Sieverts law empirical exponent for Pd-based membranes: critical analysis in pure H2 permeation

In this paper, the physical meaning of the Sieverts-type driving force exponent n is analyzed for hydrogen permeation through Pd-based membranes by considering a complex model involving several elementary permeation steps (adsorption on the membrane surface on the feed side, desorption from the surface on the permeate side, diffusion through the metal lattice, and the two transition phenomena surface-to-bulk and bulk-to-surface). First, the characteristic driving force of each step is evaluated, showing that adsorption and desorption singularly considered and the adsorption and desorption considered at the same time are characterized by driving forces depending on the ratio of feed and permeate hydrogen pressure. On the contrary, the diffusion step is found to present a driving force that is composed of two terms, one which corresponds to the original Sieverts law (with an exponent of 0.5) and the other which is the product of the pressure difference and a temperature-dependent factor. Then, the characteristic n is evaluated by applying the multistep model to two different membranes from the literature in several cases, (a) considering each permeation step as the only limiting one and (b) considering the overall effect of all steps. The results of the analysis show that for a low temperature and thin membrane thickness, the effect of the surface phenomena is, in general, a decrease of the overall exponent n toward values lower than 0.5, even though, under particular operating conditions, the n theoretical value of the surface phenomena is equal to unity. At a higher temperature and thickness (diffusion-controlled permeation), n tends to 0.5, even though the rapidity of this tendency depends strictly on the membrane diffusional parameters. In this frame, the expression developed for the diffusion step provides a theoretical reason why n values higher than 0.5 are found even for thick membranes and high temperature, where diffusion is the only rate-determining step.