## 5 Results
### Estimated Parameters Regarding Data from [PERSON] et al. (2020) [15]  
The conditions used to solve the problem and estimate the parameters are shown in Table 4.  
For the estimation of parameters using experimental data from [PERSON] et al. (2020) [15], the following condition was used: \(F=5\) mL/min, \(C_{e}=100\) mg/L, and \(L=15\) cm. A chain heating period of 3000 interactions and a number of states equal to 20,000 were used for all estimations. The estimation was also made using different step sizes for each parameter, where the values presented in Table 7 were reached. The choice was based on whether the chain of parameters was stabilized. Figure 5 shows the chains generated in the sampling process for the dimensionless parameters of each metal evaluated by [PERSON] et al. (2020) [15]. It was observed that, as well as estimating the parameters using the experimental data from [PERSON] et al. (2021) [14], the chains converged around an average value. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Parameter** & **w (Search-Step)** & **Initial Value** & **Estimated** \\ \hline \multirow{4}{*}{Cd} & \(k_{L}\) (L/mg) & 0.01 & 0.5778 & 0.5489 \\  & \(k_{L}\) (m\({}^{-1}\)) & 0.01 & 0.03053 & 0.0342 \\  & \(D_{ax}\) (cm\({}^{2}\)/min) & 0.005 & 23.1663 & 24.8497 \\  & \(k_{L}\) (L/mg) & 0.01 & 0.529 & 0.5025 \\  & \(k_{L}\) (L/mg) & 0.01 & 0.529 & [0.5015; 0.5036] \\  & \(k_{S}\) (m\({}^{-1}\)) & 0.01 & 0.03053 & 0.0256 \\  & \(D_{ax}\) (cm\({}^{2}\)/min) & 0.005 & 23.1663 & 24.3152 \\  & & & & [22.3117; 27.2322] \\ \hline \multirow{4}{*}{Cr} & \(k_{L}\) (L/mg) & 0.01 & 0.078 & 0.0743 \\  & \(k_{S}\) (m\({}^{-1}\)) & 0.01 & 0.03053 & 0.0443 \\ \cline{1-1}  & \(D_{ax}\) (cm\({}^{2}\)/min) & 0.005 & 23.1663 & 24.6374 \\ \cline{1-1}  & & & & [22.5018; 27.4753] \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parameters estimated from each metal regarding data from [PERSON] et al. (2020) [15]. Table 8 shows the model metrics of this work in relation to the experimental data and the metrics for the model by [PERSON] et al. (2021) [56] for the same data. Analyzing the metrics of this work, the highest values of R\({}^{2}\) stand out. The R\({}^{2}\) values for the three metals presented by the present work were better than those obtained by [PERSON] et al. (2021) [56]. Along with that, when evaluating the graphs of the breakthrough curves present in Figure 6, the model shows good proximity with the experimental data. That is, with this, the estimation of the parameters proves to be reliable. Figure 5: Markov chain for parameters estimation regarding data from [PERSON] et al. (2020) [15] with number of states equal a 20000. Where (**a**–**c**) correspond to the parameter chains for Cu, (**d**–**f**) for Cr, (**g**–**i**) for Cd. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Metal**} & **R\({}^{2}\)** & **R\({}^{2}\)** \\  & **(Present Work)** & **[PERSON] et al. (2021) [56]** \\ \hline Cd & 0.9950 & 0.8160 \\ Cu & 0.9972 & 0.89930 \\ Cr & 0.9163 & 0.9130 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Metric statistics for data from [PERSON] et al. (2020) [15]. Regarding the limitations of the assumptions adopted in this study, the LDF model simplifies the transport in the adsorbent solid, while a more complex model would consider the superficial and liquid diffusion present in the particle pores [25]. Therefore, it is expected that the LDF model may not be suitable for conditions where the Biot number is high and the Stanton number is low, that is, when convective transport is more relevant than diffusive transport [57]. Additionally, the model used in this study did not account for the resistance to transport in the film around the adsorbent particle, resulting in inadequate representations under the conditions of a low agitator speed in batch reactors or low flow velocity in fixed bed adsorbers [58]. Possibly, this is one of the reasons why the experimental data from [PERSON] et al. (2020) [15] are better described by the model evaluated in this study.