composition control of batch copolymerization reactors

8/13/2019 Composition Control of Batch Copolymerization Reactors

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Chemical BnsiMuinp scialce vol. 36, pp. Ml-3l5P-n pnss td., 1981. Printed in CredtBrillin

COMPOSITION CONTROL OF BATCHCOPOLYMERIZATION REACTORS

MAlTHEW TIRRELL* and KEVIN GROMLEYDepartment of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis,MN 55455. U.S.A.

Accepted 16 u ne 1980)Abstrac -Composition control of free-radically initiated batch copolymerization reactor by temperature manipula-tion is investigated through a program of calculation and experiment. The work of Ray and Gall is extended toexplicit calculation of temperature vs time policies which eliminate or minimize composition drift. Both continuousand discrete, oiecewise-constant oolicies have been computed. The former are disjoint policies which can bc found.simply by manipulating the monomer mass balances; the tatter are found through Ii ion of the discrete versionof the maximum principle. Calculations are compared to experiments done on the system styrene-acrylonitrile. It isdemonstrated that implementation of some of Ihe computed policies is feasible and quite beneficial. The time discretepolicies with smalt numbers of stages (less than -six) perform much more poorly than the continuous policiesindicating that the copolymerization reactor is much more sensitive to deviations from optimal y than batch reactorsoperating with other reaction schemes. Thus, effective implementation of the lype of control studied here wil lnecessitate use of controllers capable of closely tracking a continuous trajectory.

INTRODUCUONSimultaneous polymerization of two or more monomers isa powerful and commercially important technique forcreating polymeric materials with tailor-made proper-ties. It affords the synthetic chemist new avenues ofmacromolecular architecture and the chemical engineerwho must produce such molecules on a large scalechallenges unencountered in the design of any other sortof reactor. Many copolymer products are produced involumes small enough so that batch production is dic-tated. The dynamics of a batch reactor may interferewith production of a copolymer with uniform propertiesat high conversion. Of particular interest here is thewell-known copolymer composition drift phenomenonresulting from the fact that both monomers (in themulticomponent case, all monomers) do not react withthe growing copolymer chain at the same rate. Thus, onemonomer becomes preferentially depleted in the batchand, consequently, also the copolymer formed at anyinstant becomes progressively more depleted in thatmonomer, a situation not unlike that occurring in batchdistillation. One obvious remedy is to convert to semi-batch operation, introducing a flowrate of the morerapidly reacting monomer to nullify the drift. This mayinvolve difficult mixing operations if the polymerizingmass is very viscous.Ray and Gall[l] proposed the alternative of tem-perature control, pointing out the fact that in cases wherethe activation energies of the (four) propagation reac-tions had the proper relative values there exist tem-perature vs time policies which, in principle, eliminatedrift, without converting to semibatch operation. Theydelineated the necessary conditions on the activationenergies. The purpose of the present work is to explorethe possibilities of temperature control for real copoly-merization reactors with free radical initiation. in parti-cular, we have calculated the actual temperature vs timepolicies necessary to eliminate drift in the styrene-acry-

*Author to whom correspondence should be addressed.

lonitrile comonomer system. As we shall see, calculationof these continuous T(t) policies does not requireemploying variational calculus as the policies are dis-joint. We were also interested in studying T t)policies which were discrete in time, that is, piecewiseconstant, with a small number, N, of constant tem-perature levels and switching times between them selec-ted in some optimal fashion. Exploration of these dis-crete policies had two objectives. First, these policiesmay in some cases be simpler or more practical toimplement. Second, the difference in reactor perfor-mance between the continuous and discrete policiesaffords insight into sensitivity to deviations from opti-mality. In particular we were interested in how manysteps N need be implemented before reactor perfor-mance closely approximates the true optimum. Cal-culation of these discrete policies is, of course, a trueoptimization problem. As we shall see, the resultsobtained for the copolymerization composition controlproblem are qualitatively somewhat different than forother batch reactor optimal temperature results[2].Finally, experiments were done to assess the efficacy inpractice of these policies.

Cond i t i o n s fo r feasi b i l i t y o f temper a tu r e con t r o lOur discussion of free-radical copolymer compositionbehavior will begin with the so-called copoly-

merization eqn (I):

F, = rtfr2 tf,firtft2 + 2ftf2 + r2f2*which was originally derived more or less simultaneouslyby several groups of workers[?-51 by writing down thehatch reactor mass balances on the two monomer con-centrations and combining them thus: F, =(ddM,ldt)l[dM,/dt + dM,ldt]. Equation (1) is obtainedunder certain assumptions which have since been provenreasonable many times over[6], such as the pseudo-

CES vol. 36. NO. 2-l 367


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368 M. TIRRELL nd K. GROMLEYsteady-state of the radical population and that only theultimate unit on the growing radical effects its reactivity.Thus F, is the mole fraction monomer 1 being added tothe binary copolymer and f, is the mole fractionmonomer I in the unreacted monomer mixture. Theparameters r , and rz are known as reactivity ratios. Theyare defined in terms of the four propagation rate con-stants, kij, such that, I; = kii/kij, where kir represents therate constant for the reaction of a growing polymer chainof type i with a monomer of type j. In general. in a batchreactor both F, and f, are functions of time. The idealobjective of any copolymer composition control schemeis to maintain the instantaneous copolymer compositionF, constant at some desired value F:. Ray and Gall[l]pointed out that if a constant value of F, = F: isassumed, eqn (1) may be rewritten:

r, = acrz+ a2 (2)wherea,=(&)(Y)

anda, = _ (;;Y)(~).

If the reaction rate constants kij have exponential tem-perature dependence then the reactivity ratios willdepend on temperature, thus:

rl = rlo exp [(- AEJR) (+ - $-)I

rz = rm exp [ (- h&/R) (+ - &)]

(3)

(4)where A& = E: - Ez (i, i = 1,2). Eliminating tem-perature between eqns (3) and (4) gives:

r, = pr2 (5)where p = r,,J(r& and n = AE,IAE,. If a constant F:solution to the pair of eqns (2) and (5) exists, thentemperature control of composition is possible. The fol-lowing conditions are necessary for their to be a real

This was the contribution of Ray and Gall[l].t Inorder to assess the engineering feasibility of temperaturecontrol of composition we must yet calculate the T(t)functions which make constant F : possible.

COMPUTATION OF OPTIMAL TEMPERATURE PROGRAhlSI. Continuous policies

Given a kinetic model for the copolymerization reac-tion, the optimal temperature control policy becomesdisjoint[7]. Our computational scheme utilized eqn (7)derived first by Walling[8] for the time rate of change ofmonomer mole fraction in free radical copolymerization:

dfl= [(rl + r2 - 2)f1 + (3 - 2rz - rl )f12 + (6 - lVlluldt IW2[ YI - 24 + Y2)flZ + 2 4 - y2)fc+ Y211z(7)

where q is the rate of the initiation reaction, tl, = kJ andx,, x2, y,, y, and 4 are parameters of the propagation andtermination reactions:

and

The complete background and assumptions used inarriving at this equation are not appropriate to discusshere at length but are available in any elementary bookon polymerization[9]. Among the most importantassumptions are the quasi-steady-state approximationson radical populations and again the assumption thatonly the ultimate unit on the growing chain affects reac-tivity. We have made no attempt to examine the quan-titative accuracy of eqn (7) over a wide range of con-ditions. In fact, that is the subject of some importantcurrent research. We did find that it gave reasonablygood quantitative fits to our kinetic data.

We may also rearrange eqn (I) to give f, explicitly(noting that f, cannot be negative)

f, = 11 2FT(l- Ml + {[2Ft(l- r*) - I]- 4FtZRz(r, + rz 2))2[Ff (rl +rz-2)- r, + l] (8)positive solution to eqns (2) and (5): Equations (7) and (8) have the following form:

n


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and finally;dT-& = u(T t)[V(T, 11-1

where V= aV/aT. This, therefore, is the differentialequation which we must solve to derive the optimaltemperature program. This equation was solved[ lo]num erically using a Runge -Kutta integration with secondorder iteration to accomm odate the possibility of initiatordepletion and consequent progressive drop in ur. Wepostpone discussion of parameter values used and ofresults of these compu tations until after we describe thediscrete optimization problem.II. LXscrete poli cies

Composition control of batch copolymerization reactors 369where R,, is the rate of polymerization in terms of

(9) monomer disappearance. The rate of polymerizationexpression we have used is Wallings[8 1. It is subject tothe same limitations as those discussed in connectionwith eqn (7). This function has the desired weight pro-perty, but in any comp utational scheme leads to a trivialresult of no reaction, that is, zero duration of each stage.If no polymer is produced, no drift can occur. Therefore,the final element we need to add is a specification of theamou nt of product to be produced. Specifying the finalreaction conversion as a constraint is an option. This,however, makes for a compu tationally more difficultproblem since the last state control vector canno t bemanipu lated freely to achieve the optimization but mustalso meet the constraint.

In calculating the time-discrete profiles, the optimalcontrol policies are no longer disjoint as we can nolonger specify at very instant the objective F:. Somedegree of suboptimality is thus unavoida ble. Since thesystem under consideration is discrete in time only not amultistage, multiunit process) the strong version of thediscrete maxim um principle is applicableI I, 121 . Thus,maxim ization of the Ham iltonian with respect to thecontrol variable is necessary and sufficient to determinethe optimum of the objective function.

We chose rather to incorporate the amou nt of polymerproduced into the objective function in the followingmanner:

(max)O = 2, I [A - Ft - ~? f))~]R,, f) t tu,O+ IL Q(12)

Regarding our choice of objective function, it isdesired to maintain the instantaneous copolymer com-position F,(t) as close as possible to some specifiedcomposition F: by using N constant temperature levels,manipulating the temperature T, and duration t, of eachstage. Th is, of course, is not the same as bringing thefinal cumu lative composition FIN as close as possible toFT. This latter result may be accomplished by the simplecontrol action of choosing an initial batch comp ositionsuch that F,(O ) is different than F T and F, t) driftsthrough F: to a point where positive deviations fromF: balance negative deviations. This is not the desiredresult since it arises from a compositionally hetero-geneous mixture of polymer molecules which may haveundesirable properties as a final product[l31. (One mustfways bear in mind in polymerization reactor design thatthe true reactor performance measures are product qual-ities, usually qua lities such as strength, processability,etc. and not more refined, abstract and quantifiablemeasures such as F,.) Minimizing the instantaneouscomposition deviation will, in general, produce a moredesirable product.

The parameter A is a weighting param eter on the in-tegrated rate of monom er consumption divided by thetotal initial monom er concentration, that is, it is aweighting parameter on conversion. For the presentprobIem. it can be shown that this procedure is entirelyequivalent to constraining final conversion. For thepresent problem, it can bc shown that this procedure isentirely equivalent to constraining final conversion tosome v alue and optimizing 02 above. In that alternative,A would appear as a Lagrang e multiplier on final con-version. The advantag e gained by the choice of eqn (12)as the objective function is one of comp utational sim-plicity. The disadvantage is that it is not possible todetermine, prior to calculation, what final conversion willbe obtained for a particular choice of weight parameterA. Howev er, A can be manipu lated at will to produce anydesired final conversion. In this formulation we see expl-icitly from eqn 12) that a price will be paid in com-position drift for every increase in A (increase in con-version).

Equation (12) may be rewritten[l2]:

One possible candidate for the objective function thenis:[FT - F,tt)]R,(t) dr

(min)O* = n , 1. [F t - F,(t)J2 dt. (10)This function considers composition, only. It is actuallydesired to minimize the amount of polymer that hascomposition not equal to m. Thus, the difference [fl-F, t)]* should be weighted by the number of polymermolecules corresponding to that difference. An improvedfunction then is:

The last term is a function of the final state vector only.To apply the discrete maxim um principle we formulatethe Ham iltonian for the Nth stage:

..-. ..Y

H=J(x+u)+z .f(X_ ) (14)where x represents the vector of state variablesMl,. Mz, I) for the nth stage, II is the vector of_control Variables CT,, t,) for the nth stage and Z is the(min)O Z= 2 I (Ft-F1(t)12R P(t)dt (11)__. n


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370 M. TIRRELL nd K. GROMLEVvector of adjoint variables (aH+/ax) for the nth stage.The function f is defined by the following set of equa-tions:

x=f(x_,U) n=l,2 ,.., N (15)with initial conditions:

Equation (15) represents the integration of the differen-tial equations for each state vector component over anisothermal interval. Finally, J(x-.u) represents thefunctions:

I .4-J zz oLx_,., [Ft - F, Wlz% t) dtM, Mzo (16)where the notation for the lower limit of integration:O[x_, II], is meant to represent initial conditions forthe nth stage. The adjoint vector for the final stage isgiven by

(17)where G(x~)=A[~-(M~~+M~~)/~~~+~~~)]. Thecondition for optimality is:

$0 n=1,2,.. .,N. (18)For any given values of x0*, F:, N and A, solution ofthis system of equations gives us the optimum piecewiseconstant temperatures, T,,, and the duration of eachtemperature interval, L. These equations were solvednumerically by the successive approximation techniqueoutlined by Ray and Szekely[ 141. The procedure fol-lowed was: (1) Guess a series of control vectors, u,,n=l,2,..., N. (2) Solve eqn (15) forward in time for x0to give each xN. (3) Use eqn (14) solving backward intime to give each H. (4) Solve concurrently, z =dH+/ax, backward in time to give the adjoint vari-ables, using eqn (17) to give zN. (5) Correct uin by:

/_ aH\

(6) Go to step 2 and repeat until convergence is obtained.Sufficiently poor initial guesses can lead to unphysicalnegative T,, and t.. Given that we have the continuousoptimal profiles in hand, our initial guesses were usuallygood.III. Parameter values or alculations

The specific system which we have chosen to study indetail through computation and experiment is the styrene(I) acrylonitrile (2) copolymerization system. Due to its

commercial importance, all of the necessary kineticparameters are available from the literature with some-what more precision than is available for most systems.In particular, the activation energies for the variouspropagation reactions have been determined [t5]. Theysatisfy the conditions imposed by eqn (6). In this casen < 0. Values of all parameters used are given in Table 1.No adjustments of parameters were made to fit any data,although good agreement was obtained between cal-culated and experimental composition.

ExFlUuMENTALThe experimental system chosen to test the com-

putational results was the solution (Xylene. solvent[Mallinckrodt]) copolymerization of styrene [Aldrich] in-itiated by 2,2-azobisisobutyronitrile (AIBN) [Aldrich].Solution copolymerization was used in these experimentssince our modelling takes no account of the well-knownautoacceleration in rate occurring in polymerizing mediawith high polymer concentration. Presence of a diluentpostpones the onset of this effect, extending the expec-ted validity of our computations to higher conversion ofmonomer to polymer. Xylene was selected as solventbecause of its relatively low vapor pressure over thetemperature range of our experiments as well as its lowchain transfer constant[l6]. The monomers as receivedwere washed with base to remove inhibitors, dried, thendistilled at reduced pressure in a dried, deoxygenatednitrogen atmosphere through a 50cm Vigreaux columninto a sealed dry receiver. The xylene solvent was dis-tilled in the same manner. The purified monomers werestored at 5C until use which was always within oneweek. AIBN was purified by recrystallization frommethanol.

A 5OOml kettle-type glass reactor was used for thepolymerization reactions. The reactor was equipped witha paddle-type agitator with shaft seal, an expanded scalethermometer with O.lC precision, a thermocouplenitrogen inlet tube and a sample outlet tube. The heatsupply was an electrical heating mantle. The heatingcurrent was controlled by a simple proportional con-troller (based on an API Instruments 434-817lRl tem-perature controller) which sent a current to the mantleproportional to the difference in the signal from thereactor thermocouple and a set point. Experienceestablished the sets points necessary to produce thedesired temperatures. All tracking control was open loopwith the operator forcing the reactor as nearly as pos-sible along the desired trajectory via set point adjust-ment. The fact that these were solution polymerizationsmade the reactor somewhat easier to control since theheat generation per unit volume was reduced. All reac-tants were introduced into the thoroughly purged reactorunder a blanket of dry, deoxygenated nitrogen. Initiator,in the proper amount, was added as a solution in xyleneafter the other reactor contents had reached the desiredinitial temperature. All polymerizing media were 20% byvolume total monomer initially. Further experimentaldetails are available elsewhere [lo].As the reaction was proceeding, samples were takenperiodically and analyzed for conversion (gravimetric-


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Composition control of batch copolymerization reactors 371ally, on isolating the polymer from a known volume ofreactant mixture) and cumulative copolymer com-position, FIT, (by IR spectroscopy with a Perkin-Elmer283 spectrophotometer, utilizing the 2240 cm? CNstretching for acrylonitrife and the 16lOcm- phenylgroup skeletal mode for sturene). Using this technique,FIT could be established to within 20.01 with 95%confidence.Due to the tendency of acrylonitrile-rich copolymersto precipitate from the reaction mixture at fairly lowconversion, our experiments were largely restricted tostyrene-rich compositions. This is, however, the morecommercially-interesting region[l3].

RESULTSI. Continuous policies

As a first step in computing the temperature controlpolicies, we used eqn (I), and the necessary parametersfrom Table 1 to compute F, vs fl curves at varioustemperatures. The results are shown in Fig. I. An opti-mal temperature policy may be visualized in this figure asa horizontal line at the level of the desired Ff. To obtaina preliminary indication of the accuracy of theparameters of temperature dependence of r, and rZ, wecompared our calculations with the data of Johnson[lS].He studied the effect of polymerization temperature at afixed starting monomer concentration cf, = 0.619). heldmonomer conversion low to minimize drift, and

Table 1. Parmeter values used in computationsVALGE -EmsricE

6.02 * 1015 s-1 1631.73 kcal-ml -1 16

2.79 lOlO hlol-l-.-l10.617 kcal-rml -1

1616

(kpna) Ap22 1.56 ml* &mol-l-g-l 16E 15.45 kcal-mol -1P22 16

frlb T0 336.0 OK 17

=10 0.455 17

AE1 1.40 kEd-lhYl-1 17

(~2) Tcl 338.0 *lC 17

=20 0.033 17

*=2 -2.30 kC~1-lllO1-l 17

(km) *t11 4.839. ml3 h.a-1-s-l 16

%Ll 8.70 kcal-ml -1 16

(ktzz) At22 7.762' 1Ol3 b-m~,l-~-s -1 16Et22 9.236

$ 15. 6

1 0.60 9'2 1.0 9

h 0.6*E 0.7i=zE?

0.6

5 0.50

0.00.0 0.2 0.4 0.6 0.8 1.0MONOMER FRACTION, f,

Fig. I. Copolymer composition vs monomer composition for thestyrene-acrylonitrile system at several temperatures. Dark linesillustrate typical optimal temperature policies in this composition

space.

measured the composition F,) of the resultingcopolymer. The results are tabulated in Table 2 alongwith our predicted values. We see that the agreement isreasonable. It is possible that some unavoidable drift didoccur in the highest temperature experiment of Johnston.The discrepancy is in the right direction for this drift tobe at least partly responsible. These data reconfirm thefact that the S-AN reactivity ratios are temperaturedependent and that the literature parameter values wehave to account for this will be satisfactory for ourpurposes.

Temperature vs time policies typical for the styrene-acrylonitrile system, computed from eqn (9). are shownin Fii. 2. Various values of FT, 1, and TO wereexamined. We see that increasing temperature policiesoccur for f, below the azeotropic composition anddecreasing temperature policies occur above. It is alsoseen in Fig. 2 that the effect of increasing the initialtemperature or initiator concentration is to compress thetime scale of the T t) programs for any particular value ofFt. This was to be expected since the conversion ofmonomer to polymer increases with initiator concen-tration and temperature. Therefore, stronger compensat-ing temperature control action must be taken earlier. Forthe increasing temperature policies, this effect is stillmore pronounced due to the increased rate of poly-merization at higher temperature. Decreasing tem-perature policies mitigate this effect and thus are alwaysTable 2. Comparison of experimental [IS] and predicted styrene-acrylonitrile copolymer compositions using parameters of

Table I

5 40 0 40.965 38.2 37.1

130 37.0 34.0


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372 M. TIRXELL and K. GROMLEY

OL..,,...0 2 4 6 8 IO 12 14 16 16 20TIME hrs)

140120100

T(C) 606040200. * / 8 3 * B( 0 2 4 6 6 IO 12 14 I6 16 20

TIME (hrs)

T(C) 80604020 LL. I,, I... I I(, ..1 II0 2 4 6 6 10 I2 I4 16 I8 20TIME (hd

Fig. 2. Optimal temperature policies for maintaining constantF:. Values of Ff indicated on curves.

more slowly varying in time but, of course, also produceless polymer per unit time.

Figure 2 shows that, while profiles for certain values ofFt, appear to be very difficult to.implement, others arewell within the bounds of practicality, requiring no morethan +20 temperature changes over the time span oftwo hours. The question of how much polymer isproduced after any specified reaction time, as well asfurther questions on practical implementation, are dis-cussed in more detail below in connection with theexperimental results.II. Di screte polici es

Some representative results of calculations of discretepolicies are shown in Figs. 3 and 4. one for a decreasingT(r) policy and one for an increasing T(t) policy. Thevalues of A necessary to achieve the indicated con-versions are given in the captions. Both the discrete andcontinuous policies, corresponding to the same F: andf,, are shown, as well as the instantaneous copolymercomposition vs reaction conversion. The optimal policiesare compared in each case to isothermal operation. Thecumulative drift is proportional to the area betweeninstantaneous composition curve and the horizontal F:line. Notice that all temperature control policies have asa practical limit of use the point where one monomercompletely disappears.

INSTANTANEOUS CO-POLYMEkCOMPOSITION.60

_______________________ ^______ --_isothermal60 10 0 5 1 0TIME hrs)

Fig. 3. Comparison of continuous and discrete optimal tem-perature trajectories. Increasing T policy (A = 0.0035).

I.0 INSTANTANEOUS M-WLYMER-f

COMPOSITIONt.84

F, .82

t

,i/

.80

.76

FpO 78F;= 0.85

CONVERSIONT(t) ,isothermal_-_-_-_-_-_-_-

TIME (hrdFig. 4. Comparison of continuous and discrete optimal tem-perature trajectories. Decreasing T policy (.A= 0.0013).

Several points must be made here. First, the optimaldiscrete policy for any given set of conditions is notnecessarily the one which minimizes the area between itand the continuous T(t) curve. Rather, referring back toFig. I, it is the T(f) function which minimizes the areabetween the discrete path in F, -f, space and thedesired horizontal line. Copolymer composition is arather unusual selectivity problem, since eachmacromolecule is the result of large set of sequentialreactions and both desired and undesired speciesare incorporated irrevocably into the same productmolecule. Second, the discrete policies with between 3and 6 steps do much more poorly in meeting the objec-


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Composition control of batch copolymerization reactors 373tive F: than their continuous counterparts. This is trueof all of the extensive set of examp les we have e xaminedhere and elsewhere[lO]. This is also true irrespective ofwhether one examines cum ulative instead of instantan-eous drift or mak es the compa rison at equal reaction timeinstead of equa l conversion. Figure 4 shows how pro-gressing from isothermal to 3-stage discrete to 6-stagediscrete operation steadily improves performance butdoes not approach the true optimal performance veryclosely. In other words, the copolym erization reactor isvery sensitive to deviations from the optimal trajectory,much m ore so than has been observed previously in mostother batch reactor trajectory optimization problems.For examp le, C rescitelli and Nicoletti[2] comp ared theperformance of the continuou s with the piecewise con-stant tempe rature profiles for the maxim ization of theyield of B in the sequence of first order reactionsA 2 Bk2 C where the activation energies have therelationship _E,* < E,*. They found computationally thatvirtually no difference could be detected between thetrue optimal performance and the 3- or 6-stage discreteperformanc e. A key difference in their work from thecase considered here, besides the more complex sequen-tial nature of copolym erization is that their objective wasa function of the final state only. Thus, temp orary devia-tions from the continuou s optimal path could compen-sate one another to some extent. Tha t is not possible inthe copolym erization case.III. xperimental results

Several ex perimental tests of continuou s and discreteoptimal temperature profiles calculated as above weremade . Some illustrative results are shown in Figs. 5 and6. In Fig. 5 two isothermal policies are comp ared withtwo calculated policies. Th e lower part of the figureshows both the intended and the measured T(t). Theisothermal control was reasonably good. So also, sur-

TPC)

CUMULATIVE CO-POLYMER-84 - COMPOSIT ION

--- ol ulofed

TIME bra)Fig. 5. Experimen tal implem entation and comparisonthermal, discrete and continuou s optim al policies. IO=

f O 0.85.of iso-:o.wsM,

TIME hro)Fig. 6. Experimen tal implem entation and comp arison o f iso-thermal, discrete and continuous optima l policies. Ir, = O.O4M,

fi = 0.5.

prisingly, was the implem entation of the continuou slydecreasing T(t) policy. The approximation achieved tothe three-stage discrete policy was poor, missing onestage entirely. The measu red copolym er compo sitionover the same time period for the four policies is shownin the upper part of the figure. We see clearly thatsuperior p erformance is obtained from the closest ap-proxim ation to the continuou s optimal policy. Theattempt at implem entation of the three-stage discretepolicy also provides definite improvem ent over eitherisothermal policy but, as expected from our earlier cal-culations, is clearly inferior to the continuou s optimalcurve. In Fig. 6 results of one of the few increasingtemperatu re policies we were able to examine (due to thepolymer precipitation problem at high AN content) areillustrated. In this case we were able to follow thediscrete T t) function reasonably closely and the resul-tant improvem ent in performan ce (less drift) is illustratedin the upper part of the figure. Figures 5 and 6 make thecomparisons between the various policies at equal reac-tion time but it is of rather more interest to comp are theperformances at equal conversions. Table 3 summarizessome of those results. We see that in every case thenonisotherm al policies produc e less drift than the iso-thermal policies when the comp arison is made a t the timewhen the reaction conversions are equal. Clearly, whenincreasing T policies are possible they are most attrac-tive since they not only minimize compo sition drift butalso increase conversion per unit time. Of course, onemust also be concerned with the effect of temperature onmolecular weight distribution.

The objective of this paper has been to investigatetheoretically and experimentally the technical feasibilityof temperature control of batch copolymerization com-position. Several major points have been demonstrated.Calculations of nonisothermal continuous and piecewise-


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374 M. TIRRELL nd K. CROMLEYTable 3. Experimental results on copolymer composition drift control

Initial FinalnDnomer =z-lymernixture Find CoPnpcsition

f? F1 POliCV Convereion 2rift

D (2)

D-50 0.588

0.85 0.784

0 85 0.780

-72 -540 * .015

-72 -590.E8 .588

75 isotherm -39-65

85' isotherm -39-65

3 stage apt. -39-65

oontinuousapproximation

-39

.EOO t .OlO

.838-799.829_ 796.8Q7

8.2

0.30.0

2.06.91.95.71.52.9

1.0

D = 100 FT1 = final measured cumulative copolymercomposition

constant optimal temperature schedules lead us to theconclusion that feasible T(t) policies do exist for thestyrene-acrylonitrile system and that, while piecewise-constant nonisothermal operation affords improvementover isothermal operation, the full benefit of temperatureprogramming is only obtained by the closest possibleapproach to to the continuous optimal profile. This mar-ked sensitivity to deviations from optimality arisesfrom: (a) the necessity to minimize the instantaneouscomposition drift and (b) the unique highly non-linearsystem of consecutive reactions which is involved incopolymerization. Experiments have been describedwhich support these analytical conclusions.Given that the reactivity ratios in a copolymerizationsystem are temperature dependent and satisfy the con-ditions r,, r2 < 0 as the styrene-acrylonitrile system does,it is clear that, in principle, an alternative equally optimalpolicy to the ones we have calculated exists for eliminat-ing composition drift exists. That is, an optimal isother-mal policy where the constant reaction temperature ischosen so that the desired Ff becomes the (un-stable) azeotropic composition for the specifiedmonomer composition f,. Figure 1 shows that in prac-tice, however, very little flexibility is provided by thatapproach, compared to the continuous optimal policieswhich we have calculated, since only rather smallchanges in the azeotropic composition are produced overa broad temperature range. On the other hand, the time-varying temperature policies presented can be used inmany more practical cases as we have demonstrated. Wehave chosen to compare the results of our optimal poli-cies to the results of isothermal policies with ap-proximately the same starting temperatures. This is avalid comparison as we are here primarily interested indemonstrating the efficacy of temperature control.

The sensitivity of composition to temperature in thissystem is rather low on an absolute basis so that prac-

tically we expect this type of policy to be most usefulwhere rather small (-3-490) cumulative drifts areexpected but intolerable. These conclusions will need tobe altered for comonomer pairs with different tem-perature sensitivity.

In present and future work, we are looking experi-mentally at the effects of these time-varying temperatureprograms on copolymer molecular weight and sequencedistribution, and making direct experimental comparisonof temperature vs monomer addition control in an effortto determine which is most desirable or if both can andshould be used in concert to produce copolymers oftailor-made macromolecular architecture. It is alsonecessary to look more closely at the problem of im-plementing these policies on realistic reactors, where thethermal dynamics will be more sluggish and thus we canonly control heat input and not temperature directly aswe have done here. For this reason we are now in-corporating energy balances into the set of equationsdescribing the system and working toward modellingsituations where reaction heat generation is important.Acknowledgement-Financial assistance from the NationalScience Foundation, Grant ENG784555S is gratefully ac-knowledged.

NOIAIIONpreexponential factormole fraction monomer 1 units in copolymermole fraction monomer 1 in polymerizing massdesired constant F,Hamiltonianfree radical initiator concentrationpropagation rate constant for addition of monomerof type j to radical of type itermination rate constant for bimolecular reaction

of radicals of type i and j


9/9

Comparison control of batch copolymerization reactors 375M, concentration (molar) of monomer 1 (styrene in this REFEltEN EsL?X~~lll~~ 111 Rav W. H. and Gall C. E.. Macromolecules I%9 2 425.-------r--1Mz concentration (molar) of monomer 2 (acrylonitrilein this example)N number of piecewise constant temperature inter-

valsr l reactivity ratio (&, , , k,, I z)rz reactivity ratio (k,zz/k,zl)R, overall rate of polymerization (d( Ml + &)/dl)or rate of initiation reactionI adjoint variable (Lagrange multiplier)

i2j Cr&citelli S. and Nicoleti i., Chem. Engng Sci. 197328 463.[3] Mayo F. R. and Lewis F. M.. J. Am. Chem. See. 1944 661594.141Wall F. T., 1. Am. Chem. Sot . 1944 4 20 .[Sl Alfrey T. and Goldfinger G., J. Chum. Phy s. 1944 2 205 322.[6] L.enz R. W., Organic Chemistry o Synthetic H igh Polymers.Wiley-Interscience, New York 1967.[7] Denn M. M., I ntroduction to Optimization by Vari ationalMethods. McGraw-Hil l, New York 1969.[8] Walling C., J. Am. Cbe&. Sot. 1949 71 1930.[9] Odian G., Frinciples of Pofymerization McGraw-Hill, NewYork 1970.

Subscripts and Superscripts [ IO ]Gromley K. M., MS. Thesis, University of Minnesota 1979.[II] Halkin H., SIAMJ. Control I 6 4 90.refer to monomers or radicals of type i or j(i, j = [I21 Westerberg A. and Stephanopoulos G., I nd. Engng Chem.1 7, Fundls 1974 13 231.i, in

Pt

refers to nth pie-wise constant temperature in- [I31 Szaba T. T. and Nauman E. E., A.I .Ch.E. J. 1969 IS 575.terval [I41 Ray W. H. and Szekely J., Process Optimization. Wiley-Interscience, New York 1973.refers to last piecewise constant temperature in- [IS] Goldfinger G. and Steidiltz M., J. PolymerSci . 19483 786.terval [16] Brandrup J. and Immergut E. H. (Bds.), Polymer Handbook.

refers to initial conditions Wiley-Interscience, New York 1975.refers to propagation [17] Mayo F. R. and Walling C., Chem. Reu. 195046 191.refers to termination [18] Johnston N. W., J. Macromol. Sci.-Reu. Macromol . Chem.1976Cl4 215.

composition control of batch copolymerization reactors

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