Nonlinear Chemical Dose Controller Fall 2008-Summer 2009
Abstract
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h1. Nonlinear Chemical Dose Controller h2. Abstract {float:right|border=2px solid black|width=400px} !Non-linear CDC Design1.jpg|width=400px,align=top! h5. Figure 1: Draft nonlinear chemical dose controller design. {float} |
The
...
nonlinear
...
chemical
...
dose
...
controller
...
(CDC)
...
is
...
designed
...
for
...
turbulent
...
chemical
...
dose
...
flow
...
rates
...
to
...
be
...
used
...
in
...
conjunction
...
with
...
the
...
...
...
...
...
in
...
contrast
...
to
...
the
...
...
...
which
...
requires
...
that
...
the
...
chemical
...
flow
...
in
...
the
...
dosing
...
tube
...
be
...
laminar.
...
The
...
linear
...
CDC
...
uses
...
the
...
relationship
...
between
...
laminar
...
flow
...
and
...
major
...
losses
...
in
...
the
...
doser
...
tube
...
to
...
maintain
...
a
...
constant
...
chemical
...
dose
...
with
...
varying
...
plant
...
flow
...
rates.
...
However,
...
when
...
the
...
flow
...
in
...
the
...
dosing
...
tube
...
is
...
turbulent,
...
the
...
linear
...
relationship
...
no
...
longer
...
exists.
...
In
...
this
...
case,
...
a
...
nonlinear
...
CDC,
...
one
...
that
...
uses
...
a
...
combination
...
of
...
major
...
and
...
minor
...
losses
...
to
...
control
...
flow
...
rates,
...
can
...
be
...
used
...
to
...
maintain
...
a
...
constant
...
chemical
...
dose
...
with
...
the
...
varying
...
plant
...
flow
...
rates.
...
By
...
using
...
an
...
orifice
...
to
...
control
...
chemical
...
flow,
...
the
...
CDC
...
will
...
have
...
the
...
same
...
nonlinear
...
response
...
to
...
increasing
...
flow
...
as
...
the
...
plant
...
flow
...
rate
...
which
...
is
...
controlled
...
by
...
a
...
rectangular
...
orifice.
...
A
...
float
...
in
...
the
...
entrance
...
tank
...
controls
...
the
...
height
...
of
...
a
...
lever
...
arm.
...
The
...
dosing
...
orifice
...
is
...
located
...
at
...
the
...
opposite
...
end
...
of
...
the
...
lever
...
arm.
...
As
...
plant
...
flow
...
rates
...
increase,
...
the
...
float
...
rises,
...
and
...
the
...
dosing
...
orifice
...
falls
...
making
...
more
...
head
...
available
...
to
...
power
...
chemical
...
flow.
...
As
...
the
...
flow
...
rate
...
decreases
...
so
...
does
...
the
...
available
...
head
...
and
...
the
...
chemical
...
flow
...
rate
...
slows
...
down.
...
The
...
lever
...
arm
...
has
...
holes
...
drilled
...
into
...
the
...
top
...
of
...
it
...
so
...
that
...
the
...
dosing
...
orifice
...
may
...
be
...
inserted
...
into
...
one
...
of
...
these
...
holes
...
to
...
adjust
...
dosage
...
for
...
changes
...
in
...
turbidity.
...
Since
...
the
...
dosing
...
orifice
...
flows
...
directly
...
into
...
the
...
lever
...
arm,
...
the
...
alum
...
will
...
always
...
be
...
dispersed
...
into
...
the
...
same
...
spot,
...
no
...
matter
...
the
...
dose.
...
The
...
advantage
...
to
...
designing
...
a
...
nonlinear
...
CDC
...
is
...
that
...
higher
...
chemical
...
flow
...
rates
...
can
...
be
...
used.
...
With
...
an
...
alum
...
stock
...
solution
...
of
...
500
...
g/L
...
the
...
linear
...
flow
...
controller
...
can
...
deliver
...
a
...
maximum
...
chemical
...
dose
...
of
...
90
...
mg/L
...
to
...
plants
...
with
...
flow
...
rates
...
as
...
high
...
as
...
2000
...
L/min.
...
In
...
larger
...
water
...
treatment
...
plants
...
higher
...
chemical
...
flow
...
rates
...
are
...
necessary
...
to
...
dose
...
alum
...
and
...
chlorine.
...
A
...
flow
...
controller
...
with
...
a
...
higher
...
capacity
...
than
...
the
...
linear
...
CDC
...
flow
...
controller
...
is
...
necessary
...
for
...
the
...
nonlinear
...
CDC.
...
Head
...
loss
...
through
...
the
...
float
...
valve
...
orifice
...
limits
...
the
...
...
...
to
...
chemical
...
flows
...
less
...
than
...
400
...
mL/min.
...
Otherwise;
...
the
...
chemical
...
stock
...
tank
...
needs
...
to
...
be
...
placed
...
excessively
...
far
...
above
...
the
...
flow
...
controller
...
float
...
valve.
...
This
...
maximum
...
chemical
...
flow
...
is
...
adequate
...
for
...
...
...
,
...
...
,
...
...
,
...
and
...
...
...
given
...
their
...
flow
...
rates.
...
Two
...
flow
...
controllers
...
working
...
in
...
parallel
...
are
...
used
...
to
...
achieve
...
the
...
necessary
...
alum
...
dose
...
at
...
...
.
...
As
...
the
...
demand
...
for
...
larger
...
AguaClara
...
plants
...
grows,
...
larger
...
chemical
...
flow
...
rates
...
will
...
be
...
required.
Theory
The nonlinear doser uses a dosing orifice (minor losses) instead of a dosing tube (major losses) to control the relationship between changing plant flow rates and chemical dose. The flow rate through the CDC is related to the available head by the equation:
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h3. Theory The nonlinear doser uses a dosing orifice (minor losses) instead of a dosing tube (major losses) to control the relationship between changing plant flow rates and chemical dose. The flow rate through the 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
...
entrance
...
to
...
the
...
rapid
...
mix
...
tank
...
is
...
a
...
rectangular
...
orifice.
...
The
...
relationship
...
between
...
flow
...
rate
...
and
...
head
...
is
...
governed
...
by
...
the
...
equation:
| Latex |
|---|
}$$ Q_{Plant} = K_{orifice} \sqrt {2gh_{EtOrifice} } $${latex} where * {latex} |
where
Latex $$ Q_{Plant}$$
...
is
...
- the
...
- plant
...
- flow
...
- rate
...
Latex $$ h_{EtOrifice} $$
...
is
...
- the
...
- height
...
- of
...
- water
...
- above
...
- the
...
- entrance
...
- tank
...
- orfice
...
The
...
chemical
...
dose
...
to
...
the
...
plant
...
can
...
be
...
determined
...
by
...
a
...
simple
...
mass
...
balance:
| Latex |
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}$$C_p = {{C_c Q_{Cdc} } \over {Q_{Plant} }}$${latex} where * C ~c~ is the chemical stock concentration * C ~p~ is the chemical dose The CDC uses a lever arm to relate head above the centerline of the rectangular plant entrance orifice to head in the dosing orifice. This means that the available head for the dosing orifice is the same as the head controlling the plant flow rate. The increase in head links the chemical flow rate to the plant flow rate and the chemical dose will be constant as plant flow varies as long as the exponent of the head is the same for both the plant flow and the chemical flow. The dosing tube must be designed to minimize major losses so that the major losses that deviate from the {latex} |
where
- C c is the chemical stock concentration
- C p is the chemical dose
The CDC uses a lever arm to relate head above the centerline of the rectangular plant entrance orifice to head in the dosing orifice. This means that the available head for the dosing orifice is the same as the head controlling the plant flow rate. The increase in head links the chemical flow rate to the plant flow rate and the chemical dose will be constant as plant flow varies as long as the exponent of the head is the same for both the plant flow and the chemical flow.
The dosing tube must be designed to minimize major losses so that the major losses that deviate from the
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$$V = \sqrt {2gh}$$ |
...
| Latex |
|---|
$$V = \sqrt {2gh}$$ |
...
...
especially
...
significant
...
when
...
the
...
flow
...
through
...
the
...
dosing
...
tube
...
becomes
...
laminar.
...
The
...
dosing
...
tube
...
must
...
be
...
flexible
...
to
...
accommodate
...
the
...
lever
...
arm
...
motion
...
and
...
dose
...
adjustment.
...
The
...
flow
...
conduit
...
used
...
to
...
transport
...
the
...
chemical
...
flow
...
from
...
the
...
orifice
...
to
...
the
...
place
...
where
...
it
...
is
...
mixed
...
with
...
the
...
plant
...
flow
...
must
...
still
...
be
...
designed.
...
The
...
flow
...
conduit
...
must
...
be
...
designed
...
such
...
that
...
the
...
pressure
...
is
...
atmospheric
...
at
...
the
...
exit
...
of
...
the
...
dose
...
control
...
orifice.
Methods
Flexible Dosing Tube
The dosing tube diameter is based on the minimum available diameter that will have a head loss less than the given fraction of the maximum CDC head loss under conditions of the maximum chemical flow rate.
Dosing Orifice
The dosing orifice is designed to produce the difference in head loss between the maximum CDC head loss and the actual head loss in the flexible dosing tube:
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h3. Methods h4. Flexible Dosing Tube The dosing tube diameter is based on the minimum available diameter that will have a head loss less than the given fraction of the maximum CDC head loss under conditions of the maximum chemical flow rate. h4. Dosing Orifice The dosing orifice is designed to produce the difference in head loss between the maximum CDC head loss and the actual head loss in the flexible dosing tube: {latex}$$ h_l = K_{DoseOrifice} {{V_{DoseTube}^2 } \over {2g}} $${latex} where * h ~l~ the difference in head loss between the maximum CDC head loss and the actual head loss in the flexible dosing tube * K ~DoseOrifice~ is the required minor loss coefficient through the orifice * V ~DoseTube~ is the velocity in the dosing tube This head loss is equal to head loss in the vena contracta. Head loss in the vena contracta is modeled by the equation: {latex} |
where
- h l the difference in head loss between the maximum CDC head loss and the actual head loss in the flexible dosing tube
- K DoseOrifice is the required minor loss coefficient through the orifice
- V DoseTube is the velocity in the dosing tube
This head loss is equal to head loss in the vena contracta. Head loss in the vena contracta is modeled by the equation:
| Latex |
|---|
$$h_l = K_{Exit} {{V_{VenaContracta}^2 } \over {2g}}$${latex}
where
* h ~l~ is the head loss
* {latex} |
where
- h l is the head loss
Latex $$ K_{Exit}$$
...
is
...
- the
...
- minor
...
- loss
...
- coefficient
...
- from
...
- an
...
- exit
...
- to
...
- the
...
- atmosphere
...
- V VenaContracta is the velocity in the orifice vena contracta
| Latex |
|---|
$$ K_{Exit}$$ |
...
...
equal
...
to
...
one.
...
The
...
flow
...
rate
...
through
...
the
...
dosing
...
tube
...
is
...
known
...
and
...
velocities
...
in
...
the
...
vena
...
contracta
...
can
...
be
...
found
...
using
...
mass
...
conservation:
| Latex |
|---|
}$$A_{DoseTube} V_{DoseTube} = K_{Orifice} A_{DoseOrifice} V_{DoseOrifice} $${latex} where {latex} |
where
| Latex |
|---|
$$ K_{Orifice}$$ |
...
...
the
...
orifice
...
vena
...
contracta
...
coefficient
...
equal
...
to
...
0.63.
...
Using
...
the
...
three
...
equations
...
above
...
a
...
relationship
...
for
...
the
...
velocities
...
can
...
be
...
found:
| Latex |
|---|
}$$ K_{DoseOrifice} = \left[ {{1 \over {K_{Orifice} }}\left( {{{V_{DoseTube} } \over {V_{DoseOrifice} }}} \right)^2 } \right]^2 $${latex} {float:right|border=2px solid black|width=400px} !Chem dose error (QPlant).png|width=400px,align=top! h5. Figure 2: Dosing error from the nonlinear CDC for two maximum flow rates and for two chemical doses. The conditions were: 1 m long dosing tube, CDC head loss of 50 cm. {float} The chemical dose flow rate is not exactly proportional to the plant flow rate due to different exponential relationships between flow and head loss for the major losses. These differences arise from two sources. The most significant difference occurs when the flow in the dosing tube is laminar and thus the flow rate is proportional to the major head loss rather than to the square root of the head loss. The other source of error occurs for transitional flow where the friction factor decreases with increasing Reynolds number. The error due to the major loss contribution to the head loss is shown in figure 2. The nonlinear relationship between flow and head loss makes it more difficult to accurately control the chemical dose when the dose is low. \\ {latex} |
The chemical dose flow rate is not exactly proportional to the plant flow rate due to different exponential relationships between flow and head loss for the major losses. These differences arise from two sources. The most significant difference occurs when the flow in the dosing tube is laminar and thus the flow rate is proportional to the major head loss rather than to the square root of the head loss. The other source of error occurs for transitional flow where the friction factor decreases with increasing Reynolds number.
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{float:right|border=2px solid black|width=400px}
!Chem dose error (QPlant).png|width=400px,align=top!
h5. Figure 2: Dosing error from the nonlinear CDC for two maximum flow rates and for two chemical doses. The conditions were: 1 m long dosing tube, CDC head loss of 50 cm, major losses constrained to less than 10% of total head loss, stock concentration of 120 g/L, maximum chemical dose of 60 mg/L, and minimum plant flow that could be measured by the plant flow orifice was 20%. The [MathCAD file is attached|Nonlinear Chemical Dose Controller Fall08-Summer09^Nonlinear CD.xmcd].
{float} |
The error due to the major loss contribution to the head loss is shown in figure 2. The error is most significant for low chemical flow rates. The error can be reduced by decreasing the major losses, but that requires the use of larger dosing tubes.
The nonlinear relationship between flow and head loss makes it more difficult to accurately control the chemical dose when the dose is low.
| Latex |
|---|
$${{HL_{Cdc_{Min} } } \over {HL_{Cdc_{\max } } }} = \left( {{{Q_{Cdc_{Min} } } \over {Q_{Cdc_{\max } } }}} \right)^2 $$ |
For this reason it is recommended that the total head loss range be increased to approximately 50 cm for the nonlinear CDC. For a chemical dose that is 10% of the maximum chemical dose the head required by the dosing tube is 1% of the maximum head loss (0.5 cm). Maintaining a positioning error of less than 0.5 cm may be difficult and would require a large float to minimize errors due to changes in moment caused by the dosing tube. The CDC lever must also be accurately calibrated to minimize errors at low chemical dosing rates. Thus the maximum head loss for the CDC should be at least 50 cm.
Viscosity of Alum
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{latex}\\ For this reason it is recommended that the total head loss range be increased to approximately 50 cm for the nonlinear CDC. For a chemical dose that is 10% of the maximum chemical dose the head required by the dosing tube is 1% of the maximum head loss (0.5 cm). Maintaining a positioning error of less than 0.5 cm may be difficult and would require a large float to minimize errors due to changes in moment caused by the dosing tube. The CDC lever must also be accurately calibrated to minimize errors at low chemical dosing rates. Thus the maximum head loss for the CDC should be at least 50 cm. \\ h4. Viscosity of Alum {float:right|border=2px solid black|width=400px} [!alum viscosity.png|width=400px,align=top!|^Density and Kinematic viscosity of Alum.xmcd] h5. Figure 3: Viscosity of Alum. {float} |
The
...
...
...
...
is
...
only
...
slightly
...
larger
...
than
...
that
...
of
...
water
...
for
...
concentrations
...
used
...
in
...
stock
...
solutions
...
(Figure
...
3).
...
At
...
concentrations
...
above
...
550
...
g/L
...
the
...
viscosity
...
increases
...
rapidly
...
and
...
thus
...
concentrations
...
above
...
550
...
g/L
...
should
...
be
...
avoided
...
when
...
using
...
hydraulic
...
flow
...
control.
Lever Arm and Float
Since the head loss range required to accurately control the chemical dose at low chemical dosages is at least 50 cm, the length of the lever arm must be sufficient to handle that vertical motion.
Refer to LeverArmLength to determine a lever arm length and the maximum angle to use for a given elevation difference
The size of the float can be determined using a moment balance around the pivot of the lever arm. This is to ensure that a change of 20 cm in head in the entrance tank will cause a similar change in the relative height of the float. The float was sized using the same float sizing algorithm used by the linear CDC. For flows between 600 and 25,000 L/min the float diameter ranged between 6 and 16 in.
Flow Control module
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h4. Lever Arm and Float Since the head loss range required to accurately control the chemical dose at low chemical dosages is at least 50 cm, the length of the lever arm must be sufficient to handle that vertical motion. The size of the float can be determined using a moment balance around the pivot of the lever arm. This is to ensure that a change of 20 cm in head in the entrance tank will cause a similar change in the relative height of the float. The float was sized using the same float sizing algorithm used by the linear CDC. For flows between 600 and 25,000 L/min the float diameter ranged between 6 and 16 in. h4. Flow Control module {float:left|border=2px solid black|width=200px} !NonlinearFCM.jpg|thumbnail,align=top! h5. Figure 4: Flow Controller for Nonlinear module {float} |
When
...
choosing
...
what
...
size
...
float
...
valve
...
to
...
use,
...
the
...
constraints
...
are
...
the
...
maximum
...
flow
...
rate
...
through
...
the
...
valve,
...
the
...
allowable
...
head
...
loss
...
through
...
the
...
valve,
...
and
...
the
...
size
...
of
...
the
...
valve
...
orifice.
...
The
...
allowable
...
head
...
loss
...
through
...
the
...
valve
...
is
...
kept
...
small
...
to
...
avoid
...
excessively
...
elevated
...
stock
...
tanks
...
for
...
alum
...
and
...
chlorine.
...
Various
...
valve
...
orifices
...
sizes
...
are
...
...
.
...
Note
...
that
...
the
...
inlet
...
sizes
...
given
...
(a
...
pipe
...
thread
...
size)
...
are
...
not
...
the
...
orifice
...
dimension.
...
Functional
...
Range
...
The
...
functional
...
range
...
is
...
limited
...
by
...
the
...
float
...
valve orifice. The largest float valve manufactured by Kerick Valve has an orifice of 2.54 cm (1 inch). With a head loss of 30 cm the flow rate would be 0.77 L/s and the corresponding plant flow rate could be as high as 6000 L/s.