A revisit on intraparticle diffusion models with analytical solutions: Underlying assumption, application scope and solving method

The intraparticle diffusion plays a critical role in the adsorption of aqueous contaminants in porous adsorbents. The identification of the rate-controlling step through the intraparticle diffusion models is extremely essential to optimize the process parameters and interpret the mass-transfer mechanisms. However, with the propagation of these models, some mistakes and controversies also come along, which are bound to have an adverse impact on the future research and even draw incorrect conclusions. This review summarizes the widely used intraparticle diffusion models with analytical solutions in the literature, including their underlying assumptions, curve characteristics, application scopes and solving methods. As a close approximation of the Boyd model, the rationality of the Reichenberg and Vermeulen models is analyzed in detail. The improper forms of the Bangham–Sever and fractal-like Vermeulen models are corrected. The fractal-like kinetics provides new insights into the diffusion mechanisms in a heterogeneous system. MATLAB software can be conveniently used to solve the intraparticle diffusion models, in particular the Boyd model with the infinite series. This review also addresses some controversies and thereby prevents their further propagation in the future research.