...
Nonlinear
...
Theory
...
The
...
nonlinear
...
chemical
...
doser
...
is
...
part
...
of
...
the
...
evolution
...
of
...
alum
...
dosing
...
techniques
...
in
...
an
...
effort
...
to
...
increase
...
the
...
maximum
...
flow
...
rate
...
capacity
...
of
...
AguaClara
...
plants
...
as
...
well
...
robustness
...
of
...
design
...
and
...
implementation.
...
A
...
frequent
...
problem
...
in
...
AguaClara
...
plants
...
was
...
the
...
occurrence
...
of
...
foam
...
in
...
the
...
flocculator
...
after
...
the
...
addition
...
of
...
alum
...
to
...
the
...
entrance
...
tank.
...
It
...
was
...
postulated
...
that
...
the
...
previous
...
free
...
fall
...
method
...
of
...
dosing
...
alum
...
was
...
contributing
...
to
...
the
...
occurrence
...
of
...
foam.
...
Any
...
new
...
dosing
...
mechanism
...
would
...
need
...
to
...
have
...
alum
...
dosed
...
below
...
the
...
level
...
of
...
water
...
in
...
the
...
entrance
...
tank.
...
Previous
...
linear
...
chemical
...
dosers
...
have
...
relied
...
upon
...
the
...
major
...
head
...
losses
...
caused
...
by
...
the
...
friction
...
of
...
the
...
dosing
...
tube
...
to
...
control
...
the
...
flow
...
of
...
alum.
...
The
...
important
...
point
...
in
...
distinguishing
...
the
...
need
...
to
...
move
...
to
...
nonlinear
...
flow
...
is
...
recognizing
...
the
...
relationship
...
between
...
head
...
loss
...
and
...
the
...
flow
...
rate
...
of
...
alum
...
in
...
the
...
two
...
dosing
...
methods.
...
In
...
previous
...
linear
...
dosing
...
designs,
...
the
...
flow
...
of
...
alum
...
is
...
proportional
...
to
...
major
...
head
...
losses
...
in
...
the
...
dosing
...
tube.
...
If
...
the
...
flow
...
rate
...
of
...
alum
...
were
...
to
...
increase
...
into
...
the
...
turbulent
...
range
...
then
...
the
...
head
...
loss
...
is
...
proportional
...
to
...
the
...
flow
...
rate
...
squared,
...
as
...
shown
...
in
...
the
...
figure
...
below.
...
Since
...
the
...
relationship
...
between
...
flow
...
rate
...
and
...
head
...
loss
...
is
...
not
...
linear
...
for
...
turbulent,
...
significant
...
dosing
...
errors
...
would
...
result
...
in
...
the
...
linear
...
dosing
...
scheme.
...
In
...
order
...
to
...
allow
...
there
...
to
...
be
...
turbulent
...
alum
...
flow
...
in
...
the
...
dosing
...
tube,
...
an
...
orifice
...
controlled
...
nonlinear
...
doser
...
is
...
now
...
being
...
used.
...
In
...
the
...
nonlinear
...
orifice
...
controlled
...
doser,
...
the
...
majority
...
of
...
the
...
head
...
losses
...
is
...
due
...
to
...
the
...
minor
...
losses
...
caused
...
by
...
the
...
orifice.
...
In
...
the
...
nonlinear
...
system
...
the
...
flow
...
rate
...
of
...
alum
...
is
...
proportional
...
to
...
the
...
square
...
root
...
of
...
h
...
in
...
both
...
laminar
...
and
...
turbulent
...
ranges.
...
This
...
homogeneity
...
in
...
relationships
...
allows
...
there
...
to
...
be
...
reliable
...
dosing
...
even
...
in
...
turbulent
...
ranges.
...
 | Laminar | Turbulent | |
|---|---|---|---|
Linear doser |
|
...
|
...
Orifice |
...
doser |
|
...
|
...
where:
...
| Wiki Markup |
|---|
{latex}$$\alpha $${latex} |
...
...
proportional
...
to
...
The
...
connection
...
between
...
the
...
flow
...
control
...
and
...
the
...
flow
...
measurement
...
aspect
...
of
...
the
...
dosing
...
mechanism
...
is
...
important
...
to
...
understand
...
the
...
evolution
...
of
...
dosing
...
mechanisms
...
in
...
AguaClara
...
plants.
...
Since
...
the
...
nonlinear
...
chemical
...
doser
...
has
...
the
...
same
...
relationship
...
between
...
flow
...
and
...
head
...
at
...
the
...
turbulent
...
ranges;
...
AguaClara
...
plants
...
can
...
be
...
scaled
...
up
...
to
...
much
...
higher
...
flow
...
rates
...
without
...
being
...
limited
...
by
...
the
...
turbulence
...
in
...
the
...
dosing
...
tube.
...
This
...
is
...
a
...
huge
...
advantage
...
of
...
the
...
nonlinear
...
system
...
because
...
it
...
expands
...
the
...
AguaClara
...
plants
...
capabilities
...
to
...
serve
...
much
...
larger
...
communities.
...
The
...
size
...
of
...
AguaClara
...
plants
...
is
...
no
...
longer
...
limited
...
to
...
the
...
flow
...
limitations
...
in
...
the
...
dosing
...
tube.
...
As
...
mentioned,
...
the
...
nonlinear
...
doser
...
uses
...
the
...
minor
...
losses
...
caused
...
by
...
the
...
orifice
...
instead
...
of
...
a
...
dosing
...
tube
...
(major
...
losses)
...
to
...
control
...
the
...
relationship
...
between
...
changing
...
plant
...
flow
...
rates
...
and
...
chemical
...
dose.
...
The
...
flow
...
rate
...
through
...
the
...
Chemical
...
Dose
...
Controller
...
(CDC)
...
is
...
related
...
to
...
the
...
available
...
head
...
by
...
the
...
equation:
| Wiki Markup |
|---|
{latex}$$Q_{Cdc} = K_{orifice}\sqrt {2gh_{Cdc} } $${latex} where * |
where
Wiki Markup {latex}$$Q_{Cdc} $${latex}
...
is
...
- the
...
- chemical
...
- flow
...
- rate
Wiki Markup {latex}$$ K_{orifice} $${latex}
...
is
...
- the
...
- orifice
...
- coefficient
...
- h
...
- is
...
- the
...
- available
...
- head
...
The
...
desired
...
chemical
...
dose
...
to
...
the
...
plant
...
can
...
be
...
determined
...
by
...
a
...
mass
...
balance:
| Wiki Markup |
|---|
{latex}$$C_p = {{C_c Q_{Cdc} } \over {Q_{Plant} }}$${latex} |
where
- C c is the chemical stock concentration
- C p is the chemical dose
The influent raw water leaves the entrance tank through the Rapid Mix Tube which is the entry point for the dosing of alum. In the Rapid mix tube, an orifice is located in the tube to generate macro and micro-scale mixing. The entry point for the dosing of alum with the raw water has been redesigned to be submerged in the entrance tank in order to possibly reduce the occurrence of foam in the flocculator. After conversations with Dan Smith, the AguaClara engineer in Honduras, there doesn't appear to be foam forming in the Agalteca plant, where this dosing system has been implemented.
The flow of water through an AguaClara plant can be modeled as minor losses due to flow expansions. The relationship between plant flow rate and head loss through the plant is governed by the minor loss equation shown below.
| Wiki Markup |
|---|
* C ~c~ is the chemical stock concentration * C ~p~ is the chemical dose The influent raw water leaves the entrance tank through the Rapid Mix Tube which is the entry point for the dosing of alum. In the Rapid mix tube, an orifice is located in the tube to generate macro and micro-scale mixing. The entry point for the dosing of alum with the raw water has been redesigned to be submerged in the entrance tank in order to possibly reduce the occurrence of foam in the flocculator. After conversations with Dan Smith, the AguaClara engineer in Honduras, there doesn't appear to be foam forming in the Agalteca plant, where this dosing system has been implemented. The flow of water through an AguaClara plant can be modeled as minor losses due to flow expansions. The relationship between plant flow rate and head loss through the plant is governed by the minor loss equation shown below. {latex}$$ h{}_{plant} = K_{plant} {{Q_{plant} ^2 } \over {2g}}$${latex} |
where:
...
| Wiki Markup |
|---|
{latex}$$ K_{plant} $${latex} |
...
...
plant
...
minor
...
loss
...
coefficient
...
The
...
equation
...
above
...
is
...
important
...
since
...
it
...
defines
...
the
...
relationship
...
between
...
plant
...
flow
...
and
...
head
...
loss
...
through
...
the
...
plant,
...
which
...
is
...
important
...
as
...
well
...
for
...
our
...
dosing
...
apparatus.
...
The
...
CDC
...
uses
...
a
...
lever
...
arm
...
with
...
a
...
float
...
and
...
counterweight
...
to
...
relate
...
the
...
dosing
...
to
...
the
...
changes
...
in
...
the
...
entrance
...
tank
...
water
...
level,
...
which
...
is
...
a
...
function
...
of
...
the
...
influent
...
flow
...
rate.
...
An
...
increase
...
in
...
head
...
loss
...
links
...
the
...
chemical
...
flow
...
rate
...
to
...
the
...
plant
...
flow
...
rate
...
and
...
the
...
chemical
...
dose
...
(mg/L)
...
will
...
be
...
constant
...
as
...
plant
...
flow
...
varies.As
...
can
...
be
...
seen
...
above,
...
the
...
equation
...
which
...
describes
...
the
...
flow
...
of
...
water
...
through
...
plant
...
and
...
the
...
orifice
...
equation
...
which
...
relates
...
flow
...
of
...
alum
...
in
...
the
...
dosing
...
system
...
both
...
have
...
the
...
same
...
relationship
...
between
...
flow
...
rate
...
and
...
head
...
loss.
...
Since
...
the
...
flow
...
rate
...
for
...
both
...
the
...
plant
...
and
...
alum
...
flow
...
are
...
each
...
related
...
by
...
the
...
square
...
root
...
of
...
the
...
head
...
loss,
...
the
...
two
...
flows
...
can
...
be
...
linked
...
through
...
the
...
float
...
in
...
the
...
entrance
...
tank.
...
Any
...
rise
...
For
...
a
...
detailed
...
step
...
by
...
step
...
description
...
of
...
the
...
steps
...
involved
...
with
...
measuring
...
the
...
plant
...
flow
...
rate
...
please
...
see
...
flow
...
measurement
...
section
...
of
...
the
...
Chemical
...
Dose
...
Controller
...
Manual.
...
The
...
dosing
...
tube
...
must
...
be
...
designed
...
to
...
minimize
...
major
...
losses
...
so
...
that
...
minor
...
losses
...
dominate
...
head
...
loss.
...
The
...
amount
...
of
...
major
...
losses
...
due
...
to
...
the
...
friction
...
in
...
the
...
dosing
...
tube
...
will
...
vary
...
based
...
on
...
the
...
flow
...
of
...
alum
...
going
...
through
...
the
...
dosing
...
tube,
...
with
...
higher
...
head
...
loss
...
occurring
...
at
...
higher
...
alum
...
flow
...
rates.
...
As
...
a
...
result,
...
it
...
is
...
not
...
desirable
...
to
...
set
...
the
...
size
...
of
...
the
...
orifice
...
based
...
on
...
the
...
varying
...
major
...
losses
...
through
...
the
...
dosing
...
tube,
...
so
...
the
...
head
...
loss
...
caused
...
by
...
the
...
orifice
...
needs
...
far
...
outweigh
...
major
...
losses
...
so
...
the
...
friction
...
in
...
the
...
tube
...
can
...
be
...
ignored.
...
As
...
a
...
result,
...
the
...
dosing
...
tube
...
needs
...
to
...
have
...
a
...
large
...
enough
...
diameter
...
so
...
that
...
the
...
effects
...
of
...
major
...
losses
...
are
...
not
...
significant
...
when
...
compared
...
to
...
the
...
minor
...
losses
...
causes
...
by
...
the
...
orifice.
...
The
...
graph
...
below
...
illustrates
...
the
...
relative
...
contributions
...
of
...
head
...
loss
...
caused
...
by
...
a
...
large
...
2
...
mm
...
orifice
...
cap
...
as
...
opposed
...
to
...
the
...
major
...
losses
...
through
...
the
...
dosing tube.
| Wiki Markup |
|---|
tube. !Major loss contribution.png|border=2px solid black,align=center,width=500px|align=center,width=500px,height=350px! {center:class=myclass} h5.Figure {center} |
Figure 1:
...
Major
...
loss
...
contribution
...
to
...
the
...
total
...
head
...
loss
...
As
...
can
...
be
...
seen
...
in
...
the
...
graph
...
above,
...
the
...
relative
...
contribution
...
of
...
major
...
losses
...
to
...
the
...
total
...
head
...
loss
...
in
...
the
...
dosing
...
system
...
at
...
a
...
maximum,
...
using
...
a
...
2
...
mm
...
diameter
...
orifice
...
(the
...
large
...
orifice),
...
is
...
minor,
...
approximately
...
2%.
...
With
...
the
...
contribution
...
of
...
the
...
major
...
losses
...
less
...
than
...
2%of
...
the
...
total
...
losses,
...
so
...
it
...
can
...
effectively
...
be
...
ignored
...
in
...
the
...
sizing
...
of
...
the
...
dosing
...
orifice.
...