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Modeling Condensed Mode Operation for Ethylene Polymerization: Part I. Thermodynamics of Sorption Arash Alizadeh, Josef Chmelar, Farhad Sharif, Morteza Ebrahimi, Juraj Kosek, and Timothy F.L. McKenna Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b04288 • Publication Date (Web): 10 Jan 2017 Downloaded from http://pubs.acs.org on January 13, 2017

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Industrial & Engineering Chemistry Research

Manuscript Type: Article

Modeling Condensed Mode Operation for Ethylene Polymerization: Part I. Thermodynamics of Sorption Arash Alizadeh a,b, Josef Chmelař c, Farhad Sharif a, Morteza Ebrahimi a, Juraj Kosek Timothy F.L. McKenna b,*

c,*

,

a

Department of Polymer Engineering and Color Technology, Amirkabir University of Technology, PO Box 15875-4413, 424 Hafez Ave, Tehran, Iran b

C2P2 – LCPP Group, UMR5265 CNRS, ESCPE Lyon, Université de Lyon, 43 Bd du 11 Novembre 1918, 69616 Villeurbanne, France

c

Department of Chemical Engineering, University of Chemistry and Technology, Prague, Technicka 5, 166 28 Prague 6, Czech Republic *

Authors to whom correspondence should be addressed: [email protected] and [email protected]

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Abstract A process model based on the thermodynamic models of Sanchez-Lacombe and PC-SAFT EoS is developed to simulate and analyze the impact of vaporized n-hexane as an induced condensing agent on the rate of gas phase ethylene polymerization on supported catalyst. The simulation results of the process model indicate that the cosolubility phenomenon (i.e., the enhancement in the equilibrium concentration of ethylene in the amorphous phase of polyethylene in presence of n-hexane) cannot be the sole reason for the experimentally observed increase in the polymerization rate seen over the entire duration of reaction. At the beginning, the rate of ethylene polymerization is enhanced much more strongly than would be expected simply from the cosolubility effect alone. However, as the reaction proceeds, the enhancement magnitude gradually decreases and reaches to a steady-state value corresponding to the promotion magnitude in the equilibrium concentration of ethylene predicted by the two thermodynamic models.

Figure for Abstract:

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1. Introduction The global demand for polyolefins has reached more than 140 million tons per year worldwide, and lowpressure processes using different catalytic systems account for the production of more than 80 wt% of this global demand.1 Given the scale of production of polyolefins produced via catalytic polymerization, the development of reliable process models in order to simulate the behavior of the related process units in general and the commercial reactor units in particular is becoming more and more important. The appropriate process models can equip the user-engineer with the essential tools required for the process design, optimization, and control which could ultimately lead to significant save in time and cost during the process development and operation.2,3 During olefin polymerization on supported catalysts, the active sites are immobilized on the heterogeneous catalyst particles. As polymer accumulates in the pores, the initial porous structure of the catalyst support breaks apart in a process referred to as fragmentation. If this step proceeds as planned, the fragments become completely encapsulated within the produced semi-crystalline polymer and one catalyst particle leads to one polymer particle. Since this process leads to the complete coverage of the active sites by a layer of semi-crystalline polymer, monomer and other active species present in the reaction environment must be sorbed from the gas phase into the polymer phase, then diffuse through this layer in order to reach the active sites where the polymerization reaction takes place.4 Given the crucial role of the sorption process in determining the equilibrium concentration of monomer in polymer phase surrounding the polymerization active sites, it seems reasonable to state that the development of an appropriate model to describe the solubility of monomer under different operating conditions will constitute the core of any process models related to the gas phase polymerization of olefins on supported catalyst.5 This holds true, not just for monomers and hydrogen, but also for chemically inert compounds that can swell the polymer. The maximum productivity of heterogeneously catalyzed gas phase olefin polymerizations is limited by the amount of heat that can be removed from the reactor. Since it offers much higher gas-particle relative velocities than all other alternatives, the fluidized bed reactor (FBR) is the only type of reactor that is economically feasible for the gas phase production of polyethylene (PE). The high gas-particle velocity gives the best particle-gas heat transfer coefficients possible, and the high gas flow through allows the heat to be evacuated rapidly.4 The production capacity of FBRs during gas phase polymerizations can be substantially enhanced by adding C3 and heavier alkanes to incoming gas phase feed stream at the bottom of the reactor. If the recycle stream containing these alkanes is cooled to a temperature below its dew point, these compounds 3 ACS Paragon Plus Environment


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are usually referred to as induced condensing agents (ICAs), the reactor is said to be operating in condensed mode and the latent heat of vaporization of the liquefied ICA in the reactor bed is used to remove some of the heat of reaction.6-8 Of course, if one is using a comonomer, it too can be liquefied in the recycle stream and evaporated in the reactor. In either case, one could achieve a higher production capacity in the condensed mode than is possible in the normal dry mode operation of the FBR. Note that many processes are operated under conditions where the alkanes are not liquefied, however, their presence in the feed stream increases the gas phase heat capacity and thus also contributes to an improved production capacity.9 In the condensed mode operation, the evaporation process of the liquid phase of the feed stream inside the reactor is relatively fast.10 Consequently, even if liquefied, the ICA will be present inside the reactor as vapors for a much longer time than they are in the liquid phase. For this reason we will focus exclusively on the role of vaporized ICA in the remainder of this paper. ICAs are “chemically” inert in the sense that they do not have any influence on the behavior of the active sites. Nevertheless, recent papers from our group have clearly demonstrated that the presence of an ICA in the vapor phase can have a strong influence on the rate of both the homo- and copolymerization of ethylene.11-14 Experimental studies of sorption equilibria15-21 suggest that the presence of a heavier component (i.e., an ICA) in the gas phase enhances the solubility of the lighter component (i.e., ethylene) in the amorphous phase of polymer. This phenomenon is often referred to as cosolubility effect, and the enhancement in the rate of polymerization was primarily attributed to the cosolubility effect of the heavier ICA compounds in enhancing the local concentration of ethylene in the amorphous phase of PE surrounding the active sites. Despite the industrial importance of this topic, a limited number of theoretical and modeling studies are found to have been conducted to analyze and describe the sorption process of a multi-component gaseous mixture in polymer phase. Paricaud et al.22 attempted to explain the cosolubility effect by comparing the quality of interactions of the molecules of components composing the gas phase with each other and with the segments of polymer chains. In a different vein, Bashir et al.23 attributed the cosolubility effect of a heavier component to its higher thermal expansion coefficient which could create more free volume in the polymer phase required for the additional sorption of a lighter component. Furthermore, three types of approaches have been considered so far in modeling the cosolubility phenomenon: molecular simulation, activity coefficient approach, and equations of state. Nath et al.24 studied the solubility of a mixture of ethylene and 1-hexene in amorphous PE (modeled as C70) using united atom force field. Their molecular simulation results demonstrated that the solubility of ethylene increases when the 1-hexene partial pressure increases in the system. Yao et al.18,19 adapted the Universal Functional-group Activity Coefficient (UNIFAC) method as a model based on activity coefficient theory in order to describe the 4 ACS Paragon Plus Environment

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solubility of mixtures of ethylene/iso-pentane and ethylene/n-hexane in PE. The authors claimed that the UNIFAC model is able to provide relatively satisfactory estimates of the solubility of solutes in PE measured experimentally under various temperatures, pressures, and gas phase compositions. Equations of state (EoS) based on statistical associating fluid theory and lattice theory have also been applied recently in order to investigate their performance in predicting the solubility of a gas phase as a mixture of two components in polymer phase. Paricaud et al.22 employed the Statistical Associating Fluid Theory for potentials of Variable Range (SAFT-VR) to investigate the solubility of a mixture of two gases in PE. Their simulations showed that while 1-hexene increases the solubility of 1-butene, the 1butene solubility decreases by addition of nitrogen. Banaszak et al.25 used the molecular simulation method of united atom force field in order to parameterize the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) EoS and then extended the PC-SAFT EoS to the ternary system of two solutes and a polymer. Their simulation results predicted that by introducing 1-hexene to the gas phase composition, the solubility of ethylene in PE increases while the addition of ethylene decreases the 1hexene solubility. This, in turn, leads to an overall decrease in the gas solubility with respect to that predicted by summing results obtained for the two binary systems. Their simulation results were later validated experimentally by gravimetric sorption measurements conducted by Novak et al.21 Similar simulations were performed by Haslam et al.26 using simplified PC-SAFT and SAFT-VR predicting similar trends to the ones presented earlier by Banaszak et al.25 Sanchez-Lacombe EoS (SL EoS) as a model based on the lattice theory was employed by Bashir et al.27 to describe the solubility of a gas phase as a mixture of two α-olefins in polyolefins. The SL EoS simulation results demonstrated that this model can also adequately predict the cosolubility phenomenon as the enhancement in the solubility of a lighter α-olefin in presence of heavier one in the gas phase composition. From this rapid review, it can be deduced that all of the theoretical studies on the sorption process of gaseous mixture have been conducted for the systems of gas-polymer which are in the equilibrium state. The present modeling work has therefore been undertaken in order to investigate the impact of vaporized n-hexane (as the ICA compound) on the equilibrium concentration of ethylene in the amorphous phase of PE and consequently the rate of ethylene polymerization during the condensed mode operation – for the first time under reactive conditions. The article is organized as follows. First, the experiments performed to systematically investigate the effect of n-hexane on the polymerization rate in already published work12 are presented. Next, a brief description of the thermodynamic models used in this study (i.e., SL and PCSAFT EoS) is provided .This is followed by fitting the models to the experimental solubility data available for the binary systems of ethylene-PE and n-hexane-PE and the ternary system of ethylene-nhexane-PE. A process model based on the thermodynamic models of SL and PC-SAFT EoS is developed 5 ACS Paragon Plus Environment


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and finally, the effect of n-hexane on the ethylene polymerization rate is simulated and analyzed by implementing the process model.

2. Experimental Section 2.1. Polymerization Process The materials, experimental set-up and procedure used in the experimental investigation of the impact of different, commonly-used ICAs on the gas phase ethylene polymerization on supported Ziegler-Natta catalyst are provided in a detailed manner in reference [12], and the results analyzed herein are taken from this same reference. However, it is important that the reader understand the details of the polymerization procedure (temperature, pressure, and gas phase composition) in order to better interpret the results of the current modeling study. The ethylene pressure in the reactor was kept at a specific pre-determined value (7 or 12 bars) using a set of pressure regulators to control the inlet gas feed rate. Furthermore, the partial pressure of the ICA compound in the gas phase composition was determined by the amount of liquid ICA injected into the reactor at the room temperature. In addition to ethylene and the ICA compound, three other components were present in the gas phase composition during all the polymerization experiments in low but consistent amounts, namely argon, hydrogen and n-heptane. Argon was used to keep the reactor free of oxygen and other impurities. Hydrogen, as an agent to control the polymer molecular weight, was optionally chosen to inject the catalyst powder into the reaction environment. Finally, n-heptane was used as a solvent to dilute the co-catalyst of triethylaluminum (TEA); before starting the polymerization experiments, 1 cm3 of 1 M solution of TEA in n-heptane was introduced into reactor in order to scavenge all the remaining traces of water while also acting as co-catalyst later on, throughout the reaction. Given the experimental procedure used, it is possible to have confidence in the values of the operating pressure and composition of the gas phase during the polymerization. In contrast, constant tracking of the temperature at which ethylene polymerization takes place at the active sites appears to be quite challenging and thus needs to be treated with caution. The spherical stirred-bed gas-phase reactor set-up is presented schematically in Figure 1. Ethylene feed was introduced into the reactor at the room temperature (T1 = 25 ± 2 oC) from the experimental ballast without any preheating. The reactor was heated by circulating water in a jacket covering the external surface of the reactor. The temperature of the circulating water bath (T3) was set to 80 oC as the “desired” polymerization temperature. Meanwhile, the gas phase temperature was measured using a set of a thermocouple which was intruded obliquely from upper section of the reactor. The position, angle, and length of intrusion of 6 ACS Paragon Plus Environment

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thermocouple is designed in a manner to prevent its collision with the reactor agitator. Due to this structural limitation, one could be certain that the thermocouple is only measuring the gas phase temperature in the upper half-sphere inside the reactor. Thus, it has a limited capability to directly sense the heat being generated in the stirred bed of growing polymer particles at the lower half-sphere of the reactor – where the particles spend most of their time. This perception was supported by the fact that the temperature recorded by thermocouple (T2) was 75 ± 2 oC during all the polymerization experiments; the changes in the rate of polymerization and the associated heat release did not have any tangible impact on the measured temperature over the operating conditions listed in Table 1. The gas phase temperature measured by the thermocouple was always lower than the temperature of circulating water bath. This can be attributed to the cooling effect of the ethylene feed with lower temperature entering the reactor. Nevertheless, it should be underlined that due to the exothermic nature of the polymerization reaction and the poor heat transfer characteristics of gas phase reactions, the particle temperature (T4) is expected to be higher than the surrounding gas phase. The magnitude of temperature difference between a particle and gas phase is expected to be larger during the initial steps of the reaction and subsequently decrease during

the course of polymerization.4

Figure 1. A schematic representation of the spherical stirred-bed gas-phase reactor used in the experimental study, with T1, T2, T3, and T4 representing the temperature of ethylene feed, temperature measured by the thermocouple, temperature of circulating water bath, and particle temperature.

2.2. Kinetic Observations The effect of different ICAs on the rate of ethylene polymerization on a supported Ziegler-Natta catalyst has already been investigated in an exhaustive manner by the authors and presented in reference [12]. The experimental results corresponding to the impact of n-hexane, as one of the most commonly-used ICA 7 ACS Paragon Plus Environment


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compound, are reproduced in this section. These results will be used as the reference in this paper in order to evaluate the performance of the developed process model based on the thermodynamic models of SL and PC-SAFT EoS. The list of experiments with the partial pressure of the components present in the gas phase composition during each polymerization reaction is summarized in Table 1. The number assigned to each polymerization reaction under the specified operating condition in Table 1 will be used consistently to refer to that reaction condition throughout the current paper. In addition, it must be noted that in our analysis, the role of other components present in the reaction environment i.e., argon, hydrogen and nheptane are safely neglected. In case of argon and hydrogen, this is due to low partial pressure and much lower solubility in PE, even with respect to ethylene. In case of n-heptane, its very low partial pressure allows us to consider its impact to be negligible.

Table 1. The summary of the polymerization reactions analyzed in the current study with the partial pressure of the components (in bars) present in the gas phase composition at each reaction condition.

Component Rp1 Rp2 Rp3 Rp4 Rp5 Rp6 Rp7 Rp8

Ethylene 7.0 7.0 7.0 7.0 12.0 12.0 12.0 12.0

n-Hexane 0.3 0.6 0.8 0.3 0.6 0.8

Hydrogen ≤0.1 ≤0.1 ≤0.1 ≤0.1 ≤0.1 ≤0.1 ≤0.1 ≤0.1

n-Heptane ≤0.01 ≤0.01 ≤0.01 ≤0.01 ≤0.01 ≤0.01 ≤0.01 ≤0.01

Argon 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Figure 2 shows the influence of n-hexane on the instantaneous rate of polymerization (expressed in gr pol.

/ gr cat. hr) at the partial pressure of ethylene equal to (a) 7 and (b) 12 bars; the rate of ethylene polymerization increases in presence of n-hexane and as the partial pressure of n-hexane in the gas phase increases, the polymerization rate is enhanced more. In order to develop a better understanding about the effect of n-hexane, the instantaneous rate of polymerization in presence of n-hexane is normalized with the one without any n-hexane and presented in Figure 3. Consequently, this helps us to see that the magnitude of the increase in the polymerization rate is more pronounced at the beginning of reaction and decreases as the reaction progresses, approaching a steady-state value at the later steps. Moreover, this figure shows that the relative magnitude of increase in the reaction rate due to presence of a specific level of n-hexane are very similar (almost identical) at 7 and 12 bars of ethylene partial pressure. Thus, it can be deduced that the ethylene partial pressure has a negligible impact on the magnitude of induced enhancement in the polymerization rate under the operating conditions used. 8 ACS Paragon Plus Environment

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While the cosolubility effect of n-hexane is considered as the prime reason for the observed enhancement in the rate of ethylene polymerization, there remains a number of important questions yet to be fully understood. Is it possible to quantitatively describe the solubility of ethylene and n-hexane in the ternary system of the two solutes and PE? If so, what is the expected order of magnitude of enhancement in the polymerization rate due to the promotion in ethylene solubility in presence of n-hexane? How does partial pressure of ethylene impact the cosolubility phenomenon? Is it possible to consider the cosolubility effect as the sole reason for the observed impact of n-hexane on the polymerization rate? If not, then how can we possibly explain the evolution of enhancement magnitude in the rate during the reaction? In this study, it is attempted to answer these question by developing a process model based on the thermodynamic models of SL and PC-SAFT EoS. The process model will not only enable us to evaluate the performance of the thermodynamic models under the reactive condition, but will also serve as a useful intellectual tool to develop improved understanding about the dynamics of the polymerization process in presence of nhexane.



3000

Rp1 Rp2

2500

Rp3 Rp4

7 bars Ethylene

Rp (gr pol. / gr cat. hr)

3500

3000

Rp5 Rp6

2500

Rp7 Rp8

(b) Rp (gr pol. / gr cat. hr)

3500

(a)

2000

1500

1000

500

12 bars Ethylene

2000

1500

1000

500

0

0 0

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40

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120

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Time (minute)

60

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Time (minute)

(a)

2.0

Rp2 / Rp1

7 bars Ethylene

Rp3 / Rp1 Rp4 / Rp1

1.8

1.6

1.4

1.2

1.0

(b)

2.0

Relative Rp (dimensionless)

Figure 2. The influence of n-hexane on the instantaneous rate of gas phase ethylene polymerization on supported Ziegler-Natta catalyst at the partial pressure of ethylene equal to (a) 7 and (b) 12 bars.


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1.8

Rp6 / Rp5 Rp7 / Rp5

12 bars Ethylene

Rp8 / Rp5

1.6

1.4

1.2

1.0 0

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40

60

80

100

120

0

20

Time (minute)

40

60

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Time (minute)

Figure 3. The instantaneous rate of ethylene polymerization in presence of n-hexane normalized with the one without any n-hexane at the partial pressure of ethylene equal to (a) 7 and (b) 12 bars.

3. Equation of State Modeling


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3.1. Models Description Of the large number of equations of state available in the literature, the SL and PC-SAFT EoS can be considered today as the two major molecular-based thermodynamic models that are widely used in the polymer industry.28 This is primarily due to the promising performance of these models to study thermodynamic properties and phase behavior of polymer systems. Both models explicitly account for the size and shape of molecules. Thus, as an essential point from the perspective of a practicing engineer, they can simultaneously describe the behavior of both polymer and conventional molecules as well as their mixtures. Moreover, the relative simplicity of SL EoS and the superior predictive capability of PCSAFT EoS in comparison with the other versions of SAFT models can be considered as the most notable feature of each model which had also contributed to their present predominance in the field of polymer thermodynamics. In this section, the basic concepts on which these thermodynamic models are founded will be briefly overviewed and we will not go into developmental details of the models. Such information is available in the original references29-32 as well as the recent relevant theses of Alizadeh33 and Chmelař.34 The SL EoS is a model based on the lattice theory. In this model, in order to calculate the thermodynamic properties of a pure component, it is assumed to be broken into parts or “mers” and placed into a lattice structure. The mers are then allowed to interact with each other with a mean field type of intermolecular potential. The SL EoS is basically an extension of the classic Flory-Huggins theory. The most important improvement made in SL model is that the vacant lattice sites or holes are introduced to account for variation in density and compressibility. Thus, the system volume or density may vary by changing the fraction of holes in the lattice. In SL EoS, the thermodynamic properties of a pure component can be described by three lattice parameters of ε, v, and r which are the mer-mer interaction energy, the closedpacked molar volume of a mer and the number of sites (mers) a molecule occupies in the lattice, respectively. SL EoS has an explicit size or shape dependency through the lattice parameter r. Therefore, it takes into account the chain-like structure of polymer molecules. The three lattice parameters of ε, v, and r are used to define three macroscopic characteristic parameters for a pure component. The macroscopic parameters include T*, P*, and ρ* which are characteristic temperature, pressure, and closed-packed mass density, respectively. Either set of lattice or macroscopic parameters can be employed to characterize the thermodynamic properties of a pure component. However, the characteristic parameters in its macroscopic form is conventionally preferred and well-tabulated for a wide range of pure substances.5,30 The macroscopic characteristic parameters of SL EoS for the pure components of interest in this study are given in Table 2.



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The PC-SAFT EoS is a continuum model which pictures molecules to be moving freely in continuous space. Continuum models do not invoke the artificial lattice structure but are derived on the basis of perturbation theory. In the perturbation theory, an appropriate reference system is considered to describe the repulsive interactions of the molecules. Deviation of real molecules from the reference system are then accounted for by some correction terms which are often called perturbation terms. The PC-SAFT EoS was particularly developed for modelling chain-like molecules.31,32 In this model, the molecules are assumed to be constituted of spherical segments freely jointed and exhibiting repulsive and attractive forces among them. The repulsive interactions are described by an expression of hard chain while the attractive interactions are divided into dispersion interactions (i.e., van der Waals forces) and a contribution due to association (e.g., formation of hydrogen bonds). In PC-SAFT EoS, a pure substance without association is characterized by three parameters: the number of segments per molecule, m, the temperature-independent segment diameter, σ, and the energy related to the interaction of two segments, ε/k, where k is the Boltzmann constant. The characteristic parameters of PC-SAFT EoS for the pure components of interest in this study are provided in Table 2. Here, it must be noted that in order to estimate the parameter m for PE, the weight average molecular weight of 50 kg/mol is optionally considered in the lack of such information for polymer samples used in the solubility experiments (see Section 3.2). The recent review of experimental solubility studies of ethylene in PE by Chmelař et al.35 indicates that all PE samples used in the open literature for this purpose have had a weight average molecular weight above 50 kg/mol. In addition, it is verified that changes in the molecular weight of PE above 50 kg/mol have no impact on the calculated solubilities and consequently the performance of PCSAFT EoS. Table 2. The pure-component parameters of the thermodynamic models.

SL EoS Component

T* (K)

P* (bar)

ρ* (kg/m3)

Reference

Ethylene n-Hexane PE

283 476 653

3395 2979.1 4360

680 775 903

[5] [29] [5]

PC-SAFT EoS Component Ethylene n-Hexane PE*

m(̶) 1.5930 3.0576 1316.9

σ (Å) 3.4450 3.7983 3.9876

ε/k (K) 176.47 236.77 246.00

Reference [31] [31] [21]

* Weight average molecular weight of 50 kg/mol is considered for PE.


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Both equations of state can be extended to mixtures by introduction of one additional interaction parameter for each binary pair, kij. This parameter accounts for specific binary interaction between components (i) and (j) and is defined to correct the deviation of the geometric mean of energy parameters in both models by: 1

(1)

in which εi and εj represent the interaction energy between mers (SL) or segments (PC-SAFT) of component (i) and (j), respectively, while εij denotes the cross-energy parameter between mers (SL) or segments (PC-SAFT) of component (i) and component (j). In practice, the binary interaction parameter, kij, is employed as the adjustable parameter to fit the model to the experimental solubility data, as elucidated in details in upcoming section (3.2). A schematic demonstration of SL and PC-SAFT EoS representing the sorption equilibrium of the gas phase as a mixture of ethylene and n-hexane in the amorphous phase of PE is provided in Figure 4.

Figure 4. Schematic demonstration of SL and PC-SAFT EoS representing the sorption equilibrium of the gas phase as a mixture of ethylene and n-hexane in the amorphous phase of PE.

3.2. Models Application Procedure and Performance The thermodynamic models of SL and PC-SAFT EoS are independently fitted to the equilibrium solubility data for the binary systems of ethylene-PE and n-hexane-PE and the ternary system of ethylenen-hexane-PE. The solubility values were obtained implementing the pressure-decay technique by Yao et al.18,19 and are used here as the only source of relevant experimental data available in open literature at present time for the binary and ternary systems of interest in the current study (see Supporting



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Information for more details). The solubility measurements were performed at three equilibrium temperatures of 70, 80, and 90 oC. The PE sample used in the solubility measurements of the ternary system was of the same grade as the one used in the binary systems. The PE used in the experiments of Yao et al.18,19 is reported to have the crystallinity of 49%. The crystallinity of the samples produced in our polymerization experiments was on the order of 55% in the absence of n-hexane and close to 65% in the presence of n-hexane.14 Thus, while the crystallinity values are not identical, we will assume that the differences in the degrees of crystallinity reported herein will not have a noticeable impact on the solubility behavior of solutes in the amorphous phase of PE.35 First, SL and PC-SAFT EoS are fitted to the binary systems of ethylene-PE and n-hexane-PE by tuning the binary interaction parameter of k12. In the binary systems, component (1) and (2) are used to refer to the solute and polymer, respectively. In order to achieve the best model fitting, k12 was adjusted to minimize the percent average relative deviation (ARD) between the experimental and calculated solubility data as the objective function for the binary systems (OFbinary):

OFbinary

1 = ARD = Np

Np

∑ i =1

S iexp − S icalc S

exp i

(2)

× 100

in which Np is the number of experimental points in a given solubility isotherm while Siexp and Sicalc represent the experimental and calculated solubility values of the solute (ethylene or n-hexane) in the amorphous phase of PE at the ith point of the isotherm, respectively. Figure 5 demonstrates the best fit solubility isotherms calculated by (a) SL and (b) PC-SAFT EoS to

describe the solubility of ethylene and n-hexane obtained experimentally at 70, 80, and 90 oC in the corresponding binary systems of ethylene-PE and n-hexane-PE, respectively. The ARD for the best fitting of SL and PC-SAFT EoS to the experimental solubility data at each temperature are presented in Table 3. As can be deduced from Table 3 and Figure 5, both models, in general, can satisfactorily describe the temperature and pressure dependency of ethylene and n-hexane solubility in PE by adjusting the binary interaction parameter. However, the binary interaction parameter that provides the best fitting is found to be temperature-dependent for both thermodynamic models, as summarized in Table 4.


Page 15 of 42

(a)

0.030

0.25 o

ethylene

o

70 C 80 C o

Solubility (gr / gr am. pol.)


0.025

90 C SL EoS

0.020

0.015

0.010

0.005

0.000

0.20

SL EoS 0.15

0.10

0.05

0.00 0

500

1000

1500

2000

2500

0

20

Pressure (kPa)

(b)

n-hexane

70 C o 80 C o 90 C

o

40

60

80

100

Pressure (kPa)

0.030

0.25

ethylene

o

70 C

o

80 C

80 C

0.025


o

90 C PC-SAFT EoS

0.020

n-hexane

o

70 C

o


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.015

0.010

0.005

0.000

0.20

o

90 C PC-SAFT EoS

0.15

0.10

0.05

0.00 0

500

1000

1500

2000

2500

0

Pressure (kPa)

20

40

60

80

100

Pressure (kPa)

Figure 5. Comparison of the best fit solubility isotherms calculated by (a) SL and (b) PC-SAFT EoS for the binary systems of ethylene-PE and n-hexane-PE with the experimental data at 70, 80, and 90 oC.



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Page 16 of 42

Table 3. ARD between experimental and calculated solubility data corresponding to: (a) best fitting of SL and PC-SAFT EoS for the binary systems and (b) a priori prediction and best fitting of SL and PC-SAFT EoS for the ternary system.

SL EoS Binary System(a)

Ternary System(b)

Best Fit

Best Fit

A Priori Prediction

Best Fit

ethylene-PE

hexane-PE

ethylene-hexane-PE

ethylene-hexane-PE

T (oC)

ARD1

ARD2

ARDtot

ARD1

ARD2

ARDtot

ARD1

ARD2

70 80 90

0.7% 0.7% 0.5%

11.2% 3.9% 6.6%

64.8% 45.7% 28.7%

29.6% 15.8% 5.7%

99.9% 75.6% 51.7%

10.0% 4.4% 7.1%

12.1% 5.5% 3.5%

7.9% 3.3% 10.6%

PC-SAFT EoS Binary System(a)

Ternary System(b)

Best Fit

Best Fit

A Priori Prediction

Best Fit

ethylene-hexane-PE

ethylene-hexane-PE

ethylene-PE

hexane-PE

o

T ( C)

ARD1

ARD2

ARDtot

ARD1

ARD2

ARDtot

ARD1

ARD2

70 80 90

1.1% 0.9% 1.1%

13.7% 5.4% 5.9%

14.9% 10.4% 9.8%

18.2% 9.8% 4.7%

11.7% 11.0% 14.9%

14.4% 8.6% 7.4%

20.0% 11.7% 5.8%

8.8% 5.4% 9.0%

Table 4. Binary interaction parameters (kij) used for: (a) best fitting of SL and PC-SAFT EoS for the binary systems and (b) a priori prediction and best fitting of SL and PC-SAFT EoS for the ternary system.

SL EoS Binary System(a)

Ternary System(b)

Best Fit

Best Fit

A Priori Prediction

Best Fit

ethylene-PE

hexane-PE

ethylene-hexane-PE

ethylene-hexane-PE

T (oC)

k12

k12

k12

k13

k23

k12

k13

k23

70 80 90

-0.014 -0.022 -0.032

0.003 0.010 0.028

0.000 0.000 0.000

-0.014 -0.022 -0.032

0.003 0.010 0.028

0.000 0.000 0.000

-0.014 -0.022 -0.032

0.025 0.029 0.042

PC-SAFT EoS Binary System(a)

Ternary System(b)

Best Fit

Best Fit

A Priori Prediction

Best Fit

ethylene-hexane-PE

ethylene-hexane-PE

ethylene-PE

hexane-PE

o

T ( C)

k12

k12

k12

k13

k23

k12

k13

k23

70 80 90

0.029 0.026 0.022

0.013 0.021 0.041

0.000 0.000 0.000

0.029 0.026 0.022

0.013 0.021 0.041

0.000 0.000 0.000

0.029 0.026 0.022

0.010 0.017 0.034


Page 17 of 42

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Similarly, in the ternary system of ethylene-n-hexane-PE, the solubility measurements were conducted by Yao et al.18,19 at three equilibrium temperatures of 70, 80, and 90 oC. At each equilibrium temperature, four series of experiments were performed at the total pressure (PT) equal to 5, 10, 15, and 20 bars. A brief explanation about the experimentally observed solubility behavior of ethylene as a function of its partial pressure in the mixture might be useful here (Figure 6 and 7). Since the sorption measurements were performed at a constant total pressure, as the partial pressure of n-hexane in the mixture increased, the partial pressure of ethylene was decreased in order to keep the total pressure of gas phase, PT, constant. The higher partial pressure of n-hexane, in turn, resulted in a greater enhancement of ethylene solubility in the polymer. Consequently, we can see that as the partial pressure of ethylene in the gas phase mixture decreases, its solubility in PE remains more or less the same or even slightly increases. In order to predict the solubility of solutes in the ternary system by SL and PC-SAFT EoS, the required binary interaction parameters are initially estimated by the ones obtained in the corresponding binary systems of solute-polymer. In the ternary system, components (1) and (2) represent the solutes of ethylene and n-hexane, respectively, and component (3) refers to PE. In other words, in the a priori prediction of solubility of solutes in the ternary system, k13 and k23 are equal to the values corresponding to the best fitting of SL and PC-SAFT EoS to the experimental data in the binary systems of ethylene-PE and nhexane-PE, respectively. The value of k12 is set to 0 since the interaction between the small solute molecules in such systems are considered to be negligible (see Table 4).27 The solubilities of ethylene and n-hexane obtained experimentally at 70, 80, and 90 oC in the ternary system are compared with the values estimated by the a priori prediction of both models. It is found that SL EoS overestimates the solubility of both ethylene and n-hexane in the ternary system at all studied temperatures. Meanwhile, PC-SAFT EoS, in general, overestimates the ethylene solubility while underestimating that of n-hexane at all measurement temperatures. For the brevity purpose, the comparison of the a priori predictions of SL and PC-SAFT EoS with the ternary solubility data are presented only at 80 oC in Figure 6 and 7, respectively. The corresponding graphs at 70 and 90 oC are provided in Supporting Information. By comparing the average relative deviation between the predicted and experimental solubility values in Table 3, it can be deduced that a priori prediction of PC-SAFT EoS provides a closer estimation for the solubility of both solutes in the ternary system compared to SL EoS. Moreover, the magnitude of deviation of the predicted solubility values by both models from the ones obtained experimentally decreases by increasing the measurement temperature. In order to have an accurate estimation of ethylene concentration in the amorphous phase of PE in the ternary system, a more precise description of ethylene and n-hexane solubility in PE is required at the



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Page 18 of 42

same time. It is because while the ethylene solubility determines the mass (or mole numbers) of ethylene in the amorphous PE, this is the solubility of n-hexane which actually controls the volume and the extent of swelling of the amorphous phase of PE.5,18 Thus, as the next step, the fitting quality of both models to the ternary experimental data is improved further by tuning the binary interaction parameter of k23 as the adjustable parameter. This is because while the solubility of both ethylene and n-hexane is sensitive to k23, n-hexane solubility shows much less (almost negligible) sensitivity to k13.27 In order to have the best possible fitting of SL and PC-SAFT EoS, k23 is adjusted to minimize the total percent average relative deviation (ARDtot) as the objective function for the ternary system (OFternary) given by:

OFternary

1 1 1 = ARDtot = ( ARD1 + ARD2 ) =  2 2 Np 

Np

S iexp − S icalc

i =1

S iexp

∑

1 + Np

Np

∑ j =1

 − S calc S exp j j  × 100 exp  Sj 

(3)

in which ARD1 and ARD2 are the percent average relative deviations between the experimental and calculated solubility values of ethylene and n-hexane in the ternary system, respectively.

To achieve the best fitting of SL EoS to the experimental data, k23 is increased. As the a priori prediction of SL EoS overestimates the solubility of both ethylene and n-hexane, the increase in k23 results in a decrease in the calculated solubility of ethylene and n-hexane and consequently an improved fitting of SL EoS to the experimental solubility of both solutes in the ternary system at all measurement temperatures. On the other hand, the best fitting of PC-SAFT EoS is obtained by decreasing k23. The decrease in k23 results in an increase in calculated solubility of ethylene and n-hexane at the same time. Since the a priori prediction of PC-SAFT EoS overestimates the ethylene solubility and underestimates that for n-hexane, the best possible fitting of PC-SAFT EoS according to the procedure described above is achieved by an improved model fitting to the experimental solubility data of n-hexane and in the cost of (slight) further overestimation of ethylene solubility in the ternary system at all measurement temperatures. As the a priori prediction of PC-SAFT EoS provides a closer estimation of the solubility data, the required adjustment of k23 is smaller in the case of PC-SAFT. Nonetheless, SL EoS can provide a more accurate description of the solubility of solutes in the ternary system by adjusting k23 at all measurement temperatures.


Page 19 of 42

0.030

0.030

PT = 5 bars

ethylene

0.025

0.020



PT = 10 bars

ethylene

0.025

0.020 0.015 0.010 0.005 0.000 380

400

420

440

460

480

500

0.25

n-hexane

0.20 0.15

0.015 0.010 0.005 0.000 880

900

920

940

960

980

1000

60

80

100

120

0.25

n-hexane

0.20 0.15

0.10

0.10

0.05

0.05

0.00

0.00 0

20

40

60

80

100

120

0

20

40

Pressure (kPa)

Pressure (kPa)

0.030

0.030

PT = 15 bars

ethylene

0.025

ethylene

0.025

PT = 20 bars

0.020


0.020


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.015 0.010 0.005 0.000 1380

1400

1420

1440

1460

1480

1500

0.25

n-hexane

0.20 0.15

0.015 0.010 0.005 0.000 1880

1920

1940

1960

1980

2000

60

80

100

120

n-hexane

0.20 0.15

0.10

0.10

0.05

0.05

0.00

1900

0.25

0.00 0

20

40

60

80

100

120

0

20

Pressure (kPa)

40

Pressure (kPa)

Figure 6. Comparison of a priori prediction (‒ ‒) and best fit (―) solubility isotherms calculated by SL EoS for the ternary system of ethylene-n-hexane-PE with the experimental data (○) at 80 oC.



0.030

0.030

PT = 5 bars

ethylene

0.025

0.020



PT = 10 bars

ethylene

0.025

0.020 0.015 0.010 0.005 0.000 380

400

420

440

460

480

500

0.25

n-hexane

0.20 0.15

0.015 0.010 0.005 0.000 880

900

920

940

960

980

1000

60

80

100

120

0.25

n-hexane

0.20 0.15

0.10

0.10

0.05

0.05

0.00

0.00 0

20

40

60

80

100

120

0

20

40

Pressure (kPa)

Pressure (kPa)

0.030

0.030

PT = 15 bars

ethylene

0.025

PT = 20 bars

ethylene

0.025 0.020


0.020


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 42

0.015 0.010 0.005 0.000 1380

1400

1420

1440

1460

1480

1500

0.25

n-hexane

0.20 0.15

0.015 0.010 0.005 0.000 1880

1920

1940

1960

1980

2000

60

80

100

120

n-hexane

0.20 0.15

0.10

0.10

0.05

0.05

0.00

1900

0.25

0.00 0

20

40

60

80

100

120

0

20

Pressure (kPa)

40

Pressure (kPa)

Figure 7. Comparison of a priori prediction (‒ ‒) and best fit (―) solubility isotherms calculated by PC-SAFT EoS for the ternary system of ethylene-n-hexane-PE with the experimental data (○) at 80 oC.


Page 21 of 42

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As presented in Table 3, except for the ethylene solubility in PE in the binary system which could be described by the models in an excellent manner, the average relative deviation between the experimental and calculated solubilities based on the best fitting of both models cannot be considered to be negligible. Nonetheless, one needs to bear in mind that at each equilibrium temperature in the ternary system, 24 experimental data points of solubility (12 for each of ethylene and n-hexane) were fitted by both models covering a wide range of relevant total pressures (5 to 20 bars) and gas phase compositions. Therefore, as the seminal thrust of this paper is to provide an order of magnitude analysis on the effect of n-hexane on ethylene solubility, polymer swelling, and ultimately equilibrium concentration of ethylene in PE, it appears reasonable to consider the best fitting of SL and PC-SAFT EoS as adequate approximations for this purpose. While this presumption will lead to a small loss in accuracy, nevertheless it will be more than compensated for by extending the applicability of both models to investigate the impact of n-hexane on the rate of gas phase ethylene polymerization on supported catalyst - for the first time to the best of our knowledge.



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Page 22 of 42

4. Process Model Development In order to be able to establish a framework to construct a process model for the gas phase olefin (ethylene in this study) polymerization on supported catalysts, the development of a clear phenomenological understanding and representation of the process is obviously of the primary importance. In this process, the polymerization reaction starts at the active sites, which are chemically immobilized on the surface of a highly porous support material. The stresses generated by the polymer which successively fills the catalyst pores result in the fragmentation of the initial porous structure of the support into a finite number of smaller fragments (NF). However, the particle keeps its integrity thanks to the entangled network of produced polymer chains. By completion of the fragmentation step in the early stages of the reaction, the particle is transformed into a polymer particle where the semi-crystalline PE forms the continuous phase with the fragments of support dispersed therein, as demonstrated in Figure 8.

Figure 8. A schematic representation of a growing polymer particle by completion of the fragmentation step.

Thus, the monomer must diffuse through the macro-pores of the particle, be sorbed from the gas phase into the polymer phase, and then diffuse through the amorphous fraction of the polymer to reach the active sites on the support fragments. Since the polymerization reaction is exothermic, the heat produced at the active sites needs to be transferred the other way; through the polymer layer and structure of the particle to the particle surface; and from the surface, through the boundary layer to the bulk phase. This


Page 23 of 42

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will result in the concentration and temperature gradient through the growing particle as depicted schematically in Figure 9.

Figure 9. A schematic representation of the realistic (―) and the assumed (‒ ‒) concentration and temperature profile in the growing polymer particle.

Therefore, the apparent (or overall) rate of ethylene polymerization in a growing particle, , at each moment during the course of reaction can be expressed as: (4)

NF

R p = ∑ R loc p i i =1

in which is the local rate of ethylene polymerization at the surface of catalyst fragment (i). Since the rate of ethylene polymerization is considered to be of first order with respect to the ethylene concentration36, the local rate of polymerization can be described by: ∗ loc R loc p i = k p i Ci [ M 1 ]am. pol i

(5)

where

is the local propagation rate constant, ∗ is the local concentration of the active sites on the catalyst fragment (i), and . is the local concentration of ethylene in the amorphous phase of PE

surrounding the catalyst fragment (i). The intrinsic propagation rate constant (kp) and concentration of active sites (C*) immobilized on the surface of a catalyst support is established during the catalyst preparation step. Therefore, for a given catalytic system being used in the process,

and ∗ at each 23 ACS Paragon Plus Environment


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Page 24 of 42

moment will only be a function of the local temperature of catalyst fragment (Ti) at that moment. Meanwhile, . is determined by a) solubility or in more accurate terms the equilibrium

concentration of ethylene in the amorphous phase of PE and b) the significance of mass transfer resistance through the particle which can be described by an effective diffusion coefficient of monomer in the growing particle.37 By considering an ideal (but unrealistic) situation in which there is no temperature and concentration gradients inside the growing particle and also the concentration of the active sites on the surface of all catalyst fragments is equal, the apparent rate of polymerization given in equation (4) can be simplified into: (6)

. R p = k p C * [ M 1 ]eq am. pol

. in which . is the equilibrium concentration of ethylene in the amorphous phase of PE. In this

reference case, the temperature inside the particle is uniform and equal to the gas phase temperature while the ethylene concentration is also assumed to be homogeneous throughout the particle and equal to the equilibrium concentration of ethylene in PE (see Figure 9).

In the current analysis, the reference case is defined to represent an ideal polymerization system in which the growing polymer particle is in mass and thermal equilibrium with its surrounding gas phase during the whole period of reaction. In the reference system, the effect of an ICA compound on the rate of gas ethylene polymerization for a given catalytic system can be expressed by:

R ternary p R

binary p

[M 1 ]eqam.ternary . pol = eq .binary [M 1 ]am. pol

in which

and

(7)

are the rate of ethylene polymerization in the presence and absence of n-

. . hexane as the ICA, respectively, and . and . are the equilibrium concentration

of ethylene in the amorphous phase of PE in the presence and absence of the ICA, respectively. In the present study, the ideal system approach is adapted in order to investigate how the polymerization rate is expected to increase solely due to the promotion in the equilibrium concentration of ethylene in presence of n-hexane.


Page 25 of 42

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5. Results and Discussion 5.1. Thermodynamic Simulation Results The thermodynamic simulations based on the best fitting of SL and PC-SAFT EoS are performed to investigate the impact of n-hexane on the sorption equilibrium of ethylene in PE. To provide a succinct presentation of the simulation results, the magnitude of induced changes in solubility of ethylene .

(!,. ), volume of amorphous PE (#. ), and equilibrium concentration of ethylene ( . )

due to presence of n-hexane are defined as follows:

norm 1, am . pol

S

=

S1ternary , am. pol

(8)

binary 1, am. pol

S

where !,. and !,. are the ethylene solubility in presence and absence of n-hexane, respectively.

norm Vam . pol =

ternary Vam . pol

(9)

binary am . pol

V

where #.

and #. are the volume of amorphous PE in presence and absence of n-hexane,

respectively.

[M ]

eq .norm 1 am . pol

[M 1 ]eqam.ternary . pol = eq .binary [M 1 ]am. pol

(10)

. . where . and . are the ethylene equilibrium concentration in presence and

absence of n-hexane, respectively. -.. The magnitude of change in $%,&'.()*, +&'.()* , and ,% &'.()* as a function of partial pressure of n-

hexane (normalized with its vapor pressure, P/Pvap) based on the thermodynamic simulations of SL and PC-SAFT EoS are demonstrated in Figure 10 and 11, respectively. While the simulations presented here 25 ACS Paragon Plus Environment


are performed at 7 and 12 bars of ethylene partial pressure and 80 oC, nonetheless the very same trends are obtained by both models in the entire range of operating conditions used in the solubility measurement experiments i.e., 5 to 20 bars of ethylene partial pressure and temperature between 70 to 90 oC. These results are not presented here for the sake of brevity.

2.0

2.0

Volume of Amorphous PE 1.8

Enhancement Magnitude (dimensionless)

Equilibrium Concentration

1.6

1.4

1.2

1.0

12 bars Ethylene

Solubility of Ethylene

(b)



7 bars Ethylene


(a)


1.6

1.4

1.2

1.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

P/Pvap (dimensionless)

0.4

0.6

0.8

1.0


Figure 10. SL EoS simulation of the magnitude of change in the solubility of ethylene, volume of amorphous PE, and equilibrium concentration of ethylene as a function of normalized partial pressure of n-hexane at (a) 7 and (b) 12 bars of ethylene partial pressure and 80 oC.

2.0

2.0

7 bars Ethylene


(a)




1.6

1.4

1.2

1.0

12 bars Ethylene


(b)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 42


1.6

1.4

1.2

1.0 0.0

0.2

0.4

0.6

0.8

1.0

0.0


0.2

0.4

0.6



0.8

1.0

Page 27 of 42

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Figure 11. PC-SAFT EoS simulation of the magnitude of change in the solubility of ethylene, volume of amorphous PE, and equilibrium concentration of ethylene as a function of normalized partial pressure of nhexane at (a) 7 and (b) 12 bars of ethylene partial pressure and 80 oC.

Figure 10 and 11 show that as the partial pressure of n-hexane in the gas phase increases, the ethylene

solubility, the volume of amorphous PE, and the equilibrium concentration of ethylene in the amorphous phase of PE increase. At a given partial pressure of n-hexane, the promotion magnitude of the ethylene solubility is always higher than that of the volume of amorphous PE. This, in turn, results in the prediction of enhancement in the equilibrium concentration of ethylene in the amorphous PE. However, since the equilibrium concentration is proportional to the ratio of solubility to volume, the predicted magnitude of enhancement for the equilibrium concentration of ethylene is always lower than the one for the ethylene solubility. In addition, by comparing the simulated impact of n-hexane at 7 and 12 bars of ethylene partial pressure, it can be deduced that according to SL and PC-SAFT EoS the enhancement . magnitude in !,. , #. , and more importantly . is almost independent of the ethylene

partial pressure and majorly determined by the partial pressure of n-hexane in the gas phase. The trends simulated by SL and PC-SAFT EoS are fully in agreement with the observed influence of nhexane on the rate of gas phase ethylene polymerization on supported Ziegler-Natta catalyst presented in Section 2.2. As the partial pressure of n-hexane in the gas phase increases, the equilibrium concentration, and therefore the local concentration of ethylene in the amorphous phase of PE surrounding the active sites increases. This eventually results in the observed enhancement in the overall rate of ethylene polymerization. Moreover, the polymerization experiments showed that the promotion magnitudes in the reaction rate due to presence of a specific level of n-hexane are almost identical at 7 and 12 bars of ethylene partial pressure. According to the thermodynamic simulations of the system, this observation is consistent with and can be attributed to the negligible impact of the ethylene partial pressure on the promotion magnitude in the equilibrium concentration of ethylene in amorphous PE in presence of nhexane. After verifying the qualitative agreement between the simulation results and the kinetic observations, the process model developed in Section 4 will be implemented in order to acquire a quantitative knowledge about the enhancement magnitude in the polymerization rate solely due to the promotion in equilibrium concentration of ethylene in presence of n-hexane. One must consider that the performance of the process model depends directly on the performance of the underlying thermodynamic model. Thus, at first place, it is necessary to quantitatively compare the simulation results based on SL and PC-SAFT EoS and analyze the reason(s) for the discrepancy between them; by comparing Figure 10 and 11 we can see that at 27 ACS Paragon Plus Environment


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a given partial pressure of n-hexane and ethylene, PC-SAFT EoS predicts a higher magnitude of enhancement in the ethylene solubility while estimating a lower degree of polymer swelling. This, ultimately, results in the prediction of a larger magnitude of enhancement in the ethylene equilibrium concentration by PC-SAFT EoS in comparison with SL EoS. One needs to bear in mind that the simulation results presented here are based on the best possible fitting of SL and PC-SAFT EoS to the experimental solubility data. Consequently, the performance of each model depends not only on its inherent predictive capability but also on the quality of its fitting to the available experimental data. As presented in Section 3, both models can describe the solubility of ethylene in the binary system of ethylene-PE in an excellent manner (see Figure 5). However, in the ternary system of ethylene-n-hexane-PE, the best fitting of PC-SAFT EoS slightly overestimates the solubility of ethylene while the best fitting of SL EoS is able to describe it more accurately. Moreover, the best fitting of both models describes the solubility of n-hexane in the ternary system in more or less the same manner (compare the best fitting of SL and PC-SAFT EoS at 80 oC given in Figure 6 and 7, respectively). The overestimation of the ethylene solubility by the best fitting of PC-SAFT EoS in the ternary system leads to the prediction of a higher enhancement magnitude of the ethylene solubility. Thus, the difference between the magnitudes of ethylene solubility enhancement predicted by SL and PC-SAFT EoS results solely from their fitting quality. On the other hand, the best fitting of SL and PC-SAFT EoS describe the solubility of n-hexane in a similar manner, i.e., at a given partial pressure of n-hexane and ethylene, they estimate an almost identical value for the solubility of n-hexane in PE. However, we can see that PC-SAFT EoS predicts a lower magnitude of polymer swelling compared to SL EoS. Prediction of different levels of enhancement in the volume of amorphous PE due to the sorption of an identical amount of n-hexane can only be attributed to the difference in the inherent performance of each model to describe the polymer swelling. Given the explanation above, it should be clear that the larger magnitude of enhancement in the equilibrium concentration of ethylene predicted by PC-SAFT EoS is due to both its quality of fitting and inherent performance.

5.2. Analysis of Kinetic Observations 5.2.1. Application of Process Model Figure 12 and 13 compare the enhancement magnitude in the rate of ethylene polymerization in presence

of (a) 0.3, (b) 0.6, and (c) 0.8 bar partial pressure of n-hexane with the ones predicted by the process model based on SL and PC-SAFT EoS at 7 and 12 bars of ethylene partial pressure, respectively. The


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simulation results presented here are based on the best fitting of the models to the experimental solubility data at 80 oC as the intended temperature for the polymerization reactions (see Section 2.1). The impact of equilibrium temperature of the gas-polymer system on the cosolubility phenomenon will be analyzed separately later on in Section 5.2.2. As can be seen in Figure 12 and 13, the promotion magnitude in the rate of ethylene polymerization at the early stages of reaction is higher than the ones predicted by SL and PC-SAFT EoS. In addition, at both partial pressures of ethylene (i.e., 7 and 12 bars), as the partial pressure of n-hexane increases this discrepancy becomes more pronounced. However, as the reaction proceeds, the enhancement magnitude successively decreases, approaching a steady-state value which is in the order of the ones predicted by the two thermodynamic models. PC-SAFT EoS predicts higher magnitudes of promotion which are closer to the steady-state values obtained experimentally at the later steps of reaction. As discussed in Section 5.1, this is due to both fitting quality and inherent performance of PC-SAFT EoS. The simulation results of SL and PC-SAFT EoS indicate that the cosolubility phenomenon (i.e., the enhancement in the equilibrium concentration of ethylene in the amorphous phase of PE in presence of nhexane) cannot be the sole reason for the observed increase in the polymerization rate during the whole period of reaction. At the beginning, the rate of ethylene polymerization is enhanced much more strongly than would be expected simply from the cosolubility effect alone. This, in turn, implies that at the initial phase of reaction the system of growing polymer particle deviates significantly from the ideal system assumed in the development of the process model. However, the enhancement magnitude gradually decreases and reaches to a steady-state value which is in the order of the promotion magnitude in the equilibrium concentration of ethylene. Therefore, it can be deduced that as the reaction advances the growing polymer particles in the reactor approach to a semi-equilibrium condition in which the local concentration of ethylene and the local temperature at the active sites dispersed in each particle can be estimated by the equilibrium concentration of ethylene and the gas phase temperature, respectively. To interpret the experimentally observed impact of n-hexane, one needs to think about how the polymerization proceeds in reality and how it differs from the ideal system assumed in the process model (see Section 4). In the following we will delve into the differences between the real and ideal system while assessing their relevance and importance in determining the dynamic behavior of the polymerization reaction in presence of n-hexane.




(a)

2.0

Rp2 / Rp1

7 bars Ethylene

SL EoS 1.8

PC-SAFT EoS

1.6

1.4

1.2

1.0 0

20

40

60

80

100

120

Time (minute)


(b)

2.0

Rp3 / Rp1

7 bars Ethylene

SL EoS 1.8

PC-SAFT EoS

1.6

1.4

1.2

1.0 0

20

40

60

80

100

120

Time (minute)

(c) Relative Rp (dimensionless)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2.0

Rp4 / Rp1

7 bars Ethylene

SL EoS 1.8

PC-SAFT EoS

1.6

1.4

1.2

30 1.0 0

20

40

60

80

100

120

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Figure 12. Comparison of the enhancement magnitude in the rate of ethylene polymerization with the ones predicted by the process model based on SL and PC-SAFT EoS at 7 bars ethylene and (a) 0.3, (b) 0.6, and (c) 0.8 bar n-hexane partial pressure.


(a)

2.0

Rp6 / Rp5

12 bars Ethylene

SL EoS 1.8

PC-SAFT EoS

1.6

1.4

1.2

1.0 0

20

40

60

80

100

120

Time (minute)


(b)

2.0

Rp7 / Rp5

12 bars Ethylene

SL EoS 1.8

PC-SAFT EoS

1.6

1.4

1.2

1.0 0

20

40

60

80

100

120

Time (minute)

(c) Relative Rp (dimensionless)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2.0

Rp8 / Rp5

12 bars Ethylene

SL EoS 1.8

PC-SAFT EoS

1.6

1.4

1.2

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20

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80 31

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Figure 13. Comparison of the enhancement magnitude in the rate of ethylene polymerization with the ones predicted by the process model based on SL and PC-SAFT EoS at 12 bars ethylene and (a) 0.3, (b) 0.6, and (c) 0.8 bar n-hexane partial pressure.

5.2.2. Equilibrium Temperature In the ideal system, it is assumed that the growing polymer particle is in thermal equilibrium with its surrounding gas phase at a constant temperature. But in reality, the temperature of a particle and the immediate gas phase layer is expected to be higher at the early stages and gradually decrease during the course of reaction. This is because of the higher rate of ethylene polymerization (see Figure 2) and the smaller surface area of the particle to exchange the generated heat at this period. Thus, in this section, we will examine how the enhancement magnitude in the equilibrium concentration of ethylene evolves with the assumed equilibrium temperature for the gas-polymer system and whether or not it can account for the experimentally observed trend. Figure 14 and 15 show the impact of the equilibrium temperature on the enhancement magnitude in the

equilibrium concentration of ethylene at (a) 7 and (b) 12 bars of ethylene partial pressure predicted by SL and PC-SAFT EoS, respectively. The simulation results presented here are based on the best fitting of the models to the experimental solubility data in the binary and ternary systems of interest at 70, 80, and 90 o

C, as explained earlier in Section 3.2. Both models predict that at a given partial pressure of n-hexane

and ethylene, as the equilibrium temperature for the gas-polymer system increases, the enhancement magnitude in the ethylene equilibrium concentration decreases. PC-SAFT EoS exhibits a higher degree of sensitivity to the equilibrium temperature in comparison with SL EoS. As already discussed in Section 5.1, this discrepancy is due to both fitting quality and inherent performance of the models. The simulation results indicate that at a given partial pressure of n-hexane and ethylene, the enhancement magnitude in the equilibrium concentration of ethylene is expected to be lower at the beginning of reaction due to the higher particle temperature. Moreover, since the particle temperature decreases as the reaction proceeds, the enhancement degree in the equilibrium concentration will increase accordingly. In order to be able to evaluate the significance of this effect, one needs to be able to quantitatively describe the evolution of particle temperature under operating conditions but this falls beyond the scope of this study. However, one can note that the predicted influence of equilibrium temperature is completely opposite of the rate pattern; the polymerization rate increases more at the beginning of the reaction despite 32 ACS Paragon Plus Environment

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the expected lower degrees of enhancement in the equilibrium concentration of ethylene at this period. Thus, it is entirely reasonable to deduce that the experimentally observed trend does not originate from the effect of particle temperature on the equilibrium concentration of ethylene in the amorphous phase of PE. It is important to note that here we intended to isolate the impact of particle temperature on the ethylene solubility and concentration. We will look at the potential impact of n-hexane on the particle temperature, propagation rate constant and consequently the overall rate of polymerization in Section 5.2.4.

(a)

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7 bars Ethylene

o

SL 70 C

(b)

2.0 o

SL 80 C

SL 80 C 1.8

o

SL 90 C Experimental



1.8

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o

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o

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0.6

0.8

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0.0

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0.2

0.4

0.6

0.8

1.0


Figure 14. The effect of equilibrium temperature on the enhancement magnitude in the equilibrium concentration of ethylene in presence of n-hexane predicted by SL EoS at (a) 7 and (b) 12 bars partial pressure of ethylene. The experimental points correspond to the steady-state values of the enhancement in polymerization rate at the given partial pressures of ethylene and n-hexane.

(a)

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7 bars Ethylene

o

PC-SAFT 70 C

(b)

2.0 o

PC-SAFT 80 C 1.8

12 bars Ethylene

o

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o

PC-SAFT 80 C 1.8

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PC-SAFT 90 C Experimental



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


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Figure 15. The effect of equilibrium temperature on the enhancement magnitude in the equilibrium concentration of ethylene in presence of n-hexane predicted by PC-SAFT EoS at (a) 7 and (b) 12 bars partial pressure of ethylene. The experimental points correspond to the steady-state values of the enhancement in polymerization rate at the given partial pressures of ethylene and n-hexane.

5.2.3. Partial Pressure of n-Hexane The ideal system assumes that the growing polymer particle is in mass equilibrium with the gas phase which has a constant composition. In reality, however, the total n-hexane present in the reactor is partitioned between the gas phase and the polymer phase being generated. This, in turn, causes the partial pressure of n-hexane to drop during the course of polymerization. In Section 5.1, we showed that at a given partial pressure of ethylene, as the partial pressure of n-hexane decreases, the promotion magnitude in the equilibrium concentration of ethylene is also decreased. The gradual decrease in the partial pressure of n-hexane and consequently the ethylene equilibrium concentration leads to a successive decline of the enhancement magnitude in the polymerization rate and can obviously be considered as a phenomenological reason for the experimentally observed trend. Having said that, the question remains as to what is the significance of this effect under the operating conditions used in the polymerization experiments. To answer that, the magnitude of decrease in the partial pressure of n-hexane is estimated in the polymerization experiments in presence of 0.8 bar of n-hexane and at 7 and 12 bars of ethylene partial pressure corresponding to Rp4 and Rp8, respectively. At a given partial pressure of ethylene, the maximum drop in the pressure of n-hexane takes place in the experiment with the largest partial pressure of nhexane in the gas phase i.e., 0.8 bar. This is due to both the higher solubility of n-hexane in PE and larger mass of polymer produced. Thus, if the decrease in the partial pressure of n-hexane does impact the rate of polymerization, it must be more tangible under these operating conditions (i.e., Rp4 and Rp8). As demonstrated in Table 5, it is estimated that the partial pressure of n-hexane would decrease from 0.8 bar at the beginning to 0.72 and 0.69 bar at the end of polymerization experiments with 7 and 12 bars of ethylene partial pressure, respectively. The enhancement magnitude in the ethylene equilibrium concentration in presence of the initial and reduced levels of n-hexane is calculated by SL and PC-SAFT EoS at 7 and 12 bars of ethylene partial pressure and 80 oC and presented in Table 6. The simulation results indicate that the decrease in the partial pressure of n-hexane has a minor impact on the ethylene equilibrium concentration and clearly cannot explain the sharp drop in promotion magnitude of the polymerization rate observed experimentally. 34 ACS Paragon Plus Environment

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In addition to the analysis based on the thermodynamic modeling presented here, this perception is further supported by the kinetic results obtained. If the enhancement magnitude in the polymerization rate was actually dropping (at least in part) due to decrease in the partial pressure of n-hexane, one would expect to see a much steeper decline during the polymerization at 12 bars ethylene in comparison with 7 bars. This is because the mass of produced PE and consequently the solubilized n-hexane is considerably larger in the case of polymerization at 12 bars ethylene (for instance, compare the produced PE mass in Rp4 and Rp8, as given in Table 5). But in Figure 3, we can see that at a given partial pressure of n-hexane, the enhancement magnitude decreases in a more or less the same manner at 7 and 12 bars of ethylene partial pressure. This, in turn, independently confirms that the impact of the decrease in the partial pressure of nhexane on the polymerization rate is not significant and can be safely neglected.

Table 5. Estimation of the magnitude of decrease in the partial pressure of n-hexane due to its sorption in produced PE in the polymerization experiments in presence of 0.8 bar n-hexane and at 7 and 12 bars of ethylene partial pressure corresponding to Rp4 and Rp8, respectively.

12 × 10 -3

Catalyst Mass Injected

gr

Reaction Time

2

hr

PE Crystallinity

65%

gr crys. pol / gr pol

0.1

gr / gr am.pol

15

ml

0.655

gr / ml

Rp4

Rp8

1108.0

1662.0

gr pol / gr cat. hr

Produced PE Mass

26.6

39.9

gr

n-Hexane Mass Sorbed

0.93

1.40

gr

Solubility of n-Hexane at 0.8 bar and 80 oC o

Volume of n-Hexane Injected at 25 C o

n-Hexane Density at 25 C

Time-Averaged Polymerization Rate

Percent n-Hexane Sorbed

9.5

14.2

%

n-Hexane Pressure Drop

0.08

0.11

bar

Final Partial Pressure of n-Hexane

0.72

0.69

bar

Table 6. Comparison of the experimentally observed drop in the promotion magnitude of polymerization rate during the course of reaction with the change in the enhancement magnitude of the ethylene equilibrium concentration due to the decrease in partial pressure of n-hexane predicted by SL and PC-SAFT EoS.

7 bars ethylene

Beginning of Reaction

Pn-hexane (bar)

Rp4 / Rp1

SL EoS [M1]am.poleq.norm

PC-SAFT EoS [M1]am.poleq.norm

0.80

1.84

1.16

1.39



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

End of Reaction

0.72

1.29

1.14

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1.33

12 bars ethylene Pn-hexane (bar)

Rp8 / Rp5

SL EoS [M1]am.poleq.norm

PC-SAFT EoS [M1]am.poleq.norm

Beginning of Reaction

0.80

1.84

1.15

1.36

End of Reaction

0.69

1.43

1.12

1.29

5.2.4. Heat and Mass Transfer For a given catalytic system, the local and therefore the overall rate of ethylene polymerization at each moment during the course of reaction depend on the local temperature and concentration of ethylene at the active sites which are immobilized on the surface of catalyst fragments dispersed within each growing polymer particle (see equation 4 and 5). In the ideal system, however, it is assumed that there are no temperature and concentration gradients through the particle and its boundary layer during the whole period of reaction, as depicted schematically in Figure 9. Thus, in order to be able to interpret the discrepancy between the experimentally observed trend and the ones predicted by the process model based on SL and PC-SAFT EoS, one needs to think about how the presence of n-hexane may impact the local temperature and concentration of ethylene at the active sites during the course of polymerization. It is well-known that the heat transfer resistance is more significant on fresh catalyst particles during initial steps of polymerization. Higher rates of polymerization caused by higher ethylene concentration in the particles (due to cosolubility effect) lead to the higher rate of heat generation, which, in turn, results in further increase in the particle temperature and consequently the polymerization rate at this period. As the reaction proceeds, the rate of polymerization decreases due to the catalyst deactivation (see Figure 2). Simultaneously, the particle becomes larger providing higher surface area which facilitates the exchange of polymerization heat with the gas phase. Both effects would push the particle temperature and therefore the enhancement magnitude in polymerization rate to successively decrease. This process continues until the growing polymer particle reaches to a steady-state condition in which the particle temperature remains slightly higher than that of the surrounding gas phase and the particle-gas system can be perceived to be in thermal semi-equilibrium condition. In addition to the thermal effect, one needs to also bear in mind that during the early stages of reaction, the higher volumetric rate of polymerization induces a greater concentration gradient in the growing particle and the rate of polymerization is more diffusion-limited. As the reaction proceeds, the catalyst deactivates and the active sites become more and more diluted in the produced polymer. Therefore, the volumetric rate of monomer consumption and the magnitude of concentration gradient through the particle successively decrease. In short, the reaction becomes less diffusion-limited and the gas-particle 36 ACS Paragon Plus Environment

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system approaches to the mass semi-equilibrium condition. It is known that at constant crystallinity of polymer, n-hexane can enhance the ethylene diffusivity (in addition to its solubility) due to the plasticization of the amorphous polymer chains.38 Thus, for the same magnitude of the ICA-induced enhancement in diffusivity, it is entirely reasonable to expect a more significant effect on the polymerization rate during the initial steps of the reaction followed by progressive decrease in the magnitude of its impact during the course of polymerization. Considering the dynamics of heat and mass transfer, the growing polymer particle (both in the absence and presence of n-hexane) gradually approaches the thermal and mass semi-equilibrium condition. Therefore, the steady-state values of the enhancement magnitude in the polymerization rate reached at the later steps of reaction can be estimated by the promotion magnitude in the ethylene equilibrium concentration predicted by the process model based on SL and PC-SAFT EoS. The evolution of ethylene polymerization rate in presence of n-hexane is in agreement with the theoretically expected trend. This, in turn, suggests that it would be possible to move toward the development of a predictive process model by taking into account the heat and mass transfer phenomena taking place simultaneously during the course of polymerization.

6. Conclusion In the present study, a process model based on the thermodynamic models of SL and PC-SAFT EoS is developed to simulate and analyze the impact of vaporized n-hexane as a commonly-used ICA compound on the rate of gas phase ethylene polymerization on supported catalyst. The thermodynamic simulations demonstrate that as the partial pressure of n-hexane increases the equilibrium concentration of ethylene in the amorphous phase of PE is enhanced. This, in turn, explains the experimentally observed promotion in rate of ethylene polymerization as a function of n-hexane partial pressure. Moreover, both thermodynamic models predict that the partial pressure of ethylene has a negligible impact on the enhancement magnitude in the ethylene equilibrium concentration due to presence of a specific level of n-hexane in the gas phase composition. The simulated effect of ethylene partial pressure is consistent with and can account for the experimental results in which it is observed that at a given partial pressure of n-hexane the promotion magnitudes in the rate of ethylene polymerization are very similar at two different partial pressures of ethylene equal to 7 and 12 bars.



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The results of the process model indicate that the cosolubility phenomenon (i.e., the enhancement in the equilibrium concentration of ethylene in the amorphous phase of PE in presence of n-hexane) cannot be the sole reason for the observed increase in the polymerization rate during the whole period of reaction. At the beginning, the rate of ethylene polymerization is enhanced much more strongly than would be expected from the cosolubility effect alone. As the reaction proceeds, the enhancement magnitude gradually decreases and reaches to a steady-state value which is in the order of the promotion magnitude in the equilibrium concentration of ethylene predicted by the two thermodynamic models. It is demonstrated that the observed trend does not originate from the effect of particle temperature on the equilibrium concentration of ethylene. In addition, the decrease in the partial pressure of n-hexane (due to its sorption in the produced PE) is found to have only a minor impact on the ethylene concentration and consequently the polymerization rate. However, the process model enables us to show that the evolution of the polymerization rate in presence of n-hexane is fully in agreement with the theoretically expected trend considering the dynamics of heat and mass transfer. The process model presented in the current study is a new development in the field of modeling the process of gas phase ethylene polymerization on supported catalysts. To date such an approach does not appear to have been used to simulate and investigate the impact of process operating condition on the polymerization rate. Moreover, this approach holds considerable promise for future development of a fully predictive process model by taking into account the heat and mass transfer phenomena.

Acknowledgment Arash Alizadeh, Farhad Sharif, and Morteza Ebrahimi gratefully acknowledge the Iran's National Elites Foundation (contract number 20.618) for the partial funding of this research work. Josef Chmelař and Juraj Kosek are greatly thankful for the support from the Czech Science Foundation (project GA1607898S). Arash Alizadeh and Timothy F.L. McKenna also wish to thank Dr. Montree Namkajorn for his helpful technical input during the preparation of this paper.

Supporting Information The Supporting Information is available free of charge on the ACS Publications website.


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The procedure of solubility data extraction and processing and Comparison of a priori prediction and best fit solubility isotherms calculated by SL and PC-SAFT EoS for the ternary system of ethylene-n-hexanePE with the experimental data at 70 and 90 oC

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