Application of an Adaptive Controller to Molecular Weight Distribution Control in a Batch Polymerization Process

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APPLICA TION OF AN ADAPTIVE CONTROLLER TO MOLECULAR WEIGHT DISTRIBUTION CONTROL IN A BATCH POLYMERIZATION PROCESS T. Takamatsu*, S. Shioya**, Y. Okada* and M. Uchiyama* *Dt/,lIr/ llltll/ of C' ht lll im { E lIgilll'nillg, h yO /1I [·lIi;'l'ni/y. h lO/u 6tH;,

**DI'/)(I r/1II1' 1I/

of Fall/I'I//II/iol/ TnlllllJ{ugr. U"dw

[ ·l/h'I'nl/r.

./11/)(111 ./11/)(11/

Sui/II 565,

Abstract . This paper proposes a new control strategy of the react or tempe rature and initiator concentration in order to get a final product of polyme r which has a prescr~bed molecular weight distr ibution (MWD) in a free - radical polymerization batch reactor . First , profiles of instantaneous average chain length and polydispersity are obtained so as to get the desired MWD . Next , time profiles of reactor temperature and initiator concentration a r e determined from the profiles of instantaneous average chain length and polydispersity based on the mathematical model of the reactor . This t wo- steps method desc r ibed above does not inc l ude any ite r ative calculation . Thus , the solution can be obtained mo r e easily by this method than direct searching methods in which iterative calculations i s r equi r ed for obtaining time profiles of temperature . As the final step , the required time profile of the reactor temperature should be practically realized . Since the batch polymerization process is highly nonlinear and time - dependent , t he time pr ofile of reactor temperature was realized by the Adaptive Internal Model Controller (A I MC) developed by the authors . Application of adaptive controller . Batch polymerization process. Keywords . Molecular weight distribution control . Reactor temperature control . T~acking control .

Also , little work has been done on the development of control str ategies for broadening MWD in f r ee -radical polymerization reaction . Ar nold , Johnson and Ramsay (1980) and Couso , Alassia and Meira (1985) proposed the method to produce polymers of prescribed MWD in living anionic polymerization batch reactor by perturbing the ratio of monomer to initiator concentrati on . This method makes good use of the property that the instantaneous MWD in living anionic polymerization is Poisson- distributed . Unfo rtunately , this method can not be used in free - radica l polymerization , because in this reaction the instantaneous MWD does not have Poisson- distributi on .

INTRODUCTION The properties of the polymer product , such as the mechanica l properties and the characteristics in molding , have strong correlation with the molecular weight distr ibution (MWD) of the polymer . Martin , Nunes and Johnson (1982) found that thermal properties , stress-strain properties , impact resistance , strength and hardness of films of polymethyl methacrylate and polystyrene were all improved by narrowing MWD. It is also generally said that the polymer of long chain length gives superior mechanical properties to polymer products but has insufficient molding characteri stics . Then the molding characteristics can be improved by blending short chain length polymer into this long chain length polymer , while the good mechanical characteristics are kept . That is , the broader MWD can be obtained by this blending . Therefor e , the development of the meth odology for adjusting MWD during the reaction to a suitable one according to its use is also desired , especially in producing high quality polymers .

The two - steps calculation method based on the instantaneous MWD is proposed here . This method can determine the time profiles of reacto r temperature and initiator concentration , in a general free - radical polymerization batch reactor . By controlling temperature and initiator concentration alo qg the profiles calculated , the final polymer product with prescribed MVm , that is , the average chain length and the polyd ispersity , is produced at the preestablished monomer conversion rate . Next , the required time profile of the reactor temperatur e should be practically realized . The adaptive controller , Adaptive Internal Mode l Controller (AI MC ) developed by the authors is applied to a temperature control probl em so that the reactor temperature tracks a desired trajectory decided above. AIMC is a combined control system of Internal Model Controller (IMC) and Model Reference Adaptive System (MRAS) . IMC was developed by Garcia and Mor a r i (1982) . Parameters of the internal model used in AIMC are recursively estimated utilizing an explicit identification scheme of MRAS . For the tracking control problem considered here , the system becomes a time - varying and non- linear one . AIMC will be powerful for such a systen . Numer i cal simulations show that the react or temperature can successfully track the desired profile by the AIMC .

Several papers , Hoffman , Schreiber and Ros en (1964) , Tadmor and Biesenberger (1965) , Nishimura and Yokoyama (1968) , Hicks , Mohan and Ray (1969) , Osakada and Fan (1970) , Sachs , Lee and Biesenberger (1973) and Louie and Soong (1985) , have reported the control of MWD in batch free - radical polymerization reactors . These studies have mainly focused on the optimization problem of how to obtain the narrowest MWD. "'hen this optimization problem is solved using opt imization techniques such as the calculus of variations , maximum principle and simplex direct searching method , a great a mount of iterative calculation is required . Therefore , it is difficult to get optimal solutions whil e coping with changes of reaction conditions . Thus , instead of complex methods based on optimizat ion techniques , development of a simple calculation method is desired .

22 7

T. TakalllatslI 1'1 Ill.

MWD CONTROL IN A BATCH REACTOR Here, the number average degree of polymerization (P ) and the polydispersity (HI) which co rr esponds N to a variance of MVID are used as indices of MWD . A MWD with zer o HI is the narrowest distribution of the polymer , that is , the MWD is monodisperse . The larger HI is, the broader MWD the polymer has . According t o the practical demand of the high quality polymer product , polymers with various types of MWD are required . The precise control of MWD in a batch reactor will play more important role than before . The MWD control problems cons i de red here are ; i) t o find an operating cond i tion by which the desired product of polymers with the prespecified cumulative average chain length and the polydispersity are produced in a free-radical batc h polyme rization reactor at the final conversion rate and ii) to r ealize the operating condition by manipulating the practically available variables in a closed- l oop control system .

OPEN-LOOP CONTROL FOR MWD IN A FREE- RADICAL BATCH POLYMERIZATION REACTOR Mathematical Description of the Open- loop Control for the MWD j According to the reaction mechanism of a free -radical polymerization shown in the Appendix A, the i-th moment of the dead polymer ~i is defined as follows , /';

00

•

~. (t) = L (n)1[D (t) ]V(t)

i=0 , 1 ,2, ...

1 n=O n where [Dn(t)] is the concentr ation of the dead

(1 )

Two-steps Method f or MWD Co nt r ol j The number average degr ee of polyme ri zat i on Pn and polydispersity hi of instantane ous pol yme r are defined as foll ows, /';

Pn= ( d~1 /dt )/ (d ~O/ dt ) ~ /';

(9)

hi= ( d~O/ dt )( d ~2/dt)/(dW1/dt)

2

(10)

From Eqs.(9 ) and ( 10) , d W1/d~ 0= Pn

-

~

dW2 / d ~O =hipn

( 11) 2

(12)

In a free - radical polymeri zati on wh ic h has no cha i n transfer react ion t o polyme r, a dead polyme r does not disappea r and the final MWD of t he polymer in the batch polyme ri za t i on r ea c t io n becomes a cumulative MWD of ins t an t ane ous dead polyme r dur ing the operating pe ri od . Whe n the c ont r ol object ive shown in Eq . (8 ) is atta i ned , t he f oll owing rela ti onship between ~O f * ' ~1f * ' ~2f* ' Pn and h i ,

~ 1f*=f~Of: d~ 0 Pn 0

(13)

W * =J~Of*h '~ 2d~ 2f 0 1Pn 0

( 14)

should be he l d . The r ef ore , th e ori ginal probl em can be reduced to the probl em of finding Pn ( ~ O ) a nd

polymer of length n at t ime t , and V(t) is the reactant volume at time t . The number average degree of polymerization PN(t ) and polydispersity f HI(t ) of final product of the polymer can be f represented by ~O(tf) ' ~1 (t ) and ~2(tf) as f follows ,

h i ( ~O) whi ch sa t isfy Eqs . (13) and (14) fo r the

firs t s t ep , a nd f or t he s e cond ste p , find i ng open- loo p ope ra ti ng cond i t i on of r eactor t empera ture TR( t ) and i nitiato r co ncentration

PN(tf)=~ 1 ( tf)/~O(tf)

(2 )

[ I ( t ) ) wh ic h sat is f y Pn(~O) and hi (W ) obtai ned at O the fir st s tep . This method f or calculation of TR( t) and [ I(t)) is ne wly pr oposed here and cal l ed

HI ( tf ) =~0 (tf ) ~2(tf )/~~ (tf)

(J )

t he t wo- steps method for M10JD con t r ol .

The final conversion X(t ) is given by the f f ollowing definition, X( t )2 (V( 0)[ M(0) )-V ( t ) [M (t ) ] )/( V(0 ) [M (0) ) ) ( 4 ) f f f where [M ( t ) ] is the concent ration of the mon omer at time t . When appl ying the quasi-steady state as s umpti on , ~ 1 ( t ) is given as follows , f ~ 1 (tf)=V (O )[M( O) ) - V(tf ) [M (t ) ] (5) f Theref ore , t he final conversion i s rewritten as, X ( tf )= ~ 1 ( tf )/( V(O)[ M(O) ]) (6) Then , t he 8ontr ol objective of maki ng th e fi nal convers ion X( t ) , number average degre e of f polymerization PN ( t ) and polydispersity HI (t ) f f equal to t he desired values X* ( t ) , PN*( t ) and f f HI* (t f ) as shown in Eq . (7 ) can be r educ ed t o the control objective of making the final val ues of t he moment s of dead polymer ~O ' ~1 and ~ 2 equal t o the desired values

~ O f* ' ~1f*

and

~2f *

As the firs t step , Pn ( U ) a nd h i (W ) which satisfy o O Eq s . (13) an d ( 14 ) a r e obta i ned through the fo ll owing simple geometr ic a p pro a c~ . I t is possibl e that Pn and hi can b~ operated i nde pende nt l y by man i pulating re acto r tempe r at ure TR a nd i ni t i a tor concen t rat i on [I (t ) ]. Then f or s i mpl i ci ty of the ca l cula t i on , hi ( ~n) is fixed here J

hi . Unde r t ti s conditi on ,

at constant value

r ( U ~) n

mus t satisfy the foll owing Sqs . (1 5) and (1 6),

Pn

is, t he area s of

\..

T~at

and Pn~ f ro ~ J t o

satisfy Eqs . (1 5 ) an d (16) . U

"

w *=f Of~ dU 0

1f

;J

21'

'n

C

U * O1'~Pn

*/;:-:-J 1l l -

o

>I

2~ .

u~ : ...

One of t he s olutions of P,., ( i.': ) is a re c:angda r

as shown in

t ype (Type 1 ) a s shown in Fi g . ; .

Eq. (8) .

th is pat t ern are (7)

Pn

. ~. e ::a r a ~. e:e rs

of

ef the : irst s~ep { ~n 1) ' ~n of

the second s t e p CPn 2) and ).; 0 a t cna"g :.r.g pcir.t fr om t he first s t ep t o the s econd (U ) ' I t is ve ry Ox simpl e t o decid e Pn 1 ' Pn 2 and wa x whi ch s atisfy

229

\I ()it:c ul ar W eig h t Distr ibutio n COllt rol

"'9

"'0

~cf~' r---------------------'

~cf~.~

Io

Io

be

be

wz"'" 19W

wz"'"

~~

_ E11.2_____ .

Ci~o

wzg

W-e

>
~~N

...,:U

0

III

z

__________________~

III Z

0 L.__.L-....L._.L-....... ' iJ...:o"" x.....................

ci

Fig. 1.

0.20

0.40

0 L-...L__..L...--'__~__L-L..L---I ci

0.20

0.60

0.40

0.60

0-TH MOMENT x lO-t OF POLYMER ~O(mol) Fig . 2 . Pattern ~f Pn(W ) of Type 2 O

0-TH MOMENT x lO·t OF POLYMER }JO(mol) Pattern of Pn(W O) of Type 1

MWD CONTROL OF FDEE- RADICAL

POLYMERIZATION OF STYRENE

Eqs.(15) and (16) . In general , there are an infinite number of patterns n which satisfy

P

Eqs.(15) and (16) .

An example of

Pn

Experimental Apparatus ;

is shown in

Fig . 2 . In Fig.2 , a mixed type of O- th and first order polynomial (Type 2) is shown.

Next, TR(t) and [I(t)] which give the profiles of Pn(W O) and hi(W ) decided above should be obtained O based on the mathematical model . The differential equations of the mo ments WO ' W and W in batch 1 2 polymerization reactor are derived from a moment method . dWo/dt=V[(ktm[M]+kts[SJ) 'o+(ktc/2+ktd) A02]

(17)

dW,Idt=V[ (ktm[M]+kts[S] )+(ktc+k td ) AOA1]

(18)

dW2/dt=V[ (ktm[M]+kts[S] )+(k tc +k td ) AOA2 +k tc \ 2] (19) where the i - th moment A. of polymer radical P. · is l

J

As an example , a MWD control problem in styrene polymerization batch reactor is considered . A small lab- scale batch reactor was used in order to study styrene polymerization initiated by azobis isobutyronitrile (AIBN) with toluene as solvent . A stirred , 500- mL , three - necked distillation flask was used as a reactor. To prevent oxygen inhibition , a nitrogen blanket was applied across the top of the reactor. A reflux condenser and an agitat or shaft seal were used to minimize the loss of monomer and solvent from the reactor . A 4.7- L water bath was used to provide a well- characterized heat t r ansfer medium . Both reactor and bath temperatures were monito r ed with a data -logger interfaced to a microcomputer. The temperatures were scanned every 30 sec , and they are controlled to desi r ed values by manipulating the period of opening an electrovalve which can control the coolant flow . Conversion was determined by gas chromatography and MWD was analyzed by gel permeation chromatography (GPC) .

defined as follows , /::,

00

•

A. (t) = z: n l [P • ( t) ] i=O , 1 , 2 , • • • (20) l n=O n where [Pn ' (t)] is the concentration of polymer

11

radical of length n at time t . Substituting Eqs.(17) , (18) and (19) into Eqs.(9) and (10) and using the assumption of the quaSi-steady state,

Pn = ( 1 +2.E ) ( _2-Z 2_ ) hi= (H ) (1+~+Z) 2 1 +E are derived , where parameters E and Z are , /::,(ktm[M]+kts[S])+(ktc+ktd)AO E= k [M]

'-----0

(21 ) (22)

io E3

2

reactor heater ~ reflux condenser 4 . reactor stirrer 5. wa te r ba th 6 . thermocouple

(23)

p

Z~

k A

tc 0 (24) (ktm[MJ+kts[S])+(ktc+ktd)AO From the desired values of Pn(W ) and hi(W O) ' the O required values of E and Z are given by Eqs.(21) and (22) . Applying the assumption of the quasi-steady state to O-th moment AO of radical

Fig. 3 .

ploymer , AO is given as ,

Table 1. (25)

From values of E and Z obtained above , reactor temperature TR and initiator concentration [IJ as the open- loop ope rating condition for MWD control are calculated by Eqs . ( 23 ) , (24 ) , (25) and system equations of the solvent [SI and monomer [M] . And the two - steps method requires definitely less iterative calculations than the other methods .

Expt. No . 1 2 3

7. 8. 9. 10 . 11 .

data- logger electrovalve water tank microcomputer nitrogen flo'.

Experimental apparatus

Various r eactor conditions

T R( oc) 80 70 90

[I(O) ] (mg)

[M(O) J (mll

[S(O)] (mll

1.478 0 . 4434 0.4434

179 179 179

46 49 49

T.

I. 0

r-------------------..

1

section. Because the polymerization batch reactor is a highly nonlinear and time - variant process , and causes difficult cont r ol problems related to the removal of reaction heat . Then Adaptive Internal Model Cont ro ll er (AIMC) which has been developed by the authors (1985) is applied to this temperatur e control problem .

-

Expt 1 Expt 2 Expt 3

H U)

gj > z

o

'-'

o

REACTOR TEMP ERATURE CONTROL FOR t·n·ID CotlT!lOL BY AIMC

o

o o.S

,d.

As the successive pr ocedu re after the open- loop control of t-rHD , the required time profile of the reactor temperature should be pract i cally reali:ed . An adaptive controller will be suitable for the control system for realizin g the pattern of reactor temperature TR which is decided in the last

o " -

5

T ;t kl lll~I" lI 1'/

REACTION TIME (HR) Fig . 4 .

Conve rsi on versus reaction

ti~e

Ma thematica l Modeling ; Some experiments were performed to test several models . The init ial conditions of batch poly~erization are given in Table 1 . One of the comparisons is presented in Fig . 4 , which shows excellent agreement of date of convers i on- versus time with simulated values us i ng a mathematical model proposed by Hui and Hamielec (1968) . Their mathematical model used here is listed in Appendix

B. Open-loop MWD Control of Styrene Polyme r ization ; The desired final conversion , number average degree of polymerization and polydispersity are 0. 45 , 500 and 1. 7 , respectively. This t11tID control problem is solved using above - mentioned two - steps method . In styrene polymerization , instantaneous polydispersity hi is almost equal to 1 . 5 independently of reaction conditions . Therefore MWD control can be attained by manipu l ating only reactor temperature T • When the patterns of Pn(U ) are R O Type 1 and 2 , each pattern of TR(t) is shown in Fig . 5 and 6 . These two types are typical ones which can be practically realized . As shown in these figures , a drastic change of causes big n and sometimes discontinuous changes in reactor temperatu r e T . R

P

u 0

a...

1

(27)

n

(28) m where u(s) , y(s) are plant input , output , respectively . It is assumed that ; 1)orders of the polynomials A(s) and 5(s) are already known respectively , 2)A(s) and B(s) are coprime , 3)plant i s a stable and min i mum phase system , 4)plant input u(s) is sufficiently rich . The following transfe r function is obtained from the block diagram in Fig . 7 assuming that the controller is equal to the inverse of the model , that is , GG - 1F y (s) =

m 1\"

1 r( s) +

1+(G - G )G - F m

m

"

Gc(s)=G~

1- F

1 (s) ,

(29)

" 1 d (s )

1+(G- G )G - F f!1

m

where the process is represented by a transfer functioR G(s) wi th setpoint r(s) and disturbance d(s ) . Gm(s) is a plant mcdel which is adaptively adjusted and it is called int e rnal model . F(s) is an adjustable lo w pass fil te r and is int r od uced to make the controlle r Gc ( s)F (s) proper . F(s) is

0

0

a...

O'l

:::E W

t- O

a:

0

0

D CO tU

a: w a:

0

0

r-

2

m m-1 B(s)=bOs +b 1 s +• • • +b

0

0 O'l

D CO tU

a: w a:

where A(s)=sn+ a sn - 1+a sn- 2+ ••• +a

u 0

:::E W t- O

a:

Adaptive Internal Model Co-traIler ; A plant model which is adaptively adjusted using the input and output data , is used for an internal model in AIMC . If a first or second order lag system is adopted as an interna l model , AIMC becomes PI or PID type feedback controller with auto - tuning controller parameters . Now , the outline of AIMC will be described as fol lows . The following SISO continuous system is considered , B(s) yts) = A(s)u(s) (26)

2.0

LLD

6.0

8.0

0

0

r-

2.0

TIME (HR) ::'6 0 c .

~::..:-:-.e

~p (:)

1.1. 0 6.0 TIME (HR)

;:,:"~: :' 2.-2

.) : :-e9.c: : r

:"c:" -=-::pe ;. ( ?:'g.2 )

8.0

:e~?ere.t'J.!'e

'doleclIlar \\'cighl Di slrilllllioll COll lrnl r equired to have 1. 0 for a static gain , lim F(s)=1

Filter (30)

f---.-~Y(s)

s ~O

The controlle r output , that is , plant input u(s) is gi ve n as , &- 1 (s)F(s) u(s) = m _ (S) (r(s) - y(s)) 1 F &- 1 (s)F(s) 1\ m 1\ 1 (r( s) - d (s)) 1+(C(S) - Cm(s))C~ (s)F(s)

(31)

Fig . 7 .

Fr om Eq . (29) , the following equation y(s)=F(s)r(s)+(1 - F(s))d(s) is derived if "Cm(s)=C(s) .

Then , F(s) can be

synthesized so that the controlled system has the desi r ed dynamic and static characteristics of the r esponses y(s) with respect to setpoint r(s) and disturbance d(s) . Fo r the identification of parameters , the well known equation error method (Laundau , 1979) us i ng model refer ence adaptive technique is adopted . The identification procedure can be summarized in Appendix C.

u 0

0

0... ::E

0)

The AIt1C is used for controlling the reactor temperature so as to follow the time profile of r eactor temperature (Fig . 6) which was decided from a pattern of Pn(~O) of Type 2 (Fig . 2) . One of the simulation results fo r this problem by AIMC is shown in Fig . 8 . The controlled variable , reactor temperature TR is desired to follow the curve shown i n Fi g . 6 by manipulating the jacket temperatur e TW ' As a disturbance , the initial reactor temperature TR(O) is r aised by +10 ° c f r om the desired value . The solid line shows the reactor temperature controlled by AIMC . Fi g. 8 shows that AIMC works successfully against this disturbance . Fi nally , the HWD control is attained by the method proposed he r e and the final polymer product which has such a MWD as shown in Table 2 is obtained in the reactor under whi ch reactor temperature is controlled by AIMC .

conCLUSION First , a new two - steps calculation procedure of the reactor temperature and initiator concentration which give the desired average chain length and polydispersity of final product of polymer at the desi r ed final conversion i n polymerization batch reactor , has been proposed here . The controlled variables of MWD were represented in this paper by the average chain length and polydispe r sity . The appr oach employed here can als0 be used for control problems taking into account of the controlled variables related to higher moments , for example , skewness and kurt os is. Next , it was shown that AIMC could control the reactor te~perature so as to follow the curve determined by the strategy mentioned above very well. It should be stressed that the calculation procedure and the control system proposed here can be utilizerl f or any othe r batch reactor used for industrial free -radic9.1

~F

REACTOR TEMP. TA

W

t- O

a::::

0

0

CO

t-

U

a::::

Simulation Results for Temperatur e Cont r ol Problem ;

DESIRED VRLUE

l!l

0

a: w

polymerizations .

AIMC structure

(32)

0 0

r-

2.0

LLD

TIME

6.0

8.0

(HR)

Fig . 8 .

Reac t or temperature cont r ol by AI MC

Tabl e 2 .

Prope rties of fi na l polymer product

Real value Desi r ed value

X( t ) f

PN ( t r )

HI(t ) f

0 . 447 0 .4 5

497 . 5 500 . 0

1.746 1.70

REFERENCES Arnold , K., A. F. Johnson and J . Ramsay (1980) . Proc . IFAC PRPP 4 Automation , Chent , Bel gi um , pp . 359- 367 . Couso , D. A., L. M. Alassia and C. R. Meira ( 1985) . J . Appl . Po l ym. Sci ., 30 , pp . 3249- 3265 . Carc i a , C. E. and M. Morari (1982) . Ind . Eng. Chem . Pr ocess Des . Dev ., 21 , pp. 308- 323 . Hicks , J ., A. Mohan and ~H . Ray (1969) . Can . J . Chem . Eng ., 47 , pp . 590- 597 . Hoffma n , R. F., S . Sch r eiber and C. Rosen (1964) . Ind . Eng . CheR! ., 56 , pp . 51 . Hui , A. W., A. E. H ~ ielec (1968) . J . Po l ymer Sci . C, 25 , pp . 167- 189 . Landa~ I . D. (1979) . Ada ptive Control , Mar cel Dekke r Inc ., New York . Louie , B. R. and D. S . Soong (1985) . J . Appl. Po l ym. Sci ., 30 , pp . 3707 - 3749 . Harti n;J .R~ R. W. Nunes and J .Y. Johnson (1982) . Polym . Eng . Sci ., 22 , pp . 205- 228 . Nishimura , H. and F. Yokoyama (1968) . Ka gaku Kogaku , 32 , pp . 601 - 607. Osakada , K. and L. T. Fan (1970) . J . App!. Polym . ScL , ~ , pp . 3065 . Sachs , M. E., S . Lee a nd J . A. Biesenbe r ger (1973) . Chem . En g. Sci ., 28 , pp . 241 . Tadmor , Z. a nd J . A. Bi esenberger (1966 ) . lEC Fundamentals , 5 , pp . 336 . Takamatsu , T., S . Shioya and Y. Okada (1985) . Proe . IFAC Workshop Adaptive Control of Chemical Pr ocesses , Fr ankfurt , FRC , pp . 120- 125 .

T. TakamatslI

Appendix A.

1'1

al.

where 5 , TR, PF and PN are solvent concentration, reactor temperature , polymer weight fraction and number-average chain length, r espectively. All reaction constants and initiator efficiency are

The reaction mechanisms of a free-radical polymerization initiated by an initiator are represented generally as follows, k

I ~ R'

(i) Initiator decomposition;

(A - 1)

( B-8)

(A- 2)

( B-9 )

k.

R'+M ~ P 1

(ii) Initiation;

(B-10)

k

Pj'+M~Pj+1'

(iii) Propagation;

(A- 3)

(B-11 ) ( B-1 2)

(iv) Chain transfer reaction ; P. '+M

ktm

J

k P. '+ 5 J

k

P. · +P. ·

(v) Terminat i on;

1

ts td

J

(B-13)

(A- 4)

) D/P 1 ' ) D +P ' j 1

(A-5)

) D. +D.

(A- 6)

1

J

k tc Pi '+ P j ' • Di +j (A- 7) Initiator I is decomposed into an initiator radical R·. The initiator radical R' reacts with monomer M, a nd a radical P ' of length 1 is generated . 1 Monomer M is added onto the end of the radical P ', j forming a new radical P + The chains of radical j 1 P ' are transfered to monomer M and so l vent 5 , j forming dead polymers Dj and radicals of P •• 1 Termination by disproportionation and recombination generate dead polymer Di'

Appendix B.

2

d~0/dt=ktcA OV/2 ,

where pj

The plant model (internal model )

is assumed to be given as , m

1\

i~ObiP

m-i

y(t) = - -n- - - . u(t)

(C-2)

" n-l. p +i~1aiP n

n

That is , U decreases with monomer convers ion X. And gel effect is accounted for by allowing the termination constant k and initiator efficiency f tc to vary with viscosity of the reacting medi um . l og(f/ f i)= - 0 . 133log(1+~) (B- 5) 2

- 0 . 0777[log(1 + ~)]

(B- 6)

where fi and k

are initial values of f and k tc ' tci ~ is bulk viscosity which is assumed to be given as , l og~=17 .66- 0 . 311log(1+S) - 7 .72logTR

2

(B- 7)

(C-4)

Yf(t)=D(p)y(t) Also the filter output uf(t) is given as . 1 u ( t)= DTPl u (t) f

(C - 5)

And the generalized equation error E(t) is gi ven as . n

n "

n-i

m"

m-i

(C-6) E(t)=(p +i~1aiP )Yf(t) - (iEObi P )uf(t) The parameter identification law is given as . " a . (E . t)= 1

p

polymer , heat capacity of reaction mixture , heat of polymerization reaction , overall hea t transfer coefficient , heat transfer area and jacket temperature , respectively. The ove r all heat transfer coefficient U is assumed to be writ ten by , (B- 4) U=UO- 300exp(1-1/X)

- 10 . 23 1og(1 - P )-1 1.82[log(1 - P )] F F - 11 . 22[log(1 - PF)]3+0 . 8391ogPN

1

( B-1)

where D , c , 6H , U, A and TW are density of

Ik t Cl. )= - O . 133 1og (1+~)

(C- 3)

Then the filter output Yf(t) is given as.

d~1/dt=ktcAOA1V ,

where AO' A1 and A2 are the O- th , 1st and 2nd moments of the growing polymer chains. By using the quasi - steady state assumption , these are expressed by , 1/2 AO=(2fk d [I]/k tc ) , A1=(ktcAO+kp[M])/ktc ' (B- 2) A =A +2k [M]A /k /A 2 1 P 1 tc 0 The heat balance Dpcpd (VT R) /dt=V6H (k p [M ] AO) - UA(TR- T\,) (B- 3)

c

.

D(p)=i~OCiP l

d(V[M])/dt=- kp[M]AOV ,

d~/dt=ktc (AOA2 + A1 A1 )V

log(k t

(C-1)

For getting an effective iterative identified value , a state- variabl e filter D(s) is introduced into the identification scheme as usual,

The material balances d(V[I])/dt=-kd[I]V,

p

Appendix C.

t

J0

.

k

ai

- 1 " . (0) E(T)pnYf(T)dT+a

wh ere ka . >0

1

i=1

(C-7)

.2.···. n

1

~ . (E . t)=Jtk b E(T)pm-iuf(T)dT+~ . (0) 0

1

i

(C- 8)

1

where kb>O

i=1 . 2 .···. m

1

When the plant input u(t) is sufficiently rich. that is . u(t) contains more than (n +m+ 1 )/2 distinct frequencies . the iden tified parameter values coinside with the t rue one (Landau. 1979). Then it is pro ved that plant output y(t) coinsides with the desired command response F(p)r(t) . That is . lim [y(t) - F(p)r(t)] = 0 (C-9) t~

x

can be proved .