Role of fractional derivatives in the HPLC process: General rate model

This paper investigates solute transport in packed-bed chromatographic columns and models it using the General Rate Model (GRM), which accounts for axial dispersion, interphase mass transfer, intraparticle diffusion, and adsorption. Conventional integer-order models often fail to capture memory-dependent transport in complex porous media. In this study, a time-fractional one-dimensional GRM with the Caputo fractional derivative (CfD) was utilized to examine how fractional dynamics affect high-performance liquid chromatography (HPLC) transport and concentration profiles. CfD represents history-dependent transport arising from multiscale pore structures, tortuosity, and delayed diffusion processes within the stationary phase. Analytical and high-precision finite-difference numerical methods are used for validation and parametric analysis. Fractional model results show delayed front propagation, enhanced memory effects, and significant internal concentration dynamics. Quantitatively, reducing the fractional order from α = 1.0 to α = 0.5 delayed solute-front propagation by approximately 20–30%, while the L2-1σ scheme reduced the early-time L 2 error relative to the L1 approximation to about 0.05, confirming that fractional memory strengthens axial retardation, slows radial intraparticle penetration, and improves the reliability of internal concentration-field interpretation. When complete elution is not achieved, base-case simulations show that internal axial and radial concentration fields provide richer insights than outlet breakthrough curves. Boundary-condition analysis shows that the inlet formulation has only a weak effect on the solution, while mass-balance diagnostics indicate that conservation is a numerical challenge. A comparison of L 1 and L 2 − 1 σ discretization schemes shows that the latter offers a more reliable reference solution with comparable efficiency. Parametric analysis shows that superficial velocity accelerates transport, particle radius increases intraparticle resistance, and injection conditions mostly affect response magnitude and timing. Fractional order affects transport retardation, front spreading, and numerical robustness the most. This time-fractional GRM provides a physically meaningful and mathematically flexible framework for non-classical chromatographic transport analysis, underscoring the need for better conservation-based treatment, longer simulation horizons, and validated process design and optimization.