## 3 Results and Discussion  
### CAF Adsorption onto GAC  
The adsorption of caffeine onto GAC was studied individually to assess the potential of the technique and optimize it for subsequent application in the proposed combined process. The experimental tests and statistical analysis were based on the Box-Behnken design, enabling the determination of the optimal operating values for the following independent parameters: pH, adsorbent solid concentration, and contact time. The caffeine removal efficiency results obtained from the experimental tests based on the levels defined for the dependent and independent variables are shown in Table 3.  
According to Table 3, the final caffeine concentration in the tests ranged from 1.29 to 19.12 mg L\({}^{-1}\), with caffeine removal rates varying between 2.48% and 93.40%. The responses from the three central points showed minimal variation, indicating a high level of repeatability in the process. Consequently, mathematical models predicting the experimental results were calculated. The second-order models that correlate caffeine removal and C\({}_{\text{F}}\) as a function of the independent variables are presented in Equations (5) and (6), displaying the model in terms of coded variables. \[ \begin{array}{c}\text{R}_{\text{CAF}}(\%)=-0.315+0.004\text{ x}-0.001\text{ x}^{2}+1.511\text{ y}-0.832\text{ y}^{2}+0.010\text{ z}-4.974\times\\ 10^{-5}\text{ z}^{2}-0.012\text{ xy}+6.068\times 10^{-5}\text{ xz}+0.002\text{ yz}\end{array}  \tag{5}\]  
\[ \begin{array}{c}\text{C}_{\text{F}}=25.792-0.067\text{ x}+0.010\text{ x}^{2}-29.637\text{ y}+16.309\text{ y}^{2}-0.197\text{ z}+0.001\text{ z}^{2}+\\ 0.228\text{ xy}-0.001\text{ yz}-0.030\text{ yz}\end{array}  \tag{6}\]  
where \"R\({}_{\text{CAF}}\)\" is the caffeine removal percentage, C\({}_{\text{F}}\) is the final caffeine concentration, \"x\" is the pH, \"y\" is the adsorbent dosage in g L\({}^{-1}\), and \"z\" is the contact time in minutes. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test Runs** & **pH** & \begin{tabular}{c} **C\({}_{\text{ads}}\)** \\ **(g L\({}^{-1}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Contact Time** \\ **(min)** \\ \end{tabular} &
\begin{tabular}{c} **C\({}_{\text{F}}\)** \\ **(mg L\({}^{-1}\))** \\ \end{tabular} & **Removal (\%)** \\ \hline
1 & 7 & 10.0 & 120 & 1.91 & 90.27 \\
2 & 3 & 5.5 & 120 & 1.53 & 92.18 \\
3 & 3 & 1.0 & 65 & 14.80 & 24.56 \\
4 & 3 & 10.0 & 65 & 1.29 & 93.40 \\
5 & 11 & 10.0 & 65 & 2.97 & 84.86 \\
6 & 7 & 5.5 & 65 & 5.36 & 72.65 \\
7 & 7 & 5.5 & 65 & 5.04 & 74.32 \\
8 & 7 & 5.5 & 65 & 4.64 & 76.36 \\
9 & 7 & 1.0 & 10 & 19.12 & 2.48 \\
10 & 11 & 1.0 & 65 & 14.83 & 24.39 \\
11 & 11 & 5.5 & 120 & 2.01 & 89.73 \\
12 & 11 & 5.5 & 10 & 15.23 & 22.35 \\
13 & 3 & 5.5 & 10 & 13.70 & 30.14 \\
14 & 7 & 10.0 & 10 & 11.58 & 40.95 \\
15 & 7 & 1.0 & 120 & 12.45 & 36.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental conditions and results for design of experiments (DOE) with central composite analysis, where C\({}_{\text{ads}}\) is GAC concentration. The significance of variables, interactions, and quadratic effects was assessed using variance analysis (ANOVA). This statistical analysis was performed to verify the difference in distribution between measurements, ensure that adjustment errors are independent and normally distributed, and thus guarantee the applicability of the proposed mathematical models. Table 4 presents the results obtained, where SS represents the sum of squares, DF indicates the degree of freedom, and MS denotes the mean squares. The results of the statistical analysis revealed that the linear and quadratic terms associated with contact time and adsorbent dosage are statistically significant at a 5% level of significance, evidenced by the low \(p\)-values (0.0004, 0.0186, 0.0005, and 0.0279), all below 0.05. In contrast, the other variables do not present statistical significance (\(p\)-value \(>\) 0.05) and, therefore, do not exert a significant influence on the caffeine adsorption process on granular activated carbon.