...
Nonlinear
...
Chemical
...
Dose
...
Controller
...
Fall
...
2008-Summer 2009
Abstract
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2009 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 * |
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
...
entrance
...
to
...
the
...
rapid
...
mix
...
tank
...
is
...
a
...
rectangular
...
orifice.
...
The
...
relationship
...
between
...
flow
...
rate
...
and
...
head
...
is
...
governed
...
by
...
the
...
equation:
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{latex}$$ Q_{Plant} = K_{orifice} \sqrt {2gh_{EtOrifice} } $${latex} where * |
where
Wiki Markup {latex}$$ Q_{Plant}$${latex}
...
is
...
- the
...
- plant
...
- flow
...
- rate
Wiki Markup {latex}$$ h_{EtOrifice} $${latex}
...
is
...
- the
...
- height
...
- of
...
- water
...
- above
...
- the
...
- entrance
...
- tank
...
- orfice
...
The
...
chemical
...
dose
...
to
...
the
...
plant
...
can
...
be
...
determined
...
by
...
a
...
simple
...
mass
...
balance:
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{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 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 |
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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{latex}$$V = \sqrt {2gh}$${latex} |
...
...
do
...
not
...
cause
...
excessive
...
errors.
...
The
...
deviation
...
from
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|---|
{latex}$$V = \sqrt {2gh}$${latex} |
...
...
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: |
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:
| Wiki Markup |
|---|
{latex}$$h_l = K_{Exit} {{V_{VenaContracta}^2 } \over {2g}}$${latex}
where
* h ~l~ is the head loss
* |
where
- h l is the head loss
Wiki Markup {latex}$$ K_{Exit}$${latex}
...
is
...
- the
...
- minor
...
- loss
...
- coefficient
...
- from
...
- an
...
- exit
...
- to
...
- the
...
- atmosphere
...
- V VenaContracta is the velocity in the orifice vena contracta
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|---|
{latex}$$ K_{Exit}$${latex} |
...
...
equal
...
to
...
one.
...
The
...
flow
...
rate
...
through
...
the
...
dosing
...
tube
...
is
...
known
...
and
...
velocities
...
in
...
the
...
vena
...
contracta
...
can
...
be
...
found
...
using
...
mass
...
conservation:
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|---|
{latex}$$A_{DoseTube} V_{DoseTube} = K_{Orifice} A_{DoseOrifice} V_{DoseOrifice} $${latex} where |
where
| Wiki Markup |
|---|
{latex}$$ K_{Orifice}$${latex} |
...
...
the
...
orifice
...
vena
...
contracta
...
coefficient
...
equal
...
to
...
0.63.
...
Using
...
the
...
three
...
equations
...
above
...
a
...
relationship
...
for
...
the
...
velocities
...
can
...
be
...
found:
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|---|
{latex}$$ K_{DoseOrifice} = \left[ {{1 \over {K_{Orifice} }}\left( {{{V_{DoseTube} } \over {V_{DoseOrifice} }}} \right)^2 } \right]^2 $${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.
| Wiki Markup |
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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.
...
| Wiki Markup |
|---|
{latex}$${{HL_{Cdc_{Min} } } \over {HL_{Cdc_{\max } } }} = \left( {{{Q_{Cdc_{Min} } } \over {Q_{Cdc_{\max } } }}} \right)^2 $${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.
Viscosity of Alum
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|---|
\\ 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. Refer to [LeverArmLength|Nonlinear Chemical Dose Controller Fall08-Summer09^LeverArmLength.xmcd] 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. 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
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choosing
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what
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size
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float
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valve
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to
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use,
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the
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constraints
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are
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the
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maximum
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flow
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rate
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through
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the
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valve,
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the
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allowable
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head
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loss
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through
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the
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valve,
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and
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the
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size
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of
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the
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valve
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orifice.
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The
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allowable
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head
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loss
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through
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the
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valve
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is
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kept
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small
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to
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avoid
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excessively
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elevated
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stock
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tanks
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for
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alum
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and
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chlorine.
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Various
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valve
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orifices
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sizes
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are
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.
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Note
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that
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the
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inlet
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sizes
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given
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(a
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pipe
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thread
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size)
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are
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not
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the
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orifice
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dimension.
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Functional
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Range
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The
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functional
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range
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is
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limited
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by
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the
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float
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valve
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orifice.
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The
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...
...
...
...
...
...
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has
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an
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orifice
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of
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2.54
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cm
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(1
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inch).
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With
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a
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head
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loss
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of
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30
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cm
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the
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flow
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rate
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would
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be
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0.77
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L/s
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and
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the
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corresponding
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plant
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flow
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rate
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could
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be
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as
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high
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as
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6000
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L/s.
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