## 5 Results
### Estimated Parameters Regarding Data from [PERSON] et al. (2021) [PERSON] et al. (2021)
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Metal** & & \(\mathbf{R^{2}}\) \\ \hline Cu & & 0.9168 \\ Mn & & 0.9277 \\ Ni & & 0.9076 \\ Zn & & 0.8984 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Metric statistics for data from [PERSON] et al. (2021) [14]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Metal** & & **R\({}^{2}\)** \\ \hline Cu & & 0.9168 \\ Mn & & 0.9277 \\ Ni & & 0.9076 \\ Zn & & 0.8984 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters estimated from each metal using data from [PERSON] et al. (2021) [14]. The estimated values in this study, when compared to the initial values estimated by [PERSON] et al. (2021) [14], show a discrepancy. Several factors may explain this difference. While theyobtained the mass transfer coefficient in the external film (\(k_{f}\)) through correlations, the coefficients of surface diffusion (\(D_{S}\)) and the Langmuir constant (\(k_{L}\)) were derived from batch adsorption experiments, and the axial dispersion coefficient (\(D_{ax}\)) was determined from fixed-bed column experiments. In contrast, in our study, all three coefficients (\(k_{L}\), \(D_{ax}\), and \(k_{S}\)) were estimated exclusively using the experimental data obtained from the column. A relevant issue is the modeling of transport within the particle, where both studies did not consider diffusion transport in the pores. This suggests that other parameters may compensate for this phenomenon in their estimated values, as well as other phenomena not accounted for in the models. In chromatography, it is common to perform a variety of experiments to estimate parameters and minimize correlations. For example, column porosity and axial dispersion can be determined by passing a non-penetrating tracer through the pores, while particle porosity and film and pore diffusions can be determined using a penetrating tracer [54]. However, estimating the film diffusion coefficient through batch experiments may result in a value different from the real one for a process involving flow in a fixed-bed column [25]. While [PERSON] et al. 's approach simplifies the model and provides initial estimates, it may have limitations, such as in the accurate estimation of parameters that ideally should be determined through column experiments. On the other hand, estimating numerous parameters simultaneously using column experimental data may result in correlations between them, posing a challenge in accurately estimating their values. However, the Bayesian approach allows for the incorporation of estimated parameter values derived from these specific experiments as prior information, combined with experimental data obtained from the columns [55]. Figure 4: Breakthrough for ions regarding data from [PERSON] et al. (2021) [14]: (**a**) Cu; (**b**) Mn; (**c**) Ni; (**d**) Zn. In this regard, it is always important to assess the goals of the work when estimating the parameters of a model. If the aim is only to represent the adsorption column used to obtain experimental data, empirical models can be highly useful. However, in situations where a generalization of the model is desired, resorting to phenomenological models and assumptions close to reality is necessary, as well as an estimation method that seeks to obtain parameter values that are as close to reality as possible to avoid compromising their physical meaning. For parameter estimation, R\({}^{2}\) values are around 0.9. A factor that may have influenced their estimation was the low sensitivity of the \(D_{ax}/Pe\) and \(k_{S}/K_{S}\) parameters and even a certain linear dependence between them. In Figure 4, the region between the blue lines represents an area with a 95% confidence interval. It is crucial to highlight that the 95% confidence interval represents the range in which the model is expected to have a probability of 95% considering the uncertainty in the parameters. On the other hand, the coefficient of determination R\({}^{2}\) measures the proportion of data variability that is explained by the model.