Page 697 - Bird R.B. Transport phenomena

§22.2 Analytical Expressions for Mass Transfer Coefficients 677
Then, when the characteristic area is chosen to be the area of the interface WL, we see
that
\
KP^ABJ )
= 1.128(ReSc) 1/2 (22.2-2)

This equation expresses the Sherwood number (the dimensionless mass transfer coeffi-
cient) in terms of the Reynolds number and the Schmidt number, with Re defined in
terms of the maximum velocity v max in the film and the film length L. The Reynolds num-
ber could also be defined in terms of the average film velocity with a different numerical
coefficient.
Similarly, for the dissolution of a slightly soluble material A from the wall into a
falling liquid film of pure B, we can put Eq. 18.6-10 into the form of Eq. 22.1-3 as
follows:

- 0) s k° A Ac A (22.2-3)
c>m
©
Then, using the definition а = pgS/fx given just after Eq. 18.6-1 and the expression
for the maximum velocity in the film in Eq. 2.2-19, we find the Sherwood number as
follows:

2
C L ^ _ J(2v JS)L _ i J16 (L\(LV«»P\( H \
= = ma
'" ® © V 9а г© V 9 \s)\ м Дра J
AB r лв
= 1.017 7 £ (ReSc) 1/3 (22.2-4)

In this instance we have not only the Reynolds number and Schmidt number appearing,
but also the ratio of the film length to the film thickness.
These two problems—gas absorption by a falling film and the dissolution of a solid
wall into a falling film—illustrate two important situations. In the first problem, there is
no velocity gradient at the gas-liquid interface, and the quantity ReSc appears to the
\-power in the expression for the Sherwood number. In the second problem, there is a
velocity gradient at the solid-liquid interface, and the quantity ReSc appears to the
I -power in the Sherwood number expression.

Mass Transfer for Flow Around Spheres
Next we consider the diffusion that occurs in the creeping flow around a spherical gas
bubble and around a solid sphere of diameter D. This pair of systems parallels the two
systems discussed in the previous subsection.
For the gas absorption from a gas bubble surrounded by a liquid in creeping flow,
we can put Eq. 20.3-28 in the form of Eq. 22.1-5 thus:

- 0) s jtf дс (22.2-5)
A
г D
The Sherwood number is then

= 0.6415(ReSc) 1/2 (22.2-6)
Here the Reynolds number is defined using the approach velocity v x of the fluid (or, al-
ternatively, the terminal velocity of the rising bubble).