Page 582 - Bird R.B. Transport phenomena

562 Chapter 18 Concentration Distributions in Solids and in Laminar Flow
Fig. 18.6-1. Solid A dissolving into a falling film
of liquid B, moving with a fully developed para-
Near wall Parabolic bolic velocity profile.
velocity
v ~№)y profile of
z
fluid В
Insoluble
' wall

-Slightly soluble
wall made of A

с = saturation
А0
concentration
- c A (y, z)
=
c A

§18.6 DIFFUSION INTO A FALLING LIQUID FILM
(SOLID DISSOLUTION) 1
We now turn to a falling film problem that is different from the one discussed in the pre-
vious section. Liquid В is flowing in laminar motion down a vertical wall as shown in
Fig. 18.6-1. The film begins far enough up the wall so that v depends only on у for z > 0.
z
For 0 < z < L the wall is made of a species A that is slightly soluble in B.
For short distances downstream, species A will not diffuse very far into the falling
film. That is, A will be present only in a very thin boundary layer near the solid surface.
Therefore the diffusing A molecules will experience a velocity distribution that is charac-
teristic of the falling film right next to the wall, у = 0. The velocity distribution is given
in Eq. 2.2-18. In the present situation cos 0 = 1, and x = 8 — y, and

(18.6-1)

At and adjacent to the wall (y/8) 2 « (y/8), so that for this problem the velocity is, to a
very good approximation, v z = (pg8//x)y = ay. This means that Eq. 18.5-6, which is ap-
plicable here, becomes for short distances downstream

*£-9»^ 08.6-2,

where a = pg8//x. This equation is to be solved with the boundary conditions
B.C.I: at 2 = 0, c A = 0 (18.6-3)
B.C. 2: at у = 0, c = c (18.6-4)
A A0
B.C.3: aty=°o, c = 0 (18.6-5)
A
In the second boundary condition, c A0 is the solubility of A in B. The third boundary con-
dition is used instead of the correct one (дс /ду = 0 at у = 8), since for short contact
А
times we feel intuitively that it will not make any difference. After all, since the mole-

1
H. Kramers and P. J. Kreyger, Chem. Eng. Sci., 6,42-48 (1956); see also R. L. Pigford, Chem. Eng.
Prog. Symposium Series No. 17, Vol. 51, pp. 79-92 (1955) for the analogous heat-conduction problem.