactory relationship among the data. To approximate the mathematical relationship of the experimental matrix, two second-order polynomial equations were derived: the first relates to removal efficiency (R\({}^{\circ}\); Equation (12)), and the second pertains to adsorption capacity (Q\({}_{e}\); Equation (13)). These equations are applicable solely under the conditions employed in this study and take into account only the significant factors involved in their formulation.