Transverse dispersion in ordered pillar arrays as a Markov chain: Extension of the Galton-board model

• Published in: Journal of Chromatography A Vol. 1375; pp. 27 - 32
• Main Authors: Smirnov, Konstantin N., Shpigun, Oleg A.
• Format: Journal Article
• Language: English
• Published: Netherlands Elsevier B.V 02-01-2015
• Subjects: Arrays; Chromatography; Dispersions; Galton board; Laminar flow; Markov chain; Markov Chains; Mathematical analysis; Mathematical models; Models, Chemical; Models, Statistical; Ordered pillar array; Pillars; Random walk; Transverse dispersion; Markov chain; Galton board; Ordered pillar array; Transverse dispersion; Random walk
• ISSN: 0021-9673; 1873-3778
• DOI: 10.1016/j.chroma.2014.11.065

•Galton-board model of transverse dispersion in ordered pillar arrays was examined.•The model was extended to be more consistent with the mechanism of mass transport.•The proposed extension is based on the concept of Markov chains.•An equation that describes the available dispersion data well was suggested.
An extension of the Galton-board model of the transverse solute dispersion in laminar flow through ordered arrays of non-porous cylindrical pillars was proposed. In contrast to the original model, which describes the dispersion process as a one-dimensional random walk with independent, equally probable steps, the extended model treats the process as a Markov chain, namely as a random walk with such correlated steps that the velocity-dependent probability to make a step in the same direction as the preceding step is smaller than the probability to reverse the direction of motion. The relationship between the average squared transverse displacement of the solute and the number of steps in the chain was used to find the expression for the velocity dependence of the transverse dispersion coefficient. The obtained equation differs from the one in the Galton-board model by the multiplier that accounts for the leveling-off of the experimental dependences at high reduced velocities. Although this modified Galton-board model cannot be directly applied to low velocities, a few additional assumptions lead to the expression that fits the whole range of the recent simulated dispersion data well.