Skip to main content
SpringerLink
Log in

Dynamic Studies of Binary Adsorption System in Fixed Beds using Orthogonal Collocation on Finite Element Method

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this study, a generalized, comprehensive and feasible mathematical model was developed for the description of the adsorption of multicomponent system in a fixed bed containing solid adsorbent. It took into cognizance of all the factors considered negligible in the literature, which included intraparticle, interparticle and interphase diffusional resistances, conditions of non-equilibrium and non-linear binary adsorption of butan-2-ol and 2-methylbutan-2-ol onto the adsorbent Filtrasorb 400 (granular activated carbon). The resulting 4N hyperbolic and parabolic differential equations were numerically solved using orthogonal collocation on finite element method. Excellent agreements were achieved between the experimental data and the simulated results of breakthrough times and concentrations of the solutes in fixed beds of 0.41, 0.82 and 1.23 m. Equally, the observed chromatographic tops of the less strongly adsorbed component (butan-2-ol) and the effects of adsorber length on times of breakthrough of the components and height of peaks were precisely simulated. The results of this study are of beneficial use in the design and analysis of fixed beds employed for the treatment of multicomponent aqueous wastewater effluents from industry.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Data availability

Data are available from the corresponding author on request.

Abbreviations

ARD:

Average relative deviation

EABS:

Sum of the absolute errors

ERRSQ:

Sum of the squares of the errors

HYBRID:

Composite fractional error function

MAE:

Ean absolute error

MAPE:

Mean absolute percentage error

MPE:

Mean percentage error

MPSD:

Marquardt’s percent standard deviation

OCFEM:

Orthogonal collocation on finite element method

RMSE:

Root mean square error

SPE:

Standard prediction error

\(a_{ij}\) :

Fritz-Schlüender model constant (L/mg)

\(a_{i0}\) :

Fritz-Schlüender model constant (L/mg)

\(b_{ij}\) :

Fritz- Schlüender model exponent

\(b_{i0}\) :

Fritz- Schlüender model exponent

\(\overline{c}_{e}\) :

Mean value of experimental breakthrough data (kg/m3)

\(\overline{c}_{i}\) :

Dimensionless concentration of solute i in fluid phase of the column

\(c_{i}\) :

Concentration of solute i in fluid phase of the column (kg/m3)

\(\overline{c}_{i,\,e}\) :

Experimental breakthrough concentrations of solute for run i (kg/m3)

\(\left( {c_{i} } \right)_{0}\) :

Inlet concentration of solute i in the column (kg/m3)

\(\overline{c}_{i,\,p}\) :

Simulated (or predicted) breakthrough concentrations of solute for run i (kg/m3)

\(c_{si}\) :

Amount of solute i adsorbed per unit volume of the particle (kg/m3) at equilibrium with a liquid phase concentration, \(X_{i}\), in a solution containing N solutes

\(C_{i}\) :

Constant in Eq. (1) (dimensionless)

\(D_{Li}\) :

Axial diffusivity for component i in the fluid phase (m2/s)

\(D_{pi}\) :

Diffusivity for component i, in the fluid phase within the pore of an adsorbent (m2/s)

\(D_{si}\) :

Diffusion coefficient for the diffusion of component i through the solid phase of an adsorbent (m2/s)

\(k_{f}\) :

Freundlich isotherm constant (kg/m3)1−n

\(K_{fi}\) :

Film mass transfer coefficient of solute i (m/s)

\(n\) :

Degree of heterogeneity (dimensionless)

N:

Number of solutes in solution (dimensionless)

\(N_{e}\) :

Number of experimental runs (dimensionless)

\(Pe_{i}\) :

Peclet number of component i (dimensionless)

\(q_{i}^{*}\) :

Concentration of solute i in the solid (or adsorbed) phase (kg/m3)

\(\left( {q_{i}^{*} } \right)_{0}\) :

Concentration of solute i in the solid (or adsorbed) phase (kg/m3)

\(Q_{i}^{*}\) :

Dimensionless concentration of solute i in the solid (or adsorbed) phase

r:

Radial distance in particle, i.e., radius of the adsorbent pellet as measured from center of pellet (m)

R:

External radius of the adsorbent (m)

\(R^{2}\) :

Coefficient of determination (or regression coefficient) (dimensionless)

\(Sh_{i}\) :

Sherwood number of component i (dimensionless).

t:

Time from start of sorption process (h)

\(U_{f}\) :

Linear velocity in the positive direction of z (m/s)

\(X_{i}\) :

Concentration of solute i in pore fluid phase (kg/m3)

\(\overline{X}_{i}\) :

Dimensionless concentration of solute i in pore fluid phase

z:

Axial distance in column (m)

\(z_{T}\) :

Length of column (m)

Z:

Dimensionless axial distance

\(\alpha\) :

Geometry of the adsorbent (dimensionless)

\(\in_{b}\) :

Void fraction in the bed, i.e., volume of voids per unit volume of bed (dimensionless)

\(\in_{p}\) :

Void fraction in particles, i.e., volume of pores in the pellet per unit volume of pellet (dimensionless)

\(\sigma\) :

Dimensionless radius

\(\tau\) :

Dimensionless time

References

  1. Aksu, Z., Isoglu, I.A.: Use of agricultural waste sugar beet pulp for the removal of Gemazol turquoise blue-G reactive dye from aqueous solution. J. Hazard. Mater. 137(1), 418–430 (2006). https://doi.org/10.1016/j.jhazmat.2006.02.019

    Article  Google Scholar 

  2. Allen, S.J., Mckay, G., Porter, J.F.: Adsorption isotherm models for basic dye adsorption by peat in single and binary component systems. J. Colloid Interface Sci. 280(2), 322–333 (2004). https://doi.org/10.1016/j.jcis.2004.08.078

    Article  Google Scholar 

  3. Aribike, D.S., Olafadehan, O.A.: Modeling of fixed bed adsorption of phenols on granular activated carbon. Theor. Found. Chem. Eng. 42(3), 257–263 (2008). https://doi.org/10.1134/S0040579508030056

    Article  Google Scholar 

  4. Arora, S., Dhaliwal, S.S., Kukreja, V.K.: Solution of two-point boundary value problems using orthogonal collocation on finite elements. Appl. Math. Comput. 171(1), 358–370 (2005). https://doi.org/10.1016/j.amc.2005.01.049

    Article  MathSciNet  Google Scholar 

  5. Balzli, M.W.: Einsatz von Aktivkable zur Reingung eines Mehrkomponenten – Chemicabwassers. Ph.D. Dissertation, Eidgenossische Technische Hochschule, Zurich (1977).

  6. Bathe, K.J., Wilson, E.L.: Numerical Methods in Finite Element Analysis. Prentice-Hall Inc, Englewood Cliffs (1976). https://doi.org/10.1002/nme.1620110913

    Book  Google Scholar 

  7. Bello, V.E., Olafadehan, O.A.: Comparative investigation of RSM and ANN for multi-response modeling and optimization studies of derived chitosan from Archachatina marginata shell. Alex. J. Eng. 60(4), 3869–3899 (2021). https://doi.org/10.1016/j.aej.2021.02.047

    Article  Google Scholar 

  8. Burchard, J.K., Toor, H.L.: Diffusion in an ideal mixture of three completely miscible non-electrolytic liquids—toluene, chlorobenzene, bromobenzene. J. Phys. Chem. 66(10), 2015–2022 (1962). https://doi.org/10.1021/j100816a044

    Article  Google Scholar 

  9. Butler, J.A.V., Ockrent, C.: Adsorption from solutions containing two solutes. Nature 125, 853–854 (1972). https://doi.org/10.1038/125853b0

    Article  Google Scholar 

  10. Carey, G.F., Finlayson, B.A.: Orthogonal collocation on finite elements. Chem. Eng. Sci. 30(5–6), 587–596 (1975). https://doi.org/10.1016/0009-2509(75)80031-5

    Article  Google Scholar 

  11. Choy, K.K.H., McKay, G.: Sorption of metal ions from aqueous solution using bone char. Environ. Int. 31(6), 845–854 (2005). https://doi.org/10.1016/j.envint.2005.05.036

    Article  Google Scholar 

  12. Cooney, D.O., Strusi, F.P.: Analytical description of fixed-bed sorption of two Langmuir solutes under nonequilibrium conditions. Ind. Eng. Chem. Fundam. 11(1), 123–126 (1972). https://doi.org/10.1021/i160041a018

    Article  Google Scholar 

  13. Dehghani, M.H., Yetilmezsoy, K., Salari, M., Heidarinejad, Z., Yousefi, M., Sillanpää, M.: Adsorptive removal of cobalt (II) from aqueous solutions using multi-walled carbon nanotubes and γ-alumina as novel adsorbents: Modelling and optimization based on response surface methodology and artificial neural network. J. Mol. Liq. 299, 112154 (2020)

    Article  Google Scholar 

  14. Dursun, A.Y., Tepe, O.: Removal of Chemazol Reactive Red 195 from aqueous solution by dehydrated beet pulp carbon. J. Hazard. Mater. 194, 303–311 (2011). https://doi.org/10.1016/j.jhazmat.2011.07.105

    Article  Google Scholar 

  15. El-Azazy, M., Nabil, I., Hassan, S.S., El-Shafie, A.S.: Adsorption characteristics of pristine and magnetic biochar with resoect to clofazimine. Nanomaterials 11(963), 1–24 (2021)

    Google Scholar 

  16. Fan, L.T., Chen, G.K.C., Erickson, L.E.: Efficiency and utility of collocation methods in solving the performance equations of flow chemical reactors with axial dispersion. Chem. Eng. Sci. 26, 379–387 (1971). https://doi.org/10.1016/0009-2509(71)83013-0

    Article  Google Scholar 

  17. Finlayson, B.A.: Numerical methods for chemical engineers, 2nd edn. Wiley, Washington (2002)

    Google Scholar 

  18. Finlayson, B.A.: Nonlinear analysis in chemical engineering. McGraw-Hill Inc., New York (1980)

    Google Scholar 

  19. Fritz, W., Schlüender, E.U.: Simultaneous adsorption equilibria of organic solutes in dilute aqueous solutions on activated carbon. Chem. Eng. Sci. 29(5), 1279–1282 (1974). https://doi.org/10.1016/0009-2509(74)80128-4

    Article  Google Scholar 

  20. Fu, D., Zhang, Y., Lv, F., Chu, P.K., Shang, J.: Removal of organic materials from TNT red water by bamboo charcoal adsorption. Chem. Eng. J. 193–194, 39–49 (2012). https://doi.org/10.1016/j.cej.2012.03.039

    Article  Google Scholar 

  21. Girish, C. R., Ramachandra Murty, V.: Adsorption of phenol from aqueous solution using Lantana camara, forest Waste: kinetics, isotherm, and thermodynamic studies. Int. Sch. Res. Notices. 2014, (2014).

  22. Girish, C.R., Murty, V.R.: Mass transfer studies on adsorption of phenol from wastewater using Lantana camara, forest waste. Int. J. Chem. Eng. 2016(5809505), 11 (2016). https://doi.org/10.1155/2016/5809505

    Article  Google Scholar 

  23. Guerra, D.L., Airoldi, C.: Kinetics and modified clay thermodynamic from the Brazilian amazon region for lead removal. J. Hazard. Mater. 159(2–3), 412–419 (2008). https://doi.org/10.1016/j.jhazmat.2008.02.032

    Article  Google Scholar 

  24. https://faculty.washington.edu/finlayso/ebook/bvp/OCFE/OCFE.htm

  25. Jäger, L., Erdös, E.: Simultane adsorption von phenol und p-kresol aus verdünnten wässerigen lösungen. Czechoslov. Chem. Commun. 24, 3019–3023 (1959)

    Article  Google Scholar 

  26. Kapur, M., Mondal, M.K.: Mass transfer and related phenomena for Cr(VI) adsorption from aqueous solutions onto Mangifera indica sawdust. Chem. Eng. J. 218, 138–146 (2013). https://doi.org/10.1016/j.cej.2012.12.054

    Article  Google Scholar 

  27. Kebir, M., Tahraoui, H., Chabani, M., Trari, M., Noureddine, N., Assadi, A.A., Amrane, A., Hamadi, N.B., Khezami, L.: Water cleaning by a continuous fixed-bed column for Cr(VI) eco-adsorption with green adsorbent-based biomass: an experimental modeling study. Processes 11(2), 363 (2023). https://doi.org/10.3390/pr11020363

    Article  Google Scholar 

  28. Kumar, R.V., Moorthy, I.G., Pugazhenthi, G.: Modeling and optimization of critical parameters by hybrid RSM-GA for the separation of BSA using tubular configured MFI-type zeolite microfiltration membrane. RSC Adv. 106, 1–44 (2015). https://doi.org/10.1039/C5RA20114D

    Article  Google Scholar 

  29. Liles, A.W., Geankoplis, C.J.: Axial diffusion of liquids in packed beds and end effects. AIChE J. 6(4), 591–595 (1960). https://doi.org/10.1002/aic.690060417

    Article  Google Scholar 

  30. Mandelbaum, J.A., Böhm, U.: Mass transfer in packed beds at low Reynolds numbers. Chem. Eng. Sci. 28(2), 569–576 (1973). https://doi.org/10.1016/0009-2509(73)80054-5

    Article  Google Scholar 

  31. Mansour, A., Von Rosenberg, D.U., Sylvester, N.D.: Numerical solution of liquid-phase multi-component adsorption in fixed beds. AIChE J. 28(5), 765–772 (1982)

    Article  Google Scholar 

  32. Michelsen, M.L., Villadsen, J.: Solutions of Differential Equation Models by Polynomial Approximation, vol. 07632. Prentice-Halls, Inc., Englewood Cliffs (1978). https://doi.org/10.1080/00986447808960470

    Book  Google Scholar 

  33. Mohan, D., Pittman, C.U., Jr., Steele, P.H.: Single, binary and multi-component adsorption of copper and cadmium from aqueous solutions on Kraft lignin: a biosorbent. J. Colloid Interface Sci. 297(2), 489–504 (2006). https://doi.org/10.1016/j.jcis.2005.11.023

    Article  Google Scholar 

  34. Moreno, D., Gómez, R., Giraldo, L.: Removal of Mn, Fe, Ni and Cu ions from wastewater using cow bone charcoal. Materials (Basel). 3(1), 452–466 (2010)

    Article  Google Scholar 

  35. Myers, A.L., Prausnitz, J.M.: Thermodynamics of mixed-gas adsorption. AIChE J. 11(1), 121–126 (1965). https://doi.org/10.1002/aic.690110125

    Article  Google Scholar 

  36. Ockrent, C.: Anomalous adsorption of active charcoal. Nature 130, 206–207 (1932). https://doi.org/10.1038/130206b0

    Article  Google Scholar 

  37. Okoya, A.A., Akinyele, A.B., Amuda, O.S., Ofoezie, I.E.: Chitosan-grafted carbon for the sequestration of heavy metals in aqueous solution. Am. Chem. Sci. J. 11(3), 1–14 (2016)

    Article  Google Scholar 

  38. Olafadehan, O.A., Amoo, K.O., Oyedeko, K.F.K., Bello, A.M., Bello, V.E., Akinfolarin, S.G.: Binary adsorption of phenol and 2-chlorophenol on chitosan derived from pink shrimp shell. Pet. Petrochem. Eng. J. 6(1), 1–18 (2022). https://doi.org/10.23880/ppej-16000292

    Article  Google Scholar 

  39. Olafadehan, O.A., Susu, A.A.: Computerized solution of the dynamic sorption process for a ternary system in a heterophase medium. Theor. Found. Chem. Eng. 38(2), 140–151 (2004). https://doi.org/10.1023/B:TFCE.0000022482.57163.80

    Article  Google Scholar 

  40. Olafadehan, O.A., Susu, A.A.: Modeling and simulation of liquid-phase ternary adsorption in activated carbon column. Ind. Eng. Chem. Res. 43(25), 8107–8116 (2004). https://doi.org/10.1021/ie030876p

    Article  Google Scholar 

  41. Olafadehan, O.A., Susu, A.A.: Numerical solution of binary liquid-phase adsorption on porocel clay using linear, Freundlich and Langmuir isotherms. Adsorpt. Sci. Technol. 23(3), 195–214 (2005). https://doi.org/10.1260/0263617054353627

    Article  Google Scholar 

  42. Olafadehan, O.A.: Modelling and simulation: the under-tapped golden resource and implications for process industries, delivered at the University of Lagos J. F. Ade Ajayi Auditorium, September 18 (2019).

  43. Olafadehan, O.A.: Modelling of multi-component adsorption and reaction in porous media, Ph.D. Thesis, University of Lagos, Lagos, Nigeria (2000).

  44. Olafadehan, O.A.: Fundamentals of Adsorption Processes. LAP Lambert Academic Publishing, OmniScriptum DUE GmbH (2021). (ISBN: 978-620-3-30705-4)

    Google Scholar 

  45. Pendleton, P., Wu, S.H.: Kinetics of dodecanoic acid adsorption from caustic solution by activated carbon. J. Colloid Interface Sci. 226(2), 245–250 (2003). https://doi.org/10.1016/S0021-9797(03)00575-7

    Article  Google Scholar 

  46. Porter, J.F., McKay, G., Choy, K.H.: The prediction of sorption from a binary mixture of acidic dyes using single- and mixed-isotherm variants of the ideal adsorbed solute theory. Chem. Eng. Sci. 54(24), 5863–5885 (1999). https://doi.org/10.1016/S0009-2509(99)00178-5

    Article  Google Scholar 

  47. Radke, C.J., Prausnitz, J.M.: Adsorption of organic solutes from dilute aqueous solution of activated carbon. Ind. Eng. Chem. Fundam. 11(4), 445–451 (1972). https://doi.org/10.1021/i160044a003

    Article  Google Scholar 

  48. Sarici-Ozdemir, C., Onal, Y.: Equilibrium, kinetic and thermodynamic adsorptions of the environmental pollutant tannic acid onto activated carbon. Desalination 251(1–3), 146–152 (2010). https://doi.org/10.1016/j.desal.2009.09.133

    Article  Google Scholar 

  49. Satoh, K., Fan, H., Hattori, H., Tajima, K., Furuya, E.: Simultaneous determination of intraparticle diffusivities from ternary component uptake curves using the shallow bed technique. J. Hazard. Mater. 155(3), 397–402 (2008). https://doi.org/10.1016/j.jhazmat.2007.11.072

    Article  Google Scholar 

  50. Seida, Y., Sonetaka, N., Noll, K.E., Furuya, E.: Determination of pore and surface diffusivities from single decay curve in CSBR based on parallel diffusion model. Water 14(22), 3629–3642 (2022). https://doi.org/10.3390/w14223629

    Article  Google Scholar 

  51. Shuck, F.O., Toor, H.L.: Separation of liquids by mass diffusion. AIChE J. 9(4), 442–447 (1963). https://doi.org/10.1002/aic.690090407

    Article  Google Scholar 

  52. Sulak, M.T., Demirbas, E., Kobya, M.: Removal of Astrazon Yellow 7GL from aqueous solutions by adsorption onto wheat bran. Bioresour. Technol. 98(13), 2590–2598 (2007). https://doi.org/10.1016/j.biortech.2006.09.010

    Article  Google Scholar 

  53. Toor, H.L., Arnold, K.R.: Nature of uncoupled multicomponent diffusion equations. Ind. Eng. Chem. Fundam. 4(3), 363–364 (1965). https://doi.org/10.1021/i160015a024

    Article  Google Scholar 

  54. Villadsen, J.V., Michelsen, M.L.: Solution of Differential Equations Models by Polynomials Approximation. Prentice-Hall Inc, New Jersey (1978)

    Google Scholar 

  55. Weber, W.J., Morris, J.C.: Equilibria and capacities for adsorption on carbon. JSEDA. 90(3), 79–108 (1964)

    Google Scholar 

  56. Wilke, C.R., Hougen, O.W.: Mass transfer in the flow of gases through granular solids extended to low modified Reynolds numbers. Trans. Am. Inst. Chem. Eng. 41(4), 445–451 (1945)

    Google Scholar 

  57. Wilson, E.J., Geankoplis, C.J.: Liquid mass transfer at very low Reynolds in packed beds. Ind. Eng. Chem. Fundam. 5(1), 9–14 (1966). https://doi.org/10.1021/i160017a002

    Article  Google Scholar 

  58. Yaseen, Z.M., Ebtehaj, I., Kim, S., Sanikham, H., Asadi, H., Ghareb, M.I., Bonakdari, H., Montar, W.H.M.W., Ansari, N.A., Shahid, S.: Novel hybrid data-intelligence model for forecasting monthly rainfall with uncertainty analysis. Water 11(502), 1–23 (2019). https://doi.org/10.3390/w11030502

    Article  Google Scholar 

  59. Zienkiewicz, O.C.: The Finite Element Method, 3rd edn. McGraw Book Company, London (1977)

    Google Scholar 

Download references

Funding

This work did not receive any specific grant from funding agencies in the public, commercial or non-profit sector.

Author information

Authors and Affiliations

  1. Department of Chemical and Petroleum Engineering, University of Lagos, Akoka-Yaba, Lagos, 101017, Nigeria

    O. A. Olafadehan

  2. Department of Chemical Engineering, Lagos State University, Lagos, Nigeria

    K. O. Amoo & K. F. K. Oyedeko

  3. Department of Chemical Engineering, Covenant University, CanaanLand, Ota, Ogun State, Nigeria

    A. J. Adesina

Authors
  1. O. A. Olafadehan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. O. Amoo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. F. K. Oyedeko
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. J. Adesina
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Conceptualization: OAO; Methodology: KOA and AJA; Formal analysis and investigation: OAO; KOA, KFKO and AJA; Writing—original draft preparation: KOA and AJA; Writing—review and editing: OAO; KOA; KFKO and AJA; Resources: OAO and AJA; Supervision: OAO and KFKO.

Corresponding author

Correspondence to O. A. Olafadehan.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Olafadehan, O.A., Amoo, K.O., Oyedeko, K.F.K. et al. Dynamic Studies of Binary Adsorption System in Fixed Beds using Orthogonal Collocation on Finite Element Method. Int. J. Appl. Comput. Math 10, 108 (2024). https://doi.org/10.1007/s40819-024-01688-7

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-024-01688-7

Keywords

Search

Navigation