Journal of Engineering Mathematics

, Volume 114, Issue 1, pp 131–139 | Cite as

Effect of axial diffusion on impurity adsorption in a circular tube

  • J. S. VrentasEmail author
  • C. M. Vrentas


Circular tubes can be used to irreversibly adsorb impurities from a dilute solution of the impurity and a non-adsorbing fluid. An analysis of the unsteady adsorption process over the region \(z = - \infty \) to \(z = \infty \) can be carried out with or without axial diffusion. The transport equation in the presence of axial diffusion can be solved using the complex Fourier transform and the Green’s function method. In the absence of axial diffusion, the transport equation can be solved using the Laplace transform method. Previously reported studies over the region \(0\le z\le L\) limited the contribution of axial diffusion. Analytical solutions of the pertinent differential equations are used to determine the impurity concentration everywhere in the circular tube.


Axial diffusion Circular tube Green’s function Impurity adsorption Impurity concentration Laplace transform 



  1. 1.
    Özdural AR, Alkan A, Kerkhof PJAM (2004) Modeling chromatographic columns. Non-equilibrium packed-bed adsorption with non-linear adsorption isotherms. J Chromatogr A 1041:77–85CrossRefGoogle Scholar
  2. 2.
    Raghavan NS, Ruthven DM (1984) Dynamic behavior of an adiabatic adsorption column—II, numerical simulation and analysis of experimental data. Chem Eng Sci 39:1201–1212CrossRefGoogle Scholar
  3. 3.
    Danckwerts PV (1953) Continuous flow systems: distribution of residence times. Chem Eng Sci 2:1–13CrossRefGoogle Scholar
  4. 4.
    Aris R (1956) On the dispersion of a solute in a fluid flowing through a tube. Proc Royal Soc Lond A 235:67–77CrossRefGoogle Scholar
  5. 5.
    Smith R (1983) Effect of boundary absorption upon longitudinal dispersion in shear flows. J Fluid Mech 134:161–177CrossRefzbMATHGoogle Scholar
  6. 6.
    Mazumder BS, Das SK (1992) Effect of boundary reaction on solute dispersion in pulsatile flow through a tube. J Fluid Mech 239:523–549CrossRefGoogle Scholar
  7. 7.
    Heslop MJ, Schaschke CJ, Sefcik J, Richardson DJ, Russell PA (2008) Measurement of adsorption of a single component from the liquid phase: modelling investigation and sensitivity analysis. Adsorption 14:639–651CrossRefGoogle Scholar
  8. 8.
    Lau A, Crittenden BD, Field RW (2004) Enhancement of liquid phase adsorption column performance by means of oscillatory flow: an experimental study. Sep Purif Tech 35:113–124CrossRefGoogle Scholar
  9. 9.
    Vrentas JS, Vrentas CM (2013) Diffusion and mass transfer. CRC Press, Boca RatonzbMATHGoogle Scholar
  10. 10.
    Vianna AS Jr, Nichele J (2010) Modeling an annular tube flow reactor. Chem Eng Sci 65:4261–4270CrossRefGoogle Scholar
  11. 11.
    Spiegel MR (1968) Mathematical handbook of formulas and tables. McGraw-Hill Book Co. Inc, New YorkGoogle Scholar
  12. 12.
    Stone HA, Stroock AD, Ajdari A (2004) Engineering flows in small devices: microfluidics toward a lab-on-chip. Annu Rev Fluid Mech 36:381–411CrossRefzbMATHGoogle Scholar
  13. 13.
    Nauman EB, Nigam A (2005) On axial diffusion in laminar-flow reactors. Ind Eng Chem Res 44:5031–5035CrossRefGoogle Scholar
  14. 14.
    Nauman EB, Nigam A (2007) Mixing, flow and chemical reaction of partially miscible components in micro-scale channels. Chem Eng Res Des 85(A5):612–615CrossRefGoogle Scholar
  15. 15.
    Qamar S, Uhan FU, Mehmood Y, Seidel-Morgenstern A (2014) Analytical solution of a two-dimensional model of liquid chromatography including moment analysis. Chem Eng Sci 116:576–589CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Chemical EngineeringThe Pennsylvania State UniversityUniversity ParkUSA

Personalised recommendations