Environmental Engineering Reference
In-Depth Information
=
=
=
=
corresponding monodisperse problem (
X
1for
i
1 and
X
0for
i
2) and
30
ⓦ
, these values
ˆ
c
,
1
<ˆ
c
,
2
since the second particle is heavier. For
ʱ
=
note that
are
521 based on [
15
].
For the bidisperse system, we begin by discussing the structure of
X
ˆ
c
,
1
=
0
.
459 and
ˆ
c
,
2
=
0
.
(
s
)
and the
mixing behaviour between particle species. As
s
increases from 0 to 1,
X
(
s
)
consists
of an interval with
X
≈
0, followed by a transition region centred at
s
=
s
tr
, such
that
X
(
s
tr
)
=
1
/
2, and finally an interval with
X
≈
1. The ODE (
4.10
) can be
approximated near
s
tr
as
X
≈
C
−
1
X
(
1
−
X
),
(4.12)
−
1
exp
s
tr
−
C
−
1
.Here
C
is the
which has an explicit solution,
X
(
s
)
=
1
+
constant given by evaluating all other variables at
s
=
s
tr
:
1
9tan
ʱ
−
ˆ(
s
tr
)
ˆ
m
C
=
D
tr
(ˆ(
s
tr
))˃ (
s
tr
)
.
2
(ˁ
s
,
2
−
ˁ
s
,
1
)
In order to quantify the amount of mixing between two particle species, we define
the width of the mixing layer,
w
, to be the interval for which 0
.
05
<
X
<
0
.
95.
Based on the solution to (
4.12
), we find that
w
≈
5
.
9
C
, valid for
w
1. Since
the value of
C
primarily depends on tan
ʱ
, it can be shown that the mixing layer
width,
w
, scales with tan
(Fig.
4.3
, right), and is approximately linear where the
solution profiles are insensitive to changes in angle. This suggests that there will be
little mixing for small inclination angles. Experimentally, [
10
] observed the bidensity
slurry at low inclination angles to stratify into three layers of heavy particles, light
particles, and clear fluid. This results in three distinct fronts flowing down the plane
(Fig.
4.1
, bottom right). At higher inclination angles they observed a 'ridged' regime
with more mixing of particles, consistent with our theoretical predictions (Fig.
4.1
,
top right).
In order to investigate the bifurcation behaviour of bidensity slurries, we now
consider the total concentration
ʱ
ˆ(
s
)
and the individual concentrations,
ˆ
1
(
s
)
and
ˆ
2
(
. Analogous to the monodisperse system, we call a solution 'settled' if neither
species of particles are present up to the surface (i.e.
s
)
ˆ
=
0forsome
s
∈[
0
,
1
]
), and
'ridged' if particles (of either kind) aggregate at the surface (
1).
Like the monodisperse case, the settled regime (
S
) corresponds to the case where
ˆ
ₒ
ˆ
m
as
s
ₒ
ˆ
is monotone decreasing. Monotonicity of solutions is important for analysis of the
dynamic problem, which motivates a careful description of the equilibrium profiles
in [
26
]. For the bidisperse system,
ˆ
is not necessarily monotonic in the ridged
regime, but the individual concentrations
ˆ
2
undergo similar transitions from
decreasing to increasing as in the monodisperse case. Within the ridged regime, there
exist critical concentrations
ˆ
1
and
ˆ
A
,ˆ
B
and
ˆ
C
as functions of
X
0
, such that the profiles
for
ˆ
2
change from decreasing to mixed signs to increasing. This further
partitions the ridged regime into three distinct sub-regions (
R
A
,
R
B
, and
R
C
), as
summarized in Fig.
4.4
. We now discuss each region in greater detail.
ˆ
,
ˆ
1
, and
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