Effect of flux direction through supported metal membranes: Golden ratio as maximum benefit in pure hydrogen and concept of swap point in mixture

In this paper, the role of the flux direction in composite supported metal membranes for hydrogen purification is elucidated from both mathematical and physical point of view under pure-hydrogen and mixture conditions considering a double-layer supported Pd-based membrane. A definitive mathematical proof is given to demonstrate that, in pure-hydrogen conditions, if the metal layer obeys Sieverts' law and the diffusion in the support is controlled by the Knudsen mechanism, the permeating flux measured by feeding hydrogen from the support to the metal layer is always higher than the flux measured in the opposite direction at the same operating conditions and for whatever values of layer permeances, extending the state-of-the-art findings of the specialized literature. As a further surprising result, it is found that the maximum benefit in switching the flux direction is equal to the mathematical number called golden ratio, this representing a notable general finding. Moreover, a partial generalization of these results is extended to other particular driving forces, which leads us to propose the nondemonstrated-yet general conjecture that the highest flux be the flux established from the layer with the highest driving force pressure exponent to the layer with the lowest one. Furthermore, observing that, in mixture conditions, the permeating flux from the support side is lower than the opposite one for a sufficiently high mixture composition of non-permeating species - especially in the presence of inhibiting species like CO - the novel concept of swap point is introduced, which represents the mixture composition at which the fluxes measured in the two directions are equal. The value of the swap point depends on membrane geometrical characteristics, temperature and non-permeating species. It is highlighted that the findings of the present work hold also for other types of composite ones (polymeric, ceramic, etc.) as well as for every physical system that whose behaviour can be modelled by a sequence of non-linear resistances.