Computer Applications to Chemical Engineering - ACS Publications

13 Modeling and Control of Continuous Industrial Polymerization Reactors JOHN F. MACGREGOR Department of Chemical Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada Downloaded by UNIV OF ARIZONA on January 14, 2013 | http://pubs.acs.org Publication Date: May 30, 1980 | doi: 10.1021/bk-1980-0124.ch013

PAUL W. T I D W E L L Monsanto Company, Decatur, A L 35601

In this paper we will primarily deal with the analysis and control of continuous flow polymerization reactors although much of the methodology will be relevant to other continuous processes and some batch processes as well. Consider the situation depicted in Figure 1 where there is a single or a sequence of flow reactors and one is interested in maintaining good control over polymer properties such as viscosity, MWD, copolymer composition, particle size distribution, polymer end group composition (dye sites), etc.. This can he done by adjusting the set points of primary controllers for one or more adjustable input variables such as initiator flow, catalyst flow, monomer feed ratios and reactor temperature or pressure. It is this secondary level of control over the polymer properties with which we shall be concerned here rather than the primary level of control over the individual manipulated variable themselves; that is, over the reactor temperature or pressure, or the feedrates, etc.. These latter variables often can be measured accurately and continuously and therefore usually can be well controlled by conventional continuous pneumatic or electronic proportional-integral-derivative (PID) type controllers. The control algorithms for the control of the polymer properties will usually act by readjusting the set-points of these primary controllers. Measurement Problems in the Control of Polymer Reactors. Before proceeding it is worthwhile discussing in more detail some aspects of measurement problems which present additional difficulties in formulating control strategies for polymerization reactors. (i) Discrete data: The observations of the output polymer properties are often available only at discrete equispaced intervals of time. This, of course, rules out the use of continuous pneumatic or electronic controllers. (ii) Off-line analyses: In spite of rapid development in the area of on-line detectors a large number of polymer proper0-8412-0549-3/80/47-124-251$05.00/0 © 1980 American Chemical Society In Computer Applications to Chemical Engineering; Squires, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.

Downloaded by UNIV OF ARIZONA on January 14, 2013 | http://pubs.acs.org Publication Date: May 30, 1980 | doi: 10.1021/bk-1980-0124.ch013

252

COMPUTER APPLICATIONS TO CHEMICAL ENGINEERING

Polymer Samples

Reactor

Figure 1. Reactor control configuration

In Computer Applications to Chemical Engineering; Squires, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


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TiDWELL

Continuous Polymerization Reactors

253

ties (e.g. conversion, MWD, copolymer composition, end group compositions, etc.) are s t i l l time consuming measurements made off-line in the quality control laboratory of most plants. This usually results in introducing a considerable time lag into the feedback loop of any control scheme. A l though analytical equipment is available for on-line analysis of some of these properties, i t is a question of providing economic j u s t i f i c a t i o n for them and of making them rugged and reliable enough for in-plant use. ( i i i ) Long sampling intervals: Compounding the above problems, i t is not uncommon for many of these properties to be analyzed very infrequently, perhaps only once per 8-hour shift or once per day. This would appear to make the control problem almost impossible but, in fact, such sampling intervals are not inconsistent with the requirement of eliminating many of the major stochastic disturbances in the system. Recent work by Box and Jenkins (k) and MacGregor (lh), has emphasized that in the control of processes subject to stochast i c disturbances i t is only important to sample fast enough to pick up changes in the major forecastable disturbances i n the system. In the authors experience the stochastic d i s turbances typically affecting polymer reactors persist over periods of days and even weeks, thereby making an 8-hour sampling interval quite consistent with the achievement of good control. In fact, the contrary situation has sometimes been the case, where deterioration in the quality of the control has resulted from too frequent control action, part i c u l a r l y when the measurement noise was large and the cont r o l strategy not developed to account for i t . (iv) Large measurement error: The analysis procedures for many polymer properties such as copolymer composition are often involved and d i f f i c u l t to carry out. As a result the measurement error variance is quite large. In fact, i t i s not uncommon for the two standard deviation limits of these measurements to approach the desired specification limits for grading of the polymer. Any control procedure for polymer reactors must recognize and deal with these measurement problems. This rules out the use of conventional continuous control ideas taught to engineering students at most universities. These latter ideas are based on continuous measurements, in the presence of l i t t l e or no measurement noise, and are applicable only to systems with rather short time delays i n the feedback loop. 1

Process Modelling In order to analyze for process problems or design stochastic controllers one usually needs to run f u l l scale plant tests and develop process dynamic models from the resulting data. This section discusses several aspects related to the modelling of polymer reactors for these purposes.



254


Type of Model. Depending upon the type of problem being i n vestigated, one can employ models ranging from very complex nonlinear p a r t i a l differential equations based on theoretical polymerization kinetics and mixing theory to simple linear empirical differential or difference equations b u i l t from process data only. Both these types of models have their place in developing process control strategies and in analyzing for process problems. (i) Mechanistic (nonlinear) Models: These tend to be relatively complex models and usually require a considerable development effort to arrive at an adequate form. Their primary use has been, and probably w i l l continue to be, in the design and simulation of polymer reactors. It is usually the steady-state versions of these models that are used to size new reactor systems or to simulate operation at different conditions of throughput, temperature, feed ratios, e t c . . These steady-state models are also useful for supervisory control where they can be used in an on- or off-line mode to optimize the operation conditions of the reactors at various times and to readjust the set-points of the basic temperature or flow controllers accordingly. Such use depends very much on the specific system and objectives. Dynamic or unsteady-state forms of these mechanistic models may also be useful in simulation or on-line control. They can be used to design reactors or find operating conditions which are less sensitive to specific types of disturbances, or to develop start-up procedures for a sequence of reactors. In the control of batch polymerizers they are potentially useful in controlling the polymerization reaction to follow some desired reaction path or trajectory. For the control of continuous reactors sampled at equi-spaced intervals of time (the main topic of this presentation), they can be used in a feedback control scheme with a process control computer to compute the control action needed to bring the reactor to a desired state in some specified optimal manner, providing the control or sampling interval is long compared to the time necessary to make the necessary computations (see for example Kiparissides et a l . (13)). Unless one is interested in operating the reactors over a wide range of conditions the simple empirical models discussed below are usually better suited to this latter feedback control situation. ( i i ) Empirical (linear) Dynamic Models: These are usually d i s crete linear transfer function (difference equation) type models which provide a representation of the dynamic behaviour of the process at the discrete sampling instants. They are in general much simpler to develop than theoretical models, both their structure and parameters being empirically identified from plant data. They are very useful for developing controllers to maintain the polymer properties constant at desired target values. These control schemes can often be made simple enough (at least the univariate ones)



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that the operators can implement them using tables or a programmable calculator. Moré powerful schemes are possible i f a process minicomputer is available. One of the major drawbacks of these empirical linear models i s their limited range of applicability. They hold quite well in the regions of operation for which they were developed, but they extrapolate very poorly due to the nonlinearities present in polymer reactor systems. One method of overcoming this d i f f i c u l t y , under changing conditions, by simultaneously tracking the changing parameters and controlling w i l l be discussed later. These two general types of models represent the extremes and various levels of sophistication lying somewhere between these are possible. Given the time and manpower i t is wise to develop both types of model and compare them with one another and with plant data. The authors are aware of several situations where the simple empirically identified models have revealed important effects not accounted for i n the theoretical models, and thereby led to a considerably better understanding of the system. The remainder of this manuscript w i l l deal with the i d e n t i f i cation of linear empirical models and their use in developing cont r o l strategies. Plant Experimentation. In order to evaluate process problems or develop simple models for a process, i t is often necessary to design and run f u l l scale plant experiments under production conditions. Due to long response times, long sampling intervals and the noisy nature of the measurements, these plant tests usually last from a few days to a month or more. Therefore, they must not drive the polymer properties out of specification for any unusual period of time nor upset any other mode of operation of the plant. There is no more certain way of ensuring termination of the test than trying to obtain too much information too quickly. The test must, therefore, be planned carefully and then performed by the engineers responsible for production. Maximum use of polymer blending f a c i l i t i e s or running on only one of many parallel reactor trains can effectively be used to minimize the larger variations in polymer properties which result during the test. Even i f only steady-stage information is ultimately desired on the process, i t is often necessary to design and analyze the experiments in a manner which accounts for the dynamic or transient effect and then obtain the steady-state information by the appropriate simplification of the resulting dynamic models. The main reason behind this statement is that i t is physically d i f f i c u l t or impossible in practice to hold the process steady enough at a number of margina l l y different fixed levels of operation i n the face of major stochastic disturbances i n the system, and then to be able to d i rectly estimate meaningful steady-state effects from such data in the presence of the large measurement errors. Furthermore i f the major disturbances are of a nonstationary nature, the consequences of trying to estimate steady-state effects by running open-loop



256


factorial type experiments should "be obvious. The results obtained by performing the usual factorial regression analysis (which assumes independent, zero mean, random errors) w i l l be incorrect and the confidence limits on the estimated effects w i l l be misleading (Box and Newbold (8) ). Furthermore, different results w i l l be obtained upon any replication of the experiment. Replication and randomization procedures have been used successfully in some EVOP and response surface studies on plants subject to nonstationary disturbances. These procedures do not protect us ent i r e l y against making incorrect conclusions under those circumstances, and the cost in terms of plant experimentation time for what protection they do afford is prohibitive when compared to the dynamic experimentation route outlined below. For example, one of the authors became involved in experimentation in a petrochemical plant where factorial experiments had been done repeatedly for years in an effort to solve some severe y i e l d problems in part o f , the plant. These experiments, i t turned out, were done with a nonstationary disturbance process operating in the system and hence they yielded s i l l y results. A simple 50-hour experiment and analysis of the data with time series methods solved a problem that had existed and defied solution for about 10 years. Consider the process configuration in Figure 1. An experimental program consists of introducing designed variations into the process inputs or manipulated variables (up to about k of these variables can often be studied simultaneously), and then sampling and analyzing the polymer being produced at the desired locations along the chain of reactors. Such data can sometimes be collected under open-loop conditions, that i s , under conditions where there is no feedback control being exercised over the output polymer properties through manipulating some of these input v a r i ables. Processes on which this open-loop type of experimentation can be performed are those which are subject to f a i r l y stationary stochastic disturbances over the time period of the experiment and whose output polymer properties w i l l remain reasonably well within defined specification limits i f a l l the input variables are held fixed at suitable mean levels. In this case the design of the experiment consists of choosing when and how to impose deviations in these input variables from their mean levels. The situation i s depicted in Figure 2. Various types of input sequences can be used and some research has been done on the optimal design of these i n put sequences to maximize the amount of information obtained on the process (Box and Jenkins (h), and Mehra ( l 6 ) ) . These optimal designs rely upon one's imprecise prior knowledge of the process and disturbances, and so are rarely directly useful. However, they do point out the importance of properly choosing certain characteristics of the input sequences. The most important requirements are as follows : (i) The input sequences should be uncorrelated with one another in order to enable one to separate out the separate effects of each input in later analysis. This can be guaranteed by



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using independent random number sequences i n generating the inputs. ( i i ) The input variations must have the proper frequency content in order to obtain the maximum amount of information on the process dynamics from the small perturbations used. Too high a frequency relative to the time constants of the system w i l l result in most of the variations being damped out and hence practically unobservable in the outputs. Too low a frequency w i l l result in obtaining l i t t l e information on the important transient effects. The proper frequency must be related to the estimated time constants of the process. ( i i i ) The magnitude of variation of the inputs must be carefully chosen. They must be varied enough to reveal the important effects in the outputs, but not too much to push the polymer properties out of specification for any inordinate amount of time. This leaves very l i t t l e room for error and makes the property choice very d i f f i c u l t . One must rely on a l l prior knowledge of the process and be w i l l i n g to change the magnitude of the variation as the experimentation proceeds. It is often convenient to design the experimental programs in time blocks with adjustments to the magnitude and frequency content of the input signals being made between blocks. Because the experimentation is being performed on a plant under f u l l production conditions the magnitude of the input variations must often be chosen too small to be able to d i s tinguish the effect of each change on the process outputs from that due to the large measurement errors. As a result a reasonably sophisticated s t a t i s t i c a l analysis of the data is often necessary. Some common types of input sequences which can satisfy the above requirements and are commonly used are illustrated in Figure 2. A pseudo-random binary signal is one which takes on values at either of the two levels with the decision to switch from one level to the other being made at each time interval by a random process. This type of signal is quite satisfactory i f the switching frequency i s chosen properly. For several inputs, u n c o r r e c ted pseudo-random binary signals switching according to different random number sequences can be used. An alternative to this i s to make use of the orthogonality property of factorial designs and vary different inputs according to the pattern of these designs. This has the added property that i f the time spent at each condition is reasonably long in comparison to the process time constants and the disturbance process is stationary, a pseudo-steadystate analysis can be attempted by c l a s s i c a l factorial design and analysis methods. A third type of very useful input sequence is the class of stochastic time series inputs generated by sequences of random Normal deviates. These w i l l be discussed in more detail later. In a l l open loop experimental programs, considerable care must be taken to guard against a lurking controller, a zealous operator who manipulates a variable not included in the program to



258


X

1

PROCESS

inputs

Figure 2.

Open-loop experimental configuration

Added S i g n a l

j n n r u ~ i r

outputs (polymer p r o p e r t i e s )

*. Yi x

2

inputs

Figure 3.

outputs (polymer p r o p e r t i e s )

Closed-loop experimental configuration



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maintain the response about the target. In some situations open-loop experimentation cannot be per formed due to the presence of large or drifting disturbances in the system which necessitate that some polymer properties be main tained under control throughout the course of experimentation by manipulating one or more inputs in a feedback manner. Such a closed-loop situation is depicted in Figure 3. It i s important, in this situation, that to identify the causal relationships be tween these variables one add a designed input variation to the feedback signals and not try to identify these relationships using only the inherent process input variations resulting from the feed back control action. I f such a signal were omitted, any attempt at process model identification could lead not to a model of the process but rather to a model of the feedback controller (Box and MacGregor ( j ) ) . Empirical Model Identification. In this section we consider linear difference equation models for characterizing both the pro cess dynamics and the stochastic disturbances inherent in the pro cess. We shall discuss how to specify the model structure, how to estimate i t s parameters, and how to check i t s adequacy. Although discussion w i l l be limited to single-input, single-output process es, the ideas are directly extendable to multiple-input, multipleoutput processes. Process Transfer Function Models : In continuous time, the dynamic behaviour of an ideal continuous flow stirred-tank reactor can be modelled (after linearization of any nonlinear kinetic ex pressions about a steady-state) by a f i r s t order ordinary differ ential equation of the form x f

+

=

y

g

(1)

X

where τ is the process time constant and g is i t s gain, and X and y are deviations about the steady-state. I f this f i r s t order pro cess is sampled at discrete equispaced intervals of time ( . . . ( T - l ) h, Th, ( T + l ) h , . . . ) and the input (manipulated) variable changed at the sampling instants, but held constant over the following i n terval (the usual situation in sampled-data control schemes), then i t s dynamic behaviour at the sampling intervals is given exactly by the f i r s t order difference equation y

t

= «y

+ g(i-«) x

t x

t

(2)

1

where 6 = e""*/ and the subscript t refers to the discrete time instant. Introducing the backward shift operator z"" (where z"" y . = y-fci and z" y . = y-Jk: ) the difference equation (2) can be written as 1

7

1

k


1

260


=

g ( l - 6)

This is a f i r s t order discrete transfer function of the sampled process. In general, the dynamic behaviour between a single input variable X and an output variable y in most polymer reactor sys tems can be adequately modelled at the sampling instants by a difference equation model of the form y = 6 i y t i +

i §

i

h-" CO

266


(1 - O.Sz'Mat with a t r o l l e r would be


VX

t

=

-0.7VX

a

t 1

* 1.

The r e d e r i v e d minimum v a r i a n c e con

+ 9.21e

- k.2ke

(l8)

tmml

However, t h i s c o n t r o l l e r would r e q u i r e a p p r e c i a b l y l a r g e r and pro bably unacceptable changes i n the manipulated v a r i a b l e . T h i s i s o f t e n a problem w i t h minimum variance c o n t r o l algorithms because they e f f e c t i v e l y attempt t o c a n c e l out the f o r e c a s t e d disturbance i n the minimum time (the delay time o f the process, b ) . To over come t h i s the c o n t r o l a l g o r i t h m can be redesigned t o minimize the variance o f the output v a r i a b l e d e v i a t i o n s subject t o a c o n s t r a i n t on the v a r i a n c e o f the input v a r i a b l e manipulations. The theory i s a v a i l a b l e f o r d e s i g n i n g these c o n s t r a i n e d c o n t r o l algorithms (17, 11, 13) and i n t h i s case leads t o the same d i f f e r e n c e equat ion form of the c o n t r o l a l g o r i t h m as t h a t given i n equation ( 1 7 ) » except f o r the presence o f an a d d i t i o n a l V X 2 "term. As more con s t r a i n t i s p l a c e d on the allowable manipulated input v a r i a n c e , the performance o f the c o n t r o l l e r , as measured by the v a r i a n c e o f the output d e v i a t i o n s , w i l l i n c r e a s e . A s u i t a b l e compromise between the v a r i a n c e s o f the manipulated and c o n t r o l l e d v a r i a b l e s can be found by e v a l u a t i n g these v a r i a n c e s f o r a range o f values o f the c o n s t r a i n t parameter (λ), as i l l u s t r a t e d i n Table I f o r the c u r rent process. Table I.

Variances of e- and VXt f o r c o n t r o l l e r s with v a r i o u s degrees o f c o n s t r a i n i n g . λ 0.

2

Var(eJ/a t a 1.1+9

.01 .02 .OU .06

.10 .25 .50

1.0

1.6k 1.73 1.86 1.95 2.09

2.kh 2.82

3.35

2

Var(VX,)Aj t a 102.8

20.1 13.9

9Λ 7.6 5.8 3.6

2.5 1.7

S e l f - t u n i n g and Adaptive C o n t r o l : I t i s obvious from the preceding d i s c u s s i o n t h a t the design o f f u l l s c a l e p l a n t e x p e r i ments, the development o f s u i t a b l e dynamic-stochastic models, and f i n a l l y the design o f a s t o c h a s t i c c o n t r o l a l g o r i t h m r e q u i r e s a c e r t a i n degree o f experience and s p e c i a l s k i l l s not r e a d i l y a v a i l able i n many companies. Furthermore, these l i n e a r models and the c o n t r o l l e r s d e r i v e d from them are o n l y v a l i d i n c e r t a i n ranges o f p l a n t o p e r a t i o n and only i f the process parameters do not change a p p r e c i a b l y w i t h time. These problems have l e d t o the development


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267

of what are termed "self-timing" and "adaptive" versions of these controllers (2,18,10). The basic idea is illustrated below. Process Controller


+ ι J U p d a t e . I On-line 4. 1t Estimator Based on current knowledge of the process and i t s disturbance characteristics, one may know or choose a reasonable difference equation structure for the controller algorithm. Starting with some assumed i n i t i a l parameter values in the controller equation, the controller can be implemented on the process as shown. The control algorithm is coupled with an on-line recursive estimation algorithm which receives information on the inputs and outputs at each sampling interval and uses this to recursively estimate the optimal controller parameters on-line and to update the control ler accordingly. The idea is to use the data collected from the on-line control manipulations to tune the controller directly. Even starting off with poor i n i t i a l controller parameter values, the self-tuning algorithm usually converges quite rapidly to a stable controller. After a certain amount of data has been c o l lected one can test whether or not the assumed controller form is optimal (in the sense of the minimum variance) and then change i t i f necessary. For processes which operate over a wide range of conditions or production rates i t is to be expected that the process and d i s turbance model parameters w i l l be changing with time. With a very minor change in the recursive estimation algorithm one can use this scheme to track these slowly moving processes and thereby keep the controller well tuned at a l l times. Industrial applications of these controllers have been successfully carried out in Sweden on paper machines (9) and on an ore crusher system ( 3 ) . A good discussion of the practical as pects of the approach and an application to the control of a packed bed tubular reactor i s contained i n a paper by Harris et a l .

(12). Literature 1. 2. 3. 4.

Cited

Aström, K.J.; "Introduction to Stochastic Control Theory"; Academic Press, New York, 1970. Aström, K.J.; Wittenmark, B. Automatica, 1973, 9, 185-199. Borisson, U.; Syding R. Automatica, 1976, 12, 1-7. Box, G.E.P.; Jenkins, G.M.; "Time Series Analysis, Forecasting and Control", Holden Day, San Francisco, 1970.


COMPUTER

268 5.

APPLICATIONS

TO CHEMICAL

Box, G.E.P.; Jenkins, G.M. Applied Statistics,

ENGINEERING

1968, 17,

91-

109.

6. 7.

Box, G.E.P.; Jenkins, G.M.; MacGregor, J.F. Applied Statistics, 1974, 17, 158-179. Box, G.E.P.; MacGregor, J.F. Technometrics, 1974, 16, 391398.

8. 9. 10.

Box, G.E.P.; Newbold, P. J. Roy. Statis. Soc., 1971, Series A, 134, 229. Cegrell, T.; Hedqvist, T. Automatica, 1975, 11, 53-59. Clarke, D.W.; Gawthrop, P.J. Proc. Inst. Elec. Eng., 1975, 122,


11.

1971,

12. 13.

14. 15. 16. 17. 18.

929-934.

Clarke, D.W.; Hastings-James, R. 118,

Proc. Inst. Elec. Eng.,

1503-1506.

Harris, T.; MacGregor, J.F.; Wright, J.D. Proc. Joint Aut. Control Conf., 1978, Philadelphia, October. Kiparissides, C.; MacGregor, J.F.; Hamielec, A.E. Suboptimal Stochastic Control of a Continuous Latex Reactor, 1979, S.O.C. Report, Faculty of Engineering, McMaster University, Hamilton Canada. (Submitted for publication). MacGregor, J.F. Technometrics, 1976, 18, 151-160. MacGregor, J.F.; Tidwell, P.W. Proc. Inst. Elec. Eng., 1977, 124, 732. Mehra, R.K. IEEE Trans. Aut. Control, 1974, 19, 753-768. Wilson, G.T., 1970, Ph.D. Thesis, University of Lancaster, England. (See also Tech. Rep. No.20, Dept. of Systems Engineering.) Wittenmark, B. "A Self-Tuning Regulator", 1973, Technical Report 7311, Division of Automatic Control, Lund, Inst. of Tech., Lund, Sweden.

RECEIVED

November

5,

1979.


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