Page 287 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

manufacturer (Chicago Pump) found that the return of n i is the number concentration of particles of radius r i
3
methane gas caused a higher reaction rate. Initially, the (#=m )
3
explanation was that the methane acted as a catalyst. Soon n j is the number concentration of particles of radius r j (#=m )
after, the mixing induced by the methane gas bubbles was
determined to be the cause. Thereafter, mixing was incorpor- Step 2 Laminar Motion Equation by Smoluchowski (1918)
ated into the design of anaerobic digesters. Prior to this, the The second equation of von Smoluchowski (1918) was for
digesters were permitted to merely ‘‘sit’’ quiescently, except collisions between particles induced by fluid motion (called
for the fill or draw phases. ‘‘orthokinetic’’ motion). For laminar flow the collision fre-
quency was given as

10.2.3 EVOLUTION OF MIXING THEORY 3
N(laminar) ij ¼ (4=3)(r i þ r j ) n i n j (dv=dy) (10:2)
In 1918, Smoluchowski published a mathematical expression
for collision frequency between particles in a suspension for where
laminar flow; this was the basis for subsequent developments N(laminar) ij is the number of contacts per unit time per unit
in theory. Later, Camp and Stein (1943) applied the volume between i and j particles due to laminar fluid
3
Smoluchowski concept to turbulent mixing. motion (collisions=m =s) 1
As related to impeller–basin systems, the results of the first dv=dy is the velocity gradient due to laminar flow (s )
systematic experimental studies were by Rushton and his
associates in the late 1940s, with results published in the Step 3 The Next Step by Camp and Stein (1943) Camp and
1950s. They expressed their results in terms of dimensionless Stein, according to Argaman and Kaufman (1968, p. 5), noted
numbers, e.g., the ‘‘power number’’ vs. the Reynolds number. that turbulent conditions exist in most cases of flocculation
The theory of turbulence is the basis for another thread of and adapted Smoluchowski laminar flow equation to the case
theory that supports a further understanding of mixing, with of turbulent fluid motion,
modern ideas beginning to crystallize also about 1950 (see 3
Batchelor, 1953, for example). Then, starting about 1960, N(turbulent) ¼ (4=3)(r i þ r j ) n i n j G (10:3)
ij
with the advent of modern computers, the Navier–Stokes
equation, considered a theoretical abstraction for decades, where
became amenable to solution by finite difference techniques. N(turbulent) ij is the number of contacts per unit time per
By the 1990s, these techniques were developed into commer- unit volume between i and j particles due to laminar
3
cial software technologies for use with ‘‘work-station’’ com- fluid motion (collisions=m =s)
puters. By the end of that decade, such software found its way G is the velocity gradient averaged over volume, V,of
1
into engineering practice and was used to address various reactor (s )
‘‘what if?’’ design scenarios in water treatment.
The velocity gradient, G, was defined by Camp and Stein
10.2.3.1 Development of Collision Frequency (1943) as
Mathematics
dv
The mathematical expression for the rate of collision in a G (10:4)
dy
turbulent flow field was developed from the equations of
Smoluchowski. The enumeration of steps, presented here, is 0:5
P
from Argaman and Kaufman (1968, p. 5). ¼ (10:5)
mV
Step 1 Brownian Motion Equation by Smoluchowski (1916)
where
The first equation of von Smoluchowski (1916) was for the
v is the velocity at a point in space (m=s)
condition that collisions were due to Brownian motion (called
y is the coordinate normal to the velocity vector v (m)
‘‘perikinetic’’ motion) calculated as the diffusion flux of par-
P is the power dissipated by fluid motion, e.g., viscosity or
ticles in the radial direction around a single stationary particle,
turbulence (W)
2
m is the dynamic viscosity of fluid (N s=m )
3 3
N(diffusion) ij ¼ 4pD ij (r i þ r j ) n i n j (10:1) V is the volume in which power dissipation occurs (m )
where The derivation of Equation 10.5 is given in Section 10.2.3.2
N(diffusion) ij is the number of contacts per unit time per with further elaboration in Box 11.1. The flaw in the relation
unit volume between i and j particles due to diffusion was that in turbulent flow, the viscosity term, m, is small
3
flux (collisions=m =s) relative to inertia term (see, for example, Cleasby, 1984,
2
D ij is the combined diffusion coefficient, D i þ D j (m =s) pp. 876, 877). Nevertheless, the G parameter was widely
r i is the radius of particle i (m) accepted and is embedded in practice (Amirtharajah et al.,
r j is the radius of particle j (m) 2001, p. 162), its limitations notwithstanding.