## 5 Results
### Sensitivity Analysis  
In the sensitivity analysis using data from [PERSON] et al. (2021) [14], a disturbance value \(\varepsilon\) of 0.1 was applied to all parameters in every sensitivity analysis conducted in this study. Furthermore, the authors' experimental data points were plotted on the sensitivity graphs to assess the influence of each parameter on the model's response.  
The graphs presented in Figure 2 illustrate the sensitivity behavior of the model regarding the perturbed dimensionless parameters. Upon assessing the magnitude of the process' sensitivity to these parameters, a similar pattern was observed for all four metals, with the Langmuir constant exerting the most significant influence on the phenomenon. Conversely, the impact of the other two parameters on the model's response appeared to be relatively modest, potentially complicating the stabilization of the chain and the accurate estimation of these parameters. Ideally, the \(J_{ij}\) values should be high to mitigate the model's sensitivity to measurement errors, thereby enhancing the precision of the parameter estimates. Conversely, a low \(J_{ij}\) value implies that substantial changes in \(P_{j}\) yield minimal alterations in \(\theta_{i}\). This underscores the challenge in estimating parameter \(P_{j}\), as a broad range of \(P_{j}\) values would essentially yield the same outcome for \(\theta_{i}\)[42; 52].  
For the data from [PERSON] et al. (2020) [15], sensitivity analyses were conducted for each of the nine experiments performed for each metal, with the experimental conditions presented in Table 4. In this analysis, three heavy metals were evaluated: copper, chrome, and cadmium. The intraparticle mass transfer coefficient (\(k_{s}\)) and dispersion coefficient (\(D_{ax}\)) shown in Table 4 served as initial values, and were the mean values estimated by MCMC for the four metals observed by [PERSON] et al. (2021) [14]. Regarding the Langmuir constants, the values found by [PERSON] et al. (2017) [53] were used, which assess various isotherms, including the Langmuir isotherm, for the same metals. Finally, the \(q_{max}\) values were calculated from Equation (2) using the experimental data from the breakthrough of each metal.  
Figure 1: Breakthrough generated by MOL and pdepe for different discretizations.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & \multicolumn{4}{c}{\(\theta\) (MOL)} & \multicolumn{4}{c}{\(\theta\) (PDEPE)} \\ \hline  & n = 20 & n = 40 & n = 60 & n = 80 & n = 100 & n = 20 & n = 40 & n = 60 & n = 80 & n = 100 \\ \(\tau\) = 363.7 & 0.9993 & 0.9993 & 0.9991 & 0.9991 & 0.9988 & 0.9987 & 0.9989 & 0.9989 & 0.9989 & 0.9988 \\ \(\tau\) = 180.0 & 0.7892 & 0.7839 & 0.7820 & 0.7813 & 0.7816 & 0.7778 & 0.7777 & 0.7782 & 0.7775 & 0.77808 \\ \hline \end{tabular}
\end{table}
Table 3: Convergence analysis.  
Given the large number of scenarios, it was decided to estimate the parameters based on the most favorable experimental condition, i.e., the one where the model response showed the greatest sensitivity to the parameters (see Supplementary Materials). Unlike what was observed under the conditions of [PERSON] et al. (2021) [PERSON] et al. (2021), the intraparticle mass transfer coefficient (\(K_{S}\)) has a greater influence on the dependent variable, while the parameters \(K_{L}\) and Pe demonstrate a relatively low influence. After a visual analysis of the sensitivity graphs, the condition \(F=5\) mL/min, \(C_{e}=100\) mg/L, and \(L=15\) cm was chosen, as this proved to be the best scenario in terms of the sensitivity of the parameters.