Green's Function and Eigenvalue Representation of Time Lag for Absorptive Permeation Across a Heterogeneous Membrane

Abstract

The time- lag problem is treated for absorptive penetration across a heterogeneous membrane, where both the diffusivity D( x) and the partition coefficient K( x) depend on the coordinate x ( 0. x. h), with 0 and h being the coordinates of the upstream and downstream faces, respectively. A new concept of time- lag distribution is introduced, and the first ( time) moment t d1_ L and the second ( time) moment t d2_ L over this distribution are also difined and treated in the Lapalce domain in conjuction with the Green's function G( x, y), and eigenvalues associated with the time- independent diffusion equation subject to the absorbing boundary condition at both ends of the membrane. Our central results include t d1_ L = d h 0 Gdx, x_dx = X i 1.i and t d2_ L = d h 0 d h 0 Gdx, y_Gdy, x_dxdy = X i 1.i 2 + X i 1.2i, where.i is the ith eigenvalue of the aforementioned diffusion equation. The merits of these new resprentations and comparison with the treatments of Frisch or Eyring are also discussed.