new approach in modeling of metallocene-catalyzed olefin polymerization using artificial neural...

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New Approach in Modeling ofMetallocene-Catalyzed Olefin PolymerizationUsing Artificial Neural Networks

Mostafa Ahmadi, Mehdi Nekoomanesh,* Hassan Arabi

A new approach for the estimation of kinetic rate constants in olefin polymerization usingmetallocene catalysts is presented. The polymerization rate has been modeled using themethod of moments. An ANN has been used and trained to behave like the mathematicalmodel developed before, so that it gets polymerization rate at different reaction times andpredicts reaction rate constants. The network wastrained using modeling results in desired oper-ational window. The polymerization rates werenormalized to make the network work indepen-dent of operational conditions. Themodel has alsobeen applied to real polymerization rate data andthe predictions were satisfactory. This model isspecially useful in comparing different newmetallocene catalysts.

Introduction

Development of an accurate kinetic model plays a vital role

in the prediction of reactor behavior as well as in

maximizing the yield of desired products. It means that

all the information about the mechanism of reactions and

their unknown parameters, such as kinetic rate constants,

have to be known in order to optimize the process.

Determination of the kinetic rate constants has been the

main subject of several researches.[1–22] Earlier, it was

conventional to obtain the rate constants from time

average reaction rates and final molecular weights,[1–5]

while in recent models, instantaneous reaction rates are

M. Ahmadi, M. Nekoomanesh, H. ArabiDepartment of Polymerization Engineering, Iran Polymer andPetrochemical Institute, P.O. Box 14965/115, Tehran, IranFax: þ98 2 144580 0213; E-mail: [email protected]

Macromol. Theory Simul. 2009, 18, 195–200

� 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

being used.[6–22] In these methods, the algebraic function,

developed for desired output[7–9] (like polymerization rate)

or the nonlinear differential equations from mass con-

servation law,[10] is coupled with an optimization algo-

rithm in order to find the best rate constants. Generation of

certain rate constants needs almost numerous experi-

mental runs, which is considered as a serious limitation of

these methods.

The use of artificial neural networks (ANNs) as powerful

tools for modeling, optimization, and control of chemical

processes has become increasingly popular, especially

where the process is complex or the exact values of

parameters are unknown.[23–31] In most cases, ANNs are

used directly, for example where the network is trained to

get the reactor’s input variables (like temperature and

pressure) and reproduce its dependent outputs (like

polymerization yield or molecular weight).[23–26] In the

inverse method, they are used to find the boundaries of

DOI: 10.1002/mats.200800088 195

M. Ahmadi, M. Nekoomanesh, H. Arabi

196

reactor’s input variables required for desired outputs.[27–29]

ANNs answer much faster than a mathematical model,

especially where the mathematical model is complex. This

enhanced speed can be very useful for optimization

problems. Therefore, in some cases ANNs are used to

compensate the shortcomings of conventional mechan-

istic models such as the long time required for calcula-

tions[23,24] or the discrepancy between calculated and

measured molecular weight distribution in Ziegler-Natta

catalyzed ethylene polymerization.[25] However, these are

the most practical uses of ANNs in polymerization

modeling, but training of a reliable network needs a vast

number of experimental samples, uniformly distributed in

the entire design space that is not always available.

Hornik, Stinchcombe, and White proved that standard

multilayer feed forward networks are capable of approx-

imating any measurable function to any desired degree of

accuracy.[30] Consequently, the lack of success in many

applications arises from inadequate learning, forgetting

some effective inputs, or unsuitable network structure for

that particular task. Therefore, the necessity of being

skilled in use of ANNs is another problem in addition to the

need of numerous experimental data.

The mentioned problems associated with the optimiza-

tion of mathematical models and conventional applica-

tions of ANNs make one to look for an alternative way to

predict the polymerization outputs with a few experi-

mental results. This paper introduces a new approach

based on ANNs to study the kinetics of olefin polymeriza-

tion using metallocene catalysts. The main advantage of

this procedure is that it can be used to predict polymeriza-

tion behavior using a few experimental results, where the

neccesity of having a lot of experimental samples is the

most important shortcoming of current mathematical and

conventional ANN-based models.

Modeling

Polymerization Kinetics

At the first step, the simplest kinetic model was developed

for the prediction of the reaction rate in olefin polymer-

ization using metallocene catalysts. Despite the catalyst

being used, the basic scheme generally accepted for all

olefin polymerizations includes chain initiation Equation

(1), chain propagation Equation (2), and chain deactivation

Equation (3):

Macrom

� 2009

C� þ M �!kiP1 (1)

Pn þ M �!kpPnþ1 (2)

ol. Theory Simul. 2009, 18, 195–200

WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

kd
Pn �! Dn (3)
Site activation was assumed instantaneous, even in the

absence of premixing time. Population balance for

polymerization components is written below:

d½C��dt

¼ �ðki½M� þ kdÞ½C�� (4)

d½P �
1
dt¼ ki½M�½C�� ðkp½M� þ kdÞ½P1� (5)

d½P �
ndt
¼ kp½M�ð½Pn�1� � ½Pn�Þ � kd½Pn� (6)

d½D �
ndt
¼ kd½Pn� (7)

Using the moments of chain length distribution one can

calculate polymerization rate and yield according to the

following equations:

rate ¼ ½M� ki½C�� þ kpl0

� �(8)

yield ¼ M ðl þ m Þ (9)
0 1 1
where ln and mn are the nth order moments for the live and

dead chains, respectively and M0 is the monomer molar

mass. The following equations can be derived for the

calculation of the moments of live and dead chains:

dl0

dt¼ ki½M�½C�� kdl0 (10)

dl
1
dt¼ kiMC� � kdl1 þ kpMl0 (11)

dl
2
dt¼ ki½M�½C�� kdl2 þ kp½M�ð2l1 þ l0Þ (12)

dm
i
dt¼ kdli (13)

Training the Network

A common type of ANN with multiple layers and

supervised learning rule called feed forward was used.

Standard back propagation with Levenberg-Marquardt

algorithm was used to train the network. A set of

polymerization rates reported at 32 time points calculated

with the kinetic model proposed before was used as the

DOI: 10.1002/mats.200800088

New Approach in Modeling of Metallocene-Catalyzed Olefin Polymerization . . .

Figure 1. (a) The optimal network structure and (b) the corre-sponding training errors.

network’s input and the corresponding rate constants (ki,

kp, and kd) were used as the network’s outputs. The

considered time points were more at the beginning of

polymerization to feel the effect of initiation rate constant.

Eight levels were considered for ki between 0.001 and

10 L �mol�1 � s�1, seventeen levels for kp between 1 and

20 L �mol�1 � s�1, and seven levels for kd between 10�6

and 0.0028 s�1 to produce a total number of 952 sets of

polymerization rates. In this scenario, the network will

modify the weight of interconnections between neurons

to reproduce the given rate constants. Limits of the

rate constants were selected according to the published

data[7–10] and the information about the kinetics of our

target catalyst. It should be mentioned that the steps of

initiation and deactivation rate constant were selected

logarithmic to emphasize the effect of low values as well as

high values.

The most common problem with ANN-based models is

the lack of generalization capability which means that

they would memorize only the training set if not properly

trained. This problem can be overcome by carefully

selecting the range of data and using early stopping

method in training. Thus, early stopping method was used

in the training step to ensure the generalization. In this

method, the calculated polymerization rate sets were

divided into three subsets. The first one is used for training

(training set). The error on the second subset (validation

set) is monitored during training and the training stops if

the validation error increases. The error of the third subset

(test set) is also monitored to check whether the network

works well in the test area. In the ideal situation, the errors

of three subsets decrease concurrently during the training

course and any increase in each subset leads to incorrect

predictions. In addition, ANNs do not always predict well

near the borders of their training range. Extending the

training range so that the region of interest falls within

95% of the total training range would minimize the border

problem.

All neuron’s transfer functions were tan-sigmoid except

in the output layer where log-sigmoid function was used.

All the input and output data were coded to vary between

0.1 and 0.9 according to the following equation:[31]

Macrom

� 2009

Pcoded ¼ 0:8P � Pmin

Pmax � Pmin

� �þ 0:1 (14)

There is no rule for finding the best ANN architecture

and it depends on the specific characteristics of each

unique problem. The best ANN architecture should be

found using trial and error. This is done by testing several

topologies and comparing the three errors of training

course. Different number of hidden layers and neurons (in

each layer) were tested to get the least error. In our case,

ol. Theory Simul. 2009, 18, 195–200

WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

the optimal network contains two hidden layer with 20

neurons in the first and 10 neurons in the second layer.

Figure 1 shows the optimal network structure and the

corresponding training errors.

Results and Discussion

In order to generalize the network to work independent

from operational conditions, the calculated polymeriza-

tion rates were divided by the monomer and catalyst

concentrations [since the rate is proportional to these

parameters according to Equation (8)] before coding

Equation (14). Once the network has been trained, it can

be fed with different polymerization rate curves (experi-

mental or calculated ones) and calculate the kinetic

parameters, endlessly. ANN predictions can be used in

the mechanistic model to reproduce the input data. In this

way we have modeled an unknown experimental data

according to our desired reaction mechanism. Since there

is no comprehensive kinetic scheme for all catalysts, the

www.mts-journal.de 197


Figure 3. The experimental [32] (points) and predicted (lines)polymerization rates, polymerization conditions: pressure¼1 bar, [Zr]¼ 1.4� 10�5 mol � L�1, [Al]/[Zr]¼ 5 000, in 200 mL ofdecane.

198

best kinetic scheme and best range of rate constants

should be chosen according to the experimental results.

Four sets of rate constants were generated randomly

between 0.1 and 0.9 and transferred to the actual values to

produce network inputs using the mechanistic model,

randomly. The calculated polymerization rates were fed to

the trained network and the output rate constants were

used in the mechanistic model to reproduce the poly-

merization rates as well as polymerization yields. Figure 2

shows the calculated and predicted coded polymerization

rates and yields. It is shown that predictions are very close

to the actual values, which means that the network can

predict the outputs between the training steps. In addition,

the combined model can predict the polymerization yield

even when the network was trained using polymerization

rate data.

The network was also applied to the real case of

propylene polymerization using ansa-zirconocene catalyst

system of Me2Si(Ind)2ZrCl2/MAO in decane.[32] Details of

polymerization conditions can be found elsewhere.

Figure 3 shows the experimental rates and the model

Figure 2. The polymerization rates and polymerization yieldscalculated with the mechanistic model (lines) and predictionsof the ANN model (points).



predictions. There are some discrepancies between mea-

surements and predictions, especially at shorter reaction

times and lower temperatures. Evidently, this simple

kinetic scheme could not capture all the aspects of the real

complex behavior of metallocene catalyzed olefin poly-

merization. However the model is successful in predicting

the general behavior and this simple example reveals the

validity of the proposed method. These discrepancies can

be attributed to the simplicity of the selected kinetic

scheme and as can be found in Ref.[32] some more complex

kinetic schemes should be considered for predicting this

special decay type rate curve. In addition to the complexity

of the real kinetic behavior it should be noted that the

reaction media is not the same for all runs, because the

polymer becomes soluble as the polymerization tempera-

ture goes up and accordingly the molecular weight

becomes lower. This phenomenon causes the diffusion

limitations become more important in the prediction of

the polymerization kinetics.

Table 1 lists the kinetic rate constants at different

temperatures. When the judgment is based on observa-

tion, there are many acceptable rate constants that can

draw a specific rate curve but using this method, there is

only a unique set of rate constant that can capture the

desired rate curve. Therefore, this method is useful in

comparing kinetics of different catalysts. Two catalysts

can be compared in different operational conditions,

specifically different cocatalyst concentrations, only if

they show similar trends toward cocatalyst concentration

or the kinetic scheme should be modified to include

excessive complexation of cocatalyst molecules with

active centers.

DOI: 10.1002/mats.200800088

New Approach in Modeling of Metallocene-Catalyzed Olefin Polymerization . . .

Table 1. Kinetic rate constants at different temperatures derived by the ANN model and the corresponding Arrhenius parameters.

Run no. Temperature ki kp kd

-C L �molS1 � sS1 L �molS1 � sS1 sS1

1 40 0.056 1.240 5.520T 10S5

2 60 0.041 2.321 4.650T 10S5

3 81 0.381 5.198 1.020T 10S4

4 90 0.454 6.200 1.020T 10S4

5 105 0.126 10.186 3.170T 10S4

6 120 0.297 12.697 6.520T 10S4

7 129 5.472 16.901 1.138T 10S3

Ea/(kJ �molS1) – 17.640 13.431 16.218

log k0 – 12.226 12.081 3.860

Conclusion

This paper uses ANNs in a new way to predict the

polymerization behavior, which does not need many

experimental samples for training. The polymerization

behavior can be simulated using only one experimental

run data. The model works independent of operational

conditions such as monomer and catalyst concentrations

and polymerization temperature, so one can predict the

rate constant Arrhenius parameters if polymerization

rates at different temperatures are available. The advan-

tage of using ANNs is that the training of the network is

done only once, and after that it could be used to calculate

rate constants of different catalysts under different

operational conditions. Adding the resulted molecular

weight information, this method also has the potential of

predicting chain transfer constants.

Received: November 10, 2008; Accepted: December 22, 2008; DOI:10.1002/mats.200800088

Keywords: artificial neural networks; catalysts; chains; olefinpolymerization; polymerization kinetics; polymerization modeling

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DOI: 10.1002/mats.200800088

new approach in modeling of metallocene-catalyzed olefin polymerization using artificial neural...

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