Effectiveness Factor and Mass Transfer Coefficient in Wedge and Funnel Pores Using a Generalized Fick–Jacobs Model

Abstract

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.

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

“…The Equation (16) shows that a porous material of irregular shape increases the efficiency of an adsorption-desorption process by the combination of two factors. In first place, the corrugation and sinuosity of the pore make slower the effective diffusion process along the longitudinal direction, since the diffusion is reduced by a factor of D/D 0 compared with a cylindrical pore.…”

“…Traditionally, both factors can be taken into account in a single characteristic parameter of a chemical reactor called the Thiele modulus, which is defined as the square root of the ratio between the characteristic time of diffusion and the characteristic time of the adsorption-desorption kinetics [16] Φ 2 = characteristic time of diffusion characteristic time of adsorption .…”

“…Then, from Equation (16) in the stationary state the expression of the Thiele modulus for a cylindrical pore is (γ = 1):…”

“…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.…”

“…In those techniques, the quantities of flowing reagents and products are measured at the ends of the material. 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].…”