## 3 Results and Discussion
### Scale-Up  
Considering the criteria previously described in Section 2.3, the scale-up experiment was conducted with a large lab-scale column with a 60 cm bed height (\(H\)) and a 4 cm internal diameter (\(D\)). The flow rate was kept constant at 20 mL/min, assuming that the superficial fluid velocity was constant in both scenarios (lab and large-scale). As the determination of the large-scale bed dimensions was set based on the optimal conditions determined in the lab-scale experiments (\(Q=5\) mL/min, \(h=30\) cm), the breakthrough curves of both experiments are shown in Figure 5.  
The experimental data indicated that the breakthrough time was approximately 300 min, which is about eight times higher than that found at the lab-scale (run 2). However, the rate constants predicted by the models were of the same order in both cases (Tables 6 and 7), and the breakthrough curves have a similar shape. This scale-up experiment showed that it is possible to achieve higher \(FSB\) (increased from 0.425 to 0.764) and adsorption capacity values (increased from 70.9 to 84.1 mg/kg) than in the best conditions at the lab scale. The improvement in the \(FSB\) observed after scaling up can be attributed to factors such as enhanced mass transfer efficiency and improved bed packing in the larger column. These factors likely contributed to the more effective utilization of the adsorbent at that scale. The percentage of unused beds relative to the height of the bed decreased from 58 to 40% after scaling up the process. In addition, it was possible to maintain the effluent with final concentrations below 0.5 and 0.7 mg P/L (Urban Wastewater Treatment Directive--Directive (EU) 2024/3019) for 400 and 435 min of operation, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Logistic Equation**} & \multicolumn{2}{c}{**Bohart-Adams Model**} & \multicolumn{2}{c}{**Thomas Model**} & \multicolumn{2}{c}{**Yoon-Nelson Model**} \\ \hline \multirow{2}{*}{Run} & \(a\) & \(b\) & \(k_{ba}\) & \(q_{ba}\) & \(kth\) & \(qth\) & \(k_{ym}\) & \(\tau_{ym}\) & \(t_{50\%}\) \\  & **(-)** & **(1/min)** & **(L/mg min)** & **(mg/kg)** & **(L/mg min)** & **(mg/kg)** & **(1/min)** & **(min)** & **(min)** \\ \hline
1 & 3.479 & 0.043 & 0.004 & 27.1 & 0.004 & 94.2 & 0.043 & 80.5 & 76.7 \\
2 & 3.771 & 0.038 & 0.004 & 22.3 & 0.004 & 77.3 & 0.038 & 98.9 & 92.6 \\
3 & 4.189 & 0.026 & 0.003 & 27.6 & 0.003 & 89.9 & 0.026 & 164 & 159 \\
4 & 2.421 & 0.057 & 0.006 & 18.3 & 0.006 & 64.5 & 0.057 & 42.6 & 41.4 \\
5 & 3.127 & 0.047 & 0.005 & 21.1 & 0.005 & 71.2 & 0.047 & 66.5 & 58.9 \\
6 & 3.952 & 0.045 & 0.005 & 18.0 & 0.005 & 62.3 & 0.045 & 87.8 & 90.3 \\
7 & 1.666 & 0.098 & 0.010 & 12.2 & 0.010 & 44.5 & 0.098 & 17.1 & 13.8 \\
8 & 2.820 & 0.125 & 0.013 & 9.58 & 0.013 & 30.6 & 0.125 & 22.6 & 21.1 \\
9 & 2.944 & 0.077 & 0.008 & 11.2 & 0.008 & 36.8 & 0.077 & 38.5 & 33.9 \\ \hline
12 & 7.126 & 0.031 & 0.003 & 5.78 & 0.003 & 18.5 & 0.031 & 232 & 228 \\
13 & 3.163 & 0.171 & 0.017 & 17.9 & 0.017 & 59.1 & 0.171 & 18.4 & 18.5 \\ \hline \hline \end{tabular} \(k_{hu}\)—rate constant of [PERSON] model; \(q_{hu}\)—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; \(\tau_{ym}\)—time required to 50% of the bed saturation according to Yoon-Nelson model; \(t_{50\%}\)—experimental time required to 50% of the bed saturation.  
\end{table} 
Table 6: Parameters of the models fitted to breakthrough curves. Some studies have investigated the removal of P using other low-cost adsorbents. A literature study reported that an adsorbent derived from leftover coal materials effectively removed P from a solution with an initial concentration of 25 mg/L under a flow rate of 1 mL/min and a bed height of 8 cm. The total phosphate and sorption capacity of the coal-based material was approximately 243 mg/kg [10].