Environmental Engineering Reference
In-Depth Information
(#/cm
3
); and nj = concentration of j NPs , number of particles/L
3
(#/cm
3
). Collisions
between suspended NPs in water can occur by three mechanisms: (a) Brownian motion
(also known as microscale or perikinetic flocculation), (b) fluid shear (the gentle mixing
of the water, also known as macroscale or orthokinetic flocculation), and (c) differential
sedimentation (O'Melia, 1980; MWH, 2005). The collision frequency function Py is
contributed by the three flocculation mechanisms as follows:
frequency between particles i and j; nj = concentration of i NPs, number of particles/L
3
p.. = P
M+P(
I
+
P
(Eq. 15.25)
P
M
= i G(dj + dj)
3
(Eq. 15.26)
p^ =
^L
(-L+ l)(dj + dj)
(Eq. 15.27)
PDS = •
(
7;;
w)
g
(d, + dj)
3
d, - dj|
(Eq. 15.28)
where PM = macroscale collision frequency; p^ = microscale collision frequency; PDS =
differential settling collision frequency; dj = the particle diameter of i size class; dj = the
particle diameter of j size class; k = Boltzmann's constant, 1.3807 x 10"
2
3
J/K; T =
absolute temperature, K; and G = average velocity gradient, 1/T:
G=^7V (Eq. 15.29)
where e = local rate of energy dissipation, L
2
/T
3
; and v = kinematic viscosity, L
2
/T. For
most of NMs (1-100 nm), Brownian motion is the primary mechanism of aggregation.
However, the other two mechanisms will become dominated once NPs become larger
than about 1000 nm (1 |im) after attachment (see Fig. 15.2 and O'Melia, 1980; Logan,
1999 for detailed information).
To illustrate the application of the flocculation models, we can write a mass
balance equation around a control volume of a CSTR. In general, in eq. 15.1,
(1)= V.2£
=1
^
(Eq. 15.30)
where V = volume of the control volume, L (cm ); and ni, = concentration of NPs in size
h in control volume, M/L
3
(mg/cm
3
). To use eq. 15.30 representing the rate of NP mass
accumulation, we must assume that there is no instantaneous equilibrium
adsorption/desorption in the control volume. The assumption is valid for most aquatic
systems where suspended biomass or solids are usually very low, but is not valid for
aquatic systems containing porous media in the control volume (see eq. 15.13).
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