LT2ESWTR
Toolbox
Guidance
Manual
Proposal
Draft
D­
1
June
2003
Appendix
D
Derivation
of
Extended
CSTR
Equations
The
discussion
presented
in
the
document
used
some
key
equations
and
relied
on
specific
assumptions.
In
this
appendix,
one
key
equation
is
derived,
and
one
key
assumption
is
discussed
and
justified.

D.
1
Derivation
of
the
Equation
Used
to
Calculate
k*

In
Appendix
B,
Equation
B­
2
expressed
the
value
of
k*
between
two
points
1
and
2
as
shown
by
Equation
D­
1:

[
]











 





	
	



×
=






 
 
 
 
1
2
1
1
2
1
2
1
2
1
*
2
1
N
C
C
Volume
Q
N
k
(
D­
1)

Equation
D­
1
is
a
transformation
from
the
equation
of
first­
order
decay
across
a
series
of
N
equal­
size
CSTRs:

2
1
2
1
*
2
1
1
2
1
1
 











	
	



+
=





	
	



 
 
N
N
HDT
k
C
C
(
D­
2)

The
derivation
of
this
equation
can
be
found
in
many
reference
texts
on
modeling
chemical
reactors
(
e.
g.,
Froment
et
al.,
1990;
Levenspiel,
1999).
Since
HDT
is
equal
to
the
volume
between
locations
1
and
2,
[
Volume]
1­
2,
divided
by
the
flowrate,
Q,
then
Equation
D­
2
is
transformed
to
Equation
D­
3:

[
]
2
1
2
1
2
1
*
2
1
1
2
1
1
 













	
	



×
+
=





	
	



 
 
 
N
N
Q
Volume
k
C
C
(
D­
3)

Therefore,

[
]









=





	
	











×
+
 

 
 
 
2
1
2
1
2
1
*
2
1
2
1
1
C
C
N
Q
Volume
k
N
(
D­
4)

then,
Appendix
D
 
Derivation
of
Extended
CSTR
Equations
LT2ESWTR
Toolbox
Guidance
Manual
Proposal
Draft
D­
2
June
2003
[
]
2
1
1
2
1
2
1
2
1
*
2
1
1
 









=





	
	











×
+
 
 
 
N
C
C
N
Q
Volume
k
(
D­
5)

then,

[
]









 





	
	



=





	
	



×
 

 
 
 
1
2
1
1
2
1
2
1
2
1
*
2
1
N
C
C
N
Q
Volume
k
(
D­
6)

and
then,

[
]








 





	
	



×
=
 

 
 
 
1
2
1
1
2
1
2
1
2
1
*
2
1
N
C
C
Volume
N
Q
k
(
D­
7)

As
noted,
Equation
D­
2
is
based
on
the
fundamental
assumption
that
the
hydrodynamic
profile
through
the
volume
separating
locations
1
and
2
can
be
approximated
by
a
series
of
N
equal­
size
CSTRs.
If
equal­
size
chambers
separate
locations
1
and
2,
then
each
chamber
is
somewhat
conservatively
assumed
to
be
an
ideal
CSTR,
with
HDT
=
[
Volume]/
Q,
and
the
value
of
N
in
the
above
derivation
is
set
equal
to
the
number
of
chambers
between
locations
1
and
2.
However,
it
was
recognized
that
not
all
ozone
contactors
are
configured
with
equal­
size
chambers
in
series.
It
is
possible
to
treat
each
chamber
as
its
own
CSTR
and
have
a
series
of
unequal­
size
CSTRs.
An
expression
of
C2/
C1
similar
to
that
shown
in
Equation
D­
2
is
still
possible.
For
example,
if
locations
1
and
2
were
separated
by
three
CSTRs
with
HDT
values
of
HDTa,
HDTb,
and
HDTc,
the
ratio
of
C2/
C1
for
a
first­
order
decay
reaction
can
still
be
expressed
as:

(
)
[
]
(
)
[
]
(
)
[
]
c
b
a
HDT
k
HDT
k
HDT
k
C
C
*
2
1
*
2
1
*
2
1
1
2
1
1
1
1
1
1
 
 
 
+
×
+
×
+
=









(
D­
8)

Or
in
general
terms,

[
]
 
 
+
=









i
i
HDT
k
C
C
)
(
1
1
1
2
(
D­
9)

Unfortunately,
it
is
not
possible
transform
Equation
D­
9
to
derive
a
simple
linear
expression
of
k*
as
a
function
of
the
other
measured
parameters
when
the
number
of
chambers
is
greater
than
three.
To
maintain
a
singular
methodology
for
any
number
of
chambers,
and
to
allow
the
calculation
to
be
performed
in
conventional
spreadsheets
and
plant
computer
control
systems,
a
compromise
was
to
assume
equal­
volume
CSTRs.
With
this
assumption,
Equation
D­
1
is
used
to
calculate
the
value
of
k*
between
two
sampling
locations
regardless
of
the
number
and
sizes
of
chambers
between
the
two
locations.
Appendix
D
 
Derivation
of
Extended
CSTR
Equations
LT2ESWTR
Toolbox
Guidance
Manual
Proposal
Draft
D­
3
June
2003
The
simplifying
assumption
of
equal­
size
CSTRs
for
calculating
k*
is
non­
conservative
relative
to
a
k*
value
calculated
by
allowing
for
unequal
sized
chambers.
That
is,
for
first­
order
ozone
decay
reaction,
unequal
sized
CSTR
reactors
in
series
would
be
the
least
efficient
(
ideal)
reactor
configuration
for
promoting
ozone
decay.
Hence,
calculating
k*
based
on
equation
D­
9
gives
the
largest,
or
most
conservative,
value
of
k*.
The
model
of
equal
sized
CSTR
reactors
in
series
is
a
more
efficient
configuration
for
promoting
ozone
decay.
Hence,
calculating
k*
from
Equation
D­
1
(
based
on
equation
D­
2)
gives
a
less
conservative
estimate
of
k*.
To
take
the
comparison
to
the
opposite
extreme,
calculating
k*
based
on
a
plug­
flow
assumption
(
e.
g.,
Equation
4­
7)
gives
the
smallest,
or
a
non­
conservative,
estimate
of
k*.

The
impact
of
the
simplifying
equal­
sized
CSTR
assumption
on
the
estimate
of
k*
and
Cin
involves
several
considerations.
The
first
issue
is
the
quantitative
difference
between
the
most
conservative
estimate,
based
on
Equation
D­
9,
and
the
recommended
approach
based
on
Equation
D­
2.
This
is
essentially
an
issue
of
what
chemical
and
hydrodynamic
conditions
affect
the
efficiency
of
the
ozone
decay
reaction.
This
is
a
somewhat
complex
issue
dependent
on
the
reaction
rate
(
represented
by
the
Damkohler
I
Number,
Da1
[
Da1=
k*
×
HDT]),
the
number
of
chambers
considered,
and
the
disparity
in
volumes
among
the
unequal­
sized
chambers.
In
principal,
as
the
reaction
rate
increases,
the
number
of
chambers
approaches
two
(
the
minimum),
and
the
volume
differences
among
the
chambers
increases,
the
difference
in
reaction
efficiency
between
the
two
reactor
configurations
increases.
Some
situations
could
result
in
approximately
30%
differences
between
k*
values.
Other
situations
could
results
in
negligible
differences.
Because
of
the
many
factors
involved
it
is
difficult
to
establish
qualitative
rules
for
all
possible
cases.
However,
for
contactors
with
2­
3
chambers
with
a
large
volume
difference
and
a
large
Da1,
then
the
utility
and
the
primacy
agency
may
consider
further
analysis.

The
second,
and
perhaps
overriding,
issue
concerning
the
impact
of
the
simplifying
assumption
is
whether
or
not
it
still
provides
a
certain
element
of
conservatism
over
the
true
contactor
performance.
That
is,
an
actual
contactor
with
unequal
sized
chambers
might
have
reasonably
good
hydrodynamics
such
that
even
the
equal­
size
CSTR
assumption
is
conservative.
This
too,
however,
is
very
system
specific,
and
is
a
difficult
issue
to
resolve
due
to
the
numerous
factors
involved.
