Environmental Engineering Reference
In-Depth Information
The conservation equation for particles is given by
∂
t
ˆ
+
u
·
∇
ˆ
+
∇
·
J
=
0
,
(4.3)
which accounts for the advection of particles due to flow (
u
·
∇
ˆ
) and particle
diffusion (
J
), where
J
is the particle flux vector that is semi-empirically constructed
[
13
,
19
]. For particle-laden flows down an incline, effects of gravity and shear flow
govern the particle dynamics inside the thin film, leading to
sedimentation
[
4
] and
shear-induced migration
[
13
,
19
] of particles that opposes settling. The expressions
for
J
that account for these competing effects have been derived and experimentally
validated for the monodisperse case [
2
,
15
,
16
]. For a bidensity slurry in the same
geometry, the same physical effects of gravity and shear are present, with added
complexities due to the presence of two particle species. By combining previous
works [
7
,
15
,
16
,
25
] and recent experimental results [
10
], we construct a new
particle flux vector
J
that accounts for the mixing and sedimentation of two particle
species at varying rates.
Based on the formulation by [
22
,
25
], the flux of the
i
th particle species due to
sedimentation corresponds to
∇
·
⊡
⊤
2
g
d
2
ˆ
i
1
(ˁ
j
−
ˁ
l
)
ˆ
j
⊣
M
0
(ˁ
i
−
ˁ
l
)
+
⊦
,
J
grav
,
i
=
M
I
(4.4)
18
μ
l
ˆ
j
=
where
i
=
1
,
2.ThefirstterminEq.(
4.4
) refers to the self-mobility of particles,
M
0
∼
1
−
ˆ/ˆ
m
[
25
]. The second contribution to sedimentation comes from interaction
mobility,
M
I
−
ˆ)/μ(ˆ)
[
13
,
15
,
16
]. The total flux due to sedimentation,
J
grav
, is given by
J
grav
=
∼
f
(ˆ)
−
M
0
, where the hindrance function
f
(ˆ)
=
μ
l
(
1
J
grav
,
1
+
J
grav
,
2
.
As well as settling due to gravity, particles are subject to shear flow inside the
thin film and undergo two types of shear-induced diffusion processes [
11
,
13
]. The
first type—shear-induced “drift” diffusion—refers to the net drift of particles from
the regions of high to low total particle concentration and also from high to low
shear stress [
12
,
19
]. In the thin free-surface flows, this diffusive mechanism causes
particles to aggregate near the free surface where shear stress vanishes [
2
,
15
,
16
].
Since drift diffusion does not distinguish between particle types of equal size, we
use the empirical model for particle flux,
J
drift
,asusedin[
2
,
15
,
16
]:
K
c
∇
( ʳˆ)
−
d
2
4
K
v
ˆ ʳ
μ(ˆ)
d
μ
J
drift
=−
ˆ
∇
ˆ
,
(4.5)
d
1
u
where
K
c
and
K
v
are empirically determined constants, and
ʳ
=
4
∇
u
+
∇
is
the shear rate. The corresponding flux for each species is
J
drift
,
i
.
The second type of shear-induced diffusion is known as shear-induced “tracer”-
(or self-) diffusion [
1
,
5
,
7
,
13
,
23
,
25
]. Distinct from drift diffusion, it refers to the
=
J
drift
ˆ
i
/ˆ
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