Effective approximations for concentration-polarization in Pd-membrane separators

• Approximate models for predicting the hydrogen flux of membrane separator are derived.
• The models combine ODE equations and algebraic relations to predict 2-D profiles.
• An analytical expression for Sherwood number is derived.
• The models are validated by comparison with experimental data and CFD simulations.
• The concentration polarization effect is relatively small in most cases.

Approximate models are proposed to simulate the transmembrane hydrogen flux in an empty membrane separator. The hydrodynamic field is constructed under the assumption of constant density with a single unknown parameter – the normal wall velocity (vw). The 2-D concentration profiles are derived using a known velocity distribution. The problem is closed by definition of vw via the transmembrane flux, which admits Sievert’s law. Such an approach allows to derive two approximate models that are governed by the set of ODE equations with respect to the average variables coupled with algebraic relations to describe the radial profiles. The first model accounts for analytical concentration profiles ci(z,r), the second one presents model reduction using the mass transfer coefficient (kc) expressed as Sherwood (Sh) number. An analytical expression for local Sh is derived (Sh = 6 in a tube).We identified a parameter Γ which represents the ratio of diffusive to permeating flux and suggest that for Γ > 6 the concentration polarization effect can be neglected. We address two axisymmetric geometries (i) a tube with transport at its wall, as is the case in a membrane of an integrated reactor; (ii) an annular cylinder with transport at an inner tube, as is the case in a separator. The proposed approximations are validated by comparison with experimental and CFD simulation data.