## 6 Conclusions  
In this work, an inverse problem was solved to estimate the parameters of a model derived from the mass balance in an adsorption column using the Bayesian approach. In this methodology, a set of real experimental data was used and the estimates were evaluated through the R\({}^{2}\) metric and a comparison of the breakthrough curve generated by the model with the experimental data.  
The model used in the present work proved to be a good alternative to predict the dynamics of the adsorption process; with this, it is possible to carry out studies to size and scale up the adsorption column for pilot and industrial conditions. The solutions to the mathematical model obtained using the method of lines and using the pdepe function proved to be similar for the same conditions.  
It is possible to use this model in parameter estimation problems, such as Langmuir constants and the intra-particular mass transfer coefficient. The experimental data obtained by [PERSON] et al. (2020) [PERSON] et al. (2020) were well described by the estimates, while the estimation did not have as good an agreement with the data used in the work by [PERSON] et al. (2021) [PERSON] et al. (2021); this may be related to the initial values used for the parameters as a priori knowledge or because of the low sensitivity of the phenomenon in relation to the parameters. Another factor that may have influenced the results is the linear dependence between the \(k_{L}\) and \(k_{S}\) parameters observed in the sensitivity graphs. In this way, the Bayesian approach presented itself as a good alternative for the estimation of parameters in problems involving the adsorption process.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app14125035/s1](https://www.mdpi.com/article/10.3390/app14125035/s1)]([https://www.mdpi.com/article/10.3390/app14125035/s1](https://www.mdpi.com/article/10.3390/app14125035/s1)).  
Methodology, software, visualization, writing--original draft, W.F.S.; formal analysis, writing--review and editing, C.S.D. and C.H.R.M.; methodology, writing--review and editing, E.C.R. and D.S.D.C.; conceptualization, methodology, software, validation, writing--review and editing, supervision, D.C.E. and B.M.V. All authors have read and agreed to the published version of the manuscript.  
The authors thank FAPESPA for supporting the research through funding provided by the 15/2021-PROPESP call and the project titled \"Adsorcao De Gases Em Leito Fixo: Uso De adsorventes Produzidos A Partir De Residuos De Mineracao Em Sistema Com escala Semi Piloto\" agreement no. 13/2022.  
Not applicable.  
Upon request, the data will be made available from the corresponding author. The data are not publicly available due to privacy.  
The authors thank the Federal University of Para (UFPA) and the Amazonia Foundation for Support to Studies and Research (FAPESPA).  
The authors declare no conflict of interest.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Parameter** & **[PERSON] et al. (2021) [PERSON] et al. (2020) [PERSON] et al. (2020)** & **Present Work** \\ \hline \multirow{2}{*}{Cd} & \(k_{L}\) & 0.0005778 L/mg & 0.5778 L/mg & 0.5489 L/mg \\  & \(D_{\text{ar}}\) & 0.834 cm\({}^{2}\)/min & 2.4 cm\({}^{2}\)/min & 24.8497 cm\({}^{2}\)/min \\ \hline \multirow{2}{*}{Cu} & \(k_{L}\) & 0.000529 L/mg & 0.529 L/mg & 0.5025 L/mg \\  & \(D_{\text{ar}}\) & 0.834 cm\({}^{2}\)/min & 2.4 cm\({}^{2}\)/min & 24.3152 cm\({}^{2}\)/min \\ \hline \multirow{2}{*}{Cr} & \(k_{L}\) & 0.0008 L/mg & 0.078 L/mg & 0.0743 L/mg \\  & \(D_{\text{ar}}\) & 4.182 cm\({}^{2}\)/min & 2.4 cm\({}^{2}\)/min & 24.6374 cm\({}^{2}\)/min \\ \hline \end{tabular}
\end{table}
Table 9: Comparison of correlation coefficients for each metal.