Fractal continuum model for the adsorption-diffusion process

Abstract

In this work, we present a mathematical model to describe the adsorption-diffusion process on fractal porous materials. This model is based on the fractal continuum approach and considers the scale-invariant properties of the surface and volume of adsorbent particles, which are wellrepresented by their fractal dimensions. The method of lines was used to solve the nonlinear fractal model, and the numerical predictions were compared with experimental data to determine the fractal dimensions through an optimization algorithm. The intraparticle mass flux and the mean square displacement dynamics as a function of fractal dimensions were analyzed. The results suggest that they can be potentially used to characterize the intraparticle mass transport processes.The fractal model demonstrated to be able to predict adsorption-diffusion experiments and jointly can be used to estimate fractal parameters of porous adsorbents.Av. Playa pie de la cuesta 702, Desarrollo San Pablo, Querétaro, Qro. 76125, Mexico, (+52) 4425590385, erik.herrera@cidesi.edu.mx dimension. Segars and Piscitelle (1996) used fractal theory to generalize the BET model by proposing a power-law parameter that quantifies the surface roughness. They interpret equilibrium data finding that the isothermic adsorption strongly depends on the fractal dimension, where 2 D > . Following the same idea, the effect of fractal dimension on the adsorption process was analyzed, which allowed for modifications to the Brunauer-Emmett-Teller equation to improve predictions (Aguerre et al., 1996, Khalili et al., 1997. Other work focused on improving the isotherm predictions was made by Kanô et al. (2000) who presented a model that generalizes both the Langmuir and Freundlich isotherms. Such a model is based on the fact that the adsorbing surface depends on the amount of adsorbed mass according to A M ζ ∼ , where / 3 D ζ = quantifies the irregularities of a presumable self-similar fractal surface. Along the same line of ideas, in (Wang et al., 2007, Longjun et al., 2008, and Selmi et al., 2018, the authors present fractal models through the modification of the classical equilibrium isotherms. In such works, they discuss and apply a fractal adsorption model obtained from the Langmuir and kinetics model where the geometric characteristics of the solute are linked to the surface's fractal dimension. Additionally, they report a power-law dependence of the effective reaction order with time.On the other hand, dynamic models to interpret the adsorption-diffusion process scarcely have changed from the classical descriptions commonly used for kinetic models (Qiu et al., 2009, Foo and Hameed, 2010, and Montagnaro and Balsamo, 2014 or diffusional models (Leyva-Ramos and Geankoplis 1994, Ocampo-Perez et al., 2010, Ocampo-Perez et al., 2013and Ocampo-Pérez et al., 2017. Such approaches do not consider long-term correlations and long-range interactions appearing in diffusion processes occurring on complex domains such as self-similar (or fractal) structures. The work of Sakaguchi (2005) inc...

Paint adsorption on walls: I. Fractal-Stochastic model

“…subject to the initial condition: … (7) being the ascent -descent operator that acts on the functions of discrete variables. If it is taken into account that when an individual microscopic process takes place, the change that occurs is negligible compared to then can be considered as a continuous variable, obtaining the Fokker-Planck equation:…”

“…In the microscopic scale, the processes of adsorption and desorption occur randomly [7][8], which is manifested in the irregular patterns that form the adsorbate on the surface. These patterns are complicated to describe using Euclidean geometry, so it is more plausible to use fractal geometry [7].…”

“…In the microscopic scale, the processes of adsorption and desorption occur randomly [7][8], which is manifested in the irregular patterns that form the adsorbate on the surface. These patterns are complicated to describe using Euclidean geometry, so it is more plausible to use fractal geometry [7]. Adsorption-desorption processes are highly relevant in the film fixing industry, obtaining materials, inorganic and organic synthesis, among others [8][9][10].…”

Modelación matemática del transporte de fluidos bifásicos a través de medios fracturados: sistema de placas paralelas irregulares

“…(2) donde D β z es el operador que representa la derivada fraccional de orden igual a la dimensión fractal β del contorno longitudinal al flujo y que se relaciona con la morfología del poro (Herrera-Hernández et al, 2019;Suárez-Domínguez et al, 2020). s es un parámetro que se identifica con el grado de simetría del perfil de velocidad y su valor se encuentra entre 0 y 1.…”