Page 422 - Modelling in Transport Phenomena A Conceptual Approach

402 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

Note that the term 46 in the definition of the Reynolds number represents the
hydraulic equivalent diameter.

9.5.2.2 Short contact times
If the solubility of species A in the liquid I3 is low, for short contact times, species
A penetrates only a short distance into the falling liquid film. Under these circum-
stances, species d, for the most part, has the impression that the film is moving
throughout with a velocity equal to Vmm. Furthermore, species A does not feel
the presence of the solid wall at z = 6. Hence, if the film were of infinite thickness
moving with the velocity v,,, species A would not know the difference.
In the light of the above discussion, Eqs. (9.588)-(9.5-91) take the following
form
a2CA
8CA
vmaX - DAB - (9.5108)
=
dz 6x2
at z =O CA =CA, (9.5-109)
at x=O CA =cfi (9.5110)

at X= 00 CA =CA, (9.5111)
Introduction of the dimensionless concentration r$ as

(9.5-112)

reduces Eqs. (9.5108)-(9.5111) to

(9.5113)

at a=O (P=O (9.5114)
at z=O +=l (9.5-1 15)

at x=w 4=0 (9.5-116)
Since Eqs. (9.5-114) and (9.5116) are the same and there is no length scale, this
parabolic partial differential equation can be solved by the similarity solution as
explained in Section B.3.6.2 in Appendix B. The solution is sought in the form

4 = f (Q) (9.5-117)

where
(9.5-118)