extended observer for the polymerization of polyethylenterephthalate

7/31/2019 Extended Observer for the Polymerization of Polyethylenterephthalate

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Submitted to IFAC-World Congress 1996

(Symposium 7a: Industrial Applications / Chemical Process Control)

EXTENDED OBSERVER FOR THE POLYMERIZATION

OF POLYETHYLENTEREPHTHALATE

PAUL APPELHAUS, SEBASTIAN ENGELL

Universitaet Dortmund

Chemietechnik, Lehrstuhl Anlagensteuerungstechnik

D-44221 Dortmund

Abstract: In the batch polymerization of polyethylenterephthalate, the main reaction is an

equilibrium-reaction. The removal of ethylenglycol from the melt determines the velocity

of the polycondensation process, because it shifts the equilibrium to the side of chain

growth. In this paper, we report about the implementation of an observer, which is based

on a simple polymerization model at a pilot plant scale reactor. The observer is able to

determine two important concentrations in the polymer melt as well as the product of mass

transfer coefficient and specific surface. The knowledge of the later parameter offers new

possibilities for improved process control.

Key words: observer, polyethylenterephthalate, parameter estimation, Extended-Kalman-

Filter


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1. INTRODUCTION

Polyethylenterephthalate (PET) is a linear polyester and is well known as the most important material for man-

made fibers. It also becomes more and more popular for beverage bottles. In 1993, the world production

capacity was more than 11 million tons/year.

In recent years, considerable efforts have been made to model the PET-polycondensation process. Depending on

the aim of modeling, very complex chemical reaction models as described e.g. by Laubriet (1991) or a simple

model with two free parameters as introduced by Tomita (1973) were developed. Rafler (1983) proposed a

model which only takes the main reactions and the mass transfer of ethylenglycol into account. This model is

well suited for process control purposes. Its accuracy was found satisfactory in various experiments at a pilot

plant by Niehaus (1991) and Spies (1993). When the polymer molecular weight increases during

polymerization, the mass transfer of volatiles from the melt to the gas phase becomes the limiting factor of the

process. Laubriet (1991) and Rafler (1989) stated that the estimation of this mass transfer coefficient is still an

open problem. Efforts have been made by Rafler (1985) to determine the diffusive and convective mass transfer

coefficient by examining thin film systems. But this does not solve the problem for real operating conditions

because there the specific surface is not known. The latter is influenced very much by the development of

bubbles in the melt. Rafler and Fritsche (1989), who made experiments in a stirred laboratory reactor, report

that the size and number of the bubbles depend in a very complex manner on the rheology of the melt as well as

on stirrer geometry and speed. It is not possible to observe the bubble content and size optically under operating

conditions yet because of the high temperatures and low pressures necessary to produce PET.

In our work we tried to estimate the product of the overall mass transfer coefficient () with the specific surface

(a) between the polymer melt and the gas phase during the polycondensation step of discontinuous

polymerization of polyethylenterephthalate in a stirred tank reactor. A model of the main chemical reaction

together with a mass transfer model constitute the general process model. This is the basis of our nonlinear

observer which is extended to provide an estimate of the mass transfer parameter a. Preliminary results of our

work were presented in (Appelhaus, Engell, Grosse-Kock, 1995). In this paper we compare two observer

concepts, a nonlinear observer with fixed error feedback gain and an Extended-Kalman-Filter. We show results


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obtained at a real laboratory scale process equipped with standard industrial measurement and control

technology.

2. CHOICE OF THE PROCESS MODEL AND OBSERVABILITY ANALYSIS

Using process models from the literature as described by Tomita (1973), Rafler (1983) and Laubriet (1991), we

did experimental work on a 20 kg-polymerization reactor at AKZOs experimental plant in Obernburg to

determine the model which is best suited for our purposes. We decided to use the model described by Rafler

(1983), as it reproduced our experimental results well and has a reasonable physical background without being

too complex. It describes the equilibrium reaction, which gives rise to the formation of the polyester mac-

romolecules and the thermal decomposition as well as the mass transfer of ethylenglycol to the gas phase.

Besides the concentration of ethylenglycol [EG], the model uses the concentrations of the functional groups,

[OH] for OH-endgroups, [E] for ester groups which can be taken as the monomer groups, [COOH] and [VIN]

for COOH- and Vinyl-endgroups. The model is given below:

[ ] [ ] [ ]2

1

1

+OH

k

k E EG'

equilibrium reaction

[ ] [ ] [ ] + Ek

COOH VIN2irreversible thermal decomposition

[ ][ ] [ ]( )

d EG

dta EG EG

'*=

rate of the mass transfer of ethylenglycol from the

melt to the gaseous phase, [EG*]: concentration of

[EG] in the melt surface

( ) ( )

( )

( )

dx

dta x x k x x x

dxdt

k x x x

dx

dtk x

dx

dtk x x x k x

11 1 1 2

2

1 4

21 2

21 4

32 4

41 2

2

1 4 2 4

1

28

8

1

28

= +

=

=

=

*

( 2.1)

x1: concentration of ethylenglycol in the melt

x1*: concentration of ethylenglycol in the melt surface

x2: concentration of OH-endgroups in the melt

x3: concentration of COOH-endgroups in the melt

x4: concentration of ester-groups in the melt

k1: reaction velocity, polymerization

k1: reac. vel., depolymerization, in eq (2.1) it is

assumed that k1 = 8 * k1 (Spies 1993)

k2: reaction velocity, thermal degradation


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Pnx

mmol

g=

2000

62

192

2(2.2)

Pn: degree of polymerization, calculated from [OH]

endgroup concentration

We checked the eigenvalues of the linearized system at a degree of polymerization of 5 and equilibrium

conditions. These are realistic initial conditions. For comparison we also computed the eigenvalues in a later

stadium of the process. Only the second eigenvalue changed by one order of magnitude. The system is a stiff

one.

Eigenvalues for Pn(t=0) = 5; a = 0.01: 11

2

3

3

72 10 21 10 1 6 10= = = .4 ; . ; . ;

Eigenvalues for Pn = 90; a = 0.03: 11

2

4

3

72 7 10 3 0 10 18 10= = =

. ; . ; . ;

Local observability was checked as described by Zeitz (1977) by computation of the observability matrix of the

linearized model. Although this shows formal observability of the system as the observability matrix has the

rank of the system order, the condition number of the observability gramian matrix is 1.43 10 5. That is a hint

that it may be difficult to build a good observer for such a system.

The eigenvalues 1 - 3 can be associated with the physical phenomena of mass-transfer, equilibrium reaction

and thermal degradation. To remove the smallest pole, we removed the thermal decomposition reaction from

the model and thus obtained a 2nd-order system with eigenvalues at 1=-0.25 and 2=-0.0021 for Pn = 5. Then

the condition number of the observability Gramian is 4.5 102, three orders of magnitude smaller than for the

first model. The error made by the simplification is shown in figure 1. It is acceptable compared with other

sources of error, e. g. in the measurements.


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Fig. 1: Comparison of the full model (eq. 2.1) and the 2nd-order model

The importance of a for the process dynamics becomes clear from an analysis of the eigenvalues of the

linearized model, which was done by Spies for the range of operating conditions (1993): one eigenvalue is

dependent on a and it is two to three orders of magnitude smaller in absolute value than the eigenvalue which

results from the equilibrium reaction. As a consequence, mass transport is much slower than reaction and

therefore controls the velocity of polymerization. In our reactor we measured the stirrer torque and computed

the average chain length and the OH-endgroups content in the melt from a measurement model using torque,

temperature and stirrer speed which was calibrated for the specific reactor. The computed OH-endgroups

concentration served as the input to the observer. The observer provides an estimated state vector containing

glycol- and OH-group-concentration and the parameter a.

3. OBSERVER DESIGN

The design of nonlinear observers is a complex problem for which not much practically useful support from

mathematical control theory is available. Birk (1992) gives a good survey of the available exact methods for

nonlinear observer design. All of them require that the system dynamics have special forms which are rarely

encountered in practice. In our case, the system cannot be transformed to any of these canonical forms. A well

known approximate method is the assignment of the eigenvalues of the linearized observer error equation as


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described by Zeitz (1977). As a first order approximation it however does not provide a guarantee for success.

This approach was used to design the following observer:

d x

dta x x k x x

x xx K x x

d x

dtk x x

x xx K x x

d a

dtK x x

= + +

+

= +

+

= +

11 1 1 2

2

3020 2

1 1 2 2

2

1 2

2

3020 2

1 2 2 2

3 2 2

1

28

2

82

0

*

( )

= x f

a

2torque, stirrer speed, reactor temperature

x , x , x , : initial values for the observer (equilibrium values for Pn )10 20 30 0 0

(3.1)

(3.2)

Equation 3.1 represents the nonlinear extended observer, where the left part of the right hand side of the

differential equation represents the model from (2.1) with k2=0 and the other terms are the observer error

correction terms. Although x2 can not be measured directly, it is considered here to be the measurement as it

can be computed from torque, stirrer speed and reactor temperature. It is necessary to give initial values for the

observer states. They are computed from the initial degree of polymerization under the assumption of an

equilibrium in the melt. The error equation is linearized at x s in order to assign desired dynamics to the

observer error. The observer error is defined by (3.3) as~x and obeys the differential equation (3.4).

~x x x=

, (3.3)

( ) ( )

d x

dt

d x

dt

d x

dt f x f x Kc x

T~

~

= =

. (3.4)

Linearization yields

( )( )

dx

dt

f x

xK x c x

x ssx

s

T~

~

. (3.5)


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The eigenvalues of the matrix

f

xKc

x

T

s

are usually assigned such that the error dynamics are

considerably faster than the system dynamics. That means that the eigenvalues of the above matrix have to be

larger in absolute value than the eigenvalues of the Jacobian matrix itself.

( )( )A

f x

xeigenvalues As

x

is s is

s

= = >

; ;~ ~

maxset so that Reis is . (3.6)

During polymerization, the state x evolves on a more or less known trajectory. The eigenvalues of As were

computed along this trajectory before computing K for constant values of~is . The observer was tested by

simulating the model described by equation 2.1 and the observer with gains obtained from eigenvalue

assignment described above. For the initial vector x0 the system eigenvalues are is = [-0.36; - 0.002; 0] . The

observer was never stable with eigenvalues chosen according to (3.6). We had to use~is = [-0.4, -0.04, -0.04]

to get a reasonable starting point. The dynamic behavior still was unsatisfactory. Therefore we tried to increase

the observer gain parameter K3 to improve the convergence of the estimation of a. The observer now was

neither unstable nor too slow, but was very badly damped. The next step was to calculate the eigenvalues of the

error dynamics again and to design a new observer with reduced imaginary part of the eigenvalues. After a few

iterations a satisfactory behavior of the observer was obtained, as shown in figure 2.

Fig. 2: Observer behavior with simulated plant. Initial cond.:

x x

=0 16. 0 0; a = 0.01/ s, changed at t = 500s to 0.1 / s

eigenvalues:

~.

. .

. .

i ii

0

0 31025

00272646 0 047252

00272646 0047252

=

+

observer gains:

K =

-0.013311

0.11286

-0.05042


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We used this observer in our experiments and refer to it as the fixed gain observer in the sequel. In addition, we

tried the well known Extended-Kalman-Filter [EKF]. For the EKF we had to parametrize the covariance matrix

Q for the system noise, the scalar r for the measurement noise and the matrix P0 for the initial value error. For

P0 we set the diagonal elements equal to the absolute value of the initial state. It seamed easy to choose the

scalar r as one can estimate the relative error of the measurement. But it turned out that we had to make r much

bigger as it would follow from the error, otherwise the observer was badly damped. In the matrix Q we assigned

nonzero values only to the diagonal elements in order to get good observer results for all states with reasonable

velocity and steady-state accuracy. Because of the discretization of the observer-gain computation with a

relatively long measurement interval of 5 seconds, we introduced a damping factor d in order to reduce the

variation of the P-matrix in the EKF-algorithm. The EKF approach turned out to be more successful than the

fixed gain observer. The next figure shows a comparison of the fixed gain observer described above with a fast

and a slow EKF.

Fig. 3: Estimated a for three observers and a simulated system with 60 % initial error and step response

4. EXPERIMENTAL SET-UP

The experiments were performed in a 10 l reactor system. It consists of a stirred tank reactor with an anchor-

shaped stirrer and a jacketed heating, a condenser, a condensate separator and a vacuum system. The

polycondensation process takes place in the temperature range between 270 and 300 C at pressures from 1 to 5

mbar absolute.


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The dimer bis(2-hydroxyehtyl)-terephthalate (BHET) is filled into the reactor in solid form and is molten

therein. To inhibit reaction, melting is done under a pressure of 5 bar absolute. After reaching the temperature

of 270 C, the process is started and the pressure is decreased from 1000 to 5 mbar absolute during 20 minutes.

Then the reactor is evacuated by maximum vacuum power. The temperature is increased from 270 to 300 C

during approximately one hour, where it remains constant for 15 minutes and decreased again slowly to 290 C

at the end of polycondensation, which is approximately 130 minutes after the start. This temperature program

has the objective to optimize polymerization with as little thermal decomposition as possible. About 60 minutes

after the process has been started, a slow increase in the torque can be observed which soon becomes

significant. The process is continued until a certain momentum is reached. During the polymerization process,

samples were taken every 30 to 45 minutes with a specially designed sample valve. The relativ viscosity of the

samples was determined which is related to the degree of polymerization.

Measurement data is registered by an industrial distributed process control system (DCS) CONTRONIC P by

Hartmann & Braun. From the DCS it is transferred to a PC by blockwise spontaneous data transfer. By means

of a C-program, this data is written to the MATLAB- workspace where it serves as the data basis for the

observer algorithm and is saved for later use.

Fig. 4: Temperature and pressure during an

experiment

Fig. 5: Profile of the laboratory reactor


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5. EXPERIMENTAL RESULTS

The observer described above was used to estimate the three states [EG]-concentration, [OH]-endgroups-

concentration and a. First we tried it off-line with original data obtained from experiments as variations are

easier to implement in off-line runs. As there is no control algorithm based on the observer data, it makes at

this point no difference between off-line and on-line operation. Torque data was transferred to viscosity data

using the stirrer characteristics, stirrer speed and temperature of the melt. It was obtained using a Newton-

Reynolds law we deviated from experiments with a polyvinyl-alcohol solution in our laboratory reactor.

Ne

=112 5 1 24. Re , (5.1)

Ned

nd= =25

2

M

n2; Re (5.2)

M: torque

n: stirrer speed

d: stirrer diameter

: density

: dynamic viscosity

From the dynamic viscosity, the degree of polymerization was calculated using an empirical equation which

was suggested by Eyring [see Ludewig (1975)]. The parameters were corrected based upon measurement data

which we obtained from laboratory analysis of our polymer samples.

ln ln ; . ; .47= + + = =A B M T A Bp7000

20 20 2(5.3)

MP: mean molecular weight of the melt

T: Temperature in K


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Fig. 6: Experimental Data (Exp. 22) and EKF-estimated values of Pn and a. Parameters of EKF: diag(Q) =(0.001 0.0001 0.01), r = 124.8, d = 10, diag(P0) =( x 0 ), d = 10; Pn0 = 2.5.

Fig. 7: Experimental Data (Exp. 21) and EKF-estimated values of Pn and a. Parameters of EKF are the:same as in figure 6.

In figure 6 and 7 data from two experiments is shown. For the estimation, the measured torque (lower black

line) and the stirrer speed (dashed-dotted line) are the most important measurement values. In the first 50 min.


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the torque is nearly constant and therefore very little can be derived from it. Then it begins to rise. The stirrer

speed is 90 Rpm at the beginning and is reduced to 60 Rpm over 2 min. when torque has reached 5 Nm. The

upper solid line is the value of the degree of polymerization calculated from the meaured torque and the dotted

curve represents its estimated counterpart. The small circles are the results obtained from the analysis of sam-

ples. The degree of polymerization was analyzed by the method of relative viscosity.

The dashed line represents the estimated value of the mass-transfer-parameter a. Its estimation is hardly

possible in the first 50 min. But when the torque begins to rise, the observer results become better and the

numerical values become quite reliable. One has to accept that the true numerical value ofa is impossible to

determine. Numerical values were given by Rafler (1989) (in the range of 4 to 7.8*10-2 s-1) and determined

from own experiments by Spies (1993) (in the range of 1.2 to 6*10-2 s-1), thus are in the same range. Despite

the uncertainty in the absolute value, the change of a is an indicator of the speed of the polymerization

process, regardless of what its physical meaning is. One interesting aspect is to try to influence a, e. g. by a

variation of stirrer speed or other parameters in order to optimize the process. In figure 6 and 7 one can see the

effect of the decreased stirrer speed about 80 min. after start. Further experiments will be made in order to

quantify this influence and will be reported in the final version.

Fig. 8: Comparison of EKF and fixed gain observer for experimental data. The EKF parameters are the

same as in figure 6


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In figure 8 a comparison between the fixed gain observer and an Extended-Kalman-Filter for the estimation of

a is given. The fixed gain observer has more difficulties to overcome the mismatch of the initial value and is

more sensitive to measurement noise compared to the EKF.

6. CONCLUSIONS

Figure 6 and 7 show that it is possible to estimate the value ofa during the time interval where a significant

rise of the viscosity in the polymer melt occurs. In the beginning of the polymerization process, due to the small

changes of the measurement signal, it is impossible to estimate any state variable or parameter with the

observer. This problem could only be solved if an additional measurement signal would be available. The rate

of evaporation, obtained from measuring the weight of the condensate, could be such a signal. In the phase of

the process where the observer works well, its signals can be used to optimize the controls and to detect

irregular developments of the process. The example shows that it is possible to gain additional information

under realistic conditions with standard instrumentation for a process which is generally regarded as

complicated because of its complex chemical reaction scheme, using a carefully chosen but not too complex

model.

REFERENCES

Appelhaus, P. , Engell, S., Grosse-Kock, S. (1995). Implementation of an Extended Observer for On-Line

Estimation of Mass Transfer in the Polymerization of Polyethylenterephthalate, Conference on Process

Control 1995, Tatransk Matliare, SK.

Birk, J. , (1992). Rechnergestuetzte Analyse und Loesung nichtlinearer Beobachtungsaufgaben, Fortschritt-

Berichte VDI, VDI Verlag, Duesseldorf.

Fritzsche, P.; Rafler, G., Tauer, K., (1989). Zur Modellierung technischer Polymersyntheseprozesse, Acta

Polymerica 40 Nr. 3, 143-160.

Grosse-Kock, Stefan, (1995). Realisierung einer modellgestuetzten Prozessfuehrung fuer die Polykondensation

von Polyethylenterephthalat. Diplomarbeit, Universitaet Dortmund, Fachbereich Chemietechnik.

Laubriet, C.; LeCorre, B., Choi, K.Y., (1991). Two-Phase Model for Continuous Final Stage Melt

Polycondensation of Poly(etheylene Terephthalate). Ind.Eng.Chem.Res. 30, 2-12.

Ludewig, H.; (1975). Polyesterfasern, Chemie und Technologie. Akademie-Verlag, Berlin,

2. Auflage.

Niehaus, S. (1991). Verbesserte Prozessfuehrung der Polyester-Polykondensation mit Hilfe eines physikalisch

begruendeten Modells. Diplomarbeit, Universitaet Dortmund, Fachbereich Chemietechnik.Rafler, G.; Versaeumer, H.; Dietrich, K., (1983). Reaktion und Stofftransport bei der

Polyethylenterephtalatbildung im zwangsdurchmischten System. Acta Polymerica 36 Heft 6, 316-321.


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Rafler, G.; Reinisch, G.; Bonatz, E., (1985) Kinetics of mass transfer in the melt of polycondensation of

poly(ethylene terephthalate). J. Macromol. Sci.-Chem., A22(10), 1413-1427.

Spies, V. (1993) Modellierung der Polykondensation von Polyethylenterephthalat im diskontinuierlich

betriebenen Ruehrkesselreaktor, Diplomarbeit, Universitaet Dortmund, Fachbereich Chemietechnik.

Tomita, K.; Ida, H., (1973). Studies on the formation of poly(ethylene terephthalate): 1. Propagation and

degradation reactions in the polycondensation of bis(2-hydroxyethyl)terephthalate POLYMER. Vol 14,February, 50-54.

Weissermel, K. , Arpe, H.-J., (1994). Industrielle organische Chemie. VCH Verlagsgesellschaft mbH,

Weinheim, 4. Aufl.

Zeitz, M., (1977); Nichtlineare Beobachter fuer chemische Reaktoren, Fortschritt-Berichte der VDI-Zeitschrift,

Reihe 8, Nr. 27, VDI-Verlag, Duesseldorf.

extended observer for the polymerization of polyethylenterephthalate

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