Environmental Engineering Reference
In-Depth Information
NPs transport and removal in porous media. Important processes that can affect NMs'
fate and transport as well as associated transport equations will be considered in Sections
15.3-15.5, including agglomeration, abiotic and biotic transformations, adsorption/
desorption, attachment/filtration, and NM's transfer between different media.
Filtration of
NMs. Recently, several studies were conducted to evaluate the
behavior of NMs in clean-bed filters. The term "clean bed" refers to the assumption that
the media grains, or collectors, are homogeneous and do not contain enough deposited
particles to affect the subsequent deposition of additional particles, which translates N
s
(a surface number concentration) to zero (Logan, 1999). During the filtration process,
the number concentration of a single-sized NP can be obtained with
^
+
U
^
=D
^-
R
™
(Eq. 15.38)
3
2
N
_
fr:
8N
dN
_
1
C
TO\
where N = the number concentration of NP, #/L
3
and RTR can represent a loss due to
either mass transport [RTR = kLa
v
(N-N
s
) with ]JL = a mass transport coefficient, L/T; a
v
=
interfacial area per unit volume, L /L ; and N
s
= a surface number concentration, #/L ]
or a first-order reaction [RfR = ki(N-N
s
) with ki = a first-order rate constant, 1/T].
Obviously, eq. 15.38 is a specific form of eq. 15.13 with Rj being equal to RTR.
Assuming at steady state conditions and neglecting dispersion (D^N/St
2
), eq. 15.38
u^U-kjN or u^=-k
L
a
v
N
(Eq. 15.39)
for clean-bed filters (i.e., N
s
= 0). After integration for particle removal over a distance
L, the maximum possible rate of particle removal in a clean-bed filter is
M
K
L
a
v
L kiL
i=e-—= e—
(Eq. 15.40)
It is interesting to contrast the difference between equations derived from eq.
15.38 by assuming the particle loss being due to instantaneous adsorption, mass
transport, or a first-order reaction. In a clean-bed filter, mass transport or the first-order
reaction continuously occurs in the entire filter (and is not an instant process).
Therefore, they can't be combined with the accumulation term as in eq. 15.42. By
intuition, the rate of particle transport is a function fluid hydrodynamics; there are
physical limits to the magnitude of the mass transport coefficient, kL, while there are no
physical limits to the reaction rate constant. Therefore, the mass transport model
provides a more physically realistic picture of particle transport to a surface than a
kinetic model even though particle removal can be described with either a mass transport
or a kinetic mechanism (Logan, 1999). However, a kinetic model may provide more
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