Accepted Manuscript

Modified Two-Phase Model with Hybrid Control for Gas Phase Propylene Co-polymerization in Fluidized Bed Reactors

Ahmad Shamiri, Suk Wei Wong, Mohd Fauzi Zanil, Mohamed Azlan Hussain,Navid Mostoufi

PII: S1385-8947(14)01572-1DOI: http://dx.doi.org/10.1016/j.cej.2014.11.104Reference: CEJ 12959

To appear in: Chemical Engineering Journal

Received Date: 11 August 2014Revised Date: 19 November 2014Accepted Date: 23 November 2014

Please cite this article as: A. Shamiri, S.W. Wong, M.F. Zanil, M.A. Hussain, N. Mostoufi, Modified Two-PhaseModel with Hybrid Control for Gas Phase Propylene Copolymerization in Fluidized Bed Reactors, ChemicalEngineering Journal (2014), doi: http://dx.doi.org/10.1016/j.cej.2014.11.104

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1

Modified Two-Phase Model with Hybrid Control for Gas Phase Propylene

Copolymerization in Fluidized Bed Reactors

Ahmad Shamiria, Suk Wei Wonga, Mohd Fauzi Zanilb, Mohamed Azlan Hussaina*, Navid

Mostoufic

a Department of Chemical Engineering, Faculty of Engineering, University of Malaya, 50603

Kuala Lumpur, Malaysia

b Chemical & Petroleum Engineering, Faculty of Engineering & Built Environment, UCSI

University, 56000 Kuala Lumpur, Malaysia.

c Process Design and Simulation Research Center, School of Chemical Engineering, College of

Engineering, University of Tehran, P.O. Box 11155/4563, Tehran, Iran

ABSTRACT

In order to explore the dynamic behavior and process control of reactor temperature, a

modified two-phase dynamic model for gas phase propylene copolymerization in a fluidized

bed reactor is developed in which the entrainment of solid particles is considered. The

modified model was compared with well-mixed and two-phase models in order to investigate

the dynamic modeling response. The modified two-phase model shows close dynamic

response to the well-mixed and two-phase models at the start of the polymerization, but

begins to diverge with time. The proposed modified two-phase and two-phase models were

validated with actual plant data. It was shown that the predicted steady state temperature by

the modified two-phase model was closer to actual plant data compared to those obtained by

the two-phase model. Advanced control system using a hybrid controller (a simple designed

Takagi-Sugeno fuzzy logic controller (FLC)) integrated with the adaptive neuro-fuzzy

inference system (ANFIS) controller was implemented to control the reactor temperature and

* Corresponding author. mohd_azlan@um.edu.my (Fax: +60-379675319)

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compared with the FLC and conventional PID controller. The results show that the hybrid

controller (ANFIS and FLC controller) performed better in terms of set point tracking and

disturbance rejection compared to the FLC and conventional PID controllers.

Keywords: Olefin polymerization; Dynamic two-phase model; Entrainment; Adaptive neuro-

fuzzy inference system; Fuzzy logic controller

1. Introduction

Polymerization is an important process in the petrochemical and polymer industries. It is a

complicated process with complex chemical kinetics and physical mechanisms [1, 2], thus

making its modeling and control a very challenging task. There are a number of papers about

successful modelling and controlled of polymerization processes [3-19]. However, few

attemps have been reported on modelling and the control of polypropylene (PP)

copolymerization in fluidized bed reactors (FBR). Copolymerization is a process in the

production of polymers from two (or more) different types of monomers which are linked in

the same polymer chain.

In the industrial PP copolymerization, the most commonly used reactor configuration is

the FBR [20-22]. With this reactor configuration, shown schematically in Fig. 1, catalyst

(Ziegler-Natta and triethyl aluminium) and reactants (propylene, ethylene and hydrogen) are

fed continuously into the reactor with nitrogen as the carrier gas. Conversion of monomers is

low for a single pass through the FBR and it is necessary to recycle the unreacted monomers.

Unreacted monomer gases are removed from the top of the reactor. A cyclone is used to

separate the solid particles (i.e., catalyst and low molecular weight polymer particles) from

the gas in order to prevent them from damaging the downstream compressor or heat

exchanger. Monomer gases are then recompressed, cooled and recycled back into the FBR.

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The product (PP) is gradually removed from the bottom of the reactor as soon as a reasonable

conversion is achieved.

Several models have been proposed to describe the behavior of olefin polymerization in

FBR [23-27]. Choi and Ray [23] applied a simple two-phase model, known as the two-phase

constant bubble size model, in which the bubble phase is considered to move in plug flow

with constant bubble size and emulsion phase is completely mixed. It is also assumed in this

model that the polymerization only occurs in the emulsion phase. On the other hand, Mcauley

et al. [5] and Xie et al. [27] considered a well-mixed model for this process (known as the

well-mixed model). They found that the well-mixed model estimates temperature and

monomer concentration by 2-3 K and 2 mol %, respectively, less than the constant bubble

size model. Hatzantonis et al. [24] refined the two-phase model by including the effects of

bubble size on the steady-state and dynamic behavior of the reactor. The bubble phase is

separated in this model into a number of segments in series and the emulsion phase is

assumed to be in a perfect mixing condition. This model is known as the bubble growth

model. Later, Fernandes and Lona [25] developed a heterogeneous three-phase model (gas in

emulsion, bubble and solid particles, all in plug flow) whereas Ibrehem et al. [9] proposed a

four-phase model (bubble, cloud, and emulsion with solid phase). All these models were

developed for the production of polyethylene (PE). For homopolypropylene production,

Shamiri et al. [13] developed a simple two-phase model by considering the progress of the

polymerization reaction in both bubble and emulsion phases. They adopted this modified

two-phase model to describe the gas phase propylene homopolymerization to produce PP in a

FBR and compared its results with the two-phase constant bubble size model and the well-

mixed model. They found that the two-phase constant bubble size model overpredicted the

conversion of propylene and temperature of the emulsion phase. Meanwhile, the two-phase

model and the well-mixed model were in better agreement at the same operating condition.

4

The propylene polymerization reaction is highly exothermic. To maintain the

polypropylene production rate at the desired condition, it is essential to keep the reactor

temperature greater than the dew point of reactants in order to avoid condensation of gas

within the reactor. It is also important to keep the temperature lower than the melting point of

the polymer in order to prevent particle melting, agglomeration and subsequently reactor shut

down. Therefore, an efficient temperature control system is required to address this issue.

Choi and Ray [23] showed that a simple PI controller could only be used to control

transients with limited recycle gas cooling capacity. Ghasem [28] also investigated the

performance of a PI controller, FLC based on Mandani and Takagi-Sugeno (TS) inference

method as well as a hybrid Mamdani-PI controller and a hybrid TS-PI controller for

controlling the temperature in a polyethylene fluidized bed reactor. It was shown that the

hybrid Mamdani-PI controller and the hybrid TS-PI controller performed better compared to

the PI controller. Besides, Ibrehem et al. [10] were able to control the system with a neural

network based predictive controller. They showed that the advanced controller works better

than the PID controller. On the other hand, Shamiri et al. [29] implemented a model

predictive control (MPC) technique to control the temperature of a two-phase model for

propylene homopolymerization and compared its performance with PI controllers tuned by

using the Internal Model Control (IMC) strategy and Ziegler-Nichols (Z-N) strategy. In

another study of PP reactor, the Adaptive Predictive Model-Based Control (APMBC)

method, a hybrid of generalized predictive control (GPC) algorithm and recursive least

squares algorithm (RLS), was proposed by Ho et al. [15] to control the reactor temperature

and polymer production rate by using the model developed by Shamiri et al. [13]. The

APMBC performed excellent set point tracking and disturbance rejection of the superficial

gas velocity and monomers concentration changes when compared with IMC strategy and

PID strategy.

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In all above mentioned models, it was assumed that solid entrainment is negligible at the

top of the reactor. However, it was shown that solid entrainment did exist during the

polymerization process [30]. Therefore, in this work, the two-phase model of Shamiri et al.

[13] was modified by incorporating it with solid entrainment in order to consider the losses of

entrained catalyst and polymer particles from the fluidized bed. A two-site copolymerization

kinetic scheme for propylene and ethylene were used in this study in order to have a clearer

picture of copolymerization over a heterogeneous Ziegler–Natta catalyst in a FBR. Then, a

comparative study of the results by using the well-mixed, two-phase and modified two-phase

models for PP copolymerization in FBR was carried out. The modified two-phase model and

the two-phase model were also validated with actual plant data. Furthermore, advanced

controllers using the Takagi-Sugeno based fuzzy logic controller (FLC) controller as well as

the hybrid adaptive neuro fuzzy inference system (ANFIS) and Takagi-Sugeno based fuzzy

logic controller (FLC) were implemented on the proposed modified two-phase model to

control the reactor temperature for set point tracking and disturbance rejection, as shown in

Fig. 2, to ensure a better performance of the FBR with safety consideration. To the best of our

knowledge, this is the first time that such a modified copolymerization of propylene model

and its controlling system is implemented on the propylene copolymerization system. Lastly,

these controllers were compared with the conventional PID controller for the set-point

tracking and disturbance rejection.

2. Modeling

2.1 polymerization mechanisms

In the copolymerization reaction, there are two types of monomer forming a polymer

while in homopolymerization, only one monomer is involved in the production of the

polymer. In the current study, an extensive mechanism was employed to explain the kinetics

6

of copolymerization of propylene and ethylene over two sites of Ziegler–Natta catalyst, using

the kinetic model developed by Decarvalho et al. [4] and Mcauley et al. [31]. Reactions,

including formation, initiation, propagation, transfer and deactivation of active sites, are

listed in Table 1, based on which the moment equations, shown in Table 2, were derived. The

index j in these tables refers to the type of the active site and i refers to the type of monomer.

Rate constants of each reaction for each site type were taken from the literature and given in

Table 3.

By assuming that monomers are consumed mainly through the propagation reactions, the

consumption rate equation for each component is shown as below after the moment equations

are solved [31]:

R� = �� M�Y(0, j)k ��

�

�

��

� , k = 1,2…(1)

where ns is the number of each type of active site and m is the number of each type of

monomer. Then, the total polymer production rate can be obtained from:

R = � mw�R�

�

���(2)

2.2. Hydrodynamic modeling

2.2.1. Well-mixed model

It is assumed in the well-mixed model, proposed by Mcauley et al. [5], that the FBR is

a single-phase, continuously stirred tank reactor. They further assumed that:

1. Bubbles are small. Therefore, heat and mass transfer between bubble and emulsion phases

are fast, thus, the reactor can be considered a well-mixed reactor (single phase).

2. Temperature and concentrations throughout the bed are uniform.

3. The emulsion phase is maintained at minimum fluidization.

7

With the assumptions above, the material balance and energy balance can be written as

[24]:

V�d[M�]

dt = U#A([M�]�� − [M�]) − R&ε��[M�] − (1 − ε��)R�(3)

�[M�]C �Vε�� + V(1 − ε��)ρ #+C , #+]dTdt = U#A�[M�]C �(T�� − T-.�)

�

���

�

���

−U#A �[M�]C �(T − T-.�) − R&[�[M�]C �ε��

�

���

�

���+ (1 − ε��)ρ #+C , #+(T − T-.�)

+(1 − ε��)∆H1R2 (4)

Equations (3) and (4) can be solved with the following initial conditions:

[M�]4�5 = [M�]�� (5)

T4�5 = T�� (6)

2.2.2. Two-phase model

In the dynamic two-phase model it is assumed that the polymerization reaction

occurred in both emulsion and bubble phases. Table 4 shows the equations used for

calculating velocities in emulsion and bubble phase, heat and mass transfer coefficients as

well as other required parameters in the two-phase model. Assumptions used in deriving the

material and energy balances of the two-phase model are summarized below [13]:

1. The emulsion phase is assumed to be completely mixed and not at the minimum

fluidization condition.

2. Polymerization reactions are considered to take place in both emulsion and bubble phases.

3. The bubbles are considered to be a sphere of constant size and pass through the bed at plug

flow condition with uniform velocity.4. Heat and mass transfer resistances between solid and gas in bubble and emulsion phases

are ignored.

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5. Radial gradients of temperature and concentration in the reactor are neglected due to

severe agitation induced by the up-flowing gas. 6. Solids elutriation at the upper part of the bed is ignored. 7. Uniform particle size is considered throughout the bed.

Based on the above assumptions, the following material balances can be obtained:

For the emulsion phase:

[M�].,(��)U.A. − [M�].U.A. − R6[M�]. + ([M�]7 − [M�].)V. 8 9�:9; − (1 − ε.)R�< =

==4 (V.ε.[M�].) (7)

For the bubble phase:

[M�]7,(��)U7A7 − [M�]7U7A7 − R6ε7[M�]7 − K7.([M�]7 − [M�].)V7 −

(1 − ε7) ?@&ABC

D Ri7dz = ==4 (V7ε7[M�]7) (8)

Also, the energy balances for emulsion and bubble phases are as follows:

For the emulsion phase:

U.A.GT..(��) − T-.�I∑ [M�].,(��)C � −���� U.A.(T. − T-.�)∑ [M�].C � −���� R6(T. −

T-.�)G∑ ε.���� C �[M�]. + (1 − ε7)ρ #+C . #+I + (1 − ε.)R .∆H1 − H7.V. 8 9�:9; (T. − T7) −

V.ε.(T. − T-.�) ∑ C �==4

���� ([M�].) = (V.(ε. ∑ C ����� [M�]. + (1 − ε.)ρ #+C . #+)) ==4 (T. −

T-.�) (9)

For the bubble phase:

U7A7GT7.(��) − T-.�I∑ [M�]7,(��)C ����� − U7A7(T7 − T-.�)∑ [M�]7C ����� − R6(T7 −

T-.�)G∑ ε7���� C �[M�]7 + (1 − ε7)ρ #+C . #+I + (1 − ε7) ?@∆KC&ABC

DR =dz + H7.(T. − T7)V7 −

9

V7ε7(T7 − T-.�)∑ C �����==4 ([M�]7) = (V7(ε7 ∑ C ����� [M�]7 + (1 − ε7)ρ #+C . #+)) =

=4 (T7 −

T-.�) (10)

where

U. = LM:9L@�:9 (11)

U7 = U# − U. + u7- (12)

U7- = 0.711(gd7)�/R (13)

δ = 0.534 U1 − exp 8LM:LYZ5.[�\ ;] (14)

ε. = ε�� + 0.2 - 0.059 exp(-LM:LYZ

5.[R^ ) (15)

ε7 = 1 − 0.146exp(LM:LYZ[.[\^ )

(16)

V2. = AH(1 − ε.)(1 − δ) (17)

V27 = AH(1 − ε7)δ (18)

V. = AH(1 − δ) (19)

V7 = AδH (20)

Equations (7) to (10) can be solved by MATLAB, with the following initial conditions.

[M�]7,4�5 = [M�]�� (21)

T7(t = 0) = T�� (22)

[M�].,4�5 = [M�]�� (23)

T.(t = 0) = T�� (24)

2.2.3. The proposed modified two-phase model

In the present work, the two-phase model (described in section 2.2.2) was further

improved to consider solid entrainment at the top of the reactor for the cases where elutriation

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rate cannot be ignored. In general, coarse particles stay in the bed whereas small particles will

be entrained and pushed off from the system. However, where velocities are several times

greater than the terminal velocity, coarse particles can also be entrained from the bed [30].

Therefore, in the present study, solid entrainment was considered in the model.

Mass balances obtained based on the assumptions of this model are as follows:

For the emulsion phase:

[M�].,(��)U.A. − [M�].U.A. − R6[M�]. + ([M�]7 − [M�].)V. 8 9�:9; − (1 − ε.)R�< −

`<&<a<?<[bc]<d<

= ==4 (V.ε.[M�].)(25)

For the bubble phase:

[M�]7,(��)U7A7 − [M�]7U7A7 − R6ε7[M�]7 − K7.([M�]7 − [M�].)V7 −

(1 − ε7) ?@&ABC

D Ri7dz − `@&@a@?@[bc]@d@

=

==4 (V7ε7[M�]7)(26)

The energy balances are expressed as:

For the emulsion phase:

U.A.GT..(��) − T-.�I∑ [M�].,(��)C � −���� U.A.(T. − T-.�)∑ [M�].C � −���� R6(T. −

T-.�)G∑ ε.���� C �[M�]. + (1 − ε7)ρ #+C . #+I + (1 − ε.)R .∆H1 − H7.V. 8 9�:9; (T. − T7) −

V.ε.(T. − T-.�) ∑ C �==4

���� ([M�].) − K.A./W.(T. − T-.�)G∑ ε.���� C �[M�]. + (1 −

ε7)P #+C . #+) = (V.(ε. ∑ C ����� [M�]. + (1 − ε.)ρ #+C . #+)) ==4 (T. − T-.�)(27)

For the bubble phase:

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U7A7GT7.(��) − T-.�I∑ [M�]7,(��)C ����� − U7A7(T7 − T-.�)∑ [M�]7C ����� − R6(T7 −

T-.�)G∑ ε7���� C �[M�]7 + (1 − ε7)ρ #+C . #+I + (1 − ε7) ?@∆KC&ABC

DR =dz + H7.(T. − T7)V7 −

V7ε7(T7 − T-.�)∑ C �����==4 ([M�]7) − K7A7/W7(T7 − T-.�)G∑ ε7���� C �[M�]7 +

(1 − ε7)ρ #+C . #+) = (V7(ε7 ∑ C ����� [M�]7 + (1 − ε7)P #+C . #+)) ==4 (T7 − T-.�)(28)

In the above mass and energy balances, solid elutriation rate constant were obtained from

[30]:

K. = 23.7ρhU#?

d<exp 8:i.[jk

jl;

(29)

K7 = 23.7ρhU#?

d@exp 8:i.[Lm

LM; (30)

W. = AH(1 − ε.)ρ #+ (31)

W7 = AH(1 − ε7)ρ #+ (32)

U4 = U4∗oμρh:RGρ #+ − ρhIgq�/\ (33)

U4∗ = U18Gd ∗I:R + (2.335 − 1.744∅�)Gd ∗I:5.i]:� (34)

for 0.5 < ∅t ≤ 1,

d ∗ = d oμ:RρhGρ #+ − ρhIgqvw (35)

Similar initial conditions as shown in Equations (21) to (24) were applied and the set of

equations were solved by MATLAB.

3. Control strategy

Most of the studies on the control of temperature of the polymerization process

suggest that advanced control schemes, such as FLC (fuzzy logic controller) , MPC (model

predictive controller) and adaptive predictive model-based control (APMBC), exhibit better

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performance than conventional PI or PID controllers [15, 16, 28]. Besides these advanced

controllers, Ghasem [28] showed that a hybrid controller incorporating FLC and PI

controllers performs better than a regular PI controller. However, for such a control system

the predictive based and adaptive based methods rely heavily on the accuracy of the model

and are also tedious to be implemented online. At the same time the use of the FLC can also

be cumbersome due to trial and error methods for obtaining the fuzzy rules especially for

complex nonlinear systems. This problem can be alleviated by using the ANFIS based

controller. Therefore, advanced control employing a hybrid controller (a simple designed

Takagi-Sugeno FLC integrated with an ANFIS controller), was used in this study for

controlling the temperature of the reactor by manipulating the cooling water flow rate, Fyz,

as the manipulated variable (see Fig. 2). The hybrid controller was then compared with FLC

and conventional PID controllers.

3.1. Hybrid FLC-ANFIS controller

Fuzzy logic requires a good understanding of the process characteristic and has the

capacity to reason with the condition of the inputs and deliver the conclusion collectively.

The technique has been introduced to improve machine reasoning in decision making which

is natural for human brain to correlate the action-conclusion relation. This technique has a

tremendous influence on the various applications in engineering including process control

systems for chemical reactors.

Since the nonlinearity and uncertain complexity of this polypropylene reactor can be

appropriately handled by this methodology, the standard steps of fuzzification, fuzzy rules,

fuzzy inference system, and defuzzification mechanisms have been applied in this control

system [37].

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The diagram in detail can be seen in Fig. 3 with 2 inputs and 3 triangular membership

functions. The rules are designed in the form of IF (CONDITION) then (ACTION). This

expression correlates the relation between a set of condition parameters for the appropriate

control action. Each fuzzy rule is evaluated as shown in Table 5 based on the error and rate of

error condition.

ANFIS controllers are mainly employed in processes that encounter unpredictable

variation in process parameters where complete information of the parameters are unavailable

[38]. ANFIS uses a hybrid learning algorithm, least square method and back propagation

descent, to generate a fuzzy inference system where the membership functions are iteratively

altered according to the given input and output data. The FIS structure with 3 membership

functions for each input, as shown in Fig. 3, was generated in MATLAB.

Because of the nonlinear, process condition and model complexity, the action signals

and set-point error relationship will vary and the appropriate output signals are very hard to

determine. This will lead to a bad controller performance and therefore, an inverse correlation

is introduced inside the main controller to integrate the empirical model technique (ANFIS

controller). The ANFIS controller was designed based on the historical value of the

successful control system with several process conditions setup. The outcome of the ANFIS

controller will reflect the inverse response of set-point error (input) and the cooling water

flow rate (as an output). These integration setups are to provide a guarantee that the controller

will give a sufficient and appropriate action signal for any reactor conditions.

The propose hybrid controller requires additional inputs such as U1, U2 and U3 to the

fuzzy logic controller. Inputs are from the state parameters that significantly influence the

response of the reactor dynamics. The rules in Table 5 are shown in Fig. 3 where they are

dependent on the inputs U1, U2 and U3 concurrently. When the error e is negative, the process

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variable is actually greater than the set-point value and the three corresponding connections

will trigger three different fuzzy rules, as can be seen in Fig. 3 as well as Table 5.

System identification was used to design ANFIS controller to obtain the inverse

dynamic response of the reactor and it involves similar methodology such as neural networks

inverse plant model development [39]. For the servo system, process variable is driven to the

desired set-point when error is negative/positive and change of error is decline/incline. The

key factor for the fuzzy controller is the output value of membership function named

“GOOD”. When process error equals to zero, the inverse response from ANFIS will decide

the output value of “GOOD” and send the signal to the manipulate control variable. In the

case of error close to zero but no progression to the set-point, this approach will bring the

process variable to set-point since the other output membership function

(“CLOSE”/”OPEN”) will infer the decision of “GOOD” and adjust the final manipulated

variable signal accordingly.

Propylene concentration, superficial gas velocity, catalyst flow rate and temperature

are used as input data whereas the output data is the cooling water flow rate in this work.

Fig. 4 shows the fuzzy logic framework that was used to couple with the ANFIS inverse

response controller.

4. Results and discussion

4.1. Comparison of models

Dynamic modeling and simulation studies of the gas phase propylene and ethylene

copolymerization in the FBR was conducted using the modified two-phase model and results

were compared with results of two-phase and well-mixed models incorporated with a

comprehensive two-site kinetic scheme. Simulations were performed at the operating

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conditions given in Table 6. One of the main issues of olefin polymerization in fluidized bed

reactor is the solids entrainment. Considering particle entrainment is crucial since this

phenomenon affects the particle size distribution, agglomeration, polymer properties,

polymer production rate as well as the bed hydrodynamics. In addition, it is a key parameter

in design and control of a fluidized bed reactor. Therefore, in the present work, solids

elutriation is considered in the modified two-phase model in order to predict the dynamic

behavior of the process and control the reactor effectively. Evolutions of the emulsion phase

temperature against time for the modified two-phase, two-phase and well-mixed models are

illustrated in Fig. 5, and evolutions of propylene and ethylene concentrations in the emulsion

phase for these models are shown in Figs. 6 and 7, respectively. In this case, the reactor starts

to operate when the catalyst is fed into the reactor. It can be seen in these figures that the

response for each of these variables (temperature and concentrations) is the same at the

starting point until they reach the steady state after about 4 hours. However, the final steady

state values for each responding variable of modified two-phase, two-phase and well-mixed

models are different. The final temperatures in modified two-phase, two-phase and well-

mixed models are 354 K, 356.83 K and 336.2 K, respectively. It is shown that the proposed

modified two-phase model exhibits an emulsion phase temperature which is 2.83 K lower

than the two-phase model and 17.8 K higher than the well-mixed model. Loss of catalyst and

polymer particles by carryover in an actual or commercial polypropylene fluidized bed

reactor results in a lower reaction rate, thus, lower reactor temperature since the reaction is

exothermic. It can be seen in figure 5 that the reactor temperature predicted by the modified

model is lower than that obtained by the two-phase model. This is mainly due to considering

the solids elutriation in the modified two phase model which results in a lower reaction rate,

thus, lower reactor temperature, which is in accordance with the performance of an actual

polypropylene fluidized bed reactor. Propylene and ethylene concentration profiles for the

16

proposed modified two-phase model lie in between those of two-phase and well-mixed

models. As shown in Figs. 5, 6 and 7, the well-mixed model shows a larger deviation from

the two-phase model compared to the modified two-phase model. This is mainly due to the

simplified assumptions of the well-mixed model. The modified two-phase model shows

closer behavior to the two-phase model compared to the well-mixed model due to

considering the distribution of catalyst between emulsion and bubble phases which takes into

account polymerization reaction in both bubble and emulsion phases.

The reactor temperatures and concentrations predicted by the modified model were

lower and higher than those obtained by the two-phase model, respectively. This is mainly

due to considering the solid elutriation in this model which results in lower a reaction rate,

thus, lower monomer conversion, due to the loss of entrained catalyst and polymer particles

from fluidized bed. Generally, the modified two-phase model shows the same dynamic

behavior as the two-phase and well-mixed models at the beginning of polymerization and

starts to differ over time

Superficial gas velocity is an important operating parameter in FBR operation.

Therefore, the effect of this parameter, which is directly related to the monomer residence

time in the reactor on propylene concentration, was verified by various models and is shown

in Fig. 8. All the three models predict that propylene concentration increases with increasing

superficial gas velocity. In fact, increasing the superficial gas velocity decreases the monomer

residence time, leading to a decrease in the reaction rate and consequently the monomer

conversion. The propylene concentration as predicted by the improved two-phase model was

greater than the two-phase model. This is mainly due to considering particle entrainment in

the modified two-phase model. In addition, the high gas velocity reduces the monomer mean

17

residence time, leading to a lower reaction rate and monomer conversion per pass through the

fluidized bed. It also leads to greater elutriation of polymer particles from the bed.

4.2. Validation with actual plant data

The proposed modified two-phase and two-phase models were validated with the

steady state actual plant data. The operating conditions and gas composition conditions for

producing different polypropylene grades employed in this study are listed in Tables 7 and 8,

respectively. Comparison between results of the proposed modified two-phase and two-phase

models with the actual plant data in terms of temperature are shown in Fig. 9. As can be seen

in this figure, there is a good agreement between predicted and industrial data on

temperatures in both models. However, the data predicted by the proposed modified two-

phase model is closer to the actual plant data compared to those predicted by the two-phase

model. The maximum difference between the industrial data and the proposed modified two-

phase model prediction for the temperature is 0.96 K whereas this difference is 1.59 K for the

two-phase model. Therefore, it can be concluded that the modified two-phase model

performance is closer to the realistic condition.

4.3. Controlling

4.3.1. Non-linearity analysis of the propylene copolymerization reactor

The proposed modified two-phase model was used in this section for the control

studies since it is closer to the actual process as discussed previously. To demonstrate the

non-linear behavior of the propylene copolymerization reactor, the process was simulated for

a step change in the superficial gas velocity and catalyst feed rate, as process key parameters,

on the reactor temperature. The open-loop simulation results are shown in Figs. 10 and 11.

18

In Fig. 10 the superficial gas velocity was changed after the reactor reached the steady

state at superficial gas velocity of 0.35 m/s. The superficial gas velocity has a considerable

impact on the reactor temperature. This figure clearly indicates that negative steps in the

superficial gas velocity have more remarkable effect on the reactor temperature than the

corresponding positive steps and non-symmetric responses are produced. In other words,

reactor temperature changes nonlinearly with the superficial gas velocity. For such a

nonlinear behavior, using conventional controllers leads to poor control of the process

variables. This justifies the implementation of a more efficient control system to sufficiently

regulate the effect of superficial gas velocity on the process variable.

The effect of step changes in the catalyst feed rate on reactor temperature is illustrated

in Fig. 11. The catalyst feed rate was changed from its nominal value (0.3 g/s) by increments

of 0.05 g/s in positive and negative directions. It can be seen in this figure that a small change

in the catalyst feed rate leads to a considerable change in the reactor temperature. The slightly

symmetric nature of these profiles due to the systematic positive and negative variations in

the catalyst flow rate indicates the slightly nonlinear relation with the reactor temperature.

The open loop analysis presented in this work and in a previous work [15] reveals the

nonlinear behavior of the propylene polymerization in fluidized bed reactors, justifying the

use of an advanced control algorithm for efficient control of process variables. In this case,

the adaptive neuro-fuzzy inference system (ANFIS) controller (hybrid neuro-fuzzy model)

and combination of ANFIS and simple Takagi-Sugeno fuzzy logic controller (FLC) were

implemented to control the reactor temperature by manipulating the cooling water flow rate.

Set-point tracking and disturbance rejection were carried out to examine the performance and

feasibility of the controllers. The optimum temperature for the best performance of the

polymerization reaction is between 343K and 353K.

19

4.3.2. Set-point tracking

Fig. 12 shows the set point from 344.5 K to 351 K tracked by FLC, hybrid and PID

controllers at 30000 s. This figure shows that these three controllers are able to track the set-

point. Although the PID controller achieves the set-point almost at the same period with the

FLC controller (4000 s), but the FLC controller performance is better than the PID controller

as it does not exhibit overshoot. However, the hybrid controller exhibits a performance

superior to that of FLC and PID controllers since the system returns to the set point in half of

the time required by other two models (2000 s) with a very small overshoot.

The controller moves for PID, FLC and hybrid FLC-ANFIS controllers in tracking set

point change in the reactor temperature are shown in Fig. 13. It is found that the starting point

of cooling water flow rate for the PID controller is zero while tracking the set-point of 351 K.

This is because the temperature change is high (6.5 K). However, the PID controller exhibits

a final response almost similar to the FLC controller after the temperature is tracked. On the

other hand, the hybrid FLC-ANFIS controller shows an oscillatory behavior when the set

point is 344.5 K. This small slew rates, however, is still acceptable since the cooling water

valve for an oscillation is about 4 minutes which means that the proposed hybrid controller is

sensitive enough to operate the control valve in such a rapid opening or closing time in this

simulation. However, when dealing with a real plant, this sensitivity might not be acceptable

due to the limitation of the control valve with its small rangeability and the tolerance of the

resistor used for the data acquisition system. In order to increase the sensitivity, a resistor of

lower tolerance number is required in practical implementations. After tracking the set-point

of 351 K, the valve opening response is similar to the PID controller but the response is twice

as fast as the PID controller.

4.3.3. Disturbance rejection

20

In order to make sure that a controller can be used practically in the industry, it also

must be able to cope with regulatory problems effectively. In this study, disturbances such as

superficial gas velocity, catalyst feed rate and monomer concentration (propylene) were

imposed onto the system with an increment of 10% of each respective nominal value. Figs.

14-16 show the temperature response controlled by the three controllers with an increment of

10% of each parameter. These figures clearly show that the hybrid FLC-ANFIS controller is

able to reject the disturbance in a more efficient manner as compared to other two controllers

although it exhibits a small oscillation at the start of disturbance. As shown in Fig. 14, FLC

controller and PID controller are able to reject the disturbance within 12000 s and 18000 s,

respectively, whereas the hybrid controller brings the system back to the stable set-point

within 2500 s which is a very short time compared to the other two controllers. It can be seen

in Fig. 15 that the catalyst feed rate has the highest temperature effect on the system in the

10% increment. Therefore, all controllers take longer time to track back the set-point. The

FLC controller and PID controller are able to reject the disturbance of the catalyst feed rate

within 19000 s whereas the hybrid controller brings the system back to the stable set-point

within 7000 s. Furthermore, FLC and PID controllers are able to reject the disturbance of

propylene concentration within 14000 s and 17000 s, respectively, whereas the hybrid

controller is able to bring the system back to the stable set-point within 5000 s, as illustrated

in Fig. 16. This figure shows that the PID controller is able to track back to the normal

condition faster than the FLC controller but the response of the PID controller is larger than

the effect of FLC controller.

In the above analyses, the integral absolute error for each controller in both set-point

tracking and disturbance rejection was calculated and shown in Table 9. Error values in this

table also show that the hybrid controller exhibited a better performance compared to the

21

other two controllers since the IAE value for the hybrid controller is the lowest in both set-

point tracking and disturbance rejection studies.

5. Conclusions

A two-phase model was developed and adopted for modeling of propylene

copolymerization in FBRs. The model takes into account the entrainment of solids into the

FBR modeling. This hydrodynamic model was combined with a kinetic copolymerization

model (propylene and ethylene) to provide a better understanding of the reactor performance.

Comparative simulations were carried out using the modified two-phase model, the two-

phase model and the well-mixed model in order to investigate their dynamic responses and

the effect of different operating parameters (superficial gas velocity and catalyst feed rate) on

the performance of the reactor. The proposed modified two-phase model showed the same

response as two-phase and well-mixed models in the start of polymerization but started to

diverge over time. The modified model exhibited a steady state reactor temperature which

was 2.83 K and 17.8 K lower than the two-phase model and higher than the well mixed

model, respectively. Propylene and ethylene concentration profiles for the proposed modified

two phase model lie between those of two-phase and well-mixed models.

The proposed modified two-phase and two-phase models were validated with actual plant

data. It was shown that the performance of the modified two-phase model was closer to the

real condition. The temperature predicted by the proposed modified two-phase was closer to

the actual plant data compared to those predicted by the two-phase model. The maximum

temperature difference between the industrial data and proposed modified two-phase model

was 0.96 K. This value was lower than the temperature difference between that calculated by

the two-phase model and industrial data which was 1.59 K.

22

The modified two phase model was adopted to carry out control studies. A proper

selection of controller for industry uses was implemented in order to handle the servo and

regulatory problems effectively. Results showed that the hybrid FLC-ANFIS controller

performs better in terms of set point tracking and disturbance rejection compared to FLC and

PID controllers.

Acknowledgement

The authors would like to thank the support of the Research Council of the University of

Malaya under research grant (UM.C/HIR/MOHE/ENG/25).

23

Nomenclature

A Cross sectional area of the reactor (mR)

ALEt\ Triethyl aluminum cocatalyst

Ar Archimedes number

B� Moles of reacted monomer of type i bound in the polymer in the reactor

C � Specific heat capacity of component i (J/kg K)

C h specific heat capacity of gaseous stream (J/kg K)

C , #+ Specific heat capacity of product (J/kg K)

C b� Specific heat of component i (J/kmol K)

d7 Bubble diameter (m)

d75 Initiate bubble diameter (m)

d Particle diameter (m)

d ∗ Dimensionless particle size

Dh gas diffusion coefficient (mR/s)

D4 Reactor diameter (m)

Fy�4 Catalyst feed rate (kg/s)

f� Fraction of total monomer in the reactant gas which is monomer M�

g Gravitational acceleration (m/sR)

H Height of the reactor (m)

H7. Bubble to emulsion heat transfer coefficient (W/m\K)

H7y Bubble to cloud heat transfer coefficient(W/m\K)

Hy. Cloud to emulsion heat transfer coefficient (W/m\K)

HR Hydrogen

I� Impurity such as carbon monoxide

24

i Monomer type

J Active site type

kf(j) Formation rate constant for a site of type j

k�h�

( j )

Transfer rate constant for a site of type j with terminal monomer M�

Reacting with hydrogen

kfm�

( j)

Transfer rate constant for a site of type j with terminal monomer M�

Reacting with monomer M`

kfr�

( j )

Transfer rate constant for a site of type j with terminal monomer M�

Reacting with Aiet\

kfs�

( j )

Spontaneous transfer rate constant for a site of type j with terminal

monomer M�

kh Gas thermal conductivity (W/m K)

kh�

(j)

Rate constant for reinitiation of a site of type j by monomer M�

kh-

( j )

Rate constant for reinitiation of a site of type j by cocatalyst

ki� (

j )

Rate constant for initiation of a site of type j by monomer M�

kp��

(j)

Propagation rate constant for a site of type j with terminal monomer

Mire acting with monomer M`

kp�� propagation rate constant (m\/kmol. s)

K7 Elutriation constant in bubble phase (kgmRs:�)

K7. Bubble toemulsionmasstransfercoefficient (s:�)

K7y Bubble to cloud mass transfer coefficient (s:�)

25

Ky. Cloud to emulsion mass transfer coefficient (s:�)

K. Elutriation constant in emulsion phase (kgmRs:�)

mw� molecular weight of monomer i (g/mol)

M� Concentration of component i in reactor (kmol/m\)

[M�]�� Concentration of component i in the inlet gaseous stream

N

(j)

Potential active site of type j

N

(0, j)

Uninitiated site of type j produced by formation at sites of type j reaction

N=

( j)

Spontaneously deactivated site of type j

N=,�K

(0,j)

Impurity killed sites of type j

NK

(0.j)

Uninitiated site of type j produced by transfer to hydrogen reaction

N�

(r, j )

Living polymer molecule of length r, growing at an active site of type j ,

with terminal monomer m�

Q

(r. j )

Dead polymer molecule of length r produced at a site of type j

P Pressure (Pa)

PP Polypropylene

26

R Number of units in polymer chain

R� Instanstaneous consumption rate of monomer i (kmol/s)

R Production rate (kg/s)

R6 Volumetric outflow rate of polymer (m\/s)

Re�� Reynolds number of particles at minimum fluidization condition

T Time (s)

T Temperature (K)

T�� Temperature of the inlet gaseous stream (K)

T-.� Reference temperature

U7 Bubble velocity (m/s)

U7- Bubble rise velocity (m/s)

U. Emulsion gas velocity (m/s)

U5 Superficial gas velocity (m/s)

U�� Minimum fluidization velocity (m/s)

U4 Terminal velocity of falling particles ( m/s)

U4∗ Dimensionless terminal falling velocity coefficient

V Reactor volume (m\)

V Volume of polymer phase in the reactor (m\)

W7 Weight of solids in the bubble phase (kg)

W. Weight of solids in the emulsion phase (kg)

Y

(n,j)

N-th moment of chain length distribution for living polymer produced at a

site of type j

X

(n,j )

Nth moment of chain length distribution for dead polymer produced at a

site of type j

27

Greek letters

∆HR Heat of reaction (J/kg)

ε7 Void fraction of bubble for Geldart B particles

δ Volume fraction of bubbles in the bed

ε. Void fraction of emulsion for Geldart B particles

� Void fraction of the bed at minimum fluidization

μ Gas viscosity (Pa.s)

ρh Gas density (kg/m\)

ρ #+ Polymer density (kg/m\)

∅� Sphericity for sphere particles

Subscripts and superscripts

1 Propylene

2 Ethylene

I Component type number

In Inlet

J Active site type number

mf Minimum fluidization

pol Polymer

ref Reference condition

28

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32

Figure captions

Fig. 1. Schematic of an industrial fluidized bed polypropylene reactor

Fig. 2. Simplified schematic of the temperature control loop for the gas phase propylene

copolymerization in FBR.

Fig. 3. Fuzzy Logic Controller; 2 inputs with 3 triangular membership-functions.

Fig. 4. Simplify structure arrangement of the propose hybrid ANFIS-FLC controller.

Fig. 5. Evolution of the temperature in the emulsion phase over time for the modified two-

phase, two-phase and well-mixed models

Fig. 6. Evolution of the propylene concentration in the emulsion phase over time for the

modified two-phase, two-phase and well-mixed models.

Fig. 7. Evolution of the ethylene concentration in the emulsion phase over time for the

modified two-phase, two-phase and well-mixed models

Fig. 8. Effect of superficial gas velocity on the propylene concentration calculated by the

modified two-phase, two-phase and well-mixed models.

Fig. 9. Comparison between actual plant temperature and predicted temperature by using the

two-phase and modified two-phase models.

Fig. 10. Effect of step change in the superficial gas velocity on the reactor temperature

(catalyst feed rate (Fcat=0. 2 g/s).

Fig. 11. Effect of step change in the catalyst feed rate (Fcat) on the reactor temperature

(U0=0. 35 m/s).

33

Fig. 12. Comparison of the performance between the hybrid FLC-ANFIS controller, FLC

controller and PID controller (Kc=1.277, �� =0.0029, �� =-104.13) in tracking set point

change in the reactor temperature.

Fig. 13. Comparison between controller moves (cooling water flow rate) in percentage (a)

FLC (b) PID (c) hybrid FLC-ANFIS controller in set point tracking of reactor temperature.

Fig. 14. Comparison of the performance between hybrid FLC-ANFIS controller, FLC

controller and PID controller in rejecting the effect of superficial gas velocity on the emulsion

phase temperature. A 10% increment from 0.35m/s to 0.385m/s in the superficial gas velocity

is introduced at 60,000 s.

Fig. 15. Comparison of the performance between the hybrid FLC-ANFIS controller, FLC

controller and PID controller in rejecting the effect of catalyst feed rate on the emulsion

phase temperature. A 10% increment from 5g/s to 4.5g/s in the catalyst feed rate is

introduced at 50,000 s.

Fig. 16. Comparison of the performance between the hybrid FLC-ANFIS controller, FLC

controller and PID controller in rejecting the effect of propylene concentration on the

emulsion phase temperature. A 10% increment from 1 mol/L to 0.9 mol/L in the propylene

concentration is introduced at 50,000 s.

1

Fig. 1. Schematic of an industrial fluidized bed polypropylene reactor

Fresh Feed

Propylene

Ethylene

Hydrogen

Nitrogen

Catalyst

Product

Recycle Stream

Compressor

Reactor

Cyclone

2

Fig. 2. Simplified schematic of the temperature control loop for the gas phase propylene

copolymerization in FBR.

Coolant in, Fcw

Catalyst in

Product

Recycle Stream

Compressor

Reactor

Fresh Feed

TT

Controller

3

e

U

RULE 1

∑ (Zi)

RULE 3

RULE 2

positive

negative

zero

Z1

Z2

Z3

fuzzyfication de-fuzzyficationFuzzy

Rules

Fuzzy

Inferences

U1

U1

U2

Δe

Δe

decline

incline

unchange

e

Δee

Δee

RULE 4

RULE 6

RULE 5

Z4

Z5

Z6

U1

U2

U3

Δee

Δee

Δee

RULE 7

RULE 9

RULE 8

Z7

Z8

Z9

U2

U3

U3

Δee

Δee

Δee

Fig. 3. Fuzzy Logic Controller; 2 inputs with 3 triangular membership-functions.

4

Fuzzy Inference, Zi

i = 1,2,..9

e

Input1 MFs

Fuzzy Rules

(#9)

Δe

Input2 MFs

Fuzzy Rules Fuzzy InferenceFuzzification De-fuzzification

U2u

Inverse

Model

Response

X1

X2

X3

X4

ANFIS

RULE 3

RULE 4

RULE 2

RULE 1

OPEN

Z6U3

Δee

Umax

GOOD

Z5

Δee

U2

CLOSE

Z4

U1

Δee

Umin

RULE 5

RULE 6

RULE 8

RULE 9

RULE 7

U∑ (Zi)

Fig. 4. Simplify structure arrangement of the propose hybrid ANFIS-FLC controller.

5

Time (s)

0 5000 10000 15000 20000 25000 30000

Re

acto

r te

mp

era

ture

(K

)

318

321

324

327

330

333

336

339

342

345

348

351

354

357

Well-Mixed

Two-Phase

Modified Two-Phase

Fig. 5. Evolution of the temperature in the emulsion phase over time for the modified two-

phase, two-phase and well-mixed models

6

Time (s)

0 5000 10000 15000 20000 25000 30000

Pro

pyle

ne

co

nce

ntr

atio

n (

mo

l/lit

)

0.955

0.960

0.965

0.970

0.975

Well-Mixed

Two-Phase

Modified Two-Phase

Fig. 6. Evolution of the propylene concentration in the emulsion phase over time for the

modified two-phase, two-phase and well-mixed models.

7

Time (s)

0 5000 10000 15000 20000 25000 30000

Ety

len

e c

on

ce

ntr

atio

n (

mo

l/lit

)

0.153

0.154

0.155

0.156

0.157

0.158

0.159

0.160

0.161

Well-Mixed

Two-Phase

Modified Two-Phase

Fig. 7. Evolution of the ethylene concentration in the emulsion phase over time for the

modified two-phase, two-phase and well-mixed models

8

Superficial gas velocity (m/s)

0.25 0.30 0.35 0.40 0.45 0.50

Pro

pyle

ne

co

nce

ntr

atio

n (

mo

l/lit

)

0.945

0.948

0.951

0.954

0.957

0.960

0.963

0.966

0.969

0.972

Well-Mixed

Two-Phase

Modified Two-Phase

Fig. 8. Effect of superficial gas velocity on the propylene concentration calculated by the

modified two-phase, two-phase and well-mixed models.

9

Predicted reactor temperature (K)

351.0 351.5 352.0 352.5 353.0 353.5

Actu

al re

acto

r te

mpe

ratu

re (

K)

351.0

351.5

352.0

352.5

353.0

353.5

Two-Phase, A

Two-Phase, B

Two-Phase, C

Two-Phase, D

Two-Phase, E

Two-Phase, F

Modified Two-Phase, A

Modified Two-Phase, B

Modified Two-Phase,C

Modified Two-Phase, D

Modified Two-Phase, E

Modified Two-Phase, F

Fig. 9. Comparison between actual plant temperature and predicted temperature by using the

two-phase and modified two-phase models.

10

Time (s)

15000 20000 25000 30000 35000

Re

acto

r te

mp

era

ture

(K

)

346

348

350

352

354

356

358

360

362

364

366

368

370

U0=0.2 m/s

U0=0.25 m/s

U0=0.3 m/s

U0=0.4 m/s

U0=0.45 m/s

U0=0.5 m/s

Fig. 10. Effect of step change in the superficial gas velocity on the reactor temperature

(catalyst feed rate (Fcat=0. 2 g/s).

11

Time (s)

15000 20000 25000 30000 35000

Re

acto

r te

mp

era

ture

(K

)

346

348

350

352

354

356

358

360

Fcat=0.45 g/s

Fcat=0.4 g/s

Fcat=0.35 g/s

Fcat=0.25 g/s

Fcat=0.2 g/s

Fcat=0.15 g/s

Fig. 11. Effect of step change in the catalyst feed rate (Fcat) on the reactor temperature

(U0=0. 35 m/s).

12

Time (s)

26000 28000 30000 32000 34000 36000 38000 40000

Rea

cto

r te

mp

erat

ure

(K

)

344

345

346

347

348

349

350

351

352

353

Set point

Hybrid FLC-ANFIS

FLC

PID

Fig. 12. Comparison of the performance between the hybrid FLC-ANFIS controller, FLC

controller and PID controller (Kc=1.277, �� =0.0029, �� =-104.13) in tracking set point

change in the reactor temperature.

Time (s)

20000 25000 30000 35000 40000 45000

Co

ntr

oll

er

mo

ves

(%

)

0

5

10

15

20

25

30

FLC

(a)

Time (s)

20000 25000 30000 35000 40000 45000

Co

ntr

oll

er m

oves

(%

)

0

5

10

15

20

25

30

PID

(b)

13

Time (s)

28000 32000 36000 40000

Contr

oll

er m

oves

(%)

0

5

10

15

20

25

30

35

40

Hybrid FLC-ANFIS

28700 29400 301002224262830323436

(c)

Fig. 13. Comparison between controller moves (cooling water flow rate) in percentage (a)

FLC (b) PID (c) hybrid FLC-ANFIS controller in set point tracking of reactor temperature.

14

Time (s)

55000 60000 65000 70000 75000 80000 85000

Re

acto

r te

mp

era

ture

(K

)

349.84

349.86

349.88

349.90

349.92

349.94

349.96

349.98

350.00

350.02

350.04

Set point

Hybrid FLC-ANFIS

FLC

PID

Fig. 14. Comparison of the performance between hybrid FLC-ANFIS controller, FLC

controller and PID controller in rejecting the effect of superficial gas velocity on the emulsion

phase temperature. A 10% increment from 0.35m/s to 0.385m/s in the superficial gas velocity

is introduced at 60,000 s.

15

Time (s)40000 50000 60000 70000

Rea

cto

r te

mp

era

ture

(K

)

349.8

350.0

350.2

350.4

350.6

350.8

351.0

351.2

Set point

Hybrid FLC-ANFIS

FLC

PID

Fig. 15. Comparison of the performance between the hybrid FLC-ANFIS controller, FLC

controller and PID controller in rejecting the effect of catalyst feed rate on the emulsion

phase temperature. A 10% increment from 5g/s to 4.5g/s in the catalyst feed rate is

introduced at 50,000 s.

16

Time (s)40000 50000 60000 70000 80000

Re

acto

r te

mp

era

ture

(K

)

349.90

349.95

350.00

350.05

350.10

350.15

350.20

350.25 Set point

Hybrid FLC-ANFIS

FLC

PID

Fig. 16. Comparison of the performance between the hybrid FLC-ANFIS controller, FLC

controller and PID controller in rejecting the effect of propylene concentration on the

emulsion phase temperature. A 10% increment from 1 mol/L to 0.9 mol/L in the propylene

concentration is introduced at 50,000 s.

1

Table 1. Elementary reaction for copolymerization system [31].

Description Reaction

Formationreaction N∗�j� ��������N�0, j� Initiationreactionwithmonomers N�0, j� + M! �!"������N!�1, j� i = 1,2…

Propagation N!�r, j� + M� �)"*��������N��r + 1, j� i = k =1,2,…

Transfertomonomer N!�r, j� + M� ��."*���������N��1, j� + Q�r, j� i = k =1,2,…

Transfertohydrogen N!�r, j� + H3 �45"�������N6�0, j� + Q�r, j� i =1,2,…

N6�0, j� + M!�5"�������N!�1, j�i = 1,2,…

N6�0, j� + AlEt: �5;�������N<�1, j� Transfertoco − catalyst N!�r, j� + AlEt: ��>"�������N<�1, j� + Q�r, j� i =

1,2,…

Spontaneoustransfer N!�r, j� ��A"�������N6�0, j� + Q�r, j� i = 1,2,…

Deactivationreactions N!�r, j� �CA�������NC�j� + Q�r, j� i = 1,2,…

N�0, j� �CA�������NC�j� N6�0, j� �CA�������NC�j�

Reactionswithpoisons N!�r, j� +I. �CE�������NCE6�0, j� + Q�r, j� i = 1,2,…

N6�0, j� +I. �CE�������NCE6�0, j�

2

N�0, j� +I. �CE�������NCE�0, j�

Table 2. Moment equations derived based on Table 1.

dY�0, j�dt = GMHIJkiH�j�N�0, j� + khH�j�N6�0, j�K + kh>�j�N6�0, j�GAlEt:I

− Y�0, j� Lk4hH�j�GH3I + kfsH�j� + kds�j� + kdl�j�GIMI + RNV)P dY�1, j�dt = GMHIJkiH�j�N�0, j� + khH�j�N6�0, j�K + kh>�j�N6�0, j�GAlEt:I

+ GMHIkpHH�j�Y�0, j�+ JY�0, j� − Y�1, j�KJkfmHH�j�GMHI + kfrH�j�GAlEt:IK− Y�1, j� Lk4hH�j�GH3I + kfsH�j� + kds�j� + kdl�j�GIMI + RNV)P

dY�2, j�dt = GMHIJkiH�j�N�0, j� + khH�j�N6�0, j�K + kh>�j�N6�0, j�GAlEt:I

+ GMHIkpHH�j�2Y�1, j� − Y�0, j�K + JY�0, j� − Y�2, j�KJkfmHH�j�GMQI+ kfrH�j�GAlEt:IK− Y�2, j� Lk4hH�j�GH3I + kfsH�j� + kds�j� + kdl�j�GI.I + RNV)P

CR�S,��CQ = JY�n, j� − NH�1, j�KJkfmHH�j�GMHI + kfrH�j�GALEt:I + k4hH�j�GH3I +kfsH�j� + kds�j� + kdl�j�GI.IK − X�n, j� VWXY n = 0,1,2

CZ"CQ = R! − B! VWXY i = 1,2,…

3

Table 3. Rate constants for reactions involved in propylene copolymerization.

Reaction Rateconstant Unit Sitetype1 Sitetype2 Reference Formation k��j� s]< 1 1 [31]

Initiation ki<�j� ki3�j� kh<�j� kh3�j� kh>�j�

m:/kmol. s m:/kmol. s m:/kmol. s m:/kmol. s m:/kmol. s

9.8

14.6

1

0.1

20

9.8

14.6

1

0.1

20

[7]

[7]

[7]

[31]

[31]

Propagation kp<<�j� kp<3�j� kp3<�j� kp33�j�

m:/kmol. s m:/kmol. s m:/kmol. s m:/kmol. s

220.477

591.1098

1.701

4.561

22.0471

130.783

376.396

6.698

[32]

[32]

[32]

[32]Transfer kfm<<�j�

kfm<3�j� kfm3<�j� kfm33�j� k4h<�j� k4h3�j� kfr<�j� kfr3�j� kfs<�j� kfs3�j�

m:/kmol. s m:/kmol. s m:/kmol. s m:/kmol. s m:/kmol. s m:/kmol. s m:/kmol. s m:/kmol. s m:/kmol. s m:/kmol. s

0.006

0.0021

0.006

0.005

0.088

0.088

0.12

0.24

0.0001

0.0001

0.006

0.0021

0.006

0.005

0.088

0.088

0.12

0.24

0.0001

0.0001

[7]

[7]

[7]

[7]

[31]

[31]

[31]

[31]

[31]

[31]

Deactivation kds�j� s]< 0.0001 0.0001 [31]

4

Table 4. Equations used in the two-phase model and modified two-phase model.

Parameter Formula Reference

Minimum fluidization velocity Re.� = G�29.5�3 + 0.375ArI</3- 29.5 [33]

Bubble velocity Ud = Ue − U.� − Ud> [34]

Bubble rise velocity Ud> = 0.711�gdd�</3 [34]

Emulsion velocity Uf = u.�ε.��1 − δ� [34]

Bubble diameter dd = ddiG1 + 27�Ui − Uf�I<:�1+ 6.84H�

dde = 0.0085�forGeldartB�

[35]

Mass transfer coefficient Kdf = � 1Kdo +1Kof�]<

Kdo = 4.5 pUfddq + 5.85�Dr</3g</sddt/s �

Kof = 6.77�Drεfud>dd �

[34]

Heat transfer coefficient Hdf = � 1Hdo +1Hof�]<

Hdo= 4.5�UfρrC)rdd �

+ 5.85 �UfρrC)r�</3g</sddt/s

Hof = 6.77�ρrC)rkr�<3�εfud>dd: �</3

[34]

Bubble phase fraction δ = 0.534 w1 − exp pUe − U.�0.413 qy

[37]

5

Emulsion phase porosity εf = ε.� + 0.2 - 0.059 exp(-z{]z|}i.s3~ )

[37]

Bubble phase porosity εd = 1 − 0.146exp�Ue − U.�4.439 � [37]

Volume of polymer phase in

the emulsion phase

V�f = AH�1 − εf��1 − δ) [13]

volume of polymer phase in

the bubble phase

V�d = AH�1 − εd�δ

[13]

volume of the emulsion phase Vf = A�1 − δ�H

[13]

volume of the bubble phase Vd = AδH

[13]

Table 5. Fuzzy Logic Rules.

AND ∆e is incline AND ∆e is unchanged AND ∆e is decline

IF e : positive R#1: Action : CLOSE (U1) R#2: Action : CLOSE (U1) R#3: Action : GOOD (U2)

IF e : zero R#4: Action : CLOSE (U1) R#5: Action : GOOD (U2) R#6: Action : OPEN (U3)

IF e : negative R#7: Action : GOOD (U2) R#6: Action : OPEN (U3) R#9: Action : OPEN (U3)

6

Table 6. Operating conditions and physical parameters for modeling fluidizing bed

polypropylene reactor

Operatingconditions Physicalparameters V�m:� = 50 μ�Pa. s� = 1.14 × 10]s T>f��K�=353.15 ρr�kg/m:� = 24.17 T!S(K)=325.15 ρA�kg/m:� = 910

P�bar� = 25 d)�m� = 500× 10]� Propyleneconcentration�mol/l�

= 0.9738

∅A = 1

Ethyleneconcentration�mol/l�= 0.1602

Hydrogenconcentration�mol/l�= 0.015

Super4icialgasvelocity, Ue�m/s� = 0.35

Catalystfeedrate�g/s� = 0.5

7

Table 7. Operating conditions and physical parameters for actual plant of propylene

copolymerization.

Operating conditions Physical parameters

V (m:� = 61 μ�Pa. s� = 1.14 × 10]s T>f��K�=353.15 ρr�kg/m:� = 23.45

T>f�(K)=343.15 ρA�kg/m:� = 580

P(bar) = 14 d)�m� = 500 × 10]� Superficial gas velocity,Ue (m/s) =0.35 ε.� = 0.45

Catalyst feed rate (g/s)= 0.16 ∅A = 1

Table 8. Gas composition conditions for producing different grades of polypropylene

Gas Unit A B C D E F

Propylene

concentration

Mol % 21.54 21.059 24.248 21.317 21.437 25.05

Ethylene

concentration

Mol % 77.69 78.16 74.62 77.659 77.761 73.83

Hydrogen

concentration

Mol % 0.77 0.772 1.13 1.02 0.8 1.113

8

Table 9. Integral absolute error (IAE).

IAE Set point

tracking

Disturbance rejection

(velocity change)

1) Hybrid controller 3890 4736

2) FLC controller 3.642 × 10s 5.743 × 10s 3) PID controller 7.524 × 10s 1.613 × 10t

Coolant in, Fcw

Catalyst in

Product

Recycle Stream

Compressor

Reactor

Fresh Feed

TT

Controller

• The two-phase model was modified by incorporating it with solid entrainment.

• The proposed modified two-phase model was validated with actual plant data.

• The hybrid controller performed better compared to the FLC and PID controllers.