Dynamic simulation and control of twoseries industrial reactors producing linear lowdensity polyethylene
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Abstract
One of the leading routes for producing polyolefins is through gasphase catalytic fluidized bed reactors. In this study, the industrial gasphase ethylene polymerization reactor series of Jam Petrochemical Company has been dynamically analyzed, modeled and controlled. The copolymerization of ethylene with 1butene is defined on Zeigler–Natta catalyst, assuming a double active site mechanism. To serve this purpose, pseudokinetic rate constants and the method of moments have been employed. The proposed model is capable of predicting the unsteadystate behavior of each reactor in addition to the properties of the product such as melt flow index (MFI), dispersion index, and molecular weight distribution (MWD). The verification of the model has been conducted with plant data to prove the accuracy of the modelestimated MWD and MFI. The controllability of the process control configuration has been examined through analyzing the dynamic behavior of the process under conventional feedback PID controllers. It has been observed that the control structure delivers a convincing performance for disturbance rejection.
Keywords
Dynamic simulation Polyethylene polymerization Ziegler–Natta Reactor modeling Wellmixed model PID controlList of symbols
 A
Crosssectional area of the reactor (cm^{2})
 A_{sf}
Fraction of metal that can form “k” catalyst active sites (mol/mol Me)
 [A]
Cocatalyst concentration (mol/cm^{3})
 C_{cat}
Catalyst concentration ratio
 C_{PMi}
Specific heat of “i” monomer (cal/mol/K)
 C_{P,poly}
Specific heat capacity of the polymer product (cal/mol/K)
 C_{Pw}
Specific heat capacity of water (cal/mol/K)
 f_{cat}
Catalyst feed rate (gr/s)
 f_{i}
Mol fraction of monomer “i”
 H
Bed height (cm)
 [H_{2}]
Hydrogen concentration (mol/cm^{3})
 K_{aA}
Kinetic rate constant of activation reaction [cm^{3}/(mol s)]
 K_{0}
Kinetic rate constant of initiation reaction [cm^{3}/(mol s)]
 K_{dsp}
Kinetic constant of spontaneous deactivation reaction (1/s)
 K_{p}
Kinetic constant of propagation reaction [cm^{3}/(mol s)]
 MW
Molecular weight (g/mol)
 \(\bar{M}_{\text{w}}\)
Weight average molecular weight
 \(\bar{M}_{\text{n}}\)
Number average molecular weight
 [Me]
Active metal (Titanium) concentration, mol Me/cm^{3}
 [M_{i}]
Concentration of monomer “i” (mol/cm^{3})
 [M_{T}]
Total monomer concentration (mol/cm^{3})
 N_{m}
Total number of monomers
 N_{s}
Total number of active sites
 P_{0}
Vacant active site concentration (mol/cm^{3})
 q
Volumetric product removal rate (cm^{3}/s)
 R_{p}
Overall particle polymerization rate (gr/cm^{3} s)
 S
Concentration of potential active sites (mol/cm _{catalyst} ^{3} )
 t
Time (s)
 T
Temperature (K)
 u_{0}
Superficial gas velocity (cm/s)
 U
Overall heat transfer coefficient (W/cm^{2} K)
Greek letters
 ΔH_{rxn}
Heat of reaction (cal/gr)
 ε
Void fraction of the bed
 λ_{0}
0th moment of the total number chain length distribution of “live” copolymer chains (mol/cm^{3})
 λ_{1}
1st moment of the total number chain length distribution of “live” copolymer chains (mol/cm^{3})
 λ_{2}
2nd moment of the total number chain length distribution of “live” copolymer chains (mol/cm^{3})
 ξ_{0}
0th moment of the total number chain length distribution of “bulk” copolymer chains (mol/cm^{3})
 ξ_{1}
1st moment of the total number chain length distribution of “bulk” copolymer chains, mol/cm^{3}
 ξ_{2}
2nd moment of the total number chain length distribution of “bulk” copolymer chains, mol/cm^{3}
 ρ
Density gr/cm^{3}
 φ
Cumulative copolymer composition
Subscripts and superscripts
 1
First reactor parameter
 2
Second reactor parameter
 cat
Catalyst property
 in
Feed property
 k
Type of catalyst active site
 poly
Polymer property
 rec
Recycle property
 ref
Reference value
 i
Monomer number
Abbreviations
 LLDPE
Linear lowdensity polyethylene
 MFI
Melt flow index
 MWD
Molecular weight distribution
 PDI
Polydispersity index
Introduction
Polyolefins, the largest group of thermoplastics, are recognized to be costeffective and showing excellent characteristics in a remarkable wealth of applications mainly as packaging, machinery parts, medical applications and domestic appliances. Covering 60% of the total polyolefin production, polyethylene is considered as the dominant polymer applied in the industry. Polyethylene is practically obtained in many types of reactor configurations ranging from autoclaves and loop reactors, to fluidized bed reactors. Owing to operation at lower temperatures and pressures, no need for solvent, and better heat removal, gasphase polymerization of ethylene in catalytic fluidized bed reactors is proved to be favorable for producing a broad range of polyethylene grades [1, 2, 3].
Various mathematical models have been proposed by researchers to characterize the performance of gasphase ethylene polymerization reactors. Modeling these reactors began with the pioneering work of Choi and Ray [4], who considered both emulsion and bubble phases in modeling a polyethylene fluidized bed reactor assuming constant bubble size along the bed. Several authors proposed a wellmixed model by considering the polymerization reaction occurring in a CSTR. They indicated that the simplifications are reasonable since virtually no loss of phenomenological information is observed [5, 6, 7, 8, 9, 10, 11]. Fernandes and Lona [12] presented a threephase model which is comprised of emulsion gas, polymer particles and bubble. Hatzantonis et al. [13] divided the reactor into two sections: an emulsion phase, which is perfectly agitated and a bubble phase, which is divided into N wellmixed compartments in series. They developed a model to account for the effects of varying bubble size with respect to the bed height on the reactor dynamics and product properties. A pseudohomogenous model was proposed by Alizadeh et al. [2] assuming an average concentration of particles in the bed. They employed a tanksinseries model to characterize the flow pattern in the reactor. In fluidized bed polymerization reactors, the challenges of design and control are related to achieving adequate heat removal and production rate [14, 15, 16]. Also, other approaches were explored which dealt with grade transition, molecular weight and density control of the polymer [17, 18, 19, 20].
The process studied in the present work is the new Spherilene technology licensed by LyondellBasell which includes two gasphase reactors in series for polyethylene production. This process is designed for the production of the entire density range of polyethylene products, from linear low density polyethylene (LLDPE), to medium density polyethylene (MDPE) and high density polyethylene (HDPE). Due to the fact that many of the enduse properties of the produced polymer depend on the reaction conditions that are usually difficult to be measured online, there is a need to simulate the process to obtain an optimal estimation of the process parameters. The dynamic simulation of industrial polymerization reactor series for fluidized bed reactors are currently lacking in literature. A wellmixed model is adopted in unsteadystate conditions to simulate the dynamic behavior of the LLDPE production process. The model is developed with flexibility to be applied for the entire range of LLDPE by modifying the input values and initial conditions. Also, it is capable of predicting the crucial properties of the polymer in the reactors and the characteristics of the final product. The effects of catalyst flow rate, monomer concentration and distribution of catalyst active sites on the polymer properties and reactor behavior are also examined. Then, by modifying the model, a conventional feedback control system is applied to maintain the temperature and the height of each reactor at desired values, and the operability of the plant’s control system in the presence of common disturbances is evaluated.
Reactor modeling
Series reactor process
The feed of fluidized bed polymerization reactors includes a mixture of ethylene, 1butene as comonomer, hydrogen and the activated catalyst. Due to low singlepass conversion (3–5%), large recycle stream is provided to restore unreacted feed. The stream of unreacted gases which exits from the top of the reactor also removes the reaction heat by passing through a heat exchanger in a countercurrent flow with cooling water. The second reactor utilizes independent feed streams of ethylene, 1butene and hydrogen. The active catalyst from the first reactor is fed together with the product of the first reactor to the second one to continue the reaction at the same or different conditions. About 70–85% of the polymerization occurs in the second reactor, depending on the required properties of the effluent.
Kinetics
Since industrial production of polyethylene includes copolymerization processes, it is of essential importance to perceive kinetic behavior and polymer properties via investigating the copolymerization mechanism. For Ziegler–Natta catalyst as a multisitetype catalyst, it is generally assumed that two or more active sites are present, each with its own characteristics and kinetic rate constants. This leads to producing polymer chains with distinct properties [25].
Kinetic mechanism of ethylene copolymerization over a Ziegler–Natta catalyst [22]
Activation by aluminum alkyl  \(S_{P}^{k} + A\mathop \to \limits^{{K_{aA }^{k} }} P_{0}^{k}\) 
Chain initiation  \(P_{0}^{k} + M_{i} \mathop \to \limits^{{K_{0,i }^{k} }} P_{1,i}^{k}\) 
Propagation  \(P_{n,i}^{k} + M_{j} \mathop \to \limits^{{K_{p,ij}^{k} }} P_{n + 1,j}^{k}\) 
Spontaneous deactivation  \(P_{n,i}^{k} \mathop \to \limits^{{K_{dsp }^{k} }} C_{d}^{k} + D_{n}^{k}\) 
Chain transfer by hydrogen (H_{2})  \(P_{n,i}^{k} + H_{2} \mathop \to \limits^{{K_{tH,i }^{k} }} P_{0}^{k} + D_{n}^{k}\) 
Pseudokinetic rate constants [22]
Reaction step  Pseudokinetic rate constant 

Initiation  \(k_{0,T} = \mathop \sum \limits_{i = 1}^{{N_{m} }} k_{0,i} f_{i}\) 
Propagation  \(k_{p,TT} = \mathop \sum \limits_{i = 1}^{{N_{m} }} \mathop \sum \limits_{j = 1}^{{N_{m} }} k_{p,ij} \phi_{i} f_{j} , k_{p,Tj} = \mathop \sum \limits_{i = 1}^{{N_{m} }} k_{p,ij} \phi_{i} , k_{p,iT} = \mathop \sum \limits_{j = 1}^{{N_{m} }} k_{p,ij} f_{j}\) 
Chain transfer  \(k_{tr,T} = \mathop \sum \limits_{i = 1}^{{N_{m} }} k_{tr,i} \phi_{i}\) 
Monomer mole fraction  \(f_{i} = \left[ {M_{i} } \right]/ \mathop \sum \limits_{i = 1}^{{N_{m} }} \left[ {M_{i} } \right]\) 
Mass and energy balances

Due to the large recycle stream to fresh feed ratio, which is almost 40 in this study, the fluidized bed is approximated by a CSTR, consisting of wellmixed solid and gas phases in contact with each other.

Temperature gradient and radial concentration gradient in the reactors are neglected.

Mass and heat transfer resistance between the emulsion phase and the solid particles are considered to be insignificant.

A mean size is assumed for the polymer particles, and elutriation of solid particles at the top of the bed is assumed to be negligible.

The product removal rates from the reactors are considered such that the bed is maintained at a constant level.

In the model proposed in this section, the temperature of the recycled gas is assumed constant, and the heat exchanger dynamics is excluded from the balance equations.
Based on the above assumptions, the unsteadystate material and energy conservation equations of the first reactor are derived by following the procedure developed by Hatzantonis et al. [13].
Initial conditions of the reactors
Property  First reactor  Second reactor 

Ethylene concentration (mol/cm^{3})  6.65 × 10^{−5}  1.50 × 10^{−4} 
1butene concentration (mol/cm^{3})  0  0 
Hydrogen concentration (mol/cm^{3})  2.65 × 10^{−5}  1.20 × 10^{−4} 
Catalyst concentration (mol/cm^{3})  313  313 
Concentration of potential active sites (mol/cm _{catalyst} ^{3} )  0  0 
Temperature (K)  313  313 
Reactor control system with modified model
In this section, we assess the closedloop performance of the process in load rejection under a conventional feedback PID algorithm. Since there is a critical bed level, below which a relatively rapid polymer wash out occurs, adequate bed level control is demanded in the ethylene polymerization fluidized bed reactors. Tight temperature control is also crucial for keeping the reaction zone temperature at its desired value to prevent particle agglomeration. To implement a control structure that controls the height and temperature of each reactor, the assumptions and consequently the equations of the mathematical model proposed in the previous section require modifications.
Hence, Eqs. (10) and (11) are modified to the following equations:
The proposed control structure for industrial polyethylene reactors is depicted in Fig. 1. Based on the process experience, the cooling water flowrate of the heat exchangers is manipulated to control the bed temperature of each reactor and the polymer product is withdrawn from each reactor at a rate that controls the bed height.
Results and discussion
Operating conditions and numerical values of the physical and transport properties of the reaction mixture
Reactor parameters  Physical properties  

First reactor  Second reactor  ρ_{cat} = 2.84 g/cm^{3} 
H_{1} = 700 cm  H_{2} = 1400 cm  ρ_{poly} = 0.952 g/cm^{3} 
A_{1} = 90,746 cm^{2}  A_{2} = 180,865.8 cm^{2}  \(C_{{{\text{PM}}_{1} }} =\) 11.9 cal/mol k 
u_{0,1} = 90 cm/s  u_{0,2} = 110 cm/s  \(C_{{{\text{PM}}_{2} }} =\) 23.8 cal/mol k 
T_{in,1} = 313 k  T_{in,2} = 343 k  \({\text{C}}_{{{\text{P}},{\text{poly}}}} =\) 0.96 cal/mol k 
T_{rec,1} = 313 k  T_{rec,2} = 333 k  C_{PW} = 1 cal/gr k 
(M_{1,1})_{in} = 11,768 kg/hr  (M_{1,2})_{in} = 30,542 kg/hr  \(\Delta {\text{H}}_{{{\text{rxn}}_{1} }}\) = 25,648 cal/mol 
(M_{2,1})_{in} = 0 kg/hr  (M_{2,2})_{in} = 0 kg/hr  
(H2,_{1})_{in} = 4.9 kg/hr  (H2,_{2})_{in} = 10.2 kg/hr  \(\Delta H_{{{\text{rxn}}_{2} }}\) = 51,296 cal/mol 
f_{cat} = 1.67 g/s  U_{ex,2} = 438,151 W/cm^{2} k  
[Me] = 0.00148 mol/cm^{3}  V_{shell,2} = 2,157,870 cm^{3}  
[A] = 1.6 × 10^{−8} mol/cm^{3}  V_{tube,2} = 11,821,360 cm^{3}  
U_{ex,1} = 155,804 W/cm^{2} k  
V_{shell,1} = 6,773,770 cm^{3}  
V_{tube,1} = 5,535,120 cm^{3} 
Openloop simulation
The polydispersity index, which indicates the width of the molecular weight distribution, is defined by [27]
In the following figures, the dynamic behavior of the reactors is studied to survey the effects of changes in catalyst concentration and monomer concentration, which are the major possible effective loads of the system.
Closedloop simulation
Parameters of controllers and characteristics of closedloop response for load rejection
Controller  K _{C}  1/τ_{I} (1/s)  τ_{D} (s)  Settling time (s)  Overshoot (%)  Gain margin 

1st bed height  − 22.1  − 0.0031  0.0  10,000  20.6  60 
1st bed temperature  − 1400.9  0.0008  0.0  99,450  23.1  53 
2nd bed height  − 105.7  0.0852  0.0  11,300  19.3  59 
2nd bed temperature  − 3822.2  − 8.5099  0.0  10,550  24.9  60 
Conclusion
A dynamic model was proposed for the production of LLDPE, utilizing a serried reactor configuration in industrial scale. To serve this purpose, a wellmixed model was employed for both reactors, and a twosite kinetic model was considered for the heterogeneous Ziegler–Natta catalyst. These models contributed to a significant progress in our knowledge of the fundamental parameters of both reactors including monomer concentration, production rate, reactor temperature as well as polymer properties such as molecular weight distribution and polydispersity index. Model validation was investigated by comparing the actual plant data with the results in terms of MFI, and it was found to be satisfactory in this case. The effects of catalyst amount, monomer concentration and the catalyst active site distribution on reactor parameters and polymer properties were also surveyed in openloop state. Based on the examination of the results obtained by our model, we can state that the presented model can be adopted as a predictive tool to study the reactor behavior in the presence of disturbances. The dynamic analysis of the openloop system is employed to design a PID controller for adjusting the cooling water flow rate of heat exchangers and the reactor effluent flow rates to maintain the temperatures and the heights of each bed at their set points. For closed loop state, the model was modified for control purposes, and the dynamics of heat exchangers were taken into account. Also, the bed level of each reactor was considered as a variable. The suggested feedback closedloop system has shown a good performance for load rejection in an acceptable period of time.
Notes
Acknowledgements
We greately appreciate Jam petrochemical complex for providing plant data.
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