Pressure-dependent hydrogen permeability extended for metal membranes not obeying the square-root law

Hydrogen permeability of metal membranes is generally defined by the square-root law, as the proportional coefficient of permeation flux to the square-root difference of the pressures on both sides of the membrane. However, deviation from the law has been widely reported for palladium, niobium, etc. Although n-th power instead of the square root has often been employed to determine permeability for these membranes, it has no theoretical base. These approaches do not consider concentration dependency of hydrogen diffusivity in the membrane. This study theoretically extended the definition of permeability by taking it into account, where square root of pressure was used throughout. The resultant permeability depended on pressure. This approach had the following four characteristics. First, the permeability could be qualitatively linked with pressure-dependent solution and diffusion coefficients. For this purpose, the solution coefficient was also extended from Sieverts' law. Second, the permeability could be easily evaluated from permeation flux dependent on feed-side pressure, usually measured in membrane study. Third, this approach enabled comparison of permeation ability irrespective of obeying permeation law. Fourth, permeation flux could be estimated for any pressure conditions visually and analytically. Thus, analytically estimated values were more precise than those using the conventional square-root law. These characteristics are successfully demonstrated using experimental results obtained not only for a palladium membrane in this study but also for palladium and niobium membranes in the literature.