Page 661 - Bird R.B. Transport phenomena

§20.4 Boundary Layer Mass Transport with Complex Interfacial Motion 641

in which к is an average thickness of the concentration boundary layer. When Eq. 20.4-14
is written in terms of this new variable, we get

(20.4-15)
" 2 [dY 2 ° M
for ci) in terms of u, w, Y, and t. Since, on physical grounds, к will decrease with decreas-
A
ing 4b , the dominant terms for small 2) are those of lowest order in к—namely, all but
лв
AB
the В у and (V • n) contributions. The subdominance of the latter terms confirms the as-
o
ymptotic validity of approximations (ii) and (iii) in non-recirculating flows.
Now, the coefficients of all the dominant terms must be proportional over the range
of ЯЬ , in order that these terms remain of comparable size in the 5та11-2) лв limit. Such a
АВ
"dominant balance principle" was applied previously in §13.6. Here it gives the orders
of magnitude
ЯЬ /к 2 = and V IJ0 /K = (20.4-16,17)
АВ
for the terms of the lowest order with respect to к. Equation 20.4-16 is consistent with the
previous examples of \-power dependence of the diffusional boundary layer thickness
on % AB in free-surface flows. It also confirms the asymptotic correctness of assumption
(iv) for small values of % . Equation 20.4-17 is consistent with the proportionality of v*
AB
to ^/ЯЬ shown under Eq. 20.1-10 for the Arnold problem. Thus, the boundary layer
АВ
equation for to in either phase near a deforming interface is
A
(20.4-18)

to lowest order in к. At the next order of approximation, terms proportional to к would
appear, and these involve the tangential velocity уВц and the interfacial curvature
(V o • n). The latter term appears in Problems 20C.1 and 20C.2.
Multiplication of Eq. 20.4-18 by p/M A (a constant for the assumptions made here),
and use of z as the coordinate normal to the interface as in Example 20.1-1, give the cor-
responding equation for the molar concentration c {u, w, z, t)
A
dins
dt ' dt (20.4-19)
which allows convenient extension of several earlier examples. Another useful corollary
is the binary boundary layer equation in terms of x A and v*

(20.4-20)
dt dt dz f.
in which с and 9) have been treated as constants, as in Example 20.1-1.
AB

EXAMPLE 20.4-1 Equation 20.4-19 readily gives a generalized form of Eq. 20.1-65, by omitting the reaction
source term R A and neglecting the normal velocity term v z0 (thus assuming the interfacial
Mass Transfer with net mass flux to be small). The equation thus obtained has the form of Eq. 20.1-65, except
Nonuniform Interfacial that the total surface growth rate d In S/dt is replaced by the local growth rate, given by
Deformation д In s(u, w, t)ldt. The resulting partial differential equation has two additional space vari-
ables (u and w), but is solvable in the same manner, since no derivatives with respect to the
added variables appear.