Numerical approximation of a two-dimensional non-equilibrium model of gradient elution liquid chromatography under various gradient models and isotherms

Gradient elution chromatography is an effective separation technique for resolving complex mixtures with improved resolution and reduced analysis time compared with isocratic operation. In this study, an isothermal two-dimensional non-equilibrium kinetic model is developed for liquid chromatography under solvent gradient conditions. The model incorporates axial convection, axial and radial dispersion, nonlinear adsorption equilibria, and an additional governing equation for the spatial and temporal evolution of mobile-phase composition, yielding a strongly coupled system of nonlinear partial differential equations. Three adsorption isotherms, namely Langmuir, Bi-Langmuir, and Töth, are employed to describe nonlinear equilibrium behavior, while three gradient models, namely the QSS, Neue-Kuss, and power-law models, are used to represent solvent-gradient dynamics. The governing equations are solved numerically using a semi-discrete high-resolution finite volume scheme for accurate approximation of steep concentration fronts. The effects of gradient model, modifier concentration, adsorption isotherm, gradient start and end times, gradient duration, and key transport parameters on the elution behavior of a bi-component mixture are analyzed systematically. The results show that the proposed two-dimensional model correctly captures the influence of solvent-gradient dynamics and adsorption nonlinearity on peak shape, migration behavior, and separation performance and additionally resolves radial concentration distribution and injection effects that one-dimensional models cannot represent.