Page 625 - Bird R.B. Transport phenomena

Questions for Discussion 605

It has been observed that, if the tank is properly baffled, 1 no vortices of importance occur;
that is, the free liquid surface is effectively level. Under these circumstances, or in the absence
of a free liquid surface, the Froude number does not appear in the system description, as we
found in §3.7.
It is further found, in most operations on low-viscosity liquids, that the rate-limiting step
is the creation of a finely divided dispersion of one fluid in the other. In such a dispersion, the
diffusional processes take place over very small distances. As a result, molecular diffusion is
not rate limiting, and the Schmidt number (Sc) has little importance. It is further found that
the effect of the Reynolds number (Re) is negligible under most commonly encountered con-
ditions. This is because most of the mixing takes place in the interior of the tank where vis-
cous effects are small, rather than in the boundary layers adjacent to the tank and impeller
surfaces, where they are large. 2
For most impeller-tank combinations in common use, the Reynolds number (Re) is
4
unimportant when its value is above about 10 . This behavior has been substantiated by a
number of investigators. 3
We thus arrive, after extensive experimentation, at a surprisingly simple result. When all of
the assumptions above are valid, the concentration profile depends only on f. Hence the di-
mensionless time required to produce any desired degree of mixing is a constant for a given system
geometry. In other words, the total number of turns of the impeller during the mixing process
determines the degree of blending, independently of Re, Fr, Sc, and tank size—provided, of
course, that the tanks and impellers are geometrically similar.
For the same reasons, in a properly baffled tank, the dimensionless velocity distribution
and the volumetric pumping efficiency of the impeller are nearly independent of the Froude
4
number (Fr) and of the Reynolds number (Re), when Re > 10 .

QUESTIONS FOR DISCUSSION

1. How do the various equations of change given in Chapters 3 and 11 have to be modified for
reacting mixtures?
2. What modifications in the flux expressions given in Chapters 3 and 11 are needed to describe
chemically reacting mixtures?
3. Under what conditions is (V • v) = 0? (V • v*) = 0?
4. Equations 19.1-14 and 15 are physically equivalent. For what kinds of problems is there a
preference for one form over the other?
5. Interpret physically each term in the equations in Table 19.2-3.
6. The thermal conductivity of a mixture is defined as the ratio of the heat flux to the negative of
the temperature gradient when all the diffusional mass fluxes are zero. Interpret this state-
ment in terms of Eq. 19.3-3.

1
A common and effective baffling arrangement for vertical cylindrical tanks with axially mounted
impellers is a set of four evenly spaced strips along the tank wall, with their flat surfaces in planes
through the tank axis, extending from the top to the bottom of the tank and at least two-tenths of the
distance to the tank center.
2
The insensitivity of the required mixing time to the Reynolds number can be seen intuitively from
2
the fact that the term (l/Re)V v in Eq. 19.5-9 becomes small compared to the acceleration term Dv/Dt at
large Re. Such intuitive arguments are dangerous, however, and the effect of Re is always important in
the immediate neighborhood of solid surfaces. Here the amount of mixing taking place in the immediate
neighborhood of solid surfaces is small and can be neglected.
The insensitivity of the required mixing time to the Schmidt number can be seen from the time-
averaged equation of continuity in Chapter 21. At large Re, the turbulent mass flux is much greater than
that due to molecular diffusion, except in the immediate neighborhood of the solid surfaces.
3
E. A. Fox and V. E. Gex, AIChE Journal, 2, 539-544 (1956); H. Kramers, G. M. Baars, and
W. H. Knoll, Chem. Eng Sci, 2, 35^2 (1955); J. G. van de Vusse, Chem. Eng. Sci., 4,178-200, 209-220 (1955).