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Adsorption

, Volume 19, Issue 5, pp 1007–1018 | Cite as

Nonlinear direct inverse method: a shortcut method for simultaneous calibration and isotherm determination

  • Bijan Medi
  • Monzure-Khoda Kazi
  • Mohammad AmanullahEmail author
Article

Abstract

This work addresses a way to combine isotherm determination and nonlinear calibration. In this method, like the classical inverse method, experimental elution profiles are compared with the results of a detailed model that accounts for nonlinearity in equilibrium, axial dispersion, and mass transfer kinetics. However, unlike the classical inverse method, the calibration of detector is carried out simultaneously with isotherm determination thereby reducing cost and saving time. In this study no limitation is imposed on the linearity of the detector signal or on the overlapping of elution profiles for the separation of enantiomers. The method has been experimentally validated for the separation of a mixture of guaifenesin enantiomers over a wide range of concentration.

Keywords

Enantioseparation Inverse method Preparative chromatography Guaifenesin Genetic algorithm 

List of symbols

a

Slope of absolute calibration equation (g/L/mAU)

b

Intercept of absolute calibration equation (g/L)

c

Nonlinear parameter of absolute calibration equation (1/mAU)

c(t)

Fluid phase concentration of solute (g/L)

cTF

Total feed concentration (g/L)

D

Column diameter (cm)

Dax

Axial dispersion coefficient (m2/s)

dp

Particles diameter (μm)

E

Estimation error (–)

Hi

Henry constant of species i (–)

Ki

Equilibrium constant in Langmuir isotherm of species i (L/g)

ki

Overall mass transfer coefficient (1/s)

L

Column length (cm)

Ncy

Number of experiment cycles (–)

Nt,j

Number of discretized data points collected over time for the jth experiment (–)

n

Solid phase concentration of solute (g/L)

n*

Equilibrium solid phase concentration of solute (g/L)

S

Peak area (mAU s)

Q

Volumetric flow rate (mL/min)

qs,i

Saturation capacity of species i (g/L)

t

Time (s)

te

End time of experimental run (s)

tp

Feed pulse width (s)

v

Interstitial velocity (cm/s)

Vd

Extra column dead volume (μL)

Vinj

Injection volume (μL)

y

Absorbance (mAU)

z

Axial coordinate (cm)

Greek letters

α

Slope of analytical calibration line (mAU s L/g)

β

Intercept of analytical calibration line (mAU s)

\(\varepsilon\)

Overall void fraction of column (–)

λ

Penalty factor (-)

Subscripts and superscripts

A

More retained component

ax

Axial

B

Less retained component

exp

Experimental

F

Feed

i

Component index

inj

Injection

ns

Non-selective

sim

Simulation

Notes

Acknowledgments

The authors thank Singapore’s Ministry of Education for supporting this work through Grant No. RG24/07.

References

  1. Amanullah, M., Mazzotti, M.: Optimization of a hybrid chromatography-crystallization process for the separation of troger’s base enantiomers. J. Chromatogr. A 1107(1–2), 36–45 (2006)Google Scholar
  2. Asnin, L., Guiochon, G.: Calibration of detector responses using the shape and size of band profiles: case of a nonlinear response curve. J. Chromatogr. A 1089(1–2), 101–104 (2005)Google Scholar
  3. Asnin, L., Galinada, W., Gotmar, G., Guiochon, G.: Calibration of a detector for nonlinear chromatography. J. Chromatogr. A 1076(1–2), 141–147 (2005)Google Scholar
  4. Blumel, C., Hugo, P., Seidel-Morgenstern, A.: Quantification of single solute and competitive adsorption isotherms using a closed-loop perturbation method. J. Chromatogr. A 865(1–2), 51–71 (1999)Google Scholar
  5. Butt, J.B.: Reaction Kinetics and Reactor Design. Prentice Hall, Englewood Cliffs (1980)Google Scholar
  6. Cornel, J., Mazzotti, M.: Calibration-free quantitative application of in situ raman spectroscopy to a crystallization process. Anal. Chem. 80(23), 9240–9249 (2008)CrossRefGoogle Scholar
  7. Cornel, J., Tarafder, A., Katsuo, S., Mazzotti, M.: The direct inverse method: a novel approach to estimate adsorption isotherm parameters. J. Chromatogr. A 1217(12), 1934–1941 (2010)CrossRefGoogle Scholar
  8. Cruz, P., Santos, J.C., Magalhpes, F.D., Mendes, A.: Simulation of separation processes using finite volume method. Comput. Chem. Eng. 30(1), 83–98 (2005)CrossRefGoogle Scholar
  9. Dunnebier, G., Klatt, K.U.: Modelling and simulation of nonlinear chromatographic separation processes: a comparison of different modelling approaches. Chem. Eng. Sci. 55(2), 373–380 (2000)CrossRefGoogle Scholar
  10. Felinger, A., Cavazzini, A., Guiochon, G.: Numerical determination of the competitive isotherm of enantiomers. J. Chromatogr. A 986(2), 207–225 (2003)CrossRefGoogle Scholar
  11. Francotte, E., Richert, P., Mazzotti, M., Morbidelli, M.: Simulated moving bed chromatographic resolution of a chiral antitussive. J. Chromatogr. A 796(2), 239–248 (1998)CrossRefGoogle Scholar
  12. Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, New York (1989)Google Scholar
  13. Golshanshirazi, S., Guiochon, G.: Comparison of the various kinetic-models of nonlinear chromatography. J. Chromatogr. 603(1–2), 1–11 (1992)Google Scholar
  14. Guiochon, G.: Preparative liquid chromatography. J. Chromatogr. A 965(1–2), 129–161 (2002)Google Scholar
  15. Guiochon, G., Fellinger, A., Shirazi, D.G., Katti, A.M.: Fundamentals of Preparative and Nonlinear Chromatography, 2nd edn. Academic Press, Boston (2006)Google Scholar
  16. Juza, M., Mazzotti, M., Morbidelli, M.: Simulated moving-bed chromatography and its application to chirotechnology. Trends Biotechnol. 18(3), 108–118 (2000)CrossRefGoogle Scholar
  17. Kazi, M.K., Medi, B., Amanullah, M.: Optimization of an improved single-column chromatographic process for the separation of enantiomers. J. Chromatogr. A 1231, 22–30 (2012)CrossRefGoogle Scholar
  18. Lawrence, C.T., Tits, A.L.: A computationally efficient feasible sequential quadratic programming algorithm. Siam. J. Optim. 11(4), 1092–1118 (2001)CrossRefGoogle Scholar
  19. Levenspiel, O.: Chemical Reaction Engineering, 3rd edn. Wiley, New York (1999)Google Scholar
  20. Lisec, O., Hugo, P., Seidel-Morgenstern, A.: Frontal analysis method to determine competitive adsorption isotherms. J. Chromatogr. A 908(1–2), 19–34 (2001)Google Scholar
  21. Mazzotti, M.: Local equilibrium theory for the binary chromatography of species subject to a generalized langmuir isotherm. Ind. Eng. Chem. Res. 45(15), 5332–5350 (2006)CrossRefGoogle Scholar
  22. Medi, B., Amanullah, M.: Application of a finite-volume method in the simulation of chromatographic systems: effects of flux limiters. Ind. Eng. Chem. Res. 50(3), 1739–1748 (2011)CrossRefGoogle Scholar
  23. Phillips, M.W., Subramanian, G., Cramer, S.M.: Theoretical optimization of operating parameters in non-ideal displacement chromatography. J. Chromatogr. A 454(C), 1–21 (1988)Google Scholar
  24. Rajendran, A., Chen, W.: Binary retention time method for rapid determination of competitive langmuir isotherm parameters. Sep. Purif. Technol. 67(3), 344–354 (2009)CrossRefGoogle Scholar
  25. Ruthven, D.M.: Principles of Adsorption and Adsorption Processes. Wiley, New York (1984)Google Scholar
  26. Seidel-Morgenstern, A.: Experimental determination of single solute and competitive adsorption isotherms. J. Chromatogr. A 1037(1–2), 255–272 (2004)Google Scholar
  27. Shampine, L.F., Gordon, M.K.: Computer Solution of Ordinary Differential Equations: The Initial Value Problem. W. H. Freeman and Co., San Francisco (1975)Google Scholar
  28. Soussen-Jacob, J., Tsakiris, J., De Lara, E.C.: Adsorption of oxygen molecule in NaA zeolite: isotherms and infrared measurements. J. Chem. Phys. 91(4), 2649–2655 (1989)CrossRefGoogle Scholar
  29. Thompson, K.A., Fuller, J.E.L.: Accurate sorption isotherms using a computer-aided microgravimetric method. J. Vac. Sci. Technol. A 5(4), 2522–2525 (1987)CrossRefGoogle Scholar
  30. Wenda, C., Rajendran, A.: Enantioseparation of flurbiprofen on amylose-derived chiral stationary phase by supercritical fluid chromatography. J. Chromatogr. A 1216(50), 8750–8758 (2009)CrossRefGoogle Scholar
  31. Zabka, M., Minceva, M., Gomes, P.S., Rodrigues, A.E.: Chiral separation of R,S-α-tetralol by simulated moving bed. Sep. Sci. Technol. 43(4), 727–765 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Bijan Medi
    • 1
  • Monzure-Khoda Kazi
    • 1
  • Mohammad Amanullah
    • 2
    Email author
  1. 1.School of Chemical and Biomedical Engineering, Nanyang Technological UniversitySingaporeSingapore
  2. 2.Department of Chemical EngineeringCollege of Engineering, Qatar UniversityDohaQatar

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