Page 599 - Bird R.B. Transport phenomena

Problems 579

(a) Show by means of a shell mass balance that the pseudo-steady-state oxygen concentration
profile is described by the differential equation
1 d d X
(18B.19-1)

where \ = p 0jpo, f = r/R, and N = k 0,
(b) There may be an oxygen-free core in the aggregate, if N is sufficiently large, such that \ = 0
for f < £ 0 - Write sufficient boundary conditions to integrate Eq. 18B.19-1 for this situation. To
f
do this, it must be recognized that both \ a n d d\/d^ are zero at f = 0 . What is the physical
significance of this last statement?
f
(c) Perform the integration of Eq. 18B.19-1 and show how 0 may be determined.
(d) Sketch the total oxygen uptake rate and £ 0 a s functions of N, and discuss the possibility
that no oxygen-free core exists.
o r
f
Answer: (c) x = 1 ~~ ~z 0 ~ ^ + ~т Щ 7 ~ M ^ £ — ^o — 0, where 0 is determined as a func-
tion of N from 6 -3 V? /

18C.1. Diffusion from a point source in a moving stream (Fig. 18C.1). A stream of fluid В in lami-
nar motion has a uniform velocity v . At some point in the stream (taken to be the origin of
0
coordinates) species A is injected at a small rate W g-moles/s. This rate is assumed to be suf-
A
ficiently small that the mass average velocity will not deviate appreciably from v . Species A
0
is swept downstream (in the z direction), and at the same time it diffuses both axially and
radially.
(a) Show that a steady-state mass balance on species A over the indicated ring-shaped ele-
ment leads to the following partial differential equation if ЯЬ is assumed to be constant:
АВ
4- • 2 (18С.1-1)
dr ) dz
(b) Show that Eq. 18C.1-1 can also be written as
Sc
dC A \ _ [ l /9 / 2 -
(18С.1-2)
•dC A
2
in which s = r + z .
2
2

Uniform
stream
velocity у,

Origin of coordinates placed at
point of injection; W moles Fig.l8C.l. Diffusion of Л
A
of A are injected per second -I k Az from a point source into a
of В that moves with
stream
a uniform velocity.