## 5 Results
### Estimated Parameters Regarding Data from [PERSON] et al. (2021) [PERSON] et al. (2021)  
To estimate the uncertainty associated with the model predictions over time, using the Monte Carlo Method, a set of parameter samples was obtained from the model. These samples were then used to simulate the dependent variable (metal concentrations) at each point of interest in time. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(C_{e}\) & **100, 300 and 500 mol/L** & \(\rho_{\Phi}\) & **651.52 mg/cm\({}^{3}\)** \\ \hline Diameter of column & 2.5 cm & \(D_{\mathrm{tr}}\) & 23.1663 cm\({}^{2}\)/min \\ \hline \(L\) & 15, 30 and 45 cm & \(k_{\mathrm{r}}\) & 0.030536 min\({}^{-1}\) \\ \hline \multirow{2}{*}{\(F\)} & \multirow{2}{*}{5, 10 and 15 cm\({}^{3}\)/min} & \multirow{2}{*}{\(k_{\mathrm{L}}\)} & 0.529 L/mg (Cu) \\  & & & 0.078 L/mg (Cr) \\  & & & 0.5778 L/mg (Cd) \\ \hline \(c_{8}\) & 0.59 & Diameter of particle & 0.2 cm \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Operational conditions used by [PERSON] et al. (2020) [PERSON] et al. (2020). Figure 2: Sensitivity curve of parameters \(K_{s}\), \(K_{L}\), and Pe for each metal (**a**) copper, (**b**) manganese, (**c**) zinc, and (**d**) nickel under the experimental conditions of [PERSON] et al. (2021) [PERSON] et al. (2021) and experimental data points. For each simulated time point, the mean of the dependent variable values corresponding to the parameter samples is calculated. Subsequently, the 0.025 and 0.975 quantiles of the dependent variable values for that time point are determined. These quantiles define a 95% confidence interval for the dependent variable around the calculated mean. The lower and upper limits of this confidence interval represent the range within which it is expected that the concentration given by the model may vary, considering the uncertainty in the parameter estimates of the model. Thus, it is possible to assess the reliability of the model predictions over time and obtain a better understanding of the expected variability in the results. When estimating the parameters using MH-MCMC for each metal, a heating period of 3000 interactions was used for all chains, that is, the states after these were accounted for in the estimation. The number of states used was 10,000. The initial values used in the MCMC for the parameters are presented in Table 2 and will be taken as a reference. Figure 3 shows the chain evolutions related to each dimensionless parameter for each metal evaluated by [PERSON] et al. (2021) [14], where the red line represents the initial value used for the parameter. It is generally observed that all the chains tended to balance around an average and that they converged using few states. Figure 3: Markov chain for parameter estimation regarding data from [PERSON] et al. (2021) [14] with number of states equal a 10000. Where (**a–c**) correspond to the parameter chains for copper, (**d–f**) for manganese, (**g–i**) for nickel, and (**j,l,m**) for zinc. Table 5 shows some important information about the process, such as the step size of each parameter, the minimum and maximum values of the posterior distribution for each parameter, as well as the estimates obtained. The estimation was performed using different step sizes for each parameter and the values shown in Table 5 were reached. The choice of step size was based on whether or not the chain of parameters had stabilized. Concerning the breakthrough curve and parameter estimation, the model's ability to represent experimental data for each metal is demonstrated in the graphs of Figure 4 and the metrics of Table 6, which include the correlation coefficient. It is noted that the highest values, close to one, correspond to the manganese metal. Therefore, the model performed better in describing the experimental data for this metal. When evaluating both breakthrough curves and metrics for metals overall, we can conclude that, when using the estimated values, the model provides us with an idea of the dynamics of adsorption in the bed.