Approximate models of concentration-polarization in Pd-membrane separators. Fast numerical analysis

•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.

Several approximate solutions are derived to predict the concentration polarization effect in a shell-and-tube Pd-membrane empty separator with permeate flow either in the tube or in the shell. The first two approximations (Nekhamkina and Sheintuch (2015) [22]) are governed by ODEs describing the axial profiles of the average variables coupled with algebraic relations that describe the radial profiles of velocity components and species concentrations. The hydrogen membrane flux is approximated by a mass transfer coefficient k=D(∂cH2/∂r)w/(〈cH2〉−cH2w) expressed via the Sherwood number Sh=kd/D which is a geometry-dependent parameter (D is hydrogen diffusivity, cH2w=pH2,wret/RT is its membrane wall concentration and d a characteristic length). Here Sh is obtained by 2-D CFD simulations in a wide range of parameters and when plotted vs the separator length, it reaches an asymptotic value that matches the predicted values and is independent of operating conditions.The third approximation reduces the polarization effect to an algebraic equation by approximating the effectiveness factor η  =[(pH2,wret)0.5−(pH2per)0.5]/[〈(pH2ret)〉0.5−(pH2per)0.5] which depends mainly on the ratio of the mass-transfer coefficient to the membrane permeance parameter Γ=Γ(Sh). This expression can be used for fast design of separators that avoids or accounts for polarization effects. The three approximations offer a trade-off between accuracy and ease of application.