## 3 Results and Discussion
### Economic Assessment Based on the Proposed Tool
Thus \(\frac{C_{r,1}}{C_{w,2}}=\frac{6839.4}{12,540}=0.545\)  
\(n_{i}\)*: Optimal number of regenerations. Following the dynamics of the \(\left(\frac{C_{r,1}}{C_{w,2}}\right)\) economic function from Table 3, the economic profit increases with the number of regeneration cycles. However, a minimum value of \(\left(\frac{C_{r,1}}{C_{w,2}}\right)\) is eventually found which in turn corresponds to a maximal economic profit. The optimal number of regenerations is found to be seven, which is also confirmed using the first derivative function. The mathematical optimal point occurs when the first derivative function equals zero between six and seven regeneration cycles. However, regeneration is \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(n_{i}\) & \(t_{\text{rs}}\) (Min) & \(x_{i}\) & & \(\sum_{0}^{n_{i}}(x_{i})\) & \(\frac{C_{r,1}}{C_{w,2}}\) & \(\frac{d\left(\frac{C_{r,1}}{C_{w,2}}\right)}{d\left(\sum_{0}^{n}x_{i}\right)}\) \\ \hline
0 & 272 & 1.00 & & & 0.00 & 1.000 & \(-\)0.455 \\
1 & 257 & 0.94 & & & 1.00 & 0.545 & \(-\)0.153 \\
2 & 223 & 0.85 & & & 1.94 & 0.401 & \(-\)0.078 \\
3 & 222 & 0.82 & & & 2.79 & 0.336 & \(-\)0.049 \\
4 & 206 & 0.76 & & & 3.61 & 0.296 & \(-\)0.033 \\
5 & 155 & 0.57 & & & 4.36 & 0.271 & \(-\)0.019 \\
6 & 128 & 0.47 & & & 4.93 & 0.260 & \(-\)0.011 \\ \hline \(n_{i}\) & 7 & 90 & 0.33 & & 5.40 & 0.255 & 0.003 \\ \hline
8 & 70 & 0.26 & & & 5.74 & 0.256 & 0.014 \\
9 & 64 & 0.24 & & & 5.99 & 0.260 & 0.017 \\
10 & 58 & 0.21 & & & 6.23 & 0.264 & - \\ \hline \hline \end{tabular} \(n_{i}\)* Optimal number of regenerations.  
\end{table} 
Table 3: Parameters for the calculation of optimal number of regeneration cycles to maximize economic profit. in this case a discrete function and seven regeneration cycles can be considered as optimum number, as shown in Figure 5.  
For seven regeneration cycles, the saturation time represents about 33% of the saturation time of virgin material. According to the obtained results, the \(\left(\frac{C_{r1}}{C_{w2}}\right)\) function starts to increase after seven cycles, thus indicating that the profitability of the regeneration process is going down due to the reduced saturation time of the regenerated GAC, which is confirmed by the sign switch (- to +) of the derivative function. As the regeneration cost is quite low compared with purchasing GAC-Virgin, the function \(\left(\frac{C_{r1}}{C_{w2}}\right)\) is still under the critical value of one even after 10 regenerations. However, to keep the regeneration strategy over seven cycles, the saturation time of the regenerated GAC will drop to very low values (near to zero) and the frequency of regenerations to be performed will increase significantly (\(n_{r\rightarrow\infty}\)) and according to Equation (28) also the function \(\left(\frac{C_{r1}}{C_{w2}}\right)\). Therefore, an optimization analysis permits us to determine the optimal number of regenerations. Figure 6 presents the plots of the economic index \(\left(\frac{C_{r1}}{C_{w2}}\right)\) behaviour in function of the number of regeneration cycles (Table 3). The optimal economic profit is at \({n_{i}}^{*}=7\). The economic index is the practical outcome of the deducted model to evaluate the GAC management strategy. It can be noticed that \(0<\left(\frac{C_{r1}}{C_{w2}}\right)\)\(<\)1, therefore the regeneration is economically favourable even for higher costs of chemical regeneration cycles. In regeneration cycle seven (about 11 years of exploitation), in scenario one, just about 26% of the cost is invested compared with scenario two, which had to purchase almost six tons of virgin GAC within this period of operation. After seven regenerations, in scenario one, all the GAC-Exh had to be replaced to restart the process. The reduced saturation time of the regenerated GAC beyond seven regenerations cycles is too short to prolong the regeneration strategy.