Environmental Engineering Reference
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and density of the particles in addition to the ambient
flow velocity. The process by which suspended particles
settle to the bottom of water bodies is called
sedimenta-
tion
. Consider the case of a spherical particle of diam-
eter
D
and density
ρ
s
settling in a fluid of density
ρ
f
, the
settling velocity,
v
s
, satisfies the following particle
momentum equation
2
α
ρ ρ
(
/
−
1)
gD
s
f
v
=
(3.175)
s
18
ν
where
α
is a dimensionless
form factor
that measures
the effect of particle shape, and
α
= 1 for spherical par-
ticles. The form factor,
α
, sometimes called the
spheric-
ity
, is defined as the ratio of the surface area of a sphere
having the same volume as the particle to the surface
area of the particle. Particles in natural waters have
complex shapes and typically
α
< 1; however, beach
sand tends to be approximately spherical, in which case
α
≈ 1. The settling velocity given by the Stokes equation
(Eq. 3.175) is called the
Stokes velocity
. Settling veloci-
ties that are of interest in natural waters are given in
Table 3.2 (Burns and rosa, 1980; Chapra, 1997; Wetzel,
1975). As a general rule, settling is unimportant for par-
ticles smaller than about 1
μ
m in diameter (clays). Par-
ticles in water must be larger than about 10
μ
m in
diameter to settle through distances of several centime-
ters in time scales of an hour or less, and particles in this
class are sometimes called
settleable solids
. In cases
where the ambient water moves with a horizontal veloc-
ity,
V
, suspended particles tend to move horizontally at
the velocity,
V
, and vertically at the settling velocity.
Suspended solids in natural waters have two primary
sources: surface runoff from drainage basins and as
products of photosynthesis. The suspended-solids con-
centration in natural waters typically range from below
1 mg/L in clear waters to over 100 mg/L in highly turbid
waters. Suspended solids derived from photosynthetic
processes tend to be higher in organic matter and less
dense than suspended solids derived from surface
3
3
2
π
D dv
dt
π
D
1
2
π
D
v
s
s
ρ
=
(
ρ
−
ρ
)
g
−
C
ρ
2
s
s
f
D f
6
6
4
rate of change of momentum
submerged weight
drag force
(3.171)
where
g
is gravity, and
C
D
is the drag coefficient. Equa-
tion (3.171) equates the rate of change of momentum
of the particle to the submerged weight of the particle
minus the drag force. As the settling velocity increases,
it ultimately reaches the
terminal velocity
at which point
the submerged weight of the particle is equal to the drag
force,
dv
s
/
dt
= 0, and Equation (3.171) becomes
1
2
4
3
gD
(
ρ ρ
ρ
−
)
(3.172)
s
f
v
=
s
C
f D
where the drag coefficient,
C
D
, depends on the particle
reynolds number, re
p
=
v
s
D
/
ν
, according to the follow-
ing empirical relations
24
3
+
+
0.34 1
<
Re
<
10
4
p
Re
Re
p
p
C
D
=
(3.173)
24
Re
<
1
p
Re
p
TABLE 3.2. Typical Settling Velocities in Natural Waters
other formulas have also been suggested for cases in
which re
p
> 1, for example, Pazwash (2007) proposed
the relation
Diameter
(
μ
m)
Settling
velocity
a
(m/d)
Particle Type
Phytoplankton:
Cyclotella meneghiniana
2
3
2
0.08 (0.24)
Re
=
24
p
Thalassiosira nana
4-5
0.1-0.28
C
D
1
+
,
for
1
<
Re
<
1000
(3.174)
p
Re
6
Scenedesmus quadricauda
8
0.27 (0.89)
p
Asterionella formosa
25
0.2 (1.48)
19-34
0.39-0.21
Thalassiosira rotula
For very large granular particles, such as coarse sand
and gravel, where re
p
> 1000, the drag coefficient,
C
D
approaches an asymptote of
C
D
= 1.2. It is apparent that
the determination of the settling velocity requires a
simultaneous solution of Equations (3.172) and (3.173).
Cases where re
p
< 1 roughly correspond to particle
diameters less than or equal to 0.1 mm (fine sand and
silt), and in these cases, the combination of Equations
(3.172) and (3.173) yields the so-called
Stokes equation
,
which is given by
Coscinodiscus lineatus
50
1.9 (6.8)
Melosira agassizii
55
0.67 (1.87)
Rhizosolenia robusta
84
1.1 (4.7)
Particulate organic carbon
1-10
0.2
10-64
1.5
>64
2.3
Silt
10-20
3-30
Clay
2-4
0.3-1
a
Numbers in parentheses are for stationary phase of microbial
growth.
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