FEMLAB - KTH

conditions because there are no concentration gradients in the direction of their
normals. Hence, for any species φi at those boundaries, the flux is given by:
n · (D ▽ φi) = 0 (2.9)
where n is the normal to the boundary.
So far, I have mentioned only external boundaries. I shall consider internal
boundaries now: The first internal boundary from the left is that between the rectangular
subdomain and the curved boundary. In reality, the whole domain is continuous
here, but I decided to split them because of the different scaling factors
applied in the rectangular subdomain in the radial direction. To avoid polar coordinates,
and taking advantage of the fact that there are no concentration gradients
in the θ-direction, I have straightened that subdomain into a rectangle, as opposed
to its original arc shape. Then I introduced coupling variables between its right
boundary and the left boundary of the remaining part of the geometry.
Generalising, if a certain species φi only stays within a certain subdomain Ωi
and does not diffuse through, then (2.9) holds for that species. However, if the
species diffuses through the boundary separating say Ω1 and Ω2, then the flux will
be a function of φ1 and φ2 at the boundary. Moreover, if the partition coefficient for
φ is Kp between Ω1 and Ω2, where Kp < 1, then the flux into Ω1 is given by
n · D ▽ φi = M (φ2 − Kpφ1) (2.10)
and that into Ω2 through that boundary is simply the negative of the flux into Ω1. M
is the mass transfer coefficient, and it is a measure of the resistance to the transport
of any given species between the two given subdomains. A high M implies a small
resistance to mass transfer, and vice-versa.
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