Real-time estimation of the chain length distribution in a polymerization reactor

Chemical [email protected] Science. Vol. 40. No. Printed in Great &i&n.

10, pp. 1891-1904,

1985. 0

ooo9~2So9/85 53.00 + 0.00 1985. Pergamon Press Ltd.

REAL-TIME ESTIMATION OF THE CHAIN LENGTH DISTRIBUTION IN A POLYMERIZATION REACTOR HANS Institut fiir Systemdynamik

SCHULER+

and ZHANG

SUZHEN*

und Regelungstechnik, University of Stuttgart, Pfaffenwaldring 9, D-7000 Stuttgart 80, F.R.G. (Received

17 September

1984)

method of Kalman filtering has been used to obtain on-line stateestimatesof a batch styrene polymerizationreactor. The temperatures of the plant and the refractive index of the reaction mixture are

Abstract-The

given as measurement signals to a Kalman filter realized in a process computer. Based on mathematical process models, one can reconstruct, from these measurement variables, the conversions of reactants and the chain length distribution of the produced polymer. These (calculated) variables provide a better insight into the actual process state and enable the characterization of product quality during production.

1. INTRODUCTION

Industrially-produced polymers are composed of macromolecules of different lengths. The distribution of these molecules over their chain length determines important technological properties of the polymeric product. It is clear that the characterization of product quality will be facilitated significantly when this chain length distribution is known. For this reason much effort has been undertaken in analytical techniques to measure the distribution function. The most important result of this development is gel permeation chromatography (GPC). This measuring technique has been successfully applied to problems in laboratory practice. The gel chromatographic determination of the chain length distribution works relatively precisely, but it requires a considerable amount of sophisticated equipment and is corrupted by certain sampling lags. For these reasons, continuous supervision of plants by use of GPC measurements has not yet been usual in industrial practice. In industrial production, as opposed to laboratory use, there is a need for simple and disturbanceinsensitive measurement techniques applied in the context of an automation system already installed at the plant. Easily available and undelayed measurement variables are preferred. Starting from these requirements, an attempt is made considering as much model information on the process as possible in order to obtain the conversions of the reactants and the chain length distribution of the produced polymer. With the help of such a model-based measuring technique [ 11, actual information on product quality is available during plant operation. A suitable procedure for reconstructing badly accessible, but interesting process states from easily

+Current address: BASF AG, Abt. Technische Entwicklung, D-6700 Ludwigshafen,F.R.G. *Current address: East China Institute of Chemical Technology, Meilung Road 130, Shanghai, The People’s Republic of China.

available measurement variables, is provided by the method of Kalman filtering[l-31. This state estimation technique has as yet found relatively few applications in chemical engineering. In the field of polymerization reactors, the drift of a vinyl acetate reactor was able to be determined from the pressure drop of the reaction mixture pumped through a capillary tube[4]. For an isothermal reactor it was possible to calculate important characteristic variables (conversion, mean molecular weight) from refractive index and viscosity data [S]. Turbidity measurements were used as a basis of state estimation and optimal control of an emulsion polymerization reactor [6,7]. For the solution of a control problem as well, the effective initiator concentration from temperature measurements and a process model was also determinable [S]. In the context of adaptive control, important parameters (e.g. the heat transfer coefficient) were determined from viscosity data [9, lo]. The energy balance was used in [ll, 121 for the optimal control of PVC-autoclaves, whereby the conversion was estimated. Similar methods are described in [ 133. Time series models were used for the estimation and control of polymerization reactors [ 141, as well as for the detection of fouling in a PE-reactor [ 151. Nevertheless, there seems to be no experimental work published in which the whole state vector of a polymerization reactor is estimated during nonisothermal operation. An essential requirement for the application of Kalman filtering is the knowledge of a mathematical model suitable to describe the process behaviour with satisfactory accuracy. In this paper the free radical solution polymerization of styrene initiated by AIBN was chosen, as the kinetics of this reaction system are relatively well-known and quantitatively describable. For discontinuous stirred tank reactors, one can take well-known process models from the literature and adapt them to the specific laboratory plant. In the following the styrene polymerization models will be described according to the extent of their use in the design of the Kalman filter. After a description of the

HANS

1892

SCHULER

and ZHANG

laboratory plant, the special filter algorithms will be treated. Adjoining this is the presentation and discussion of results. 2. MODELS

OF THE

FREE

RADICAL

STYRENE

POLYMERIZATION

Under appropriate conditions, styrene polymerizes in a free radical mechanism which has the following elementary steps [16-l 81: dissociation of the initiator: ZR”-2Z*

+ N,t

(1)

chain start by the initiator radical: Z*+MR”_P:

(2)

chain start induced thermally: Rm

2M--+=P:

(3)

SUZHEN

from [22] (case b) was used. In a simplified description, this reaction can be disregarded (case a). Monomer is added in the chain propagation step to a radical chain. This forms an active chain which is one unit longer. The reaction rate of this step must be corrected by a term cp, considering the influence of solvent on the reaction rate. The factor cp was determined from the theory of EDA-complexes [23] (case b). The chain termination reaction is inhibited by the gel effect. The correction factor gt. considering the diffusion limitation of the termination step, was taken from [24] (case b). At low conversions. this effect can be disregarded (case a). Alternative assessments on the basis of the free volume theory [25-281 were not considered. The individual reaction rates appear on the righthand sides of the balance equations of the species involved. Assuming quasi-stationary balances of radical species, one obtains the following equations for the mole numbers n of the reactants and products:

dn Wf)

chain propagation:

-=

P:+MRP-Pf+‘,,

Rf

(17)

dn(Z)

P:

+ Pi

dt

=

dn(S) -= dt

(5)

chain transfer to solvent: p:+s-

-(2f‘Rd+2R,+R,+R,-)V

(4)

chain transfer to monomer: P:+M-

-r(M)V=

dt

dn (pi) -= Rs P:+Pi

-R,V

-r(S)V=

-RR,

+r(PJV=

dt

(6)

-r(Z)v=

P:

+P@!+Pi+p

i = I, 2, (7)

(19) -cx)#-‘)

1V Y

1)(1 -@a+‘)

3, . . .

(20)

The chain propagation probability CLis given by a = R,/(R,

Table 1 contains the corresponding reaction rates. Besides the rates derivable from elementary steps (case a), the table also encloses some correction terms resulting from detailed modelling ideas (case b). They will be commented on in the following. The initiator AIBN precisely follows a first order rate law [18-201. The rate constant k,, depends, to a certain degree, on the solvent in which the dissociation occurs [18]. In eqs (8b-d), the sum of individual rate constants was used, weighted by the volume fractions of monomer M (styrene) and solvent S (toluene). In addition to their main reaction (2), the initiator radicals Z* areable to react in side reactions to inert byproducts. The initiator efficiency factor f is the fraction of initiator radicals reacting in the desired step (2). In most cases, f is taken as a constant (case a). A correlation (9b) taken from [22]--considering the complicated reactions-is given as a figure in Table 1 (case b). Styrene is the only important monomer capable of a thermally-initiated polymerization reaction. For the thermally induced chain start, correlation (I 0)

v

[(R,+R)(~ s

+ )R,(ichain termination by combination:

(18)

+ R, + R, + R,).

(21)

The eqs (20) for the polymer chains Pi are the basis for the calculation of the chain length distribution. Their mean values can be determined from the moments n(lzi) =

2

ijn(P,);

j = 0, 1, 2,

_ . . .

(22)

i=l

The following differential equations are obtained for the first three moments:

dn&)

-=

dt

+r(&)V=

[(R,+R,)+&R,-i]v

W&) dt

-

+r(A,)V=

[(R,+R,)+fR,.2]v&

dn(&) ~

=

+r(&)v=

[(R,-+R,)(l

dt

(23)

(24)

x

1

(2+a)lvcI _aj2.

+a)+

+R,.2

(25)

From these moments, one can determine the number average chain length iii,, the weight average chain

Chain Table Initiated

1. Reaction

chain

(case a):

start:

k, =

XS = nW/(n

R,

solution

polymerization

1893

of styrene

(8) @a)

+ kd, xs)

exP

+ MS));

(W

+ n(S))

factor/:

f = f. = const. _f=ftCMl,T)

b):

reactor

= k,,[1]

(k,,x,

efficiency

(case a): @se

in a polymerization

rates of the free radical

x&f = NM)/(n(M)

Initiator

distribution

kd=kd,-,e~p{-~}

b):

(case

length

(94

(9W

0.3

0

2

L

6

[Ml /mol.L-’

Thermally

induced

chain

(case a):

k, s 0

(case b):

k,

Chain

a):

(casebk

R,

=k,,,exp{-s)CM]’

R,

(lob) =k.exp{-&}[M][P*]q

(11)

cp=l

(1 la)

w = (CMl+bb,Csl)/(CM~+b,CSl)

(lib)

&=??!x wm b2 =-

(1 lc)

KS

tlld)

KM

Chain

transfer

to monomer:

Chain

transfer

to solvent:

Chain

termination g, ZE 1

(caseb):

gt = exp{D,

Sum

R,

= k,exp(-

$$]IMlCP*l

= k, exp (-

R,

by combination:

(case a):

xc =

(f0) (lcw

= const.

propagation:

(case

start:

~}cw*1

R, = k,exp{-&][P*]‘g,

chains:

(13)

(14)

(144 +D2xp+Djx~+D4x~+T(DS+D6xp+D7x~+D8xp3)}

1 -_n(~,)/{n(M)+n(S)+n(l)+n(R1)}

of radical

(12)

[P*]

=

2 i=l

(14b) (146

CP:]

=

(15)

HANSSCHULER

1894

and ZHANGSUZHEN

Table 1. (Conrd.) Reaction enthalpy ( -Ah,): (case a):

(-Ah,)

= (-Ah,),

b):

(-Ah,)

=(-A/z,)(T)

(case

=

2.

const.

(-Ah,)

16 L-_; 18 0

/ kcal

50

mol-’

100

150

T I OC

length iii,, and the polydispersity

There is a remarkable temperature variation of the densities p as well as of the volume contraction factor E.

I&:

(26) (27)

mm

In order to achieve sufficient accuracy, one must solve a relatively large number of balance equations for chains Pi. In this work, 20 chain lengths were selected such that the chain length distribution seemed to be a smooth function of i. An approximation by Laguerre’s polynomials [29, 301 gave unsatisfactory results under unstationary operating conditions. When calculating the reaction rates as a function of the concentrations. =

!P+);

[Z] = y:

[S] = Fete.

(29)

the reactor volume V is needed. In a first approximation it is constant for batch processes, and equal to the volume of the reactants (case a). In a rigorous treatment, the volume is a function of the composition and the temperature (case b): Y = E(M)n(M) + l?(S)n(S) + zqP)n(P).

(30)

The specific molar volumes V can be calculated from the densities p and the molar masses M of the components: iqh4)=-.

M(M) ~0’4) ’

(31)

M(S) P(S)

B(S) = -;

D(P) = a(M)(l

(34a)

P(S) = PO(s)+ @(S)(T--T,)

(34b)

E = &#J + &‘(T-_a).

_

u=m,__I. -

[M]

P(M) = Pa(M) + P’(M)(T--T,)

-E).

(33)

(344

Using these relations, one can describe isothermal polymerization processes in batch stirred tank reactors. The polymerization of styrene is a strongly exothermic reaction. The reaction enthalpy of the overall reaction rate r(M) is often given as aconstant (-Ah,), (case a). In a detailed treatment, the reaction enthalpy is a function of the temperature (case b). The relation depicted in Table 1 was used in the calculations. It was taken from [ 171. The temperature of the reaction mixture can be obtained from the heat balance.

Cs

= k, F,(T, - T) + (PC,) i/Ret,(Tserr-T + (-

Ah,)r(M)V.

) (35)

T Refris the temperature of the refractometer installed at

the laboratory plant, which is passed by the circular stream Va,t, of the reaction mixture. The overall heat capacity C is the sum of that of the reaction mixture and the heat capacity Wof the empty reactor c=mc,+FK

(36)

In a first approximation, C is a constant (case a). But the specific heat cP of the reaction mixture can vary due to fluctuations of composition and temperature (case b) CP= c,(M)x(M)

+ c,(S)x(S) + c,(P)x(P).

(37)

1895

Chain length distribution in a polymerization reactor The temperature-dependent components c,(M)

specific

heats

of

= c&l(M), +c;(W(T--T,)

(38a)

c,(S)

= CP(S)cI + c’,(S)(T--T,)

(38b)

c,(P)

= cp(P),

(38~)

are weighted x(M)

+cZ(W(T---T,)

The signal flow diagram of the simple model (a) is shown in Fig. 1. It is composed of two subsystems in a serial structure. As a consequence of this signal structure, one can treat the equations for the temperature and the conversions of monomer and initiator separately. The state variables of this first subsystem act as input variables to the second subsystem. The state variables of this subsequent subsystem describe the chain length distribution and its characteristic parameters. No feedback exists from its output to the dynamics of the temperature and conversions. The detailed model (b) has additional signal paths resulting from lesser effects of modelling. Because of their smallness, they do not influence the principal dynamic behaviour of the system. The signal flow diagram of Fig. 1 can therefore be used as a suitable basis describing the essential system structure and the important features of its dynamic behaviour.

by their molar fractions = n(M)/@(M)

+ n(S) + n(& ))

x(S) = n(S)/(WW x(P)

all effects discussed previously are considered. This model thus describes the process behavior-u in a very detailed manner. Similar models were used in [2+28, 38, 391.

the

(39a)

+ n(S) + n(M)

= n(&)l(+W

(39b)

+ n(S) + n(& )).

(396

The mass m of the reactants and the heat capacity Wof the empty reactor have constant values to be determined experimentally. Given the enthalpy balance, the entire set of equations is available that is necessary for modelling the free radical styrene polymerization in a well-stirred nonisothermal reactor. The initial conditions no(M), n,,(1), no(S), n,,(P,), ttr,(.Xj)and To of the differential equation system are given by the composition of the recipe and the starting temperature. These balance equations are able to predict the behaviour of the nonisothermal polymerization reactor within the model accuracy. The derived model equations are special cases of the general nonlinear state equation dx = f(x, u, t)+Gv; dt

x(0)

= x,,,

3. EXPERIMENTAL For experimental work, a laboratory plant was constructed. Its flow diagram is shown in Fig. 2. As a reaction vessel, a glassy stirred pot with a 2 1 capacity was used. It is covered by a Teflon plate which contains several cuts. The reactor content is mixed by a blade stirrer of 100 mm @. The reaction mixture is covered by a nitrogen atmosphere. The coolant is pumped from a thermostat through the cooling jacket. Destabilized styrene is stored in a refrigerator. Azobis-isobutyro-nitrile (AIBN) is used as an initiator, toluene (in the quality pro andysi) as a solvent. The reactants are transported by a pump from a storage vessel, covered by nitrogene, into the reactor. Another way is to pour them directly into the reactor via a funnel. Ni/Cr/Ni-thermocouples are installed at different locations of the plant, providing information on the present reactor temperature, the refractometer temperature, as well as the inlet- and outlet-temperature of the cooling jacket. After their preamplification, these signals are selected by a scanner. Having passed

(40)

which represents a formal equation for describing the behaviour of dynamic processes. Depending on the sophistication of modelling, two models of different complexity are used in this paper: (case a): the simple model (a) has the state vector xr = {T, EM], [I], [Pi]. [Ai]}. On the right hand sides of the model equations (40), all lesser effects and sophisticated approximations are disregarded. This frequently used model [31-371 possesses the minimal structure required for describing the typical phenomena of free radical polymerization. (case b): the complex model (b) has the state vector xr = (T, n(M), n(l), u(S), n(P,) n(Aj)}e On the right-hand sides of the corresponding eq. (40)

TO

"1

-

V2

-

y

-

g

[Ml,

[II,

= f, LT,IMl.[II)

"1

T

a _dlAl]

TK

TRefr

+

w

= f,(T.[MI.tII)

l VT

w

= f~(T,IMl,tI1)

+

[MI

“I

[II

"I

*

d

1 St

subsystem

v3

dt

5 9

=

f,(T.IMI.[Il)

= f,(T.[MI,[II)

+v,

“l’-

+ v, [P,]

L

2 “d

subsystem

Fig. 1. Signal flow diagram of the process model (a) describing a polymerization reactor.

1896

HANS SCHULER

and

ZHANG

I

I

._-

1

SUZHEN

nitrogen

I stirrer

I

I I

‘Ltom&ij

L-,-_-J,

refrqemtor

I

hi

’ ice Lbth

01,

temperature

signals

scanner

I

refmctive

index

signal

Fig. 2. Flow sheet of the laboratory plant.

another amplifier, they are given to the A/D-converter at the process computer (PDP 1 l/34). Their sampling time is 5 s. In order to supervise the conversion, the refractive index of the reaction mixture is measured with a process refractometer. The fluid is pumped through this instrument thermostatted to 25°C. The amplified measurement signal is given to the A/D-converter of the process computer every 60 s. The values of the specific parameters (k,F,) and W were determined from cooling and heating experiments in an operation range with negligible reaction rates. The experiments with reaction were carried out with the same total reactant mass and stirrer speed. A typewriter serves as a communication device to the computer, as well as an output device for the results computed on-line in equidistant time steps during the runs. The actual estimations of the chain length distribution are drawn on a plotter.

4.

MEASUREMENT EQUATIONS

The actual values of the measured temperatureTand the refractive index x make access possible to a part of the state vector. From a calibration with standardized mixtures of monomer, solvent and polymer, the empirical relation

CMlo--CM1 = Cl+c*(x-XJ CWo

(41)

were found valid at constant solvent concentrations. Using a thermostatted refractometer, one can determine, from eq. (41), the actual monomer concentration [M]. The measurement equations y, =T+w,

(42)

y,=CMl+w,

(43)

are special cases of the general vectorial measurement

1897

Chain length distribution in a polymerization reactor

update of the measurement

equation y = h(x)+w.

(44)

The temperatures r,,,,, TKaus,Tserr, which were also measured, must be treated as input variables in our context. The measurement signals y do not provide the exact values of the state variables. The noise w, added in signal transmission, amplifier drifts, etc. will cause, as well as statistical errors in the empirical measurement equations, a certain disturbance of measurement signals. Considering this fact, the following problem arises: calculation of a best actual estimate 2 of the whole state vector x from the measurements y which are corrupted by the noise w, under consideration of the given process model {f, h} disturbed by the modelling uncertainties v. 5. KALMAN

FILTERING

The problem of the reconstruction of inaccessible state variables from easily available but disturbed measurement variables can be solved with the help of Kalman filtering [l-3]. Hereby it is supposed that the dynamic behaviour of the process can be predicted by the system eqs (40). These equations contain modelling errors v having the following statistical properties: E(v(t)}

= 0;

E { v(t)+(t’)}

= Q (r)6(t, t’). (45)

The measurements of y contain information on the actual process state due to the discrete measurement eqs (44). The following statistical properties of the measurement noise w will be assumed: E{w(k)j

= 0;

E(w(k)w=(k’)}

= R(k)b(k,

k’). (46)

At the beginning of the operation the initial conditions 2, = E { x0} are available from the specific recipe. These initial conditions are given with limited accuracy, due to the reproducibility of volumetric measurements determining the recipe components PO = E((x,

-a,,(XO-%e),}.

(47)

Using this information, one can realize the state estimation with the help of the following algorithm (extended Kalman filter using a process model continuous in time, and given time discrete measurements) C&33: predictions in measurements:

the

period

dS = f(x^, II, r); dr dP dt

= FP+

PF’+

GQGr;

(t&, fk+l)

between

the

2 (t*) = x^(k/k)

(48)

P(r3 = P(k/k)

(49)

a priori estimates: %(k+

l/k) = %?(&+A?)

(50)

P(k + l/k) = P(t, +Ar)

(51)

f(k+

1) = h(ii(t,+At})

(52)

y(k+

1) = y(k+

1)-y^(k+

y(k + 1):

1)

(53)

S(k+l)=H(k+l)P(k+l/k)HT(k+l)+R(k+l) (54) K(k+

1) = P(k+

l/k)H=(k+

l)S-‘(k+

1)

(55)

Q posteriori estimates: ~(k+l/k+l)=~(k+l/k)+K(k+l)y(k+l) P(k+l/k+l)=

(56)

(I-K(k-+l)H(k+l))P(k+l/k). (57)

In these equations

the Jacobians (58)

of system and measurement equations are used. As a result of Kalman filtering, one obtains the expected state vector and its covariance matrix: < = E(x);

P = E((x-2)(x-%r}.

(59)

The estimates are said to have an approximately normal probability distribution. A necessary condition for the application of this filtering algorithm is the observability of the state vector from the available measurement variables. The analysis of local observability [40] shows that the rank of the observability matrix I-‘=

{H,

HF,

HF’,

. . . , HF’“-“1

(60)

is three when taking the simple process model (a) as a basis. A detailed examination shows that one can reconstruct from temperature (and refractive index) measurements only the states of the first subsystem according to Fig. 1. The other state variables, particularly those describing the polymer and its chain length distribution, are not directly observable. Their values can be determined excepting a constant bias. The detailed model (b) considers only modelling effects of a higher order, so that the interesting variables of the CLD are practically not observable in this model, either. The limited observability is a consequence of the system structure depicted in Fig. 1: the measurement variables are part of the input variables which enter the second subsystem describing the chain length distribution. A variation of the distribution has no feedback effect on the input of the first subsystem. Therefore, they are not observable from the measurement variables resulting from this first subsystem. Because of this special system structure, a filter of reduced order must be designed. The state vector to be updated contains only the observable states (temperature, monomer and initiator). Parallel to this reduced order filter one must integrate the balance equations of the solvent, the chain length distribution and the moments without correcting these variables in the update step. Figure 3 shows the signal flow diagram of this special filtering structure. The predictions are realized using the detailed model (b) in order to obtain good a priori estimates of the

1898

HANS SCHULER and ZHANG SUZHEN

TK

_

control variables

PROCESS

TRefr

FILTER

g

= f,~i,~hl.~il~+K,,ly,-il+K,2~yt-t~11.

-9

__c

= fi(i,[~l,[il)+K,,Iy,-i)+K,,Iyz-I~l) -

y

^

= f,li.~lil,~il,

T : qp

rF11

= f,~i,r1;1l,til)

I

+K~,(Y,-~)+KJ~(.v~-I~~II state

i rfil ril

estimates

1

Ir

rq1 $1

Fig. 3. Signal flow diagram of the special filter structureadapted to the polymerizationreactor model in Fig. 1.

state vector. The calculation of the a posteriori estimates is carried out suboptimally based on the simple process model (a). Doing so, one reduces the sensitivity of the algorithm due to disturbances; another effect is improved stability and shorter computing times. The algorithm was realized on a process computer PDP 11/34 in the RT-11 operating system. All programs were written in Fortran. 6.

RESULTS

In the following, some results of the on-line experiments are presented. (a) State estimation during heating up of the process Figure 4(a) shows the measured variables T, Tk, [M] during heating up of the reaction mixture to the reaction temperature of approx. 85°C. In this run of 2 h length, one can see the onset of the polymerization reaction. The converted quantities of reactants and the characteristic properties of the polymeric molecules produced are to be estimated. The difference between the input and output temperature of the cooling jacket results from the heat of reaction. Both jacket temperatures are higher than the reaction temperature. This fact is caused by the stream passing the thermostatted refractometer of 25°C. Steps in the course of the measured monomeric mole number are explained by its lower sampling frequency. Figure 4(b) shows the estimated values of the observable state variables. The estimates L?and 6(M) are practically identical to the corresponding measurements. The estimated mole number of the initiator decreases continuously because of the onset of reaction

at higher temperatures. The lu-confidence intervalls depicted additionally are very small for the measured states. The estimate of the initiator mole number is less accurate because this estimation is based mainly on model information. The predictions of states of the second subsystem provide the desired information on product quality. Figure 4(c) shows the courses of the estimated moments Q,), n^(n,), A(&); in Fig. 4(d) the estimated means &, r%, and u^of the chain length distribution are depicted. One recognizes from the moments that only few polymer are produced at lower temperatures. The chains of these polymers are remarkably long and have a large polydispersity. At higher temperatures, these quantities change only to a limited degree. This can also be stated from the chain length distributions, depicted in Fig. 4(e), in time intervals of 5 min. These chain length distributions are estimates of the real distribution. They are obtained under unstationary conditions on the basis of the underlying model information of the process. In addition to the reconstruction of state variables, the Kalman filtering technique allows supervision of other quantities which have been inaccessibIe till now. As an example, Fig. 4(f) shows the course of the reaction rate r^((M)describing the actual conversion of monomers. The onset of the reaction can easily be detected as well as its diminishing as a consequence of reactant consumption. Seen as a whole the estimated states allow deeper insight into the actual process behaviour. They enable continuous supervision of variables which were inaccessible from the original measurement signals.

Chain length distribution in a polymerization

measurements

T/V_

:

1899

reactor

; n(M) lmol

80

01 state

60 t /min

estimales

O! 1, 120 0

I

0

T/Y

i?(I) I ma1

I

1

80

0

60 t I min

0

0

1

60 t I min L

500

120

0,000

L

mw,mn

U

0,7

iiT, ^ m,

300

,;

dz

0.6 0,5 fLl

0

i.fl(Pi)

60

I fi(X7)

i.ii(P,l/

3.10-3

30 ml”

t=l,ZO

i?(hjl mbn

@

I!!& ;5

2-1o-3

120

t/min

t I mln

t I min

120

60 t/min

0

3.1o-4 2

1-10-j

^r(M) / mol. l-‘-s“

1o-4

0 ,iCM)

1-1o-4

0

LOO n

800

0

0

MO

0 IEd 0

000

n

60 t I min

120

Fig. 4. State estimation during heating up of the polymerization process: (a) measurement variables, (b) estimates of the observable state variables, (c) estimates of the moments of the chain length distribution, (d) estimates of the means (numberaveragechain lengthiiz,,weightaveragechainlengthiii,, polydispersityU) of the chainlengthdistribution,(e) estimatedchain lengthdistributionsin equidistant time intervals of 5 min, (f) estimated reaction rate r(M) of monomer conversion.

(b) State estimation under unstationary operating conditions In another experiment the response of the filter to a temporary change in operating conditions is examined: the cooling temperature is elevated and, after some time, decreased to its former value. In Fig. 5(a), the measurement values of the reactor submitted to these operating conditions are depicted. unstationary Figure 5(bb(f) shows the responses of all important variables estimated by the filter. From these calculations, one can see that shorter chains with a narrow

distribution are produced at higher temperatures. The filter follows the changing conditions to a reasonable extent. (c) State estimation exclusively measurement

from

temperature

signals

In the fundamental work [40], the estimation of the conversion of chemical reaction from temperature measurement signals alone is proposed. Following this proposition, the temperature signals are considered exclusively when updating the measurements. For

1900

SCHULERand ZHANG SUZHEN

HANS

:

measurements

T/Y

;n(M)lmol

,.

I;=:

0 state

:

estimates

@I 4 120

I

0

60 tlmin

I,iqg

‘[ZiJ 0

60 t I mln

0

120

60 t /min

t I min 3.10-3

i-h(P, I/ I?

120

‘0 tl

tlmin

60 min

120

t/min

,.,,-3~p 2.10-3

70

^r(M) I mol

3.1o-L

I-’ s-’ 0

2 10-4 _^r(Ml

l.lo-4 0

1 .10-3 0 I!?& 0 n

800 n

Ezl

0

60 t t min

120

Fig. 5. State estimation of a polymerizationprocess submitted to unstationaryoperation conditions (for legend see Fig. 4).

purposes of illustration, the calculations were done with the data of the second experimental run depicted in Fig. 5(a). In a comparison of monomer measurements with its calculated estimates one can judge the accuracy of the estimation. Figure 6 shows the monomer measurement signal and its estimate based on the filtered temperature signal. The size of the estimation error remains limited during this experimental run. The filter algorithm based exclusively on temperature signals showed in some cases, a certain sensitivity to parameter and modelling errors. In experimental runs covering a broad range of operating conditions a monotonic filter divergence is sometimes observed.

(d) Influence of model complexity on state estimation In order to examine the influence of model complexity, one can use different process models in the prediction step of the filter algorithm. The simple process model (a)and the complex model (b) are compared here. In the update step, only the temperature measurements of the following experimental run have been used. During the experimental run shown in Fig. 7, the refractometer had been switched off ( pRCRcfr -= 0). The reaction temperature is then higher than the cooling temperature. As an intensive disturbance the cooling stream passing the jacket was decreased to a very low value. Up to this moment, marked by an

Chain length distribution in a polymerization

1901

reactor

creased parametric sensitivity of the complex model when applying it in a Kalman filter.

n(M),fi(M)/mol 10

7. DISCUSSION

8n(M)

6-

4

I 60

0

tl

120 min

Fig. 6. Estimation exclusively from temperature measurement signals. Comparison of estimated and measuredmole number of monomer.

arrow in Fig. 7, all predictions are performed with wrong parameters. Figure 7(b) shows the filtering results using the complex model (b). As soon as a noticeable modelling error exists, a filter divergence occurs, leading to unreasonable estimates. The filter operation based on the simple model (a) remains in a stable range. This comparison indicates also an in-

By applying Kalman filtering techniques to experimental polymerization runs, it has been possible to calculate--from easily available measurement signals (temperature, refractive index jthe actual estimates of the chain length distribution and its characteristic values during operation. In addition to these polymer properties, one obtains the quantities of all species present in the reactor. They enable the calculation of conversions at each time of the production cycle. Based on kinetic rate laws, one can obtain on-line, the reaction rates, the composition of the reaction mixture and other interesting variables. With the Kalman filter a powerful instrument is available for gaining deeper insight in the process during its operation. This procedure will allow a supervision of variables which were not accessible previously. An implementation of this model-based measuring aid can therefore lead to more efficient production and to safer operation of the process. An application of this technique to the supervision of the exothermic reaction was described earlier in [41].

120

60 t I min

4-

41 120

0,020

i?(I)/

0

60

120

t I mln

t lmin

i?(I) / mol

mol 0,020 a

0,010 E

/ R(I)- Gr

C(I)+ Gl ”

0,010

Fill) 0,000 0

60 t I min

120 t I min

Fig. 7. Comparison of estimates resulting from models of different complexity. Predictions on the basis of a simple (a) and a complex model (b).

1902

HANS

SCHULER

and ZHANG

The runs have confirmed the expectation that a consideration of additional measurement signals will lead to better results of filtering. GPC-measurements can also be taken into account in the filtering concept [42]. Detailed models will lead to better estimates as well, but they require accurate knowledge of their numerous parameters. Besides, they sometimes show a certain sensitivity to disturbances and wrong parameters. In contrast to this are simple models not always able to describe all details of process behaviour. The estimates of the chain length distribution correspond to published data to a sufficient degree of accuracy. At the time this work was done the estimates could not be compared with measurements because suitable measurement devices were not available. In the meantime a GPC has been installed in our laboratory. We hope to present a comparison of CLDestimates and measurement data in a subsequent contribution. 8. CONCLUSIONS

Acknowledgements-The authors thank the “Deutsche Forschungsgemeinschaft” for its financial support. They are obliged to Prof. Gilles for his valuable stimulating ideas and discussions. Mrs Dreis is thanked for writing essential parts of the programs. NOTATION

All concentration of species A will be denoted by [A]. The expected value of a variable x is 9. total heat capacity C constants of gel effect Di activation energy expectation of the variable x %xi F jacobian of the system equation f cooling surface F1, disturbance matrix (G = I) G H jacobian of the measurement equation b identity matrix I initiator I initiator radical I* K correction gain matrix equilibrium constants of EDA-complexes K M’ KS M(A)

P

P,

P*

Q R

W

mass

of

species

gas constant; reaction rate of individual reaction steps covariance matrix of the innovation process y solvent temperature volume volumetric flow rate heat capacity of the empty reactor constants in the solvent correction factor cp constants in the calibration relation for

R S s T V ti W b l/Z Cl/Z

the

A

covariance matrix of the state estimation error (x - 2) dead chain of length i

refractometer

specific

CP

heat

of

the

reaction

mixture

heat of species A temperature coefficient of cp vector of modelling equations initiator efficiency factor specific

CP

.:

gel

SC

effect

b (Ah,)

vector

i

chain

of measurement

reaction

equations

enthalpy

length

number

j k

of

number

the

moment

of

the

rate

measurement;

constant heat

kw m m, m, n

transfer

coefficient

mass number

average

weight mole

chain

average

dimension

n(A) r(A) t

of

number

reaction

rate

chain

length length

the state of of

vector

A A

species species

v(A)

time sampling time vector of the input variables polydispersity vector of the process noise (modelling error) specific molar volume of species A

W

vector

X

state

At u u V

; 6 E : p(A) P’

I7

the

measurement

noise

fraction

of

species

A

measurement vector

Y

Greek a

of

vector

molar/volume

~(-4)

monomer molar

radical chain of length i sum of all radical chains covariance matrix of process noise v covariance matrix of measurement noise

P:

G(A)

In this contribution, a Kalman filter was applied to state estimation of a reaction system in which styrene is polymerized in a free radical mechanism. Because of the complexity of this chemical process, the results obtained have not seemed to be trivial. It was possible to estimate the interesting process variables from easily accessible measurement variables. The Kahnan filter can be interpreted as a model-based measurement method for the conversion of the reaction and the chain length distribution of the produced polymers. The method enables a calculation of these variables from easily measurable quantities on the basis of process models.

M

SUZHEN

letters

chain propagation probability innovation vector observability matrix &function contraction factor temperature coefficient of E moment of the chain length distribution density of species A temperature coefficient of p standard deviation

1903

Chain length distribution in a polymerizationreactor

solvent correction factor refractive index

cp

x

Cl41 MacGregor Cl51

Indices I K Kein,aus

M P R

Refr s r” i, i’ i k Y P S t W

Polymers,

York 1952.

1958 80 779. l-*01 Kulkarni M. G., Mashelkar

L. K.,

13.

E71 Kiparissides C., MacGregor J. and Hamielec A. E., Can. J. them.

Engng

1980

1969

58 48, 56, 65.

159

Instrument.

A.I.Ch.E.

Symp. Ser. 1976

72, 222.

Cl01 Notton A. R. M. and Choquette P., Proc. Symp.,

Cl11 Gran [12]

Technol.

57.

c91 Ahlberg D. T. and Cheyne I., Toronto, p. 2, 1968. H., Andersen J. A.

and

M.

En&g

to

Gran H., Grande H. and Lange S., Proc. oj’4th ISCRE VIII-363, 1976. [ 131 Amrehn H., Automatica 1977 13 533.

J. 1967

k &#=3.105x10-~s-‘;

fo = 0.5; k, = 3.95 x to9 Imol-‘s-‘; k, =

K.fK, E,

1.051 x lO’lmol-‘s-‘;

PI c405

E dO = 30.78

kcal

k ds = 2.479

x 10-e

R = E,

mol- ’ ;

1.9864 x IO-’

= 27.98

E, = 7.064

s-1;

kcalmol-‘K-l;

kcal mol- ’ ; kcal mol- ‘;

k, = 2.31 x 10” lmol-i

= 12.667 kcalmol-‘;

k, = 5.58 x 10’ lmol-‘s-i;

17.164kcalmol-‘;

R. A. and Doraiswamy

J. appl. Polym.

Sci. 1972

16 749. Kirchner K. and Riederle K., Angew. Makromol. Chem. 1983 111 1. Zeitz M., Fortschr.-Ber. Reihe 8, Nr. 27, VDI-Z 1977. Schuler H.. Fartschr.-Ber. Reihe 8, Nr. 52, VDI-Z 1982. Schuler H., 4th IFAC Con& Ghent/Belgien, p. 369, 1980.

= 1.28; Es =

Sot.

13 1081.

DATA

,

J. Am. them.

Sci. 1980 35 823. Radical Polymerization.

(381 Hui A. W. and Hamielec A. E.,

J.. 4th

Applicat&ns

k, = 1.58 x 1Ol5 s-i.

S., Polymerisation. VerlagChemie,

Academic Press. London 1961. c**3 Ho&ink R. and Stavermann A. J. (Eds), Chemie und Kunststofe, Band 1. Akad. Technoloaie der Verlagsg&llschaft, Leipzig 1962. c*31 Henrici-Olive G. and Olive S., Z.fiir phys. Chem. Neue Folge 1966 48 35, 51. c*41 Jainsinghani R. and Ray W. H., Chem. Engng Sci. 1977 32 811. c*51 Marten F. L. and Hamielec A. E., ACS Symp. Ser. 1979 104 43. WI Ross R. T. and Laurence R. L., ACS Symp. Ser. 1976 160 72, 74. c*71 Schmidt A. D. and Ray W. H., Chem. Engng Sci. 1981 36 1401. Hamer J. W., Akramov T. A. and Ray W. H., Chem. PI Engng Sci. 1981 36 1897. c*91 Rav W. H.. J. Macromol. Sci. Revs. Macromol. Chem. 19;2 CE (lj, 1. c301 Rav W. H. and Laurence R. L., in Chemical Reactor Theory-A Review, p. 532. Prentice-Hall, Englewood Cliffs. _ _ c311 Kenat T. A.. Kermode R. I. and Rosen S. L., Ind. Ennnn Chem. Proc: Des. Develop. 1967 6 363. c3*1 Thiele R., Herbrich K.-D. and Fischmann J., Chem. Tech. 1970 22 455. c333 Thiele R., Chem. Tech. 1967 19 221. 1967 15 __ U.. Reaelunastechnik c341 Gilles E. D. and Kn&tY 199, 262. c353 Zamani H., Daroux M., Greffe J. L. and Bordet J.,Chem. Engng Commun. 1982 17 297. C361 Wittmer P., Ankel T., Gerrens H. and Romeis H., Chem. Ing. Tech. 1965 37 392. I371 Duerksen J. H., Hamielec A. E. and Hodgins J. W., A.I.Ch.E.

IFAC/IFiP

Hessen

IFACf IFIP Cant on Digital Computer Process Control, Zurich, p. 312, 1974.

Chem.

c*11 Bevington J. C..

V., Nichtlineare Filterung. Oldenbourg, Miinchen, 1980. c41 Hyunn J. C., Graessley W. W. and Bankoff S. G., Chem. Engng Sci. 1976 31 945, 953. Jo J. H. and Bankoff S. G., A.1.Ch.E. J. 1976 22 361. g; Kiparissides C., MacGregor J. and Hamielec A. E.,

Adams P. G. and Schooley A. T.,

Oliv6

Its

New

Weinheim 1969.

Cambridge 1974.

PI

and

Cl91 Van Hook J. P. and Tobolskv A. V.,

c31 Krebs

27

G.

Katalyse-Kinetik-Mechanismen.

REFERENCES

J. 1981

and Derivates. Reinhold,

Copolymers

El’31 Henrici-Olive

C’l Gilles E. D., Technisches Messen 1979 225, 271. PI Gelb A. (Ed.), Applied Optimal Estimation. MIT Press,

A.I.Ch.E.

52.

Cl61 Dhlinger H., Polystyrol. Springer, Berlin 1955. Cl71 Boundy R. H. and Boyer R. F. (Eds), Styrene;

reference value initiator cooling system input/output of cooling jacket monomer polymer reaction refractometer solvent dissociation of initiator chain transfer to monomer chain lengths number of the moment number of the measurement thermally initiated chain start initial condition chain propagation chain transfer to solvent chain termination weight

B

J. and Tidwell P. W., Computer ripplications to Chemical Engineering. ACS, p. 251,198O. Hwu M. C. and Foster R. D., Chem. Engng Prog. 1982,

k, =

s-‘;

1.255 x lo9 lmol-‘s-l;

HANS

1904

ZHANG

SUZHEN

1.68 kcalmol-‘;

D,

=

3.000541

02

=

DB =

-

Ds

=

-38.48330;

0, D,

x 1O-3 K-‘; -88.93976 1.430559 x 1o-2 K-‘;

04

=

6.61756;

= = 290,

p,,(M) p0 (S)

x 10-2K-‘;

3.537124

x lo-=

20, 50, 85, 120,

i =

320, 380. 440,

=

0.9238

kg I ~ ’

=

0.8855

kg 1- ’ ;

Eg =

140, 160, 180, 200, 500, 600.

= 0.39 kcalK-‘kg-‘;

p’(S) c,(M) c;(S)

= 0.29 kcalK-‘kg-‘;

( -A/I~)~

= W=

V,(M)

1 I; V,(S)

ct =

-0.05839;

T Rerr =

=

1 1;

5 s;

=

0.2 K;

=

0.7K; 2.2 x 10e4 10-6moll-‘s-‘;

=

c+ =

K-‘;

kgl-’

4.436

, kcalK-‘kg-

x 10_4K_‘-

I-03 x 1O-3

= 0.80 x 1O-3 = 0.965

K-l;

x lO-3

kcalK-*

=

3.55 x IO-3

kcalK-‘s-l;

=

3.28 x lo-”

kg;

31.571;

Xs = At2 6

1.5200;

=

4x

=

60 s;

lo-Jls-‘;

=

I x 10-3molI-‘;

‘; kg-‘;

kcalK-2kg-1;

0°C;

k,F,

‘Rc‘r

mol;

kgl-’

9.55 Y 1O-4

ma(l)

25°C;

=

=

TB =

0.25 kcalK-‘; =

Ar,

c;(P)

17.8 kcalmol-‘;

looo;

= 8.766 x lo-“

p’(M)

E’ =

Cam

-30.48225; 42.34226;

230, 260,

700, 800, 900,

0.1506;

= 0.37 kcalK_‘kg-‘;

Cam

K-‘;

;

cp(M),

&

and

D5 =

E, =

&I,

SCHULER