Page 127 - Separation process principles 2

92 Chapter 3 Mass Transfer and Diffusion

From (3-1 ll), NSh,,, = 3.414, which is the smallest value The error function is defined as
that the Shenvood number can have for a falling liquid film.
The average mass-transfer flux of A is given by

Using the Leibnitz rule with (3-116) to differentiate this in-
tegral function,
For values -q < 0.001 in (3-loo), when the liquid-film I
flow regime is still laminar without ripples, the time of con-
tact of the gas with the liquid is short and mass transfer is
confined to the vicinity of the gas-liquid interface. Thus, the Substituting (3-119) into (3-117) and introducing the Peclet
film acts as if it were infinite in thickness. In this limiting number for mass transfer from (3-102), we obtain an expres-
case, the downward velocity of the liquid film in the region sion for the local mass-transfer coefficient as a function of
of mass transfer is just uym,, and (3-96) becomes distance down from the top of the wall:

a CA a2CA
~y,, - DAB - (3-1 14)
=
a~ az2 snys (3-120)
Since from (3-90) and (3-91), u,,, = 3iiy/2, (3-114) can be
rewritten as The average value of kc over the height of the film, L, is ob-
tained by integrating (3-120) with respect to y, giving
(3-115)
(3-121)
where the boundary conditions are
Combining (3-121) with (3-112) and (3-102),
CA = CA~ for z > 0 and y > 0
CA = CA, for z = 0 and y > 0 = /
s = 11-12?)
CA = C& for large z and y > 0
Equation (3-1 15) and the boundary conditions are equivalent where, by (3-108), the proper mean to use with kc,% is the log
to the case of the semi-infinite medium, as developed above. mean. Thus,
Thus, by analogy to (3-68), (3-73, and (3-76) thesolution is
(cAi - CA)rnean = (cAi - ?A)LM
(3-123)
E=I-O= CA, - CA erf z (3-116) - (CA, - CAO) - (CA~ - CAL)
-
CA, - CA~ 2 ln[(c~, - CA~)/(CA, - ;A,>]
Assuming that the driving force for mass transfer in the film When ripples are present, values of kc=,, and NSh,,, can be
is CA, - c&, we can use Fick's first law at the gas-liquid considerably larger than predicted by these equations.
interface to define a mass-transfer coefficient: In the above development, asymptotic, closed-form solu-
tions are obtained with relative ease for large and small
values of q, defined by (3-100). These limits, in terms of the
=kc(c~,-c~,) (3-117)
average Sherwood number, are shown in Figure 3.13. The

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L I ( 1 1 11111 I I 1 1 1 1 1 1 1 I 1 1 1llll1 I I I I I Ilr
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- -
-
- -
- -
-
-
-
-
-
- -
-
- -
- -
-
Long residence-time solution-
- -
Eq. (3-1 11)
I I 1 111111 I I 1 1 1 1111 I I 1 111111 I I 111111.
10
Figure 3.13 Limiting and general solutions for
mass transfer to a falling, laminar liquid film.