Analytical solution of a two-dimensional model of liquid chromatography including moment analysis

Abstract

This work presents an analysis of a two-dimensional model of a liquid chromatographic column. Constant flow rates and linear adsorption isotherms are assumed. Different sets of boundary conditions are considered, including injections through inner and outer regions of the column inlet cross section. The finite Hankel transform technique in combination with the Laplace transform method is applied to solve the model equations. The developed analytical solutions illustrate the influence and quantify the magnitude of the solute transport in radial direction. Comparing Dirichlet and Danckwerts boundary conditions, the predicted elution profiles differ significantly for large axial dispersion coefficients. For further analysis of the solute transport behavior, the temporal moments up to the fourth order are derived from the Laplace-transformed solutions. The analytical solutions for the concentration profiles and the moments are in good agreement with the numerical solutions of a high resolution flux limiting finite volume scheme. Results of different case studies

Analysis of two‐dimensional models for liquid chromatographic reactors of cylindrical geometry

“…Our research group has solved analytically the linear one-dimensional (1D) and two-dimensional (2D) models of nonreactive and reactive chromatography. [11][12][13][14][15][16] The current work extends our previous analysis on 1D-GRM 17 to the analysis of linear two-component reactive 2D-GRM considering both axial and radial concentration gradients. Analytical solutions of the model for irreversible and reversible reactions are derived by applying the Hankel transformation, the Laplace transformation, the eigendecomposition technique, and the conventional solution technique for ordinary differential equations (ODEs).…”

Analytical solution of non-isothermal two-dimensional general rate model of liquid chromatography

“…Two specific conditions of injection are considered for triggering the radial mass and heat transfer effects. A new parameter r is introduced for dividing the inlet cross sectional area of the column into an inner cylindrical core and into an outer annular ring, see Qamar et al (2014 and David et al (2018). Thus, sample can be injected to the column either through an inner, through an outer ring, or through the whole cross section.…”

“…The latter case is possible when r is set equal to the radius of the column denoted by R c . The classical equation of mass balance for a single-solute in the bulk phase of the fluid is given as Qamar et al (2014, David et al (2018) In the equation above, c represents the solute concentration in the bulk phase of the fluid, q represents averaged concentration in the solid particles, b is the external porosity, u represents the velocity, D z denotes the dispersion coefficient in the axial direction and D r denotes the dispersion coefficient in the radial direction. The averaged concentration in the spherical solid particles of volume V p and radius R P is defined as where p is the internal particle porosity, c p is the solute concentration in the particle pores and q p is the local equilibrium concentration of solute in stationary phase.…”

“…The boundary conditions for Eqs. 17and 19, which correspond to the nonporous walls of the column and the symmetry of radial profile, are used along the radial coordinate at = 0 and = 1: 17and 19, we apply the Dirichlet and Danckwerts boundary conditions considering two types of injections (inner region and outer region injections) along the axial coordinate, see Qamar et al (2014 and David et al (2018) for more details. We give Danckwerts BCs and provide conditions under which these BCs reduce to Dirichlet BCs.…”

“…The derived solutions are transformed back to their original coordinates numerically because they involve some complicated functions (Durbin 1974;Rice and Do 1995). Keeping in view the usefulness of temporal moments Suzuki 1973;Miyabe 2009;Qamar et al 2014;Leweke and von Lieres 2016;Parveen et al 2016;David et al 2018), we numerically calculate the moments, as the analytical expressions of temporal moments are difficult to calculate. Analysis of the effects of different kinetic and thermodynamic parameters can be done by using these moments.…”