## 3 Results and Discussion
### Scale-Up
In another study, calcium-modified attapulgite achieved a maximum adsorption capacity of 13.5 mg/g (equivalent to 13,500 mg/kg), as estimated by the [PERSON] model [32]. However, these values may be overestimated, as the experiments were conducted using synthetic wastewater, which lacks the complexity and competing interaction characteristics of real wastewater matrices considered in the present study. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{3}{c}{**Experimental Results**} & \multicolumn{3}{c}{**Predicted Results**} \\ \hline \(t_{bp}\) (min) & 334 & Logistic Equation & \\ \(t_{ex}\) (min) & 570 & \(a\) (-) & 17.33 \\ \(t_{st}\) (min) & 438 & \(b\) (1/min) & 0.035 \\ \hline \(\Delta t_{ads}\) (min) & 236 & [PERSON] model & \\ \(FSB\) & 0.764 & \(k_{bat}\) (L/mg.min) & 0.003 \\ \(q_{e}\) (mg P/kg) & 84.1 & \(q_{bat}\) (mg/kg) & 61.29 \\ \hline \(MTZ\) (cm) & 24.8 & [PERSON] model & \\ \(LUB\) (cm) & 14.2 & \(kth\) (L/mg.min) & 0.003 \\  & & \(q_{ths}\) (mg/kg) & 121 \\ \hline \multicolumn{3}{c}{Yoon-Nelson model} \\  & & \(k_{ym}\) (1/min) & 0.035 \\ \(\tau\) (min) & 495 \\ \(t_{50\%}\) (min) & 506 \\ \hline \hline \end{tabular} \(t_{bp}\)—breakthrough time; \(t_{ex}\)—ehaustion time; \(t_{st}\)—stoichiometric time; \(\Delta t_{ab}\)—difference between \(t_{ex}\) and \(t_{bp}\); \(FSB\)—fraction of saturated bed; \(q_{e}\)—adsorption capacity; \(MTZ\)—mass transfer zone; \(LUB\)—length of unused bed; \(k_{bat}\)—rate constant of [PERSON] model; \(q_{bat}\)—maximum adsorption capacity of [PERSON] model; \(kth\)—rate constant of [PERSON] model; \(qth\)—maximum adsorption capacity of [PERSON] model; \(k_{ym}\)—rate constant of Yoon-Nelson; \(\tau\)—time required to 50% of the bed saturation.  
\end{table} 
Table 7: Parameters calculated to assess the process performance in a large lab-scale column, both experimentally and predicted [\(Q\) = 20 mL/min; \(H\) = 60 cm; 20 \(\pm\) 1 \({}^{\circ}\)C]. Figure 5: Breakthrough curves of lab-scale columns under optimum conditions (\(Q\) = 5 mL/min, \(h\) = 30 cm) and a large-lab-scale column (\(Q\) = 20 mL/min, \(H\) = 60 cm), and fitted with Logistic Equation (7) [10 mg P/L, 20 \(\pm\) 1 \({}^{\circ}\)C, pH 7.33]. According to the results of both scales, the goal is to predict whether it is viable to operate CES700 in an industrial column and for how long. The _LUB_ method, Equations (8)-(11), was used to simulate this behavior, assuming that the _LUB_ and \(u_{z}\) at the industrial scale are equal to those found in the large-lab-scale experiment. If a bed height of 1 m was intended for use on an industrial scale, the expected \(t_{bp}\) and \(t_{st}\) values would be around 600 and 730 minThe diameter of an industrial column can also be estimated. For that, the superficial velocity (0.01592 m/min) was maintained under optimal conditions (Run 2) and large-lab-scale conditions, and a treated flow rate of about 50 m\({}^{3}\)/day was assumed after the anaerobic digestion process [19]. Due to the reasonable scale-up reasons, it was assumed that this flow rate would be treated in five parallel columns. According to these considerations, the estimated column diameter was 0.75 m. These results and lab-scale models offer useful insights. However, to improve the accuracy of full-scale design, this information should be complemented by pilot-scale on-site experiments and the application of mechanistic models to achieve a realistic interpretation of mass transfer mechanisms.