Fractal-Like Kinetic Models for Fluid–Solid Adsorption

Abstract

The journey of an adsorbate molecule from the bulk of the fluid phase to the active centre of the adsorbent particle is characterised by complex transport and capture mechanisms . These need to be described by accurate mathematical expressions for a proper design of a fluid–solid adsorption unit. In turn, this is a fundamental prerequisite to comply with strict emission limits of pollutants in fluid streams.

Consider a chemical process evolving in a system with homogeneous spatial distribution of reactants. Classical rate equations reported in literature include time-invariant kinetic constants and transport parameters. This is not appropriate for diffusion-limited heterogeneous processes. Actually, they take place into spaces where the reactants are constrained by phase boundaries or force fields such as it happens in porous solids. Rate coefficients can show temporal memories with a decrease in time that leads to the definition of fractal-like kinetic expressions. This scenario is likely to be applied when adsorption of a pollutant from fluid phase on a porous sorbent is considered.

In this article, we investigate and discuss adsorption from liquid phase. This with an original focus on interrelationships among pollutants and adsorbents characteristics, dynamic adsorption results and fractal-like modelling of a selection of experimental data present in literature.

Among all models taken into consideration, fractal pseudo-first-order was the most statistically accurate. The value of the hybrid fractional error function ranged between 3 × 10⁻⁴ and 1 × 10⁻². It resulted in general one order of magnitude lower than what obtained for a canonical pseudo-first-order model, that appeared to overestimate the values of degree of surface coverage by 10–20% under definite case-studies. Correspondingly this model predicted a faster attainment of equilibrium conditions, up to three-times earlier than experimentally recorded.

The fractal model involves the presence of an instantaneous rate coefficient. It ranged between orders 10⁻³ and 1 min⁻¹. Adsorbents showing different features have been analysed, for example with variations in particle size (order 100 μm), specific surface area (order 10² m² g⁻¹), total pore volume (order 10⁻¹ cm³ g⁻¹). For short adsorption times, the instantaneous rate coefficient was positively affected by (i) a decrease in particle size; (ii) an increase in specific surface area and total pore volume; (iii) an increase in specific mesopore volume; (iv) an increase in average pore size.

The heterogeneity parameter of the fractal model influences the time-decay rate of the former coefficient. This parameter ranged between 0.138 and 0.478, and it was higher when: (i) the pore space is more crowded by already-adsorbed molecules; (ii) a larger degree of surface chemical heterogeneity is determined; (iii) the mean micropore size is smaller; (iv) the specific volume of ultramicropores is larger; (v) the pore size distribution is more polydispersed. Higher values for this parameter means that the exploration of the intraparticle environment is more hindered for the adsorbate.