Environmental Engineering Reference
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rates, can also lead to anomalous results that require
specialized fate and transport models (e.g., Salamon
et al., 2006).
where
d
g
is the median grain size diameter of the porous
medium, and
η
is the
collection efficiency
defined as the
probability that a colloid approaching a grain surface
will come into physical contact with the surface.
5.5.4 Biocolloids
5.5.4.2 Modified Colloid Filtration Theory.
many
microorganisms have negatively charged surfaces, and
the surfaces of quartz grains that comprise many sandy
aquifers are also negatively charged thus creating unfa-
vorable conditions for attachment. In contrast, iron
oxides that coat some sedimentary grains are predomi-
nantly positively charged creating favorable conditions
for attachment. This mixture of favorable and unfavor-
able conditions for microbial attachment may coexist in
many sandy aquifers (Scheibe et al., 2011), and this
leads to the following modifications to colloid filtration
theory when applied to microbes: (1) incorporation of
a
collision efficiency factor
,
α
c
, which represents the
probability that collision between a microbe and a
mineral surface will result in the attachment of the
microbe (dimensionless), and (2) incorporation of a
detachment rate to represent dislodgement of microbes
from secondary minimum attachment sites. In the modi-
fied colloid filtration theory, the straining rate (∂
S
/∂
t
) is
a fixed fraction of the mass flux and can be expressed
in the form (Scheibe et al., 2011)
Waterborne pathogens, such as
Cryptosporidium
parvum
,
Escherichia coli
,
Giardia duodenalis
spp., are
classified as
biocolloids
, and the fate and transport of
biocolloids in groundwater is commonly modeled using
the colloid filtration theory originally proposed by Yao
et al. (1971). The transport of biocolloids in porous
media is attributed to advection, dispersion, straining,
and sorption.
Straining
is defined as the permanent
physical trapping of colloidal particles in pore throats
that are too small to allow passage of colloidal particles.
Straining is a major removal process in porous media
with grain diameters smaller than 20-100 times the bio-
colloidal diameter (Xu et al., 2006).
5.5.4.1 Conventional Colloid Filtration Theory.
Conventional colloid filtration theory was originally
developed for application to engineered systems, such
as sand filters, but has also been applied to describe
the transport of microorganisms in groundwater. The
one-dimensional advection-diffusion-filtration equa-
tion can be expressed as
ρ
∂
∂
S
t
3
2
(
−
n
)
b
=
ηα
c
−
rS
(5.76)
c
n
d
g
∂
∂
c
t
ρ
b
∂
∂
S
t
∂
∂
c
x
∂
∂
2
c
(5.73)
+
+
v
=
D
2
n
x
where
r
is the detachment rate coefficient (mL−3T−1).
−3
T
−1
).
The coefficients
α
c
and
r
in Equation (5.76) depend on
the electrochemical properties of the cell membrane,
extracellular structures, mineral surfaces, and hydro-
dynamic forces; these parameters are typically esti-
mated from experimental observations by model fitting.
The combination of Equations (5.76) and (5.73)
describes the concentration of microbes in the aqueous
environment.
where
c
is the concentration of colloids suspended in
the pore water (mL
−3
),
t
is time (T),
ρ
b
is the bulk density
(mL
−3
),
n
is the porosity (dimensionless),
S
is the con-
centration of strained colloids (mm
−1
) (e.g., mg colloid/g
solid matrix),
v
is the seepage velocity (LT
−1
),
x
is the
coordinate parallel to flow (L), and
D
is the dispersion
coefficient (LT
−2
). In conventional colloidal filtration
theory, the straining rate (∂
S
/∂
t
) is a fixed fraction of the
mass flux and can be expressed in the form
5.5.4.3 Accounting for Dieoff.
The advection-
diffusion-filtration equation as given by Equation (5.73)
does not account for removal of biocolloids by dieoff.
To incorporate dieoff in both the conventional and
modified colloid filtration theory, dieoff coefficients are
typically included, and Equation (5.73) becomes (Ojha
et al., 2011)
ρ
b
n
∂
∂
S
t
=
fvc
(5.74)
where
f
is called the
filtration coefficient
(L
−1
), which is
assumed to be constant in time and space. In accordance
with the conventional colloid filtration theory, the filtra-
tion coefficient,
f
, can be related to the physical proper-
ties of the colloid and the porous medium by
∂
∂
c
t
ρ
∂
∂
S
t
∂
∂
c
x
∂
∂
2
c
λ
ρ
b
b
(5.77)
+
+
v
=
D
−
λ
c
−
n
S
c
s
n
x
2
3
2
(
−
n
)
where
λ
c
and
λ
s
are the decay coefficients (T) for the
aqueous phase and sorbed biocolloids, respectively.
f
=
η
(5.75)
d
g
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