Two-dimensional non-equilibrium model of gradient elution chromatography incorporating dual-mode gradients and core-shell particles.

A two-dimensional non-equilibrium (lumped kinetic) model is developed for non-isothermal, dual-mode gradient elution chromatography in columns packed with core-shell particles. The model couples mobile- and stationary-phase mass balances, fluid- and solid-phase energy balances, and a solvent volume fraction transport equation with a competitive multicomponent Langmuir isotherm, yielding nonlinear, convection-dominated partial differential equations. Three retention models, linear solvent strength (LSS), quadratic solvent strength (QSS), and Neue-Kuss, are incorporated with van't Hoff temperature dependence, and a dual-mode gradient programs mobile-phase composition and column temperature as piecewise-linear profiles. Sample is injected through the inner cylindrical inlet zone, resolving radial dispersion explicitly. The equations are solved by a high-resolution finite volume scheme with the Koren flux limiter and a second-order total-variation-diminishing Runge-Kutta integrator. Parametric studies relate core-radius fraction, adsorption enthalpy, isotherm nonlinearity, transfer coefficients, Péclet numbers, and gradient timing to cut time, cycle time, productivity, and yield. The core-radius fraction is the dominant design variable: an intermediate value (ξcore≈0.4) roughly halves analysis time while preserving resolution. QSS provides the fastest elution and highest productivity, whereas the Neue-Kuss model provides the broadest preservation of selectivity. The gradient start time is an effective control parameter for separation performance, whereas variations in the gradient end time have a comparatively limited effect. The results guide intensification of dual-mode gradient protocols in preparative high-performance liquid chromatography.