Page 296 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Mixing 251

λ(eddy)>d(separation)
λ(eddy)>>d(particle)
and
λ(eddy)<d(separation)
d(particle)
λ(eddy)
λ(eddy)
λ(eddy)<<d(particle)
λ(eddy)
d(particle)
d(separation)
d(separation)
(a) (b) (c)

FIGURE 10.11 Eddy sizes relative to effective mixing. (a) Eddy too large. (b) Eddy too small. (c) Effective eddy. (Adapted from Argaman, Y.
and Kaufman, W.J., Turbulence in orthokinetic flocculation, SERL Report No. 68-5, Sanitary Engineering Research Laboratory, University of
California, Berkeley, CA, 1968.)

form will generate large inertial eddies with an energy Solution
spectrum E(k) 1 shifted to the left, i.e., with small The tabular summary indicates the outcome based upon
wave numbers as illustrated by Figure 10.11a. But a the foregoing guidelines, i.e., as outlined in Figure 10.11.
smaller form drag will result in a shift to the right,
which distributes more energy toward the higher wave
G l* a d(floc)
numbers. At the same time, the higher the energy input,
1
Case Case Description (s ) (mm) (mm) Outcome
the lower the Kolmogorov microscale, l*. Forms that
generate suitable turbulence include a grid across a 1 Floc smaller 10 316 250 Floc will not
pipe. For example, Stenquist and Kaufman (1972) than microscale break since floc
is smaller than
showed that mixing was more effective with a grid
smallest
rather than with a back-mix reactor. The grid size
turbulent eddies
may be designed to be compatible with the scale of
2 Floc larger 30 200 250 Floc will interact
turbulence desired. The same is true for flocculation;
than microscale with turbulent
the paddle blades should be not too large so the E(k) 1 flow field and
energy does not have to cascade from very large eddy rupture
sizes to the e(k) 2 energy distribution.
a Calculated by Equation 10.14.
In adsorption–destabilization, much of the process
occurs at the small length scales. The more energy
added to the water in a localized region, the smaller
the microscale of turbulence. Therefore, the higher the Discussion
For Case #1, the particles are smaller than the Kolmo-
energy input per unit mass, the more the energy spec-
gorov microscale, i.e., d(floc) < l*. A particle in this size
trum is shifted to the right and the more effective is the
range is like a mosquito inside a car; the mosquito does
mixing for small primary particles. On the other
not know when the car turns a corner. And so it is for a
hand, if the particles are larger in size, e.g., d(particle) particle inside a vortex tube; the particle with size,
1 mm, the energy input may have some upper limit d(particle) < l*, does not feel the inertial interactions
for effective mixing, i.e., such that d(eddy) > d(par- of the vortex (simile from discussions with Professor
ticles), so that Case 3, Figure 10.11c prevails. Adrian Hanson, September 10, 2001); the vortex does
not cause mixing.
For Case #2, the particles are larger than the Kolmo-
Example 10.1 Shear of Floc Particles with
gorov microscale, i.e., d(floc) > l*. A particle in this
Respect to Kolmogorov Microscale (from Hanson
size range is subject to the shear field generated by the
and Cleasby 1990)
smallest turbulence (but does not feel the turbulence of
the larger vortices).
Given
Two cases are summarized in the following table: (1) floc
is smaller than the Kolmogorov microscale, and (2) floc is 10.3.1.2.4 Diffusing Substances, Diffusion
larger than the microscale. Rate, and Sinks
Required Figure 10.12 illustrates in (a), (b), and (c), respectively,
For each case, determine whether the floc will rupture. substances that diffuse, a graphic depiction of mathematical