Geoscience Reference
In-Depth Information
n
k
considering the aggregation and disaggregation of flocs is (see Tsai
et al
., 1987)
ε
2
j
−
∂(ω
sf
,
k
δ
i
3
n
k
)
∂
n
k
∂
+
∂(
u
i
n
k
)
∂
=
∂
∂
s
∂
n
k
1
+
k
α
β
lj
n
l
n
j
lj
t
x
i
∂
x
i
x
i
∂
x
i
+
l
=
∞
−
1
α
jk
β
jk
n
j
n
k
+
k
γ
kj
n
j
j
=
j
>
−
k
γ
jk
n
k
(11.27)
<
j
β
ij
is the frequency function for collision between flocs of size fractions
i
and
where
j
,
ij
is the frequency function of
disaggregation of flocs from size fraction
j
to
i
. The second term on the right-hand side
of Eq. (11.27) accounts for the formation of size fraction
k
by collision of smaller flocs.
The third term quantifies the loss of size fraction
k
due to collision with other flocs.
The fourth term represents the generation of size fraction
k
due to disaggregation of
larger flocs. The fifth term denotes the loss of size fraction
k
due to disaggregation to
smaller flocs.
Particle collision may be caused by Brownian motion, fluid shear, and differential
settling. The original collision theory was due to Smoluchowski (1917), and additional
work was done by Camp and Stein (1943). The frequency function
α
ij
is the probability of cohesion after collision, and
γ
β
ij
for different
collisions is presented below (see Tsai
et al
., 1987).
For Brownian motion,
2
2
3
kT
µ
(
d
i
+
d
j
)
β
=
(11.28)
ij
d
i
d
j
10
−
23
N
·
◦
K
−
1
),
T
is the absolute tem-
where
k
is the Boltzmann constant (1.38
×
·
m
perature (
◦
K),
is the dynamic viscosity of the fluid, and
d
i
and
d
j
are the diameters
of the two colliding particles.
For fluid shear,
µ
G
6
(
3
β
=
d
i
+
d
j
)
(11.29)
ij
where
G
is the mean velocity gradient in the fluid. For a turbulent fluid,
G
can
be approximated by
1
/
2
, in which
(ε/ν)
ε
is the energy dissipation rate and
ν
is the
kinematic viscosity of the fluid (Saffman and Turner, 1956).
For differential settling,
=
π
g
72
2
d
i
d
j
)
β
µ
(ρ
−
ρ
f
)(
d
i
+
d
j
)
(
−
(11.30)
ij
s
Fig. 11.11 shows the frequency functions for particles of different sizes colliding
with a 1-
µ
m particle and a 25-
µ
m particle due to Brownian motion, fluid shear, and
