Page 295 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

250 Fundamentals of Water Treatment Unit Processes: Physical, Chemical, and Biological

complex than indicated here. But the short answer is that the viscous sub-range, is rapidly dissipated by viscos-
whileheconfirmed that the N m=kg units were correct ity. From the foregoing, it follows that ultimately
(and are consistent with the development shown), other the big eddies will supply most of the energy
alternative units may be preferred by those in the field. of the turbulence (Batchelor, 1953, p. 92). The
practical outcome is that the advective motion of
. Kolmogorov microscale: A particular wave number large eddies is essential to distribute a given species
indicated on curve e(k) 2 is k*, called the ‘‘Kolmo- to different neighborhoods of the fluid bulk where
gorov microscale.’’ A characteristic is that in the smaller eddies, i.e., vortices, can do the job of
range k ¼ k*, then R ¼ 1; in other words, the inertia final mixing.
forces equals the viscous forces. For k > k*, the As related to Figure 10.10, as G increases, the
energy function drops sharply with the energy spectra distribution extends to higher frequencies
being dissipated to heat. The Kolmogorov micro- (Argaman and Kaufman, 1968, pp. 84–94, 1970).
scale is the practical limit of eddy size. At such size These results are in accordance with the theory
scales, diffusion, the final transport step, becomes which predicts the relationship between the power
dominant. The length scale, l*, and the vortex input and the Kolmogorov microscale of turbulence,
3
velocity, v*, may be calculated as (Batchelor i.e., Equation 10.14, l* ¼ (n =« m ) 1=4 , resulting in
(1953, p. 1115; Argaman and Kaufman, 1968, decreasing l* with increasing « m (or G); therefore,
p. 32; Frisch, 1995, p. 91; Clark, 1996, p. 207; the E(k) 2 curve shifts to the right at higher G values.
Logan, 1999, p. 196), Also, the area under the E(k) 1 curve is larger with
increasing G, since more energy must be distributed
over the E(k) 2 energy spectrum.
1=4
3
n . Particle transport by eddies: Only those eddies
, (10:14)
e within certain size ranges will influence particle
l* ¼
transport (Argaman and Kaufman, 1968, p. 19).
Some of the ideas underlying particle transport
and
mechanisms are illustrated by Figure 10.11a
through c, respectively, i.e.,
v* ¼ (n e) 1=4 (10:15) 1. In Figure 10.11a, i.e., where l(eddy) d(separ-
ation), the eddy may transport particles, but the
scale is such that the particles are entrained to
where
move with the eddy rather than to be effective in
l is the size of eddy at Kolmogorov microscale
collisions with other particles.
(m) 2. In Figure 10.11b, i.e., where l(eddy) < d(par-
2
n is the kinematic viscosity (m =s)
ticle), the eddy is too small to transport the par-
e is the rate of energy dissipation (N m=s=kg)
ticle. Eddies that are smaller than the particle
v is the peripheral velocity of vortex at Kolmo-
size, do not contribute substantially to turbulent
gorov microscale (m=s)
diffusion (p. 19).
3. In Figure 10.11c, i.e., where l(eddy) d(particle)
The Reynolds number for the Kolmogorov micro-
and l(eddy) < d(separation), the eddy is of such
scale, R*, has a value of unity, i.e., R* ¼ 1 and is
size as to cause effective particle transport and
calculated (Argaman and Kaufman, 1968, p. 32),
collisions with other particles. In other words,
the eddy-induced motion of two particles relative
v*l* to one another will be governed by eddies that
(10:16)
n are larger than their size and smaller than their
R* ¼
separation.
. Role of large eddies: Regarding the big slow eddies, To summarize, from Hanson and Cleasby (1990),
they interact very weakly with the remainder of the process of vortex stretching will create myriad
the turbulence and preserve their energy virtually localized shear fields, which drive the process of
intact (Batchelor, 1953, p. 91). The energy spectrum contacts between primary particles and coagulants
of very small wave numbers (the larger eddies) (and also between like particles as in flocculation).
suffers very little modulation during the whole of . Design implications: Eddies sizes about the size of the
the decay process; the reason is that most of the energy primary particles (or slightly larger) are most effective
of the E(k) 1 inertial sub-range is associated with in causing collisions, as illustrated in Figure 10.11c.
smaller wave numbers, as seen in Figure 10.10. On Thus the source of turbulence should not be too large
the other hand, the energy in higher wave numbers (e.g., in rapid mix, the blade width of a radial-flow
(the smaller eddies) of the spectrum, E(k) 2 , i.e., in impeller would be narrow rather than wide). A large