Analysis of transient permeation fluxes into and out of membranes for adsorption measurements

Abstract

A relationship between the integrated fluxes into and out of a membrane following a positive or negative step change in feed concentration was derived. This analysis allows adsorption isotherms in the transport pathways through membranes to be determined from transient permeation responses to step concentration changes in the feed without measuring the retentate response. For Fickian diffusion through a membrane with Langmuir adsorption and zero coverage at the permeate boundary, the difference between the time-integrated flux into and flux out of the membrane is shown to be three times the time-integrated difference between the steady-state flux and the flux out. For Maxwell–Stefan diffusion, this ratio of integrated flux differences is 3 at low coverages and decreases towards 2 at saturation coverage. Mass transfer resistance at the permeate boundary increases the Fickian ratio above 3, and the ratio increases with decreasing Sherwood number. The ratio of integrated flux differences is shown to be identical to the steady-state replenishment time divided by the time lag. Thus, the ratio can be calculated directly from the steady-state concentration profile and the concentration-dependent diffusion coefficient. Surface diffusion through zeolite membranes was analyzed to demonstrate the calculation of the flux relationship for specific adsorption and diffusion models, but the method developed can be applied to membrane permeation in general.