Pressure-Dependent Hydrogen Permeability Extended for Metal Membranes Not Obeying the Square-Root Law

Abstract

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.

Application of Extended Permeability to a Thick Palladium Membrane

“…The pressure dependency is the smaller at the lower pressures and the higher temperatures. These characteristics are the same as those for the 50 µm membrane reported previously [6].…”

“…These definitions are explained in detail in a previous report [6]. Conventional square-root law, power law and extended law in this study are compared in Table 1 On the other hand, the power law can often describe permeation behavior more precisely than the square-root law.…”

“…However, thus determined permeability is usually difficult to analyze connecting diffusion and solution behavior due to lack of theoretical base. The permeability could not be compared directly with other permeability with different n. The extended law can deal with the power law also, that is, the power low can be converted to the extended law using the following formula [6]: …”

Developments in the Ni–Nb–Zr amorphous alloy membranes

“…Summary of reported hydrogen permeability values for Ni-based amorphous alloy membranes and selected Pd-based alloys as reference materials equilibrium hydrogen concentration did not obey Sieverts law, however, and was nearly linear with the quarter power determined using Kirchheim theory. Hara et al [39] established a theory of permeability as a function of hydrogen pressure and developed equations for pressure dependent permeability from permeation tests. They also reported that the dissolution of hydrogen did not obey Sieverts' law (i.e., proportional to P 0.5 ).…”

Consistent description of hydrogen permeation through metal membrane based on hydrogen chemical potential and its application to alloy design

“…To understand these exceptional cases, Hara et al have recently proposed a concept based on the pressure dependent hydrogen permeability. [17][18][19] It is well known that the gradient of hydrogen concentration, dc/dx, is not always the driving force for hydrogen diffusion. Strictly speaking, the hydrogen diffusion is driven by the gradient of the hydrogen chemical potential, dl/dx.…”