...
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:
...
| Latex |
|---|
...
$$\alpha $$ |
...
...
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:
| Latex |
|---|
}$$Q_{Cdc} = K_{orifice}\sqrt {2gh_{Cdc} } $${latex} where * {latex} |
where
Latex $$Q_{Cdc} $$
...
is
...
- the
...
- chemical
...
- flow
...
- rate
...
Latex $$ K_{orifice} $$
...
is
...
- the
...
- orifice
...
- coefficient
...
- h
...
- is
...
- the
...
- available
...
- head
...
The
...
desired
...
chemical
...
dose
...
to
...
the
...
plant
...
can
...
be
...
determined
...
by
...
a
...
mass
...
balance:
| 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. {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.
| Latex |
|---|
}$$ h{}_{plant} = K_{plant} {{Q_{plant} ^2 } \over {2g}}$${latex} |
where:
...
| Latex |
|---|
...
$$ K_{plant} $$ |
...
...
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.
| 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.