Page 662 - Bird R.B. Transport phenomena

642 Chapter 20 Concentration Distributions with More Than One Independent Variable

SOLUTION Rewriting Eq. 20.1-66 with a boundary layer thickness function 8(u, w, t) leads by analogous
steps to the relation

2
5(w,w?,f)= *®AB\ [s(u,w,t)/s{u,w,t)] dt (20.4-21)
V Jo
and the corresponding generalizations of Eqs. 20.1-71 and 72:
z
(20.4-22)
2
l[s{u,w,~t)/s(u,w,t)] d~t
_ \ - 1 / 2
' >
N ,o = CAO I ^ T ( 7 I [ s ( w w f)/s(w ' w > ^ d ~ f (20.4-23)
A
\ TTf \t J o )
These solutions, unlike Eq. 20.1-71 and Eq. 20.1-72, include the spatial variations of the
boundary layer thickness and interfacial molar flux N that occur in nonuniform flows.
Az0
Local stretching of the interface (as at stagnation loci) thins the boundary layer and enhances
N . Local interfacial shrinkage (at separation loci) diminishes N , but also ejects stale fluid
Az0 Az0
from the boundary layer, allowing its mixing into the interior of the same phase. Observa-
tions of mass transfer enhancement by such mixing have been interpreted by some workers
as "surface renewal/ 7 even though creation of new surface elements in an existing surface is
not permitted in continuum fluid mechanics.
These results, and others for negligible v , are obtainable conveniently by introducing
z0
the following new variables into Eq. 20.4-19:
=
Z = zs(u, w, t) and r • = s\u, w, w, t)dt (20.4-24,25)
s\u,
Jo Jo
In the absence of chemical reactions, the resulting differential equation for the concentration
function c (u, w, Z, T) becomes
A
5 = ^ǤJ (20.4-26)

This is a generalization of Fick's second law to an asymptotic relation for forced convection in
free-surface flows.

EXAMPLE 20.4-2 Show how to generalize Example 20.1-2 to flow systems, by using Eq. 20.4-26 for the two re-
action-free zones.
Gas Absorption with
Rapid Reaction and SOLUTION
Interfacial Deformation
Using Eq. 20.4-26, we get the following replacements for Eqs. 20.1-26 and 27:
^ = ЯЬ А5 д -Ц for 0 < Z < Z R (20.4-27)

z <
-~
S = % s § for ZR ~ °° (20 4 28)
Now the reaction plane z = z R of the original example is a time-dependent surface, Z = Z , or
R
z (u, w, t) = Z /s(u, w, t). The initial and boundary conditions remain as before, subject to this
R
R
generalization of the reaction-front location.
The solutions for c and c then take the forms in Eqs. 20.1-35 and 36, with z/Vf replaced
A B
by Z/VT, and z /Vt by Vy. The latter constant is again given by Eq. 20.1-37. The enhance-
R
ment of the absorption rate by the chemical reaction accordingly parallels the expressions that
will be given in Eq. 22.5-10, and simplified in Eqs. 22.5-11 through 13.