Missouri University of Science and Technology
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Materials Science and Engineering Faculty Research
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Materials Science and Engineering
1-1-1995
Modeling and Control of a Batch Condensation
Polymerization Reactor
D. G. C. Robertson
Missouri University of Science and Technology, [email protected]
Stephen A. Russell
Jay H. Lee
Babatunde A. Ogunnaike
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Recommended Citation
D. G. Robertson et al., "Modeling and Control of a Batch Condensation Polymerization Reactor," Proceedings of the 1995 American
Control Conference, Institute of Electrical and Electronics Engineers (IEEE), Jan 1995.
The definitive version is available at http://dx.doi.org/10.1109/ACC.1995.529807
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please contact [email protected].
N
ELING AND CONTROL OF A BATC
POLYMERIZATION REAC
Douglas G. Robertson, Stephen A. Russell, Jay
Department of Chemical Engineering, Auburn University, Auburn, AL 36849
Babatunde A. Ogunnaike
E.I. DuPont de Nemours and Company, Wilmington, DE 19880
Abstract
the process is such that returningthem to the nominal trajectory
m a y not yield acceptable product.
In this paper we examine the control of a condensation polymuisation reactor. The nylon 6,6 system is chosen as a representative polymerization process. A model for a nylon autoclave
tor is presented which captures the important process behavior.
The model is subsequently used to evaluate the performance of
control schemes based on tracking nominal pressure and temperature trajectories. In particular, the ability of this type of control
strategy to compensate for disturbancesas well as variations in
feed conditions is studied.
Due to the availability of published information [3]-[12](e.g., kinetics, physical properties, process description), the nylon 6,6
process was chosen as a representative cxample of a polycondensation system. Nylon 6,6 polymerization is a reversible reaction of the A-A/B-B type (two monomers, each of which has
the same h c t i o n group on both ends) where hexamethylcnediamine (HMD) and adipic acid react to form a polymer link and
a molecule of water. In this paper we develop a model of the
process designed to achieve a reasonable compromise between
model complexity and ability to capture the important process
behavior. The model is then used to assess the ability of control schemes based on following nominal trajectories to reduce
the variability in product polymer properties created by disturbances and Veriations in the initid conditions.
1. Introduction
Condensation polymerisation involves the reaction of functional
groups on adjacent molecules to form polymer chains with the
evolution of a low molecular weight by-product (condensation
product) such as water. To produce a polymer product with a
high number average molecular weight, the extent of reaction
must be well above 0.99. To achieve this high extent of reaction,
the polymeri5ation is often carried out in a batch reactor with
the excess condensation product being continuously vaporised to
drive the reaction towards completion [l]. The result is that in
addition to the usual challenges associated with batch polymerizations (highly nonlinear behavior, lack of on-line measurements
of polymer properties and end-use performance characteristics,
poor models) we must also be concerned with the relationship
between the liquid reaction phase and the vapor phase which
will contain the condensation product as well M some of the reactants. Another complication is that the maximum achievable
number average molecular weight is very sensitive to the stoichiometric ratio of the two monomers. Small deviations of this
ratio from unity can result in a drastic reduction in the average
molecular weight of the product (see [2],Figure 1.8).
2. Autoclave Model
2.1. Kinetic Model
The kinetic model was taken &om [lo, 111. The kinetics are
in tapur of the equivalent U n i t s involved in the reaction (i.e,
water, polymer linhs,and end groups). The reaction mechanism
is modeled as follows:
C .+ SE + W
L-,SE+A
C
polyamidation A
L W
degradation
+
+
Where A is amine end groups (2 per HMD), C is carboxyl end
groups (2 per adipic acid), L is a polymer link, W is water and
SE is a stabili5ed end group. The degradation reactions have
One method for controlling batch condensation polymerization
reactors is to use historical pressure and temperature trajectories from acceptable runs as the setpoints for profile tracking
controllers [2]. The advantage of this method is that it does not
require a model of the process, only historical data. The dieadvantage is that it caxtnot compensate for deviations from the
historical operating conditions. Batch-to-batch variations in the
feed composition or heat transfer characteristics generally result
in variations in product quality. For example, reactor foulingreduces the heat transfer capabilities from one batch to the next.
Of particular interest is the case when batch runs deviate si&%cantly from their trajectories due to disturbances. The nature of
been included in the model since they aflect the temperature
dependence of the achievablenumber average molecular weight.
The rate equations for the above reactions are
where C< b the concentration of species i (GTis total concentration). The lcinetic parametas can be found in [lo,111. Other
physical properties used in the model were developed from the
references [3]-[12]and an not listed due to space limitations.
"To whom all correspondence should be addressed phone
(334)8442060,fax (334)8442063,e-mail:jhlQeng.auburn.edu
t 746
2.2.
Mole Balance for the Liquid Phase
the two monomers, and p , the extent of reaction of the limiting
end group.
To determine the composition of the mass vaporising, we 0sume that it contains only water and HMD and is in equilibrium
with the liquid phase. The total mass flowrate of vaporisation
is modeled as K(PvQP P ) [13], where K is a pseudo overell
mass transfer coefficient, PvaP is the vapor pressure of the liquid phase, and P is the reactor pressure. The relative amounts
of water and HMD in the vapor an determined from the VLE
relationship
==aXHMD
A. problem that arises in the application of the Flory distribution
to the nylon 6,6 system is that HMD is being removed through
vaporisation while carboxyl end groups are subject to degradation by forming stabilised end groups which can PO longer react.
To account for these factors we make the following modifications
to Flory’s original derivation-the reasoning is that, in terms of
the current chain length distribution, the moles of HMD that
have left the reactor can be treated as if they were never there,
while the moles of stabilised end groups that an formed can be
treated as carboxyl end groups that have not reacted. Equation (30) of [15] becomes
-
vw
XW
(4)
whcre x; is the mole fraction of species iin the liquid phase, g;
is the corresponding equilibriummole fraction of species iin the
vapor phase, and a is taken to be the ratio of the pure component
vapor pressures. From this equation, the mass fraction of HMD
and water ( w b and tu&,, respectively) in the vapor can be
determined.
Based on these assumptions, the mole and overdl mass balances
for the autoclave an
dvcc
- -
-VRa-VRi
-
VRa-VRa
dt
#CLdt
where nl is the mole fraction of HMD,p is the extent of reaction
of the limiting end group, r is the feed ratio (< l), (VC;)is the
moles of species i in the liquid phase (the subscript 0 denotes
initial conditions), and the integral term is the total amount
of HMD that has vaporieed. These equations are defined for
t < 1 (excess HMD in the feed). At some point (due to the
vaporisation of HMD),A becomes the limiting end group and r
becomes greater than one. When this happens Equation ( 2 9 ) of
[15] is modified accordingly:
when: V is the volume of the liquid phase, w;’ is the vapor phase
mass fraction of species i that is in equilibrium with the liquid
phase, M; is the molecular mass of species i,and p is the density
of the liquid phase. The mass of the initial charge to the reactor
was assumed to be 2000 kg with 25 weight percent water. The
“8hhg-8
Was assumed to contain equbOl8r amounts of
HMD and adipic acid. Two additional kilograms of HMD were
added to the feed to compensate for losses due to vaporisation
[SI*
The concentrationof HMD can then be caldatedfrom the mole
fraction CHMD= n l ( . s C ~ .5Cc . ~ C S I ~ ) .
The vapor phase mole balance has been neglected for the following reasons: (1) Only the total reactor pressure in the vapor
space &ects the liquid phase dynamics. (2) A pressure controller
is used to manipulate the reactor pressure by adjusting a vent
valve. (3) The dynamics of the pressure control loop an assumed
sufliciently fast to allow them to be neglected. hrthumore, we
assume that the pressure control is good enough that the reactor
pressure can be takm as an input.
+
2.4.
+
Energy Balance Around the Liquid
Phase
A s s u m i n g l u m p e d p a t c r s and negligible heat loss, the energy
balance for the liquid phase is
2.3. Modifications to the Flory
Distribution
- A H , H ~ D w ; ~ ~ ~ K ( P * c P - -P ) A H , WM~Z*pKC( PP-V- P
MHMD
POPV
One additional M c u l t y is that the VLE relationship (4) requires the mole fraction of HMD in the liquid phase; whereas,
the mole balances are in t a n u of the species A-cquivalent
amine end groups. This was necessery to avoid more complicated moment-balance equations [14], but do- not distinguish
between an amine group on the end of a HMD molecule and
an amine group on the end of a polymer molecule. To determine the concentrationof HMD in the reactor, some assumptions
muat be made about the distributionof molecule u s u . A us&
method based on the qual reuctiritity assumption (reaction rate
doe0 not depend on molecule &te)b the Florp distribution[15].
The IFlory distribution for A-A/B-B polymerisation characterises the distribution of polymer molecule lcqtha (the monomer
has length one) in tof two parameters: t, the feed ratio of
)
+
where AH; is the heat of reaction for reaction a, AH; is the heat
of vaporisation for species a, Q is the heat input to the reactor
supplied by a steam jacket, and Cp is the specific heat capacity.
The heat input to the reactor was modeled as Q = UA(Tj T)
where the jacket temperature Tj is determined from the jacket
pressure by the Antoine equation for saturated steam [12].
-
2.5.
Nominal Trajectories
The nominal trajectories are defined as those which result in
the desired polymer properties under open loop operation with
1747
P
:I
P
150
/
'\\
0
20
4b
60
1
L
80
120
140
. Pressore
.
160
180
0
200
9
0
Time (min)
n
wherc G r is the Grashof number and Pr is the Prandtl number. The subscript f denotes properties evduated at the film
temperature T f , the subscript j denotes the jacket temper at^,
and the subscript b denotes the bulk fluid temperature. k is the
thcnnal conductivity of the solution, L is the height of the heat
transfer surface, g is the acceleration due to gravity, p is density,
p is viscosity, and fl is the volumetric coefficient of expansion.
The j&t
temperature Tj is 330 OC. The constants C1 and 71
were chosen so that the model matched the desired temperature
trajectory during the initial heating stage (time 0 to 60 min).
combining the above equations, absorbing 811 constant parameters and conversion factors into Cl,and simplifying yields
The second region of heat transfer is the boiling phase and is
modeled by a relation given in Section 10 of [17]:
Time (min)
Figure 1: Nominal Trajectories for Nylon Autoclave
Model
Rea
DG
= -
-
Prr,.= CpfW
kf
where Re is the Reynolds number, P r is the Prandtl number,
D is the autoclave diameter, and G is the vapor mass velodty
(proportional to PvaP - P). After simplifying and absorbing
constant parameters and conversion factors into Ca, the equation
is
for
no disturbances and no&d initial conditions. The trajectories of interest for this model are reactor temperature, reactor
pressure, and vent rate. The n o d profiles are presented in
Figure 1. These profiles were determined from the combination
[6],
knowledge of industrial
of those in Jacobs and Z-ennan
nylon reactors, and the dynamics of the current polymer model.
To construct the n o d trajectories, Equations (5)-(9) were
simulated with temperature and reactor pressure as inputs. The
trajectories were determined as follows:
The constants Ca, m, and 7s were chosen so that the model
matched the reference trajectory during the boiling phase (time
60-160minutes).
1. The feed to the autodavewaa assumedto be 150 OC. During
the initial heating phase of the reactor (before boiling begins),
the rate of increase in reactor temperature is nearly constant. A
reasonable rate of 1.6O C/min waa chosen.
2. The initial setpoint for the reactor pressure was chosen to
be 250 psis.
3. When the vapor pretmure of the liquid phase exceeds that
of the reactor pressure, the mixture begins to boil. The heat
of vaporieation being removed from the liquid phase causes the
change in slope seen in the temperature prof& around time 60.
Recall that vaporbtion rate is modeled aa K ( P v a P - P ) . The
overd mass transfer coefficient was chosen to give a reasonable
rate of vaporization during this period.
4. As the rete of vaporisation begins to drop, the reactor
pressure is ramped down to induce water removal and shift the
equilibrium towards high molecular weight polymer.
5. The reactor is held at 280 OC and atmospheric prcssurc to
complete the polymuisation.
By the end of the boiling phase most of the water has vaporized
and the reactor contents are large, viscous polymer molecules.
The heat transfer in this region is modeled by conduction
U, = C&f
(15)
with Cs chosen to match the reference temperature trajectory.
2.7. Polymer Properties
Two importsnt characteristics of the polymer product are the
number average mole&
weight Mn and end group concentretions [SI. The number average molecular weight and the concentration of amine end groups will be used ss representative polymer characteristics for evaluating the controller performance.
The number average molecular weight can be calculated by assuming the !?lory distribution [15] and making the same modifications as before. In this case, Equation (32) becomes
2.6. Heat Transfer Model
The modding of the overall heat transfer coefficient U was divided into three stages corresponding to the stages of a u t o c ~ v e
operation [SI. In the initial stage of the batch, the mixture is
heated until boiling be+.
The heat transfer coefficient in this
region,V,,, is modeled by the €allowing natural convenctionrelation [MI:
dt tMa-c(VOc)o
e K ( p w m p - p )dt+(vcc)o
MA-A ( V C A ) O -t ~ea ~K
d ( P " p - P )
MO=
(vCA)O-]o*
wherc MO is the mass of one unit of the polymer chain, MA-A
is the mass of an A A segment within the polymer molecule,
ancl MC-Cis the mass of a C - c segment within the polymer
molecule.
-
The concentration of amine end groups is typically measured
in gram-equivalents of amine ends/lOegrama of polymer and is
given by the following equation:
1748
3. Reactor Control
heat transfer coefficient U ;(3) a +5 psi bias in steam pressure
measurement; (4) a twqty percent increase in the initial water
concentretion; and (5)a ten degree decrease in the initial reactor
temperature. The number average molecular weight and a m h e
ends concentrationfor these runs are presented in Table 1, The
The model described in Section 2 will now be used to evaluate
possible reactor control strategies. The control objective for this
process is to obtain the target values for number average molecular weight and concentration of amine end groups at the end
of the batch. However, the direct regulation of polymer characteristics is not possible due to the lack of on-line measurements
related to these or any other end-use properties. A feasible control objective is the regulation of temperature or other secondary
process variables correlated with the polymer properties.
Case
1
2
3
4
5
Historically, a typical method for controlling this type of reactor has becn to utiliee nominal controlled variable trajectories
from acceptable runs as setpoints for profile tracking controlla
[a]. Based on process understanding gained from developing the
model as well as industrial practices, two strategies of this type
were chosen for evaluation: controlling (1) reactor and jacket
pressure and (2) reactor pressure and reactor temperature. The
simulated pcIformanct of these strategies will be evaluated in
tcrms of their ability to produce desirable polymer properties in
the presence of typicalprocess disturbancesand variationsin the
initial batch conditions.
Disturbance
Tawet
U:-G%
Pj: +5 bias
WO:
+20 %
TO:-10’
MW
13128
13060
12997
12707
13058
Amine Ends
54.07
57.02
57.16
48.63
53.23
Table 1: Polymer properties for PID tracking of nominal reactor and jacket pressure trajectories
target values correspond to those generated by the model following the nominal trajectories. Cases 2-3 and 4-5 demonstrate
the variability in polymer properties due to disturbancesin heat
transfer and variations in feed conditions, respectively. The results are not suprising since this strategy does not employ feedback of the reactor temperature information and therefore has
limited ability to reject these disturbance types.
3.1. Implementation of Trajectory
Tracking PID
A discretetime PID controller [18]was used to track the nominal
trajectories discussed above. The controlleroutput WM in terms
of the deviation from a reference input trajectory. The following
modifications were made to ensure proper performance: Filtering of the error signal was necessary to reduce the sensitivity
to measurement noise. Saturation limits were employed on both
the abaolute value and the rate of change of the controller output
to d c c t process constraints. A standardanti-windup algorithm
was also included to account for the input constraints.
3.3. Tracking of Reactor Pressure and
Temperature Trajectories
Many of the most common disturbances concern heat transfer
to the reactor and, therefore, directly affect temperature. Also,
the final polymer properties are strongly dependent on the temperaturc history of the process. It is reasonable to expect that a
control scheme which incorporatesfeedback of temperature measurements can reduce the variability in the polymer properties.
To examine this possibility, a control scheme based on tracking
nominal reactor temperature and reactor pressure profiles was
implemented. Temperature tracking was accomplished by a cascaded PID controller with the jacket pressure setpoint as the
manipulated variable. The reactor pressure and jacket pressure
loops were unchanged from the first scheme.
The nonlinear, time-varying nature of the process presents a significant d “ g e to fixed-parameter control strategies. To overcome these difficulties,we assumed that the dynamicschangedatively slowly. Based on this assumption, a gain scheduled controller was implementedwith tuning parametera for each stage of
operation determinedfrom the local dynamics. As a first approximation, the PID parameters were determined from a relay test
[18].An approximatedrelayinput was used to produce austained
oscillations in the measured output variable. The ultimate gain
Ku and period Tu Of the Syat- W-C determined by anslysing
the amplitude and frequency of the resulting oscillations. Using
Ku and Tu,the initial controller settings w u z determined by
Zicglcr-Nichols criteria [18]. The controller settings were then
fine-tuned through simulation.
Table 2 su”ari5es the performance of this control strategy for
the same conditions as the simulations recorded in Table 1. It
Case
1
2
3
4
5
3.2. Tracking of Reactor and Jacket
Pressure Trajectories
To serve as a base case for our study, PID control of reactor
and steam pressure profiles was implemented. Because of the
fast vapor dynamics noted earlier, this amounts to specifying
the reactor and j&t pressure profiles. Note that the nominal
pressure profiles will r e d t in the nominal reactor temperature
profile in the absence of disturbances. Based on this fact, control of reactor and jacket pressure can be interpreted as open
loop control of reactor texnpcraturc. DUGto the lack of temperature feedback, tbia arrangement h not expected to be robust
to disturbances ai€ectingheat transfer. The utility of this strategy is in characterieing the sensitivity of the process to vrvioru
disturbances.
Disturbance
Target
u:-io%
Pj: +5 bias
WO:
+20 %
TO:-10’
MW
13128
13145
13130
12633
13158
Amine Ends
54.07
55.00
54.70
47.70
54.97
Table 2: Polymer properties for PID tracking of reactor temperature and pressure trajectories
apparent that robustness to heat transfer disturbances is increaaedconsiderablyfrom the first strategy (see Cases 2-3). However. Case 4 demonstraten that sensitivity to variatiansin the initial water content has not been improved. As can be seen from
Figure 2, the PID controller tracks the deaired temperature proBe despite variations in the initial water content. However, Table 2 indicates that in terms of polymer properties, variations in
the initial water content lead to considerable variations in polymer properties despite the improvements in temperaturecontrol.
The reason ia that the cxccss water shifts the equilibrium in the
M
The control scheme was simulatedfor the following disturbances:
(1) baseline no disturbance (2) a ten percent decrease in the
-
1749
Acknowledgement
The authors are grateful to Dr. Mael Sela and Dr. Tony W. Liu
at the DuPont Experimental Station for their valuable support
in preparing this paper.
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im
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The results of this study suggest that tracking pre-determined
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the process conditions change. The open-loop strategy based on
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any of the disturbances considered. Although feedback of autoclave temperature measurements was able to improve the robustness to certain types of disturbances, this strategy docs not
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to the autoclave temperature trajectory, the polymer properties
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start of the batch (especially water to monomer ratio) and correctly adjusting temperature and pressure/vent rate trajectories
to obtain better quality polymer. Because of the high level of
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and control schemes for the nylon process. The results presented
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