Environmental Engineering Reference
In-Depth Information
flexibility to eq. 15.38 when reactions other than first-order kinetics are involved in the
filtration process.
Clean-Bed Filter Theory.
Due to their high surface areas, NMs released into
the soil can be strongly sorbed to the soil and, thus, be immobile. However, NMs are
small enough to fit into smaller spaces between soil particles, and might therefore, travel
farther than larger particles before being trapped in the soil matrix (USEPA, 2007).
Recently, models developed on the basis of the clean-bed filter theory have been used to
evaluate the transport of NMs. Logan (1999) introduced and compared the models
developed based on the clean-bed filter theory, including the Yao model, the pore
velocity model, the Happel model, the Yao-Habibian model, and the RT model (Table
15.3); these models are based on a filtration equation:
— =e-
aX
L
(Eq. 15.41)
N
where a = sticking coefficient (or the collision efficiency or attachment efficiency
factor), 0 < a < 1, unitless; A, = the filtrate-ion constant; and L = distance in the filter, L.
If the NMs have a particle distribution no(d
p
), then the effluent particle size will be n(d
p
)
= n
0
(dp)e"
ca
L
according to eq. 15.41.
Filtration models developed based on clean-bed filter theory
3
.
X
3(l-9)V(2d
c
)
3(l-9)V(2d
c
9)
3(l-9)V(2d
c
)
3(l-9)V(2d
c
)
T, = 4.04 Pe~
2
/
3
+ 3R*
2
/2 + S*
Overall
r\ = T\
O
+ Kir +
T\
S
T, = 4.04 Pe~
2
/
3
+ 3R*
2
/2 + S*
T, = 4.04 y
2
/
3
Pe-
2
/
3
+ 3R'
2
/2 + S*
•n = 4.04 bJ/
3
Pe-
2
/
3
+ hnLo
1
/
8
^
15
/
8
,, = 2.44A
1
/
3
V-
081
Pe-
0
-
715
N
v
0
dW + 0.55A
s
Ni-
675
N°-
12
5
+ 0.00338b
H
S'
12
R'
0
4
3(1 - 0) Ti/(2d
c
)
_
a
Adapted based on Logan (1999) and Tufenkji and Elimelech (2004); Pe (Peclet number)
—
Ud
c
/d
p
; R* (dimensionless radius, also called interception number)
—
d
p
/d
c
; S* (gravitational
number) = u
p
,/U; 7 (solid fraction in a porous medium) = (1- 0)
1
'
3
; b
H
(Happel cell constant) =
(2(l-7
5
)/(2-37 + 3~/
5
-2~/
6
); Lo (London number)
=
4A/(97iud
p
2
U); A (Hamaker constant)
=
p(pL
A
7i/m
a
)
2
; where U = groundwater (Darcy) velocity; d
c
-
diameter of the porous medium; d
p
-
diameter of NPs; u
p
,
s
—
the NP's settling velocity; 0
—
porosity; |i
—
water dynamic viscosity; p
—
a
composite van der Waals parameter related to the induced or permanent dipole moments; p
—
particle density; L
A
—
the Avogadro number; m
a
—
the molar mass of the particle materials; in the
TE model: A
s
—
b
H
, N
R
= R*, N
vd
w
—
A/kT (k
—
Boltzmann constant, T
—
absolute temperature),
N
A
—
attraction number
—
A/(6 7i|id
p
U);
r\ —
overall collector efficiency, unitless; T|
D
—
collision
efficiency due to particle diffusion; T|I
-
collision efficiency due to interception; and T|
s
-
collision efficiency due to gravitational sedimentation.
b
PV = Pore Velocity; YH = Yao-
Habibian; RT
—
Rajagopalan and Tien; and TE
—
Tufenkji and Elimelech (2004).
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