## 3 Results and Discussion
### Economic Assessment Based on the Proposed Tool  
Based on the experimental results, the economic criteria \(\left(\frac{C_{r1}}{C_{w2}}\right)\) by applying Equation (28) resulted in the data depicted in Table 2. From Table 2, the first value of \(\left(\frac{C_{r1}}{C_{w2}}\right)=1\), since both scenarios one and two need to purchase the same amount of GAC-Virgin and any regeneration has not been performed yet (\(n_{i}=0\)). In addition, both scenarios are using GAC-Virgin till exhaustion after two years of exploitation and cost for both scenarios are the same within this period. The function \(\sum_{0}^{n_{i}}x_{i}\) can be calculated by Equation (29). \[\sum_{0}^{n_{i}}x_{i}=x_{ni-1}+\sum_{0}^{n_{i}-1}x_{i} \tag{29}\]  
Equation (29) mathematically corrects the loss in exploitation duration (no loss when \(x_{i}=1\)) when regenerated GAC is used. This loss is defined by calculating the ratio of the exploitation time of a GAC-Exh after a regeneration cycle \(n_{i}\) and the exploitation time of GAC-Virgin. It is clear that after each regeneration, this \(x_{i}\) value decreased. In Table 3, the calculation process of \(\sum_{0}^{n_{i}}x_{i}\) is indicated by arrows as a simplified scheme for the first case. When \(n_{1}\), \(x_{1}=0.94\), this value represents the ratio of breakthrough time for GAC-Exh\({}_{1}\) divided by the breakthrough time for GAC-Virgin (Equation (25)). Figure 4: Breakthrough curves after different number of regeneration cycles (n\({}_{i}\)) and virgin GAC exhaustion. c(Ca\({}^{2+}\))\({}_{i}\)/c(Ca\({}^{2+}\))\({}_{0}\) (water hardness after filtration)/(initial water hardness). Analysing the behaviour of the \(\left(\frac{C_{r,1}}{C_{w,2}}\right)\) function at studied conditions, the regeneration investment is quite low compared with the purchasing cost of the virgin material (just around 9.1% of the purchasing cost of virgin material => 569.4/6270) for ten regenerations cycles and the ratio \(\left(\frac{C_{r,1}}{C_{w,2}}\right)\) is still below the breakout line \(\left(\frac{C_{r,1}}{C_{w,2}}\right)=1\) thus suggesting that the process is economically favourable even when the performance of the regenerated GAC after ten regenerations is significantly deteriorated. However, a deeper analysis must be conducted on the \(\left(\frac{C_{r,1}}{C_{w,2}}\right)\) ratio to determine the optimal number of regeneration cycles. In order to find the optimal number of regeneration cycles (\(n_{i}\)*), the first derivative of \(\left(\frac{C_{r,1}}{C_{w,2}}\right)\) in terms of \(\sum_{0}^{n_{i}}x_{i}\) and \(n_{i}\)* can be determined having \(\frac{d\left(\frac{C_{r,1}}{C_{w,2}}\right)}{d\left(\sum_{0}^{n}x_{i}\right)}=0\). The optimal number of regenerations (\(n_{i}\)*) was numerically determined and presented in Table 3. Based on the found results comparing both strategies (with and without regeneration) in terms of the ratio \(\left(\frac{C_{r,1}}{C_{w,2}}\right)\), at the beginning of the process (filling up the GAC filters with virgin material) both scenarios have to spend the same amount of money for purchasing the GAC, therefore \(\left(\frac{C_{r,1}}{C_{w,2}}\right)=1\). However, after the first two years, in scenario two, all GAC-Exh is replaced and an investment of USD 6270 to buy fresh GAC-Virgin is required. Meanwhile, in scenario one, an investment of just USD 569.4 is needed for regenerating one ton of GAC-Exh. That means that in the first two years for scenario one, an investment of only USD 6270+569.4 is required and for scenario two, this increases to USD 6270 + 6270. However, due to regeneration, a small loss of GAC material could be possible. It will be proposed to do this regeneration in situ for future installation, which reduce this loss to a minimum. On the other side, an extra cost has to be included in the construction of the GAC filter as the actual GAC filters in the WTU are made from carbon steel. The use of HCl during regeneration demands special construction materials, e.g., stainless steel (see supplementary information).