Page 391 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

whereas the hydroxide floc data with sand and anthra-
A interception
cite are more useful as reference for practical applica- B sedimentation
tion. Note that Equation 12.13, that is, l ¼ l o þ cs,is C diffusion
included first for historical reasons since it was
B
included in Iwasaki’s paper, and second it acknow-
ledges that the filter coefficient changes as the media
becomes ‘‘clogged’’ with solids. The c coefficient is
affected, however, by a number of variables and
A
cannot be quantified easily.
C
12.3.3.2 Filter Coefficient
Collector
O’Melia and Stumm (1967) recognized that the filtration
process has two steps: (1) transport (affected by physical Streamlines
factors) and (2) attachment (affected by chemical factors). A
particle to be removed must reach a sand grain, that is, a
‘‘collector,’’ and then it must attach. These steps explain
then, in simple terms, the process of depth filtration. Later,
Yao et al. (1971) disaggregated the Iwasaki filter coefficient FIGURE 12.19 Transport mechanisms. (Adapted from Yao, K.M.
(see Equation 12.14) mathematically in terms of these two et al., Environ. Sci. Technol., 11(5), 1106, 1971.)
steps, that is, transport and attachment, that is,
one-half particle diameter distance from a collector). Two
3 1 e
ah (12:17) other mechanisms are inertia and shear, which are considered
l ¼
2 d
not important.
where
3 3 1. Interception: The transport of a coagulated particle to
e is the filter bed porosity (m of void volume=m of filter
a spherical collector by advection along a streamline
bed volume)
is interception. The particle at A in Figure 12.19
d is the grain diameter (m)
illustrates.
a is the number of contacts which produce a particle col-
2. Diffusion: Random motion due to thermal energy is
lector adhesion divided by the number of particle
superimposed upon the advective motion within the
collector collisions, called attachment efficiency (number
filter media, as defined by a given streamline. The
of particle collector attachments=number of particle
particle at C in Figure 12.19 illustrates.
collector collisions)
The contact frequency between particles and col-
h is the rate at which particles strike a collector divided by
lectors depends upon the number of random ‘‘steps’’
the rate at which particles flow toward the collector,
per unit time, which is proportional to temperature
called the transport efficiency (particle collector colli- 2
(i.e., from molecular theory of gases, 1=2mv ¼ kT,
sions=particle flux associated with a given collector)
which is not perfectly transferred to particles in
liquids). If N ¼ number of steps=s, then the number
The first group of terms, that is, (3=2) (1 e)=d, is the
of steps per unit length along a streamline is N= v.
grain surface area per unit volume of filter bed, that is,
Therefore, the lower the interstitial velocity, the
the ‘‘specific surface’’ (Ives and Sholji, 1965, p. 3) with the
more steps per unit distance, which in turn increases
3=2 referring to a spherical particle shape. The transport and
the probability of particle-filter grain contact by dif-
attachment coefficients, that is, a and h, respectively, are
fusion. Thermally induced random motion can be
delineated further in the sections that follow.
observed microscopically, for example, for Staphylo-
coccus aureus bacteria, which is about 1 mm in size
12.3.3.3 Transport Coefficient (Hendricks et al., 1970, p. 19).
The transport step involves getting a coagulated particle to 3. Sedimentation: The third major transport mechan-
a collector (a term used often by theoreticians in referring to a ism, sedimentation, is described mathematically by
grain of the filter medium). The three transport mechanisms Stoke’s law. Adding the gravitational velocity vector
are interception, diffusion, sedimentation (Yao et al., 1971). (as defined by Stoke’s law) to the advective velocity
Figure 12.19 illustrates the path of a single particle for each vector, which is tangent to a streamline at any
mechanism. The diffusion and sedimentation mechanisms given point, gives a resultant particle trajectory that
cause particles to cross streamlines and thus be transported incorporates the influence of gravity. The particle at
to the proximity where attachment with a media grain could point B in Figure 12.19 illustrates how the particle
occur. With interception, the particle follows the streamline trajectory is modified from its advective path along a
and may ‘‘brush’’ a collector (for streamlines that pass within streamline to a path influenced by gravity.