## 2 Materials and Methods
### Fixed-Bed Experiments  
Fixed-bed experiments were performed using a laboratory column with an internal diameter (d) of 2 cm and height (h) of 40 cm. The bed was filled with CES700 at a fixed height for each experiment, and the remaining space was filled with small glass spheres (inert material). The wastewater was percolated through the column in ascending flow (defined for each experiment) at room temperature (20 \(\pm\) 1 \({}^{\circ}\)C) using a peristaltic pump (Minipuls3-Gilson, Gilson SAS, Sarcelles, France). Several samples were collected from the top of the column at predetermined time intervals. The remaining soluble reactive P concentration in the liquid was measured using the ascorbic acid method (EPA Method 365.3) [19]. The plot of the normalized concentration (\(C_{t,p}\)/\(C_{0,p}\)) versus the time of the experiment allowed the determination of the breakthrough curves, and several parameters were estimated to be important for scaling up the process at the industrial level. In particular, the breakthrough time (\(t_{bp}\)) and exhaustion time (\(t_{ex}\)) correspond to the time at which the concentration of P at the column outlet was approximately 5% and 95% of the initial P concentration, respectively. The stoichiometric time was calculated according to Equation (1) using numerical integration of the breakthrough curve [25]. \[t_{st}=\int_{0}^{t_{ex}}\left(1-\frac{C_{t,p}}{C_{0,p}}\right)dt \tag{1}\]  
where \(C_{0,p}\) and \(C_{t,p}\) (mg P/L) are the P initial and concentrations at time t.  
The adsorption time (\(\Delta t_{abs}\)) is the difference between \(t_{ex}\) and \(t_{bp}\):  
Another relevant parameter is the fraction of saturated beds (\(FSB\)), which represents the ratio of the total mass of P adsorbed in the column until the breakthrough and saturation times (Equation (2)) [25]. \[FSB=\frac{\int_{0}^{t_{bp}}\left(1-\frac{C_{t,p}}{C_{0,p}}\right)dt}{\int_{0} ^{t_{ex}}\left(1-\frac{C_{t,p}}{C_{0,p}}\right)dt} \tag{2}\]  
The adsorption capacity (\(q_{e}\) in mg P/kg adsorbent), residence time (\(\tau\)), mass transfer zone length (\(MTZ\)), and length of the unused bed (\(LUB\)) under dynamic conditions were calculated using Equations (3)-(6), respectively [26; 27]. \[q_{e}=\frac{QC_{0,p}}{m}\int_{t=0}^{t_{ex}t}\left(1-\frac{C_{t,p}}{C_{0,p}}\right) \tag{3}\]  
\[\tau=\frac{\epsilon_{b}V}{Q} \tag{4}\]  
\[MTZ=h\left(1-\frac{t_{bp}}{t_{ex}}\right) \tag{5}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{d_{p}}\) (mm) & \(\epsilon_{b}\) & \(\rho_{b}\) (g/cm\({}^{3}\)) & pH & pH\({}_{2 pc}\) & \(\mathbf{S_{BET}}\) (m\({}^{2}\)/g) & \(\mathbf{V_{pore}}\) (cm\({}^{3}\)/g) \\ \hline
0.5–1.19 & 0.52 & 0.70–0.75 & 9.84 & 8.43 & 1.83 & 0.003 \\ \hline \hline \end{tabular} \(\mathrm{d_{p}}\)—particle diameter; \(\epsilon_{b}\)—bulk porosity; \(\rho_{b}\)—bulk density; pH\({}_{2 pc}\)—pH of zero-point charge; \(\mathrm{S_{BET}}\)—BET specific area; \(\mathrm{V_{pore}}\)—powe volume.  
\end{table} 
Table 1: Characterization of the adsorbent (CES700) [18; 24]. \[LUB=h(1-FSB) \tag{6}\]  
where \(Q\) (L/min) is the flow rate, \(m\) (kg) is the adsorbent mass, \(\varepsilon_{b}\) (-) is the bed void fraction, \(V\) (cm\({}^{3}\)) is the bed volume, and \(h\) (cm) is the bed height. The Bohart-Adams [28], [PERSON] [29], and Yoon-Nelson [30] models are normally used to estimate kinetic parameters related to the adsorption process in dynamic systems. However, these models are mathematically equivalent and can be expressed in terms of a logistic equation (Equation (7)), as discussed by [PERSON] [31]. These models are used to overcome some difficulties arising from the mathematical solution of the intraparticle diffusion models. Indeed, these models were developed based on simplifying hypotheses, making them less complex and time-consuming while still providing accurate predictions. \[\frac{C_{t,P}}{C_{0,P}}=\frac{1}{1+\exp(a-bt)} \tag{7}\]  
where \(a\) and \(b\) are general parameters of the logistic equation.