Abstract
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.
Similar content being viewed by others
References
Özdural AR, Alkan A, Kerkhof PJAM (2004) Modeling chromatographic columns. Non-equilibrium packed-bed adsorption with non-linear adsorption isotherms. J Chromatogr A 1041:77–85
Raghavan NS, Ruthven DM (1984) Dynamic behavior of an adiabatic adsorption column—II, numerical simulation and analysis of experimental data. Chem Eng Sci 39:1201–1212
Danckwerts PV (1953) Continuous flow systems: distribution of residence times. Chem Eng Sci 2:1–13
Aris R (1956) On the dispersion of a solute in a fluid flowing through a tube. Proc Royal Soc Lond A 235:67–77
Smith R (1983) Effect of boundary absorption upon longitudinal dispersion in shear flows. J Fluid Mech 134:161–177
Mazumder BS, Das SK (1992) Effect of boundary reaction on solute dispersion in pulsatile flow through a tube. J Fluid Mech 239:523–549
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 14:639–651
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 35:113–124
Vrentas JS, Vrentas CM (2013) Diffusion and mass transfer. CRC Press, Boca Raton
Vianna AS Jr, Nichele J (2010) Modeling an annular tube flow reactor. Chem Eng Sci 65:4261–4270
Spiegel MR (1968) Mathematical handbook of formulas and tables. McGraw-Hill Book Co. Inc, New York
Stone HA, Stroock AD, Ajdari A (2004) Engineering flows in small devices: microfluidics toward a lab-on-chip. Annu Rev Fluid Mech 36:381–411
Nauman EB, Nigam A (2005) On axial diffusion in laminar-flow reactors. Ind Eng Chem Res 44:5031–5035
Nauman EB, Nigam A (2007) Mixing, flow and chemical reaction of partially miscible components in micro-scale channels. Chem Eng Res Des 85(A5):612–615
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 116:576–589
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Vrentas, J.S., Vrentas, C.M. Effect of axial diffusion on impurity adsorption in a circular tube. J Eng Math 114, 131–139 (2019). https://doi.org/10.1007/s10665-018-09985-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-018-09985-z