Page 680 - Bird R.B. Transport phenomena

660 Chapter 21 Concentration Distributions in Turbulent Flow

soluble in the fluid (a liquid B) flowing through the tube. Material A dissolves in liquid В
and then disappears by a first-order reaction. We shall be particularly interested in the
behavior with high Schmidt numbers and rapid reaction rates.
For tube flow with axial symmetry and with c A independent of time, Eq. 21.2-8
becomes

•k"'c A (21.4-1)

Here we have made the customary assumption that the axial transport by both molecu-
lar and turbulent diffusion can be neglected. We want to find the mass transfer rate at
the wall
) (21.4-2)

where c A0 and c A axi s are the concentrations of A at the wall and at the tube axis. As
pointed out in the preceding section, the turbulent diffusivity is zero at the wall, and
consequently does not appear in Eq. 21.4-2. The quantity k c is a mass transfer coefficient,
analogous to the heat transfer coefficient h. The coefficient h was discussed in Chapter 14
and mentioned in Chapter 9 in connection with "Newton's law of cooling." As a first ap-
proximation 1 we take с л axi s to be zero, assuming that the reaction is sufficiently rapid
that the diffusing species never reaches the tube axis; then dcjdr must also be zero at
the tube axis. After analyzing the system under this assumption, we will relax the as-
sumption and give computations for a wider range of reaction rates.
We now define the dimensionless reactant concentration С = c /c . Then under
A
A0
the further assumption 1 that, for large z, the concentration will be independent of z,
Eq. 21.4-1 becomes
: (21.4-3)

This equation may now be multiplied by r and integrated from an arbitrary position to
the tube wall to give
ас f R
rC(r)dr (21.4-4)

Here the boundary conditions at r = 0 have been used, as well as the definition of the
mass transfer coefficient. Then a second integration from r = 0 to r = R gives

k R Г — dr-\ = к"' Г - — Г гСШг dr (21.4-5)
c
Here we have used the boundary conditions С = 0 at r = 0 and С = 1 at r = R.
Next we introduce the variable у = R — r, since the region of interest is right next to
the wall. Then we get

k R Г - -—dy- 1 =k"' \ R 1 — [ У (R-y)C(y)dy\dy
c
J
h (R - y№ AB + Э<&) ^ J o (R - у)(Э л в + S O L o У У У ] У
(21.4-6)
in which C(y) is not the same function of у as C(r) is of r. For large Sc the integrands are
important only in the region where у « R, so that R — у may be safely approximated
by K. Furthermore, we can use the fact that the turbulent diffusivity in the neighborhood