Environmental Engineering Reference
In-Depth Information
runoff. Caution should be used in applying the Stokes
equation to calculate the settling velocity of living par-
ticles, since some phytoplankton, such as blue-green
algae, can become buoyant due to the development of
internal gas vacuoles.
Many contaminants sorb strongly onto suspended
particles, so that prediction of the fate and transport of
suspended sediments is essential for describing the fate
and transport of these sorbing contaminants in natural
waters. Heavy metals and hydrophobic organic com-
pounds, such as PCBs, are two classes of contaminants
that sorb strongly onto suspended sediments. Microbial
indicators of fecal contamination such as
Enterococcus
spp. and
Escherechia coli
also sorb significantly to sus-
pended particles.
removal rate of suspended particles
=
v c
s
1 25 0 05
0 0625
=
( .
)( .
)
=
.
kg/d m
⋅
2
Since heavy metals are attached to the sediment at
the rate of 1 g/kg, the removal rate of heavy metals
is given by:
removal rate of heavy metals
=
=
( )( .
1 0 0625
0 0625
)
2
.
g/d m
⋅
3.5 TURBULENT DIFFUSION*
Thus far, it has been asserted that mixing at the molecu-
lar level and mixing at the macroscopic level caused by
turbulent velocity fluctuations can both be described by
Fick's law. As a consequence, the advection-diffusion
equation can be used to describe both processes. In this
section, we take a closer look at the nature of turbulence
and the rationale for assuming that mixing in turbulent
flows is Fickian.
Turbulent flows are characterized by random velocity
fluctuations and so must necessarily be described in
statistical terms. Typically, local velocities in turbulent
flows can be characterized by a mean and standard
deviation, with the standard deviation of velocity
fluctuations commonly referred to as the
turbulence
intensity
.
The general diffusion equation given by Equation
(3.15) applies to all tracers in all incompressible fluid
flows, regardless of whether the flow is laminar or tur-
bulent. However, in order to apply the diffusion equa-
tion, the ambient velocity field must be known, and
therein enters the role of turbulence. For most flows, and
particularly in the case of turbulent flows, a detailed
description of the flow field is not possible either because
of uncertainty or randomness in the velocity field. under
these conditions, the tracer concentration,
c
, the ambient
fluid velocity,
V
, and the source mass flux,
S
m
, in the
general diffusion equation must be treated as random
variables. These quantities can be expressed in terms on
their expected values and perturbations as follows
EXAMPLE 3.21
Analysis of water from a lake indicates a suspended-
solids concentration of 50 mg/L. The suspended parti-
cles are estimated to have an approximately spherical
shape with an average diameter of 4
μ
m and a density
of 2650 kg/m
3
. (a) If the water temperature is 20
°
C,
estimate the settling velocity of the suspended particles.
(b) If the suspended particles are mostly clay, compare
your estimate of the settling velocity with the data in
Table 3.2. (c) If there is 1 g of heavy metal ion per kilo-
gram of suspended particles, determine the rate at which
heavy metals are being removed from the lake by
sedimentation.
Solution
(a) From the data given,
α
= 1 (spherical particles),
ρ
s
= 2650 kg/m
3
,
ρ
w
= 998 kg/m
3
at 20°C,
D
= 4
μ
m =
4 × 10
−6
m, and
v
w
= 1.00 × 10
−6
m
2
/s. Substituting into
the Stokes equation (Eq. 3.175) gives
α
ρ ρ
(
/
−
1)
gD
2
s
w
v
=
s
18
ν
w
−
6 2
( )
(
2650 998 1 9 81 4 10
18 1 00
/
−
)( .
)(
×
)
=
1
( .
×
10
−
6
)
=
1 44 10
1 25
.
.
×
−
5
m/s
=
m/d
c
( , )
x
t
=
c
( , )
x
t
+ ′
c
( , )
x
t
(3.176)
(b) This result is consistent with the settling velocities
for clay-sized particles shown in Table 3.2, which
indicates that a 4
μ
m clay particle will have a set-
tling velocity on the order of 1 m/day.
(c) Since the concentration,
c
, of suspended particles is
50 mg/L = 0.05 kg/m
3
, the rate at which sediment is
accumulating on the bottom of the lake is given by:
(3.177)
V
( , )
x
t
=
V
( , )
x
t
+ ′
v
( , )
x
t
i
i
i
(3.178)
S
( , )
x
t
=
S
( , )
x
t
+ ′
S
( , )
x
t
m
m
m
*
This section contains advanced fundamentals that can be omitted in
a first course on fate and transport.
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