## 3 Results and Discussion
### Reaction Kinetics  
Experimental results indicated that the MnCe-GAC processes could effectively remove ammonium, MEA, and TMAH from aqueous solutions. In general, the removal of ammonium and MEA by these processes were better than that of TMAH as shown in Figure 10. Since the MnCe-GAC/O\({}_{3}\)/H\({}_{2}\)O\({}_{2}\) process had the highest removal efficiency among these processes, reaction kinetics of the process was investigated. A suitable model could help in future design and implementation of the process. Many sophisticated models such as pore diffusion, homogeneous surface diffusion, and heterogeneous diffusion models have been developed to describe various kinetics processes. However, these models might not be convenient for practical use due to the mathematical complexity of these models [41; 42]. Therefore, four simplified models (pseudo-first-order equation or generalized first order equation [41; 42; 43], pseudo-second-order equation [2], intra-particle diffusion model [2; 38; 44; 45; 46], and liquid film diffusion [46]) were applied in this research. Figure 9: Degradation of TMAH by the MnCe-GAC/O\({}_{3}\)/H\({}_{2}\)O\({}_{2}\) process. (**a**) synthetic (Co = 100 mg/L) and (**b**) real wastewater samples (C\({}_{\rm o}\) = 100 mg/L, V = 0.5 L, solid 1 g, 0.25 mmol O\({}_{3}\)/min, 0.125 mmol H\({}_{2}\)O\({}_{2}\)/min). Pseudo-first-order equation:  
\[\frac{dq_{t}}{dt}=K_{1}\left(q_{e}-q_{t}\right), \tag{2}\]  
\[\frac{1}{q_{t}}=\left(\frac{K_{1}}{q_{e}}\right)\left(\frac{1}{t}\right)+\frac{1 }{q_{e}}. \tag{3}\]  
In the above equations, \(q_{t}\) is the amount of adsorbate adsorbed (mg g\({}^{-1}\)) at time \(t\) (min), \(q_{e}\) is the maximum adsorption capacity (mg g\({}^{-1}\)), and \(K_{1}\) is the pseudo-first-order rate constant (min\({}^{-1}\)) for the process. Pseudo-second-order equation:  
\[\frac{dq_{t}}{dt}=K_{2}(q_{e}-q_{t})^{2}, \tag{4}\]  
\[\frac{t}{q_{t}}=\frac{1}{K_{1}q_{e}^{2}}+\frac{t}{q_{e}}. \tag{5}\]  
In the above equations, \(K_{2}\) is the pseudo-second-order rate constant (g mg\({}^{-1}\) min\({}^{-1}\)). Intra-particle diffusion model (or Weber and Morris' method):  
\[q_{t}=K_{p}\times t^{1/2}+C_{p}. \tag{6}\]  
In the above equation, \(C_{p}\) is the intercept (mg g\({}^{-1}\)) and \(K_{p}\) is the intra-particle diffusion rate constant (mg min\({}^{-0.5}\) g\({}^{-1}\)). Liquid film diffusion model:  
\[\ln\left(1-\frac{q_{t}}{q_{e}}\right)=-K_{L}t. \tag{7}\]  
In the above equation, \(K_{L}\) is the liquid film diffusion rate constant (min\({}^{-1}\)) for the adsorption process. Experimental and simulation results are presented in Figure 11 and Table 2. To evaluate these models, root mean square error (RMSE) was calculated as follows:  
\[\text{RMSE}=\sqrt{\frac{\sum\left(q_{exp}-q_{pre}\right)^{2}}{N}}. \tag{8}\]  
Figure 9:0 Degradation of ammonium, MEA, and TMAH by the MnCe-GAC/O\({}_{3}\)/H\({}_{2}\)O\({}_{2}\) process (Co = 100 mg/L, V = 0.5 L, solid 1 g, 0.25 mmol O\({}_{3}\)/min, 0.125 mmol H\({}_{2}\)O\({}_{2}\)/min). In the above equation, \(q_{exp}\) and \(q_{pre}\) are model experimental and predicted values, respectively, and \(N\) is the number of data points. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Adsorbate**} \\ \cline{3-5}
**Models** & & **Ammonium** & **MEA** & **TMAH** \\ \hline Pseudo-first-order model & K\({}_{1}\) & 0.161 & 0.101 & 0.056 \\  & RMSE & 1.31 & 0.43 & 2.61 \\ \hline Pseudo-second-order model & K\({}_{2}\) & 0.00383 & 0.00196 & 0.00117 \\  & RMSE & 12.81 & 8.89 & 8.15 \\ \hline Intra-particle diffusion model & K\({}_{\text{P}}\) & 4.54 & 4.60 & 3.68 \\  & C\({}_{\text{P}}\) & 10.51 & 5.91 & 0.70 \\  & RMSE & 60.25 & 22.28 & 0.38 \\ \hline Liquid film model & K\({}_{\text{L}}\) & 0.0698 & 0.0437 & 0.0247 \\  & RMSE & 1.31 & 0.43 & 2.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation results with various kinetic models.