Page 684 - Bird R.B. Transport phenomena

664 Chapter 21 Concentration Distributions in Turbulent Flow

Then both Eqs. 21.5-1 and 2 take the following form over the whole system:

(21.5-8)

Here the subscript i can represent either solute A or solute B, and

Г = 0 for (a) the entering Л-rich stream, or
(b) initially Л-rich region (21.5-9)
Г = 1 for (a) the entering B-rich stream, or
(b) initially B-rich region (21.5-10)
It follows that, for equal diffusivities, the time-smoothed concentration profiles,
Г(х, у, z, t) are identical for both solutes, where
p _ C A0 C (21.5-11)
c
B0
However, the fluctuating quantities Г' are also of interest, as they are measures of "un-
mixedness." These can be equal only in a statistical sense. To show this, we subtract Eq.
21.5-11 from Eq. 21.5-7, and then square the result and time-smooth it to give

(21.5-12)

Here d(x, y, z, t) is a dimensionless decay function, which decreases toward zero at large z
[for the motionless mixer in Fig. 21.5-1 (я)], or at large t [for the mixing tank of Fig. 21.5-
l(b)]. Cross-sectional averages of this quantity can be measured, and are shown in Fig.
21.5-2.
It remains to determine the functional dependence of the decay function, and to do
this we introduce the dimensionless variables:
v t (21.5-13)
o

Then Eq. 21.5-8 becomes
Р Г _ 1 , (21.5-14)
Dt R e S c
in which Re = v p//Ji.
l
o o
In order to be able to draw specific conclusions, we now focus our attention on mix-
ing tanks [see Fig. 21.5(b)], and further assume low-viscosity liquids and low-molecular-
weight solutes. For these systems / is normally chosen to be the diameter of the
0
impeller, and v 0 to be 1 N, where N is the rate of impeller rotation in revolutions per unit
O
time.

L
A-+-
Г Fig. 21.5-1. Two types
B-+- of mixers: (я) a baffled
mixer with no moving
(a)
parts; (b) a batch mixer
(6) with a stirrer.