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Inverse analysis applying the Levenberg–Marquardt method for simultaneously estimating parameters of the adsorption of CO2 on activated carbon in a fixed-bed adsorber

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Abstract

An inverse approach (IA) was used to estimate unknown physical parameters selected of a mathematical model built up (MMB) for the adsorption process of CO2 on an activated carbon (AC) bed in a fixed-bed adsorber (FBA). The MMB was reported as the direct problem (DP) for the FBA and is used as the working model of this article. The DP is defined for mass balance equations of CO2 in gaseous and solid phases inside the FBA. The implicit finite volume method and explicit finite difference method were used to solve the DP of this work. These methods were used to test the better method regarding to processing time of the DP. The IA has been applied to the DP to adjust the axial dispersion coefficient of CO2, gas–solid mass transfer coefficient, and adsorption constant of CO2 on the AC bed of the FBA using the Levenberg–Marquardt (LM) method. The solution procedures applied in this work for direct and inverse problems have increased the reliability of the estimation of the chosen parameters from the DP. The results of this work show that the LM method was successfully applied to an adsorption problem to estimate the parameters chosen of the MMB. The numerical simulations have reported that the MMB predicts concentration distribution of CO2 in the gaseous and solid phases, and therefore, the simulated results enable us further generalization of the MMB to adsorption of CO2 on the AC inside the FBA.

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Abbreviations

\(\mathrm{AC}\) :

Activated carbon

\(\mathrm{CCS}\) :

Carbon capture and storage

\(\mathrm{CI}\) :

Confidence interval

\(\mathrm{CPDEs}\) :

Coupled partial differential equations

\(\mathrm{GSHMM}\) :

Gas–solid heterogeneous mathematical model

\(\mathrm{DP}\) :

Direct problem

\(\mathrm{DAE}\) :

Discretized algebraic equation

\(\mathrm{EFD}\) :

Explicit finite difference

\(\mathrm{ES}\) :

Experimental system

\(\mathrm{FBA}\) :

Fixed-bed adsorber

\(\mathrm{GD}\) :

Gradient descent

\(\mathrm{GHGs}\) :

Greenhouse gases

\(\mathrm{GN}\) :

Gauss–Newton

\(\mathrm{IA}\) :

Inverse approach

\(\mathrm{IFV}\) :

Implicit finite volume

\(\mathrm{LI}\) :

Langmuir isotherm

\(\mathrm{LM}\) :

Levenberg–Marquardt

\(\mathrm{MMB}\) :

Mathematical model built up

\(\mathrm{MRD}\) :

Maximum relative deviation

\(\mathrm{NAEs}\) :

Nonlinear algebraic equations

\(\mathrm{NLAI}\) :

Nonlinear adsorption isotherm

\(\mathrm{NR}\) :

Newton–Raphson

\(\mathrm{PDEs}\) :

Partial differential equations

\(\mathrm{RMSE}\) :

Root mean square error (–)

\(\mathrm{SO}\) :

Simultaneous optimization

\(\mathrm{SQR}\) :

Objective function

\(\mathrm{UA}\) :

Uncertainty analysis

\(C_{{\text{g}}}\) :

Concentration of CO2 in the gas phase \(\left( {{\text{mol|m}}^{3} } \right)\)

\(C_{{\text{g,0}}}\) :

Feed concentration of CO2 in the gas phase \(\left( {{\text{mol|m}}^{3} } \right)\)

\(C_{{\text{p}}}\) :

Concentration of CO2 in the solid phase \(\left( {{\text{mol|m}}^{3} } \right)\)

\(D_{{{\text{ax}}}}\) :

Axial dispersion coefficient of CO2 \(\left( {{\text{m}}^{2} {\text{|min}}} \right)\)

\(d_{{\text{c}}}\) :

Absorber diameter \(\left( {\text{m}} \right)\)

\(D_{{{\text{CO}}_{2} - {\text{N}}_{2} }}\) :

Binary molecular diffusivity \(\left( {{\text{m}}^{2} {\text{|min}}} \right)\)

\(D_{{{\text{m}},{\text{CO}}_{2} }}\) :

Mixture molecular diffusion coefficient \(\left( {{\text{m}}^{2} {\text{|min}}} \right)\)

\(D_{{{\text{k}},{\text{CO}}_{2} }}\) :

Knudsen diffusivity \(\left( {{\text{m}}^{2} {\text{|min}}} \right)\)

\(J\left( {p_{j} } \right)\) :

Jacobian or sensitivity matrix \(\left( {j = 1, 2, \ldots , N} \right)\)

\(J_{ij}\) :

Sensitivity coefficients \(\left( {i = 1,2, \ldots , M;j = 1, 2, \ldots , N} \right)\)

\(d_{{\text{p}}}\) :

Particle average diameter \(\left( {{\text{mm}}} \right)\)

\(d_{{\text{c}}}\) :

Inner diameter (m)

\(D_{{{\text{p}},{\text{CO}}_{2} }}\) :

Effective pore diffusion coefficient of CO2 \(\left( {{\text{m}}^{2} {\text{|min}}} \right)\)

\(d_{{{\text{pore}}}}\) :

Diameter of the pores (μm)

\({K}_{{\mathrm{CO}}_{2}}\) :

Adsorption constant of CO2 \(\left({\mathrm{m}}^{3}|\mathrm{mol}\right)\)

\({k}_{\mathrm{gs}}\) :

Gas–solid mass transfer coefficient \(\left(\mathrm{m}|\mathrm{min}\right)\)

\({k}_{\mathrm{gs},\mathrm{eff}}\) :

Effective mass transfer coefficient \(\left(\mathrm{m}|\mathrm{min}\right)\)

\(L\) :

Bed length \(\left(\mathrm{m}\right)\)

\({M}_{{\mathrm{CO}}_{2}}\) :

Molecular weight of CO2 \(\left(\mathrm{kg}|\mathrm{mol}\right)\)

\({M}_{{\mathrm{N}}_{2}}\) :

Molecular weight of component j \(\left(\mathrm{kg}|\mathrm{mol}\right)\)

\({n}_{\mathrm{p}}\) :

Point numbers (–)

\({P}_{\mathrm{op}.}\) :

Operating pressure (kPa)

\({p}_{j}\) :

Unknown parameters to be estimated \(\left(j=1, 2, \dots , N\right)\)

\({Q}_{\mathrm{g}}\) :

Gas flow rates \(\left({\mathrm{m}}^{3}|\mathrm{min}\right)\)

\({q}_{\mathrm{max}}\) :

Maximum amount adsorbed of CO2 in the solid phase \(\left(\mathrm{mol}|\mathrm{kg}\right)\)

\({q}_{\mathrm{p}}\) :

Absolute amount adsorbed of CO2 in the solid phase \(\left(\mathrm{mol}|\mathrm{kg}\right)\)

\(R\) :

Radial coordinate (m)

\({R}^{2}\) :

Determination coefficient (–)

\({R}_{\mathrm{gas}}\) :

Gas constant (J mol−1 K−1)

\({S}_{\mathrm{p}}\) :

Specific surface area (m2 g−1)

\(t\) :

Time (min)

\({T}_{\mathrm{op}.}\) :

Operating temperature (K)

\({V}_{\mathrm{sp}}\) :

Superficial gas velocity (m min−1)

\(z\) :

Spatial coordinate (m)

\({\varepsilon }_{\mathrm{b}}\) :

Bed porosity \(\left({\mathrm{m}}^{3}\mathrm{ gas}/{\mathrm{m}}^{3}\mathrm{ absorber}\right)\)

\({\varepsilon }_{\mathrm{p}}\) :

Particle porosity \(\left({\mathrm{m}}^{3}\mathrm{ pores}/{\mathrm{m}}^{3}\mathrm{ particle}\right)\)

\({\varepsilon }_{{\mathrm{CO}}_{2}-{\mathrm{N}}_{2}}\) :

Binary characteristic energy of molecules (–)

\({\varepsilon }_{{\mathrm{CO}}_{2}}\) :

Characteristic energy of CO2 (–)

\({\varepsilon }_{{\mathrm{N}}_{2}}\) :

Characteristic energy of N2 (–)

\({\rho }_{\mathrm{p}}\) :

Density of AC particles (kgcat m−3)

\({\sigma }_{\mathrm{m}}\) :

Standard deviation of the measurement errors

\({\sigma }_{{\mathrm{CO}}_{2}-{\mathrm{N}}_{2}}\) :

Binary characteristic length of molecules (–)

\({\sigma }_{{\mathrm{CO}}_{2}}\) :

Characteristic Lennard–Jones length of CO2 (–)

\({\sigma }_{{\mathrm{N}}_{2}}\) :

Characteristic Lennard–Jones length of N2 (–)

\({\mu }^{\rm k}\) :

Positive scalar called relaxation parameter

\({\Psi }^{\rm k}\) :

Diagonal matrix

\({\tau }_{\mathrm{p}}\) :

Pore tortuosity (–)

\({\chi }^{2}\) :

Chi-square (–)

\({\Omega }_{{\mathrm{CO}}_{2}-{\mathrm{N}}_{2}}\) :

Binary diffusion collision integral (–)

\(\mathrm{ax}\) :

Axial

\({\mathrm{CO}}_{2}\) :

Carbon dioxide

CO2 − N2 :

Carbon dioxide–nitrogen

\(\mathrm{g}\) :

Gas

\(\mathrm{g},0\) :

Feed

\(\mathrm{gs}\) :

Solid–gas

\(\mathrm{gs},\mathrm{eff}\) :

Effective solid–gas

\(ij\) :

Row and column

\(i\) :

Row

\(j\) :

Column

\(\mathrm{k},{\mathrm{CO}}_{2}\) :

Knudsen of CO2

\(\mathrm{m}\) :

Molecular

\(\mathrm{m},{\mathrm{CO}}_{2}\) :

Mass of CO2

\(\mathrm{max}\) :

Maximum

\({\mathrm{N}}_{2}\) :

Nitrogen

\(\mathrm{op}\) :

Operating

\(\mathrm{p}\) :

Particle

\(\mathrm{pore}\) :

Pore identification

\(\mathrm{p},{\mathrm{CO}}_{2}\) :

Pore effective

\(\mathrm{sp}\) :

Superficial

Reg :

Reynolds number, \({\mathrm{Re}}_{\mathrm{g}}= \frac{{\rho }_{\mathrm{g}}{V}_{\mathrm{sg }}{d}_{\mathrm{p}}}{{\mu }_{\mathrm{g}}}\)

Scg, i :

Schmidt number, \({\mathrm{Sc}}_{\mathrm{g},i}= \frac{{\mu }_{\mathrm{g}}}{{\rho }_{\mathrm{g}}{D}_{\mathrm{m}}}\)

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Acknowledgements

The authors of this paper would like to thank the CNPq (National Council of Scientific and Technological Development) for the financial support given (Process 57354/2018).

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  1. Laboratory of Environmental and Energetic Technology, Polytechnic School - UPE, Rua Benfica - 455, Madalena, Recife, PE, CEP 50750-470, Brazil

    Jornandes Dias da Silva

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da Silva, J.D. Inverse analysis applying the Levenberg–Marquardt method for simultaneously estimating parameters of the adsorption of CO2 on activated carbon in a fixed-bed adsorber. J Braz. Soc. Mech. Sci. Eng. 44, 460 (2022). https://doi.org/10.1007/s40430-022-03756-9

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