Page 654 - Bird R.B. Transport phenomena

634 Chapter 20 Concentration Distributions with More Than One Independent Variable

according to Eq. A.7-16. The diffusion equation for the concentration boundary layer is then

where Eqs. A.7-15 and 17 have been used. In writing these equations it has been as-
sumed that: (i) the x- and z-components of the diffusion flux are negligible, (ii) the
boundary layer thickness is small compared to the local interfacial radii of curvature,
and (iii) the density and diffusivity are constant. We now want to get formal expressions
for the concentration profiles and mass fluxes for two cases that are generalizations of
the problems solved in §18.5 and §18.6. When we get the expressions for the local molar
flux at the interface, we will find that the dependences on the diffusivity ({-power in
§18.5 and the f-power in §18.6) correspond to cases (a) and (b) below. This turns out to be
of great importance in the establishment of dimensionless correlations for mass transfer
coefficients, as we shall see in Chapter 22.

Zero Velocity Gradient at the Mass Transfer Surface
This situation arises in a surfactant-free liquid flowing around a gas bubble. Here v x does
not depend on y, and v can be obtained from the equation of continuity given above.
y
Therefore, for small mass-transfer rates we can write general expressions for the velocity
components as
v x = v (x,z) (20.3-3)
5

where у depends on x and z. When this is used in Eq. 20.3-2, we get for the diffusion in
the liquid phase
]_^A__^A. ^ ^ л (20.3-5)
V =
which is to be solved with the boundary conditions

B.C.I: atx = 0, c A = 0 (20.3-6)
B.C. 2: at у = 0, c = c (20.3-7)
A A0
B.C.3: as у o o , CA -> 0 (20.3-8)
^
The nature of the boundary conditions suggests that a combination of variables treatment
might be appropriate. However, it is far from obvious how to construct an appropriate di-
mensionless combination. Hence we try the following: let c /c = fit]), where 17 = y/8 (x, z),
A A0 A
and 8 (x, z) is the boundary layer thickness for species A, to be determined later.
A
When the indicated combination of variables is introduced into Eq. 20.3-5, the equa-
tion becomes

with the boundary conditions: /(0) = 1 and /(00) = 0. If, now, the coefficient of the
7]{df/dj]) term were a constant, then Eq. 20.3-9 would have the same form as Eq. 4.1-9,
which we know how to solve. For convenience we specify the constant as

= 2 (20.3-10)

Next we insert the expression for у from Eq. 20.3-4 and rearrange the equation thus:

m
2
2
8
T" A + I T " (h-v,) 2 ]8 = — ^ - ^ (20.3-11)
A
dx A \dx - 5 / A v 5