Analytical Solution and Temporal Moments of the Tempered Prabhakar Fractional General Rate Model for Linear Liquid Chromatography

Abstract

Abstract Intraparticle transport in packed adsorption and chromatography columns is routinely represented by Fickian pore and surface diffusion, yet disordered pore networks and energetically heterogeneous surfaces frequently produce transport with long-lived memory that a single diffusivity does not capture. We formulate a general rate model (GRM) in which the intraparticle accumulation term carries a tempered−Prabhakar fractional operator, and we solve the single-solute linear problem in closed form in the Laplace domain for Danckwerts and Dirichlet inlet conditions under both pulse and step injection. The complete set of temporal moments follows analytically through a cumulant construction, and the first four central moments are reported explicitly in terms of the fractional parameters and the transport groups. Time-domain band profiles are recovered by the de Hoog−Knight−Stokes algorithm. The exponential tempering restores finite moments that the untempered operator forfeits, and the closed-form moments supply a Pearson realizability criterion, κ ≥ S2 − 2, that separates parameter sets producing physically positive elution from those that do not. Setting μ → 1 with γ = 0 recovers the classical linear GRM, and the reduction reproduces published moment values to within 0.13%. A parameter study over eighteen cases quantifies how the operator reshapes retention, plate count, peak asymmetry, and the plate-height dependence on velocity. Fitted to measured butanol breakthrough on two activated carbons, the fractional model returns μ = 1.000 for Norit ROW 0.8, for which no additional memory contribution is supported, and μ = 0.874, 0.902, and 0.924 for AC F-400 at three flow rates, for which the fractional model is favored by the Akaike information criterion. The Laplace-domain construction is meant to serve, further out, as scaffolding for fractional formulations that bring in energy balances and nonlinear adsorption equilibria.

The authors' abstract, as published at the source. Analytical Chemistry, 2026 · DOI ↗