Page 687 - Bird R.B. Transport phenomena

Questions for Discussion 667

It has been suggested 4 that Eq. 21.5-20 is also true for the micromixing stage. Where this
can be assumed (e.g., in the common situation where macromixing is rate controlling), it
follows that reactive and nonreactive processes lead to identical descriptions of solute
fluctuations.
In practice, fast reactions (e.g., neutralization of acids with bases) are often used to
determine the effectiveness of mixers, as these are much easier to follow experimentally
than nonreactive mixing. Frequently one can use simple macroscopic measures such as
temperature rise or an indicator color change. However, the measurement of concentra-
tion fluctuations can provide more insight into the nature and the course of the mixing
process.
Slow reactions are also important, and we consider the special case of irreversible sec-
ond-order kinetics, defined by
(21.5-21)
When this is time-smoothed, we get

R A = ~k"\c c + с~У ) (21.5-22)
в
A B
We find, therefore, that the fluctuations in solute concentration increase the time-
smoothed reaction rate relative to that when a simple product of time-smoothed concen-
trations is used. It is, however, difficult to assess the practical importance of this effect.
We illustrate this point by a simple order-of-magnitude analysis, beginning with the
definition of a reaction time constant t for one of the reactants, here solute A:
A
— ''ДО' A \£. L .\J £.J/
i A
To a first approximation, we may write
t A~\/k' 2"c B0 (21.5-24)
Fast and slow reactions may then be defined as those for which
'mix » t A fast reaction (21.5-25)
'mix « t A slow reaction (21.5-26)
We have already discussed the case of fast reaction. For slow reactions, turbulence has
no significant effect, because fluctuations become negligible before any appreciable reac-
tion has taken place.
If the mixing and reaction time constants are of the same order of magnitude, a
deeper analysis than the above is needed. Such an analysis must include a model for the
turbulent motion, and does not appear to be presently available.

QUESTIONS FOR DISCUSSION
1. Discuss the similarities and differences between turbulent heat and mass transport.
2. Discuss the behavior of first- and higher-order reactions in the time-smoothing of the equa-
tion of continuity for a given species. What are the consequences of this?
3. To what extent are the turbulent momentum flux, heat flux, and mass flux similar in form?
4. What empiricisms are available for describing the turbulent mass flux?
5. How can eddy diffusivities be measured, and on what do they depend?
6. Would you expect to get trustworthy results for mass transfer in turbulent tube flow without
chemical reaction just by setting Rx = 0 in Eq. 21.4-8?

4
K.-T. Li and H. L. Toor, Ind. Eng. Chem. Fundam., 25, 719-723 (1986).