Article

Robust optimal control of polymorphic transformation in batch crystallization

AIChE Journal (Impact Factor: 2.75). 10/2007; 53(10):2643 - 2650. DOI: 10.1002/aic.11266
ABSTRACT
One of the most important problems that can arise in the development of a pharmaceutical crystallization process is the control of polymorphism, in which there exist different crystal forms for the same chemical compound. Different polymorphs can have very different properties, such as bioavailability, which motivates the design of controlled processes to ensure consistent production of the desired polymorph to produce reliable therapeutic benefits upon delivery. The optimal batch control of the polymorphic transformation of L-glutamic acid from the metastable α-form to the stable β-form is studied, with the goal of optimizing batch productivity, while providing robustness to variations in the physicochemical parameters that can occur in practice due to variations in contaminant profiles in the feedstocks. A nonlinear state feedback controller designed to follow an optimal setpoint trajectory defined in the crystallization phase diagram simultaneously provided high-batch productivity and robustness, in contrast to optimal temperature control strategies that were either nonrobust or resulted in long-batch times. The results motivate the incorporation of the proposed approach into the design of operating procedures for polymorphic batch crystallizations. © 2007 American Institute of Chemical Engineers AIChE J, 2007

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Available from: Richard D Braatz
PROCESS SYSTEMS ENGINEERING
Robust Optimal Control of Polymorphic
Transformation in Batch Crystallization
Martin Wijaya Hermanto and Min-Sen Chiu
Dept. of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117576
Xing-Yi Woo
Dept. of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 11576;
and Dept. of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, IL 61801
Richard D. Braatz
Dept. of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, IL 61801
DOI 10.1002/aic.11266
Published online August 27, 2007 in Wiley InterScience (www.interscience.wiley.com).
One of the most important problems that can arise in the development of a pharma-
ceutical crystallization process is the control of polymor phism, in which there exist dif-
ferent crystal forms for the same chemical compound. Different polymorphs can have
very different properties, such as bioavailability, which motivates the design of con-
trolled processes to ensure consistent produc tion of the desired polymorph to produce
reliable therapeutic benefits upon delivery. The optimal batch control of the polymor-
phic transformation of L-glutamic acid from the metastable a-form to the stable b-form
is studied, with the goal of optimizing batch productivity, while providing robustness
to variations in the physicochemical param eters that can occur in practice due to var-
iations in contaminant profiles in the feedstocks. A nonlinear state feedback controller
designed to follow an optimal setpoint trajectory defined in the crystallization phase
diagram simultaneously provided high-batch productivity and robustness, in contrast
to optimal temperature control strategies that were either nonrobust or resulted in
long-batch times. The results motivate the incorporation of the propos ed approach into
the design of operating procedures for polymorphic batch crystallizations.
Ó 2007
American Institute of Chemical Engineers AIChE J, 53: 2643–2650, 2007
Keywords: T-control, robust T-control, C-control, polymorphic transformation, phar-
maceutical crystallization
Introduction
Polymorphism, in which multiple crystal forms exist for the
same chemical compound, is of significant interest to the phar-
maceutical industry.
1–5
According to Ostwald’s Rule of Stages,
in a polymorphic system, the most soluble metastable form
appears first, followed by more stable polymorphs. This rule
holds for most polymorphic systems, which implies that care
must be taken to avoid the formation of metastable crystals
when trying to crystallize the most stable crystal form. Some-
times a relatively small shift in the operating conditions can
result in the appearance of crystals of an undesired polymorph.
Metastable crystals have appeared during the production of
specialty chemicals, such as pharmaceuticals, dyestuffs, and
pesticides. The variation in physical properties, such as crys-
tal shape, solubility, hardness, color, melting point, and
chemical reactivity makes polymorphism an important issue
for the food, specialty chemical, and pharmaceutical indus-
Correspondence concerning this article should be addressed to Min-Sen Chiu at
checms@nus.edu.sg.
Ó 2007 American Institute of Chemical Engineers
AIChE Journal October 2007 Vol. 53, No. 10 2643
Page 1
tries, where products are specified not by chemical com-
position only, but also by their performance.
2
As a result,
controlling polymorphism to ensure consistent production of
the desired polymorph is important in those industries, includ-
ing the drug manufacturing industry where safety is paramount.
Deliberate isolation of metastable phases is sometimes desired
when they have advantageous processing or application proper-
ties, such as increased dissolution rate. In most cases, however,
the formulation of a product as a metastable phase is undesired
due to potential subsequent phase transformation during drying
or storage, which would change product characteristics.
6
Although the crystallization control literature is vast, to
the authors’ knowledge there are no articles on the optimal
control of crystallization processes in which more than one
polymorph occurs. The vast majority of articles on nonpoly-
morph crystallization have considered the optimal control of
only one or two characteristics of the crystal-size distribu-
tion, such as weight mean size. The most widely studied
approach is to determine a temperature profile (T-control)
that optimizes an objective function based on an offline nom-
inal model.
7–11
Although T-control is simple to implement, it
has become well-known in recent years that T-control can be
very sensitive to variations in the kinetic parameters.
12,13
This motivated the development of robust T-control, which
explicitly includes the impact of uncertainties in the objec-
tive, while determining the optimal temperature-time trajec-
tory to be followed during batch operation.
14–16
With advan-
ces in sensor technologies, another control strategy developed
to provide improved robustness to model uncertainty is C-
control, which follows an optimal or nearly optimal concen-
tration-temperature trajectory.
3,13,17–19
Motivated by the industrial need to control polymor-
phism,
2,20
this article evaluates and compares the perform-
ance of these optimal control strategies for the polymorphic
transformation of L-glutamic acid from the metastable a-
form to the stable b-form. The Process Description section
describes the process model for the polymorphic transforma-
tion of L-glutamic acid. The T-control, robust T-control, and
C-control sections discuss the three control strategies investi-
gated in this article. The simulation results in the Results
section are followed by the conclusions.
Process Description
The population balance equations for the polymorphic
transformation of L-glutamic acid from the metastable a-
form to the stable b-form are
21
:
@f
a
@t
þ
@ðD
a
f
a
Þ
@L
¼ 0; (1)
@f
b
@t
þ
@ðG
b
f
b
Þ
@L
¼ B
b
dðL L
0
Þ; (2)
where
f
0
a
ðLÞ¼f
a
ðL; 0Þ; (3)
D
a
¼ k
d;a
ðC C
sat;a
Þ
100
ða-dissolution rateÞ; (4)
G
b
¼ k
g;b
L
ðC C
sat;b
Þ
100
ðb-growth rateÞ; (5)
C
sat;i
¼ a
i
expðb
i
TÞ; i 2fa; bsolubilitiesÞ; (6)
k
d;a
¼ exp
8:98703 10
7
T
2
6:0761 3 10
5
T
þ1:0141310
3

; (7)
k
g;b
¼ 0:410 exp
10900
8:314 T

: (8)
f
i
is the number density for the i-form crystals. The initial
crystal-size distributions of polymorph i, Y
i
(L,0), is described
by the sum of three log-normal distributions, the n
th
moment
of the i-form crystals is
l
i;n
¼
Z
1
0
L
n
f
i
dL; (9)
and the nucleation rate for b-form crystals is
B
b
¼ k
b;b
ðC C
sat;b
Þ
100
l
b;3
; (10)
k
b;b
¼ 7 3 10
7
: (11)
Applying the method of characteristics to Eq. 1, and
method of moments to Eq. 2, gives
f
a
ðL; tÞ¼f
0
a
L
Z
t
0
D
a
dt

; L 0 (12)
d l
b;n
dt
¼ nG
0
b
l
b;n
þ B
b
L
n
0
; n ¼ 0; 1; 2; ::: (13)
where G
0
b
5 G
b
/L. The aforementioned equations are aug-
mented by the solute mass balance:
dC
dt
¼300
q
c
V
slurry
m
solution
1
C
100

k
v;a
D
a
l
a;2
þ k
v;a
D
a
f
a
ðL
0
; tÞL
3
0
3
þ k
v;b
G
0
b
l
b;3
þ k
v;b
B
b
L
3
0
3
; ð14Þ
where q
c
is the crystal density, m
solution
is the mass of solution,
V
slurry
is the total volume of crystals and solution, k
v,i
is the vol-
umetric shape factor for polymorphic form i,andL
0
is the size
of the nucleated crystals. The parameter values are in Ref. 21.
In this study, the following uncertain parameters are assumed:
k
0
g;b
¼ k
g;b
ð1 þ h
1
Þ exp
10900
8:314T
h
2

; 0:2 h
1
; h
2
0:2
(15)
k
0
d;a
¼ k
d;a
ð1 þ h
3
Þ; 0:2 h
3
0:2 (16)
C
0
sat;a
¼ C
sat;a
ð1 þ h
4
Þ; 0:05 h
4
0:05 (17)
C
0
sat;b
¼ C
sat;b
ð1 þ h
5
Þ; 0:05 h
5
0:05 (18)
where y
1
and y
2
are the uncertainties in the growth parame-
ters for the b-form crystals, y
3
is the uncertainty in the disso-
lution kinetics of the a-form crystals, and y
4
and y
5
are the
uncertainties in the solubility curves of the a and b forms,
respectively. The nominal model corresponds to y
i
5 0, i 5
1, ..., 5. While uncertainties in parameters quantified from
2644 DOI 10.1002/aic Published on behalf of the AIChE October 2007 Vol. 53, No. 10 AIChE Journal
Page 2
experimental data are typically correlated,
22,23
uncorrelated
uncertainties were used here so as to separately assess the
robustness to different types of uncertainties.
T-control
The first journal articles on the control of batch polymor-
phic crystallization implemented temperature control (e.g.,
see
21
). The most widely studied approach for the optimal
control of nonpolymorphic crystallization processes has uti-
lized T-control, in which the temperature trajectory has been
computed from the optimization of an objective function,
based on an offline model with nominal parameters.
9
In this
study, the objective function that is maximized is the yield
of the b-form minus a penalty on the time required for the
mass of the a-form to be below some tolerance
I
nominal
¼ m
b
ðt
f
Þw
n
t
a
ðcÞ
¼ q
c
k
v;b
V
slurry
l
b;3
ðt
f
Þw
n
t
a
ðcÞ (19)
where t
f
is the batch time, m
b
(t
f
) is the yield of b-form at the
end of the batch, t
a
(c) is the time taken to reduce the mass
of a -form below c, and w
n
is a weighting parameter. The val-
ues of t
f
, w
n
, and c are 7 h, 1 3 10
23
, and 5 3 10
24
g,
respectively. The second term t
a
(c) is included to increase
the productivity of the batch crystallizer (shorter batch times
lead to more batches per day).
To implement this strategy, the temperature-time trajectory
is parameterized as a first-order spline with 64 time intervals
(6.56 min). Examination of the temperature trajectories indi-
cated that this temporal resolution is fine enough for this
crystallization process. The temperature trajectory was con-
strained to be within the region where crystals of the a-form
dissolve, while crystals of the b-form grow, that is, the tem-
perature is constrained to be between the saturation tempera-
tures of the a- and b-forms (T
sat,a
T T
sat,b
).* In addition,
the minimum and maximum temperature can be achieved by
cooling and heating are 258C and 508C. These constraints
were handled by parameterizing the temperature-time trajec-
tory, such that the decision variables were fractions between
0 and 1, with 0 and 1 indicating the lower and upper bounds
on the temperature at each time instance, respectively. A
genetic algorithm was used to determine an initial tempera-
ture trajectory, which was further optimized using sequential
quadratic programming. The resulting concentration-tempera-
ture trajectory for nominal model is shown in Figure 1. There
are two sections in the trajectory (heating followed by cool-
ing) where the supersaturation of b-form is maximized due
to its growth kinetics being the rate-limiting step.
Robust T-control
The solubility curves and nucleation and growth kinetics
can vary somewhat from batch to batch due to impurities in
the feed. Further, any model parameters obtained from
experiments have uncertainties due to measurement noise
and unmeasured disturbances that occur during the collection
of the experimental data used to estimate parameters. Assum-
ing all the uncertainties are independent from each other,
h
min;i
h
i
h
max;i
; (20)
the uncertain parameters may be expressed as:
e
h
¼
n
h : h ¼
^
h þ dh; kW
h
dhk
1
1
o
; (21)
^
h ¼
h
min
þ h
max
2
; (22)
ðW
h
Þ
jj
¼
2
h
max;j
h
min;j
; (23)
where y 5 [y
1
, y
2
, ...,y
n
]
T
,
^
h ¼½
^
h
1
;
^
h
2
; ...;
^
h
n
T
, y
min
5
[y
min,1
, y
min,2
, ...,y
min,n
]
T
, and y
max
5 [y
max,1
, y
max,2
, ...,
y
max,n
]
T
are the actual, nominal, minimum, and maximum
values of the uncertain model parameters, respectively, W
y
is
a weight matrix quantifying the magnitude of the uncertainty
in each parameter, and kk
?
is the vector ?-norm.
In robust T-control the objective function to be maxi-
mized appends a term to include the impact of uncertain
parameters
14–16
I
robust
¼ I
nominal
w
r
dI
w:c
(24)
where w
r
[ [0,1] is a weighting parameter and dI
w.c.
is the
worst-case deviation in the objective due to model uncertain-
ties. If w
r
5 1, I
robust
is the objective function obtained for
the worst-case perturbation at the cost of potential degrada-
tion in nominal performance. The value of w
r
can be selected
to be smaller than one depending on the desired trade-off
between the nominal and worst-case performance. In this
paper, w
r
5 0.6 provided the best balance between nominal
and worst-case performance. In robust T-control the objective
(24) is maximized subject to the condition that the con-
straints in the previous section hold for all parameters within
the uncertainty description.
Figure 1. Solubility curves and concentration-tempera-
ture trajectories of T-control and C-control
with parameters p
1
5 36.56, p
2
5 11.5, p
3
5
0.4340, and p
4
5 0.03656.
*Note that a slight violation of this constraint occurs initially due to the initial
concentration being outside this range. According to Ref. 21 the a-form crystals
grow for a short time before crossing the a solubility curve.
AIChE Journal October 2007 Vol. 53, No. 10 Published on behalf of the AIChE DOI 10.1002/aic 2645
Page 3
With the widely-used approximation
22,24,25
dI ¼
@I
@h
dh ¼ Ldh; (25)
where L is the objective function sensitivity row vector, the
worst-case deviation in the objective function is
15,23
max
kW
h
dhk
1
1
jdIj¼kLW
1
h
k
1
(26)
and a worst-case parameter uncertainty vector is
dh
w:c
¼W
1
h
v; with v
k
¼
ðLW
1
h
Þ
k
LW
1
h
Þ
k
j
¼ sgn ðLW
1
h
Þ
k
;
(27)
where kk
1
is the vector 1-norm and the objective function
sensitivity vector can be computed from
L ¼
dI
dh
t
f
¼
dI
dx
t
f
dx
dh
t
f
¼
@I
@x
1
@I
@x
2
...
@I
@x
n

@x
1
@h
1
@x
1
@h
2
...
@x
1
@h
p
@x
2
@h
1
@x
2
@h
2
...
@x
2
@h
p
.
.
.
.
.
.
.
.
.
.
.
.
@x
n
@h
1
@x
n
@h
2
...
@x
n
@h
p
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
(28)
where x 5 [x
1
, x
2
, ..., x
n
]
T
is the vector of the states
involved in the simulation; for this study,
x ¼½l
a;0
; l
a;1
; l
a;2
; l
a;3
; l
a;4
; l
b;0
; l
b;1
; l
b;2
; l
b;3
; l
b;4
; C; h
T
(29)
with h 5 $
t
0
D
a
(s)ds.
The system Eqs. 9, 12, 13, and 14 can be represented as a
system of differential-algebraic equations (DAE):
M
_
x ¼ fðt; x; hÞ (30)
where M is an n 3 n mass matrix of constant coefficients of
the form
M ¼
I
ðsÞ
0
00
ðnsÞ

(31)
and I
(s)
is the s 3 s identity matrix, and 0
(n2s)
is the (n 2 s)
3 (n 2 s) matrix of zeroes. In this study, s 5 7 and n 5 12.
Differentiating the system equation with respect to y gives
the sensitivity equation
M
@
_
x
@h
¼
@f
@x
@x
@h
þ
@f
@h
: (32)
The following steps were used to compute the robust tem-
perature profile:
(1) The first iteration (j 5 1) was initialized with random
parameters for the temperature profile. For j [ 1, the temper-
ature parameters were determined from a genetic algorithm
applied to the optimization of the robust objective function
subject to all operating constraints for the full range of
uncertain parameters.
(2) The temperature profile from Step 1 was applied to
the nominal model (y
i
5 0 for i 5 1, ... , 5) by integrating
the system Eq. 30, and the sensitivity Eq. 32. At the final
batch time (t
f
), the nominal objective function (I
nominal
) was
obtained from Eq. 19.
(3) The objective function sensitivity vector (L) was com-
puted from Eq. 28, and the worst-case parameter uncertain-
ties obtained from Eq. 27.
(4) The temperature profile from Step 1 was applied to the
model with the worst-case model parameters calculated in
Step 3. I
robust
was calculated from Eq. 24 at the batch time t
f
.
(5) Steps 1 to 4 were repeated until there was no signifi-
cant change in the temperature profile.
This algorithm uses Eq. 25 only for estimating the worst-
case parameters Eq. 27 with the full dynamic simulation
used to compute I
robust
(Step 4).
C-control
In many experimental and simulation studies of nonpolymor-
phic batch crystallizations, the C-control strategy (Figure 2) has
resulted in low-sensitivity of the product quality to most practi-
cal disturbances and variations in kinetic parameters.
3,13,17–19,26
In the last two years, the C-control strategy has been applied
experimentally to several polymorphic crystallizations, to pro-
duce large crystals of any selected polymorph.
27
C-control can
be interpreted as nonlinear state feedback control,
26,28
in which
the nonlinear master controller acts on the concentration C as a
measured state
29
to produce the setpoint temperature T
set
as its
manipulated variable.
The difference between the calculated
T
set
, and the measured temperature T is used by the slave con-
troller to manipulate the jacket temperature T
j
, so that the devia-
tion between T
set
and T is reduced. Because the slave controller
is just temperature control of a mixed tank, and the batch dy-
namics are relatively slow, any reasonably tuned proportional-
integral controller will result in accurate following of T
set
.
The setpoint concentration-temperature trajectory was
parameterized as follows:
(1) Heating section:
(a) This time-dependent relation forces the temperature into
the region between both solubility curves in the first 5.25 mins.
Figure 2. Prefered implementation of C-control for a
batch cooling crystallizer.
26
For this application, the nonlinear master controller is given by Eqs. 35–38.
2646 DOI 10.1002/aic Published on behalf of the AIChE October 2007 Vol. 53, No. 10 AIChE Journal
Page 4
T
0
¼ T
o
þ
p
1
T
o
ð5:25Þð60Þ
t (33)
T
set
¼ maxfT
min
; minfT
0
; T
max
gg (34)
where p
1
(8C) is the first decision variable, T
o
5 258C is the
initial temperature, T
min
5 258C and T
max
5 508C are the
minimum and maximum temperatures achievable by cooling
and heating the water bath, respectively, T
set
is the setpoint
temperature to a lower level controller, and t is time in sec-
onds.
(b) Assuming a linear concentration-temperature trajectory
when heating
T
0
¼ T
p
þ p
2
ðC C
p
Þ (35)
T
set
¼ maxðT
min
; minðT
0
; T
max
ÞÞ (36)
where p
2
C
g solute
=100 g solution

is the second decision
variable, and T
p
(8C) and C
p
(g solute/100 g solution) are the
temperature and concentration at time equal to 5.25 mins;
(2) Cooling section:
After the mass of a-form decreases below a certain
value (chosen to be 0.5 g), the assumed trajectory is fol-
lowed
T
0
¼
lnðC=p
3
Þ
p
4
(37)
T
set
¼ maxðT
min
; minðT
0
; T
max
ÞÞ (38)
where p
3
(g solute/100 g solution), and p
4
(1/8C) are the third
and fourth decision variables, respectively.
The rationale behind the structure of Eqs. 33 to 38 is to
obtain the best fit to the concentration-temperature trajectory
obtained by applying the optimal temperature-time trajectory
from T-control to the nominal model. Then, the values for p
1
to p
4
were fitted accordingly. The lower level controllers for
all control strategies are assumed to have very fast response
compared to the overall batch time, which is a good assump-
tion for this process, which has a relatively long batch time.
Results
This section compares the performance and robustness of
the three control strategies to the parameter perturbations
shown in Table 1. The yields and purities of b-form at
the end of batch for all control strategies are tabulated in Ta-
ble 2. The concentration-temperature trajectories in Figure 1
for T-control applied to the nominal model and the corre-
sponding C-control obtained from Eqs. 33 to 38 are coinci-
dent, indicating that the parameterization of Eqs. 33 to 38, is
suitable for representing the C vs. T setpoint used in C-con-
trol. The growth kinetics of the b crystals are relatively slow,
which results in the optimal control trajectories being very
close to the solubility curve for the a-form, to maximize the
supersaturation with respect to the solubility of the b-form
while operating between the two solubility curves.
The nominal temperature trajectory produced by T-control
is highly nonrobust to perturbations in the physicochemical
parameters, as seen in Figure 3b–d, with the temperature
constraints violated for parameter sets 2 and 3. The tempera-
ture trajectories reoptimized for the perturbed model parame-
ters indicate that the optimal temperature trajectory is very
sensitive to shifts in the model parameters. The mass profiles
in Figure 3e,f indicate that the crystals of the a-form com-
pletely dissolve within 2 h for the two feasible parameter
sets, whereas crystals of the b-form continue to grow for 5 h,
which is consistent with the notion that the growth rate of
crystals of the b-form is rate-limiting for the design of the
batch temperature trajectory for this polymorph transforma-
tion.
The temperature trajectories produced by robust T-control
are robust in terms of satisfying the operating constraints for
the whole set of perturbed parameters, but are very conserva-
tive in terms of having very long batch times, and poor pro-
ductivity for all values of the physicochemical parameters
(see Figure 4a–d). Comparing the reoptimized T-control and
robust T-control mass profiles in Figure 4e,h indicates that
robust T-control leads to unnecessarily long batch times for
some values of the physicochemical parameters. Designing a
batch control trajectory to satisfy the operating constraints
for the whole set of potential perturbed model parameters
can result in very sluggish performance irrespective of what
the actual physicochemical parameters happen to be in a par-
ticular batch run. While such approaches have been heavily
studied in the batch design and batch control literature (for
example, see articles cited in
16
), these approaches can result
in very poor performance when applied to practical batch
processes.
Although C-control does not explicitly include robustness
in its formulation, C-control nearly satisfies all of the operat-
ing constraints for all sets of model parameters (see Figure
5a–d), demonstrating nearly the same robustness as robust T-
control.
{
Further, C-control results in much faster batch times
and higher productivity than robust T-control for some sets
of physicochemical parameters (see Figure 5a and d). In
addition, C-control results in the batch productivity similar to
that obtained by T-control reoptimized for each parameter
set, as seen by the closeness of the C-control and reoptimized
T-control trajectories in Figure 5a–d. C-control has nearly
the same performance as that of the best T-control trajectory,
with the batch times obtained by C-control are large only
when necessitated by the particular values of the physico-
chemical parameters. This performance is obtained by C-con-
Table 1. Case 1 has no Uncertainties (the Nominal Model),
Case 2 has the Worst-case Parameter Values, Case 3 is the
same as Case 2, but only Includes Variations in the Kinetic
Parameters, and Case 4 are Parameter Variations with Fast
Growth Rate for Crystals of the b-form (Which is the
Rate-limiting Step)
Cases y
1
y
2
y
3
y
4
y
5
1 0.0 0.0 0.0 0.0 0.0
2 20.2 0.2 0.2 20.05 0.05
3 20.2 0.2 0.2 0.0 0.0
4 0.2 20.2 0.2 20.05 0.05
{
There is a slight violation of the lower bound on temperature bound for the pa-
rameter set for Case 4, which is due to the shift in the solubility curve of the a-
form. This violation can be removed by shifting the nominal concentration-temper-
ature trajectory slightly away from the a-solubility curve (see Figure 6).
AIChE Journal October 2007 Vol. 53, No. 10 Published on behalf of the AIChE DOI 10.1002/aic 2647
Page 5
Table 2. The Yields and Purities of b-form at the End of Batch for Different Control Strategies
Cases
Optimal T-control Robust T-control C-control
b-yield (g) b-purity (%) b-yield (g) b-purity (%) b-yield (g) b-purity (%) b-yield (g) b-purity (%)
1 24.15 100.0 24.15 100.0 16.86 100.0 24.15 100.0
2 9.68 97.50 n.a.
^
n.a.
^
9.02 69.80 9.65 95.10
3 13.70 100.0 n.a.
^
n.a.
^
13.21 100.0 13.41 100.0
4 23.88 100.0 23.88* 100.0 16.42 100.0 n.a.
n.a.
^
These values would be meaningless for comparison purposes due to temperature constraint violations.
*From Figure 3f, this value is approached at a much later time (4.67 h) compared to the optimal one (3 h).
The constraint violation can be removed as mentioned in the text and Figure 6, and the resulting b-yield and purity are 23.86 g and 99.9%, respectively, for
this case.
Figure 3. Temperature profiles applied to: (a) case 1,
(b) case 2, (c) case 3, and (d) case 4, for T-
control (2), the temperature trajectory reop-
timized for the perturbed model parameters
(2 2 ), and the shaded region showing the
constraints on the temperature for T-control.
Mass profiles for: (e) case 1 and (f) case 4, for T-control
(M
a
, 2;M
b
, 2 2 ), and for the temperature trajectory
reoptimized for the perturbed model parameters (M
a
, *;
M
b
, 1). (The mass profiles for cases 2 and 3 are not shown
since the temperature trajectories do not satisfy the operat-
ing constraints).
Figure 4. Temperature profiles applied to: (a) case 1,
(b) case 2, (c) case 3, and (d) case 4, for ro-
bust T-control (2), the temperature trajectory
reoptimized for the perturbed model parame-
ters (2 2 ), and the shaded region showing
the constraints on the temperature for robust
T-control.
Mass profiles for: (e) case 1, (f) case 2, (g) case 3, and (h)
case 4, for robust T-control (M
a
, 2;M
b
, 2 2 ), and for
the temperature trajectory reoptimized for the perturbed
model parameters (M
a
, *;M
b
, 1).
2648 DOI 10.1002/aic Published on behalf of the AIChE October 2007 Vol. 53, No. 10 AIChE Journal
Page 6
trol without having the poor robustness of T-control com-
puted from the nominal model. In summary, C-control has
all the respective advantages of T-control and robust T-con-
trol, without any of their respective disadvantages.
C-control uses nonlinear state feedback control of the con-
centration measurement to follow a desired path in the phase
diagram.
26
This article, for the first time, demonstrates the
improved robustness of C-control for the solvent-mediated
transformation from one polymorph to another. There is the-
oretical and simulation support that relatively simple nonlin-
ear state feedback controllers can be derived that provide
nearly optimal performance and robustness for batch pro-
cesses.
16
For a polymorphic crystallization, this paper derives
such a nonlinear state feedback controller, motivated by and
interpreted within the context of the crystallization phase dia-
gram, which has the desired performance and robustness
properties.
Conclusions
The robustness and performance of T-control, robust T-
control, and C-control strategies were compared for maxi-
mizing the batch productivity during the solvent-mediated
polymorphic transformation of L-glutamic acid fr om the
metastable a-form to the stable b-form crystals. Operating a
batch polymorphic crystallization using the existing
approach based on control along a temp erature vs. time tra-
jectory
21
is shown to be very sensitive to variations in the
nucleation and gr owth kinetic s and shifts in the solubility
curve, resulting in violations of the operating constraints.
For the polymorphic transformation fro m a-form to b-form
crystals, these constraint violations can result in the nuclea-
tion and re-growth of undesired a-form crystals. Robust T-
control resulted in satisfaction of the operating constraints
for a full range of variations in the physicochemical param-
eters for the kinetics and thermodynamics of the polymor-
phic transformation, but resulted in v ery poor batch produc-
tivity ( long batch times) for parameters in which short batch
times are possible.
A nonlinear state feedback controller designed to follow
an optimal trajectory in the concentration-vs-temperature-
Figure 5. Temperature profiles applied to (a) case 1, (b)
case 2, (c) case 3, and (d) case 4, for C-con-
trol (2), the temperature trajectory reopti-
mized for the perturbed model parameters
(2 2 ), and the shaded region showing the
constraints on the temperature for C-control.
Mass profiles for: (e) case 1, (f) case 2, and (g) case 3, for
C-control (M
a
, 2;M
b
, 2 2 ), and for the temperature
trajectory reoptimized for the perturbed model parameters
(M
a
, *;M
b
, 1). (The mass profiles for case 4 are not
shown since the temperature trajectory (slightly) violates
the T lower limit).
Figure 6. (a) Temperature profile applied to case 4 for
C-control with parameters p
1
5 36.56, p
2
5
11.5, p
3
5 0.4185, and p
4
5 0.03656 (2), the
temperature trajectory reoptimized for the
perturbed model parameters (2 2 ), and
the shaded region showing the constraints
on the temperature for C-control; (b) the cor-
responding mass profiles for C-control (M
a
,
2;M
b
, 2 2 ), and for the temperature tra-
jectory reoptimized for the perturbed model
parameters (M
a
, *;M
b
, 1).
AIChE Journal October 2007 Vol. 53, No. 10 Published on behalf of the AIChE DOI 10.1002/aic 2649
Page 7
phase diagram was highly robust to variations in the kinetic
parameters, while providing batch productivity nearly as high
as optimal control applied to batch crystallization with known
parameters, as illustrated in Figure 5c,g and many simulation
studies (not shown here) with variations in y
1
to y
3
(while
maintaining y
4
5 y
5
5 0). Although not explicitly included in
the optimization formulation, the operating constraints were
satisfied for the entire range of physicochemical parameters
(see Figure 5), except for a small constraint violation due to
variation in the solubility of a-form crystals that was removed
by slightly shifting the concentration setpoint trajectory away
from the a-solubility curve (Figure 6). Alternatively, shifts in
any solubility curve can be accounted for by updating meas-
urements of the solubility curve whenever there are significant
changes in feedstocks between batch runs. Automated systems
exist for measuring such solubility curves.
17,19,26
Published results,
2,20
as well as one of the author’s experi-
ence consulting with industry on their polymorphic crystalli-
zations suggest that the solubility curves of most polymorphs
are typically much closer together than for the a and b poly-
morphs of L-glutamic acid (Figure 1).
§
If this is true, then
the desired operating region for most polymorphic crystalli-
zations is typically much smaller than for the system investi-
gated in this study, making the robustness of batch control
strategies of much greater importance for most polymorphic
crystallizations. The results in this article indicate that the
design of operating procedures for future polymorphic crys-
tallizations should implement C-control.
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§
This is especially true for enantiomeric polymorphs, in which the solubility
curves intersect.
2650 DOI 10.1002/aic Published on behalf of the AIChE October 2007 Vol. 53, No. 10 AIChE Journal
Page 8
  • Source
    • "This is the case of microstructured reactors [4]. Unfortunately, channel blocking issues limit, for the moment, the industrial application [4]. From a more fundamental point of view, the complex interaction of the physical chemistry (nucleation, crystal growth rates) and chemical engineering (hydrodynamics, transport processes, scale up), which controls the polymorphic form, crystal stability and CSD, is a key topic. "
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