## 3 Results and Discussion
### Economic Tool to Evaluate the Advantages of Regeneration
Considering that the purchasing cost of GAC-Virgin is the same; thus,  
\[P_{o_{1}}=P_{o_{2}}=\dots P_{o_{n}} \tag{18}\]  
If the inlet water quality to the filter and the filter operation parameters do not change significantly then the saturation/exhaustion time will not change;  
\[P_{r}=V_{HCl,37\%}\cdot P_{HCl} \tag{10}\]0  
Under that condition, Equation (17) can be rewritten as:  
\[P_{r}=V_{HCl,37\%}\cdot P_{HCl} \tag{10}\]1  
where  
\(n_{c}\): number of replacement cycles (dimensionless)  
\(t_{sv}\): saturation time of GAC-Virgin (in years)In this specific case, the saturation or exhaustion time \(t_{sv}\) is two years for the GAC-Virgin under study. Therefore, for the strategy of scenario two, the cost \(C_{w,2}\) has a constant value according to Equation (21). \[P_{r}=V_{HCl,37\%}\cdot P_{HCl} \tag{10}\]2  
In contrast, for the chemical regeneration strategy implemented in scenario one, the cost \(C_{r}\) can be then expressed by Equation (22). \[P_{r}=V_{HCl,37\%}\cdot P_{HCl} \tag{10}\]3  
For scenario one, there is an initial investment process of purchasing GAC-Virgin (\(P_{o}\)) to fill the industrial adsorber of 1 ton of capacity. According to the virgin condition of the GAC, the saturation/exhaustion time is \(t_{sv}\). After the first two years of exploitation, the GAC in the filter becomes exhausted. However, a regeneration strategy is implemented in scenario one, and the investment to perform the first chemical regeneration process of GAC-Exh is \(P_{r_{1}}\) (in USD); thus \(P_{r_{n}}\) is the investment for \"n\" regeneration cycles. Correspondently, \(t_{sr_{1}}\) to \(t_{sr_{n}}\) are the saturation/exhaustion times of the GAC regarding the first to \"n\" chemical regeneration cycles. If the regeneration process occurs under the same operational conditions of HCI-GAC contact, HCI flow, retention time and the amount of HCI solution applied is the same, investment cost will also remain the same; thus,  
\[P_{r}=V_{HCl,37\%}\cdot P_{HCl} \tag{10}\]4  
Therefore,  
\[P_{r}=V_{HCl,37\%}\cdot P_{HCl} \tag{10}\]5  
where  
\(P_{r_{i}}\): Investment cost for chemical regeneration in an \"i\" regeneration cycle (in USD)  
\(n_{i}\): Number of chemical regeneration cycles (dimensionless)  
On the contrary, compared with GAC-Virgin where saturation time (\(t_{sv}\)) is a constant parameter, the saturation/exhaustion time after each regeneration cycle will be different. After each chemical regeneration cycle, the efficiency of adsorbates removal from the GAC pores decays which in turn progressively reduces the adsorption capacity of the adsorbent. In that case, \(t_{sr_{1}}>t_{sr_{2}}>t_{sr_{3}}>\ldots t_{sr_{n}}t_{sr_{i}}\): Saturation time of the regenerated GAC after the \"i\" regeneration cycle (in years)  
However, taking as reference the saturation time of the virgin GAC, the exploitation time of a regenerated GAC after an \"i\" regeneration cycle (\(t_{sr_{i}}\)) can be expressed as a fraction of the saturation time of virgin material (\(t_{sv}\)), and it can be defined using Equation (25):  
\[P_{r}=V_{HCl,37\%}\cdot P_{HCl} \tag{10}\]6  
where  
\(x_{i}\): Fractional saturation time of a regenerated GAC in the regeneration cycle \"i\" in terms of the saturation time of the virgin material. Then,  
\[P_{r}=V_{HCl,37\%}\cdot P_{HCl} \tag{10}\]7  
Taking into account the progressive loss of adsorption capacity after each regeneration cycle and also the higher adsorption capacity of GAC-Virgin compared to the regenerated GAC, it is logical to expect that the saturation time of the regenerated GAC will progressively decrease in comparison with the saturation time of GAC-Virgin.