...
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 |
|---|
\large\[V_c = \frac{{dV_{up} }}{{L_{tube} \sin (\alpha )\cos(\alpha ) + d}}\] |
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\[\alpha \] |
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.