Page 341 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

11.3.2.2.2 Orthokinetic Motion 11.4.1.1 Frequency of Particle Collisions
The second equation of Smoluchowski (1918) was for colli- The general equation for collision frequency is (O’Melia,
sions between particles induced by fluid motion (called 1978, p. 227) given by
‘‘orthokinetic’’ motion). For laminar flow, the collision fre-
quency was given as J(i, j) ¼ g(i, j) n i n j m, (11:1)
where

4 3 dv J(i, j) is the frequency of collision per unit volume
(r i þ r j ) n i n j (10:2)
3 dy of suspension between i species and j species (i–j
N(laminar) ij ¼
3
collisions=m =s)
3
g(i, j) is the kinetic rate constant (m =s)
where
n i is the concentration of particles of species i (particles
N(laminar) ij is the number of contacts per unit time per unit 3
i=m )
volume between i and j particles due to laminar fluid n j is the concentration of particles of species j (particles
3
motion (collisions=m =s) j=m )
3
1
dv=dy is the velocity gradient due to laminar flow (s )
In other words, the frequency of collisions is proportional
11.3.2.3 Camp’s G to the concentrations of particles that are subject to interaction,
Camp and Stein (1943) derived an expression for velocity i.e., as used in the Smoluchowski models. The frequency
gradient called G (Camp and Stein, 1943), which is based on coefficient, g(i, j), however, incorporates considerable com-
the forces acting on a fluid element: plexity; basic tenants are reviewed in paragraphs that follow.
11.4.1.1.1 The Kinetic Rate Constant
s ffiffiffiffiffiffiffi
P The kinetic rate constant, g(i, j), is the product
(10:5)
G
mV
g(i, j) ¼ a(i, j) k(i, j) (11:2)
where where
1
G is the average velocity gradient for basin (s ) a(i, j) is the collision frequency factor, i.e., fraction of
P is the power applied to paddle wheel (N m=s) particle collisions that result in collisions for curvilinear
2
m is the dynamic viscosity of water (N s=m ) model as compared to the rectilinear model (collisions
3
V is the volume of flocculation basin (m ) that occur by curvilinear model=collisions that occur by
rectilinear model)
3
Later, in a 1955 paper (Camp, 1955), Camp delineated k(i, j) is the rate constant for rectilinear model (m =s)
how to apply G to flocculation for diffused aeration, paddle
11.4.1.1.2 The a Factor
wheels, and reciprocating blades. The procedure that he
described remains current. Following Langelier’s suggestion The traditional form of Equation 11.2 is for a(i, j) ¼ 1, which
for tapered flocculation, Camp suggested higher G values, i.e., describes the ‘‘rectilinear’’ model of collision frequency (Ives,
1 1
G 70 s in the first compartment, tapering to G 20 s in 1978, p. 45). The a(i, j) term is a ‘‘correction’’ factor to take
the third or fourth, based upon data that he compiled from into account that the ‘‘i’’ particle path is not rectilinear, but
operating plants, as described in Table 11.1. curvilinear, and also as the particles come into the proximity
of one another they are subject to ‘‘short-range’’ forces, e.g.,
van der Waals (Han and Lawler, 1992, p. 83). The a(i, j) term
11.4 THEORY OF FLOCCULATION is defined as the ratio
The theory of flocculation has to do with the rate of aggregation 2
x c
of flocs. The starting point has been, nearly always, the classic a(i, j) ¼ 2 (11:3)
(d i þ d j )
Smoluchowski equations in 1916 and 1918 for Brownian
motion, i.e., perikinetic, and fluid motion, i.e., orthokinetic,
where
respectively (see Sections 10.2.3.1 and 11.3.2.2). x c is the ‘‘critical’’ diameter just outside the trajectory
‘‘shadow’’ of particle i beyond which particle j will not
collide with particle i (m)
11.4.1 KINETICS
d i is the diameter of particle i (m)
For flocculation, the kinetic model given by O’Melia d j is the diameter of particle j (m)
(1978), an adaptation of the Smoluchowski model (Section
10.2.3.1), incorporates terms that can be expanded to explain The values of a(i, j) are given additional resolution by Han
the rate of aggregation of flocs. This section deals with and Lawler (1992) in separate equations for Brownian
those factors. motion, shear, and settling, respectively.