On the apparent dispersion coefficient of the equilibrium dispersion model: An asymptotic analysis

Abstract

To model chromatography, researchers have developed several approaches. These cover a broad range of applications and, depending on the assumptions adopted, have different levels of accuracy. In general, the most suitable modelling approach is the simplest that can describe a process with the desired accuracy. A model that often meets this criterion is the equilibrium dispersion model (EDM). This features one mass balance equation per analyte, including an axial dispersion term, and assumes the analyte concentrations in the mobile and stationary phases to be in local equilibrium. To account for the finite mass transfer rate between the phases, the model employs an apparent dispersion coefficient. Two expressions are available for this coefficient, one being used much more frequently than the other. In this paper, we aimed to clarify which one should be favoured. A desirable feature of simple models is that they can be derived from more general ones with appropriate physical assumptions and rigorous mathematical methods. Thus, to answer our research question, we derived the EDM from the more general pore diffusion model (POR), using an asymptotic method. The expression obtained for the apparent dispersion coefficient does agree with one of the two reported in the literature – the less frequently used. To test the validity of this expression, we simulated elution profiles using the two versions of the EDM and compared the results against those from the POR model. The simulations were conducted in the range where the POR and EDM models should be essentially equivalent, their results confirming the outcome of the asymptotic analysis. This work offers a solid theoretical grounding for the EDM, clarifies which formulation of the model is correct, and provides usable applicability conditions for the model.