...
A
...
common
...
parameter
...
used
...
to
...
describe
...
sedimentation
...
is
...
the
...
capture
...
velocity.
...
Capture
...
velocity
...
is
...
the
...
velocity
...
of
...
the
...
floc
...
that
...
is
...
only
...
barely
...
settled-out,
...
or
...
"captured",
...
during
...
the
...
sedimentation
...
process.
...
To
...
gain
...
a
...
more
...
profound
...
understanding
...
of
...
this
...
crucial
...
parameter,
...
we
...
will
...
walk
...
through
...
the
...
derivation
...
of
...
the
...
capture
...
velocity
...
equation.
...
The
...
figure
...
below
...
illustrates
...
the
...
longest
...
path
...
that
...
a
...
floc
...
is
...
able
...
to
...
travel
...
in
...
a
...
sedimentation
...
tube
...
and
...
still
...
be
...
captured.
...
Figure 1.
...
Longest
...
possible
...
path
...
of
...
a
...
floc
...
through
...
a
...
sedimentation
...
tube.
...
The
...
capture
...
velocity
...
is
...
the
...
velocity
...
that
...
the
...
particle
...
must
...
travel
...
to
...
fall
...
a
...
distance
...
'x'
...
(see
...
figure
...
above)
...
in
...
the
...
same
...
amount
...
of
...
time
...
that
...
a
...
particle
...
traveling
...
at
...
| Latex |
|---|
...
\large\[V_\alpha\] |
...
...
travel
...
| Latex |
|---|
...
\large\[L + b\tan (\alpha )\] |
...
Now,
...
we
...
know
...
that
...
one
...
way
...
to
...
determine
...
the
...
travel
...
time
...
of
...
a
...
particle
...
is
...
to
...
divide
...
the
...
distance
...
travelled
...
by
...
the
...
particle
...
by
...
the
...
particle
...
velocity.
...
Therefore,
...
we
...
can
...
equate
...
the
...
following
...
travel
...
times:
| Latex |
|---|
}\large\[\frac{d}{{V_c \cos (\alpha )}} = \frac{{L + d\tan(\alpha )}}{{V_\alpha }} = \frac{{L\sin (\alpha ) + \frac{b}{{\cos (\alpha )}}}}{{V_{up} }}\]{latex} |
Solving
...
for
...
the
...
capture
...
velocity
...
by
...
equating
...
the
...
first
...
and
...
third
...
expressions,
...
we
...
find
...
that:
| Latex |
|---|
{latex}\large\[V_c = \frac{{dV_{up} }}{{L_{tube} \sin (\alpha )\cos(\alpha ) + d}}\]{latex} |
Where
...
L
...
is
...
the
...
length
...
of
...
the
...
tube,
...
d
...
is
...
the
...
inner
...
diameter
...
of
...
the
...
tube,
...
Vup
...
is
...
the
...
vertical
...
velocity
...
through
...
one
...
tube,
...
and
...
| Latex |
|---|
...
\large |
...
\[\alpha \] |
...
...
equal
...
to
...
sixty
...
degrees.
...
From
...
the
...
capture
...
velocity
...
equation
...
just
...
derived,
...
it
...
is
...
evident
...
that
...
capture
...
velocity
...
is
...
dependent
...
on
...
the
...
diameter,
...
length,
...
and
...
angle
...
of
...
the
...
tube
...
settlers,
...
as
...
well
...
as
...
the
...
flow
...
rate
...
through
...
the
...
settlers.
...