Generalized Fick–Jacobs Approach for Describing Adsorption–Desorption Kinetics in Irregular Pores under Nonequilibrium Conditions

Abstract

We present a study exploring the range of applicability of a generalized Fick-Jacobs equation in the case when diffusive mass transport of a fluid along a pore includes chemical reactions in the bulk and pore's surface. The study contemplates nonequilibrium boundary conditions and makes emphasis on the comparison between the predictions coming from the projected Fick-Jacobs description and the corresponding predictions of the original twodimensional mass balance equation, establishing a simple quantitative criterion of validity of the projected description. For the adsorption-desorption process, we demonstrate that the length and the local curvature of the pore are the relevant geometric quantities for its description, allowing for giving very precise predictions of the mass concentration along the pore. Some schematic cases involving adsorption and chemical reaction are used to quantify with detail the concentration profiles in transient and stationary states involving equilibrium and nonequilibrium situations. Our approach provides novel and important insights in the study of diffusion and adsorption in confined geometries.

Review on the Macro-Transport Processes Theory for Irregular Pores able to Perform Catalytic Reactions

“…The detailed analysis on the properties of the bulk and surface contributions of the total diffusion current at the boundaries, is provided in Ref. [21].…”

“…Here, we have introduced the transversal area of the pore A(z) ≡ πR 2 (z). For non-twisted and large enough (L >> R) pores we can assume C(R, z, t) C b (z, t), meaning that homogeneity along the radial direction is rapidly reached [21]. Thus, after integrating and performing some algebraic manipulations at the right hand side of Equation 7, we obtain…”

“…Assuming that the adsorbed layer does not modify in an noticeable extent the available volume of particles diffusing in the bulk, one can proceed along the same lines of the last section in order to obtain (see Ref. [21] for details)…”

“…The present article generalizes to three dimensionsa novel approach to this problem that was recently formulated for the case of non-regular domains in two dimensions [16,18,21,30,31]. In this alternative or generalized macrotransport process theory, the non-regularity of the geometry of the pores is explicitly taken into account when the description is contracted.…”

“…These properties have to reflect several processes taking place at the micro-scale, defined in terms of the transversal characteristic length of the pores, the real length and other internal properties [15][16][17][18][19][20]. The relation between the microscopic details of the shape of the pores and the longitudinal effective mass transport across porous materials can be established accurately when the pores are long enough compared with their average width [21]. When this assumption holds, the microscopic scale description based on the three-dimensional diffusion equation can be simplified onto a coarse grained description along the main transport direction, that is, onto a one dimensional diffusion equation having effective transport coefficients [22][23][24][25].…”