Environmental Engineering Reference
In-Depth Information
random motion of particles in shear that occurs even in the absence of concentration
gradient and leads to zero net drift of particles. While tracer diffusion does not affect
ˆ
, it governs how one particle species mixes with the other in bidensity suspensions,
resulting in the flux of individual species:
ˆ
i
ˆ
d
2
4
=−
ʳ
J
tracer
,
i
D
tr
(ˆ)ˆ
∇
,
(4.6)
where
D
tr
(ˆ)
is the tracer diffusivity and dictates the extent of mixing in our model.
In the limit of dilute suspensions, [
11
,
13
] proposed the empirical expression
D
tr
=
ˆ
2
2. For large concentrations, numerical simulations and experiments [
23
] suggest
that the tracer diffusivity becomes constant beyond a value
/
ˆ
tr
≈
0
.
4. Therefore, we
1
2
tr
}
use the expression:
D
tr
(ˆ)
=
.
Combining Eqs. (
4.4
)-(
4.6
) yields the total flux of the
i
th species,
J
i
,
2
min
{
ˆ
,ˆ
J
grav
,
i
+
ˆ
i
ˆ
J
i
=
J
drift
+
J
tracer
,
i
,
(4.7)
=
J
grav
+
and the total flux
J
of both species simply corresponds to
J
J
drift
. Notably,
tracer diffusion drops out of the total particle flux (i.e.
J
tracer
,
1
+
0),
justifying its neglect in modelling monodisperse slurries [
2
,
15
,
16
,
19
]. In addition,
Brownian diffusion is not included in particle fluxes by assuming a large Péclet
number, or Pe
J
tracer
,
2
=
d
2
=
ʳ
/
D
1, where
D
is the solvent diffusivity.
4.2.1 Thin Film Approximations and Equilibrium Theory
A thin film geometry [
18
] gives us the following dimensionless variables:
J
x
ʴ
J
z
u
H
2
d
2
U
0
1
H
(ʴ
1
U
0
w
ʴ
J
(
x
,
z
)
=
x
,
z
) ,
u
=
,
,
=
,
,
H
μ
l
U
0
μ
μ
l
,
=
ˁ
i
−
ˁ
l
ˁ
l
p
=
p
,
μ
=
ˁ
s
,
i
,
where
H
and
L
are the characteristic film thickness and axial length scale, respec-
tively, and
U
0
H
2
g
sin
ʱ/ʽ
l
. Hats denoting the dimensionless quantities will be
subsequently dropped for brevity. In the thin film limit of
=
ʴ
≡
H
/
L
1, the
momentum equation in the axial direction reduces to
˃
=−
(
1
+
ˁ
s
,
1
ˆ
1
+
ˁ
s
,
2
ˆ
2
),
(4.8)
where
z
is the dimensionless shear stress, and the prime denotes
the derivative with respect to
z
. In addition, we assume
˃
=
μ(ˆ)∂
u
/∂
2
ʴ
(
d
/
H
)
1, which
Search WWH ::
Custom Search