Page 686 - Bird R.B. Transport phenomena

666 Chapter 21 Concentration Distributions in Turbulent Flow

many different mixer geometries, that the product of the required mixing time f mix and
rotation rate N is a constant independent of mixer size and Reynolds number:
M = K(geometry) or t = K/N (21.5-16)
m i x mix
That is, the required mixing time t mix corresponds, for a given tank geometry, essen-
tially to the required number of turns of the impeller. This expectation is confirmed by
experience.
This finding is consistent with observations 2 that both the dimensionless volume
3
5
3
flow rate through the impeller, Q/ND , and the tank friction factor, P/pN D , are con-
stants, depending only on the tank and impeller geometries (see Problem 6C.3). Here Q
is the volumetric flow in the jet produced by the impeller, and P is the power required to
turn it.
Similar remarks usually apply to motionless mixers, where increasing the flow ve-
locities typically has little effect on the degree of mixing. However, approximations like
this must be tested, and such tests should be considered as first steps in an experimental
program. As a practical matter, these approximations are almost always reliable on
scale-up, since Reynolds numbers normally increase with equipment size.

Reaction Occurring
We next consider the effects of a homogeneous, irreversible chemical reaction, and for
simplicity we write this as A + В -> products. Again we assume dilute solutions, so that
the heat of reaction and the presence of reaction products have no significant effect. In
addition, we assume equal diffusivities for the two solutes.
We next define
г . = <мо ~ fc'A ~ c )
1 B
reaction i \£l.D I/)
r C r C
A0 ^ B0
Then when we subtract Eq. 21.5-2 from Eq. 21.5-1, we find that the description of r reaction
is identical to that for its nonreactive counterpart. Hence
o-(c A-c B)\ _fc AQ-c A\ (c B
c Ao + reactive \ A( > /nonreactive \ йи /nonreactive
By subtracting from this its time-smoothed counterpart, we find that an equation like Eq.
21.5-18 must hold for the fluctuations:

C C C
A0 + B0/ reactive \ Ao) nonreactive
2
The time-smoothed mean square of the quantity on the right side is equal to d , which is
measurable as illustrated in Fig. 21.5-2, and therefore we have a way of predicting the
corresponding quantity for reacting systems.
Equation 21.5-19 suggests that the fluctuations in c and c in reactive problems
A
B
occur on the same time and distance scales as for nonreactive problems. Note that this is
true for arbitrary geometry, flow conditions, and reaction kinetics. We are now ready to
consider special cases.
We begin with a fast reaction, for which the two solutes cannot coexist, and the rate
of the reaction is controlled by the diffusion of the species toward each other. Then, for
the first (macromixing) stage of the blending process, where diffusion is very slow com-
pared to the larger-scale convective processes, there is no significant reaction. In this,
typically dominant, stage of the blending process

С А \ _ ( C A
с l \c ! (21.5-20)
C A0 /rpartivP \ A0 nonreactive
L