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Optimization of semibatch polymerization reactions
Computers chem. Engng, Vol. 13, No. 1/2, pp. 63-72, 1989 Printed in Great Britain.All rights reserved
0098-1354/89 $3.00+ 0.00 Copyright© 1989Pergamon Press plc
OPTIMIZATION OF SEMIBATCH POLYMERIZATION REACTIONS G. D. CAWTrIONand K. S. KNAEBELaf Department of Chemical Engineering, The Ohio State University, Columbus, OH 43210, U.S.A. (Received 27 June 1988; received for publication 6 July 1988)
Abstract--Dynamic chemical processes often require a multiobjective formulation for complete process optimization. Specifically, semibatch copolymerization systems are analyzed using vector-objective techniques to determine the tradeoffs between the different goals: narrow the copolymer composition and polydispersity distributions, and minimize the time of reaction. Due to the complexities of the detailed kinetic models, simpler tendency models are evaluated by fitting proposed reaction networks to experimental data. Simulation studies of an acrylonitrile-styrenecopolymer and a step-wise oligomer are then used to determine optimal production schemes.
INTRODUCTION
only at specific operating conditions since it may be unrelated to the actual process mechanism. The "tendency" model approach (Filippi et al., 1986) is a compromise between the detailed mechanism and a totally empirical reaction model. This technique employs engineering insight into the proposed kinetic structure, whose parameters are calculated using available experimental data. Furthermore, since simulation of this model usually requires less computational time, real-time implementation for adaptive optimal control is possible. This study develops a optimization strategy for both free-radical and step-wise polymerization in semibatch reactors. Processes are analyzed as multiobjective optimization problems to determine the effects of and tradeoffs between the polydispersity, composition distribution and minimum end-time using various combinations of temperature and reactant flowrate profiles. Also, tendency models are generated for two examples to illustrate the models' applicability and accuracy.
Chemical processes typically have multiple performance objectives which require simultaneous inspection for best process operation. The optimal process design of semibatch reactors allows manufacture of product with the desired physical properties in the minimum amount of a time. The molecular weight distribution is ~the commonly measured variable, representing the variety of molecules comprising the polymeric product. In copolymerization, a significant problem to be considered is the composition distribution in the composite. Molecules which have similar molecular weights can and usually have different proportions of monomers due to changes in reaction conditions. Even for molecules having the same ratio of molecules, there can be a different ordering of monomer units which can appreciably change the product properties. Movement of the chemical industries towards manufacture of specialty chemicals has made batch operations more popular. Because these dynamic systems have small production levels and complex operating procedures, thorough understanding of reactor performance and reaction kinetics is usually not economically justified. Nevertheless, accurate modelling of these systems is a very valuable tool for achieving optimal system performance. The common approach to modelling a reactor system requires specification of the individual physical and chemical phenomena (such as heat transfer, stoichiometry and kinetics) to provide a detailed mechanism. This technique requires a significant effort in identification of many unknown parameters resulting from the complex set of equations. The alternative method consists of empirically fitting experimental data; the model represents the system
MULTIOBJECTIVE OPTIMIZATION PROBLEM Because of the several performance objectives, process optimization of polymeric systems requires a multiobjective formulation. Two possible alternatives for the solution are available: combine all performance measures into one objective to produce the single optimal control policy, or generate a series of control strategies which relate the tradeoffs between the objectives and the control parameters. Priority is requested for the first technique to form a single objective function although the weighting factors are difficult to specify a priori. The second method leads to a much better understanding of the process operation and provides valuable information for subjective decision-making regarding the different control
tAuthor to whom all correspondence should be addressed. 63
64
G.D. CAWTnONand K. S. KNAEBEL
strategies. The general vector optimization problem is formulated as follows (Chankong and Haimes, 1983): min F(x) = min{Fl(X), F 2 ( x ) , . . . , subject to:
initiation: kDl
(1)
F~ (x)},
dx ~-~ -- l(x, u)
x(0) = Xo,
gk(X, U) ~<0,
k = 1, 2 . . . . . m
,2R'
R" + M l
) PI,0
R.+M2
) Q0,1,
propagation: kpll
Pn,m+ MI
where F is the objective function vector, x is the decision vector, u is control vector arid g is the constraint vector. The common objectives for semibatch copolymerization reactions are the composition error (i.e. the difference between the specified and the instantaneous compositions), the polydispersity and the time of reaction:
P. + 1,~ kpl2
P.,m + M2
Q.,m + 1, kp21
Q..~+MI
'Pn+l,m
kp22
Qn,m + M2
min F(u) =
[CC* - CC(t)] 2 dt, D(tf), tr.
Qn,m+ 1,
(2) termination (by combination):
The first step in a multiobjective problem consists of determining where a tradeoff exists between competing objectives. These compromises can be determined by using the concept of noninferiority. Noninferior solutions are control policies which no improvement can be made in one objective without degrading another objective. These solutions will allow the decision maker to decide between the different operating regimes so that the optimal control policy can be chosen. A popular method for determination of the noninferior sets is the E-constraint approach (Faber, 1986). It is based on treating all but one of the objective functions as parameterized constraints, thus solving the vector optimization problem using a scalar approach. The original problem is then reformulated as:
en,m q- Pr,q
kcll
' Mn +r,m+q kcl2
Pn.m q- Qr,q
' Mn +r,m+q, kc21
Qn,m -+- Pr.q
' Mn+r,m+q kc22
Q.,m + Qr,q
' Mn+r.m+q,
chain transfer (to monomer): P.,,. + M 1
kfl 1
, M.,,. + Pl.o
kfl2
P.,~ + M2
~ M.,m + Qo,l,
kf21
subject to:
Q,,m + M1
(3)
min F 1(x) dx dS = I(x, u)
x(0) = Xo
~ ( x ) ~
j = 2, 3 . . . . .
gk(X) ~<0.
k=l,2 ..... m
Ej-= F* + E*,
j=2,3 ..... n
~*~<0,
j=2,3 ..... n
, M,,m + PI,o k f22
Q.,m + M2 n
The minimum value of the j t h objective when all other objectives are ignored is F* and the deviation from this minimum value is e*. The values of E* are parametrically in order to determine the noninferior set. FREE-RADICAL COPOLYMERIZATION MODEL
The standard reaction model for homogeneous solution or (low-conversion bulk) free-radical copolymerization consists of the following steps:
' M.,m + Qo,1.
For application of this model, the pseudosteady-state assumption and the long-chain hypothesis are valid concepts used for mathematical simplification. Also, phenomena such as volume contraction, thermal initiation and gel-effects are neglected for this study. Employing these simplifications, a set of mass balances may be written as (Tsoukas et al., 1982): dV d--t = QM,(t) + QM,(t) + QM2(t ),
(4)
dI dt
(5)
koII -~
CzQMI(t) V
I dV V dt '
dM~ dt = [(kpll + knl)P° + (kp21 + kf2OQo]Ml CiQMl(t) q ~
Ml d V V dt'
(6)
Optimization of polymerization reactions dM2 dt = [(kpl2-}- kfl2)P° d- (kp22+ kn2)Qo]M2
C2QM2(t) q
STEP-WISE OLIGOMER MODEL
M 2 dV
V
V dt"
(7)
The parameter used to represent the radical concentrations are the zeroeth order moments of the radical distributions Po and Qo:
n=0m=0
'
I
kcll_t_2flke12+f12kc22 kpl2 q- kf12 M2
(9)
The molecular weight distribution of the radical and dead polymeric molecules can be solved by applying generating functions and the method of moments described by Ray (1972). The concentration of the dead polymeric product is represented by the moments (o, ~1 and ~2. d(° = ~ 2 p02 +
The step-wise polymerization system to be examined is an oligomeric compound used as a precursor to many copolymerization reactions. Due to its proprietary characteristics, only a framework of the process chemistry will be presented. The oligomer is produced by the reaction of an initiator with two differing monomers. The resulting kinetic mechanism is represented by the three steps:
, (8)
Qo = flPo fl - kp21 + kol MI"
--dt
65
k~2~ 2 k¢12P°Q° + T Qo + (kfllMl
+ kn2M2)Po + (knlMx + knzM2)Qo
I + MI ~ P1,0.
(17)
P .... , + M 2 ~ Q ....
(18)
Q...+ M I ~ P . + I , ..
(19)
The subscripts for variables P..._ ~ and Q.,. indicate the number of each constituent making up each product molecule. Because the composition follows a unique sequence of alternating units, all monomer 1 terminated units are made up of one less molecule of monomer 2 than component 1. Products terminated with monomer 2 have an equal number of both components. The general configuration is denoted by the structure: I - [ - MI M 2 - ], - M~.
~0 dV V dr"
(20)
(10)
d~l -_ kcllPoP1 + kc12(PiQo + PoQO + k¢22QoQ1 q- ( k n l M l q- kfl2Mz)P l q- (kf21M1
~,1 dV + kf22M2)Ql - v d--t-'
(11)
t = kcHPoP2 + kc,~P~ + kc12(PzQo + PoQ2) + 2kc12P1Ql + kc22QoQ2 + k¢22Q12 + (kmM1 + Kn2M2)P2 + (kmMl ~2 dV + knzM~)Q2 - V d--t-'
~o~2
(13)
The state equations can now be represented in vector form as: dx
d t (t) = f[x(t), u(t)],
P0 = (M? -- M,) - (M ° - M2),
(12)
The total number average molecular weight (g.) and the polydispersity index D are given by: ~L
The molecular weight and its distribution are key variables in the product's performance. Typically, smaller molecular weight materials with low polydispersity indices are desired for their reduced viscosities and better casting properties. Due to the nature of the reaction mechanisms, analogies can be made in relationship to the freeradical polymerization example presented earlier. Instead of the terminal radicals of either type 1 or type 2, the end-groups for the step mechanism are the functional groups for each monomer. Designation of the oligomeric product can be described by the end group compositions through the relationships:
(14)
Q0 = - ( M 7 - M~) + (M~ - M2) + (I ° - / ) .
(15)
(22)
Generating functions may also be used for step mechanisms in determining the three moments of the molecular weight distribution. A variation of the formula is needed to account for the weight of initiator in the molecular weight and its distribution. The resulting relationship is depicted as:
~k = ~
~ (nw, q-- mw 2 q- Wl)kMn,,n
n=lm=l
k = 0, 1, 2 . . . .
where the state variables of the process at time t are: x r = [V(t), l(t), Ml(t), M2(t), (0(t), (l(t), (2(/)]
(21)
(23)
The total number average molecular weight (#.) and the polydisperity index D are then given by:
and the control inputs to the process at time t are: u r = [T(t), QMI(t), QMI(t), QMz(t)].
(16)
#" = (0 + ('0 D -
((1 + ~'1)2
(24)
66
G.D. C~,WTaONand K. S. KNAEBEL
where the (is are the molecular weight moments of the monomer 1 endgroups and the ~s are the moments of the monomer 2 terminated molecules. In modelling the dynamic behavior of the oligomer process, the required differential-algebraic equations resemble the set used for the free-radical system. Typically, industrial reactions are limited by their heat removal, requiring the process to operate in a suboptimal fashion for safety and product quality considerations. To include this phenomenon into the simulation study, an additional equation representing the system's heat transfer properties is required. TENDENCY MODEL The tendency models are developed by using fundamental material and energy balances along with an approximate overall reaction network. The model's stoichiometric coefficients and the kinetic parameters are estimated by least-squares regression of experimental data. Because the kinetics assumed in the tendency model may not closely represent the real kinetic network, several models may be needed for process simulation and optimization. Although the proposed model is usually less accurate than the detailed mechanism, it may provide important insight into the true kinetics and guide the process to a more optimal operation. The first stage of the model development is the identification of a simple stoichiometric network (Filippi et al., 1986). Needed information includes experimental initial [C(j, m, 0)] and final [C(j, m, tO] compositions of the significant reactants for each experiment. The index j denotes one of the n¢ components used in the approximate reaction network while m represents a specified batch run from a total of ne experiments. The transformation of the compositions into dimensionless quantities is performed using: nc
Ct(m) = ~ C(j, rn, 0), j-I
yU, m, t) = CU, m, t)/Ct(m), Yij, m , t ) = y ( j , m , t ) - y ( j , m , O ) .
(25) (26)
Generally, byproducts are not analytically measured in industrial reactors; these species can be combined into pseudocomponents. Also, polymeric products can be represented by this technique. Since polymers are often measured in weight as opposed to molar concentrations, the tendency model may use experimental weight data in determining the reaction network. Thi~ presents no problem except that rate constrants are combinations of molar and mass quantities. The two sets of unknowns must satisfy the following equations: nr
Y(j, m) = ~ vijz(i, m)
for
i=1
j = l . . . . . no, m = l . . . . . n~.
(27)
In the algorithm's initial step, the ith reaction is assumed to represent the process behavior. The corresponding stoichiometric coefficients are determined by the minimization of the error function: nr
error = E
ne
•
nc
E [Y(J, m ) - v•z(i , m)]2.
(28)
t=lra=lj=l
The result typically depicts the stoichiometric coefficients as decimal values (e.g. -1.125, -0.987, 0.824, 0.111) for a one-reaction/four-component system. Since coefficients are expected as integers or simple fractions, the user can select approximate values (such as - 1.000, - 1.000, 1.000, 0.000 for the above coefficients). When the recalculated error is unacceptable, an additional reaction may be introduced into the model. In the example, another reaction is needed since the fourth component is uninvolved in the one-reaction mechanism. The stoichiometric coefficients of all previous reactions are not allowed to change when an additional reaction is introduced. The new reaction's coefficients and the other reaction steps' extents of reaction are calculated to minimize the error function. The resulting kinetic network is unique in that the calculation of the stoichiometric coefficients are userdefined. Parameters, such as the maximum allowable error and the number of reactions, are also selected in applying available heuristic information in the model. If an appropriate solution is unattainable, the reaction may be divided into stages with each section represented by a distinct kinetic model. Identification of the tendency model's Arrhenius constants is performed with experimental temperature/concentration profiles. Each reaction's preexponential factor and activation energy (for nonisothermal processes) are evaluated by minimizing the deviations between the model and experimental compositions over the course of reaction. The heats of reaction are then estimated by comparing the experimental heat generation with values calculated using the kinetic network. The final product is a process model which can be used in simulation and optimization studies. RESULTS AND DISCUSSION
A. Free-radical copolymerization The free-radical example tested was the semibatch solution copolymerization of acrylonitrile-styrene in toluene initiated by benzoyl peroxide. The experimental data and reaction parameters are listed in Table 1 ~and were similar to the values used by Lin et al. (1979) and Chen and Lee (1985). The temperaturedependent reactivity ratios were obtained from Tsoukas et al. (1982). Constraints on the reaction conditions included a temperature range from 60 to 120°C and a 50°/'o final conversion (of the initially charged monomers). The process objective was to maintain a copolymer composition of 28% acrylonitrile while minimizing the reaction time and the
67
Optimization of polymerization reactions Table 1. Detailed model parameters for acrylonitrile-styrenesystem (cal, reel, 1, K, min)
2.4
120 I-
1.6
Initiation: Aa = 1.93 x 1016
Ed
= 30590
f = 0.0956
Acrylonitrile: Apn = 1.37 x 10t3 At1~= 1.737 x 1027 Am = 1.5 x 10 4kp]t
/
(M l + M2)
g 8o I- o~
EpH = 10199 EelI = 23606
E /,v
~- 60 L
Styrene: Ap22= 2.837 x 109 At22= 7.530 x 10I° ,4r22=4.16 x 108
Ep22= 7068 Ec22= 1677 El22= 11598
.--
Copolymerization: kvl2= 1.50 x 104 x kpu exp(-2184/T) kp2 = 3.91 x 10 t x k_22exp(599/T )
/---0/ 0
¢. o
1,9
~" 0.28
1.8
°°
1.7
E 0.26
1.6
o 0
Temperature I I 50 60
V CopoLymer composlhon P°trydispersity j
ktl 2 = 160(ktn x kt22)I/~ kfl2 = 30 X krl 1
kf21 = 5 X kf22
•
I 120
I 90
30
0
J
60
I 120
90
Time (min)
Reactor parameters: Initiator concentration Acrylonitrile concentration Styrene concentration Reactor volume
1° = 3.5 10 -3 mol I i M~ = 1.0 mol lM~ = 3.0 mol 1 t V = 0.41
Composition specification
CC* = 0.28
polydispersity. Process operation was controlled by the manipulation of temperature, reactant flowrates and time of reaction. The optimal control policies were calculated with Pontryagin's minimum principle (Bryson and He, 1975) using a number average molecular weight constraint of 107,700 (obtained by an average chain length of 1200 at the desired composition). Figure 1 depicts the increasing temperature policy .which minimized the reaction time but caused compositional drifts. The resulting polydispersity was reduced although chain transfer reactions prohibit values lower than 1.750 with the selected conditions. A decrease in reactor temperature was needed to maintain good control of the 28% acrylonitrile composition at the expense of a larger polydispersity. The acrylonitrile deficient mixture required lower temperatures to alter the temperature-senstttve reactivity ratios. Tsoukas et al. (1982) reported that m o n o m e r flow could be effectively used to control copolymer composition while the polydispersity remained
Fig. 2. Dual control for minimum time and composition error. relatively unaffected. Also, temperature control was found to be the significant method for maintaining the lowest polydispersity. Therefore, a dual-control problem (as seen in Fig. 2) was found to minimize both objectives. Due to the behavior of the two objectives, the problem was decoupled by allowing m o n o m e r flow to maintain the composition only while temperature was adjusted for the minimum reaction time. Figures 3 and 4 represent the noninferior sets for the process objectives under various control strategies. The first plot indicates that both temperature and m o n o m e r flowrate may be used to control the copolymer composition. The polydispersity was most affected by temperature changes while m o n o m e r flows had no significant effect. Optimization of the decoupled dual control problem provided a synergistic relationship between both controls. Figure 4 illustrates the tradeoffs between the minim u m end time and the polydispersity. With temperature control, the time of reaction and polydispersity were minimized at higher temperatures. Initiator flow allowed more flexibility in the product polymer's polydispersity; the polydispersity decreased with 2.0 F
9_.~ 120 F =
100 r
1.9[--
60
I 30
0 2.0 -
Temperature Monomer fLow Dual control policy
-
I 60
I 90
I 120
~,
1.8.~--'~.
....
8 0.28
.;-=_ 0 D-
_~ 0.26 -
1.8
~_ 1.7 1.6
8
1.7 f
1.6
~0.24
Copotymer
~ o
---
0 0.22 0
composition
1.5
Potya,~persity [ 30
I 60 Time
0 I 90
I 120
(rain)
Fig. 1. Temperature control to minimize the end-time for acrylonitrile-styrene production.
I 0.4
I 0.8
Composition
I
I
I
1.2
1.6
2.0
error
Fig. 3. Noninferior set for minimum polydispersity and composition error using optimal temperature and monomer flowrate policies.
G. D. CAWTHONand K. S. KNAEBEL
68 240
--
200 -
---• .
I
~
Temperature Initiator flow D u a L - rain end time D u a t - rain poLydispersity
2
120 F ."
Fl
~
AcrytonitriLe
---
Temperature
160
\
\
\
1.6
30
2.o[.~ 0.3; \
"i
\\
120 -
80
..,-- . . - . . - ~ - -
I
\ '~10 C LU
fLow
60
90
120
150
1.91-o-
\
I
I
I
I
1.7
1.8
1.9
2.0
1.6 L o~ 0.2E
---
" CopoLymer composition P=°Lydlsper slty J 30
0
PoLydispersity
60
90
I
I
120
150
Time of reaction (rain)
Fig. 4. Noninferior set for minimum end-time and polydispersity using optimal temperature and initiator flowrate policies.
Fig. 5. Dual control policy for optimal process performance with CC* = 0.28, D(tr) = 1.85 and minimum end-time.
increasing initiator concentration. Since the chain transfer rate was proportional to the radical concentration while termination was proportional to its square, increasing the initiator concentration diminished the fraction of polymer produced by chain transfer mechanisms. Decreasing the temperature to the lowest limit allowed for the highest initiator composition and the lowest polydispersity. The best end time was found at the maximum temperature level and the lowest possible initiator flow. Utilizing the experimental data, the tendency modelling approach was implemented to form a simpler reaction network representing the system's behavior. The proposed model followed the structure:
Figure 5 shows an optimal control policy obtained using the numerical techniques with the detailed model. The objectives include minimized end-time and copolymerization composition drift while specifying that the polydispersity must be 1.85 at the final conditions. The unusual conditions of the temperature having to decrease initially and then increase is due to the initial concentrations of monomers being t o o large so that the excess acrylonitrile has to be reacted for composition control. The resulting concentration trends for both the theoretical and tendency models are shown in Fig. 6. The profiles deviated from each other due to the new process operating conditions. Adjustments in the tendency model's parameters correct for this error and allow accurate process modelling for a selected operating range. With widely varying process conditions, more than one model was needed for process simulation.
M] ~ PI ,
2M2 ~ P2, P( + P2 = PT,
(29)
The first simplification of the kinetic network was the omission of the initiator from the reaction network. The product of the first reaction was termed pseudocomponent 1 and the result of the second reaction being pseudocomponent 2. These additional polymeric compounds were included in weight units as opposed to the typical molar units since their unknown molecular weight changes throughout the course of reaction. By totalling the weight of the two pseudocomponents, a combined product weight was introduced into the reaction scheme, allowing calculation of the tendency model's kinetic constants. The tendency models adequately represented the reactants' compositions over the course of the reactions. The best kinetic parameters for the linear temperature rise operation were (with units of 1mol- 1min- ():
{ [; 1]}
kl =0.725 x lO-3exp --7375
k 2 = 0 . 9 8 2 x 10-4exp - 9 6 5 0
--3~
-3-~
'
(30)
" (31)
B. Oligomer production Oligomer manufacture was performed with an industrial 9501 stirred tank reactor designed for 3.2~'<4 • .. ~ - , ~ ' o oo ° -~. -,.Oo .~ 2.4-
°°° "~ ~ , .
-6 E
o c u c
". ° o % ° •~.
°°° "~- ,~..
1.6 -- ~ --.... .......
%°.o
AcrytonitriLe(theoreticaL modeL) Styrene (theoretical model. ) AcryLonitrRe ( t e n d e n c y modeL) Styrene (tendency m o d e l )
~ ~
0.8--
0
I 30
I 60
I 90
I 120
I 150
Time of reoction (min) Fig. 6. Concentration profiles for the acrylonitrile-styrene system resulting from dual control policy.
69
Optimization of polymerization reactions semibatch operation. Real-time evaluations of heat transfer properties were determined through online analysis of process conditions. The reactor was equipped with both cooling coils and an external condenser (for reflux cooling) to assist in the reaction heat removal. High performance liquid chromatography was used to determine the concentration of the reactants and reaction intermediates. Molecular weight distributions were obtained using gel permeation chromatography. The rate constants for the three-step mechanism were 'estimated from several temperature-programmed experiments. Tile resulting relationships were (with units in lmo1-1 min-l):
{
k l = 0.0535 exp - 6 2 5 0
[; l l} - ~
,
(32)
kl = 0.150 e x p { - 1850F1 - 1-~-l~ Lr 386JJ'
(33)
k2 = 0.680 e x p { - 2 2 0 0 [ 1 1 3 ~ 1 }.
(34)
for this reactor system were typically less than 5% of the heat transferred by the coils. The deviation between the experimental and the calculated heat release was typical to pilot plant/commercial production. Initially, the heat of reaction assisted in the elevation of the reaction temperature to the maximum value. A 9 min time lag was found for the removal of heat by the coils. The external condensers had a larger time delay (ca 25 min) due to two phenomena: the rate of material/heat flows between equipment and the residence time constant for the condenser's liquid holdup. Therefore, predictive real-time models are needed for adequate control of such pilot plant/ commercial reactors since large time lags can lead to runaway reactions. Utilizing the experimental data, the tendency modelling approach was implemented to form a simpler reaction network representing the system's behavior. The proposed model followed the structure:
I + M2--* P1, Ml + Mz ~ P2,
the heats of reaction found were - 13,800, - 10,600 and -21,600 cal mol -~ for the initiation, second and third reactions, respectively. Figure 7 depicts the experimental and predicted energy evolved by the reaction. The maximum heat release was located at the point where the highly-exothermic third reaction was most prominent. Also, the figure illustrates the portion of heat extracted using the cooling coils as related to the total amount removed (by the cooling coils, the external total condenser and the heat loss to the surroundings). The coil heat transfer coefficients for this system were found to range from 2.0 to 0.80 cal min -~ cm -2 °C -~ as the reaction proceeded. This effect was primarily due to changes in physical properties of the reacting mixture. Since most of the condensate was the low-boiling monomer 1, reflux cooling was not as significant for this system as in other types of polymerization processes. Heat losses
<-
A Heat Loss by COD,S Tatar. heat Loss
12u •
.E
k2
E
(35)
This model combined the three reactants into polymer structures which were not simply described by monomer 1 and monomer 2 terminated molecules. The product of the first reaction was termed pseudocomponent 1 and the result of the second reaction being pseudocomponent 2. These additional compounds were included in weight units as opposed to the typical molar units since their unknown molecular weight changed throughout the course of reaction. By totalling the weight of the two pseudocomponents, a combined product weight was introduced into the reaction scheme allowing calculation of the tendency model's kinetic constants. The tendency model adequately represented the reactants' compositions over the course of the reaction. The kinetic parameters which best fit the concentration profiles were (with units in I real-l min- ~):
1 5 V E x p e r i m e n t a k run 1
x
7
P1 + P2 = Pr"
l Ol Oex+,8,0[ll]}
36)
=0.205expt-1200]a- ] }
(37)
9
g o ,=
,~ G "r
0 0
7O
~'°. I 140
I 210
I 280
Time (rain)
Fig. 7. Reaction heat release at experimental conditions (with the detailed model simulation depicted by the solid line).
The detailed and tendency models deviated at the start of the reaction; this trend was expected since the propagation step (reaction 2) of the tendency model began immediately while the detailed model required the initiation step to occur. This effect caused the initial heat release calculated by the tendency model to be significantly larger since there was no restriction to the more highly-exothermic second reaction. The heat of reaction values for the two reactions were - 15,000 and - 34,000 cal real i. The minimum time problem for step polymerization systems with all reactants initially charged has a
70
G.D. CAWTHONand K. S. KNAEBEL
relatively trivial control policy. The reaction temperature should be at the maximum possible for the highest rate of reaction for irreversible step polymerization mechanisms (Hicks et al., 1969). With the charge of all readtants, the zeroeth and first moments (and the resulting number average molecular weight) are constant; ,the molecular weight ratio was 1.09 at the selected experimental conditions. This occurs with any temperature history. One degree of freedom was available for optimization of these problems. The control objective chosen was the minimization of the second moment which was identical to the reduction of the weight average molecular weight and the polydispersity. With the experimental composition and a maximum reactor temperature limit of 113°C, conversion is finished in 40 min. The corresponding polydispersity is reduced to a value of 1.33, which was a significant improvement from the theoretical value of two. Accompanying the high reaction rate was an equally large heat release (ca 155 × 105calmin-1). Since these conditions were unrealistic for this reactor, a heat transfer limitation of 10 × 10Scalmin -~ was selected for safe operation. Taking into account the heat transfer constraint, the minimum time problem using temperature control provided a reaction time of 132 min (Fig. 8). The resulting polydispersity was 1.72, lower than the restricted isothermal case at 33°C (i.e. 1.91). Temperature control was also used in minimizing the final polydispersity index although no significant improvement in product quality (final polydispersity index equal to 1.64) was found while the final endtime increased to approx. 235 min. With this reaction network, the best temperature profile began at the maximum limit which directly conflicted with the heat release constraint. Therefore, temperature controls alone were inadequate for maintaining a minimum polydispersity. Dual control policies which regulate both temperature and monomer flowrates simultaneously were 120
-
45
x I0C
----
I. . . . I
/
12
-
/
\\ /111/!~.111/
G o
I
~
8c-
9
a.
6(1-
6
40-
3
E
Heat r e l e a s e Temperature
b-
20-
T
0 0
I 40
I 80
120
I 160
Time (rain)
Fig. 8. Temperature control to minimize reaction time for oligomer production (with a maximum heat release rate of 106ca1 min- ~).
550 t
\
280
\ \
210 "o c
- -
Temperature
---
M o n o m e r 2 flow
--.--
Duat control
'
/
140
70
o
t.o
I tz
I 1.4
I 1.6
I 1.8
I 2.o
PoLydispersity
Fig. 9. Noninferior set for minimum end-time and polydispersity. also examined. Using monomer 2 flow control, process performance was improved over that of the temperature control alone. The final conditions were a reaction time of 122min and a polydispersity of 1.35. Since one degree of freedom was eliminated by adding the other control variable, the same end-time value was found using either monomer 1 or 2 flow control. The polydispersity index does not follow this trend. For the lowest polydispersity, dual controls (with monomer 2 flow) were found to provide significant improvements (final value being 1.26 with an end time of 124min). Therefore, dual control procedures attain process objectives in a shorter time of reaction. Noninferior sets for end-time and polydispersity are represented by Fig. 9. This illustrates the range of process objectives which may occur using a selected process control strategy. Also, the advantages of the different control mechanisms can be compared in evaluating the best operating procedure achieving a selected performance criteria. The figure illustrates that for systems which may or may not have heat transfer limitations, no real improvement in polydispersity can be made using temperature control alone. Monomer 2 flow can be controlled to achieve the range of polydispersity values. Dual control problems which manage both temperature and monomer 2 flows guide the process to a selected final condition, improving on a single control's performance. Lastly, the tendency model was evaluated for predicting process behavior as related to the detailed model's responses. The optimal control policy which minimized the reaction time with monomer 2 flow (temperature = 56°C) was the prescribed operating procedure. The resulting concentration profiles favorably compared with the trends obtained using the detailed model. Deviations were found for the initiator conversion due to the large difference in reaction temperature dependencies. Therefore, a new model could be utilized in the high conversion range to better represent these trends. Lastly, both models
7I
Optimization of polymerization reactions 25
b
DetaiLed kinetic model 20
E
Tendency model
- ~ -
15
10 t~
"6 (l) "1-
0
0
60
120
180
240
Time (rain)
Fig. 10. Heat generation using the tendency model with optimal control policy. followed the same heat release behavior (as seen in Fig. 10), confirming the tendency model's ability to predict heat generation characteristics in real-time process control. CONCLUSIONS
The purpose of this study was to investigate the effects and tradeoffs of optimizing dynamic reaction systems. Techniques were developed which applied to both chain and step reactions. Optimal control policies included temperature and reactant addition, .along with combinations forming dual controls. Multiobjective analysis was then used to obtain the ranges of process objectives attainable using the different control strategies. For the acrylonitrile-styrene system, m o n o m e r addition was a more effective means of controlling the copolymer composition during the reaction than was temperature. Conversely, adjustment of the reactor temperature was a better method than m o n o m e r flow control for maintaining the polydispersity. Because of this behavior, the optimization problem was decoupled into two separate problems:, using m o n o m e r flow to correct the composition and temperature to correct for the molecular weight and end time. N o significant process performance was lost with this approach. For initiator flowrate operation, the best policy allowed constant flowrate (and constant concentration) over the course of reaction. Minimum polydispersity indices were found at lower temperatures and greater initiator concentrations. Again, the dual control policy of initiator flowrate and temperature were used for better production than with a single control adjustment. Oligomer production was optimized using selected process control strategies. Isothermal operation at the maximum temperature produced a product with a minimum value of the polydispersity in the shortest amount of time. Since this process was unacceptable due to heat transfer limitations, constraints were added and new operational procedures
were evaluated. Selection of m o n o m e r flowrates allowed a variety of product qualities, although dualcontrol policies of both temperature and m o n o m e r flow functioned more rapidly. Noninferior sets were presented to describe the dependence of the product's polydispersity index with the time of reaction. Lastly, the tendency model was used with an optimal operating procedure and was found to model adequately the system's behavior. Adequate predictions o f conversion were obtained under optimal control conditions using the tendency model. Because the model can be augmented or utilized over a specific operating range, better fits are expected for other complex processes. Therefore, this method would be directly applicable to adaptive control of semibatch reactor systems.
NOMENCLATURE
C, = Concentration of reactant i in feed stream CC = Copolymer composition distribution D = Polydispersity index of total number molecular weight distribution F = Vector of objective functionals (components F,, i=1,2 .... ) f = Initiator efficiency g = Vector of constraint functions kcu = Termination rate constant by combination reaction of radicals of type i and j kDl = Rate constant of initiator decomposition keu = Chain transfer to monomer rate constant from a radical of type i to a monomer of type j kv,j = Propagation rate constant for addition of monomer of type j to radical of type i M, = Concentration of monomer M..,. = Concentration of dead polymer containing n units of M I and m units of M 2 P..m = Concentration of growing polymer containing n units of M~ and m units of M2 and ending with an M t unit P, = ith moment of the total number molecular weight distribution of radicals of type 1 Q.... = Concentration of growing polymer containing n units of M~ and m units of M2 and ending with an M 2 unit Qi = ith moment of the total number molecular weight distribution of radicals of type 2 QM, = Volumetric flowrate of reactant i R- = Concentration of free radical initiator rM, = Concentration rate of change of monomer M, t = Time tf = Final time of reaction T = Temperature u = Vector of control inputs V = Reactor volume wi = Molecular weight of species i x = Vector of reactor states x0 = Initial value of state vector
Greek symbols E~= Constraint to the i objective functional ~ = ith moment of the total number molecular weight distribution #. = Number average molecular weight v0 = Stoichiometric coefficient for the ith reaction and the jth component X = Extent of reaction
G. D. CAWTHONand K. S. KNAEBEL
72 REFERENCES
Bryson A. and Y. Ho, Applied Optimal Control. Blaisdell, Waltham, Mass. (1975). Chankong V. and Y. Haimes, Multiobjective Decision Making: Theory and Methodology. North-Holland, New York (1983). Chen S. and S. Lee, Poly. Engng Sci. 25, 987 (1985). Farber J., Poly. Engng Sci. 26, 499 (1986).
Filippi C., J. Greffe, J. Bordet, J. Villermaux, J. Barney, P. Bonte and C. Georgakis, Chem. Engng Sci. 41, 913 (1986). Hicks J., A. Mohan and W. Ray, Can. J. Chem. Engng 47, 590 (1969). Lin C., W. Chiu and C. Wang, J. Appl Poly. Sci. 23, 1203 (1979). Ray W., Macromolec. Chem. 1, 1 (1972). Tsoukas A., M. Tirrell and G. Stephanopoulos, Chem. Engng Sci. 37, 1785 (1982).
0098-1354/89 $3.00+ 0.00 Copyright© 1989Pergamon Press plc
OPTIMIZATION OF SEMIBATCH POLYMERIZATION REACTIONS G. D. CAWTrIONand K. S. KNAEBELaf Department of Chemical Engineering, The Ohio State University, Columbus, OH 43210, U.S.A. (Received 27 June 1988; received for publication 6 July 1988)
Abstract--Dynamic chemical processes often require a multiobjective formulation for complete process optimization. Specifically, semibatch copolymerization systems are analyzed using vector-objective techniques to determine the tradeoffs between the different goals: narrow the copolymer composition and polydispersity distributions, and minimize the time of reaction. Due to the complexities of the detailed kinetic models, simpler tendency models are evaluated by fitting proposed reaction networks to experimental data. Simulation studies of an acrylonitrile-styrenecopolymer and a step-wise oligomer are then used to determine optimal production schemes.
INTRODUCTION
only at specific operating conditions since it may be unrelated to the actual process mechanism. The "tendency" model approach (Filippi et al., 1986) is a compromise between the detailed mechanism and a totally empirical reaction model. This technique employs engineering insight into the proposed kinetic structure, whose parameters are calculated using available experimental data. Furthermore, since simulation of this model usually requires less computational time, real-time implementation for adaptive optimal control is possible. This study develops a optimization strategy for both free-radical and step-wise polymerization in semibatch reactors. Processes are analyzed as multiobjective optimization problems to determine the effects of and tradeoffs between the polydispersity, composition distribution and minimum end-time using various combinations of temperature and reactant flowrate profiles. Also, tendency models are generated for two examples to illustrate the models' applicability and accuracy.
Chemical processes typically have multiple performance objectives which require simultaneous inspection for best process operation. The optimal process design of semibatch reactors allows manufacture of product with the desired physical properties in the minimum amount of a time. The molecular weight distribution is ~the commonly measured variable, representing the variety of molecules comprising the polymeric product. In copolymerization, a significant problem to be considered is the composition distribution in the composite. Molecules which have similar molecular weights can and usually have different proportions of monomers due to changes in reaction conditions. Even for molecules having the same ratio of molecules, there can be a different ordering of monomer units which can appreciably change the product properties. Movement of the chemical industries towards manufacture of specialty chemicals has made batch operations more popular. Because these dynamic systems have small production levels and complex operating procedures, thorough understanding of reactor performance and reaction kinetics is usually not economically justified. Nevertheless, accurate modelling of these systems is a very valuable tool for achieving optimal system performance. The common approach to modelling a reactor system requires specification of the individual physical and chemical phenomena (such as heat transfer, stoichiometry and kinetics) to provide a detailed mechanism. This technique requires a significant effort in identification of many unknown parameters resulting from the complex set of equations. The alternative method consists of empirically fitting experimental data; the model represents the system
MULTIOBJECTIVE OPTIMIZATION PROBLEM Because of the several performance objectives, process optimization of polymeric systems requires a multiobjective formulation. Two possible alternatives for the solution are available: combine all performance measures into one objective to produce the single optimal control policy, or generate a series of control strategies which relate the tradeoffs between the objectives and the control parameters. Priority is requested for the first technique to form a single objective function although the weighting factors are difficult to specify a priori. The second method leads to a much better understanding of the process operation and provides valuable information for subjective decision-making regarding the different control
tAuthor to whom all correspondence should be addressed. 63
64
G.D. CAWTnONand K. S. KNAEBEL
strategies. The general vector optimization problem is formulated as follows (Chankong and Haimes, 1983): min F(x) = min{Fl(X), F 2 ( x ) , . . . , subject to:
initiation: kDl
(1)
F~ (x)},
dx ~-~ -- l(x, u)
x(0) = Xo,
gk(X, U) ~<0,
k = 1, 2 . . . . . m
,2R'
R" + M l
) PI,0
R.+M2
) Q0,1,
propagation: kpll
Pn,m+ MI
where F is the objective function vector, x is the decision vector, u is control vector arid g is the constraint vector. The common objectives for semibatch copolymerization reactions are the composition error (i.e. the difference between the specified and the instantaneous compositions), the polydispersity and the time of reaction:
P. + 1,~ kpl2
P.,m + M2
Q.,m + 1, kp21
Q..~+MI
'Pn+l,m
kp22
Qn,m + M2
min F(u) =
[CC* - CC(t)] 2 dt, D(tf), tr.
Qn,m+ 1,
(2) termination (by combination):
The first step in a multiobjective problem consists of determining where a tradeoff exists between competing objectives. These compromises can be determined by using the concept of noninferiority. Noninferior solutions are control policies which no improvement can be made in one objective without degrading another objective. These solutions will allow the decision maker to decide between the different operating regimes so that the optimal control policy can be chosen. A popular method for determination of the noninferior sets is the E-constraint approach (Faber, 1986). It is based on treating all but one of the objective functions as parameterized constraints, thus solving the vector optimization problem using a scalar approach. The original problem is then reformulated as:
en,m q- Pr,q
kcll
' Mn +r,m+q kcl2
Pn.m q- Qr,q
' Mn +r,m+q, kc21
Qn,m -+- Pr.q
' Mn+r,m+q kc22
Q.,m + Qr,q
' Mn+r.m+q,
chain transfer (to monomer): P.,,. + M 1
kfl 1
, M.,,. + Pl.o
kfl2
P.,~ + M2
~ M.,m + Qo,l,
kf21
subject to:
Q,,m + M1
(3)
min F 1(x) dx dS = I(x, u)
x(0) = Xo
~ ( x ) ~
j = 2, 3 . . . . .
gk(X) ~<0.
k=l,2 ..... m
Ej-= F* + E*,
j=2,3 ..... n
~*~<0,
j=2,3 ..... n
, M,,m + PI,o k f22
Q.,m + M2 n
The minimum value of the j t h objective when all other objectives are ignored is F* and the deviation from this minimum value is e*. The values of E* are parametrically in order to determine the noninferior set. FREE-RADICAL COPOLYMERIZATION MODEL
The standard reaction model for homogeneous solution or (low-conversion bulk) free-radical copolymerization consists of the following steps:
' M.,m + Qo,1.
For application of this model, the pseudosteady-state assumption and the long-chain hypothesis are valid concepts used for mathematical simplification. Also, phenomena such as volume contraction, thermal initiation and gel-effects are neglected for this study. Employing these simplifications, a set of mass balances may be written as (Tsoukas et al., 1982): dV d--t = QM,(t) + QM,(t) + QM2(t ),
(4)
dI dt
(5)
koII -~
CzQMI(t) V
I dV V dt '
dM~ dt = [(kpll + knl)P° + (kp21 + kf2OQo]Ml CiQMl(t) q ~
Ml d V V dt'
(6)
Optimization of polymerization reactions dM2 dt = [(kpl2-}- kfl2)P° d- (kp22+ kn2)Qo]M2
C2QM2(t) q
STEP-WISE OLIGOMER MODEL
M 2 dV
V
V dt"
(7)
The parameter used to represent the radical concentrations are the zeroeth order moments of the radical distributions Po and Qo:
n=0m=0
'
I
kcll_t_2flke12+f12kc22 kpl2 q- kf12 M2
(9)
The molecular weight distribution of the radical and dead polymeric molecules can be solved by applying generating functions and the method of moments described by Ray (1972). The concentration of the dead polymeric product is represented by the moments (o, ~1 and ~2. d(° = ~ 2 p02 +
The step-wise polymerization system to be examined is an oligomeric compound used as a precursor to many copolymerization reactions. Due to its proprietary characteristics, only a framework of the process chemistry will be presented. The oligomer is produced by the reaction of an initiator with two differing monomers. The resulting kinetic mechanism is represented by the three steps:
, (8)
Qo = flPo fl - kp21 + kol MI"
--dt
65
k~2~ 2 k¢12P°Q° + T Qo + (kfllMl
+ kn2M2)Po + (knlMx + knzM2)Qo
I + MI ~ P1,0.
(17)
P .... , + M 2 ~ Q ....
(18)
Q...+ M I ~ P . + I , ..
(19)
The subscripts for variables P..._ ~ and Q.,. indicate the number of each constituent making up each product molecule. Because the composition follows a unique sequence of alternating units, all monomer 1 terminated units are made up of one less molecule of monomer 2 than component 1. Products terminated with monomer 2 have an equal number of both components. The general configuration is denoted by the structure: I - [ - MI M 2 - ], - M~.
~0 dV V dr"
(20)
(10)
d~l -_ kcllPoP1 + kc12(PiQo + PoQO + k¢22QoQ1 q- ( k n l M l q- kfl2Mz)P l q- (kf21M1
~,1 dV + kf22M2)Ql - v d--t-'
(11)
t = kcHPoP2 + kc,~P~ + kc12(PzQo + PoQ2) + 2kc12P1Ql + kc22QoQ2 + k¢22Q12 + (kmM1 + Kn2M2)P2 + (kmMl ~2 dV + knzM~)Q2 - V d--t-'
~o~2
(13)
The state equations can now be represented in vector form as: dx
d t (t) = f[x(t), u(t)],
P0 = (M? -- M,) - (M ° - M2),
(12)
The total number average molecular weight (g.) and the polydispersity index D are given by: ~L
The molecular weight and its distribution are key variables in the product's performance. Typically, smaller molecular weight materials with low polydispersity indices are desired for their reduced viscosities and better casting properties. Due to the nature of the reaction mechanisms, analogies can be made in relationship to the freeradical polymerization example presented earlier. Instead of the terminal radicals of either type 1 or type 2, the end-groups for the step mechanism are the functional groups for each monomer. Designation of the oligomeric product can be described by the end group compositions through the relationships:
(14)
Q0 = - ( M 7 - M~) + (M~ - M2) + (I ° - / ) .
(15)
(22)
Generating functions may also be used for step mechanisms in determining the three moments of the molecular weight distribution. A variation of the formula is needed to account for the weight of initiator in the molecular weight and its distribution. The resulting relationship is depicted as:
~k = ~
~ (nw, q-- mw 2 q- Wl)kMn,,n
n=lm=l
k = 0, 1, 2 . . . .
where the state variables of the process at time t are: x r = [V(t), l(t), Ml(t), M2(t), (0(t), (l(t), (2(/)]
(21)
(23)
The total number average molecular weight (#.) and the polydisperity index D are then given by:
and the control inputs to the process at time t are: u r = [T(t), QMI(t), QMI(t), QMz(t)].
(16)
#" = (0 + ('0 D -
((1 + ~'1)2
(24)
66
G.D. C~,WTaONand K. S. KNAEBEL
where the (is are the molecular weight moments of the monomer 1 endgroups and the ~s are the moments of the monomer 2 terminated molecules. In modelling the dynamic behavior of the oligomer process, the required differential-algebraic equations resemble the set used for the free-radical system. Typically, industrial reactions are limited by their heat removal, requiring the process to operate in a suboptimal fashion for safety and product quality considerations. To include this phenomenon into the simulation study, an additional equation representing the system's heat transfer properties is required. TENDENCY MODEL The tendency models are developed by using fundamental material and energy balances along with an approximate overall reaction network. The model's stoichiometric coefficients and the kinetic parameters are estimated by least-squares regression of experimental data. Because the kinetics assumed in the tendency model may not closely represent the real kinetic network, several models may be needed for process simulation and optimization. Although the proposed model is usually less accurate than the detailed mechanism, it may provide important insight into the true kinetics and guide the process to a more optimal operation. The first stage of the model development is the identification of a simple stoichiometric network (Filippi et al., 1986). Needed information includes experimental initial [C(j, m, 0)] and final [C(j, m, tO] compositions of the significant reactants for each experiment. The index j denotes one of the n¢ components used in the approximate reaction network while m represents a specified batch run from a total of ne experiments. The transformation of the compositions into dimensionless quantities is performed using: nc
Ct(m) = ~ C(j, rn, 0), j-I
yU, m, t) = CU, m, t)/Ct(m), Yij, m , t ) = y ( j , m , t ) - y ( j , m , O ) .
(25) (26)
Generally, byproducts are not analytically measured in industrial reactors; these species can be combined into pseudocomponents. Also, polymeric products can be represented by this technique. Since polymers are often measured in weight as opposed to molar concentrations, the tendency model may use experimental weight data in determining the reaction network. Thi~ presents no problem except that rate constrants are combinations of molar and mass quantities. The two sets of unknowns must satisfy the following equations: nr
Y(j, m) = ~ vijz(i, m)
for
i=1
j = l . . . . . no, m = l . . . . . n~.
(27)
In the algorithm's initial step, the ith reaction is assumed to represent the process behavior. The corresponding stoichiometric coefficients are determined by the minimization of the error function: nr
error = E
ne
•
nc
E [Y(J, m ) - v•z(i , m)]2.
(28)
t=lra=lj=l
The result typically depicts the stoichiometric coefficients as decimal values (e.g. -1.125, -0.987, 0.824, 0.111) for a one-reaction/four-component system. Since coefficients are expected as integers or simple fractions, the user can select approximate values (such as - 1.000, - 1.000, 1.000, 0.000 for the above coefficients). When the recalculated error is unacceptable, an additional reaction may be introduced into the model. In the example, another reaction is needed since the fourth component is uninvolved in the one-reaction mechanism. The stoichiometric coefficients of all previous reactions are not allowed to change when an additional reaction is introduced. The new reaction's coefficients and the other reaction steps' extents of reaction are calculated to minimize the error function. The resulting kinetic network is unique in that the calculation of the stoichiometric coefficients are userdefined. Parameters, such as the maximum allowable error and the number of reactions, are also selected in applying available heuristic information in the model. If an appropriate solution is unattainable, the reaction may be divided into stages with each section represented by a distinct kinetic model. Identification of the tendency model's Arrhenius constants is performed with experimental temperature/concentration profiles. Each reaction's preexponential factor and activation energy (for nonisothermal processes) are evaluated by minimizing the deviations between the model and experimental compositions over the course of reaction. The heats of reaction are then estimated by comparing the experimental heat generation with values calculated using the kinetic network. The final product is a process model which can be used in simulation and optimization studies. RESULTS AND DISCUSSION
A. Free-radical copolymerization The free-radical example tested was the semibatch solution copolymerization of acrylonitrile-styrene in toluene initiated by benzoyl peroxide. The experimental data and reaction parameters are listed in Table 1 ~and were similar to the values used by Lin et al. (1979) and Chen and Lee (1985). The temperaturedependent reactivity ratios were obtained from Tsoukas et al. (1982). Constraints on the reaction conditions included a temperature range from 60 to 120°C and a 50°/'o final conversion (of the initially charged monomers). The process objective was to maintain a copolymer composition of 28% acrylonitrile while minimizing the reaction time and the
67
Optimization of polymerization reactions Table 1. Detailed model parameters for acrylonitrile-styrenesystem (cal, reel, 1, K, min)
2.4
120 I-
1.6
Initiation: Aa = 1.93 x 1016
Ed
= 30590
f = 0.0956
Acrylonitrile: Apn = 1.37 x 10t3 At1~= 1.737 x 1027 Am = 1.5 x 10 4kp]t
/
(M l + M2)
g 8o I- o~
EpH = 10199 EelI = 23606
E /,v
~- 60 L
Styrene: Ap22= 2.837 x 109 At22= 7.530 x 10I° ,4r22=4.16 x 108
Ep22= 7068 Ec22= 1677 El22= 11598
.--
Copolymerization: kvl2= 1.50 x 104 x kpu exp(-2184/T) kp2 = 3.91 x 10 t x k_22exp(599/T )
/---0/ 0
¢. o
1,9
~" 0.28
1.8
°°
1.7
E 0.26
1.6
o 0
Temperature I I 50 60
V CopoLymer composlhon P°trydispersity j
ktl 2 = 160(ktn x kt22)I/~ kfl2 = 30 X krl 1
kf21 = 5 X kf22
•
I 120
I 90
30
0
J
60
I 120
90
Time (min)
Reactor parameters: Initiator concentration Acrylonitrile concentration Styrene concentration Reactor volume
1° = 3.5 10 -3 mol I i M~ = 1.0 mol lM~ = 3.0 mol 1 t V = 0.41
Composition specification
CC* = 0.28
polydispersity. Process operation was controlled by the manipulation of temperature, reactant flowrates and time of reaction. The optimal control policies were calculated with Pontryagin's minimum principle (Bryson and He, 1975) using a number average molecular weight constraint of 107,700 (obtained by an average chain length of 1200 at the desired composition). Figure 1 depicts the increasing temperature policy .which minimized the reaction time but caused compositional drifts. The resulting polydispersity was reduced although chain transfer reactions prohibit values lower than 1.750 with the selected conditions. A decrease in reactor temperature was needed to maintain good control of the 28% acrylonitrile composition at the expense of a larger polydispersity. The acrylonitrile deficient mixture required lower temperatures to alter the temperature-senstttve reactivity ratios. Tsoukas et al. (1982) reported that m o n o m e r flow could be effectively used to control copolymer composition while the polydispersity remained
Fig. 2. Dual control for minimum time and composition error. relatively unaffected. Also, temperature control was found to be the significant method for maintaining the lowest polydispersity. Therefore, a dual-control problem (as seen in Fig. 2) was found to minimize both objectives. Due to the behavior of the two objectives, the problem was decoupled by allowing m o n o m e r flow to maintain the composition only while temperature was adjusted for the minimum reaction time. Figures 3 and 4 represent the noninferior sets for the process objectives under various control strategies. The first plot indicates that both temperature and m o n o m e r flowrate may be used to control the copolymer composition. The polydispersity was most affected by temperature changes while m o n o m e r flows had no significant effect. Optimization of the decoupled dual control problem provided a synergistic relationship between both controls. Figure 4 illustrates the tradeoffs between the minim u m end time and the polydispersity. With temperature control, the time of reaction and polydispersity were minimized at higher temperatures. Initiator flow allowed more flexibility in the product polymer's polydispersity; the polydispersity decreased with 2.0 F
9_.~ 120 F =
100 r
1.9[--
60
I 30
0 2.0 -
Temperature Monomer fLow Dual control policy
-
I 60
I 90
I 120
~,
1.8.~--'~.
....
8 0.28
.;-=_ 0 D-
_~ 0.26 -
1.8
~_ 1.7 1.6
8
1.7 f
1.6
~0.24
Copotymer
~ o
---
0 0.22 0
composition
1.5
Potya,~persity [ 30
I 60 Time
0 I 90
I 120
(rain)
Fig. 1. Temperature control to minimize the end-time for acrylonitrile-styrene production.
I 0.4
I 0.8
Composition
I
I
I
1.2
1.6
2.0
error
Fig. 3. Noninferior set for minimum polydispersity and composition error using optimal temperature and monomer flowrate policies.
G. D. CAWTHONand K. S. KNAEBEL
68 240
--
200 -
---• .
I
~
Temperature Initiator flow D u a L - rain end time D u a t - rain poLydispersity
2
120 F ."
Fl
~
AcrytonitriLe
---
Temperature
160
\
\
\
1.6
30
2.o[.~ 0.3; \
"i
\\
120 -
80
..,-- . . - . . - ~ - -
I
\ '~10 C LU
fLow
60
90
120
150
1.91-o-
\
I
I
I
I
1.7
1.8
1.9
2.0
1.6 L o~ 0.2E
---
" CopoLymer composition P=°Lydlsper slty J 30
0
PoLydispersity
60
90
I
I
120
150
Time of reaction (rain)
Fig. 4. Noninferior set for minimum end-time and polydispersity using optimal temperature and initiator flowrate policies.
Fig. 5. Dual control policy for optimal process performance with CC* = 0.28, D(tr) = 1.85 and minimum end-time.
increasing initiator concentration. Since the chain transfer rate was proportional to the radical concentration while termination was proportional to its square, increasing the initiator concentration diminished the fraction of polymer produced by chain transfer mechanisms. Decreasing the temperature to the lowest limit allowed for the highest initiator composition and the lowest polydispersity. The best end time was found at the maximum temperature level and the lowest possible initiator flow. Utilizing the experimental data, the tendency modelling approach was implemented to form a simpler reaction network representing the system's behavior. The proposed model followed the structure:
Figure 5 shows an optimal control policy obtained using the numerical techniques with the detailed model. The objectives include minimized end-time and copolymerization composition drift while specifying that the polydispersity must be 1.85 at the final conditions. The unusual conditions of the temperature having to decrease initially and then increase is due to the initial concentrations of monomers being t o o large so that the excess acrylonitrile has to be reacted for composition control. The resulting concentration trends for both the theoretical and tendency models are shown in Fig. 6. The profiles deviated from each other due to the new process operating conditions. Adjustments in the tendency model's parameters correct for this error and allow accurate process modelling for a selected operating range. With widely varying process conditions, more than one model was needed for process simulation.
M] ~ PI ,
2M2 ~ P2, P( + P2 = PT,
(29)
The first simplification of the kinetic network was the omission of the initiator from the reaction network. The product of the first reaction was termed pseudocomponent 1 and the result of the second reaction being pseudocomponent 2. These additional polymeric compounds were included in weight units as opposed to the typical molar units since their unknown molecular weight changes throughout the course of reaction. By totalling the weight of the two pseudocomponents, a combined product weight was introduced into the reaction scheme, allowing calculation of the tendency model's kinetic constants. The tendency models adequately represented the reactants' compositions over the course of the reactions. The best kinetic parameters for the linear temperature rise operation were (with units of 1mol- 1min- ():
{ [; 1]}
kl =0.725 x lO-3exp --7375
k 2 = 0 . 9 8 2 x 10-4exp - 9 6 5 0
--3~
-3-~
'
(30)
" (31)
B. Oligomer production Oligomer manufacture was performed with an industrial 9501 stirred tank reactor designed for 3.2~'<4 • .. ~ - , ~ ' o oo ° -~. -,.Oo .~ 2.4-
°°° "~ ~ , .
-6 E
o c u c
". ° o % ° •~.
°°° "~- ,~..
1.6 -- ~ --.... .......
%°.o
AcrytonitriLe(theoreticaL modeL) Styrene (theoretical model. ) AcryLonitrRe ( t e n d e n c y modeL) Styrene (tendency m o d e l )
~ ~
0.8--
0
I 30
I 60
I 90
I 120
I 150
Time of reoction (min) Fig. 6. Concentration profiles for the acrylonitrile-styrene system resulting from dual control policy.
69
Optimization of polymerization reactions semibatch operation. Real-time evaluations of heat transfer properties were determined through online analysis of process conditions. The reactor was equipped with both cooling coils and an external condenser (for reflux cooling) to assist in the reaction heat removal. High performance liquid chromatography was used to determine the concentration of the reactants and reaction intermediates. Molecular weight distributions were obtained using gel permeation chromatography. The rate constants for the three-step mechanism were 'estimated from several temperature-programmed experiments. Tile resulting relationships were (with units in lmo1-1 min-l):
{
k l = 0.0535 exp - 6 2 5 0
[; l l} - ~
,
(32)
kl = 0.150 e x p { - 1850F1 - 1-~-l~ Lr 386JJ'
(33)
k2 = 0.680 e x p { - 2 2 0 0 [ 1 1 3 ~ 1 }.
(34)
for this reactor system were typically less than 5% of the heat transferred by the coils. The deviation between the experimental and the calculated heat release was typical to pilot plant/commercial production. Initially, the heat of reaction assisted in the elevation of the reaction temperature to the maximum value. A 9 min time lag was found for the removal of heat by the coils. The external condensers had a larger time delay (ca 25 min) due to two phenomena: the rate of material/heat flows between equipment and the residence time constant for the condenser's liquid holdup. Therefore, predictive real-time models are needed for adequate control of such pilot plant/ commercial reactors since large time lags can lead to runaway reactions. Utilizing the experimental data, the tendency modelling approach was implemented to form a simpler reaction network representing the system's behavior. The proposed model followed the structure:
I + M2--* P1, Ml + Mz ~ P2,
the heats of reaction found were - 13,800, - 10,600 and -21,600 cal mol -~ for the initiation, second and third reactions, respectively. Figure 7 depicts the experimental and predicted energy evolved by the reaction. The maximum heat release was located at the point where the highly-exothermic third reaction was most prominent. Also, the figure illustrates the portion of heat extracted using the cooling coils as related to the total amount removed (by the cooling coils, the external total condenser and the heat loss to the surroundings). The coil heat transfer coefficients for this system were found to range from 2.0 to 0.80 cal min -~ cm -2 °C -~ as the reaction proceeded. This effect was primarily due to changes in physical properties of the reacting mixture. Since most of the condensate was the low-boiling monomer 1, reflux cooling was not as significant for this system as in other types of polymerization processes. Heat losses
<-
A Heat Loss by COD,S Tatar. heat Loss
12u •
.E
k2
E
(35)
This model combined the three reactants into polymer structures which were not simply described by monomer 1 and monomer 2 terminated molecules. The product of the first reaction was termed pseudocomponent 1 and the result of the second reaction being pseudocomponent 2. These additional compounds were included in weight units as opposed to the typical molar units since their unknown molecular weight changed throughout the course of reaction. By totalling the weight of the two pseudocomponents, a combined product weight was introduced into the reaction scheme allowing calculation of the tendency model's kinetic constants. The tendency model adequately represented the reactants' compositions over the course of the reaction. The kinetic parameters which best fit the concentration profiles were (with units in I real-l min- ~):
1 5 V E x p e r i m e n t a k run 1
x
7
P1 + P2 = Pr"
l Ol Oex+,8,0[ll]}
36)
=0.205expt-1200]a- ] }
(37)
9
g o ,=
,~ G "r
0 0
7O
~'°. I 140
I 210
I 280
Time (rain)
Fig. 7. Reaction heat release at experimental conditions (with the detailed model simulation depicted by the solid line).
The detailed and tendency models deviated at the start of the reaction; this trend was expected since the propagation step (reaction 2) of the tendency model began immediately while the detailed model required the initiation step to occur. This effect caused the initial heat release calculated by the tendency model to be significantly larger since there was no restriction to the more highly-exothermic second reaction. The heat of reaction values for the two reactions were - 15,000 and - 34,000 cal real i. The minimum time problem for step polymerization systems with all reactants initially charged has a
70
G.D. CAWTHONand K. S. KNAEBEL
relatively trivial control policy. The reaction temperature should be at the maximum possible for the highest rate of reaction for irreversible step polymerization mechanisms (Hicks et al., 1969). With the charge of all readtants, the zeroeth and first moments (and the resulting number average molecular weight) are constant; ,the molecular weight ratio was 1.09 at the selected experimental conditions. This occurs with any temperature history. One degree of freedom was available for optimization of these problems. The control objective chosen was the minimization of the second moment which was identical to the reduction of the weight average molecular weight and the polydispersity. With the experimental composition and a maximum reactor temperature limit of 113°C, conversion is finished in 40 min. The corresponding polydispersity is reduced to a value of 1.33, which was a significant improvement from the theoretical value of two. Accompanying the high reaction rate was an equally large heat release (ca 155 × 105calmin-1). Since these conditions were unrealistic for this reactor, a heat transfer limitation of 10 × 10Scalmin -~ was selected for safe operation. Taking into account the heat transfer constraint, the minimum time problem using temperature control provided a reaction time of 132 min (Fig. 8). The resulting polydispersity was 1.72, lower than the restricted isothermal case at 33°C (i.e. 1.91). Temperature control was also used in minimizing the final polydispersity index although no significant improvement in product quality (final polydispersity index equal to 1.64) was found while the final endtime increased to approx. 235 min. With this reaction network, the best temperature profile began at the maximum limit which directly conflicted with the heat release constraint. Therefore, temperature controls alone were inadequate for maintaining a minimum polydispersity. Dual control policies which regulate both temperature and monomer flowrates simultaneously were 120
-
45
x I0C
----
I. . . . I
/
12
-
/
\\ /111/!~.111/
G o
I
~
8c-
9
a.
6(1-
6
40-
3
E
Heat r e l e a s e Temperature
b-
20-
T
0 0
I 40
I 80
120
I 160
Time (rain)
Fig. 8. Temperature control to minimize reaction time for oligomer production (with a maximum heat release rate of 106ca1 min- ~).
550 t
\
280
\ \
210 "o c
- -
Temperature
---
M o n o m e r 2 flow
--.--
Duat control
'
/
140
70
o
t.o
I tz
I 1.4
I 1.6
I 1.8
I 2.o
PoLydispersity
Fig. 9. Noninferior set for minimum end-time and polydispersity. also examined. Using monomer 2 flow control, process performance was improved over that of the temperature control alone. The final conditions were a reaction time of 122min and a polydispersity of 1.35. Since one degree of freedom was eliminated by adding the other control variable, the same end-time value was found using either monomer 1 or 2 flow control. The polydispersity index does not follow this trend. For the lowest polydispersity, dual controls (with monomer 2 flow) were found to provide significant improvements (final value being 1.26 with an end time of 124min). Therefore, dual control procedures attain process objectives in a shorter time of reaction. Noninferior sets for end-time and polydispersity are represented by Fig. 9. This illustrates the range of process objectives which may occur using a selected process control strategy. Also, the advantages of the different control mechanisms can be compared in evaluating the best operating procedure achieving a selected performance criteria. The figure illustrates that for systems which may or may not have heat transfer limitations, no real improvement in polydispersity can be made using temperature control alone. Monomer 2 flow can be controlled to achieve the range of polydispersity values. Dual control problems which manage both temperature and monomer 2 flows guide the process to a selected final condition, improving on a single control's performance. Lastly, the tendency model was evaluated for predicting process behavior as related to the detailed model's responses. The optimal control policy which minimized the reaction time with monomer 2 flow (temperature = 56°C) was the prescribed operating procedure. The resulting concentration profiles favorably compared with the trends obtained using the detailed model. Deviations were found for the initiator conversion due to the large difference in reaction temperature dependencies. Therefore, a new model could be utilized in the high conversion range to better represent these trends. Lastly, both models
7I
Optimization of polymerization reactions 25
b
DetaiLed kinetic model 20
E
Tendency model
- ~ -
15
10 t~
"6 (l) "1-
0
0
60
120
180
240
Time (rain)
Fig. 10. Heat generation using the tendency model with optimal control policy. followed the same heat release behavior (as seen in Fig. 10), confirming the tendency model's ability to predict heat generation characteristics in real-time process control. CONCLUSIONS
The purpose of this study was to investigate the effects and tradeoffs of optimizing dynamic reaction systems. Techniques were developed which applied to both chain and step reactions. Optimal control policies included temperature and reactant addition, .along with combinations forming dual controls. Multiobjective analysis was then used to obtain the ranges of process objectives attainable using the different control strategies. For the acrylonitrile-styrene system, m o n o m e r addition was a more effective means of controlling the copolymer composition during the reaction than was temperature. Conversely, adjustment of the reactor temperature was a better method than m o n o m e r flow control for maintaining the polydispersity. Because of this behavior, the optimization problem was decoupled into two separate problems:, using m o n o m e r flow to correct the composition and temperature to correct for the molecular weight and end time. N o significant process performance was lost with this approach. For initiator flowrate operation, the best policy allowed constant flowrate (and constant concentration) over the course of reaction. Minimum polydispersity indices were found at lower temperatures and greater initiator concentrations. Again, the dual control policy of initiator flowrate and temperature were used for better production than with a single control adjustment. Oligomer production was optimized using selected process control strategies. Isothermal operation at the maximum temperature produced a product with a minimum value of the polydispersity in the shortest amount of time. Since this process was unacceptable due to heat transfer limitations, constraints were added and new operational procedures
were evaluated. Selection of m o n o m e r flowrates allowed a variety of product qualities, although dualcontrol policies of both temperature and m o n o m e r flow functioned more rapidly. Noninferior sets were presented to describe the dependence of the product's polydispersity index with the time of reaction. Lastly, the tendency model was used with an optimal operating procedure and was found to model adequately the system's behavior. Adequate predictions o f conversion were obtained under optimal control conditions using the tendency model. Because the model can be augmented or utilized over a specific operating range, better fits are expected for other complex processes. Therefore, this method would be directly applicable to adaptive control of semibatch reactor systems.
NOMENCLATURE
C, = Concentration of reactant i in feed stream CC = Copolymer composition distribution D = Polydispersity index of total number molecular weight distribution F = Vector of objective functionals (components F,, i=1,2 .... ) f = Initiator efficiency g = Vector of constraint functions kcu = Termination rate constant by combination reaction of radicals of type i and j kDl = Rate constant of initiator decomposition keu = Chain transfer to monomer rate constant from a radical of type i to a monomer of type j kv,j = Propagation rate constant for addition of monomer of type j to radical of type i M, = Concentration of monomer M..,. = Concentration of dead polymer containing n units of M I and m units of M 2 P..m = Concentration of growing polymer containing n units of M~ and m units of M2 and ending with an M t unit P, = ith moment of the total number molecular weight distribution of radicals of type 1 Q.... = Concentration of growing polymer containing n units of M~ and m units of M2 and ending with an M 2 unit Qi = ith moment of the total number molecular weight distribution of radicals of type 2 QM, = Volumetric flowrate of reactant i R- = Concentration of free radical initiator rM, = Concentration rate of change of monomer M, t = Time tf = Final time of reaction T = Temperature u = Vector of control inputs V = Reactor volume wi = Molecular weight of species i x = Vector of reactor states x0 = Initial value of state vector
Greek symbols E~= Constraint to the i objective functional ~ = ith moment of the total number molecular weight distribution #. = Number average molecular weight v0 = Stoichiometric coefficient for the ith reaction and the jth component X = Extent of reaction
G. D. CAWTHONand K. S. KNAEBEL
72 REFERENCES
Bryson A. and Y. Ho, Applied Optimal Control. Blaisdell, Waltham, Mass. (1975). Chankong V. and Y. Haimes, Multiobjective Decision Making: Theory and Methodology. North-Holland, New York (1983). Chen S. and S. Lee, Poly. Engng Sci. 25, 987 (1985). Farber J., Poly. Engng Sci. 26, 499 (1986).
Filippi C., J. Greffe, J. Bordet, J. Villermaux, J. Barney, P. Bonte and C. Georgakis, Chem. Engng Sci. 41, 913 (1986). Hicks J., A. Mohan and W. Ray, Can. J. Chem. Engng 47, 590 (1969). Lin C., W. Chiu and C. Wang, J. Appl Poly. Sci. 23, 1203 (1979). Ray W., Macromolec. Chem. 1, 1 (1972). Tsoukas A., M. Tirrell and G. Stephanopoulos, Chem. Engng Sci. 37, 1785 (1982).