Effective medium theory expressions for the effective diffusion in chromatographic beds filled with porous, non-porous and porous-shell particles and cylinders. Part I: Theory.
Using the permeability analogue of the diffusion and partitioning processes occurring in a chromatographic column, the different Effective Medium Theory (EMT) models that exist in literature for the electrical and thermal conductivity have been transformed into expressions that accurately predict the B-term band broadening in chromatographic columns. The expressions are written in such a form that they hold for both fully porous and porous-shell particles, and both spherical and cylindrical particles are considered. Mutually comparing the established EMT-expressions, it has been found that the most basic variant, i.e., the Maxwell-based expression, is already accurate to within 5% for the typical conditions encountered in liquid phase chromatography, independently of the exact microscopic morphology of the packing. For most typical values of the intra-particle diffusion rate and the species retention factors, it is even accurate to within 1%. If even higher accuracies are needed, more elaborate EMT-expressions are available. The modelling accuracy of all explicit EMT-expressions is much better than the residence time weighted (RTW) B-term expressions that have been used up to now in the field of chromatography, where the error is typically on the order of 10% and more. The EMT-models have also been used to establish expressions for the obstruction and tortuosity factor in packings of non-porous particles. The EMT has also been applied to the meso-porous zone only, yielding an expression for the intra-particle diffusion coefficient that can be used without having to specify any obstruction factor. It has also been shown that the EMT also provides a very simple but exact expression to represent the way in which the solid core obstructs the effective intra-particle diffusion in the case of porous-shell particles. This obstruction factor is given by γ(part)=2/(2+ρ³) for spherical particles and γ(part)=1/(1+ρ³) for cylinders. Back-transforming the obtained expressions, a set of simple explicit expressions has been obtained that allow to directly obtain the intra-particle diffusion coefficient (D(part)) from peak parking or B-term constant measurements. Using these expressions, it could be demonstrated that the traditionally employed RTW-model yields D(part)-values that display an erroneous retention factor dependency, even in cases where the RTW-model appears to be able to closely fit the peak parking measurements.
Vrije Universiteit Brussel, Department of Chemical Engineering, Pleinlaan 2, 1050 Brussels, Belgium. gedesmet@vub.ac.be
Gert
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Desmet
Sander
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Deridder
D000465:Algorithms; D055598:Chemical Phenomena; D002853:Chromatography, Liquid; D004058:Diffusion; D008962:Models, Theoretical; D016062:Porosity