Page 288 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Mixing 243

10.2.3.2 Derivation of G t ¼ (m þ h) dv (10:10)
The starting point in the derivation of Camp and Stein’s G is dn
the well-known relation between shear and velocity gradient, and
given here in one dimension, i.e.,
h
(10:11)
dv e v r
t ¼ m (10:6)
dn
where
where h is the eddy viscosity of the fluid for turbulent conditions,
2
2
t is the overall shear stress (N=m ) i.e., analogous to m (N s=m )
2
m is the dynamic viscosity of fluid (N s=m ) e v is the kinematic eddy viscosity for turbulent conditions,
2
v is the local velocity of fluid (m=s) i.e., analogous to n (m =s)
n is the coordinate normal to velocity vector and coincident
with velocity gradient, r~ v (m) The eddy viscosity, h, is a characteristic of the fluid flow but
it has the advantage of being analogous to the molecular
Multiplying both sides of Equation 10.6 by dv=dn, and since viscosity, m; but dividing by r gives another parameter, e v ,
02 1=2
t(dv=dn) ¼ dP=dV (recalling that shear is force per unit area where e v ¼ l(v ) , which depends only on the eddy size, l,
and power is force times velocity), and the root mean square of the velocity deviations from the
02 1=2
mean, i.e., (v ) (Rouse, 1946, p. 178). The eddy viscosity, h,
2 is dominant in turbulent flow, i.e., at high R; on the other hand,
dP dv
¼ m (10:7) the dynamic viscosity, m, is a property of the fluid and is
dV dn
dominant in laminar flow, i.e., for low R, e.g., R 2000.
Eddy viscosity is important in understanding the role of turbu-
In terms of a finite volume, the average power expended is
lence in collisions between particles, but cannot be evaluated
quantitatively (as contrasted with m the dynamic viscosity).
2
P dv Cleasby (1984, p. 894) recommended that (1) G is valid
¼ m (10:8)
V dn only for particles smaller than the Kolmogorov microscale,
which is not typical, but may be applicable, or rapid mixing of
Since Equation 10.8 is the same as Equations 10.4 and 10.5 short duration in which the initial phases of aggregation of
combined the derivation is established. particles smaller than microscale occur; (2) otherwise, the
2=3
Dividing both sides of Equation 10.8 by the fluid density, parameter e m is more appropriate for practice because the
r, gives a variation in Equation 10.8 from Saffman and Turner turbulent eddies are larger than the Kolmogorov microscale,
(1956), which is power expended per unit mass, i.e., and is of independent temperature.
2
P dv 10.2.3.4 Empirical Parameters
e m ¼ n (10:9)
rV dn Empirical parameters include P=Q, P=V, G, q, and e m ;
Table 10.1 gives some representative guidelines from the
where literature. The G parameter remains in water treatment prac-
e m is the work of shear per unit mass per unit time tice largely because of its history, because guidelines are
(N m=s=kg) available, and it has not been supplanted. The water treatment
3
r is the density of fluid (kg=m ) research community has criticized the G criterion increasingly
2
v is the kinematic viscosity of fluid (m =s) since the work of Argaman and Kaufman (1968, 1970), who
n m=r noted that G calculates average energy dissipation over the
volume of the whole reactor and does not take into account
The e m parameter and Equation 10.9 are given because that the velocity field of a reactor is nonuniform. The P=V
they are used often in the literature, i.e., as opposed to P=V. parameter is used most frequently in chemical engineering
0.5
The relation between G and e m is, G ¼ [e m =n] . and is used also in water treatment, often in lieu of G.
10.2.3.3 Modifying Camp and Stein’s G 10.2.3.5 G and u
The most obvious issue with the Camp and Stein G is that the The effect of G and u on settled water turbidity was deter-
dynamic viscosity of the fluid is incorporated in Equation mined for various pH and alum dosages that covered the
0.5
10.5, i.e., G ¼ [P=mV] (Cleasby, 1984, p. 875). Another adsorption–destabilization and sweep-floc zones of the alum
issue identified by Clark (1985, p. 759) was that a root- coagulation diagram (Amirtharajah and Mills, 1982, pp. 214–
mean-square average velocity gradient is not representative 216). The span of Gu covered was 16,000 Gu 18,000,
1
of the complex structure of a turbulent flow regime. Regard- with G 300, 1,000, 16,000 s , and u ¼ 60, 20, 1 s, respect-
ing the viscosity issue, Cleasby (1984, p. 876) proposed, for ively. For the adsorption–destabilization zone, their
fluid shear term, findings showed markedly lower settled water turbidities at