Estimation of Kinetic Parameters of a Polymerization Reactor using Real Data

Viviane Botelho*, Isadora de Souza Moreira**, Jorge Otávio Trierweiler*,

Gustavo Alberto Neumann**, Marcelo Farenzena*

*GIMSCOP – Group of Intensification, Modeling, Simulation, Control and Optimization of Process ** Braskem S.A.

Federal University of Rio Grande do Sul, Chemical Engineering Department R. Eng. Luis Englert, s/n. Campus Central. CEP: 90040-040 - Porto Alegre - RS - BRAZIL,

E-MAIL: {vivirb, isasm, jorge}@enq.ufrgs.br

Abstract: This paper presents the comparison of some method related with the most important steps in the parameter estimation procedure to fit kinetic parameters of an HDPE polymerization reactor using real data from an industrial plant. An identifiability analysis was performed prior to the parameters estimation. For this analysis, two methods for scaling the sensitivity matrix and two methods for parameter screening are considered. This paper also compares the minimizing of the goal functions considering the square error (SE) and the square of the derivative of the error (SDE).

Keywords: Identifiability, Modeling, Optimization, Parameters Estimation, Polymerization.

1. INTRODUCTION

Phenomenological modelling of chemical reactors is widely used in the polymer industry. The dynamic models have several application such as: process improvement; operator training and development of advanced process control strategies. This paper presents the modelling, simulation and comparison of different techniques of sensitivity analysis and kinetic parameters estimation applied to a real polymerization CSTR (Continuous Stirred Tank Reactor).

The two most commonly used techniques for process models are empirical and phenomenological modelling. The empirical model has the following advantages: ease and speed in obtaining the model data. On the other hand, the drawback of empirical models is based on the fact that they are only valid for the operating range in which they were identified. Phenomenological models are valid for a greater operating range and are able to predict several process characteristics. However, phenomenological models require a higher time investment and knowledge about the plant.

Fitting the model parameters to real data is crucial for developing a good model. For that it is necessary to verify which parameters should be selected to fit the model. This procedure is called Identifiability Analysis. After we know which parameters should be used to improve the model, an optimization problem must be formulated and solved. This work applies all these steps to HDPE polymerization reactor introduced in section 2. The identifiability analysis is studied in section 3, and the optimization problem is formulated in section 4. Section 5 compares all the possibilities in the case study and then, the main conclusions are presented.

2. HDPE POLYMERIZATION REACTOR

2.1 Process description

The reactor is fed through the bottom with hexane, responsible for maintaining the formed polymer in suspension. The ethylene is the reaction monomer and is fed in gaseous phase. The catalyst is injected in the form of a solid suspended in hexane and Hydrogen is injected to regulate the size of polymer’s chains, serving as controller of the resin melt index (MI). It is also required the injection of an alkyl-aluminium, whose function is to react with any impurities and keep the catalyst active.

Temperature control is done through the cooling jacket and from external heat exchangers to the reactors. The reactor level is controlled by the outlet flowrate and the pressure inside the reactor by the injection of ethylene and catalyst into the vessel.

2.2 Process model development

Polyethylene is formed through chain reaction, which is divided into four main steps: initiation, propagation, chain transfer and termination. The initiation and chain propagation steps have the same kinetic constant, K���. The rate constant for monomer and hydrogen are, respectively, K��� e K��.

There are also spontaneous transfer reactions and reactions to cocatalyst, defined by constants K�� and K���. Finally, the constant of catalyst deactivation is represented by K. The catalyst is the Ziegler-Natta and it was considered that there is only one site type.

Poisson’s statistical moments method was used to perform mathematical modelling of mass balances for the polymer. Moments of order zero, one and two were used. To perform the mass balance in the reactor is necessary to establish phase distribution. The polymerization process in slurry is composed of two phases: a gaseous phase and a phase containing the polymer and hexane solvent. In this

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model construction, the existence of a gaseous phase and a homogeneous liquid + solid phase were considered. In the gas phase, ethylene and hydrogen are fed into reactor, whilst in the liquid + solid phase, hexane, catalyst and cocatalyst are injected.

Fig. 1. Reactor phases structure

In the liquid + solid phase, mass balances were performed for the moments of live and dead polymer and for reactants.

The general form for mass balance of polymer moments is presented in (1): � . �(���)/�� = ����. � −���� . ��� (1)

Where ! is the slurry volume, "#$is the ik order moment for live polymer. %"#$ corresponds to the reaction rate, and &'() is the slurry outlet flow.

The reaction rates equations for moments and reactants are presented in detail in (Fontes, 2001).

For reactants, a global mass balance was accomplished. The general form is presented in (2).

�(*�)/�� = �++�� . ,-� −.�. ���� −�� . � (2)

where *� is the total number of moles of specie i inside the reactor. �++�� and ,-� are the volumetric flow and the molar concentration of the specie i in the feed stream, respectively..� is the molar concentration of the specie inside the liquid + solid phase. As the concentrations of reagents are measured in the gas phase, it was necessary to establish a relationship between the two phases, in order to obtain the reactants concentration on the surface of catalyst particles. On this process, Henry's Law was used to for ethylene and hydrogen. The general form of the Henry’s Law equation is presented in (3). /. 0� = 1� . ℎ� (3) where P is the reactor pressure, 0� is the molar fraction of the specie i in gas phase and 1� is the molar fraction of the specie i in liquid + solid phase. ℎ� is the Henry constant, evaluated at reactor temperature.

Raoult’s Law was used for hexane. The equation is presented in (4):

/. 034 = 134 . /_67834 (4)

where /_67834 is the vapour pressure of hexane evaluated at reaction’s temperature.

Henry's Law constants, hexane vapour pressure and kinetic constants used in the simulation were obtained based on the data and equations presented by (Fontes, 2001).

In addition to the mass balances for products and reagents, total pressure and global mass balance calculations were developed for the reactor. The global mass balance considers a perfect reactor’s level control, in other words, a static mass balance for the liquid + solid phase was performed.

�++�34 . ,-34 . 9934 +�;�� . � . 99;�� = <.34 . 9934 +.;�� . 99;��=. ���� (5)

where 9934 and 99;�� are the molar masses of hexane and polymer, respectively. .34 and .;�� are the molar concentrations of hexane and polymer in liquid + solid phase.

The yield rate is obtained from ethylene’s reaction rate, as presented in (6).

>?+@��7�+ = 993A . �3A . � (6)

where 993A is the molar mass of ethylene, and �3A is the reaction rate of ethylene.

3. PRACTICAL IDENTIFIABILITY ANALYSIS

In general, the estimation of all parameters of a model is not a feasible process, since some of these parameters may be strongly correlated, leading to a high degree of uncertainty. Based on this problem, practical identifiability is an essential step to be performed before the parameter estimation. The main objective of this step is to determine which parameters can be estimated reliably from a data set under certain operating conditions

To perform practical identifiability analysis, is necessary to determine the system local sensitivity matrix (S). Assuming that a model can be described by a general set of equations dx/dt== f(x, u; θ) and y = g(x, u; θ) where x correspond to all states of the model, y represent the measured states, u represent the inputs and θ represent the parameters, the sensitivity of each output (yi) related with each parameter (θj) can be described as following: B�C = (D0� DEC)⁄ (7)

Gas Phase

Liquid + Solid Phase

Ethylene

Hydrogen

HexaneCatalyst

Cocatalyst

Outlet

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A key issue in this step is the scaling of this matrix. In general, parameters and states of a model have different magnitudes, which directly influence the sensitivity analysis result. Therefore, it is necessary to make sensitivity functions on the same order of magnitude. (Brun et al., 2001) provides some practical rules for scaling the system. In this paper, we consider the following:

B�C = (EG,C 0B�)⁄ (D0� DEC)⁄ IJKJL (8)

where θ0,j is the initial estimative for the corresponding parameter j and ysi as the average of the corresponding output i values.

Another technique to scale the system is presented by Botelho et al. (2012), in which the system is scaled by determining the minimizing condition number of the sensitivity matrix, since this indicator is strongly dependent on the scaling of the matrix. One way to calculate the scaling which leads to minimum conditioning is the use of LMI (Linear Matrix Inequality), which leads to the following optimization problem (Boyd et al. 1994). minP,Q R(ST�)B. �.S ∈ ℜ;W;, �?7XY*7@+*ãYB?*X[@7\� ∈ ℜ]W] , �?7XY*7@+*ãYB?*X[@7\ (9)

Where R represent the conditioning number of the matrix (ST�), S and � are diagonal matrices containing thescalingfactorsforeachoftheelementsofthematrixS. After scale the sensitivity matrix, the parameters to be estimated can be selected. In this paper, the methodology based on NSRGA of sensitivity matrix was used. In this method, first we calculate de NSRGA matrix, according to (10) , where Tl represents the pseudo-inverse of sensitivity matrix. Then the elements of each column are added and the parameters selected are the ones with the highest value of corresponding column sum in NSRGA matrix. More details about this method could be found in Botelho et al. (2012).

mT�no = T1Tlp (10)

Another method used is proposed by Brun et al. (2001). In this method, the parameter are ordered considering their influence in the model, according to (11) then the sensitivity matrix is normalized and the collinearity between parameters Once the most important parameters are defined, the columns of sensitivity matrix should be normalized and a subset of parameters (k) selected, giving priority to those with greater importance index. Afterwards, the normalized Sensitivity Matrix is restructured, keeping the columns of the selected subset. Finally, the subset collinearity index is calculated by (12), where λkmin is the smallest eigenvalue of resulting sensitivity matrix.

qC = r*stuvB̅�Cxyz�Ku (11) {� = 1 }�~��y⁄ (12)

It is important to emphasize that identifiability analysis is an iterative process that must be performed before and each estimation step, until the desired compliance is achieved.

4. PARAMETER ESTIMATION

Since the studied model is time-variant, is necessary to use a dynamic optimization method to perform the parameter estimation. In this paper we consider the single-shooting approach. This method is classified as a sequential method. In this formulation, given the initial conditions and a set of parameters, the algebraic-differential system is integrated and solved in an intern loop while the parameters values are updated on the outside part of the loop, using an NLP (Biegler et al., 1997). Mathematically, the problem can be defined according to (13) and (14).

,?*J �[ℎ(1)] (13)

ℎ(1) = �o@X+�\7?- − �?��+\+*�?7@B0B�+,.Y*B�\7?*B (14)

Single-shooting method is relatively simple to build. On the other hand, it may fail if the system is unstable or ill-conditioned (Biegler et al., 1997). In this case study, any NLP solver may be used, since constraints could be incorporated into the system integration phase, so that the optimization step must be composed only by minimizing the objective function. NLP solvers that do not use the gradient of objective function and constraints are highlighted, since in many cases, the complexity associated with this type of calculation affects the convergence of the system and increases the computational effort. In this article, we have used BOBYQA solver, developed by (Powell, 2009) due to the high speed of convergence and stability associated with it.

Another important issue related with parameter estimation is the objective function formulation used in the optimization problem. Since the identification of phenomenological models aims to generate representative models of real systems, the most efficient way to accomplish such estimative is to minimize the discrepancy between the measured values of the states and the model prediction. One of the more traditional formulations for this optimization problem is to minimize the square error, according to (15).

SE = minJ� vv+xy�CKu

ys�Ku (15)

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where θ' is the subset of parameters to be estimated, selected by the identifiability analysis, ns is the number of samples and ny is the number of states measured. The error (e) is defined as the difference between the measured states and the predicted states. Thus, for the problem under study, the objective function is defined according to (16).

T� = minJ� �v�>� � >���xy

�Ku:v��x/.x � �x/.x� �x

y

�Ku� (16)

were YR represent the Yeld Rate and H2/C2 the relation between hydrogen and ethylene, both described in section 2. Another possible approach is to consider the minimization of the square of the derivative of the error (SDE) according to (17). This type of formulation is indicated for cases where real plant data are used, since it has a greater ability to reject unmeasured disturbances, generating a dynamic model capable to capture the trend of the process, rather than trying to adjust each of the experimental points and has been proposed in (Trierweiler and Machado, 2004).

T�� � minJ� vv��+���xy�

CKu

ys

�Ku (17)

For the problem under study, the objective function at the derivative criterion is defined according to (18). It is important to emphasise that in this paper de derivative was calculated considering the difference between the errors of two consecutives measurement points.

T�� �minJ� vv���>� � >����� �xy�

CKu

ys

�Ku: ����x.x ��x.x��

�� �x (18)

5. RESULTS AND DISCUSSIONS

From the described model and proposed methodologies, parameters estimation of the system was performed. The estimation procedure was implemented in Python using the package JModelica and Nlopt. For the simulations and sensitivity calculations, the Jmodelica language, developed by (Åkesson et al., 2010) was used, and for nonlinear optimization, the package Nlopt developed by (Johnson, 2000). Initially, a set of real plant data was selected by performing a dynamic simulation using the values presented by (Fontes, 2001) for the following kinetic parameters: Kpee, Kts, Kth, Ktme, Ktcc e Kkd. The SE and SDE in these conditions were, respectively, equal to 32.48 and 3.5 E4. Fig. 2 shows the simulation results.

(a) (b)

Fig. 2. Simulation results using original parameters values

The graphics shows that the original values of the kinetic parameters of the model adequately fit the production rate, but are unable to adequately describe the relationship H2/C2.

The study proceeded with the identifiability analysis, considering the methods presented in section 3 and the parameters specified in section 2.2. Fig. 3 shows the parameter ordering, considering both scaling criteria for NSRGA method and Table 1 present this results for the method of identifiability presented by Brun et al. (2001).

(a) (b)

Fig. 3. Parameters order by partial NSRGA identifiability method: (a) Brun scaling and (b) Minimizing condition

number scaling

Table 1. Importance Rank of Parameters

Scaling Suggest by Brun et al. (2001)

Scaling using Minimizing Condition Number

Parameter Importance Index (�� Parameter

Importance Index (��

Kpee 0,6616 Kts 0,5155 K th 0,6402 Kth 0,4793 K ts 0,5055 Kd 0,4105 Kd 0,3018 Ktcc 0,4100

K tme 7,295E-3 Ktme 0,3142 K tcc 5,799E-8 kpee 0,2691

The best subsets of parameters according with each presented method of scaling and selection is presented in selected. Table 2.

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Table 2. Selected Subsets

Scaling Method Identifiability

Method Selected Subset

Brun et al. (2001)

Brun et al. (2001)

---

NSRGA Kpee, Kth

Minimizing Condition Number

Brun et al. (2001)

K ts, K th

NSRGA Kts, K th

Table 2 illustrates that the Brun et al. (2001) complete method (i.e. considering the scaling and selection criteria define by these authors) are incapable to indicate a subset to estimation. However, if the scaling by minimizing condition number is used, both identifiability criteria (NSRGA and Brun) conduct to the same results. Model parameters estimation was performed for the subsets defined on Table 2. Single-shooting method using a free- derivative NLP solver was stable in most cases and showed high speed of convergence for the optimization problem, demonstrating to be the most suitable for this problem. Table 3 presents the numerical results obtained after the estimation, where the values between brackets correspond to the parameter variation after the estimation (EG). Fig. 4 and Fig. 5 illustrate then.

Table 3. Achieved Values for SE and SDE.

Parameter SE goal function SDE goal function

Kpee, Kth 80.33

(17.86%;85.95%)

0.23

(52.53%;92.41%)

K ts, K th 79.12

(25.22%;85,96%)

10.60

(17.15%;84.70%)

(a) (b)

(c) (d)

Fig. 4. Results after Kpee and Kth estimation (a) and (b) by SE criterion and (c) and (d) SDE criterion.

(a) (b)

(c) (d)

Fig. 5. Results after Kts and Kth estimation (a) and (b) by SE criterion and (c) and (d) SDE criterion.

Table 3 demonstrates that both methods of sensitivity scaling yielded similar results, when comparing the value of square error. The square error derivative shows that the parameters selected according to (Brun et al., 2001) scaling lead to a better adherence to the plant real data, but comparing Fig. 4(c) and Fig. 5(c) we verify that this difference are not significant in practical terms. Another important issue demonstrated by Fig. 4 and Fig. 5 is that, when using the SDE, the estimated model tends to better capture the tendency of the system, which is a desirable characteristic for usage of the studied model to implement advanced process control on the plant.

Once the estimation of most influential parameters is achieved, a new identifiability analysis was performed, fixing the parameters estimated in the previous step and selecting a new subset for estimation. After optimization, the SE and SDE were reduced to less than 1% of their values, in both subset select, concluding that in the problem studied, only one step of estimation is enough.

To test the model quality, the new parameters have been simulated and the corresponding prediction compared with new dataset not used for parameter estimation. The analysis was performed only considering the relationship H2/C2 because as shown in Fig. 2, Fig. 4 (b and d) and Fig. 5, the production rate has satisfactory adherence, and is slightly sensitive to variations in the parameters. Table 4, Fig. 6 and Fig. 7 show the results.

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Table 4. Achieved Values for SE and SDE

Estimated Parameter

SE SDE

None 28598,9 32,34

Kpee, Kth 546,17 0,71213

K ts, K th 427,50 10,592

All 1025,9 7,603

(a) (b)

Fig. 6. Validation of Kpee and Kth estimated (a) by SE criterion and (b) SDE criterion.

(a) (b)

Fig. 7. Validation of Kts and Kth estimated (a) by SE criterion and (b) SDE criterion.

Table 4 and Fig. 6 and Fig. 7 demonstrate that both subsets of parameters estimated are capable to describe the system in a different operational region, confirming that the estimation is valid. Additionally, we verify that, the estimative of all parameters, increase the computational time, increase the parameter uncertainty what is responsible to reduce the model predictability and usefulness (cf. see last row in Table 4) demonstrating that identifiability analysis is an essential step to be performed before the parameter estimation.

6. CONCLUSION

In the case of polymerization reactors models applied to real systems, the estimation of kinetic parameters is essential, because they govern both gains and the dynamics of the process. The practical identifiability analysis is an important tool in identification of dynamic models of high complexity, since it provides an overview of possible combinations of parameters that can be estimated. It is based on sensitivity matrix of the system, so the proper scale is essential. Both scaling strategies of the sensitivity matrix have produced good and similar results, since the parameters indicated are capable to rightly represent the behavior of the system, for different subsets of data.

The SDE criterion in the objective function formulation could better capture the tendency of the system. This result was expected, given that the capture of trends is the main goal of this formulation. It is important to emphasize that it is possible to formulate an optimization problem combining SE and SDE terms, what can make possible to balance both desired goasl.

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