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.

In this work we study the diffusion–adsorption process in porous media and analyze the effect that the irregular geometry of the pores has on the efficiency of two types of adsorption processes: (a) when there is a net flux along the pore and (b) when the pore is completely saturated. In the first case, we measured the mass transfer coefficient, which is the constant of proportionality between the net flux and the difference of concentration. In the second case, we measure the effectiveness factor, which is the ratio between the actual rate of adsorption and the rate which would be achieved if the entire surface were at the same external concentration. In order to perform this analysis, we use a generalized Fick–Jacobs equation that considers the net effect of diffusion and adsorption along the direction of transport. For this analysis we have used wedge-shaped and conical pores, due to the simplicity of the treatment and their importance in the elaboration of a new brand of artificial materials. We have proved that the enhancement or diminution of the mass transfer coefficient or the effectiveness factor depend upon the specific rate of adsorption; therefore, they can be controlled using our model as a prediction tool in order to build artificial materials with a specific output flux of material. Additionally, our work allows to find how the Thiele modulus locally depends on the geometry of the pore for a linear reaction.

In this work we show how the effective diffusion coefficient of a membrane, in which heterogeneous reaction-diffusion processes are present, is related to the geometric irregularity of the pores and how it is affected by the surface diffusion of an adsorbed phase. A theoretical expression for this effective membrane diffusion coefficient is deduced starting from a recent generalization of the so-called Fick–Jacobs approximation. Our analysis comprises the interrelation of bulk and surface diffusion with the heterogeneous catalysis and adsorption/desorption interchange of matter between the bulk and the walls of the pore. Therefore, our theoretical framework is a very useful tool in modeling the process of heterogeneous catalysis inside porous materials when the shape of the pores is known. Through some illustrative examples involving the classical Langmuir processes as a reference, we provide a useful methodology correlating the local properties of transport inside a pore of irregular shape with the effective diffusion coefficient of a membrane. This coefficient can be measured experimentally by fitting the spatio-temporal concentration profiles in adsorption experiments. We show the different dependencies of this coefficient with the surface diffusion, the shape of the pore, and the average loading of adsorbed particles. A reverse methodology is also sketched, in which we suggest how to use the data emerging from experiments in order to determine the intensity of the surface diffusivity.

In this work, we provide a theoretical relationship between the spatial-dependent diffusion coefficient derived in the Fick-Jacobs (FJ) approximation and the macroscopic diffusion coefficient of a membrane that depends on the porosity, tortuosity, and the constriction factors. Based on simple mass conservation arguments under equilibrium as well as in nonequilibrium conditions, we generalize previous expressions for the effective diffusion coefficient of an irregular pore, originally obtained by Festa and d'Agliano for horizontal and periodic pores, and then extended by Bradley for tortuous periodic pores, to the case of pores with arbitrary geometry. Through a formal definition of the constrictivity factor in terms of the geometry of the pore, our results provide very clear physical interpretation of experimental measurements since they link the local properties of the flow with macroscopic quantities of experimental relevance in the design and optimization of porous materials. The macroscopic diffusion coefficient as well as the spatiotemporal evolution of the concentration profiles inside a pore have been recently measured by using pulse field gradient NMR techniques. The advantage of using the FJ approach is that the spatiotemporal concentration profile inside a pore of irregular geometry is directly related to the pore's shape and, therefore, that the macroscopic diffusion coefficient can be obtained by comparing the spatiotemporal concentration profiles from such experiments with those of the theoretical model. Hence, the present study is relevant for the understanding of the transport properties of porous materials where the shape and arrangement of pores can be controlled at will.

By using a mass balance equation, we deduce an effective equation for the concentration of adsorbed particles that considers the surface diffusion of the particles and an adsorption–desorption process inside a pore of irregular shape. This equation, together with the generalized Fick–Jacobs equation for the bulk diffusion, allows us to quantify the augment in the effective flux through a porous material due to the migration of the gas material along the surface. The equation for the surface concentration has a similar structure to the well-known Fick–Jacobs equation and it takes explicitly into account the shape of the pore through the width and length of the walls, making our model an important tool in the understanding of the interaction of diffusion and adsorption in porous materials where the length of the pores is greater than its width. In this work we predict the profile for the fractional surface coverage as a function of the geometry, the surface and bulk diffusion coefficients, and the isotherm of the process in several illustrative situations that permit us to prove that the effective diffusion coefficients augments with the surface diffusion, that the surface diffusion can give place to internal fluxes in opposite directions between bulk and surface particles, and finally that the diffusivity of adsorbed particles is greater in the narrow regions of the pore, in contrast with what happens to bulk particles. Our description predicts very interesting couplings between bulk and surface diffusion that occur locally and are regulated by the adsorption–desorption kinetics.