Theoretical Investigation of Liquid Chromatography in Cylindrical Columns

Abstract

Liquid chromatography is one of the most widely used separation and puri¯cation tech- niques utilizing cylindrical columns. Generally, one-dimensional chromatographic models are used to study concentration gradients along the column axis only, while gradients along the radial direction are ignored. However, consideration of radial concentration gradients are necessary when sample injection at inlet of the column is imperfect, that is when a radial pro¯le is inserted at the column entrance, and radial dispersion is rate limiting. In such a scenario, the two-dimensional (2D) models of chromatography have to be consid-ered. This work focuses on the derivation of analytical solutions and temporal moments of linear 2D models of liquid chromatography in cylindrical geometry. In liquid chromatogra-phy, the injected sample size is small and diluted, thus, the current assumptions of linear adsorption and reaction in the models are valid and practically applicable. The models are formed by convection-dominated partial di®erential equations coupled with some algebraic and di®erential equations. Both non-reactive and reactive chromatographic models, two sets of boundary conditions, as well as injections through inner and outer regions of the column inlet cross section are considered. The Laplace and Hankel transformations are successively applied to solve the model equations. To further analyze the process, tempo-ral moments up to the fourth order are derived from the Hankel and Laplace transformed solutions. The correctness of analytical results are veri¯ed by comparing them with the numerical solutions of a high resolution ¯nite volume scheme. Results of di®erent test prob-lems are presented and discussed covering wide ranges of kinetics in°uencing mass transfer and reaction rates. The developed analytical solutions and moments provide useful tools to quantify combined longitudinal and radial dispersion e®ects, to perform sensitivity anal-ysis, to analyze numerical algorithms, as well as to understand, design and optimize the process.