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Synthesis of open-loop controls for semibatch copolymerization reactors by inverse feedback control method
0005-1098/89 $3 00 + 0 O0 Pergamon Press plc ~) i989 intemauonal Federatton of Automauc Control
Automaaca Vol 25, No 6 pp 917-923 1989 printed m Great Bntam
Bnef Paper
Synthesis of Open-loop Controls for Semibatch Copolymerization Reactors by Inverse Feedback Control Method* K
Y
CHOI?:~ and D
N
BUTALA~"
Key Words--Industnal control,opttm~.auon, suboptimal control,polymenzauon reactor
Alam'act--In th~ paper a new approxunauon method for the synthests of open-loop control systems m batch and semlbatch free ra&cal copolymenzaUon processes Is presented Bong characterized by the complexRy of polymenzaUon iuneUcs and reactor behavmr, many batchtype copolymenzaUon reactors pose &ilicult quality control problems In the proposed method, the open-loop control pohcles for both molecular weight and copolymer composluon controls are derived from the tranaent response of the fictmous feedback control system obtamed by the dynamic stmulaUon of a detaded reactor model The proposed method offers rapid and accurate solutions to open-loop batch and senubatch copolymenzauon process control problems The apphcablhty of the proposed method is dlustrated through numencal examples of the styreneacrylomtnle (SAN) copolymenzaUon process
P,
Notatton
t
Ac FC/, f f~ AH, hc 1 If kd
kf,p
kp,, kin/
M, M,f Ms MNd Mw P
P~,,n
PD Q Q~ Q. ,. Rp
r~ T
r, u
heat transfer area per reactor volume [cm2 1- I ] spectfic heat of reactton mvtture [cal g-i K-t] mole fracaon of tth monomer m copolymer, ~ = 1, 2 inittator efiictency mole fraction of tth monomer m reaction mixture, t ffi 1,2 heat of copolymenzaUon [cal mol - t ] overall heat transfer coefficient [cal cm- 2 s- 1 K - 1] lmaator concentratmn m reactmn mtxture |moi I -I] mltaator concentration m feed [mol !-I] m R i a t o r decomposition rate constant [s -!] chmn transfer rate constant, ~, I = 1, 2[1 m -1 s -1] propagatton rate constant, t, I = 1, 2 [I m- 1 s- 1] combination termmauon rate constant, t, 1 = 1, 2 [Im-I s -t ] tth monomer concentrauon tn reaction mixture, I - 1,2 [moll -t] tth monomer concentration m feed, ! = 1, 2 [mol I-I] total number average molecular weight demred value of total number average molecular weight total weight average molecular weight total growing polymer concentrat,on of type 1 [mol I- l]
ul
V Wl
tth moment of the total number molecular weight distnbuUon of radicals of type 1 concentrauon of growing polymer contatmng n units of monomer 1 and m umts of monomer 2 [mol I-I] polychspermy total growing polymer concentration of type 2 [moll -t ] ~th moment of the total number molecular weight dlstnbuuon of radicals of type 2 concentraUon of growing polymer containing n units of monomer 1 and m units of monomer 2 and ending m monomer 2 [mol !-t] copolymertzauon reacUon rate [mol 1-t s -t] monomer reacUvity ratio reactor temperature [K] jacket me&a temperature [K] monomer feed temperature [K] reacuon ume [s] monomer feed rate [l s -t] tth mampulated vanable, t--1, feed rate, t = 2 , reactor temperature reactor volume [i] molecular weight of lth monomer, t = 1,2 [g mol -t ]
Greek letters ~.~ p
¢ ef es ~t
kth moment of the dead copolymer total number molecular weight &stnbutmn; k = 0, 1, 2 denstty of reaction mtxture [gl -I] molar raUo of monomers m reaction mixture monomer mole ration m feed stream desired value of molar ratio of monomers m reactmn mixture cross termmatmn factor
1 lntroducuon PRECISE control of polymer propemes ~s one of the most ~mportant objecuves m controlhng mdustnal polymenzatton processes Although continuous polymenzatmn processes, m general, produce polymers of more conststent quahty m large volume than batch polymenzaUon processes, the latter are stdi mdustnally very tmportant and particularly well stated for the production of polymers of varying grades m a rap,dly changing market environment In parttcular, many specmhty polymers are produced mostly m batch-type polymenzauon reactors The producUvtty of such polymenzauon processes ts closely related to the accurate control of polymer property parameters Therefore, tt ts extremely important to develop e~c~ent computatmnal methods for the synthesis of batch polymenzaUon reactor control strategies Thts paper ~s concerned wRh the development of a new method for the design of open-loop controls m batch and semtbatch copolymenzauon processes In free radtcal copolymenzatton processes, controlhng both copolymer composmon and molecular weight (MW) or molecular wetght dtstnbuuon (MWD) is of pnmary
* Received 20 September 1987, revtsed 20 April 1988, revised 2 November 1988, revased 10 January 1989, received in final form 3 March 1989 The ongmal vers,on of this paper was not presented at any IFAC meeting Thts paper was recommended for publication m revised form by Assooate Editor Y Arkun under the dtrectton of EdRor H Austin Spang Ill ? Department of Chemical and Nuclear Engmeermg and Systems Research Center, Umvemty of Maryland, College Park, MD 20742, U S A ~:Author to whom all correspondence should be addressed 917
918
Brief Paper
~mportance For instance, two styrene-acrylonltrtle (SAN) copolymers ddIenng by more than 4% m acrylonltrde level are mcompattble, resulting m poor physical and mechamcal properties (Molau, 1965) It ~s also well known that s~gmficant batch-to-batch vanauon m product polymer properties is quite common m many mdustnal processes and the resulting offspeofication products are often wasted Moreover, when the product grades are to be changed, the reactor engineers should be able to change the reactor operating pollcles properly so that desired polymer properties can be obtained with minimal offspeoficatton products Many of the ddficult problems m destgnmg the reactor control systems for batch or sem~batch polymerization processes are due to the lack of accurate on-hne polymer property sensors Although some promising progress has been made m recent years m developing on-hne sensors for certain polymerization systems (Schork and Ray, 1983), on-hne sensors for the measurement of many ~mportant polymer properues (e g MW, MWD, copolymer composition, conversion) are not readily available at th~s ttme If such on-hne sensors are available, the polymer properties can be preosely controlled by using feedback controllers Unfortunately, this ~s not qmte possible with present sensor technology Some recent works show that the extended Kalman filter can be used satisfactorily for the estimation of MW m batch free radical homopolymenzatton processes (Papadopoulou and Gdles, 1986, Taylor et al, 1986, 1988) Due to the different reactlvtttes of the comonomers in free radical copolymenzat~on processes, compos~tton drift occurs unless a more reactive monomer is added to the reactor In order to mamtam a constant mole ratio In controUmg the copolymer properties m batch or semlbatch processes, one needs to determine how the monomers should be added to the reactor to maintain constant bulk phase compos~tton and how the reactor temperature should be vaned to produce the polymers of destred MW Since the on-hne measurements of these property parameters are not available, one must determine the reactor operating policies "a prtort" To do so, we need to have good process models Therefore, batch or semlbatch copolymenzatlon reactor control problems have always been viewed as open-loop controller design problem and the following general destgn procedure has been used (I) Synthesis of open-loop ttme-varylng control pohc~es which minimize properly defined performance index (Task Level Control) Reactor temperature, feed rates of catalyst or reactants are some examples of manipulative variables Inevitably, th~s step requires an accurate process model and deep physical insight into the process behavior (2) Design of a control system which will drive the plant to follow the open-loop control pohc~es obtained m step (1) as closely as possible (Execution Level Control)
happening during the batch period as long as he obtains the polymers whtch meet the final property speoficattons The second approach is to control polymer properties during the entire course of potymenzatton In free radical copolymertzauon, most of the polymer property parameters to be controlled (e g copolymer composmon, MW and MWD) can be maintained at their target values from the beginning to the end of the reactor operation This ~s because the polymer MW increases very rapidly and the copolymer composition can be mamtamed at the desired level from the beginning of polymerization by adding more reactive monomer or monomer mtxture to maintain constant bulk phase monomer composmon Using these umque charactensacs of free radical copolymerlzatton processes, one can design the batch type copolymenzatlon reactor operatmg pohctes m which both MW and composmon are controlled at the destred values (target values) during the entire course of polymerization In thts case the specifications of these property parameters are always m target values (except for a short tmtml period) and only the monomer conversion changes w~th batch reacnon rime The reactor operauon ~s terminated when the monomer conversion reaches the prespeofied desired value or the reactor reaches ~ts full volume One potentml advantage of this approach ~s that even when the batch Is terminated for some reason before the final prespeofied monomer conversion ts reached, the product wdl still have the desired properttes and the batch products wdl not be wasted but saved In this paper, we shall introduce a new approximation method to tackle the second type of control problems described above For the first type of control design problem (1 e end-pomt control), the proposed method ~s not apphcable and other optimization methods such as multtobjectwe dynamtc opttmtzatlon should be used (Butala et al . 1988)
2 Design of open-loop control strategy Let us consider a batch dynamic process represented by the following non-hnear modeling equation x = t(x, u)
(1)
x(0) = Xo
(2)
wnth mmal condltnons
where x IS an n-state vector and u an m-control vector (n>-m, m - l ) Suppose that we wish to find open-loop controls that would bnng certain components of x exactly to some fixed desired values 0 e set points) dunng the ennre
processmg pertod I e x,(O<--t<--tf)=X,d,
t=l,
,q(q<--n)
(3)
,q
(4)
or
This paper describes the development of the task level control scheme for free radical batch and semJbatch copolymenzat~on reactors Here, the concept of reverse feedback control technique is introduced to derwe the tmme-varylng monomer addmon and/or reaction temperature profiles which are reqmred to yield styrene-acrylommle copolymers of desired composltton and MW When one attempts to develop optimal open-loop control policies for batch or semtbatch copolymenzatton processes, one must first define the design objectives In most cases, the control of copolymer composmon, MW and MWD are of utmost importance Even after the ultimate process goals such as those described above have been tdenttfied, one may have to define more speofic control objectives For example, it may be desirable to control only the end-time properties. e copolymer properties at the end of batch or semJbatch reactor operations This would be a reasonable and practical goal since what one ts concerned about may be the final properties of the polymers obtained after the polymerization ts completed In this case, the design of open-loop control becomes an end-pomt control problem Moreover, the reactor operator does not have to be concerned about what ~s
x,=O.
O<--t<=h. t = l ,
where h t s the total process (batch) ume, x, the state variable being controlled, and X,d the desired target value of the state variable x, In batch or semibateh copolymenzatton processes, x, may be the bulk phase monomer eomposttton (or copolymer composmon) or copolymer MW The potential control variables are the reactor temperature and the monomer feed rate as a funetton of time The perfect control up may be found tf there exists u which satisfies equation (4) If such a n can be represented exphc~tly as a funcuon of state variables, i e a = V(x)
then, the
perfect open-loop
(5)
control tralectones, ,,p
(t,O<-t<-tf) wdl be detenmned completely by solving equaUons (I) and (5) In most pracUcal muaUons m batch or semlbatch copolymenzatJon process control such perfect controls (up) may be extremely difflcultor tmposslble to derive and express m usable forms due to the complexity of the process model A classlcalapproach to opumal control syntbests ts to use
Bnef Paper the necessary cond,tmns of Pontryagm's maximum pnncmple (Bryson and He, 1975, Denn, 1969), however, when the d,menston of the process model ms large and the process model is very complex (e g copolymenzat,on reactor model) it may be ,mpract~cal and computatlonally too ddticuit to derive opt,real control pohoes using the maximum pnnctple Moreover, when the Hamfltoman ~s a hnear functmn m control a, one faces a singular control problem which reqmres special solutmn techmques For ted-batch type (sere,batch) reactor operauons, such a singular control problem occurs frequently One of the alternative methods for the synthes,s of the opumal control strategy ~s to use approximate methods such as the control vector parametenzat,on techn,que (Ray, 1981) m which the parametenzat,on of the opUmal control is made by expanding the control variable(s) m a set of real functmns, (~,j), of the form m
u,(t) = ~ a,,~k,,(t)
(6a)
I=l or
m
u,(x) = ~ b,,tp,,(x)
(6b)
l=i
Then, any parameter optmmmzatmn techmques may be used to determine the optlmal set of coellic~ents a,j or b,~ Thins approx,matton method ms attract,re m that no adjomt equatmns need to be solved However, mt ts reqmred that the functmnal form of the opt,real control be prespeclfied and proper we,ghtmg factors chosen when more than one vanable are controlled Obvlously, th~s method needs a consmderable amount of computing time and deep physlcal insight rote the process behavmr being consldered Recently, computer alded des,gn (CAD) techniques have been successfullyapphed to these multmbjecuve and multlvanable open-loop copolymenzatlon reactor control problems using the method of feaslble d,rectmn algonthm (Butala et al, 1988) In what follows, we shall present a new and slmple method of des,gmng open-loop control pohoes for free radical copolymenzatton processes w,th dynamlc process models Let us firstconslder a feedback control system shown m Frog I Here, the vanables to be controlled (x,) are momtored on-hne The objective of thls feedback control system ,s to mmmm~ze the set point dewatlons, ~,= IX,d--X,J(X,d=fixed, O
x,~~
_x.±tx -~1
I
F~(~ I F,ctmous feedback control system
x,
919
denved In other words, the open-loop control pohcy ms deduced by reverting the transient response of the fict,tlous feedback control Th,s ,dea, inverting the transient feedback response, of obtaining the open-loop control pohcy has not been utilized for pract,cal copolymenzatlon reactor control problems to date Then our next questmon ,s how to obtain such feedback control profiles u~, (t, 0 ~ t "~tf) "a prmn' Thins can be accomphshed relat,vely easily by performing the dynamic simulation of the process control model Here, the feedback controller G c of the fictmous feedback control system ,s first designed unal the best closed-loop response ,s obta,ned Then, the trans,ent trajectones of the mampulated variables, uc, (0 <-t-< tf), are recorded Since such control u¢, Is what has been needed to obtain the good serve response of the fictmous feedback control system, it is now clear that the use of this control trajectory m the actual open-loop process should result m the same control performance as the fictltmus closed-loop system from which uc, was obtained In order to obtain the most satisfactory performance of feedback control, one may use any feedback control algorithms available There are many advantages of this ,nverse feedback control techmque (1) The open-loop control pohcy msdetermined d,rectly by solving the process model of fictmous closed-loop control system (2) Since the control pohcles are determined by the dynamic smmulatmn of the process model, very soph,sucated and thus perhaps more accurate process models can be used for the open-loop controller design (3) Although for this approach we,ghtmg factors are not needed exphotly for the control obleeUves, they may be used when more than one control vanable and advanced controllers are used (4) The effect of various operating conditmns such as m, tml reactor condmons and man,pulatlve vanable characteristics can be analyzed easily This mspart,cularly ,mportant when one wants to redesign open-loop control pohc, es frequently m order to produce the polymers of ddterent grades The factors wh,ch influence the quahty of the open-loop control pohcles determined by thts reverse feedback control techmque are O) the accuracy of the process model and (n) the goodness or the quahty of the feedback controllers used m the numerical s,mulaoon of the fict~tmus closed-loop control system The mteracttons between the feedback loops for mult,vanable systems may also affect the quahty of control pohCleS In the next sectmn, we shall use the proposed techmque to find open-loop control pohoes for the control of batch and sem,batch SAN copolymenzatmn reactors
3 Semtbatch copolvmertzauon reactor model The control object,ve here ,s to mamtam constant copolymer compos, tmn and constant polymer MW during the course of polymenzatlon by man,pulatmg the reactor variables such as monomer add,tmn rate and reactor temperature In what follows we shall illustrate the proposed design method through the numerical s,mulatmon of the solut,on copolymenzatmn process to produce SAN copolymers The solvent and m,tlator used are xylene and AIBN, respectively The reasons for selecting th,s system are SAN copolymer ,s commeroally ,mportant and opt,real control ms necessary if copolymer composmon has to be mamma,ned at points other than the azeotrop,c po,nt where the bulk phase compos,tmn 0 e monomer mole ratio) is the same as the copolymer composition and no composlt,on dnft occurs The kinetic model and the numerical values of kmet,c parameters are described m the Appendix The modehng equatmns for semJbatch free radical copolymenzatmn processes take the following form dM l u -~ = ~ ( M I f - M l ) - [(kpll + kt I1)P +
(kp2 i +
kf21)Q]M l
(7)
020
Brtef Paper
dM2 = u (M2f dt V
-
X 105 15
M2) -
[(kp22+ kf22)Q + (kp~2 + kn2)P]M2 (8) dl
MN 0 75 "
u
-.~ = -~ (If - I) - kd!
(9)
dV
d-7= u
where the temperature dependence of the rate constants are expressed In Arrhenms form Here, M1 and M2 are molar concentrations of monomer 1 (styrene) and monomer 2 (acrylomtnle), respectwely, I the mmator concentraUon, and u the volumetric monomer adchUon rate P and Q are the total concentratmns of live polymers having monomer M~ and monomer M2 as the end units, respectively Other parameters are defined m the Notaaon For the calculauon of polymer MWs, the following moment equauons for dead polymers are used d•
2
~.
1
+(kmMl+kfl2M2)P+(kfaT.M2+kt21MOQ---~u
(11)
I I l [ t l l l l
+ (ktc22Q + ktd22Q + ktcl2P+ ktdl2P + kf22M2+ k f21Mt)Q! --~u
[ t l t l l l r
04 ~°~
10
15
05
10
O0
O0
=(ktcllP + ktdtlP + ktct2Q+ ktd12Q+ kftlMt + kf12M2)P!
l
FI 06~
MN
2
-~f = (2ktcll + ktdll)P + (2ktc22 + kto22)Q + (ktcl2 + 2ktdl2)PQ
~
O0 08
(10)
I000
2000
3000
4000
TIME (MIh0
FIG 2 Isothermal batch copolymenzatmn (330 K, f~ = 0 25, 1o = 0 05 mol I - i )
(12)
d~td
- ~ = ( k,oH P + ktdH P + ktcI2Q + k,m2Q + kfH Ml + kf12M2)~ + (kt,22 Q + kto.2 Q + kt,12P + ktdt2P + kf22M2 + kf21M1)Q2
+ktcflPl +ktc22Ql 2ktcl2PlQ1--~ u
(13)
Notice that the reactor volume V vanes with ume due to the addmon of monomers The instantaneous copoiymer composmon is descnbed by the Mayo-Lewis equation
FI
=
ri ¢2 + $ rtq~2+ 2~ + r2
(14)
which can be held constant by m m n t a m m g the bulk phase monomer mole ratm (~) constant In equaUon (14), r~ and r2 represent reactivityratios The conversmn of monomer I (styrene) ~s defined as VoMto + I ~u(t)Mtf dt - VMt(t)
Xt =
-'o
r,
(15)
V°Mm + J0 u(t)M~fdt where Vo ~s the msual reaction volume, M m the mRzal monomer concentration m the reactor, Mtf the feed monomer concentrataon, and u(t) the monomer feed rate 4 Dtscussmn o f results In our numencal simulations the mmal solvent mole fraction is 0 25 and the Imtzal imtaator concentration ~s 0 05 tool !-1 For all the senubatch copolymenzatlon cases the volume of the total mmal reactants (monomers, solvent and mttmtor) is 1 hter, except for the batch copolymenzatlon case where the starting reactor volume is 3 hters The mmal monomer mole ratio (¢p) In the reactant mixture is 1 0 (or F1 -- 0 58) and the mmal reaction temperature for all cases is 60°C For all the copolymenzatton composRton control problems the feed includes 0 25 mole fractmn of solvent, 0 05moll -~ of mmator and monomers wRh a mole ratm (Mlf/M2f) of 1 5 4 1 Isothermal copolymenzanon To justify the need for proper control systems for the productmn of SAN
copolymers of desired composmon and MW, the numerical ssmulaUon of isothermal batch solutmn copolymenzatton has been conducted and the results are shown m Fig 2 Here, F~ IS the instantaneous styrene concentration in the copolymer, M N the number average molecular weight As expected, slgmficant vanaUons in composlUon and ~ occur as the monomer (styrene) conversion increases Such nonumformlty In polymer properties IS not acceptable for the practical apphcatlons of the polymers 4 2 Control o f copolymer composition tn Isothermal semzbatch reactor The inverse feedback control method descnbed in Section 2 has been used to synthesize the open-loop copolymer composition control pohcy m the ,sothermal copolymenzaUon process In our s,mulaaons, PI controllers are used to demonstrate the appheabdlty of the proposed method The PI controllers are tuned empmcally to obtam the best output response For our examples, constraints on the control variables are incorporated in the stmulatmn scheme Here, the copolymer composmon ,s controlled by adding the reactant mixture contmmng solvent, mmator and monomers to mmntam the constant bulk phase monomer mole ratio ( ¢ = 1 0, FI = 0 58) Since styrene is more reactive, the feed stream ,s ncher in styrene ( e l = 1 5) Thus, a t,me-varymg monomer add,tmn policy is sought to maintain constant instantaneous monomer mole ratm in the bulk phase The controller demgn procedure ts as follows (1) Assume that a perfect fictitious copolymer composmon sensor is avadable Then, perform the dynanuc stmulaUon of the ficUUous feedback control system and find the best feedback controls wh,ch rmmmtze the servo errors (1¢ - edl, 0 ~ t ~ If) Any constraints on the control variables of the form [~mm< Uc ----Umax can be incorporated into the slmulanon scheme The final batch time ts reached when the desired conversion ,s attmned or when the reactor is full (2) Record the time-varying trajectory of the resulting feedback control signal uc(t ) from t = 0 to tf (3) Implement the entire trajectory of u c obtmned from (2) into the actual open-loop control system where no on-hne composmon sensor is available In practice, the original control signal uc(t ) which resulted directly from the dynamic stmulaUon of the fictmous feedback control system may be nmsy due to controller actmn and impractical for the implementation into the actual
Bnef Paper
921 38O0
X 105 1 5
X 105 1 5
- 3500
M N 0 75 •
M N 0 75 -
~M N O0 08
3200 500
00 O8
200
'i F FI
.... -100
06.
FI
, , .F_1 / /
.
.
.
.
. 25
06
0 t.~
/.1 04 20
MN
,,,I
,TI
,,,I,
00 10
,,
z u~
f_~jl
15
$
-05
~ ~ , , [ ~ ~ , , I , , ~ , I .'r. ~ ~ ~
04 2O
MW
_
_
-05
MN
> Z
00 10
o3 re W
oo 10
00
O0 1000
O0
2O00
3000
00
4000
Tnvt~(MIN)
open-loop process Such a problem can be carcumvented by smootlung or filtenng the ongmal feedback signals and fitting them into the properly parametenzed forms F~gure 3 dlustrates the performance of the open-loop controller d e u g n e d by the proposed method Note that copolymer composmon control ts excellent Although the total amount of added monomers is not great, such composmon adlustment results m a stgmficant improvement of the copolymer composmon control However, note that the MW (MN) vanes considerably with the monomer converston X 105 1 5
3800 A
/°2
3500 p-
MN
' ~ ' ~ I ] ' ' ' i ' ' ' ' L ' ' ' '
oo 08
F1
06
3200
~
20
lO
~ f
//
u~
z
MN
O
15"
-05
8 10
i 00
i
1 l
] 1000
i
i
i
i
l
,
i
2000
2000 "r~E
FIo 3 Open-loop composmon control m an isothermal semlbatch reactor (333 K, f , = 0 25, Io = 0 05 tool 1-1, K~l = - 0 7 I s - ' , ¢n = 1000 0s)
.. o,, ~
1000
i
i
[ 3000
i
i"ii
O0
3000
4000
(MIN)
FIG 5 Open-loop ¢omposmon and MW control in a scmlbatch reactor ( 0 - < u l _ < 7 0 m l m m - I , 323 _< u2 < 363 K, f~=025, 10=005moll -I, Kcl= - 0 7 I s -I, Kc2 = 0 001 cal s -1 K -1, rll = r2 = 1000 0 s) Clearly, thts is because only one mampnlated vanable was used for this non-hnear multtvanable interacting system 4 3 Control of copolymer M W m batch reactor If the copolymer MW (MN) is the property parameter to be controlled, the reaction temperature wdl be a better control vanable than the monomer addmon rate In th~s case, the copolymenzauon reactor is operated m a batch mode and a time-varying temperature set point program ~s reqmred to obtain MN = 30 000 Figure 4 shows the performance of the batch reactor controller designed by the inverse feedback control method Note that the reactor temperature must be increased dunng the mmal reacuon period m order to lower the polymer MW near to the destred value Again, very good MW control is obtained with only slight offset The temperature profile shown m Fig 4 can be easily implemented into any process control computer Note, however, that this ttme the polymers of poor copolymer composmon are obtained 4 4 Control of copolymer composttlon and MW in sermbatch reactor Figures 3 and 4 clearly illustrate that both composmon and MW should be controlled by mampulatmg the monomer feed rate (u~) and the reactor temperature (u2) The target values of ~ = 1 0 (or F I = 0 58) and Mr~ = 30 000 are chosen The open-loop control pohcles for both composition and MW controls have been determined by the proposed method m which Pl controllers were used m both composmon control loop and MW (MN) control loop to obtain sausfactory fictmous feedback control Figure 5 ~hows that excellent open-loop control of composmon and MW has been obtained The slowly varymg monomer addmon rate and the temperature set point profiles shown m Fig 5 can easdy be implemented into the modem high-speed process control computers Notsce that MWD is also maintained at a constant level In implementing the transient response of u~ and u2 obtained from the slmulaUons of the fictmous feedback control system, the original feedback control signals were filtered and fit into the mnth-order polynomials
4000
TIME OvtIN)
FIG 4 O p e n - l o o p c o p o l y m e r M W c o n t r o l m a batch reactor (323 ~ u : s 363 K . A = 0 25. 1o = 0 05 tool I - l. Kc2 = 0 0 0 5 c a l s - I K -1, T n = 1000 0 s )
5 Concluding remarks In this paper, a new approxtmatlon method has been introduced for the synthesis of open-loop control pohcles for batch and semlbatch free radical copolymenzatlon processes
922
Brief Paper
m which both copolvmer composition and MW are controlled at thew target values during the entire course of polymerization The method is based on the discovery that the tlme-varwng trajectories of the mampulated variables obtained from the fictitious feedback control system are equwalent to the trajectories of the control variables or t~me-varymg set point programs reqmred for the control of an open-loop control system m which no on-l,ne sensors are avadable Our slmulatton results for the SAN copolymerlzatJon process indicate that this method ~s simple to use and very powerful to synthestze the open-loop control pohoes for the control of both copolymer composition and MW m batch and semlbatch reactors In particular, the proposed method will be very useful when one wants the rapid computation of monomer feed rate and temperature profles for the production of polymers having different properties The quahty of the open-loop control pohoes developed using the proposed method may depend upon the following factors (1) The quahty of process models The proposed method, hke any other open-loop control pohcy developed based on the process models cannot cope wtth unmodeled dtsturbances and modeling errors Therefore, it is ~mportant that the copolymenzauon kinetic model must be tested and vahdated through expenmental data Thts constraint apphes to any model-based open-loop opttmal control designs (2) The quahty of feedback controllers used m the vtmulattons When s~mple PI controllers are used as in our examples, the designer must find the best controller tuning parameters which y~eld the best feedback controls for the "fictmous" closed-loop system However, one can use any feedback control algorithms other than a conventional PID algorithm to obtain the best feedback control trajectortes For example, opttmal PID controllers or adaptwe controllers may be useful to obtain the tmproved performance of the "fictmous ' closed-loop control It may be poss~bte to observe different open-loop control pohctes which may yteld similar output responses with d~fferent sensRw~es to model/plant mismatch
(1988) Est~matton of the molecular weight distribution m batch polymenzatmn A 1 Ch E J , 34(8), 1341-1353 Tsoukas, A , M Tirrell and G Stephanapoulos (1982) Multlobjectwe dynamic optlmlzatton of semlbatch copolymenzatlon reactors Chem Engng Sct, 37, 1785-1795
Appendix
Kmettcscheme and moment equattons
The following kinetic model ts used to describe the homogeneous solution free radical copolymenzat~on of styrene wRh acrylomtrtle At high solvent volume fraction, the effect of diffusion controlled termmauon (gel effect) is not s~gmficant The penultimate effect is also assumed neghgtble (These assumpuons do not resmct the apphcabdlty of the proposed controller destgn techmque, however ) The numerical values of kinetic constants are hsted m Table A1
Intttatton 1"-" 2R
kll R + Mf-"*Plo kj2
R + M2-'-'Qo, Propagatton P.,,, + M1
enm+M2
kpll
kpl2
'P.+I., ,Qnm+l
kp21
Q,,m+Mt
~P,,+tm
Q,, ,,, + M2
kp22 ,Q.,,,+,
Combtnanon termination kt¢ll
P . . + P,.q
Acknowledgements--This
work was supported by the Systems Research Center at the UntversRy of Maryland, College Park The authors would hke to thank the anonymous reviewers whose comments helped clanfy some issues described m this paper
)M,, . . . . q
ktcl2 P. m + Q . q
'M. ....
q
ktc22 Q. m + Q, q
' Mn+r ra+q
References Bryson, A
E
and Y
C
Ho (1975)
Apphed Opumal
Control, 2nd ed Blatsdeil, Waltham Butala, D N , K Y Chol and M K H Fan (1988) Mult~objectwe dynamtc opttmtzatton of semzbatch free radical copolymenzatton process with interactive CAD tools Comput Chem Engng, 12(11), 1115-1127 Denn, M M (1969) Opumtzatton by Vartattonal Methods McGraw-Hall, New York Molau, G E (3965) Heterogeneous polymer systems--Ill Phase separation m styrene-acrylomtrfle copolymers Polym Lett , 3, 1007-1015 Papadopolou, S A and E D Gdles (1986) Continuous esumauon of the chain length dtstnbutmn m polymerization reactor considering ume dtscrete gel permeaUon chromatography measurements In K H Retchert and W Gelsler (Eds), Polymer Reactton Engineering, pp 243-260 Huthlg and Weph, Basel Ray, W H (1981) Advanced P~.ocess Control McGrawHill, New York Schork, F J and W H Ray (1983) On-hne measurements of surface tension and densRy w~th apphcat~on to emulsion polymerization J Appl Polym Set, 28, 407-430 Taylor, T W , V Gonzales and K F Jensen (1986) Modehng and control of molecular weight dtstnbutlon m methyl methacrylate polymerization In K H Relchert and W Gelseler (Eds), Polymer Reacuon Engineering, pp 263-274 Huthlg and Weph, Basel Taylor, T W . M F Elhs, V Gonzalez and K F Jensen
TABLE A1 NUMERICAL VALUES OF KINETIC PARAMETERS AND REACTION CONDITIONS FOR STYRENE-ACRYLONITRILE COPOLYMERIZATION
Initiator AIBN Monomer Styrene Acrylomtnle
Parameter
Preexponential factor
kd f
6 02 x 101~s -1 06
Act,vanon energy
(calg-1 m o l - i ) Reference
31 730 --
I tool -J s - l
kpl I k,. kfl !
1 06 X 10"~ 1 25 x 10'~ 2 31 x 106
kp2: kt22 kf22 rI r2 kfl2 kt2!
3 3 x 1012 6 93 x 106 2 56 6 67 x 10-5 30 x kfl I 5 X kfl I
et
3 0 x 10"~
5
x
7067 1677 12670 4100 5400 5837 1190 -4340 12 670 5837
kr~
Tsoukas et al (1982)
--
23 Reactor parameters 0 05 mol l - I
J's= 0 2 5 ,
tm~ = 6 h ,
Vm. x = 4 0 1 `
Io =
Brief Paper where
DtsproportIonatIon termmatton
P...+t'~
ktdll
fl
,M.,.+M,~
ktdl2
P."+Q.o
923
kpl. + kf,.
M,
The application of the pseudosteady state approxlmauon to live polymers leads to the followmg live polymer moment equations
,Mnm+M,.q
ktd22
Qnm+Qrq
'Mnm+Mrq
wlC,a, + a , y
Cham transfer
( 1 - tr0
P,, ,.,,+ M,
kill
~M. .. + Pio
Yr2 '
Q,= P,, ,,, + M 2
Q,,,,, + M,
Q,,,,, + M2
kfl2
kf21
k f22
1 on
the
end
/
-\-
yr2
(1 ---'~2)
(A7)
' M,, ,, + Qo,
w2Ctoq + °qY Q. + 2wloqP, rl
J'M,,,.,,,+ P1o
~
+ 2Witl'lY QI +wl(otlP r1
Similarly,
Q . ,., r e p r e s e n t s
+ £~'1'~ rl
Q)
P2 =
'M,,,,, + Qo,
(A8) (1 - 0q)
where Pn ,~ represents a growmg copolymer chain with n units of monomer 1 and m units of monomer 2, and monomer
)
w~.C2a2 + ~2 P2 + 2w2°r2 P,
]0"2
the
yr2
+ 2w.tr.Q, +w2( ~
growing copolymer chain with monomer 2 on the end M. ,~ denotes inactive (or dead) polymer The copolymer MW and MWD are computed by using three leadmg moments of the total number average copolymers The instantaneous kth moment is gwen by
" "
P+o:.O)
)'2
-
(A9)
Q2-
(1 - tr2)
where
,I.~= ~, ~ (nwl+mw2)kM,,,,,, n~l
k=0,1,2
(AI)
Cj = (kfllP + kt21Q)Mt/kpltM ,
m=,
where w, and w2 are the MWs of monomer 1 (styrene) and monomer 2 (acrylonimle), respectively The total number average chain length (Xn). the total weight average chain length (Xw) and the polydlspersity Index (PD) which is a measure of MWD broadening are expressed as XN = ~
(A2)
Xw = ).--~
(A3)
C2=
(kf22Q + kfl2P)M 2 kp22M2
kpll rl = kpl2'
7,~ = kp22 ~
lip21 kp21
Y=kpl 2
XN
(A4)
The total concentration of live polymers P is given by
p=((ktc, t+ktd,,)
2fka I
+ 2fl(ktcl2 + kid,2) + ]~2(k,c22 + kta22)
(All) (Al2) (A13)
k p I I MI
(A14)
OtI
PD = X__~w
(Al0)
[(kpll + ktlt)Mt + (kpl2 + ktl2)M 2 + (k,cN + k,du)P + (ktcl2 + ktd12)Q] kp22M2
),/2 (A5)
(A15) [(kp22 + kf22)M2 + (kp2 ! + kf21)M I + (ktc22 + ktd22)Q + (ktcl2 + ktal2)P]
Automaaca Vol 25, No 6 pp 917-923 1989 printed m Great Bntam
Bnef Paper
Synthesis of Open-loop Controls for Semibatch Copolymerization Reactors by Inverse Feedback Control Method* K
Y
CHOI?:~ and D
N
BUTALA~"
Key Words--Industnal control,opttm~.auon, suboptimal control,polymenzauon reactor
Alam'act--In th~ paper a new approxunauon method for the synthests of open-loop control systems m batch and semlbatch free ra&cal copolymenzaUon processes Is presented Bong characterized by the complexRy of polymenzaUon iuneUcs and reactor behavmr, many batchtype copolymenzaUon reactors pose &ilicult quality control problems In the proposed method, the open-loop control pohcles for both molecular weight and copolymer composluon controls are derived from the tranaent response of the fictmous feedback control system obtamed by the dynamic stmulaUon of a detaded reactor model The proposed method offers rapid and accurate solutions to open-loop batch and senubatch copolymenzauon process control problems The apphcablhty of the proposed method is dlustrated through numencal examples of the styreneacrylomtnle (SAN) copolymenzaUon process
P,
Notatton
t
Ac FC/, f f~ AH, hc 1 If kd
kf,p
kp,, kin/
M, M,f Ms MNd Mw P
P~,,n
PD Q Q~ Q. ,. Rp
r~ T
r, u
heat transfer area per reactor volume [cm2 1- I ] spectfic heat of reactton mvtture [cal g-i K-t] mole fracaon of tth monomer m copolymer, ~ = 1, 2 inittator efiictency mole fraction of tth monomer m reaction mixture, t ffi 1,2 heat of copolymenzaUon [cal mol - t ] overall heat transfer coefficient [cal cm- 2 s- 1 K - 1] lmaator concentratmn m reactmn mtxture |moi I -I] mltaator concentration m feed [mol !-I] m R i a t o r decomposition rate constant [s -!] chmn transfer rate constant, ~, I = 1, 2[1 m -1 s -1] propagatton rate constant, t, I = 1, 2 [I m- 1 s- 1] combination termmauon rate constant, t, 1 = 1, 2 [Im-I s -t ] tth monomer concentrauon tn reaction mixture, I - 1,2 [moll -t] tth monomer concentration m feed, ! = 1, 2 [mol I-I] total number average molecular weight demred value of total number average molecular weight total weight average molecular weight total growing polymer concentrat,on of type 1 [mol I- l]
ul
V Wl
tth moment of the total number molecular weight distnbuUon of radicals of type 1 concentrauon of growing polymer contatmng n units of monomer 1 and m umts of monomer 2 [mol I-I] polychspermy total growing polymer concentration of type 2 [moll -t ] ~th moment of the total number molecular weight dlstnbuuon of radicals of type 2 concentraUon of growing polymer containing n units of monomer 1 and m units of monomer 2 and ending m monomer 2 [mol !-t] copolymertzauon reacUon rate [mol 1-t s -t] monomer reacUvity ratio reactor temperature [K] jacket me&a temperature [K] monomer feed temperature [K] reacuon ume [s] monomer feed rate [l s -t] tth mampulated vanable, t--1, feed rate, t = 2 , reactor temperature reactor volume [i] molecular weight of lth monomer, t = 1,2 [g mol -t ]
Greek letters ~.~ p
¢ ef es ~t
kth moment of the dead copolymer total number molecular weight &stnbutmn; k = 0, 1, 2 denstty of reaction mtxture [gl -I] molar raUo of monomers m reaction mixture monomer mole ration m feed stream desired value of molar ratio of monomers m reactmn mixture cross termmatmn factor
1 lntroducuon PRECISE control of polymer propemes ~s one of the most ~mportant objecuves m controlhng mdustnal polymenzatton processes Although continuous polymenzatmn processes, m general, produce polymers of more conststent quahty m large volume than batch polymenzaUon processes, the latter are stdi mdustnally very tmportant and particularly well stated for the production of polymers of varying grades m a rap,dly changing market environment In parttcular, many specmhty polymers are produced mostly m batch-type polymenzauon reactors The producUvtty of such polymenzauon processes ts closely related to the accurate control of polymer property parameters Therefore, tt ts extremely important to develop e~c~ent computatmnal methods for the synthesis of batch polymenzaUon reactor control strategies Thts paper ~s concerned wRh the development of a new method for the design of open-loop controls m batch and semtbatch copolymenzauon processes In free radtcal copolymenzatton processes, controlhng both copolymer composmon and molecular weight (MW) or molecular wetght dtstnbuuon (MWD) is of pnmary
* Received 20 September 1987, revtsed 20 April 1988, revised 2 November 1988, revased 10 January 1989, received in final form 3 March 1989 The ongmal vers,on of this paper was not presented at any IFAC meeting Thts paper was recommended for publication m revised form by Assooate Editor Y Arkun under the dtrectton of EdRor H Austin Spang Ill ? Department of Chemical and Nuclear Engmeermg and Systems Research Center, Umvemty of Maryland, College Park, MD 20742, U S A ~:Author to whom all correspondence should be addressed 917
918
Brief Paper
~mportance For instance, two styrene-acrylonltrtle (SAN) copolymers ddIenng by more than 4% m acrylonltrde level are mcompattble, resulting m poor physical and mechamcal properties (Molau, 1965) It ~s also well known that s~gmficant batch-to-batch vanauon m product polymer properties is quite common m many mdustnal processes and the resulting offspeofication products are often wasted Moreover, when the product grades are to be changed, the reactor engineers should be able to change the reactor operating pollcles properly so that desired polymer properties can be obtained with minimal offspeoficatton products Many of the ddficult problems m destgnmg the reactor control systems for batch or sem~batch polymerization processes are due to the lack of accurate on-hne polymer property sensors Although some promising progress has been made m recent years m developing on-hne sensors for certain polymerization systems (Schork and Ray, 1983), on-hne sensors for the measurement of many ~mportant polymer properues (e g MW, MWD, copolymer composition, conversion) are not readily available at th~s ttme If such on-hne sensors are available, the polymer properties can be preosely controlled by using feedback controllers Unfortunately, this ~s not qmte possible with present sensor technology Some recent works show that the extended Kalman filter can be used satisfactorily for the estimation of MW m batch free radical homopolymenzatton processes (Papadopoulou and Gdles, 1986, Taylor et al, 1986, 1988) Due to the different reactlvtttes of the comonomers in free radical copolymenzat~on processes, compos~tton drift occurs unless a more reactive monomer is added to the reactor In order to mamtam a constant mole ratio In controUmg the copolymer properties m batch or semlbatch processes, one needs to determine how the monomers should be added to the reactor to maintain constant bulk phase compos~tton and how the reactor temperature should be vaned to produce the polymers of destred MW Since the on-hne measurements of these property parameters are not available, one must determine the reactor operating policies "a prtort" To do so, we need to have good process models Therefore, batch or semlbatch copolymenzatlon reactor control problems have always been viewed as open-loop controller design problem and the following general destgn procedure has been used (I) Synthesis of open-loop ttme-varylng control pohc~es which minimize properly defined performance index (Task Level Control) Reactor temperature, feed rates of catalyst or reactants are some examples of manipulative variables Inevitably, th~s step requires an accurate process model and deep physical insight into the process behavior (2) Design of a control system which will drive the plant to follow the open-loop control pohc~es obtained m step (1) as closely as possible (Execution Level Control)
happening during the batch period as long as he obtains the polymers whtch meet the final property speoficattons The second approach is to control polymer properties during the entire course of potymenzatton In free radical copolymertzauon, most of the polymer property parameters to be controlled (e g copolymer composmon, MW and MWD) can be maintained at their target values from the beginning to the end of the reactor operation This ~s because the polymer MW increases very rapidly and the copolymer composition can be mamtamed at the desired level from the beginning of polymerization by adding more reactive monomer or monomer mtxture to maintain constant bulk phase monomer composmon Using these umque charactensacs of free radical copolymerlzatton processes, one can design the batch type copolymenzatlon reactor operatmg pohctes m which both MW and composmon are controlled at the destred values (target values) during the entire course of polymerization In thts case the specifications of these property parameters are always m target values (except for a short tmtml period) and only the monomer conversion changes w~th batch reacnon rime The reactor operauon ~s terminated when the monomer conversion reaches the prespeofied desired value or the reactor reaches ~ts full volume One potentml advantage of this approach ~s that even when the batch Is terminated for some reason before the final prespeofied monomer conversion ts reached, the product wdl still have the desired properttes and the batch products wdl not be wasted but saved In this paper, we shall introduce a new approximation method to tackle the second type of control problems described above For the first type of control design problem (1 e end-pomt control), the proposed method ~s not apphcable and other optimization methods such as multtobjectwe dynamtc opttmtzatlon should be used (Butala et al . 1988)
2 Design of open-loop control strategy Let us consider a batch dynamic process represented by the following non-hnear modeling equation x = t(x, u)
(1)
x(0) = Xo
(2)
wnth mmal condltnons
where x IS an n-state vector and u an m-control vector (n>-m, m - l ) Suppose that we wish to find open-loop controls that would bnng certain components of x exactly to some fixed desired values 0 e set points) dunng the ennre
processmg pertod I e x,(O<--t<--tf)=X,d,
t=l,
,q(q<--n)
(3)
,q
(4)
or
This paper describes the development of the task level control scheme for free radical batch and semJbatch copolymenzat~on reactors Here, the concept of reverse feedback control technique is introduced to derwe the tmme-varylng monomer addmon and/or reaction temperature profiles which are reqmred to yield styrene-acrylommle copolymers of desired composltton and MW When one attempts to develop optimal open-loop control policies for batch or semtbatch copolymenzatton processes, one must first define the design objectives In most cases, the control of copolymer composmon, MW and MWD are of utmost importance Even after the ultimate process goals such as those described above have been tdenttfied, one may have to define more speofic control objectives For example, it may be desirable to control only the end-time properties. e copolymer properties at the end of batch or semJbatch reactor operations This would be a reasonable and practical goal since what one ts concerned about may be the final properties of the polymers obtained after the polymerization ts completed In this case, the design of open-loop control becomes an end-pomt control problem Moreover, the reactor operator does not have to be concerned about what ~s
x,=O.
O<--t<=h. t = l ,
where h t s the total process (batch) ume, x, the state variable being controlled, and X,d the desired target value of the state variable x, In batch or semibateh copolymenzatton processes, x, may be the bulk phase monomer eomposttton (or copolymer composmon) or copolymer MW The potential control variables are the reactor temperature and the monomer feed rate as a funetton of time The perfect control up may be found tf there exists u which satisfies equation (4) If such a n can be represented exphc~tly as a funcuon of state variables, i e a = V(x)
then, the
perfect open-loop
(5)
control tralectones, ,,p
(t,O<-t<-tf) wdl be detenmned completely by solving equaUons (I) and (5) In most pracUcal muaUons m batch or semlbatch copolymenzatJon process control such perfect controls (up) may be extremely difflcultor tmposslble to derive and express m usable forms due to the complexity of the process model A classlcalapproach to opumal control syntbests ts to use
Bnef Paper the necessary cond,tmns of Pontryagm's maximum pnncmple (Bryson and He, 1975, Denn, 1969), however, when the d,menston of the process model ms large and the process model is very complex (e g copolymenzat,on reactor model) it may be ,mpract~cal and computatlonally too ddticuit to derive opt,real control pohoes using the maximum pnnctple Moreover, when the Hamfltoman ~s a hnear functmn m control a, one faces a singular control problem which reqmres special solutmn techmques For ted-batch type (sere,batch) reactor operauons, such a singular control problem occurs frequently One of the alternative methods for the synthes,s of the opumal control strategy ~s to use approximate methods such as the control vector parametenzat,on techn,que (Ray, 1981) m which the parametenzat,on of the opUmal control is made by expanding the control variable(s) m a set of real functmns, (~,j), of the form m
u,(t) = ~ a,,~k,,(t)
(6a)
I=l or
m
u,(x) = ~ b,,tp,,(x)
(6b)
l=i
Then, any parameter optmmmzatmn techmques may be used to determine the optlmal set of coellic~ents a,j or b,~ Thins approx,matton method ms attract,re m that no adjomt equatmns need to be solved However, mt ts reqmred that the functmnal form of the opt,real control be prespeclfied and proper we,ghtmg factors chosen when more than one vanable are controlled Obvlously, th~s method needs a consmderable amount of computing time and deep physlcal insight rote the process behavmr being consldered Recently, computer alded des,gn (CAD) techniques have been successfullyapphed to these multmbjecuve and multlvanable open-loop copolymenzatlon reactor control problems using the method of feaslble d,rectmn algonthm (Butala et al, 1988) In what follows, we shall present a new and slmple method of des,gmng open-loop control pohoes for free radical copolymenzatton processes w,th dynamlc process models Let us firstconslder a feedback control system shown m Frog I Here, the vanables to be controlled (x,) are momtored on-hne The objective of thls feedback control system ,s to mmmm~ze the set point dewatlons, ~,= IX,d--X,J(X,d=fixed, O
x,~~
_x.±tx -~1
I
F~(~ I F,ctmous feedback control system
x,
919
denved In other words, the open-loop control pohcy ms deduced by reverting the transient response of the fict,tlous feedback control Th,s ,dea, inverting the transient feedback response, of obtaining the open-loop control pohcy has not been utilized for pract,cal copolymenzatlon reactor control problems to date Then our next questmon ,s how to obtain such feedback control profiles u~, (t, 0 ~ t "~tf) "a prmn' Thins can be accomphshed relat,vely easily by performing the dynamic simulation of the process control model Here, the feedback controller G c of the fictmous feedback control system ,s first designed unal the best closed-loop response ,s obta,ned Then, the trans,ent trajectones of the mampulated variables, uc, (0 <-t-< tf), are recorded Since such control u¢, Is what has been needed to obtain the good serve response of the fictmous feedback control system, it is now clear that the use of this control trajectory m the actual open-loop process should result m the same control performance as the fictltmus closed-loop system from which uc, was obtained In order to obtain the most satisfactory performance of feedback control, one may use any feedback control algorithms available There are many advantages of this ,nverse feedback control techmque (1) The open-loop control pohcy msdetermined d,rectly by solving the process model of fictmous closed-loop control system (2) Since the control pohcles are determined by the dynamic smmulatmn of the process model, very soph,sucated and thus perhaps more accurate process models can be used for the open-loop controller design (3) Although for this approach we,ghtmg factors are not needed exphotly for the control obleeUves, they may be used when more than one control vanable and advanced controllers are used (4) The effect of various operating conditmns such as m, tml reactor condmons and man,pulatlve vanable characteristics can be analyzed easily This mspart,cularly ,mportant when one wants to redesign open-loop control pohc, es frequently m order to produce the polymers of ddterent grades The factors wh,ch influence the quahty of the open-loop control pohcles determined by thts reverse feedback control techmque are O) the accuracy of the process model and (n) the goodness or the quahty of the feedback controllers used m the numerical s,mulaoon of the fict~tmus closed-loop control system The mteracttons between the feedback loops for mult,vanable systems may also affect the quahty of control pohCleS In the next sectmn, we shall use the proposed techmque to find open-loop control pohoes for the control of batch and sem,batch SAN copolymenzatmn reactors
3 Semtbatch copolvmertzauon reactor model The control object,ve here ,s to mamtam constant copolymer compos, tmn and constant polymer MW during the course of polymenzatlon by man,pulatmg the reactor variables such as monomer add,tmn rate and reactor temperature In what follows we shall illustrate the proposed design method through the numerical s,mulatmon of the solut,on copolymenzatmn process to produce SAN copolymers The solvent and m,tlator used are xylene and AIBN, respectively The reasons for selecting th,s system are SAN copolymer ,s commeroally ,mportant and opt,real control ms necessary if copolymer composmon has to be mamma,ned at points other than the azeotrop,c po,nt where the bulk phase compos,tmn 0 e monomer mole ratio) is the same as the copolymer composition and no composlt,on dnft occurs The kinetic model and the numerical values of kmet,c parameters are described m the Appendix The modehng equatmns for semJbatch free radical copolymenzatmn processes take the following form dM l u -~ = ~ ( M I f - M l ) - [(kpll + kt I1)P +
(kp2 i +
kf21)Q]M l
(7)
020
Brtef Paper
dM2 = u (M2f dt V
-
X 105 15
M2) -
[(kp22+ kf22)Q + (kp~2 + kn2)P]M2 (8) dl
MN 0 75 "
u
-.~ = -~ (If - I) - kd!
(9)
dV
d-7= u
where the temperature dependence of the rate constants are expressed In Arrhenms form Here, M1 and M2 are molar concentrations of monomer 1 (styrene) and monomer 2 (acrylomtnle), respectwely, I the mmator concentraUon, and u the volumetric monomer adchUon rate P and Q are the total concentratmns of live polymers having monomer M~ and monomer M2 as the end units, respectively Other parameters are defined m the Notaaon For the calculauon of polymer MWs, the following moment equauons for dead polymers are used d•
2
~.
1
+(kmMl+kfl2M2)P+(kfaT.M2+kt21MOQ---~u
(11)
I I l [ t l l l l
+ (ktc22Q + ktd22Q + ktcl2P+ ktdl2P + kf22M2+ k f21Mt)Q! --~u
[ t l t l l l r
04 ~°~
10
15
05
10
O0
O0
=(ktcllP + ktdtlP + ktct2Q+ ktd12Q+ kftlMt + kf12M2)P!
l
FI 06~
MN
2
-~f = (2ktcll + ktdll)P + (2ktc22 + kto22)Q + (ktcl2 + 2ktdl2)PQ
~
O0 08
(10)
I000
2000
3000
4000
TIME (MIh0
FIG 2 Isothermal batch copolymenzatmn (330 K, f~ = 0 25, 1o = 0 05 mol I - i )
(12)
d~td
- ~ = ( k,oH P + ktdH P + ktcI2Q + k,m2Q + kfH Ml + kf12M2)~ + (kt,22 Q + kto.2 Q + kt,12P + ktdt2P + kf22M2 + kf21M1)Q2
+ktcflPl +ktc22Ql 2ktcl2PlQ1--~ u
(13)
Notice that the reactor volume V vanes with ume due to the addmon of monomers The instantaneous copoiymer composmon is descnbed by the Mayo-Lewis equation
FI
=
ri ¢2 + $ rtq~2+ 2~ + r2
(14)
which can be held constant by m m n t a m m g the bulk phase monomer mole ratm (~) constant In equaUon (14), r~ and r2 represent reactivityratios The conversmn of monomer I (styrene) ~s defined as VoMto + I ~u(t)Mtf dt - VMt(t)
Xt =
-'o
r,
(15)
V°Mm + J0 u(t)M~fdt where Vo ~s the msual reaction volume, M m the mRzal monomer concentration m the reactor, Mtf the feed monomer concentrataon, and u(t) the monomer feed rate 4 Dtscussmn o f results In our numencal simulations the mmal solvent mole fraction is 0 25 and the Imtzal imtaator concentration ~s 0 05 tool !-1 For all the senubatch copolymenzatlon cases the volume of the total mmal reactants (monomers, solvent and mttmtor) is 1 hter, except for the batch copolymenzatlon case where the starting reactor volume is 3 hters The mmal monomer mole ratio (¢p) In the reactant mixture is 1 0 (or F1 -- 0 58) and the mmal reaction temperature for all cases is 60°C For all the copolymenzatton composRton control problems the feed includes 0 25 mole fractmn of solvent, 0 05moll -~ of mmator and monomers wRh a mole ratm (Mlf/M2f) of 1 5 4 1 Isothermal copolymenzanon To justify the need for proper control systems for the productmn of SAN
copolymers of desired composmon and MW, the numerical ssmulaUon of isothermal batch solutmn copolymenzatton has been conducted and the results are shown m Fig 2 Here, F~ IS the instantaneous styrene concentration in the copolymer, M N the number average molecular weight As expected, slgmficant vanaUons in composlUon and ~ occur as the monomer (styrene) conversion increases Such nonumformlty In polymer properties IS not acceptable for the practical apphcatlons of the polymers 4 2 Control o f copolymer composition tn Isothermal semzbatch reactor The inverse feedback control method descnbed in Section 2 has been used to synthesize the open-loop copolymer composition control pohcy m the ,sothermal copolymenzaUon process In our s,mulaaons, PI controllers are used to demonstrate the appheabdlty of the proposed method The PI controllers are tuned empmcally to obtam the best output response For our examples, constraints on the control variables are incorporated in the stmulatmn scheme Here, the copolymer composmon ,s controlled by adding the reactant mixture contmmng solvent, mmator and monomers to mmntam the constant bulk phase monomer mole ratio ( ¢ = 1 0, FI = 0 58) Since styrene is more reactive, the feed stream ,s ncher in styrene ( e l = 1 5) Thus, a t,me-varymg monomer add,tmn policy is sought to maintain constant instantaneous monomer mole ratm in the bulk phase The controller demgn procedure ts as follows (1) Assume that a perfect fictitious copolymer composmon sensor is avadable Then, perform the dynanuc stmulaUon of the ficUUous feedback control system and find the best feedback controls wh,ch rmmmtze the servo errors (1¢ - edl, 0 ~ t ~ If) Any constraints on the control variables of the form [~mm< Uc ----Umax can be incorporated into the slmulanon scheme The final batch time ts reached when the desired conversion ,s attmned or when the reactor is full (2) Record the time-varying trajectory of the resulting feedback control signal uc(t ) from t = 0 to tf (3) Implement the entire trajectory of u c obtmned from (2) into the actual open-loop control system where no on-hne composmon sensor is available In practice, the original control signal uc(t ) which resulted directly from the dynamic stmulaUon of the fictmous feedback control system may be nmsy due to controller actmn and impractical for the implementation into the actual
Bnef Paper
921 38O0
X 105 1 5
X 105 1 5
- 3500
M N 0 75 •
M N 0 75 -
~M N O0 08
3200 500
00 O8
200
'i F FI
.... -100
06.
FI
, , .F_1 / /
.
.
.
.
. 25
06
0 t.~
/.1 04 20
MN
,,,I
,TI
,,,I,
00 10
,,
z u~
f_~jl
15
$
-05
~ ~ , , [ ~ ~ , , I , , ~ , I .'r. ~ ~ ~
04 2O
MW
_
_
-05
MN
> Z
00 10
o3 re W
oo 10
00
O0 1000
O0
2O00
3000
00
4000
Tnvt~(MIN)
open-loop process Such a problem can be carcumvented by smootlung or filtenng the ongmal feedback signals and fitting them into the properly parametenzed forms F~gure 3 dlustrates the performance of the open-loop controller d e u g n e d by the proposed method Note that copolymer composmon control ts excellent Although the total amount of added monomers is not great, such composmon adlustment results m a stgmficant improvement of the copolymer composmon control However, note that the MW (MN) vanes considerably with the monomer converston X 105 1 5
3800 A
/°2
3500 p-
MN
' ~ ' ~ I ] ' ' ' i ' ' ' ' L ' ' ' '
oo 08
F1
06
3200
~
20
lO
~ f
//
u~
z
MN
O
15"
-05
8 10
i 00
i
1 l
] 1000
i
i
i
i
l
,
i
2000
2000 "r~E
FIo 3 Open-loop composmon control m an isothermal semlbatch reactor (333 K, f , = 0 25, Io = 0 05 tool 1-1, K~l = - 0 7 I s - ' , ¢n = 1000 0s)
.. o,, ~
1000
i
i
[ 3000
i
i"ii
O0
3000
4000
(MIN)
FIG 5 Open-loop ¢omposmon and MW control in a scmlbatch reactor ( 0 - < u l _ < 7 0 m l m m - I , 323 _< u2 < 363 K, f~=025, 10=005moll -I, Kcl= - 0 7 I s -I, Kc2 = 0 001 cal s -1 K -1, rll = r2 = 1000 0 s) Clearly, thts is because only one mampnlated vanable was used for this non-hnear multtvanable interacting system 4 3 Control of copolymer M W m batch reactor If the copolymer MW (MN) is the property parameter to be controlled, the reaction temperature wdl be a better control vanable than the monomer addmon rate In th~s case, the copolymenzauon reactor is operated m a batch mode and a time-varying temperature set point program ~s reqmred to obtain MN = 30 000 Figure 4 shows the performance of the batch reactor controller designed by the inverse feedback control method Note that the reactor temperature must be increased dunng the mmal reacuon period m order to lower the polymer MW near to the destred value Again, very good MW control is obtained with only slight offset The temperature profile shown m Fig 4 can be easily implemented into any process control computer Note, however, that this ttme the polymers of poor copolymer composmon are obtained 4 4 Control of copolymer composttlon and MW in sermbatch reactor Figures 3 and 4 clearly illustrate that both composmon and MW should be controlled by mampulatmg the monomer feed rate (u~) and the reactor temperature (u2) The target values of ~ = 1 0 (or F I = 0 58) and Mr~ = 30 000 are chosen The open-loop control pohcles for both composition and MW controls have been determined by the proposed method m which Pl controllers were used m both composmon control loop and MW (MN) control loop to obtain sausfactory fictmous feedback control Figure 5 ~hows that excellent open-loop control of composmon and MW has been obtained The slowly varymg monomer addmon rate and the temperature set point profiles shown m Fig 5 can easdy be implemented into the modem high-speed process control computers Notsce that MWD is also maintained at a constant level In implementing the transient response of u~ and u2 obtained from the slmulaUons of the fictmous feedback control system, the original feedback control signals were filtered and fit into the mnth-order polynomials
4000
TIME OvtIN)
FIG 4 O p e n - l o o p c o p o l y m e r M W c o n t r o l m a batch reactor (323 ~ u : s 363 K . A = 0 25. 1o = 0 05 tool I - l. Kc2 = 0 0 0 5 c a l s - I K -1, T n = 1000 0 s )
5 Concluding remarks In this paper, a new approxtmatlon method has been introduced for the synthesis of open-loop control pohcles for batch and semlbatch free radical copolymenzatlon processes
922
Brief Paper
m which both copolvmer composition and MW are controlled at thew target values during the entire course of polymerization The method is based on the discovery that the tlme-varwng trajectories of the mampulated variables obtained from the fictitious feedback control system are equwalent to the trajectories of the control variables or t~me-varymg set point programs reqmred for the control of an open-loop control system m which no on-l,ne sensors are avadable Our slmulatton results for the SAN copolymerlzatJon process indicate that this method ~s simple to use and very powerful to synthestze the open-loop control pohoes for the control of both copolymer composition and MW m batch and semlbatch reactors In particular, the proposed method will be very useful when one wants the rapid computation of monomer feed rate and temperature profles for the production of polymers having different properties The quahty of the open-loop control pohoes developed using the proposed method may depend upon the following factors (1) The quahty of process models The proposed method, hke any other open-loop control pohcy developed based on the process models cannot cope wtth unmodeled dtsturbances and modeling errors Therefore, it is ~mportant that the copolymenzauon kinetic model must be tested and vahdated through expenmental data Thts constraint apphes to any model-based open-loop opttmal control designs (2) The quahty of feedback controllers used m the vtmulattons When s~mple PI controllers are used as in our examples, the designer must find the best controller tuning parameters which y~eld the best feedback controls for the "fictmous" closed-loop system However, one can use any feedback control algorithms other than a conventional PID algorithm to obtain the best feedback control trajectortes For example, opttmal PID controllers or adaptwe controllers may be useful to obtain the tmproved performance of the "fictmous ' closed-loop control It may be poss~bte to observe different open-loop control pohctes which may yteld similar output responses with d~fferent sensRw~es to model/plant mismatch
(1988) Est~matton of the molecular weight distribution m batch polymenzatmn A 1 Ch E J , 34(8), 1341-1353 Tsoukas, A , M Tirrell and G Stephanapoulos (1982) Multlobjectwe dynamic optlmlzatton of semlbatch copolymenzatlon reactors Chem Engng Sct, 37, 1785-1795
Appendix
Kmettcscheme and moment equattons
The following kinetic model ts used to describe the homogeneous solution free radical copolymenzat~on of styrene wRh acrylomtrtle At high solvent volume fraction, the effect of diffusion controlled termmauon (gel effect) is not s~gmficant The penultimate effect is also assumed neghgtble (These assumpuons do not resmct the apphcabdlty of the proposed controller destgn techmque, however ) The numerical values of kinetic constants are hsted m Table A1
Intttatton 1"-" 2R
kll R + Mf-"*Plo kj2
R + M2-'-'Qo, Propagatton P.,,, + M1
enm+M2
kpll
kpl2
'P.+I., ,Qnm+l
kp21
Q,,m+Mt
~P,,+tm
Q,, ,,, + M2
kp22 ,Q.,,,+,
Combtnanon termination kt¢ll
P . . + P,.q
Acknowledgements--This
work was supported by the Systems Research Center at the UntversRy of Maryland, College Park The authors would hke to thank the anonymous reviewers whose comments helped clanfy some issues described m this paper
)M,, . . . . q
ktcl2 P. m + Q . q
'M. ....
q
ktc22 Q. m + Q, q
' Mn+r ra+q
References Bryson, A
E
and Y
C
Ho (1975)
Apphed Opumal
Control, 2nd ed Blatsdeil, Waltham Butala, D N , K Y Chol and M K H Fan (1988) Mult~objectwe dynamtc opttmtzatton of semzbatch free radical copolymenzatton process with interactive CAD tools Comput Chem Engng, 12(11), 1115-1127 Denn, M M (1969) Opumtzatton by Vartattonal Methods McGraw-Hall, New York Molau, G E (3965) Heterogeneous polymer systems--Ill Phase separation m styrene-acrylomtrfle copolymers Polym Lett , 3, 1007-1015 Papadopolou, S A and E D Gdles (1986) Continuous esumauon of the chain length dtstnbutmn m polymerization reactor considering ume dtscrete gel permeaUon chromatography measurements In K H Retchert and W Gelsler (Eds), Polymer Reactton Engineering, pp 243-260 Huthlg and Weph, Basel Ray, W H (1981) Advanced P~.ocess Control McGrawHill, New York Schork, F J and W H Ray (1983) On-hne measurements of surface tension and densRy w~th apphcat~on to emulsion polymerization J Appl Polym Set, 28, 407-430 Taylor, T W , V Gonzales and K F Jensen (1986) Modehng and control of molecular weight dtstnbutlon m methyl methacrylate polymerization In K H Relchert and W Gelseler (Eds), Polymer Reacuon Engineering, pp 263-274 Huthlg and Weph, Basel Taylor, T W . M F Elhs, V Gonzalez and K F Jensen
TABLE A1 NUMERICAL VALUES OF KINETIC PARAMETERS AND REACTION CONDITIONS FOR STYRENE-ACRYLONITRILE COPOLYMERIZATION
Initiator AIBN Monomer Styrene Acrylomtnle
Parameter
Preexponential factor
kd f
6 02 x 101~s -1 06
Act,vanon energy
(calg-1 m o l - i ) Reference
31 730 --
I tool -J s - l
kpl I k,. kfl !
1 06 X 10"~ 1 25 x 10'~ 2 31 x 106
kp2: kt22 kf22 rI r2 kfl2 kt2!
3 3 x 1012 6 93 x 106 2 56 6 67 x 10-5 30 x kfl I 5 X kfl I
et
3 0 x 10"~
5
x
7067 1677 12670 4100 5400 5837 1190 -4340 12 670 5837
kr~
Tsoukas et al (1982)
--
23 Reactor parameters 0 05 mol l - I
J's= 0 2 5 ,
tm~ = 6 h ,
Vm. x = 4 0 1 `
Io =
Brief Paper where
DtsproportIonatIon termmatton
P...+t'~
ktdll
fl
,M.,.+M,~
ktdl2
P."+Q.o
923
kpl. + kf,.
M,
The application of the pseudosteady state approxlmauon to live polymers leads to the followmg live polymer moment equations
,Mnm+M,.q
ktd22
Qnm+Qrq
'Mnm+Mrq
wlC,a, + a , y
Cham transfer
( 1 - tr0
P,, ,.,,+ M,
kill
~M. .. + Pio
Yr2 '
Q,= P,, ,,, + M 2
Q,,,,, + M,
Q,,,,, + M2
kfl2
kf21
k f22
1 on
the
end
/
-\-
yr2
(1 ---'~2)
(A7)
' M,, ,, + Qo,
w2Ctoq + °qY Q. + 2wloqP, rl
J'M,,,.,,,+ P1o
~
+ 2Witl'lY QI +wl(otlP r1
Similarly,
Q . ,., r e p r e s e n t s
+ £~'1'~ rl
Q)
P2 =
'M,,,,, + Qo,
(A8) (1 - 0q)
where Pn ,~ represents a growmg copolymer chain with n units of monomer 1 and m units of monomer 2, and monomer
)
w~.C2a2 + ~2 P2 + 2w2°r2 P,
]0"2
the
yr2
+ 2w.tr.Q, +w2( ~
growing copolymer chain with monomer 2 on the end M. ,~ denotes inactive (or dead) polymer The copolymer MW and MWD are computed by using three leadmg moments of the total number average copolymers The instantaneous kth moment is gwen by
" "
P+o:.O)
)'2
-
(A9)
Q2-
(1 - tr2)
where
,I.~= ~, ~ (nwl+mw2)kM,,,,,, n~l
k=0,1,2
(AI)
Cj = (kfllP + kt21Q)Mt/kpltM ,
m=,
where w, and w2 are the MWs of monomer 1 (styrene) and monomer 2 (acrylonimle), respectively The total number average chain length (Xn). the total weight average chain length (Xw) and the polydlspersity Index (PD) which is a measure of MWD broadening are expressed as XN = ~
(A2)
Xw = ).--~
(A3)
C2=
(kf22Q + kfl2P)M 2 kp22M2
kpll rl = kpl2'
7,~ = kp22 ~
lip21 kp21
Y=kpl 2
XN
(A4)
The total concentration of live polymers P is given by
p=((ktc, t+ktd,,)
2fka I
+ 2fl(ktcl2 + kid,2) + ]~2(k,c22 + kta22)
(All) (Al2) (A13)
k p I I MI
(A14)
OtI
PD = X__~w
(Al0)
[(kpll + ktlt)Mt + (kpl2 + ktl2)M 2 + (k,cN + k,du)P + (ktcl2 + ktd12)Q] kp22M2
),/2 (A5)
(A15) [(kp22 + kf22)M2 + (kp2 ! + kf21)M I + (ktc22 + ktd22)Q + (ktcl2 + ktal2)P]