Document Type : Full Lenght Research Article
Authors
^{1} Graduate Program in Process Engineering, Federal University of Pará, Pará, Brazil
^{2} Graduate Program in Natural Resource in the Amazon, Federal University of Pará, Pará, Brazil
^{3} Faculty of Chemical Engineering, Federal University of Pará, Pará, Brazil
^{4} Faculty of Biotechnology, Federal University of Pará, Pará, Brazil
Abstract
Keywords
Main Subjects
Semnan University 
Journal of Heat and Mass Transfer Research Journal homepage: https://jhmtr.semnan.ac.ir 

Research Article
Parameter Estimation in Mass Balance Model Applied in Fixed Bed Adsorption Using the Markov Chain
Monte Carlo Method
Rhaisa Sousa Tavares ^{a}, Camila Santana Dias ^{b}, Carlos Henrique Rodrigues Moura ^{b},
Emerson Cardoso Rodrigues ^{c}, Bruno Marques Viegas ^{d } ,
Emanuel Negrão Macêdo ^{c } , Diego Cardoso Estumano [*]^{,}^{d }
^{a} Graduate Program in Process Engineering, Federal University of Pará, Pará, Brazil.
^{b }Graduate Program in Natural Resource in the Amazon, Federal University of Pará, Pará, Brazil.
^{c }Faculty of Chemical Engineering, Federal University of Pará, Pará, Brazil.
^{d }Faculty of Biotechnology, Federal University of Pará, Pará, Brazil.
Paper INFO 

ABSTRACT 
Paper history: Received: 20220810 Revised: 20230215 Accepted: 20230221 
In this work, a mathematical model is adopted to predict the breakthrough curve in a fixed bed adsorption process, neglecting radial dispersion effects in the bed, with properties such as interstitial velocity and porosity being constant, linear adsorption kinetics and equilibrium relationship represented by the Langmuir isotherm. The resulting partial differential equation is numerically solved by the Method of Lines (MOL), while the Markov Chain Monte Carlo method is employed to estimate the model parameters, using simulated measures and a priori Gaussian probability distribution for the parameters, varying the mean and standard deviation. A convergence analysis was performed to look for numerical convergence between the number of nodes (N) used and the computational cost (CPU time) and it was observed that N = 100 obtained the lowest computational cost (less than 0.2 s). The estimated values of Peclet's number (Pe) and Langmuir's constant (KL) showed deviations of 7% and 0.01%, respectively, compared to their exact value which shows that the estimates were accurate, i.e., the parameters are close to the exact value. Also, the estimated values were within the credibility interval of 99 % established, which shows precise estimates. The information taken from these estimates has become of fundamental importance in predicting the behavior of the breakthrough curve at different points in the bed, showing that the MOL in combination with the MCMC are efficient tools in the direct and inverse analysis of models of breakthrough curves. 



Keywords: Adsorption; Breakthrough curve; MCMC; Parameter estimation; Convergence analysis. 


© 2022 Published by Semnan University Press. All rights reserved. 
Pollution of water resources by the recalcitrant presence of emerging contaminants stimulates environmental concern about this topic and makes efforts aimed at remedying its negative effects on the environment relevant. In this sense, studies to develop effective methodologies for the treatment of these contaminants have been carried out around the world [19]
Among the already known methods that are effective for environmental remediation purposes are photocatalysis [10], membrane ultrafiltration [11] ,ozonation [12], photoFenton reaction [13] and ion exchange [14]. In this scenario, adsorption stands out as an attractive alternative for presenting relative simplicity of execution, low implementation cost compared to other approaches, and considerable level of effectiveness [1517].
Adsorption can be performed in batch; however, this mode of operation is inappropriate when dealing with largescale wastewater treatment [1820]. Thus, for its industrial application, the use of fixed bed columns is suitable since they allow the continuous passage of effluents with a high load of pollutants through a column filled with adsorbent [18;2122].
Before the elaboration of a project of an industrial scale fixedbed column, a mathematical model capable of successfully representing the dynamics of experimentally obtained breakthrough curves is needed [2324]. Many analytical models have been widely used to describe the breakthrough curves in fixed bed column adsorption systems [2529].
Despite their importance, these models fail to identify mechanisms such as axial dispersion in the bed and mass transfer between phases. In addition, they need experimental curves for their parameters to be estimated, limiting the scope of the analysis to this specific curve [3034].
To overcome the limitations of analytical models in representing breakthrough curves, several works have used more complex approaches in relation to the topic of adsorption in a fixed bed, some of which are highlighted in Table 1. In order to contribute to previous studies a model obtained from a mass balance for the fluid phase was used in this work, which is based on the conservation of mass in the system, kinetics, and conditions adsorption equilibrium[23].
The parameter estimation was performed from Bayesian inference perspective's to cover the effect that uncertainties intrinsic to the experimental execution would exert on the value found for the parameters. Since such effects can cause incompatibilities in the modeling by allowing physically improbable parameters to be obtained and, thus, affecting the design and operation of the process[35].
The Bayesian method of Markov Chain Monte Carlo (MCMC) was used in the estimation process. Simulated measurements were used to verify the elaborated code, and different levels of uncertainty were assigned to evaluate the effect produced on the results. The prior probability distribution, the one that contains previously available information about the analyzed system, was evaluated here by changes made to its characteristic metrics such as mean and standard deviation. The measurements that are usually obtained only at the exit of the bed were used to estimate information on the adsorptive process at other points in the column.
Table 1. Different approaches applied to the study of adsorption in a fixed bed column.
Breakthrough curve models 
Equilibrium Isotherms 
Kinetic 
Solution Method 
Parameter Estimation 
Reference 
Thomas BohartAdams Yan 
 
 
Nonlinear adjustment 
Origin Pro 8 
[36] 
Computational fluid dynamics (CFD) 
Nonlinear Langmuir 
Linear Drive Force (LDF) 
COMSOL Multiphysics 
Empirical correlations 
[24] 
Thomas, YoonNelson, AdamsBohart and Wolbourska 
 
 
Microsoft Excel's Solver Extension 
Linear and Nonlinear Regression 
[37] 
Mass balance in fluid phase 
Langmuir and BET 
Linear Drive Force (LDF) 
RungeKuttaFehlberg Method 
Minimization of an objective function using the downhill simplex optimization method 
[38] 
Logistic Model (BohartAdams, YoonNelson and Thomas), Wolborska, Modified Doseresponse, Clark, Gompertz and LogGompertz. Mass Transfer Model 
Langmuir 
Linear Drive Force (LDF) 
Finite Elements 
Comsol Multiphysics V5.4. 
[34] 
Hydrus1D, Thomas, YoonNelson and BohartAdams 
General sorption model that, depending on the value of the parameters, may fall into the Langmuir, Freundlich or linear isotherm 
 
Hydrus1D 
LevenbergMarquardt and operational parameters by experiments 
[39] 
The representation of physical model is presented in Figure 1 and demonstrates in a simplified way an adsorption column with ascending feed of initial concentration C_{0} and output current C.
The mass balance in a differential element of the bed was carried out assuming the following hypotheses: negligible radial dispersion, the significant variation was considered only in the axial direction, the solid/fluid interfaces establish a thermodynamic equilibrium state, porosity and interstitial velocity are constant [38].

(1.a) 

(1.b) 
Langmuir Isotherm:

(1.c) 
Initial conditions:

(1.d) 

(1.e) 
Boundary conditions:

(1.f) 

(1.g) 
where is the fluid velocity, is the dispersion coefficient, is the bed void fraction, is the kinetic constant, is the maximum capacity of adsorption and is the Langmuir parameter.
The dimensionless groups presented in Equation 2 were adopted; thus, the mass balance model in dimensionless form is shown in Equation 4.

(2.af) 

(2.gk) 
where q_{max}, the maximum adsorption capacity in the column, is given by Equation (3):

(3) 
where Q is the volumetric flow rate (cm³/min) and W is the adsorbent mass (g).

(4.a) 

(4.b) 

(4.c) 
Initial conditions:

(4.d) 

(4.e) 
Boundary conditions:

(4.f) 

(4.g) 
The numerical procedure used to solve the nonlinear partial differential equation (PDE) was the method of lines. This method is used to solve the mass balance model carried out in an adsorption column, to discretize the domain of the dependent variable in space, transforming the obtained PDE into a system of timecontinuous ordinary differential equations (ODEs) [40].
The schematic representation shown in Figure 1 summarizes the discretization of the dimensionless domain in the range of . In the method of lines, the domain is discretized into equal lengths, where and N is the number of nodes in the spatial domain. Therefore, N ODEs are developed by discretizing the governing PDE and boundary conditions [40].
Equations 4.ac describe the dynamics that occur in the bed and correspond to the internal points of mesh in the domain interval. Equations 5.ac show the discretized PDE.

(5.a) 

(5.b) 

(5.c) 
At and Equation 4.a takes the form described in Equation 5.d:

(5.d) 
The boundary condition at the bed entrance described by Equation 4.f was used to determine , using the central difference on , the arrangement as shown by Equation 5.

(5.e) 
To determine at PDE in (Equation 5.f), the backward finite difference was applied to the boundary condition of Equation 4.g, which assumed the form shown in Equation 5.g.

(5.f) 

(5.g) 
It is important to emphasize that the Method Of Lines is a methodology to obtain an approximate solution of the PDE given by the Equations (4.ag). Therefore, the limitation of the method tends to be in the amount of precision needed to approximate the exact solution.
This accuracy is related to the number of grid points and available computing power. In this way, a convergence analysis becomes necessary to provide the computational power necessary to reach adequate precision to approximate the solution.
Figure 1. Model of an adsorption column and schematic representation
of the domain discretized in the interval
In this work, the algorithm used for convergence analysis is shown below:
where is the value of the dependent variable obtained for N nodes, θ_{2} the value of the dependent variable obtained for N + ΔN nodes, N is the number of nodes in the mesh, tol is the tolerance and tol_{SC }is the initial defined tolerance.
In many cases, different prior probability densities can be assumed for the parameters and thus, it is impossible to obtain an analytical treatment for a posterior probability distribution. In this scenario, the Markov Chain Monte Carlo method, an iterative version of traditional Monte Carlo methods, is used to extract samples of all possible parameters so that posterior probability inference turns into sample inference [4145].
The MCMC combines the properties of Monte Carlo and the Markov chain. The first is estimating the properties of distribution by examining random samples from the distribution. On the other hand, the second aims at the idea that a given sequential process generates random samples, where each random selection is used as a step to develop the next one. A particular property is that, although each new choice depends on the previous one, new samples do not rely on any instance before the last one.
In the present work, the MetropolisHastings algorithm is used to estimate the parameters of the mathematical model of the breakthrough curve[41;4650], which following the following steps:

(6) 
where is a random variable N(0,1) and w is the search step.

(7) 
A flowchart of the MetropolisHastings algorithm is illustrated in Figure 2 below.
Figure 2. Sequential flowchart of the MCMC method using MetroplisHastings
In this work, simulated measures are used to carry out parameter estimates. Such measures are generated by adding noise, υ, so the measures by Equation (8):

(8) 
where is the solution of the direct problems with knowing reference parameters and .
The numerical tests carried out concerned the direct model's evaluation (presented in section 2) using the technique shown in section 3 to solve the inverse problem of parameter estimation involved in the adsorption phenomenon in a fixed bed column
The system of ordinary differential equations originated from the discretization of the partial differential equation and was solved by the ode15s function of the Matlab R2021a software. The same software programmed code for the MetropolisHastings algorithm.
The mesh convergence analysis is performed to verify the number of nodes (N_{i}) sufficient for model discretization when applying the method of lines and reaching a satisfactory convergence.
The number of nodes in the discretized domain varied, and how this variation influenced the computational time was observed. The relevance of this analysis for this work aim at the need that the MCMC method must solve the direct model several times (10.000 states of Markov Chain), therefore defining the number of nodes and the desirable computational cost to obtain precision in the solution becomes desirable.
Simulated measurements were used to apply the estimates of the parameters of the analyzed model. The prior probability distribution of the parameters was adopted as Gaussian, and its influence and the influence of the acquisition frequency of measurements are evaluated. The simulations were performed using as a reference for the parameters the following values: Pe = 10.00, Ks = 1.00, Q_{max} = 7.00,
K_{L} = 1.00 and ε = 0.40. The choice of these parameters was to simulate a breakthrough without numerical instabilities, since the Peclet number (Pe) can be important to characterize the transport of solutes by advective or diffusive means, the numerical stability depends on the number of Pe. If they reach some critical limits the numerical solution begins oscillating in space and time.
The parameters Q_{max}, K_{s} can be calculated, and the porosity ε can be obtained experimentally. In this sense, the parameters estimated here were Pe because they included operational and diffusivity information, and K_{L} referring to the Langmuir isotherm.
Figure 3 presents the results of the convergence analysis performed. It has been found that increasing the number of nodes increases the computational cost
(a) 
(b) 
(c) 
(d) 
Figure 3. Convergence analysis regarding Q_{max }and ε were kept constant in 7.00 and 0.40, respectively.
(a) Pe = 10.00 ; K_{L} = 1.00, (b) Pe = 10.00 ; K_{L} = 3.00, (c) Pe = 2.00 ; K_{L} = 1.00 e (d) Pe = 20.00 ; K_{L} = 1.00.
The prior probability distribution of the parameters contains the information previously known. In this sense, variations in the mean of the prior probability distribution of the parameters, shown in Table 1, were performed to evaluate whether the estimated values would approach the exact value when the mean value of this distribution is changed.
The graph of the prior probability distribution function for these case studies is shown in Figure 4. It is possible to notice the displacement of the mean of the probability density distributions of the parameters Pe and K_{L} for each case concerning the value adopted as reference (green line).
Figure 5 shows the estimation result for case studies 1 and 5(see Table 2). It is observed that the prior distribution mean is far from the exact value. After the estimation process, the result shown by the posterior probability distribution converges to a determined region close to the reference, suggesting that despite the displacement carried out in the prior mean, adequate information regions to select candidate parameters could be reached, demonstrating the robustness of the MCMC method.
Table 2 shows the results of the estimates in all cases in which the influence of the displacement of the mean on the prior probability distribution is evaluated. It was possible to observe that there was precision for all cases since the estimated parameters are within the 95% credibility interval and accuracy since the estimates are close to the exact value of the parameters. It is also found that the deviations from the parameters' estimates were low and had a low level of uncertainty.
(a) 
(b) 
Figure 4. Prior probability distribution function (pdf) evaluating the influence
of the mean for the parameters: (a) Pe e (b) K_{L}.
(a) 
(b) 
Figure 5. Prior and posterior probability distribution function evaluating the influence
of the mean displacement on the parameters' prior: (a) Pe e (b) K_{L}
Table 2. Influence of the mean on the prior probability distribution.
Caso 
Parameter 
Exact 


1 
Pe 
10.00 
N(3, 3) 
9.84 (9.39;10.38) 
2 
10.00 
N(5, 3) 
9.79 (9.37;10.18) 

3 
10.00 
N(15, 3) 
10.02 (9.31;10.66) 

4 
10.00 
N(20, 3) 
9.39 (8.96;9.74) 

5 
K_{L} 
1.00 
N(0.3, 0.3) 
1.00 (0.99;1.01) 
6 
1.00 
N(0.5, 0.3) 
1.00 (0.99;1.02) 

7 
1.00 
N(1.5, 0.3) 
1.00 (0.99;1.02) 

8 
1.00 
N(2.0, 0.3) 
1.01 (0.99;1.02) 
In addition to analyzing the influence of the mean, another critical assessment is to observe the effect that the variation in the standard deviation can have on the estimates since high standard deviation values can lead to uninformative priors. Even though they lead to own posteriors, poorly informative priors can generate a certain instability when posterior is obtained numerically [51].
Figure 6 shows the influence of different standard deviation values on the prior probability distribution function of the parameters. It is possible to observe that the search for candidate parameters falls within a reduced range of values, whereas for more significant deviations, this range is extended.
Table 3 shows the results (mean and credibility interval 99%) obtained for the estimates by varying the standard deviation of the prior probability distribution of the parameters. It is observed that the increase in deviation decreases the accuracy of the estimates since the estimated values are far from the exact value.
(a) 
(b) 
Figure 6. Prior probability distribution function evaluating the influence
of standard deviation on the prior for parameters: (a) Pe e (b) K_{L}.
Table 3. Influence of the standard deviation on the prior probability distribution.
Parameter 
Exact 


Pe 
10.00 
N (10,1 ) 
10.05 (9.47; 10.68) 
10.00 
N (10,2) 
9.34 (8.87; 10.10) 

10.00 
N (10,3) 
9.37 (8.65; 9.94) 

10.00 
N (10,4) 
9.33 (8.95; 9.70) 

K_{L} 
1.00 
N(1,0.1) 
1.00 (0.99; 1.01) 
1.00 
N(1,0.2) 
1.01 (0.99; 1.02) 

1.00 
N(1,0.3) 
1.01 (0.99; 1.02) 

1.00 
N(1,0.4) 
1.01 (1.00; 1.03) 
Figure 7 seeks to represent the acquisition frequency of measurements, dτ, and how uncertainties associated, , influence the data dispersion. As shown in Figure 7ad, it is possible to observe that as dτ increases, the number of measurements obtained decrease. On the other hand, measurements are placed close to the breakthrough curve at low values attributed to uncertainty, such as 1%. Higher values of were used to represent more realistic scenarios and interfered in the increase of dispersion around the reference curve, as observed in Figure 7eg.
a)

b)

; 
; 
c)

d)

; 
; 
e)

f)

; 
; 
g)


; 
Figure 7. Acquisition frequency of measurements and influence of the increase in uncertainty.
The parameters estimated from the use of the MCMC Bayesian method allowed that measurements obtained in the output current could offer the possibility of predicting curves that form at different points along the column. In this work, the analyzed points were close to the inlet θ = 0.25, in the middle
θ = 0.5 and at the exit of the column θ = 1. The results presented in Figure 8 showed the dynamics of the advance of the adsorptive phenomenon.
Figure 8a), c) and e) show the comparison between the exact solution of the model (black line) and the one obtained from the prior probability distribution of the parameters (solid blue line).
Figure 8b), d) and f) show the results for the exact solution of the model (black line) and the one obtained from the a posteriori probability distribution of the estimates made with the MCMC method (solid blue line). You can see that the exact and estimated curves overlap and remain within a 95% confidence interval (dashed blue line). It is evident that an excellent agreement was achieved between the estimated and exact measurements.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
Figure 8. Estimation of the breakthrough curve at different points in the fixed bed column. a) from ,
b) from , c) from , d) from ,
e) from f) from
Conclusion
The parameters estimation of the mathematical model of the adsorption of a chemical species in a fixed bed column was performed using the Markov Chain Monte Carlo method. In addition, simulated measurements generated from Gaussian noises were used to verify the developed algorithm.
The obtained model is solved by the Method of Lines, and a mesh convergence study was carried out to determine a sufficient value of discretization. A high tolerance value demands many discretization, which is not desirable since this increases the computational cost and causes numerical and approximation errors to occur. However, in the present work, the increase in computational cost was not significant.
The estimate analysis explored two scenarios: the influence of the mean and standard deviation on the prior probability distribution of the Peclet number, Pe, and the Langmuir isotherm constant, K_{L}. In these scenarios, we observed that even changing the prior distribution of the parameters, the posterior distribution samples converge to values close to exact. Thus, the estimated results were satisfactory, and there was both precision and accuracy in the inferences.
From the Bayesian inference, it was possible to use the information at a certain point in the column and thus obtain estimates with considerable precision in other places of interest where measurements were not available. The simulated scenario also allowed the observation of the influence of experimental uncertainties in obtaining measurements, showing that they tend to present greater dispersion the greater the uncertainties associated with their acquisition.
Thus, from the results presented, it is shown that the application of the Bayesian technique of MCMC is robust and presents itself as an excellent tool to understand the dynamics of the fixed bed adsorption process and other mass transfer processes.
Nomenclature
C 
Adsorbate concentration at bed outlet, (mg.L^{1}) 
q 
Adsorbate concentration at solid phase, (mg.g^{1}) 

Interstitial velocity, (cm.min^{1}) 

Axial dispersion coefficient, (cm^{2}.min^{1}) 

Langmuir constant, (L.mg^{1}) 

Maximum adsorption capacity, (mg.g^{1}) 

Global mass transfer coefficient, (min^{1}) 

Equilibrium concentration in the solid phase, (mg.g^{1}) 

Porosity 
t 
Time, (min) 

Bed density, (g.L^{1}) 

Dimensionless concentration of adsorbate at bed outlet 

Dimensionless concentration of adsorbate in the solid phase 

Peclet number 

Dimensionless Langmuir Constant 

Dimensionless maximum adsorption capacity 

Dimensionless global mass transfer coefficient 

Dimensionless length 

Dimensionless equilibrium concentration in the solid phase 

Dimensionless time 
Conflicts of Interest
The authors declare that have no conflict of interest regarding the publication of this article.
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[*]Corresponding Author: Diego Cardoso Estumano.
Email: dcestumano@ufpa.br