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Adaptive Control of the Molecular Weight-Distribution in Living Anionic Polymerization Processes
PLASTICS : COMPUTERS AND MODELLING (2) Copyright
© (FAC
PRP 4 Automation, Gh~nt , B~lgium 1980
ADAPTIVE CONTROL OF THE MOLECULAR WEIGHT-DISTRIBUTION IN LIVING ANIONIC POLYMERIZATION PROCESSES K. Arnold 1, A. F. Johnson2 and J. Ramsay Postgraduate Schools of Polymer Science and Control Engineering, University of Bradford, Bradford BD7 lDP, West Yorkshire, England
Abstract. A model reference control strategy is discussed for the production of monodisperse polymers and polymers of any pre-defined molecular weight distribution fram 'living' anionic polymerizations carried out in a smallscale tubular reactor system. Initially it was assumed that when a fixed (monomer) /(i nitiator ) ratio was introduced into the reactor that it was possible to produce monodisperse polymer with a predictable average molecular weight and further that by perturbing the (monomer)/(initiator) ratio in a prescribed monomer with time that an integral product of any molecular weight distribution (MWD) could be achieved. Experimental results indicate that the limitations of this simple approach were (i) the (initiator) was severely altered by adventitious side reactions, (ii) axial dispersion in the reactor and a finite reaction initiation rate resulted in an undesirable broadening effect on the MWD. To overcome these limitations a spectrophotometer has been incorporated into an input adaptive loop to control the livi ng chain end concentration in the reactor and the axial dispersion effects on the MWD as deduced from gel permeation chromatography measurements have been used to adjust the input feed rate of the reactants; both modifications are currently being tested and are outlined here. In principle perturbing the feed to a polymerization reactor can be applied to any polymer ization process to alter the MWD envelope of the product but the efficacy of this approach is sensitive to the mechanism of the polymerization reaction and the reactor configuration. Keywords. Feed-forward control; anionic polymerization process; molecular weight distribution control; model reference control; gel permeation chromatography; tubular reactor; adaptive loop control.
reacti on is much faster than the propagation rate, (b) propagation occurs exclusively by the addition of monomer to the liv ing polymer chain ends , (c) there are no terminations , chain transfer or other secondary reactions, (d) that the reaction is carried out in a perfectly mixed batch reactor or an ideal plug-flow tubular reactor (Szwarc, 1968). When all these conditions pertain it follows that the average molecular weight of the po lymer will be given by [M] / [11 and MjM" ~ 1. 0 , where [M] and [lJ are the re ac tor n feed concentrations of monomer and initiator respectively and M and M are the weight and number ave~ageWmo~ecu~ar weights of the product. The M and M may be defined in the usual way a~ shownnin Eqns. (3) and (4).
INTRODUCTION The measured molecular weight distribution of any synthetic polymer is a c omp lex function of the chemistry of the polymerization process, the polymerization reacto r configuration employed for the synthesis and the mode of operation and control of the reactor system. Unless care is taken, the observed molecular weight distributi on may also be influenced by the technique employed for its measurement. For the case of 'living' polymerization reactions which involve initiation (1) and propagation reacti ons (2) only k. M + I ~ IMl IM
n
k + M - L IM n+l
(1)
n
=
2,3 .••
00
( 2) M
w
a Poisson molecular weight distribution will be observed if the following requirements are fulfilled; (a) the rate of the initiation
A (2)
A (1)
LN .M. 2 ~
~
LN.M. ~
~
LW.M. ~
1 20n leave of absence from R. and D. Centre, Chemoautomatyka, Warsaw, Poland. To whom correspondence should be addressed.
359
~
LW.
~
LW.M. ~
~
(3)
360
K. Arnold, A. F. J o hnson and J. Ramsay
M
n
~) A (0)
k
EN .M. ~
~
EN.
~
Termination I + X (killing)
(4)
Ln . M .
~
~
where A(O) , A( l) and A(2) are the zero, first, and second moments of the density distribution of molecular weights (Fig. 1) and N, M, Wand n are the number, molecular weight , weight fraction and number fracti ons respectively of the ith pol~e£ chain (see Figs. 2 and 3). The ratio M /Mn is a convenient measure of the bread~h of the density distribution curve and is frequently represented as Dn' the dispersity index.
LE
The basis of our approach has been to take 'living' anionic polymerizations which are thought to conform to the conditions (a), (b) and (c) mentioned above and to conduct the polymerizations in a tubular reactor taking the appropriate steps to ensure , as far as possible, that ideal plug flow conditions are achieved in the reactor. Under steady- state operating conditions mono disperse polymers (D ~ 1.0) should be produced. Provided thi~ is the case and that monodisperse material of any molecular weight can be produced by altering the feed ratio [M] / [I ) then the reactions may serve to produce polymers of any molecular weight distri bution. To achieve this latter objective the monomer and initiator feeds to the reactor have to be subjected to periodic oscillations in such a way as t o give a sequence of steadystate operating conditions each of which gives rise to a known instantaneous molecular weigh~ When polymerization is allowed t o go to completion, the product collected over a period of time should have a molecular weight distribution which is quantitatively related to the input feed conditions of the reactor. The instantaneous average molecular weight can be readily seen to be related to the reactor feeds. If the reacti on mechanism is more fully expressed by Eqns. ( I ) to (I V) , k. Initiation I + M ~ LEl k Propagation LEn + M ---R...;,
LEn+l
(ii)
(iii)
NA
+ X
NA
(IV)
when steps (iii) and (IV) do not take place the reaction may be described as ' living'. Provided k . » k and b o th are large , when no killing~occur~
[I] o
(5)
and R
We have been examining the behaviour of living polymerizations in tubular reactors for the production of mono-disperse polymers , but, more particularly for the production of polymers with any pre-selected molecular weight distributions. Our general objective has been to establish laboratory reactor systems which are capable of producing polymers with specifically defined molecular size characteristics in sufficient quantity to undertake realistic processing tests on the materials. As far as we are aware there have been no previous attempts to establish an automated process for the production of polymer with any desired molecular weight distribution which has been the specific aim of this work .
r
a
~
k [ M1
P
(6)
The average number of monomer molecules i~a polymer chain (degree of polymerization, DP) at 100% conversion of monomer is given by DP
[ M] o / [IJ 0
M
DP x (Molecular weight of M) + (molecular weight of initiator fragment)
(7)
and n
(8)
When there is perfect mixing of the reagents the spread of DP values will be small and W. = J
[ Y/(Y+l~ j e- Y
j 2
y - /(j_l)
(9)
where y =
[M] o /
[LE]
0
= DP
(lO)
In practice the 'killing' reactions (3) and (4) readily occur . The species X need only be traces of moisture and both the initiator and living polymer chain would be rendered non-active (NA). The rate constants ka and k are generally both large and greater tRan k . or k • ~ p We have found that there is not a simple relationsh i p between the input and output conditions of the reactor. Here we wish to outline the control techniques which we are endeavouring to implement to improve the performance of the reactor.
EXPERIMENTAL Experimental work has been conducted using either sec- butyllithium or a -methylstyrene dianion~s initiators , styrene as the monomer and tetrahydrofuran as solvent. It is known that under appropriate conditions that the initiators will induce polymerization which conforms closely to the model described by Eqns. (1) and (2). The secbutyllithium was used as supplied (Aldrich Chemical Co.) . The a -methylstyrene dianion initiator was prepared by the reaction o f a -methylstyrene with an excess of sodium metal in tetrahydrofuran. The styrene was dried over calcium hydride and vacuum distilled prior to use. The solvent was predried with calcium hydride. It was then
Adaptive Control of the Molecular Weight-Distribution
conditioned with sodium metal and anthracene or a-methylstyrene and distilled from the 'living' anion solution. All purification steps and the metering of reagent were carried out under argon in a purpose built rig. The overall process is shown schematically in Fig. 4. The vessels I and M are used for the storage of initiator and monomer solutions. Each reagent is pumped, using stainless steel positive displacement pumps (Precision Metering Co. Ltd.), at a flow rate which is determined by the set-points of the pump controllers. The controller setpOints are adjusted manually or by computer. Good mixing of the initiator and monomer solutions is essential. In the initial experiments a simple impingement mixing device was used but was not very satisfactory. The most effective mixing chambers were similar in design to those used in stopped-flow equipment. A stainless steel tubular reactor was attached directly to the mixing chamber. Ideally there should be turbulent flow in the reactor but the viscosity of the polymer solutions is high and changes along the length of the reactor. In general flow rates of approximately 100 ml/min were used and about 4m of O.3cm diameter tubing were used to ensure complete conversion of monomer. The polymer living ends (LE) have a characteristic absorption spectrum in the UV/visible range and the Amax varies with the counter~on and to some extent the solvent. The LE in the reactor was measured (at any selected point along the length of the reactor) in one of two ways. Either the tubular reactor was made to flow directly through a UV/ visible spectrophotometer cell compartment (Cecil Instruments Ltd.) or a specially designed cell was attached to the reactor and connected to the spectrophotometer (Unicam, SP800) by quartz light tubes (Schott Glass Ltd., W.Ger.). The reactor was immersed in a water bath and the temperature was o maintained constant in the range 2S C to 0 60 C. The output from the reactor could be collected in one of two tanks the valves to which were automatically controlled. Polymer samples were taken from the reception tank and analysed by gel permeation chromatography (Waters Associates, HPLC). Crosslinked styrene-divinyl benzene columns were used and those found to produce the least broadening in the analysis were PL-columns (Polymer Laboratories Ltd.). The experimental rig and associated equipment was under the supervisory control of an ARGUS700 (Ferranti Ltd.) process control computer. The computer accepted operating instructions via a teletype, e.g. desired MWD, and using the appropriate algorithms
361
undertook all control functions and data analysis including that for the gpc which was on-line for data analysiS only. Many aspects of the reactor operation, instrumentation, the role of the computer and data analysiS are too complex to present briefly and will be reported in full elsewhere.
INITIAL CONTROL STRATEGIES If the assumption is made that the process under investigation is an 'ideal living' reaction and that a plug flow regime exists in the reactor, the response to any changes in the input conditions are without transient processes and a pure transport lag only has to be taken into account. In this respect the system considered here differs greatly from those modelled in the literature «Spitz, 1976) where, in general, the applied oscillations in the polymerization systems kept the reactors in a transient state which could not be described using steady-state equations. The molecular weight of a polymer produced in a living polymerization may be related to the flow rates of monomer and initiator into the reactor through the DP. FM [M] 0 DP .
(11)
~
where F
F I [1]0
is the flow rate of the monomer
solutio~ and F that of the initiator and CM] and [11 ~re the molar concentrations
of ~he monom~r and initiator solutions respectively. Monomer conversion is assumed to be 100%. If the distribution is narrow then the following statement is justified. ( 12)
DP.
DP.
~
~
When [M] and [IJ are maintained constant then Eqn~. (11) agd (12) permit the calculation of FM/FI for each molecular species ~n any selected polymer MWD. For the case where F is 'l.rbitrarily maintained constant then (iJnoring '; :he initiator fragment in the polymer) F
I
[IJ
0
DP.
CM] o
~
(13)
When F is constant then the same number of I . du ce d at any ~nstant . polymer cha~ns are pro irrespective of DP .. The time for which the are maintain~d constant controls the qualIty of material of DP . which is produced. In a cyclic process the p?oduction time of a particular chain length species can be related to the mole fraction of those chains in the product thus
F~F
t.
~
T.n. (DP.) ~
~
(14)
K. Arnold, A. F. Johnson and J. Ramsay
362
where T
Lt.
(15)
~
Here T is the period of an oscillation, t. is a fraction of that period and n. is th~ ~ mole fraction of polymer of DP • i These equations may be represented graphically as shown in Fig. Sa. To avoid transients resulting from the rapid change in F at the end of each cycle the period of o~cillation was divided into five. During the period> T/2 the FM profile was reversed. The desired MWD is shown in Fig. 6a as a simple square distribution which can be readily related to the desired cumulative distribution as shown in Fig. 6b. The necessary monomer flow profile for the production of this MWD in a time T is shown in Fig. 6c. Any value of T may be selected, i.e. F can be made to change slowly over severa~ hours or in only minutes (the cycle being repeated many times in the latter case). Short cycle times have many advantages when dealing with very labile species. Normally T = 20 mins has been used and found to be satisfactory.
the sample to the machine. Although our online data handling fram this machine removes time delays for MWD calculations, at present, there is no feasible way of incorporating the gpc into a feedback loop during one cycle period T, but it may be used to correct for longer term drifts. It is possible to monitor [LE] spectrophotometrically. The spectrophotometer, therefore, provides a fast, continuous and automatic means of moni toring [LE] in the reactor. Furthermore, the recent developments in model reference control (MRC) (Jones, 1975), where some form of on-line model of the process is used to predict what the process output will be if existing conditions are maintained at their present value, offers an interesting prospect for application for MWD control. As before it has been assumed that any deviations from plug flow were negligible. Fig. 9 shows recent results of steady-state runs which have been made after making improvements in our operating conditions. There is no doubt that, under appropriate conditions, the assumption is valid. If the system is without dynamics and transport delays only are considered then yet)
A simple feedforward model reference control strategy may therefore be adopted for the production of polymers as shown in Fig. 7. The results for a typical steady state experiment are shown in Fig. S. When the desired molecular weight distribution approaches Dn = 1.0 then the F and F flows approach constant values the r~tios of which give It is obvious that the observed outputnwhere D = 1.5 + 0.25 (Meira, 1975) is far from th~t wanted and there is a tendency for M to be higher than that desired as seen innFig. S. The broadening in the output response was largely a function of poor reagent mixing and poor reactor conditions, e.g. too high a solids content producing a complex relationship between output signal and reaction m~xture viscosity. The higher than expected M of the product can be directly attributab£e to extensive 'killing' reactions.
M.
where yet) is the output signal from the reactor (DP. of the polymer), X(t) the input signal to tRe reactor (F IF ); o(t-T ) is M I d the 'delta' function and * stands for convolution. In the initial experiments (F ). was I arbitrarily retained constant. ~TO calculate the monomer and initiator flows for a particular MWD in the product, one might retain (FM)' or the total flow rate constant (Eqn. 17) . ~ constant
F
The MWD of the product is the polymer parameter of greatest interest. However the measurement of moecular weight distribution takes from 5-15 min. by gpc and this time does not include that necessary for the preparation of the sample for presentation of
(17)
The latter is more general in application but gives slightly more complex equations for FI and FM' namely (lSa)
F/(l + Ri) R.F/(l + R.)
Developments in Control Strategies As a result of these early observations considerable modifications were made to the reactor system, particularly the mixing cell and to the operating conditions. We wish to continue here by the control strategies which have been developed (and are currently being tested experimentally) in order to effect better overall control of the process.
(16)
~
~
(lSb)
where R. = (F ). I (F ). = KM. and K = [I] I ~ M.~ I ~ ~ . 0 M [J M . M be~ng the molecular we~ght ot t~e mgname~. In the case of constant monomer flow the same weight of each species is produced at any instant, i.e. t.
~
T w(M.) ~
(19)
This means that the weight distribution function is proportional to the time distribution of the applied ratios R .. For the case where the total flow F is~retained constant
363
Adaptive Control of the Molecular Weight-Distribution
(20)
t,
~
where F ~s the monomer flow rate calculated for theMweight average molecular weight. To synchronize the solenoid valves to the reception tank, the reactor transport lag, T , has to be calculated, d L.S F
(21)
where Land S are the length and crosssectional area of the reactor.
significant since the LE's cannot be formed if i t occurs; further impurities do not enter the reactor from the external atmosphere once the LE's have been produced. The reaction of impurities with the initiator and living polymer chains of lo~ molecular weight (i.e. r small in (4)) can be surrmarized as AC + X
The feed forward model reference control strategy which was applied in the initial experiments made use of an inverse model algorithm for the set point control (SPC) of the flow controllers. Success was entirely dependent upon a knowledge of upstream conditions in the feed, i.e. [M~and [I] • It was presupposed that there were no 'k~lling' impuri ties whic:h might influence )~o and CM] and that LIJ itself could be precisely mea~ured. This cgntrol scheme is illustrated in Fig. 10 where the process block represents the reactor together with the pumps and their controllers and is assumed to be described by Eqn. (16); Y(t) is the MWD of the product; Yd(t) is the desired MWD which is the input signal to the feed forward controller; the controller block represnets the inverse model algorithm for the feed forward controller. X(t) is the controlled input signal to the process; Z(t) are uncontrolled and are the [MJ and [11 inputs to the feed streams of 0 o the process. Unfortunately, the estimated values of [11 o could not be made sufficiently precise to guarantee the successful performance of the system. A further input signal acting on the system was found to be the 'killing' impurities in the monomer solution which could not be estimated in any way. Of the 'killing' reactions represented by Eqns. (3) and (4), the former is the more
(22)
NA
where AC represents all active material and X and NA are impurities and dead material as before. Hence 'LE] ~
The main advantages of retaining F constant are that the range of DP, values which can be produced in one oscillation is larger for this case. If, say, R, can vary 100 fold 2 with one flow rate fix~d, R, will vary 100 fold when the two flows osc111ate to their maximum amplitude. With constant F the transport lag is also constant facilitating the solenoid value operation to the reception tanks. When F is maintained constant F becomes non-li~ear when corrections for M 'killing' have to be made but the varying transport lag arguably reduces wall build up effects.
-------;>
=
((F ), [IJ 0 I ~
(F ), [IM] M ~
0
)/F (23)
where [IM] is the concentration of impurities in th~ monomer solution. Since [Mt» CI1a and [I] »[IM} had to pertain for polymerizati8n to occ~r, the magnitude of [M1 could be considered to be constant becauseOthe 'killing' of LE's was so fast that the DP was necessarily very small. Modifications can be made to the inverse process model to take into account the 'killing' reactions as shown in (24). DP,
(FM)'~ [M1 0
(24)
~
Taking this equation into account means that when (F ), was arbitrarily made constant in the conEr61 strategy, as was the case in the initial experiments, there was a loss of linearity with respect to the average molecular weight. If FM is maintained constant then ( 25) where K3 = F
~s convenien~
[IM] 0/ [11 and_ K because wRen
F /K which K3 = O.
LIMj o = t3,
If F is maintained constant then Ri
KM/K 4
(26)
whereK = (1 + K M,) andK = [IM] /MM[Ml 5 when ~~J = 0 i t i~plies t~at K =0 1 and 0 K4 is theOcorrection factor for ~killing' reactions. In order to evaluate the apparent initiator concentration by Eqn. (23) use was made of the spectrophotometer (Bhattacharyya, 1965) and one of several computation methods. For example, a least squares method may be employed. If [LE1 is the measured concentration of LE' s an~ [LE} that calculated by Eqn. (23) then if N is tfie number of points taken during a scan of the absorption spectrum then the sum of squares is given by Eqn. (27) •
364
K. Arnold, A. F. Johnson and J. Ramsay
N
ss
([LE}
1:
j=l If Eqn.
. - [LE] .) 2 m, J c, J
(27)
(23) is written in the form:
[LEJ
A . [I]
.
c, J
J
0
+ B. [LM1 J
''b,l
(28)
0
where A. = (FI).lF and B. = -(FM) ./F then minimization ofJSS with tespect td [IJ and UM] yields, in matrix notation 0 o
c
(29)
where [IJ
c
[LE] m, 1
o
x
y
iJ,E] m, 2 [LE]
m,N
Alternatively, if Eqn. (23) is written for two consecutive measurements with different values of A. and B. (i.e. different (F ). I and (F ) . wHich isJthe normal situation J duringMa;\ experiment using oscillatory feeds), a set of equations is formed which may be solved for [r} and [IM] . For N spectrophotometriC ob~ervationsoin a spectrum there will be N pairs of [I] and [LM] i f Eqn. (23) is solved for j =0 1 with th~ assumption that [LMl- 0 = O. Application of the least squares method to [I} 0 and [IM1o eventually yields a recursive, moving average, type of equation.
.
[J 0ap(j-l)
(J-l) I
[11 ap
+
state experiments, this approach is inappropriate and adaptive action is necessary after each spectrophotometric scan and initial action is taken with the assumption that ~M~P = 0 as shown in Fig. 14.
[1 c I aj
Assuming that the adaptive control of the reactor feeds are successful, the correct instantaneous DP. of the product will only be achieved provided that ideal plug flow conditions are sustained under all feed conditions to the reactor. In the case of polymerization reactions this is unlikely to be the case because of the larger changes in viscosity which occur even with very low monomer conversions. In the case of living anionic polymerizations the viscosity of the reaction mixture will not only be a function of F and F but -.M. . I there will be a true depen d ence of v~scos~ty on conversion for any particular value of each. Time dependent variations in viscosity and the consequent deviations fran plug flow behaviour cannot be discussed within the scope of this article but it is possible to consider the attempts which we have made to correct for broadening effects based on the performance of the reactor under steady state conditions. In the steady state the desired distribution for perfectly monodisperse polymer is a pulse. The idea of a pulse response function defined in the molecular weight domain has been used and a convolution type equation has been developed to describe the broadening Eqn. (32) .
j=l, ... N (30)
OJ
oo
W(M) where ap indicates the apparent value and c the value calculated from Eqn. (28). The variance of the evaluated apparent concentration may also be readily deduced. Similar expressions can be arrived at for [j:M] • o As can be seen fran Eqn. (3), as j increases the additional information obtained by making (j + 1) measurement becomes insignificant. In practice the number of measurements is restricted to a convenient number of wavelength scans for each period of an oscillation in the feeds. At the end of each oscillation j is reset and counting commences at j = 1 once more followed by
rI] -
ap
---;>
o,N
( 31)
for the next oscillation. The technique described above worked well in the case of experiments where F and F vary I M. continuously. In this case the adaptat~on of the set points for a forthcoming oscillation can be made on the basis of the newly evaluated conditions for [1]0 and ernlo at the end of each period. However, when FI and F are constant, as they are during steady M
l
f ( \1 ).G (M-\1)d\1
(32)
o
In Eqn. (32) W(M) is the MWD of the product of reaction, f( \1 ) the input MWD from which the input flow program is calculated and G the broadening or pulse response function. The G (r-l.-\1) broadening function may be evaluated from analytical models or, as in the case of this work, from steady state experiments. Knowing the desired MWD, W(M) and the broadening function, the required input MWD can be calculated by deconvolution. This approach formed the basis of the algorithm of a feed forward control policy designed in the molecular weight domain. A representation of the adaptive model reference control scheme for the up dating of the parameters of the feed forward controller are shown in Fig. 12. If it is assumed that the broadening function is Gaussian (Eqn. (33))
1
a (2 7T )~
e
( 33)
Adaptive Control of the Molecular Weight-Distribution
where a is the standard deviation then a may be calculated from a steady state experiment using Eqn. (34)
D
D
n
w
(34)
CONCLUSIONS 1.
A fully computer controlled reactor system has been developed for the production of monodisperse polymers and polymers of any molecular weight distribution from 'living' anionic polymerization processes carried out in a tubular reactor. The major limitations of the process appear to stem from 'killing' reactions and non-plug flow conditions in the reactor. The performance of the reactor is monitored using an on-line gel permeation chromatograph.
2.
Adaptive loop control schemes have been described which, in theory, modify the input signals and achieve products with a molecular weight distributions with those required. One loop is based on an on-line spectrophotometric measurement of 'living' polymer chain concentration and a second makes use of the assumption that the form of the reactor broadening function is known and its parameters can be identified on-line by gel permeation chromatography. Preliminary experiments appear to be validating the control concepts. REFERENCES
Bhattacharyya, D.N., C.L. Lee, J. Smid and M. Szwarc (1965). Reactivities and Conductivities of Ions and Ion Pairs in Polymerization Processes. J. Phys. Chem., 69, 612-623. Jones, R.E. and B.G. Freedman (1975). Applying Model Reference to Process Control. Control Engineering, Mar, 48-51. Spitz, J.J., R.L. Lawrence and D.C. Chappelear (1976). Periodic Operation of a Polymerization Reactor. In T.C. Bouton and D.C. Chappelear (Eds), Continuous Polymerization Reactors, Vol. 72, No. 160, AIChE, New York, pp 86-101. Szwarc, M. (1968). Carbanions , Living Polymers and Electron Transfer Processes. Interscience, New York. ACKNOWLEDGEMENTS The authors wish to acknowledge the work of Dr. G. Meira and Dr. N. Bourikas in this area which has also been submitted for publication recently.
365
366
K. Arnold, A. F. Johnson and J. Ramsay
z
,....-------f
I
Target
~
Z(t)
I- - - - -
I nv. Model
K=1/ KM
4_
~
Process I-~
S im r le F"ed f orw ard Mo ·;el
;·· e ~- :.o ren c, :
I
KM
Controller
;- ,
.
. .-.
S im?l e
c.') n tr (~ l .
~eed!c r~a rd
--I
- - - -
M o~rl
I
S(t-lJ
I
Pe~crcnce
Co:"t r ol.
[M}0
I nput
Output
r_-----A- - - - -__,
r--A--
r
)'
Orf1· 5~·2 5 4 t
0
M·
W(M)
W(M)
~
Target
1nv. Model
18ab;26 ~~
10
I-
.
Proc,--=
,
,r
Estim . Time ~ Lag [I J:'JIMr!
T(min)
J - •. . :....t 1:-' - . .
~ r : l·.:. i?~
tale
~-' :: - ~ ri:"1 e nta ~
:·\..:;- .. :.. . . s
).
Input
_--J.---_
Output
Target
A. " r--~ W M)
r
W(M)
W(M)
°n:108:t·03 1
4
~-
10
T(min)
M x 10-4
Oeconvolution
FM~- - - - , Process
+~...
IAL
Time Evaluation it Lag pg~~~i G(M-p)L--_ _ _ -J
367
Adaptive Control of the Molecular Weight-Distribution
..C
::J
o E
Mol. Wt. : 1. , : *
1.
n·I
Density !J istr i b ut i o n '] : :-!u lecular ';iei -o hts.
r i '.
I.
S chemii'_ ic :< epres e ntati ') n
0 '
t he
~ r o c e ss .
Differential W·I
n·I
n·I
Cum.
F
Cum.
LW·I
Mol.Wt.
Mol Wt
T
Mol Wt 19.
~.
Mo l e cu l ar
~ei ~hl
Di s tri b uti )ns -
::~
...... C
::J
Time
o
E
~
Mol. Wt . ii i:codai "J is tr ib'J li or.
Time 0:
Mo lecu ~ar
: yp ical
S ~m c: crica l
Os c ill ii t io ns.
© (FAC
PRP 4 Automation, Gh~nt , B~lgium 1980
ADAPTIVE CONTROL OF THE MOLECULAR WEIGHT-DISTRIBUTION IN LIVING ANIONIC POLYMERIZATION PROCESSES K. Arnold 1, A. F. Johnson2 and J. Ramsay Postgraduate Schools of Polymer Science and Control Engineering, University of Bradford, Bradford BD7 lDP, West Yorkshire, England
Abstract. A model reference control strategy is discussed for the production of monodisperse polymers and polymers of any pre-defined molecular weight distribution fram 'living' anionic polymerizations carried out in a smallscale tubular reactor system. Initially it was assumed that when a fixed (monomer) /(i nitiator ) ratio was introduced into the reactor that it was possible to produce monodisperse polymer with a predictable average molecular weight and further that by perturbing the (monomer)/(initiator) ratio in a prescribed monomer with time that an integral product of any molecular weight distribution (MWD) could be achieved. Experimental results indicate that the limitations of this simple approach were (i) the (initiator) was severely altered by adventitious side reactions, (ii) axial dispersion in the reactor and a finite reaction initiation rate resulted in an undesirable broadening effect on the MWD. To overcome these limitations a spectrophotometer has been incorporated into an input adaptive loop to control the livi ng chain end concentration in the reactor and the axial dispersion effects on the MWD as deduced from gel permeation chromatography measurements have been used to adjust the input feed rate of the reactants; both modifications are currently being tested and are outlined here. In principle perturbing the feed to a polymerization reactor can be applied to any polymer ization process to alter the MWD envelope of the product but the efficacy of this approach is sensitive to the mechanism of the polymerization reaction and the reactor configuration. Keywords. Feed-forward control; anionic polymerization process; molecular weight distribution control; model reference control; gel permeation chromatography; tubular reactor; adaptive loop control.
reacti on is much faster than the propagation rate, (b) propagation occurs exclusively by the addition of monomer to the liv ing polymer chain ends , (c) there are no terminations , chain transfer or other secondary reactions, (d) that the reaction is carried out in a perfectly mixed batch reactor or an ideal plug-flow tubular reactor (Szwarc, 1968). When all these conditions pertain it follows that the average molecular weight of the po lymer will be given by [M] / [11 and MjM" ~ 1. 0 , where [M] and [lJ are the re ac tor n feed concentrations of monomer and initiator respectively and M and M are the weight and number ave~ageWmo~ecu~ar weights of the product. The M and M may be defined in the usual way a~ shownnin Eqns. (3) and (4).
INTRODUCTION The measured molecular weight distribution of any synthetic polymer is a c omp lex function of the chemistry of the polymerization process, the polymerization reacto r configuration employed for the synthesis and the mode of operation and control of the reactor system. Unless care is taken, the observed molecular weight distributi on may also be influenced by the technique employed for its measurement. For the case of 'living' polymerization reactions which involve initiation (1) and propagation reacti ons (2) only k. M + I ~ IMl IM
n
k + M - L IM n+l
(1)
n
=
2,3 .••
00
( 2) M
w
a Poisson molecular weight distribution will be observed if the following requirements are fulfilled; (a) the rate of the initiation
A (2)
A (1)
LN .M. 2 ~
~
LN.M. ~
~
LW.M. ~
1 20n leave of absence from R. and D. Centre, Chemoautomatyka, Warsaw, Poland. To whom correspondence should be addressed.
359
~
LW.
~
LW.M. ~
~
(3)
360
K. Arnold, A. F. J o hnson and J. Ramsay
M
n
~) A (0)
k
EN .M. ~
~
EN.
~
Termination I + X (killing)
(4)
Ln . M .
~
~
where A(O) , A( l) and A(2) are the zero, first, and second moments of the density distribution of molecular weights (Fig. 1) and N, M, Wand n are the number, molecular weight , weight fraction and number fracti ons respectively of the ith pol~e£ chain (see Figs. 2 and 3). The ratio M /Mn is a convenient measure of the bread~h of the density distribution curve and is frequently represented as Dn' the dispersity index.
LE
The basis of our approach has been to take 'living' anionic polymerizations which are thought to conform to the conditions (a), (b) and (c) mentioned above and to conduct the polymerizations in a tubular reactor taking the appropriate steps to ensure , as far as possible, that ideal plug flow conditions are achieved in the reactor. Under steady- state operating conditions mono disperse polymers (D ~ 1.0) should be produced. Provided thi~ is the case and that monodisperse material of any molecular weight can be produced by altering the feed ratio [M] / [I ) then the reactions may serve to produce polymers of any molecular weight distri bution. To achieve this latter objective the monomer and initiator feeds to the reactor have to be subjected to periodic oscillations in such a way as t o give a sequence of steadystate operating conditions each of which gives rise to a known instantaneous molecular weigh~ When polymerization is allowed t o go to completion, the product collected over a period of time should have a molecular weight distribution which is quantitatively related to the input feed conditions of the reactor. The instantaneous average molecular weight can be readily seen to be related to the reactor feeds. If the reacti on mechanism is more fully expressed by Eqns. ( I ) to (I V) , k. Initiation I + M ~ LEl k Propagation LEn + M ---R...;,
LEn+l
(ii)
(iii)
NA
+ X
NA
(IV)
when steps (iii) and (IV) do not take place the reaction may be described as ' living'. Provided k . » k and b o th are large , when no killing~occur~
[I] o
(5)
and R
We have been examining the behaviour of living polymerizations in tubular reactors for the production of mono-disperse polymers , but, more particularly for the production of polymers with any pre-selected molecular weight distributions. Our general objective has been to establish laboratory reactor systems which are capable of producing polymers with specifically defined molecular size characteristics in sufficient quantity to undertake realistic processing tests on the materials. As far as we are aware there have been no previous attempts to establish an automated process for the production of polymer with any desired molecular weight distribution which has been the specific aim of this work .
r
a
~
k [ M1
P
(6)
The average number of monomer molecules i~a polymer chain (degree of polymerization, DP) at 100% conversion of monomer is given by DP
[ M] o / [IJ 0
M
DP x (Molecular weight of M) + (molecular weight of initiator fragment)
(7)
and n
(8)
When there is perfect mixing of the reagents the spread of DP values will be small and W. = J
[ Y/(Y+l~ j e- Y
j 2
y - /(j_l)
(9)
where y =
[M] o /
[LE]
0
= DP
(lO)
In practice the 'killing' reactions (3) and (4) readily occur . The species X need only be traces of moisture and both the initiator and living polymer chain would be rendered non-active (NA). The rate constants ka and k are generally both large and greater tRan k . or k • ~ p We have found that there is not a simple relationsh i p between the input and output conditions of the reactor. Here we wish to outline the control techniques which we are endeavouring to implement to improve the performance of the reactor.
EXPERIMENTAL Experimental work has been conducted using either sec- butyllithium or a -methylstyrene dianion~s initiators , styrene as the monomer and tetrahydrofuran as solvent. It is known that under appropriate conditions that the initiators will induce polymerization which conforms closely to the model described by Eqns. (1) and (2). The secbutyllithium was used as supplied (Aldrich Chemical Co.) . The a -methylstyrene dianion initiator was prepared by the reaction o f a -methylstyrene with an excess of sodium metal in tetrahydrofuran. The styrene was dried over calcium hydride and vacuum distilled prior to use. The solvent was predried with calcium hydride. It was then
Adaptive Control of the Molecular Weight-Distribution
conditioned with sodium metal and anthracene or a-methylstyrene and distilled from the 'living' anion solution. All purification steps and the metering of reagent were carried out under argon in a purpose built rig. The overall process is shown schematically in Fig. 4. The vessels I and M are used for the storage of initiator and monomer solutions. Each reagent is pumped, using stainless steel positive displacement pumps (Precision Metering Co. Ltd.), at a flow rate which is determined by the set-points of the pump controllers. The controller setpOints are adjusted manually or by computer. Good mixing of the initiator and monomer solutions is essential. In the initial experiments a simple impingement mixing device was used but was not very satisfactory. The most effective mixing chambers were similar in design to those used in stopped-flow equipment. A stainless steel tubular reactor was attached directly to the mixing chamber. Ideally there should be turbulent flow in the reactor but the viscosity of the polymer solutions is high and changes along the length of the reactor. In general flow rates of approximately 100 ml/min were used and about 4m of O.3cm diameter tubing were used to ensure complete conversion of monomer. The polymer living ends (LE) have a characteristic absorption spectrum in the UV/visible range and the Amax varies with the counter~on and to some extent the solvent. The LE in the reactor was measured (at any selected point along the length of the reactor) in one of two ways. Either the tubular reactor was made to flow directly through a UV/ visible spectrophotometer cell compartment (Cecil Instruments Ltd.) or a specially designed cell was attached to the reactor and connected to the spectrophotometer (Unicam, SP800) by quartz light tubes (Schott Glass Ltd., W.Ger.). The reactor was immersed in a water bath and the temperature was o maintained constant in the range 2S C to 0 60 C. The output from the reactor could be collected in one of two tanks the valves to which were automatically controlled. Polymer samples were taken from the reception tank and analysed by gel permeation chromatography (Waters Associates, HPLC). Crosslinked styrene-divinyl benzene columns were used and those found to produce the least broadening in the analysis were PL-columns (Polymer Laboratories Ltd.). The experimental rig and associated equipment was under the supervisory control of an ARGUS700 (Ferranti Ltd.) process control computer. The computer accepted operating instructions via a teletype, e.g. desired MWD, and using the appropriate algorithms
361
undertook all control functions and data analysis including that for the gpc which was on-line for data analysiS only. Many aspects of the reactor operation, instrumentation, the role of the computer and data analysiS are too complex to present briefly and will be reported in full elsewhere.
INITIAL CONTROL STRATEGIES If the assumption is made that the process under investigation is an 'ideal living' reaction and that a plug flow regime exists in the reactor, the response to any changes in the input conditions are without transient processes and a pure transport lag only has to be taken into account. In this respect the system considered here differs greatly from those modelled in the literature «Spitz, 1976) where, in general, the applied oscillations in the polymerization systems kept the reactors in a transient state which could not be described using steady-state equations. The molecular weight of a polymer produced in a living polymerization may be related to the flow rates of monomer and initiator into the reactor through the DP. FM [M] 0 DP .
(11)
~
where F
F I [1]0
is the flow rate of the monomer
solutio~ and F that of the initiator and CM] and [11 ~re the molar concentrations
of ~he monom~r and initiator solutions respectively. Monomer conversion is assumed to be 100%. If the distribution is narrow then the following statement is justified. ( 12)
DP.
DP.
~
~
When [M] and [IJ are maintained constant then Eqn~. (11) agd (12) permit the calculation of FM/FI for each molecular species ~n any selected polymer MWD. For the case where F is 'l.rbitrarily maintained constant then (iJnoring '; :he initiator fragment in the polymer) F
I
[IJ
0
DP.
CM] o
~
(13)
When F is constant then the same number of I . du ce d at any ~nstant . polymer cha~ns are pro irrespective of DP .. The time for which the are maintain~d constant controls the qualIty of material of DP . which is produced. In a cyclic process the p?oduction time of a particular chain length species can be related to the mole fraction of those chains in the product thus
F~F
t.
~
T.n. (DP.) ~
~
(14)
K. Arnold, A. F. Johnson and J. Ramsay
362
where T
Lt.
(15)
~
Here T is the period of an oscillation, t. is a fraction of that period and n. is th~ ~ mole fraction of polymer of DP • i These equations may be represented graphically as shown in Fig. Sa. To avoid transients resulting from the rapid change in F at the end of each cycle the period of o~cillation was divided into five. During the period> T/2 the FM profile was reversed. The desired MWD is shown in Fig. 6a as a simple square distribution which can be readily related to the desired cumulative distribution as shown in Fig. 6b. The necessary monomer flow profile for the production of this MWD in a time T is shown in Fig. 6c. Any value of T may be selected, i.e. F can be made to change slowly over severa~ hours or in only minutes (the cycle being repeated many times in the latter case). Short cycle times have many advantages when dealing with very labile species. Normally T = 20 mins has been used and found to be satisfactory.
the sample to the machine. Although our online data handling fram this machine removes time delays for MWD calculations, at present, there is no feasible way of incorporating the gpc into a feedback loop during one cycle period T, but it may be used to correct for longer term drifts. It is possible to monitor [LE] spectrophotometrically. The spectrophotometer, therefore, provides a fast, continuous and automatic means of moni toring [LE] in the reactor. Furthermore, the recent developments in model reference control (MRC) (Jones, 1975), where some form of on-line model of the process is used to predict what the process output will be if existing conditions are maintained at their present value, offers an interesting prospect for application for MWD control. As before it has been assumed that any deviations from plug flow were negligible. Fig. 9 shows recent results of steady-state runs which have been made after making improvements in our operating conditions. There is no doubt that, under appropriate conditions, the assumption is valid. If the system is without dynamics and transport delays only are considered then yet)
A simple feedforward model reference control strategy may therefore be adopted for the production of polymers as shown in Fig. 7. The results for a typical steady state experiment are shown in Fig. S. When the desired molecular weight distribution approaches Dn = 1.0 then the F and F flows approach constant values the r~tios of which give It is obvious that the observed outputnwhere D = 1.5 + 0.25 (Meira, 1975) is far from th~t wanted and there is a tendency for M to be higher than that desired as seen innFig. S. The broadening in the output response was largely a function of poor reagent mixing and poor reactor conditions, e.g. too high a solids content producing a complex relationship between output signal and reaction m~xture viscosity. The higher than expected M of the product can be directly attributab£e to extensive 'killing' reactions.
M.
where yet) is the output signal from the reactor (DP. of the polymer), X(t) the input signal to tRe reactor (F IF ); o(t-T ) is M I d the 'delta' function and * stands for convolution. In the initial experiments (F ). was I arbitrarily retained constant. ~TO calculate the monomer and initiator flows for a particular MWD in the product, one might retain (FM)' or the total flow rate constant (Eqn. 17) . ~ constant
F
The MWD of the product is the polymer parameter of greatest interest. However the measurement of moecular weight distribution takes from 5-15 min. by gpc and this time does not include that necessary for the preparation of the sample for presentation of
(17)
The latter is more general in application but gives slightly more complex equations for FI and FM' namely (lSa)
F/(l + Ri) R.F/(l + R.)
Developments in Control Strategies As a result of these early observations considerable modifications were made to the reactor system, particularly the mixing cell and to the operating conditions. We wish to continue here by the control strategies which have been developed (and are currently being tested experimentally) in order to effect better overall control of the process.
(16)
~
~
(lSb)
where R. = (F ). I (F ). = KM. and K = [I] I ~ M.~ I ~ ~ . 0 M [J M . M be~ng the molecular we~ght ot t~e mgname~. In the case of constant monomer flow the same weight of each species is produced at any instant, i.e. t.
~
T w(M.) ~
(19)
This means that the weight distribution function is proportional to the time distribution of the applied ratios R .. For the case where the total flow F is~retained constant
363
Adaptive Control of the Molecular Weight-Distribution
(20)
t,
~
where F ~s the monomer flow rate calculated for theMweight average molecular weight. To synchronize the solenoid valves to the reception tank, the reactor transport lag, T , has to be calculated, d L.S F
(21)
where Land S are the length and crosssectional area of the reactor.
significant since the LE's cannot be formed if i t occurs; further impurities do not enter the reactor from the external atmosphere once the LE's have been produced. The reaction of impurities with the initiator and living polymer chains of lo~ molecular weight (i.e. r small in (4)) can be surrmarized as AC + X
The feed forward model reference control strategy which was applied in the initial experiments made use of an inverse model algorithm for the set point control (SPC) of the flow controllers. Success was entirely dependent upon a knowledge of upstream conditions in the feed, i.e. [M~and [I] • It was presupposed that there were no 'k~lling' impuri ties whic:h might influence )~o and CM] and that LIJ itself could be precisely mea~ured. This cgntrol scheme is illustrated in Fig. 10 where the process block represents the reactor together with the pumps and their controllers and is assumed to be described by Eqn. (16); Y(t) is the MWD of the product; Yd(t) is the desired MWD which is the input signal to the feed forward controller; the controller block represnets the inverse model algorithm for the feed forward controller. X(t) is the controlled input signal to the process; Z(t) are uncontrolled and are the [MJ and [11 inputs to the feed streams of 0 o the process. Unfortunately, the estimated values of [11 o could not be made sufficiently precise to guarantee the successful performance of the system. A further input signal acting on the system was found to be the 'killing' impurities in the monomer solution which could not be estimated in any way. Of the 'killing' reactions represented by Eqns. (3) and (4), the former is the more
(22)
NA
where AC represents all active material and X and NA are impurities and dead material as before. Hence 'LE] ~
The main advantages of retaining F constant are that the range of DP, values which can be produced in one oscillation is larger for this case. If, say, R, can vary 100 fold 2 with one flow rate fix~d, R, will vary 100 fold when the two flows osc111ate to their maximum amplitude. With constant F the transport lag is also constant facilitating the solenoid value operation to the reception tanks. When F is maintained constant F becomes non-li~ear when corrections for M 'killing' have to be made but the varying transport lag arguably reduces wall build up effects.
-------;>
=
((F ), [IJ 0 I ~
(F ), [IM] M ~
0
)/F (23)
where [IM] is the concentration of impurities in th~ monomer solution. Since [Mt» CI1a and [I] »[IM} had to pertain for polymerizati8n to occ~r, the magnitude of [M1 could be considered to be constant becauseOthe 'killing' of LE's was so fast that the DP was necessarily very small. Modifications can be made to the inverse process model to take into account the 'killing' reactions as shown in (24). DP,
(FM)'~ [M1 0
(24)
~
Taking this equation into account means that when (F ), was arbitrarily made constant in the conEr61 strategy, as was the case in the initial experiments, there was a loss of linearity with respect to the average molecular weight. If FM is maintained constant then ( 25) where K3 = F
~s convenien~
[IM] 0/ [11 and_ K because wRen
F /K which K3 = O.
LIMj o = t3,
If F is maintained constant then Ri
KM/K 4
(26)
whereK = (1 + K M,) andK = [IM] /MM[Ml 5 when ~~J = 0 i t i~plies t~at K =0 1 and 0 K4 is theOcorrection factor for ~killing' reactions. In order to evaluate the apparent initiator concentration by Eqn. (23) use was made of the spectrophotometer (Bhattacharyya, 1965) and one of several computation methods. For example, a least squares method may be employed. If [LE1 is the measured concentration of LE' s an~ [LE} that calculated by Eqn. (23) then if N is tfie number of points taken during a scan of the absorption spectrum then the sum of squares is given by Eqn. (27) •
364
K. Arnold, A. F. Johnson and J. Ramsay
N
ss
([LE}
1:
j=l If Eqn.
. - [LE] .) 2 m, J c, J
(27)
(23) is written in the form:
[LEJ
A . [I]
.
c, J
J
0
+ B. [LM1 J
''b,l
(28)
0
where A. = (FI).lF and B. = -(FM) ./F then minimization ofJSS with tespect td [IJ and UM] yields, in matrix notation 0 o
c
(29)
where [IJ
c
[LE] m, 1
o
x
y
iJ,E] m, 2 [LE]
m,N
Alternatively, if Eqn. (23) is written for two consecutive measurements with different values of A. and B. (i.e. different (F ). I and (F ) . wHich isJthe normal situation J duringMa;\ experiment using oscillatory feeds), a set of equations is formed which may be solved for [r} and [IM] . For N spectrophotometriC ob~ervationsoin a spectrum there will be N pairs of [I] and [LM] i f Eqn. (23) is solved for j =0 1 with th~ assumption that [LMl- 0 = O. Application of the least squares method to [I} 0 and [IM1o eventually yields a recursive, moving average, type of equation.
.
[J 0ap(j-l)
(J-l) I
[11 ap
+
state experiments, this approach is inappropriate and adaptive action is necessary after each spectrophotometric scan and initial action is taken with the assumption that ~M~P = 0 as shown in Fig. 14.
[1 c I aj
Assuming that the adaptive control of the reactor feeds are successful, the correct instantaneous DP. of the product will only be achieved provided that ideal plug flow conditions are sustained under all feed conditions to the reactor. In the case of polymerization reactions this is unlikely to be the case because of the larger changes in viscosity which occur even with very low monomer conversions. In the case of living anionic polymerizations the viscosity of the reaction mixture will not only be a function of F and F but -.M. . I there will be a true depen d ence of v~scos~ty on conversion for any particular value of each. Time dependent variations in viscosity and the consequent deviations fran plug flow behaviour cannot be discussed within the scope of this article but it is possible to consider the attempts which we have made to correct for broadening effects based on the performance of the reactor under steady state conditions. In the steady state the desired distribution for perfectly monodisperse polymer is a pulse. The idea of a pulse response function defined in the molecular weight domain has been used and a convolution type equation has been developed to describe the broadening Eqn. (32) .
j=l, ... N (30)
OJ
oo
W(M) where ap indicates the apparent value and c the value calculated from Eqn. (28). The variance of the evaluated apparent concentration may also be readily deduced. Similar expressions can be arrived at for [j:M] • o As can be seen fran Eqn. (3), as j increases the additional information obtained by making (j + 1) measurement becomes insignificant. In practice the number of measurements is restricted to a convenient number of wavelength scans for each period of an oscillation in the feeds. At the end of each oscillation j is reset and counting commences at j = 1 once more followed by
rI] -
ap
---;>
o,N
( 31)
for the next oscillation. The technique described above worked well in the case of experiments where F and F vary I M. continuously. In this case the adaptat~on of the set points for a forthcoming oscillation can be made on the basis of the newly evaluated conditions for [1]0 and ernlo at the end of each period. However, when FI and F are constant, as they are during steady M
l
f ( \1 ).G (M-\1)d\1
(32)
o
In Eqn. (32) W(M) is the MWD of the product of reaction, f( \1 ) the input MWD from which the input flow program is calculated and G the broadening or pulse response function. The G (r-l.-\1) broadening function may be evaluated from analytical models or, as in the case of this work, from steady state experiments. Knowing the desired MWD, W(M) and the broadening function, the required input MWD can be calculated by deconvolution. This approach formed the basis of the algorithm of a feed forward control policy designed in the molecular weight domain. A representation of the adaptive model reference control scheme for the up dating of the parameters of the feed forward controller are shown in Fig. 12. If it is assumed that the broadening function is Gaussian (Eqn. (33))
1
a (2 7T )~
e
( 33)
Adaptive Control of the Molecular Weight-Distribution
where a is the standard deviation then a may be calculated from a steady state experiment using Eqn. (34)
D
D
n
w
(34)
CONCLUSIONS 1.
A fully computer controlled reactor system has been developed for the production of monodisperse polymers and polymers of any molecular weight distribution from 'living' anionic polymerization processes carried out in a tubular reactor. The major limitations of the process appear to stem from 'killing' reactions and non-plug flow conditions in the reactor. The performance of the reactor is monitored using an on-line gel permeation chromatograph.
2.
Adaptive loop control schemes have been described which, in theory, modify the input signals and achieve products with a molecular weight distributions with those required. One loop is based on an on-line spectrophotometric measurement of 'living' polymer chain concentration and a second makes use of the assumption that the form of the reactor broadening function is known and its parameters can be identified on-line by gel permeation chromatography. Preliminary experiments appear to be validating the control concepts. REFERENCES
Bhattacharyya, D.N., C.L. Lee, J. Smid and M. Szwarc (1965). Reactivities and Conductivities of Ions and Ion Pairs in Polymerization Processes. J. Phys. Chem., 69, 612-623. Jones, R.E. and B.G. Freedman (1975). Applying Model Reference to Process Control. Control Engineering, Mar, 48-51. Spitz, J.J., R.L. Lawrence and D.C. Chappelear (1976). Periodic Operation of a Polymerization Reactor. In T.C. Bouton and D.C. Chappelear (Eds), Continuous Polymerization Reactors, Vol. 72, No. 160, AIChE, New York, pp 86-101. Szwarc, M. (1968). Carbanions , Living Polymers and Electron Transfer Processes. Interscience, New York. ACKNOWLEDGEMENTS The authors wish to acknowledge the work of Dr. G. Meira and Dr. N. Bourikas in this area which has also been submitted for publication recently.
365
366
K. Arnold, A. F. Johnson and J. Ramsay
z
,....-------f
I
Target
~
Z(t)
I- - - - -
I nv. Model
K=1/ KM
4_
~
Process I-~
S im r le F"ed f orw ard Mo ·;el
;·· e ~- :.o ren c, :
I
KM
Controller
;- ,
.
. .-.
S im?l e
c.') n tr (~ l .
~eed!c r~a rd
--I
- - - -
M o~rl
I
S(t-lJ
I
Pe~crcnce
Co:"t r ol.
[M}0
I nput
Output
r_-----A- - - - -__,
r--A--
r
)'
Orf1· 5~·2 5 4 t
0
M·
W(M)
W(M)
~
Target
1nv. Model
18ab;26 ~~
10
I-
.
Proc,--=
,
,r
Estim . Time ~ Lag [I J:'JIMr!
T(min)
J - •. . :....t 1:-' - . .
~ r : l·.:. i?~
tale
~-' :: - ~ ri:"1 e nta ~
:·\..:;- .. :.. . . s
).
Input
_--J.---_
Output
Target
A. " r--~ W M)
r
W(M)
W(M)
°n:108:t·03 1
4
~-
10
T(min)
M x 10-4
Oeconvolution
FM~- - - - , Process
+~...
IAL
Time Evaluation it Lag pg~~~i G(M-p)L--_ _ _ -J
367
Adaptive Control of the Molecular Weight-Distribution
..C
::J
o E
Mol. Wt. : 1. , : *
1.
n·I
Density !J istr i b ut i o n '] : :-!u lecular ';iei -o hts.
r i '.
I.
S chemii'_ ic :< epres e ntati ') n
0 '
t he
~ r o c e ss .
Differential W·I
n·I
n·I
Cum.
F
Cum.
LW·I
Mol.Wt.
Mol Wt
T
Mol Wt 19.
~.
Mo l e cu l ar
~ei ~hl
Di s tri b uti )ns -
::~
...... C
::J
Time
o
E
~
Mol. Wt . ii i:codai "J is tr ib'J li or.
Time 0:
Mo lecu ~ar
: yp ical
S ~m c: crica l
Os c ill ii t io ns.