Mathematical Evaluation of Direct and Inverse Problem Applied in Breakthrough Models of Metal Adsorption

Abstract

The treatment of industrial effluents has great environmental and human health importance. The purification of water from polluting components, such as metals and organic compounds, can be considered one of the main applications in this field, with adsorption being one of the main treatment methods. Therefore, with the objective of describing the dynamics of the process in an adsorption column and estimating the parameters involved, in this work, an algorithm for the Method of Lines (MOL) was used in order to numerically solve the model formed by the mass balance in liquid phase, the linear driving force equation (LDF), and the Langmuir isotherm for equilibrium. In addition, a sensitivity analysis of the phenomenon was carried out in relation to the parameters and a subsequent estimation of these was made through the Monte Carlo technique via the Markov chain (MCMC). The validation algorithm was created using data from actual breakthrough curves found in the literature. The experimental data were obtained from the literature for the adsorption of Cadmium (Cd), Copper (Cu), Nickel (Ni), Zinc (Zn), and Chrome (Cr) ions. Among all the estimates, the one that had the lowest adjustment to the data was that related to zinc metal, which had an R² equal to 0.8984. For the other metals, the correlation coefficient had a value closer to unity. This demonstrates that, in general, the estimates were good enough to represent the dynamics of adsorption.

1. Introduction

Industrial growth and the increase in the scale of production in societies have had a major impact on the quality of water available to the population. Several contaminating sources can be cited as causing this threat to the drinkability of water, including contamination by heavy metals through waste effluents from industries such as battery manufacturing, foundry, mining, and electroplating, among others [1,2,3].

Such effluents may contain a mixture of heavy metals, such as Pb, Cd, Ni, Cu, etc., which, if disposed of improperly, present a threat to human health [4]. These metal ions, aside from being toxic to humans, are non-biodegradable, meaning they can accumulate in the human body and throughout the entire food chain, potentially causing various health issues such as headaches, stomach pain, and cancer [2,5,6,7,8].

Various treatment methods have been proposed in the literature to address this issue. Among them are chemical precipitation [9,10], ion exchange [11,12,13], adsorption [14,15], membrane filtration [16,17,18], reverse osmosis [19,20,21], and electrochemical treatment [22]. Many of these methods have high capital and operating costs. In this context, adsorption stands out as one of the most widely used methods [23] due to its high efficiency, low cost, and ease of operation.

Adsorption is an important tool for treating contaminated effluents, which, as observed by Patel et al. [24], has excellent performance if used in a continuous process in fixed bed columns, where the adsorbent works better. During the bed adsorption process, each adsorbent particle in the column will adsorb the solute until it reaches the saturation state. This process proceeds successively, layer by layer, from the input to the output of the column [25]. Thus, describing the rupture curve of the adsorption processes is of great importance for evaluating the dynamics of the phenomenon, since it contains a lot of information, such as the rupture point and saturation point.

Furthermore, mathematical models are used to predict the behavior of the system and relate these to the data and process parameters. The dynamics of adsorption in the column are based on phenomena such as axial dispersion, resistance to diffusion in the outer film, and resistance to intra-particle diffusion [26]. Therefore, it is important that the model manages to account for all the mechanisms that occur within the fixed bed [27]. There are two ways to develop mathematical models: through differential equations obtained through mass balances or through empirical approximations [28]. Several mathematical models used to evaluate adsorbent–adsorbate interactions can be found in the literature, such as the model by Thomas, Adams and Bohart, Yoon–Nelson, Clark, and Wolbourska [13].

However, the models that are developed from mass balance are less limited than the empirical ones, since they are based on the fundamental principle of the conservation of mass and are not limited by the specific experimental conditions of the process. Therefore, once the model is shown to be reliable, the challenge is to estimate the parameters that are part of it. When we deal with inverse problems and we have some knowledge of the structure of the model, solving these problems involves estimating the parameters that constitute them. Thus, using information from the experimental data and the mathematical structure of the model, it is possible to estimate the values of the parameters that best link the model to the data. There are several approaches to this estimation, including statistics using prior knowledge (Bayesian approach), that is, using the prior knowledge we have about a certain parameter, which proves to provide a great advantage over other estimation techniques [29,30].

Thus, in this work, a Bayesian statistical approach is used to estimate the parameters involved in the model, obtained through the mass balance in the column, Langmuir isotherm, and LDF hypothesis, for the experimental data of Ryu et al. (2021) [14] and Renu et al. (2020) [15], who describe the monocomponent adsorption of heavy metals. The estimation is made using the Metropolis–Hastings algorithm in the Monte Carlo method via the Markov Chain (MH-MCMC) and the direct problem is solved through the Line Method (MOL).

2. Direct Problem

The direct model adopted in this work, which represents the dynamics of the ion concentration at the outlet of the fixed bed, has three important components: the partial differential equation representing mass transfer in the column, the model used to describe the equilibrium assumed at the interface between the bulk phase and the adsorbent particle (isothermal model), and the hypothesis used to describe mass transport within the solid particle.

2.1. Isotherm Model

The isotherm proposed by Langmuir is presented by Equation (1):

$$q_s=\frac{q_{\mathit{max}}{k}_{L}C}{1+k_{L}C}$$

(1)

where $k_L$ (mg⁻¹) is the Langmuir constant that represents the ratio between the adsorption and desorption rate and $q_{max}$(mgg⁻¹) is the maximum adsorption capacity of the adsorbate by the adsorbent. To calculate $q_{max}$ from the failure curve data, use Equation (2).

$$q_{\mathit{max}}=\frac{F}{1000m_a}\left(C_e{t}_s-{\displaystyle \underset{0}{\overset{t_s}{\int }}Cdt}\right)$$

(2)

where (mL/min) is the volumetric flow of the effluent to be treated, $C_e$(mg/L) is the adsorbate concentration in the inlet stream, $m_a$(g) is the adsorbent mass, and $t_S$ (min) is the saturation time.

2.2. Kinetics

The transfer of metal ions from the solution to the adsorption sites within the adsorbent particles is restricted by mass transfer resistance, which determines the time required to reach the equilibrium state. In the temporal progress of this process in the Linear Driving Force (LDF) model approach, it is formally assumed that the transportation of the adsorbate at the adsorption sites inside the particle occurs in a fictitious film, in a comparable process to what happens externally in the diffusion film. Consequently, the solid phase concentration gradient is replaced by a linear difference between the equilibrium amount on the outer surface of the particle and the amount adsorbed by the particle [25,31]. Thus, the equation used to describe this mechanism is approximated by Equation (3):

$$\frac{\partial q}{\partial t}=-k_s\left(q-q_s\right)$$

(3)

where $q$ (mg/g) is the amount adsorbed by the particle, $q_S$ (mg/g) is the equilibrium amount on the outer surface of the particle, and $k_S$ (min⁻¹) is the intra-particle mass transfer coefficient. Glueckauf [31] found the following relationship (Equation (4)) between the mass coefficient, $k_S$ and the surface diffusion coefficient, $D_S$ (cm²/min):

$$k_s=\frac{15D_s}{r_p^2}$$

(4)

2.3. Mass Balance

In this present work, the adsorption process in a fixed bed column is modeled considering the following hypotheses [32,33,34]: this is an isothermal system; no chemical reaction occurs; there is negligible radial dispersion; the adsorbent particles are uniform; the flow rate is constant; there is a constant axial dispersion coefficient; there is constant porosity.

Thus, the mass balance that describes the breakthrough in the bed can be organized as shown by Equation (5) [25]:

$$\stackrel{\mathrm{Accumulation}}{\overbrace{\frac{\partial C(z,t)}{\partial t}}}+\stackrel{\mathrm{Adsorption}}{\overbrace{\frac{{\rho }_B}{{\epsilon }_B}\frac{\partial q}{\partial t}}}=\stackrel{\mathrm{Axial} \mathrm{dispersion}}{\overbrace{D_{ax}\frac{{\partial }^{2}C(z,t)}{\partial z^2}}}-\stackrel{Advection}{\overbrace{u\frac{\partial C(z,t)}{\partial z}}}$$

(5)

where $C$ is the solute concentration (mg/L), $t$ is time (min), ${\rho }_B$ is bed density (g/L), ${\epsilon }_B$ is porosity, $D_{ax}$ is the dispersion coefficient (cm²/min), $z$ is the spatial coordinate (cm), and $u$ is the interstitial fluid velocity (cm/min).

Equation (5) is subject to the following initial conditions:

$$C(z,t=0)=0$$

(6)

$$q(z,t=0)=0$$

(7)

These initial conditions express the fact that there is no solute inside the adsorption column at time zero. The Dankwertz boundary conditions for Equation (5) are as follows:

$$D_{ax}\frac{\partial C(z=0,t)}{\partial z}=-u\left[C_e-C(z=0,t)\right]$$

(8)

$$\frac{\partial C(z=L,t)}{\partial z}=0$$

(9)

2.4. Dimensioneless Model

The mathematical model represented by Equations (1)–(9) was dimensionless according to the dimensionless groups shown in

Table 1. Dimensionless parameters.

$\theta =\frac{C}{C_e}$	$\eta =\frac{z}{L}$	$\tau =\frac{t}{t_{ref}}$	$Pe=\frac{uL}{D_{ax}}$	$Q=\frac{q}{q_r}$
$K_s=k_s{t}_{ref}$	$K_L=k_L{C}_e$	$q_r=\frac{C_e}{{\rho }_B}$	$Q^{*}=\frac{q_s}{q_r}$	$t_{ref}=\frac{L}{u}$

.

Based on the dimensionless groups presented in Table 1, it is possible to obtain Equations (10)–(12) for mass transfer, adsorption kinetics, and equilibrium relation, respectively:

$$\frac{\partial \theta }{\partial \tau }+\frac{\partial \theta }{\partial \eta }=\frac{1}{Pe}\frac{{\partial }^2\theta }{\partial {\eta }^2}-\frac{1}{{\epsilon }_B}\frac{\partial Q}{\partial \tau }$$

(10)

$$\frac{\partial Q}{\partial \tau }=K_s(Q^{*}-Q)$$

(11)

$$Q^{*}=\frac{Q_{\max }{K}_L\theta }{1+K_L\theta }$$

(12)

$$\theta (\eta ,0)=0$$

(13)

$$Q(\eta ,0)=0$$

(14)

$$-\frac{\partial \theta (\eta =0,\tau )}{\partial \eta }=Pe(1-\theta (\eta =0,\tau ))$$

(15)

$$\frac{\partial \theta (\eta =1,\tau )}{\partial \eta }=0$$

(16)

3. Inverse Problem

The Bayesian approach in statistics aims to decrease uncertainty in inference and decision-making problems by utilizing all available information. As new data emerge, they are integrated with prior data to support statistical processes. In Bayesian theory, this information is expressed in terms of probability, frequently depicted by a probability density function. The Bayes theorem establishes a formal method for combining new information with prior information [35,36,37,38,39].

$${\pi }_{Posterior}\left(P\right)=\pi \left(P|Y\right)=\frac{{\pi }_{Prior}\left(P\right)\pi \left(Y|P\right)}{\pi \left(Y\right)}$$

(17)

where ${\pi }_{Posterior}\left(P\right)$ is the posterior probability density; ${\pi }_{Prior}\left(P\right)$ is the a priori density of the parameters, that is, the encoded information for the parameters available before the measurements; $\pi \left(Y\right|P)$ is the likelihood function, which expresses the probability density of the $Y$ measurements given the $P$ parameters; $\pi \left(Y\right)$ is the marginal probability density of the measurements, which plays the role of a normalization constant.

In several applications, one is more interested in the form presented in Equation (18) of the Bayes equation, where the marginal probability distribution of the measures is considered irrelevant, since what is expected is the optimization of the posterior distribution, which basically depends on the numerator of Formula (17) [30,40,41]:

$${\pi }_{\mathit{Posterior}}\left(P|Y\right)\propto {\pi }_{\mathit{Prior}}\left(P\right)\pi \left(Y|P\right)$$

(18)

One of the ways to determine the posterior density, ${\pi }_{Posterior}\left(P\right)$, is through sampling techniques, such as the Monte Carlo methods via Markov Chain. One of the most used algorithms for the implementation of the Monte Carlo method with Markov chains is the Metropolis–Hastings algorithm. In this, from the initial value for the parameter, through the transition kernel, a new candidate parameter value is generated, which is evaluated before either being accepted or not using the ratio between the posterior probability densities of the current and candidate states. Thus, the Metropolis–Hastings algorithm describes a category of Monte Carlo methods that construct a Markov Chain in stages through the random sampling of the posterior distribution described by the Bayes relation [42].

3.1. Sensitivity Analysis

The sensitivity coefficient $J_{ij}$ describes the sensitivity of the state variable in relation to a disturbance in parameter $P_j$ and has an important function in parameter estimation. Small magnitude values of this coefficient indicate that large variations in $P_j$ cause small changes in the dependent variable. The sensitivity coefficient formula is shown in Equation (19) [30].

$$J_{ij}=\frac{\partial {\theta }_i}{\partial P_j}$$

(19)

For situations where the analytical solution of the problem is not known, it is possible to determine the sensitivity coefficient by approximating the first-order derivative using finite differences. The forward difference approximation given by Equation (20) was used.

$$J_{ij}\cong \frac{{\theta }_i(P_1,P_2,\dots .,P_j+\epsilon P_j,\dots ,P_{Npar})-{\theta }_i(P_1,P_2,\dots .,P_j,\dots ,P_{Npar})}{\epsilon P_j}$$

(20)

However, to compare the values of the sensitivity coefficients for parameters with different units and orders of magnitude, the reduced-sensitivity coefficient is presented in Equation (21).

$$J_{Pj}=P_j\frac{\partial {\theta }_i}{\partial P_j}$$

(21)

3.2. Markov Chain Monte Carlo

In many cases it is not possible to obtain an analytical solution for the posterior probability distribution within the Bayesian structure. In such cases, simulation methods, which use a sampling process to obtain information about the probability density posteriori, can be used, as is the case with the Markov chain Monte Carlo (MCMC) method [43].

The MCMC combines the properties of Monte Carlo and the Markov chain. The first is estimates the properties of a distribution by examining random samples from the distribution. The second is based on the idea that a special sequential process generates random samples, where each random sample is used as a step to generate the next one. A special property of the chain is that, although each new sample depends on the previous one, new samples do not depend on any sample before the previous one [44,45,46,47,48,49,50].

One of the most widely used algorithms for implementing the Markov chain Monte Carlo is the Metropolis–Hastings algorithm. This algorithm uses a proposal distribution, $\overline{q}\left({\mathit{P}}^{*}|{\mathit{P}}^{(\mathrm{i})}\right)$, to generate a vector of candidate parameters, ${\mathit{P}}^{*}$, based on the current state of the chain, ${\mathit{P}}^{(\mathrm{i})}$. The Markov chain then moves to ${\mathit{P}}^{*}$ with the probability of acceptance given by Equation (22).

$$\alpha \left({\mathit{P}}^{(\mathrm{i})},{\mathit{P}}^{*}\right)=\min \left[1,\frac{\pi \left({\mathit{P}}^{*}|Y\right)\overline{q}\left({\mathit{P}}^{(\mathrm{i})}|{\mathit{P}}^{*}\right)}{\pi \left({\mathit{P}}^{(\mathrm{i})}|Y\right)\overline{q}\left({\mathit{P}}^{*}|{\mathit{P}}^{(\mathrm{i})}\right)}\right]$$

(22)

The Metropolis–Hastings algorithm can be specified using the following steps [44,51]:

• Start the iteration counter, i = 1, and specify an initial value, ${\mathit{P}}^1$;
• Generate a vector of candidate parameters from the proposal distribution, ${\mathit{P}}^{*}$.
• Calculate the probability of acceptance, $\alpha \left({\mathit{P}}^{(\mathrm{i})},{\mathit{P}}^{*}\right)$, given by Equation (22);
• Generate an auxiliary random sample of a uniform distribution, $\overline{u}~U(0,1)$;
• If $\overline{u}\le \alpha$, then accept the new value and update the parameter vector, ${\mathit{P}}^{(\mathrm{i}+1)}={\mathit{P}}^{*}$. Otherwise, ${\mathit{P}}^{(\mathrm{i}+1)}={\mathit{P}}^{(\mathrm{i})}$;
• Increment the counter from i to i + 1 and go back to step 2.

4. Methodology

4.1. Direct Problem Solution Method

The dimensionless model, consisting of Equations (10)–(16), was addressed using two distinct methods. Firstly, the Method of Lines (MOL) was employed, which entails converting the partial differential equation (PDE) into a set of ordinary differential equations (ODEs) (refer to Supplementary Materials). These ODEs were then numerically solved using the ode15s subroutine within a computational code developed in Matlab (R2022a). Secondly, the pdepe subroutine of the Matlab program was utilized, enabling the PDE to be directly solved without the requirement for prior reduction to a system of ODEs.

4.2. Sensitivity Analysis

A sensitivity analysis was conducted of the parameters to be estimated, following the conditions specified by Ryu et al. (2021) [14] and Renu et al. (2020) [15]. Renu et al. (2020) [15] require that each of the three metals assessed by the authors (copper, chromium, and cadmium) are subjected to nine experimental conditions, involving variations in bed length, volumetric flow rate, and initial metal concentration. Consequently, by analyzing the sensitivity plots, the condition exhibiting the highest sensitivity to the parameters was identified, and the optimal scenario among the conditions was selected for estimation.

It was decided to employ the Method of Lines (MOL) for both the sensitivity analysis and parameter estimation, as using pdepe incurred higher computational costs.

4.3. Parameter Estimation

The parameter estimation was conducted for the datasets provided by Ryu et al. (2021) [14] and Renu et al. (2020) [15], encompassing various experimental conditions and metals. The parameters slated for estimation are outlined in Equation (23). They were selected due to their direct association with several key assumptions, including axial dispersion, Langmuir-prescribed conditions, and the LDF approach. Furthermore, no equation exists to compute their values that is not empirical.

$$P=\left[k_L,k_s,D_{ax}\right]$$

(23)

For the estimation, it is assumed that the measurement noise follows a Gaussian distribution with a zero mean and is both additive and independent of the parameter vector P. Given that W is the covariance matrix, which becomes a diagonal matrix when considering a constant standard deviation in the measurements, the likelihood function can then be expressed by Equation (24). For all estimations, the standard deviation taken for measurement errors was considered constant at 5% relative to the largest experimental measurements.

$$\pi \left(Y|P\right)={\left(2\pi \right)}^{-\frac{I}{2}}|W{|}^{-\frac{1}{2}}\exp \left\{-\frac{1}{2}{\left[Y-Y\left(P\right)\right]}^T{W}^{-1}\left[Y-Y\left(P\right)\right]\right\}$$

(24)

It was also proposed that the prior probability distribution of the parameters, as shown in Equation (25), follows a Gaussian distribution, with a mean μ equal to 95% of the parameters’ reference value (that is, the initial value) and a covariance matrix V. For all estimations, the standard deviation associated with the parameters is held constant, representing 5% of the reference value.

$$\pi \left(P\right)={\left(2\pi \right)}^{-\frac{N}{2}}|V{|}^{-\frac{1}{2}}\exp \left\{-\frac{1}{2}{\left[P-\mu \right]}^T{V}^{-1}\left[P-\mu \right]\right\}$$

(25)

To generate a candidate P* from the current state Pⁱ⁻¹, the transition kernel used in the present work has the form shown in Equation (26):

$${\mathit{P}}^{*}={\mathit{P}}^{(\mathrm{i})}\left(1+\mathbf{w}\xi \right)$$

(26)

where $\xi$ is a variable N(0,1) and w is the random walk.

5. Results

5.1. Verification of the Direct Model Solution

To assess the spatial mesh discretization and compare the results for the pdepe function and the MOL, we utilized the experimental condition and parameters related to manganese adsorption provided by Ryu et al. (

Table 2. Experimental condition and parameter values estimated by Ryu et al. (2021) [14].

${\mathit{C}}_{\mathit{e}}$	0.5 mmol/L	${\mathit{\rho }}_{\mathit{B}}$	248 g/L
Diameter of collum	1 cm	$D_{ax}$	24 cm2/min
$L$	10 cm	$k_s$	0.02736 min−1(Cu) 0.01512 min−1 (Mn) 0.02556 min−1 (Ni) 0.0225 min−1 (Zn)
F	1 cm3/min	$k_L$	0.01461 L/mg (Cu) 0.0191 L/mg (Mn) 0.01339 L/mg (Ni) 0.005169 L/mg (Zn)
Ɛ	0.84	qmax	97.4795 mg/g (Cu) 39.2806 mg/g(Mn) 66.5583 mg/g (Ni) 77.4753 mg/g (Zn)

). The conditions and parameter values for each metal in a fixed bed were taken from the experiment conducted by the same authors. The intra-particular mass transfer coefficients were calculated based on the surface diffusivity values also estimated by these authors.

Figure 1 shows the breakthrough generated by the solutions of the model obtained using MOL and pdepe for discretizations that varied from 20 to 100 nodes. Thus, by comparing the results generated by the two methodologies, it is possible to confirm that there is an agreement between them.

A convergence analysis of the solutions was also carried out. In

Table 3. Convergence analysis.

	$\mathit{\theta }$ (MOL)					$\mathit{\theta }$ (PDEPE)
	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

, the dimensionless concentration values are represented, setting the spatial variable at η = 1, in two spatial times for different numbers of points (n). When analyzing Table 3, it is noted that there are small variations in some values with the change in the discretization; however, these variations showed the same behavior for larger values regarding the number of points. Therefore, we opted for a mesh with n = 40 to generate the results in this work, since there was an increase in computational time with the increase in the number of points in the discretization.

5.2. Sensitivity Analysis

In the sensitivity analysis using data from Ryu et al. (2021) [14], a disturbance value ε 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 θ_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 θ_i [42,52].

For the data from Renu 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. Operational conditions used by Renu et al. (2020) [15].

${\mathit{C}}_{\mathit{e}}$	100, 300 and 500 mol/L	${\mathit{\rho }}_{\mathit{B}}$	651.52 mg/cm3
Diameter of collum	2.5 cm	$D_{ax}$	23.1663 cm2/min
$L$	15, 30 and 45 cm	$k_s$	0.030536 min−1
F	5, 10 and 15 cm3/min	$k_L$	0.529 L/mg (Cu) 0.078 L/mg (Cr) 0.5778 L/mg (Cd)
ƐB	0.59	Diameter of particle	0.2 cm

. 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 Ryu et al. (2021) [14]. Regarding the Langmuir constants, the values found by Renu 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.

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 Ryu et al. (2021) [14], 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 $F=5 \mathrm{m}\mathrm{L}/\mathrm{m}\mathrm{i}\mathrm{n}$, $C_e=100 \mathrm{mg}/\mathrm{L}$, and $L=15 \mathrm{cm}$ was chosen, as this proved to be the best scenario in terms of the sensitivity of the parameters.

5.3. Estimated Parameters Regarding Data from Ryu et al. (2021) [14]

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.

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 Ryu 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.

Table 5. Parameters estimated from each metal using data from Ryu et al. (2021) [14].

	Parameter	w (Search-Step)	Initial Value	Estimated
Cu	$k_L$ (L/mg)	0.005	0.01461	0.009163 [0.008525; 0.009782]
	$ks$ (min−1)	0.008	0.02736	0.031971 [0.021550; 0.042861]
	$D_{ax}$ (cm2/min)	0.1	24	22.65535 [20.06766; 25.85996]
Mn	$k_L$ (L/mg)	0.005	0.0191	0.010707 [0.009727; 0.011828]
	$ks$ (min−1)	0.008	0.01512	0.025464 [0.016410; 0.037095]
	$D_{ax}$ (cm2/min)	0.1	24	24.353081 [21.220490; 28.850910]
Ni	$k_L$ (L/mg)	0.005	0.01339	0.009391 [0.008674; 0.010199]
	$ks$ (min−1)	0.008	0.02556	0.033893 [0.022761; 0.045974]
	$D_{ax}$ (cm2/min)	0.1	24	23.165294 [20.017246; 27.514774]
Zn	$k_L$ (L/mg)	0.005	0.005169	0.009217 [0.008538; 0.009946]
	$ks$ (min−1)	0.008	0.0225	0.030815 [0.020114; 0.0433]
	$D_{ax}$ (cm2/min)	0.1	24	22.49183 [19.45121; 26.43147]

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. Metric statistics for data from Ryu et al. (2021) [14].

Metal	R2
Cu	0.9168
Mn	0.9277
Ni	0.9076
Zn	0.8984

, 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.

The estimated values in this study, when compared to the initial values estimated by Ryu 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 Ryu 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].

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, R² 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 R² measures the proportion of data variability that is explained by the model. In the context of this study, R² was calculated based on the mean values of the parameters obtained through MCMC sampling. Therefore, although a high R² may indicate a good ability of the model to explain the variation in the experimental data, it does not guarantee that the experimental data will fall within the 95% confidence interval.

5.4. Estimated Parameters Regarding Data from Renu 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 Renu et al. (2020) [15], the following condition was used:$F=5 \mathrm{mL}/\min , { C}_e=100 \mathrm{mg}/\mathrm{L}, \mathrm{and} L=15 \mathrm{cm}$. 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. Parameters estimated from each metal regarding data from Renu et al. (2020) [15].

	Parameter	w (Search-Step)	Initial Value	Estimated
Cd	$k_L$ (L/mg)	0.01	0.5778	0.5489 [0.5478; 0.5500]
	$ks$ (min−1)	0.01	0.03053	0.0342 [0.0299; 0.0396]
	$D_{ax}$ (cm2/min)	0.005	23.1663	24.8497 [22.9914; 27.2044]
Cu	$k_L$ (L/mg)	0.01	0.529	0.5025 [0.5015; 0.5036]
	$ks$ (min−1)	0.01	0.03053	0.0256 [0.0225; 0.0293]
	$D_{ax}$ (cm2/min)	0.005	23.1663	24.3152 [22.3117; 27.2322]
Cr	$k_L$ (L/mg)	0.01	0.078	0.0743 [0.0734; 0.0753]
	$ks$ (min−1)	0.01	0.03053	0.0443 [0.0369; 0.0532]
	$D_{ax}$ (cm2/min)	0.005	23.1663	24.6374 [22.5018; 27.4753]

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 Renu et al. (2020) [15]. It was observed that, as well as estimating the parameters using the experimental data from Ryu et al. (2021) [14], the chains converged around an average value.

Table 8. Metric statistics for data from Renu et al. (2020) [15].

Metal	R2 (Present Work)	R2 Danish et al. (2021) [56]
Cd	0.9950	0.8160
Cu	0.9972	0.89930
Cr	0.9163	0.9130

shows the model metrics of this work in relation to the experimental data and the metrics for the model by Danish et al. (2021) [56] for the same data. Analyzing the metrics of this work, the highest values of R² stand out. The R² values for the three metals presented by the present work were better than those obtained by Danish 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.

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 Renu et al. (2020) [15] are better described by the model evaluated in this study. This is because the flow rates used in their experimental conditions were higher than those employed by Ryu et al. (2021) [14], resulting in a thinner film thickness around the adsorbent particle allowing for higher flows. Thus, these conditions are closer to the assumptions adopted in the present study, at least compared to Ryu et al.’s (2021) conditions.

In the literature, there are some simplifying assumptions that can be adopted to facilitate modeling, such as in the case of adsorption processes at high flow rates, where the dispersion term can be ignored. This is because its impact on the breakthrough curve dispersion is insignificant compared to the influence of slower mass transfer processes [59,60,61]. Additionally, in some situations, the accumulation term dc/dt is also disregarded, assuming that the accumulation in the liquid phase is small compared to the adsorption process [25]. With these simplifications, the phenomenon can be described by a first-order partial differential equation and no longer requires the estimation of a parameter ($D_{ax}$).

Regarding the estimated values, these remained relatively close to the values used as the initial value. In order to compare the estimated values for the parameters,

Table 9. Comparison of correlation coefficients for each metal.

	Parameter	Danish et al. (2021) [56]	Renu et al. (2020) [53]	Present Work
Cd	$k_L$	0.0005778 L/mg	0.5778 L/mg	0.5489 L/mg
	$D_{ax}$	0.834 cm2/min	2.4 cm2/min	24.8497 cm2/min
Cu	$k_L$	0.000529 L/mg	0.529 L/mg	0.5025 L/mg
	$D_{ax}$	0.834 cm2/min	2.4 cm2/min	24.3152 cm2/min
Cr	$k_L$	0.0008 L/mg	0.078 L/mg	0.0743 L/mg
	$D_{ax}$	4.182 cm2/min	2.4 cm2/min	24.6374 cm2/min

shows some parameter values obtained by Danish et al. (2021) and Renu et al. (2020) [15] for the same experimental data. As in the studies by Danish et al. (2021) and Renu et al. (2020) [15], the film diffusion approach was used as a mass transfer model, with a similar linear difference to LDF, and there are no $k_S$ values to compare. The differences in values may derive from differences in the model hypotheses used by each author.

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 R² 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 Renu et al. (2020) [15] were well described by the estimates, while the estimation did not have as good an agreement with the data used in the work by Ryu et al. (2021) [14]; 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.