1. 1
D r . C h r i s S w a r t z
Y a n a n C a o
Air
Separation
Unit
Monica
Salib
There
is
no
doubt
that
optimization
is
the
key
for
the
new
engineering
future.
It
has
a
significant
impact
in
chemical
engineering
research.
It
is
the
tool
that
allows
us
to
improve
the
existing
processes
in
terms
of
cost,
energy
and
overall
efficiency.
As
chemical
engineers,
we
are
expected
to
develop
new
computational
and
mathematical
algorithms
that
can
cope
up
with
the
world’s
needs.
This
paper
studies
different
ASU
(Air
Separation
Unit)
optimization
strategies
with
gPROMS
and
addresses
some
of
the
ASU
industrial
issues.
Monica
Salib
Chemical
Engineering
Spring-­‐2015

2. 2
Table
of
Contents
1.
CSTR
Steady
State
Optimization
.................................................................................
1
2.
Batch
reactor
Dynamic
Optimization
........................................................................
2
2.1
Single
Scenario
Dynamic
Optimization
..........................................................................
2
2.2
Multi-­‐scenario
Dynamic
Optimization
...........................................................................
4
3.
Air
Separation
Unit
..........................................................................................................
5
3.1
Overview
...................................................................................................................................
5
3.2
Air
Separation
Process
.........................................................................................................
6
3.3
Dynamic
Simulation
..............................................................................................................
7
4.
Steady
State
Optimization
..........................................................................................
10
4.1
Objective
function
formulation
.......................................................................................
10
4.2
Constraints
.............................................................................................................................
12
4.3
Results
.....................................................................................................................................
13
5.
Dynamic
Optimization
................................................................................................
14
6.
Dynamic
Optimization
Under
Uncertainty
...........................................................
15
6.1
Overview
.................................................................................................................................
15
6.2
Objective
function
formulation
.......................................................................................
15
6.3
Observations
.........................................................................................................................
16
7.
Economics
Dynamic
Optimization
..........................................................................
16
8.
Conclusion
.......................................................................................................................
17

3. 1
1. CSTR
Steady
State
Optimization
Optimization
is
highly
required
to
ensure
that
any
given
process
is
operating
at
its
best
feasible
conditions.
In
this
example,
four
isothermal
CSTRs
are
connected
in
series.
Steady
state
is
assumed
throughout
the
process
because
of
having
a
fixed
flow
rate.
The
task
was
to
find
the
most
economical
operation
while
staying
in
the
plant
operation
limits
(constraints),
which
was
ensuring
that
the
summation
of
all
the
volumes
is
20
m!
.
To
do
so,
our
controls
(decision
variables)
were
the
volumes
of
each
CSTR.
The
optimization
task
is
summarized
in
Table
1
below.
Objective
Maximize
yield
of
product
Constraint
ΣV
=20
Controls
V
of
each
CSTR
Two
solution
approaches
were
implemented
in
order
to
solve
this
problem
in
gPROMS.
The
starting
point
is
the
same,
which
requires
defining
the
parameters
and
the
variables.
First,
mole
balance
equations
(one
for
each
CSTR)
were
written
in
the
MODEL
section
in
gPROMS
as
shown
below
in
Eqn1-­‐Eqn4.
Eqn 1. Fc! − Fc! − kc!
!.!
V! = 0
Eqn 2. Fc! − Fc! − kc!
!.!
V! = 0
Eqn 3. Fc! − Fc! − kc!
!.!
V! = 0
Eqn 4. Fc! − Fc! − kc!
!.!
V! = 0
Second,
four
MODELS
were
created
(one
for
each
CSTR)
and
then
they
were
linked
together
to
the
upper
layer
model
by
defining
the
inlet
concentration
of
each
CSTR
as
the
outlet
of
the
previous
one
in
series.

4. 2
The
optimization
file
was
created
using
the
information
in
Table
1
and
the
solver
successfully
converged
giving
us
same
results
with
both
solution
strategies.
2. Batch
reactor
Dynamic
Optimization
2.1
Single
Scenario
Dynamic
Optimization
A
lot
of
research
has
been
going
on
dynamic
optimization
since
it
is
the
more
realistic
and
applicable
type
in
the
existing
plants.
The
Ramirez
control
problem
was
examined
and
the
task
was
to
maximize
the
yield
of
species
B
at
the
final
time
by
calculating
the
optimal
temperature
profile,
T
(t)
for
a
batch
reactor
with
the
consecutive
reactions
shown
below
are
carried
out.
A
!!
B
!!
C
The
material
balances
for
species
A
(concentration,x!) and B (concentration, x!)
are
shown
below
in
Eqn5-­‐Eqn8.
Eqn
5.
!!!(!)
!"
= −k!(t) ∗ x!(t)
Eqn
6.
!!!(!)
!"
= k! t ∗ x! t − k!(t) ∗ x!(t)
Eqn
7.
k! t = k!" ∗ e
!!"
!"(!)
Eqn
8.
k! t = k!" ∗ e
!!"
!"(!)

5. 3
The
desired
goal
was
to
maximize
the
yield
of
species
B
at
the
final
time,
i.e.,
x! t! − x!(t!).
After
setting
up
this
optimization
problem,
it
was
solved
in
two
different
ways:
(1) Piecewise
constant
(2) Piecewise
linear
The
main
difference
between
both
methods
is
in
the
controlled
variables.
In
the
piecewise
constant
method,
it
is
the
temperature
that
is
being
changed
till
it
reaches
the
optimum
value.
In
the
piecewise
linear
case,
it
is
the
slope
of
the
temperature
that
changes
till
it
shapes
the
optimum
temperature
profile
for
the
process.
The
results
of
both
strategies
are
shown
below
in
Figure
1
and
Figure
2.
Figure
1.
Piecewise
constant
optimum
Temperature
profile
Figure
2.
Piecewise
linear
optimum
Temperature
profile

6. 4
It
was
concluded
by
looking
at
the
results
that
the
piecewise
linear
is
a
better
approach
as
it
converges
at
a
higher
yield
value
even
if
it
takes
a
little
while
longer.
2.2
Multi-­‐scenario
Dynamic
Optimization
There
is
no
doubt
that
a
design
is
more
optimum
if
it
can
tolerate
different
reactions
with
different
parameters
and
rates.
Therefore,
taking
uncertainty
into
consideration
in
the
design
stage
helps
optimize
the
dynamic
performance
of
the
plant.
Using
the
previous
example,
nine
scenarios
were
implemented
each
with
a
different
combination
of
k
values.
If
the
changing
parameters
are
the
reaction
rate
constants
k!
and
k!
where:
k!
∈
[k!
!"#
, k!
!"#
]
k!
∈
[k!
!"#
, k!
!"#
]
Then,
a
theta
(θ)
variable
is
introduced
as
θ
=
k!
k!
and
each
scenario
is
assigned
to
a
different
θ
combination
as
illustrated
below.
Scenario
1:
θ
=
k!
!"#
k!
!"#
Scenario
2:
θ
=
k!
!"#
k!
!"#
Scenario
3:
θ
=
k!
!"#
k!
!"#
Scenario
4:
θ
=
k!
!"#
k!
!"#
Scenario
5:
θ
=
k!
!"#
k!
!"#
Scenario
6:
θ
=
k!
!"#
k!
!"#
Scenario
7:
θ
=
k!
!"#
k!
!"#
Scenario
8:
θ
=
k!
!"#
k!
!"#
Scenario
9:
θ
=
k!
!"#
k!
!"#

7. 5
Generally,
the
optimization
problem
follows
the
same
formulation
as
the
single
scenario
case.
The
objective
function
is
the
only
thing
that
changes,
as
it
has
to
account
for
all
the
scenarios’
constraints
at
the
same
time.
The
most
common
technique
for
the
objective
function
reformulation
is
assigning
it
to
the
average
of
the
summation
of
the
scenarios
as
shown
in
Eqn
9.
Eqn9.
New_objective_function
=
!
!
∗ ∑!!!
!!!
objective (i)
Piecewise
linear
and
piecewise
constant
were
again
used
to
solve
the
optimization.
Piecewise
linear
proved
to
be
a
better
approach
in
solving
multi-­‐scenario
dynamic
optimization
problems
as
it
converged
at
a
higher
yield.
The
only
disadvantage
it
was
the
time
it
needed
to
converge.
Overall,
both
methods
worked
successfully
in
gPROMS
and
settled
at
very
close
values.
3. Air
Separation
Unit
3.1
Overview
The
air
separation
industry
is
essential
as
it
plays
an
important
role
in
a
lot
of
markets
such
as
food
processing,
petrochemicals
and
healthcare.
For
a
long
time
in
the
air
separation
industry,
the
dynamic
performance
of
the
plant
was
assessed
by
how
capable
it
is
in
rejecting
disturbances.
The
idea
of
switching
the
operation
points
was
not
relevant
due
to
the
usual
stability
of
electricity
prices.
Recently,
this

8. 6
has
not
been
the
case
because
of
the
fluctuations
in
electricity
prices
in
many
regions.
Those
deregulations
in
the
price
cause
unexpected
rapid
changes
in
the
plant
cost
since
electricity
is
the
main
operating
cost
for
the
air
separation
plant.
Therefore,
further
techniques
that
take
dynamic
rapid
changes
into
account
should
be
adopted
as
steady
state
simulation
has
their
limitations.
In
this
paper,
all
the
studies
and
results
are
based
on
the
cryogenic
approach
of
air
separation.
It
produces
large
gas
phase
quantities
of
air
components
(Nitrogen,
Oxygen
and
Argon)
and
operates
at
a
low
temperature
distillation.
3.2
Air
Separation
Process
A
simple
schematic
of
the
main
process
equipment
is
shown
in
Fig
3.
Further
description
of
the
process
can
be
found
in
Roffel
et
al.
[2000]
and
Miller
et
al.
[2008a].

9. 7
Fig
3.
A
cryogenic
air
separation
plant
that
produces
argon,
oxygen
and
nitrogen.
LAr
=
Liquid
Argon;
L02
=
Liquid
Oxygen;
G02
=
Gas
Oxygen;
LN2
=
Liquid
Nitrogen;
GN2
=
Gas
Nitrogen;
PHX
=
Primary
Heat
Exchanger;
LC
=
Lower
Column;
UC
=
Upper
Column.
3.3Dynamic
Simulation
Step
tests
were
conducted
to
investigate
the
effect
of
certain
input
changes
on
the
dynamic
performance
of
the
process.
Fig
4.
Shows
the
output
results
of
the
product
impurity,
air
feed,
reflux
and
GN2
production
upon
-­‐5%
change
in
inlet
volumetric
air
feed.
Fig
5.
Shows
the
dynamic
response
of
the
product
impurity,
gas
draw
fraction
rate,
reflux
and
GN2
production
after
being
subjected
to
a
positive
step
change
in
the
gas
draw
rate.

10. 8
Figure
4.
Dynamic
response
of
selected
scaled
variables
to
a
negative
step
change
in
the
air
feed

11. 9
Figure
5.
Dynamic
response
of
selected
scaled
variables
to
a
negative
step
change
in
the
air
feed.
While
keeping
all
other
inputs
fixed
in
the
system,
as
the
gas
draw
fraction
increases,
GN2
production
increases
and
reflux
rate
decreases.
The
system
tries
to
reach
a
new
equilibrium
steady
state
but
since
the
reflux
rate
is
much
lower
than
before
relative
to
the
air
feed
at
the
new
steady
state,
the
product
impurity
increases
significantly.

12. 10
4. Steady
State
Optimization
4.1
Objective
function
formulation
This
paper
focuses
on
optimizing
the
air
separation
unit
so
that
it
can
handle
the
frequent
changes
in
demand
and
electricity.
Normally,
the
plant
runs
steadily
under
certain
operating
points.
When
a
change
in
demand/electricity
price
occurs,
the
plant
is
adjusted
to
operate
at
a
new
set
of
operating
conditions.
Therefore,
the
plant
is
expected
to
operate
at
steady
state
all
the
time
except
that
time
during
the
transition
between
the
operating
points.
Since
the
plant
is
assumed
to
be
steady,
we
have
to
ensure
that
all
operation
points
are
feasible
and
optimum.
Hence,
a
steady
state
economical
optimization
is
performed,
not
only
to
make
sure
that
all
operation
point
are
economically
optimum,
but
also
to
serve
as
a
guideline
in
the
dynamic
transition
optimization
as
well.
The
steady
state
optimization
takes
the
form
shown
in
Eqn
10.
Eqn. 10 max Φ!! = C!"# F!"# !"#$ + F!"#$ − C!"!#W!"#$ − C!"#$F!"#$
Subject
to:
f
(x
=
0,
x,
z,
u
,
p)
=0
g(x,
z,
u
,
p)
=0
h
(x
,
z,
u
,
p)
<
0

13. 11
Where:
x
=
differential
state
vector
z
=
algebraic
state
vector
u
=
control
input
vector
• u!=
Inlet
volumetric
air
flow
rate
under
standard
conditions
• u!=
Liquid
molar
air
flow
rate
to
the
column
• u!=
Liquid
Nitrogen
production
rate
(Distillate)
• u!=
Gas
draw
fraction
• u!=
Evaporation
rate
of
liquid
nitrogen
for
unsatisfied
demand
p
=
parameter
vector
C!"#=
Sales
price
of
gas
nitrogen
C!"!#, C!"#$=
Costs
associated
with
compression
and
evaporation
F!"# !"#$=
Flow
rate
of
GN
2
produced
F!"#$=
Rate
of
evaporation
of
pre-­‐stored
liquid
N2
W!"#$=
Power
consumption
of
the
compressor
A
detailed
plant
configuration
with
labeled
variables,
parameters
and
inputs
is
shown
in
Figure
5
below.

14. 12
Figure
5.
Plant
configuration
with
labeled
decision
variables
4.2Constraints
There
are
different
types
of
constraints
that
we
need
to
put
in
consideration
to
make
sure
that
the
plant
runs
without
violating
the
physics/chemical
laws
or
the
operational
conditions
along
with
satisfying
the
customer’s
needs.
All
the
constraints
are
categorized
and
summarized
in
Table
2
below
Table
2.
Constraints
for
steady
state
optimization
Operational
Constraints
Product
Specification
Modeling
Constraints
Compressor
Surge
Demand
Satisfaction
Pressure
in
PHX
Flooding
No
Overproduction
Temp
diff.
in
IRC
Product
Purity

15. 13
Each
constraint
mentioned
above
is
modelled
by
an
equation
that
has
a
certain
tolerance.
The
goal
is
always
to
minimize
the
tolerance
to
approach
zero
4.3Results
It
was
observed
that
the
initial
guess
plays
an
important
role
in
the
optimization
process
in
terms
of
the
converging
time
and
value.
That
is
due
to
the
non-­‐
convexity
from
the
nonlinear
model.
Therefore,
different
initial
guesses
were
provided
and
the
best
point
was
reported.
Also,
the
system
responds
differently
in
terms
of
active
constraints
(when
the
final
value
is
very
close
to
one
of
the
bounds)
depending
on
the
change
it
was
subjected
to.
For
instance,
as
demand
increases,
both
flooding
and
impurity
constraints
are
active
because
the
system
has
to
settle
at
its
maximum
level
of
impurity.
However,
as
demand
decreases,
the
compressor
surge
constraint
is
the
active
one
because
of
the
decrease
in
the
flow
rate.
The
steady
state
optimization
results
upon
demand
fluctuations
(-­‐30%
to
+30%)
are
reported
in
Table
3.
Those
output
results
were
used
again
as
a
target
to
dynamic
optimization.
Table
3.
Steady
state
optimization
results
for
demand
fluctuations
**
Data
in
the
Table
are
scaled
values
for
company
confidentiality
-­‐30%
-­‐20%
-­‐10%
0%
10%
20%
30%
LN2
production
rate
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
0.0002
Gas
draw
rate
fraction
0.058
0.068
0.0754
0.0758
0.0758
0.076
0.076
Air
feed
volumetric
flow
rate
29.998
30
30.4
33.8
37.075
38.2
38.212
Liquid
air
to
the
column
6.84
6.5434
6.21
5.356
4.6202
2.5288
2.5288
Evaporation
rate
0
0
0
0
0
2.176
5.64

16. 14
5. Dynamic
Optimization
Steady
state
optimization
provided
us
with
feasible
optimal
operating
points.
However,
it
did
not
account
for
the
transitions
between
those
points.
Hence,
dynamic
optimization
is
conducted
to
switch
from
the
base
optimal
case
to
the
new
operation
point
upon
demand
fluctuations.
The
objective
function
is
formulated
differently
as
it
is
no
longer
a
cost
based
one
but
rather
a
trajectory
demand
tracking
function
as
shown
in
Eqn
11.
Eqn 11. min Φ = t! 1 −
F!"# !"#$ t
F!"# !"#$
∗
!!!
!!
dt + w! 1 −
u! t!
u!
∗
!!!
!!!
Where
𝑤!
represents
the
weights
assigned
to
the
manipulated
variables
as
shown
below:
w!"# !""# = w!"# !"# = w!"# !"#$ = 1
w!"# = 0.1
The
problem
was
solved
using
5
control
intervals
with
a
tolerance
of
1E-­‐5.
The
number
of
control
intervals
is
critical
in
the
optimization
problem
formulation.
It
has
to
be
big
enough
for
capturing
the
control
behaviour
but
not
too
big
for
unwanted
oscillations
in
the
results.
The
time
was
divided
into
three
periods
with
an
input
slope
of
0
to
the
first
and
last
period.
This
ensures
that
both
the
start
and
end
point
operate
at
optimal
feasible
steady
state.
Also,
a
dummy
variable
was
introduced
to
represent
the
slowest
variable
in
the
process.
By
minimizing
that
variable,
we
guarantee
that
our
system
converge
at
a
steady
state.

17. 15
The
solution
time
in
the
air
separation
unit
is
too
long
compared
to
the
one
in
the
batch
reactor.
That
concludes
that
the
complexity
of
the
problem
is
a
key
variable
that
affects
the
solution
time.
6. Dynamic
Optimization
Under
Uncertainty
6.1
Overview
The
bigger
picture
comes
into
place
after
the
individual
demand
dynamic
optimization
cases
have
successfully
converged.
The
task
gets
more
complicated
because
our
objective
is
not
only
to
optimize
the
operating
conditions
and
the
trajectory
for
certain
demand
percentage
change;
instead
it
is
finding
the
conditions
that
are
optimum
overall
regardless
of
the
fluctuations.
The
approach
used
to
tackle
the
uncertainty
is
very
similar
to
the
“Multi-­‐
scenario
dynamic
optimization”
one
in
section
2.2
regarding
the
batch
reactor.
6.2
Objective
function
formulation
Since
the
solution
time
increases
drastically
in
uncertainty
problems,
only
two
cases
(0%
demand
change
+
10%
demand
change)
were
solved
simultaneously
to
better
understand
the
capabilities
and
limitations
of
gPROMS
in
handling
those
complex
problems.
The
problem
formulation
was
very
similar
to
the
single
case
dynamic
optimization.
The
difference
was
mainly
including
both
cases
into
one
single
process
and
having
the
new
objective
function
(Φ∗
)
capturing
both
of
them
as
explained
by
Eqn
12.

18. 16
Eqn
12.
Φ∗
=
Φ! + Φ!
Where:
Φ!
=
Objective
function
of
the
0%
demand
change
case
Φ!
=
Objective
function
of
the
10%
demand
change
case
All
the
optimization
elements
were
handled
in
the
same
manner
as
the
single
case
with
only
the
difference
of
having
two
of
each
manipulated
variable/constraint,
one
representing
each
case.
6.3
Observations
Multiple
factors
were
observed
to
affect
the
optimization
under
uncertainty
more
than
just
the
number
of
the
control
intervals.
Based
on
the
results
reported
in
Table
4,
relaxing
the
bounds
seems
to
be
an
important
key
that
affects
the
ability
of
the
optimization
to
converge.
Also,
using
the
dynamic
optimization
results
as
initial
guesses
plays
an
important
role
in
terms
of
the
solution
time.
Table
4.
Observations
in
uncertainty
dynamic
optimization
trials
with
fixed
control
intervals
Provided
initial
guess
from
previous
optimization
Relaxed
bounds
Optimization
converged
Solution
time
Trial
1
✖
✖
✖
-­‐
Trial
2
✓
✖
✖
-­‐
Trial
3
✖
✓
✓
6.6
h
Trial
4
✓
✓
✓
2.6
h
7. Economics
Dynamic
Optimization
Another
economical
base
approach
was
tried
out
to
optimize
the
dynamic
process.
The
objective
function
was
not
formulated
to
track
the
demand

19. 17
trajectory
as
before
but
rather
to
maximize
the
accumulation
of
the
profit.
The
constraints
and
controls
were
handled
in
the
same
manner
as
the
demand
trajectory
case.
The
optimization
successfully
converged
in
30
min
for
the
10%
demand
change
case
and
that
is
promising
as
the
solution
time
is
very
short
compared
to
the
demand
based
one.
8. Conclusion
The
research
is
still
ongoing
to
better
optimize
the
air
separation
process,
as
it
is
the
primary
supplier
for
our
everyday
vital
chemicals.
The
design
has
got
a
lot
of
interesting
possibilities
that
should
be
examined
in
future
work.
For
instance,
the
introduction
of
external
liquid
nitrogen
and
the
addition
of
a
vent
stream
after
the
compressor
can
be
some
promising
alternatives.
Also,
more
cases
should
be
solved
simultaneously
to
achieve
a
better
operating
point.