Page 667 - Bird R.B. Transport phenomena

Questions for Discussion 647

tions and precise experiments. Their experiments showed that the form of radial av-
eraging is important at times shorter than the recommended range shown in Fig.
20.5-2 for the Taylor-Aris formula. Depending on the type of analyzer used, the data
may be better described either by a cup-mixing average p Ab or by the area average
(p ) used above.
A
Hoagland and Prud'homme 12 have analyzed laminar longitudinal dispersion in
tubes of sinusoidally varying radius, R(z) = R (l + e sin(27rz/A)), to model dispersion in
0
packed-bed processes. Their results parallel Eq. 20.5-19, when the variations have small
relative amplitude e and long relative wavelength A/K . One might think that the axial
o
dispersion in a packed column would be similar to that in tubes of sinusoidally varying
radius, but that is not the case. Instead of Eq. 20.3-19, one finds К « 2.52) Рё , with the
лв
ЛБ
first power of the Peclet number appearing, instead of the second power and with К in-
dependent of %b . u Brenner and Edwards 14 have given analyses of convective disper-
AB
sion and reaction in various geometries, including tubes and spatially periodic packed
beds.
Dispersion has also been investigated in more complex flows. For turbulent flows in
straight tubes, Taylor derived and experimentally verified the axial dispersion formula
15
K/Rv* = 10.1, where v* is the friction velocity used in Eq. 5.3-2. Bassingthwaighte and
Goresky 1 investigated models of solute and water exchange in the cardiovascular sys-
tem, and Chatwin and Allen 2 give mathematical models of turbulent dispersion in rivers
and estuaries.
Equations 20.5-1 and 19 are limited to the conditions of Eqs. 20.5-2 and 4. Therefore,
they are not appropriate for describing entrance regions of steady-state reactor opera-
tions or systems with heterogeneous reactions. Equation 20.5-1 is a better starting point
for laminar flows.

QUESTIONS FOR DISCUSSION

1. What experimental difficulties might be encountered in using the system in Example 20.1-1 to
measure gas-phase diffusivities?
2. What problems do you foresee in using the Taylor dispersion technique of §20.5 for measur-
ing liquid-phase diffusivities?
3. Show that Eq. 20.1-16 satisfies the partial differential equation as well as the initial and
boundary conditions.
4. What do you conclude from Table 20.1-1?
5. Why are Laplace transforms useful in solving the problem in Example 20.1-3? Could Laplace
transforms be used to solve the problem in Example 20.1-1?
6. How is the velocity distribution in Example 20.1-4 obtained?
7. Describe the method of solving the variable surface area problem in Example 20.1-4.
8. Perform the check suggested after Eq. 20.1-74.
9. What effects do chemical reactions have on the boundary layer?
10. Discuss the Chilton-Colburn expressions in Eq. 20.2-57. Would you expect these same rela-
tions to be valid for flows around cylinders and spheres?

D. A. Hoagland and R. K. Prud'homme, AIChE Journal, 31, 236-244 (1985).
12
13
A. M. Athalye, J. Gibbs, and E. N. Lightfoot, /. Chromatog. 589, 71-85 (1992).
14
H. Brenner and D. A. Edwards, Macrotransport Processes, Butterworth-Heinemann, Boston (1993).
G. I. Taylor, Proc. Roy. Soc, A223,446-467 (1954).
15