Beyond Sieverts' Law: A Comprehensive Model of Permeation In Palladium Membranes

There is much ongoing research on developing dense metal membranes based on palladium and its alloys for producing high purity hydrogen. Such membranes are almost exclusively permeable to hydrogen in a feed gas mixture via the following steps: s₁) dissociative adsorption on the surface, followed by: s₂) solution and then: s₃) diffusion of hydrogen atoms within the bulk metal matrix, and finally: s₄) evolution from the bulk to the surface, and: s₅) associative desorption on the permeate side. However, the hydrogen flux is widely described by the so-called Sieverts’ law, although there is much evidence that its application is limited to high temperatures and low hydrogen pressures, when diffusion (step, s₃) is controlling, and hydrogen concentration in bulk metal is low. We describe a comprehensive microkinetic model that includes all five steps in the general case. Using an electrical analogy and comparing the step resistances allows a systematic reduction to a three-step (adsorption, diffusion, and desorption control) model that adequately describes hydrogen permeation over a broad range of temperatures and pressures. Further, we describe the case of non-ideal hydrogen solution involving phase-transition within the membrane, which nicely explains the sharp peak at intermediate temperatures in hydrogen flux observed experimentally, but not yet explained theoretically. The model also includes metal surface roughness, not typically accounted for in the literature.