Heterogeneous Catalysts for Olefin Polymerization: Mathematical

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Heterogeneous catalysts for olefin polymerization: mathematical model for catalyst particle fragmentation Adriano Giraldi Fisch, João Henrique Zimnoch Dos Santos, Argimiro Resende Secchi, and Nilo Sérgio Medeiros Cardozo Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b03740 • Publication Date (Web): 17 Nov 2015 Downloaded from http://pubs.acs.org on November 24, 2015

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Industrial & Engineering Chemistry Research

Heterogeneous catalysts for olefin polymerization: mathematical model for catalyst particle fragmentation Adriano G. Fisch1*, João H. Z. dos Santos2, Argimiro R. Secchi3, Nilo S. M. Cardozo4 1

Chemical Engineering Department. Universidade Luterana do Brasil. Av. Farroupilha 8001, 92425-900, Canoas, RS, Brazil. 2

Chemistry Institute. Universidade Federal do Rio Grande do Sul.

Av. Bento Gonçalves 9500. 91501-970 Porto Alegre, RS, Brazil. 3

PEQ/COPPE, Universidade Federal do Rio de Janeiro.

Centro de Tecnologia G-116, Cx.P. 68502, 21941-972, Rio de Janeiro, RJ, Brazil. 4

Chemical Engineering Department, Universidade Federal do Rio Grande do Sul. Rua Eng. Luis Englert s/n. 90040-040 Porto Alegre, RS, Brazil.

*Corresponding author. E-mail: [email protected]. Tel.: +55 51 3477 4000.

Abstract- A model for studying the catalyst fragmentation in the early stages of olefin polymerization is presented. The model is based on measurable and observable parameters of the catalyst and on the energy balance for the fragmentation phenomenon. The model allows the fragmentation behaviors to be discriminated as regards the influence of particle size, polymerization rate, and active site distribution. The results are supported by experimental studies available in the literature indicating the deterministic nature of the model and its capabilities of prediction. The performance of the model allows the optimization of the catalyst synthesis in terms of support nature as well as particle and pore morphology. Keywords: fragmentation, polyolefin, heterogeneous catalyst, modeling, polymerization

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Introduction The fragmentation of the heterogeneous catalyst that occurs at the early moments of olefin polymerization reaction plays an important role in the performance of the process. Two main problems in olefin polymerization technology arise from the catalyst fragmentation.1-2 One is the generation of very small particles of polymer (fines) during polymerization, which causes fouling in the reactor and obstruction of transfer lines, for instance. The formation of fines has been frequently connected with a severe fragmentation of the support, where parts, or pieces, of the growing polymer particle, displace from the catalyst. The latter problem, which is the opposite of the former one, is the poor fragmentation of the support. In this case, the pressure inside the pore, which is due to the nascent polymer, is not high enough to break the pore wall and, consequently, the catalyst particle skeleton. As a result, the polymer plugs the pore, leading to the reduction of the polymerization reaction rate by hindering monomer access to active sites. Several studies on fragmentation of catalyst particles during polymerization of olefins have been accomplished following either an experimental and/or a theoretical (e.g., modeling) approach.1-13 These studies allowed understanding the fragmentation mechanism and its effects on the polymerization processes and polymer properties. The modeling approaches that are used in the studies of catalyst fragmentation and polymer particle growth have evolved from the multigrain model (MGM)14, in which the growing polymer particle is discretized as an agglomerate of micro-particles. A complete model was proposed by Kittilsen et al. and further extended by Grof et al.15a-d, in which the particle was composed by two domain, polymer, and support, and a viscoelastic approach modeled the interaction among them. Following a similar concept, Horackova et al.16 presented a model for catalyst fragmentation based on the particle discretization of the MGM and the viscoelastic interactions among the micro-particles. ACS Paragon Plus Environment

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Such model describes the particle as an agglomeration of several micro-particles, extending the discretization criteria of the multigrain model for polymer (live and dead), support (with and without active sites) and pores. Following this methodology, it is necessary to set the quantity and size of the micro-elements as model parameters, which makes this approach less attractive in terms of experimental validation. In other studies17, the polymer particle growth is modeled incorporating the fragmentation as an empirical factor. An extensive number of variables is involved in the catalyst synthesis, such as support nature (e.g., magnesium dichloride and silica), support porosity, pore morphology, active sites reactivity, and active sites distribution on the particle. Despite the efforts3-17, the influence of these variables on the fragmentation is still addressed only qualitatively. Consequently, the development of heterogeneous catalysts takes a long time dealing with the support characteristics and supporting methodologies for the transition metal. From the point of view of catalysis, it is crucial to understand the interaction among the different variables in order to develop adequate heterogeneous catalysts for a given polymerization process. In this sense, this paper presents a deterministic model for the catalyst particle fragmentation, which is modeled considering the internal stress on pore wall and the respective strength of the catalyst support. The proposed model was built using measurable or observable characteristics of the catalyst and the particle was not discretized into small sub-elements to model the polymerization and fragmentation process. These considerations allow generating a deterministic model on the light of available experimental data.



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Mathematical model The model was built by coupling (i) catalyst fragmentation, (ii) kinetic of polymerization reaction, and (iii) mass balance considering a single catalyst particle. The process was modeled as isothermal, i.e., negligible internal and external heat transfer, and the polymer particle expansion was not taken into account. In fact, the approach considers the simultaneous effects of (i) reaction conditions, (ii) transport of the reactants, (iii) dynamic of polymer properties, (iv) particle and pore morphology of the catalyst, and (v) active site nature (reactivity) and its distribution through the catalyst particle. Catalyst fragmentation A proper description of the fragmentation mechanism is possible from an energy balance approach.18 The work used to strain the particle skeleton is stored in the solid as mechanical energy of stress, just as energy stored in a coiled spring. The polymer growing inside the pores is the source of such mechanical energy. As additional energy is applied to the just stressed pore wall by the continual polymerization reaction, it is distorted beyond its ultimate strength and eventually it ruptures into fragments. At this point, a new surface is generated. Since a unit of area of solid has a finite amount of surface energy, the creation of new surface requires work, which is supplied by the energy of stress stored in the particle skeleton just before the fragmentation. By conservation, all energy of stress in excess with respect to that required to the new surface creation must appear as thermal energy. In addition, it is known19 that the fracture starts from cracks previously present in the material structure. Considering the aforementioned fragmentation process and that the polymer inside the pore mimics the free pore volume, the energy (or work) stored by the particle skeleton (E) could be obtained as a function of the pore volume (Vpore), as given by Equation 1:


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∂E =P ∂V pore

(1)

where P is the pressure on the particle skeleton produced by the polymer. As polymers are viscoelastic materials, the pressure P could be assessed from the bulk (compressive reciprocal) modulus (Equation 2a) that can be rewritten as given in Equation 2b.20 In Equation 2, Vpol is the polymer volume.

K=

−1 1  ∂V pol    V pol  ∂P T

(2a)

∂P −K = ∂V pol V pol

(2b)

It is worth mentioning that Equations 1 and 2 are valid when pores are full of polymer, i.e., there is not free space inside the pore for the polymer to be accommodated without compaction. Before the pore is full of polymer, pressure, and energy on the wall are considered null. As the volume available to the polymer accommodation is constant, since it is the pore volume of the catalyst particle, the compaction of the polymer occurs due to the continuous polymer production inside the pore. In this case, the specific mass of polymer, ρ pol , is a proper variable to account for this compaction. Considering K as a constant, for sake of simplification, it is possible to integrate Equation 2b, resulting in P as a function of ρ pol , as given by Equation 3:

ρ   V final   = − K ln inital  P = − K ln  ρ   Vinitial   pol 

(3)

where the initial specific mass, ρinitial, is that of the polymer obtained without any spatial restriction.



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Using the relation of Grüneisen-Tobolsky20 (Equation 4), the bulk modulus (K) of a semicrystalline polymer could be estimated as a function of the following properties: (i) cohesive energy (Ecoh), (ii) molecular volume of the structural unit (υ, Equation 5 for polyethylene), (iii) polymer fusion heat (∆Hm), and (iv) polymer crystallinity content (χ, Equation 6).20

 E + χ∆H m  K = 8.04 coh  υ  

(4)

υ = 28 / ρ pol

(5)

 ρ pol − ρ a  ρ c    χ =     ρ c − ρ a  ρ pol 

(6)

In Equation 6, ρpol is the specific mass of polymer, ρa is the specific mass of 100 % amorphous polymer, and ρc is the specific mass of 100 % crystalline polymer. The specific mass of the polymer, ρpol, is defined in Equation 7, based on the polymer mass (mpol) that is produced by the active site. In order to assess ρpol, it is necessary to define an auxiliary variable, ρ, as a ratio of polymer mass to pore volume (Equation 8).

ρ pol =

ρ=

∂m pol

(7)

∂V pol

∂m pol

(8)

∂V pore

When the pore is full of polymer, the polymer volume Vpol is the pore volume Vpore and ρ is equivalent to the specific mass of polymer, that is, ρ = ρpol when Vpol = Vpore. When the pore is not full of polymer, ρpol is ρinitial. Equation 9 assesses ρpol as a case function considering a maximum limit, ρmax, which is defined in order to maintain the physical sense for the system. ACS Paragon Plus Environment

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ρ pol = MIN(ρmax , MAX(ρ , ρinitial ))

(9)

The relationship between pore volume, Vpore, and the particle radius, r, is obtained according to Equation 10, taking into account an average catalyst porosity,φ, and a spherical catalyst particle.

4 V pore = φV part = φ πr 3 3

(10)

It is possible to assess the porosity from the product of the specific pore volume, ϑcat, to the specific mass of catalyst, ρcat, as given in Equation 11. It is important noticing that ρcat is also dependent on ϑcat and on the specific mass of the pore-free solid (ρskeleton) as given in Equation 12.

φ = ϑcat ρ cat

(11)

1

ρ cat =

1

ρ skeleton

(12)

+ ϑcat

Similarly to Equation 1, the maximum energy supported by the particle skeleton (E’) is evaluated as a function of its skeletal volume (Vw), according to Equation 13:

∂E' =λ ∂Vw

(13)

where λ is the compressive strength of the material used as support in the catalyst design. The skeletal volume Vw could be evaluated according to Equation 14.

4 Vw = (1 − φ )V part = (1 − φ ) πr 3 3

(14)

It is well-known21 that the mechanical properties of porous solids depend on the porosity of the material. Therefore, many models have been developed22-25a,b in order to predict the mechanical



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properties of porous materials as a function of its porosity. In the present work, the model developed by Chen et al.25a is employed, which is an extension of the Griffith’s model25b for fracture. Equation 15 expresses the compressive strength as a function of the support porosity, φ, as given:  φ − φ  c   φc 

λ = λo 

1+ m

 1 − φ  

2 3

   

1 2

(15)

where λo, φc, and m are constants related, respectively, to the compressive strength of the pore-free material, the critical porosity at failure threshold, and the degree of the randomness of the pores (m = 1 for pores are randomly distributed; m < 1 for pores are clustered). Table 1 lists typical values of compressive strength for some materials used as support in catalyst synthesis. Among these, amorphous silica (SiO2) and magnesium dichloride (MgCl2) are the most employed26. Unfortunately, the references in Table 1 did not cite the material porosity. Thereby, the experimental value of compressive strength of fused amorphous SiO2 (λ = 1.1×109 Pa) can be used as an approximation of λo for SiO2. Magnesium dichloride is more friable than SiO226-27 and then it is expected to have a lower λo than SiO2. A value of λo for MgCl2 is not possible from the available literature, but it is speculated that it is roughly λo = 108 Pa. Table 1: Typical values of compressive strength. Compressive strength Material (× ×10-9 Pa) Al2O3

4.5

SiO2 (crystalline)

2.5

MgO

1.4

SiO2 (fused; amorphous)

1.1

MgO.MgCl2.H2O

0.07

Data from Munz and Fett28 and Liu et al. 29.


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Two different approaches could be accounted for particle fragmentation. The first is a differential approach, which explains the break of the pore walls in a given radial position by the local increase of the energy stored in the particle skeleton beyond its resistance. The second one is an integral approach, which considers the total energy accumulated in an inner domain of the catalyst particle (inner sphere) surrounded by a respective outer shell. According to this approach, the outer shell is fragmented when the energy accumulated in the inner sphere is greater than the maximum energy supported by the shell. In such a way, the shell is able to break even without the presence of nascent polymer in its porosities and regardless of whether the inner sphere is already fragmented or not. This integral approach is important in the sense that it is able to predict particle explosion and the consequent release of small parts. In the differential approach, the fragmentation is assessed by equating Equations 1 and 13,

 ∂E ∂E '  =   , where these derivatives can be obtained from Equations 10 and 14. Therefore, a ∂r   ∂r differential fragmentation index (ΓD) can be defined as follows:

∂E ∂r  φ  P  ΓD = ∂E ' =    1 − φ  λ  ∂r

(16)

so that the particle skeleton will fragment when ΓD ≥ 1. Similarly, the condition for fragmentation in the integral approach is evaluated by equating E and E’ (Equations 1 and 13, respectively). The resulting integral fragmentation index (ΓI) is given by Equation 17:



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r

ΓI =

4φπ ∫ r 2 Pdr

E 0 = E ' 4 (1 − φ )πλ R 3 − r 3 3

(

(17)

)

where E is the integrated energy of an inner sphere of radius r, which is obtained from the integration of Equation 1 within limits 0 to r. E’ is the maximum energy supported by the particle shell, which is obtained from the integration of Equation 13 within the limits r to R. Again, fragmentation will take place when ΓI ≥ 1, which indicates that the inner sphere of radius r has higher energy than the shell formed between r and R. As two different approaches for fragmentation are considered, a global index is formulated using a case function as described in Equation 18. For ΓG < 1, the particle is not fragmented and, for ΓG ≥ 1, the particle is fragmented either due to the differential or due to integral approach.

ΓG = MAX(ΓD , ΓI )

(18)

Kinetic mechanism In order to evaluate the dynamic of pore filling by the nascent polymer and its relationship with fragmentation, it is necessary to couple the fragmentation model to the kinetic of polymerization reaction. The reaction rate of olefin polymerization was modeled following the kinetic mechanism described in Table 2, which is typical for a metallocene-based catalyst in the polymerization of ethylene.1,30


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Table 2: Kinetic mechanism considered in the polymerization reaction of ethylene using a single-site catalyst. Reaction constanta

Reaction Monomer insertion:

S+M  → P1

kp= 6.7×102 L.mol-1.s-1

Propagationb:

Pr + M  → Pr +1

kp

Termination by β-elimination:

Pr  → Dr + S

ktβ= 5.7×10-1 s-1

Termination by hydrogen:

Pr + H  → D r + S

ktH= 1 L.mol-1.s-1

S+I → S d

kd= 2.1×102 L.mol-1.s-1

Pr + I  → Dr + S d

kd

Deactivation by impurities:

a- data from Chien and Wang30. Reaction constants evaluated at 60 °C for ethylene polymerization using Cp2ZrCl2. b- r ≥ 1.

Mass balances The mass balance was performed considering the catalyst particle limits, in a unidirectional model, as depicted in Figure 1. The transport of reactants from bulk reaction medium through the catalyst particle, taking into account diffusion and advection as the transport mechanisms, is given by Equation 19:

∂[X ] 1 ∂  ∂[X ]  ∂ = 2  DX r 2  + (U [ X ]) − RX (0 < r < R ; 0 < t ≤ ∞) ∂t r ∂r  ∂r  ∂r

(19)

with the following initial and boundary conditions:

[X ] t=0 (0 < r < R) = 0

(20a)

∂[X ] (0 ≤ t ≤ ∞) = 0 ∂r r =0

(20b)



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DX

∂[X ] (0 ≤ t ≤ ∞) = (kL a)([X ]bulk − [X ]r=R ) ∂r r =R

(20c)

where [X] is the reactant concentration (monomer M, hydrogen H and impurities I); R is the radius of the catalyst particle, Xbulk is the reactant concentration in the bulk reaction medium (outside the catalyst particle), r is the radial position of the particle, RX is the reaction rate of the reactant, DX is the reactant diffusivity, U is the medium velocity, and t is the time of simulation.

Figure 1: Definition of spatial domains of the catalyst particle. Diffusion coefficient depends on the medium characteristics. Therefore, it is necessary to characterize the pore environment along the polymerization time. Polymerization process takes place inside the support pores, which are full of fluid at the beginning of the reaction. Along the polymerization, this environment changes and the polymer occupies a fraction of space previously occupied by the fluid. Additionally, a fraction of the fluid is involved in polymer swelling. Thus, two phases could be present inside the pores during support fragmentation: (i) fluid and (ii) swollen polymer. Considering this environment change, the diffusion coefficients inside the pore are calculated as a linear relationship of the diffusion coefficients in the solvent and in the swollen polymer, as described in Equation 21: ACS Paragon Plus Environment

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DX =

φDeff φ ((1 − f ) D X − solv + fD X − pol ) = τ τ

(21)

where Deff is the effective diffusivity, D X − solv is the reactant diffusivity in the solvent, D X − pol is the reactant diffusivity in the swollen polymer, f is the filling factor, i.e., the volume fraction of pore occupied by the polymer (defined in Equation 22), and τ is the pore tortuosity (calculated31 by Equation 23). V pol f =

V pol V pore

=

m pol ρ = V pore ρ pol m pol

(22)

τ = 1 − 0.49 log (φ )

(23)

The reactant diffusivity in the solvent, DX-solv, was calculated following the correlation of Wilke–Chang32 (Equation 24a). Typically, the diffusion coefficients through toluene are ca. 10-8 m2.s-1. On the other hand, the diffusion coefficients through the swollen polyethylene, DX-pol, are ca. 10-11 m2.s-1, i.e., 1000-fold lower than the respective value in the solvent.1,33 Based on that, the diffusion coefficients through the swollen polymer were estimated using Equation 24b. 7.4 × 10 −8 (ϕM solv ) T 0. 5

D X − solv =

(24a)

η solvν X 0.6

D X − pol = 10 −3 D X − solv

(24b)

where Msolv is the molar mass of the solvent, T is the temperature (K), ηsolv is the solvent viscosity, νX is the molar volume of X at boiling point and, ϕ is the interaction parameter between solvent and diffusing component X (ϕ = 1 for non-polar solvents).



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For the evaluation of the advection transport inside the pore, the average velocity of the reaction medium, U, could be estimated as a function of the volumetric growth of the polymer inside the pore. The reference value of this variable, Uo, corresponds to that obtained when the pores are empty, as expressed in Equation 25: Uo =

M M Vr RM ρ pol A

(25)

where MM is the molar mass of the monomer, Vr is the reaction medium volume, RM is the reaction rate of the monomer, ρpol is the polymer specific mass, and A is the total pore transverse area available for flow in a given particle radius, which is defined in Equation 26: A=

r 3 2φπ R τ

(26)

where r is the radial position, R is the particle radius, φ is the porosity, and τ is the pore tortuosity. Considering the reduction of the pore volume available for the medium flow as the soon as polymer starts being produced, the expression in Equation 27 was proposed for the calculation of U as a function of the filling factor f (defined in Equation 22). U = U o (1 + 2 f )(1 − f

)

(27)

The right-hand term (1 + 2 f )(1 − f ) in Equation 27 is intended to represent the increase of velocity that occurs at the initial stage of the pore filling and the subsequent decay at the final stages of the process when the resistance of the flow increases significantly. The active sites (S), inactive sites (Sd) and growing polymer chains (Pr) are not transported along the particle radius during the fragmentation process (the particle is not expanding its volume). The respective population balances of these reactants involve polymer chains of a broad range of ACS Paragon Plus Environment

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lengths (Pr with 1 ≤ r ≤ ∞), which results in a system of infinite equations. In order to reduce the system, the concept of moments of distribution was employed to solve the polymer chain length distribution.34a,b Equation 28 defines the moments of distribution of zero order and the balances for active sites (S), inactive sites (Sd) and growing polymer chain (Pr) are described by Equations 29-31, with the initial conditions given by Equation 32-34, respectively. ∞

µ o = ∑ [Pr ]

(28)

∂[S ] = k tβ µ o + k tH [H ]µ o − k p [S ][M ] − k d [S ][I ] ∂t

(29)

∂[S d ] = k d [S ][I ] + k d [I ]µ o ∂t

(30)

∂µ o = k p [M ][S ] − ktβ µ o − ktH [H ]µ o − k d [I ]µ o ∂t

(31)

[S ] t =0 (0 ≤ r ≤ Rcat ) = S0

(32)

[S d ] t =0 (0 ≤ r ≤ Rcat ) = 0

(33)

µ o t =0 (0 ≤ r ≤ Rcat ) = 0

(34)

r =1

The active site distribution along the catalyst radius at the beginning of polymerization, S0, is expressed according to Equation 35a for well- and ill-distributed active sites. The parameter F is the load of the organometallic complex in the catalyst, and a and b are additional parameters dealing with the distribution of active site through the particle radius according to Equation 35b-d. r   3+b  R

 r  S 0 = a  R

4πR 3 ρ cat F 3Vr

(35a)



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a = 1 Well-distributed case  b = 0

(35b)

a = 1 Ill-distributed case 1  , for active sites concentrated at the outer particle b = 1

(35c)

 a = 0 .9 Ill-distributed case 2  , for active sites concentrated at the inner particle b = −1.2

(35d)

Equations 36-38 assess the reaction rate of the monomer, hydrogen, and impurities, respectively, along the particle radius r.

RM = k p [M ][S ] + k p [M ]µo

(36)

RH = ktH [H ]µ o

(37)

RI = k d [S ][I ] + k d [I ]µ o

(38)

The polymer mass (mpol) produced along the time for a given radius r is calculated by Equation 39, where MM is the monomer molar mass.

∂m pol ∂t

= Vr M M RM

(39)

Implementation of the model The dependence of the fragmentation mechanism on the support parameters was investigated using the steady-state model composed by Equations 3-6 and 15-18 (single fragmentation model). This model was implemented in Matlab and the influence of the support parameters on the support fragmentation was evaluated in terms of the specific mass of polymer (ρpol) versus the differential fragmentation index ΓD (Equation 16).


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The dynamic effects of the pore filling by the nascent polymer on catalyst fragmentation were investigated using the differential-algebraic model formed by Equations 3-9, 11, 12, 15-27 and 29-39, which was implemented in gPROMS simulator (dynamic fragmentation model). The spatial domain was discretized using central finite differences (spatial grid= 120) and its integration was performed using the gPROMS method dasolv, with default parameters (relative and absolute tolerances of 10-5). The material and process parameters used in the simulations are listed in Table 3. The final set of dimensionless equations implemented in the simulator is available in the Supporting Information. Table 3: Set of material and process parameters used in the simulations. Parameter

Value

Immobilized transition metal (Zr)

0.5 %w

[Zr] in reaction medium

5×10-5 M

[impurities] in reaction medium

10-15 M

Al/Zr (molar ratio)

1000

Temperature

60 °C

Partial pressure of ethylene

1 bar

Solvent

Toluene

Particle radius of catalyst (Rcat)

25 µm

Specific pore volume of catalyst (ρcat)

0.7×10-3 m3.kg-1

Pore-free compressive strength of support (λo)

109 Pa

Global coefficient for mass transfer (kLa)

10-6 m.s-1

Pore-free specific mass of support (ρskeleton)

2.2×103 kg.m-3



Page 18 of 46

18

Results and Discussion The fragmentation was evaluated regarding support parameters using the single fragmentation model. The results are presented as a relationship between the polymer specific mass versus the differential fragmentation index (ρpol × ΓD). The material resistance (λ) is dependent on parameters which are related to the nature of the material and to the pore morphology of the catalyst particle, as calculated by Equation 15. The influence of the pore-free compressive strength (λo) on the differential fragmentation index (ΓD) is shown in Figure 2, assuming constant catalyst porosity (φ).

The polymer specific mass (horizontal

axis) relates to the quantity of polymer that is needed to get enough energy to break up the catalyst particle. 1.0

7

λo= 10 Pa

0.9

Diff. fragmentation index [-]

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

8

2x10

3x10

8

0.8 5x10

0.7

8

0.6 0.5 0.4

9

10

0.3 0.2 0.1 0.0 940

950

960

970

-3

980

990

Polymer specific mass [kg.m ]

Figure 2: Influence of pore-free compressive strength on the diff. fragmentation index for φ = 0.45. According to the results in Figure 2, it is necessary to accumulate more energy to complete the fragmentation of materials with higher λo, which is evidenced by the decrease in the slope of the curve ΓD vs ρpol with the increase of λo. The pore-free compressive strength is inversely related to the material friability. For instance, magnesium dichloride (MgCl2) has λo ~ 107 Pa and it is more friable


Page 19 of 46


19

than amorphous silica (SiO2) that has λo ~ 109 Pa.26-27 In this sense, the results in Figure 2 predicted by single fragmentation model resemble experimental reports35 in which more friable material were easily fragmented. Figure 2 also presents a limiting λo for which fragmentation does not occur (ΓD < 1) within a consistent range of polymer density (e.g., compare the results of λo = 5×108 and 109 Pa).36-37 In such a case, reduction of the compressive strength of the particle is necessary in order to complete the fragmentation. Figure 3 shows the influence of particle porosity on the differential fragmentation index. It can be seen that the proposed differential fragmentation index was able to correlate qualitatively to the fact that the compressive strength is lower for less dense, i.e., more porous particle.25a,b,37 So, for example, although fragmentation is not achieved for catalyst particles of φ = 0.45 for λo = 109 Pa (as noticed in Figure 2), it occurs when particle porosity increases to φ ~ 0.60, as denoted in Figure 3. Additionally, data of Figure 2 and Figure 3 also reflect the fact that supports of higher compressive strength will require a more porous structure in order to fragment. This result is in agreement with the fact that, for example, the particle porosity of SiO2-based Ziegler-Natta catalysts ranges between 0.7 and 0.9 while for MgCl2-based catalysts the porosity ranges from 0.40 to 0.6.26 1.0

φ= 0.75

0.70

0.9


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

0.65 0.60

0.8

0.55

0.7 0.6 0.5

0.50

0.4 0.3 0.2 0.1 0.0 940

950

960

970

-3

980

990


Figure 3: Influence of particle porosity on the diff. fragmentation index for λo = 109 Pa.



Page 20 of 46

20

The influence of pore distribution in the particle volume on the fragmentation index is presented in Figure 4. The results show that particles of well-distributed pores (m = 1) are more friable than those particles of clustered pores (m < 1) and fragmentation is easier for the former. 1.0

m= 1 0.7

0.9


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

0.8

0.9

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 940

945

950

955

960

-3


965

970

Figure 4: Influence of pore distribution on the diff. fragmentation index for φ = 0.65 and λo = 109 Pa. The single fragmentation model was simulated in order to evaluate the possibility of particle explosion, as depicted in Figure 5 for φ = 0.65, λo = 109 Pa, and ρpol = 945-970 kg.m-3. There is a crossover point in the radial position where the integrated energy of the inner particle (E) is equal to the resistance of the respective shell (E’). The crossover point restricts a fragmented region to the outer shell, i.e., the fragmented shell is formed from this point to the outer particle limits. As ρpol increases due to the continuous compression of the polymer from ρinitial= 940 kg.m-3, the crossover point is reduced suggesting that a higher volume of the particle shell fragments according to the integral approach. In fact, the fragmented shell is prone to displace from the inner particle generating fines only if there is not enough quantity of polymer occupying the pore volume of the shell, which serves as connection medium among the fragments.38 It is proposed that an ill distribution of active sites through the catalyst particle or, even, a preferential deactivation of sites that are located in the particle periphery, could result in the lack of polymer in the shell. From Figure 5 it is also possible to notice


Page 21 of 46


21

that the shell fragmentation could occur with the inner particle fragmented previously or not. The pore wall fragmentation of the inner particle happens when ρpol is around 960 kg.m-3, according to the differential fragmentation approach (see Figure 3). -6

3.5x10

E'

-6

3.0x10

-6

2.5x10

-6

E, E' (J)

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

2.0x10

970

-6

1.5x10

960

-6

1.0x10

E

-7

5.0x10

950 -3

0.0

945 kg.m

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Radial position [-]

Figure 5: Energy build up within the inner sphere (E) and the corresponding outer shell resistance (E’) for φ = 0.65, λo = 109 Pa, and ρinitial = 940 kg.m-3. The arrow indicates crossover point. The effects of the particle pore filling by polymer on catalyst fragmentation were studied using dynamic simulations of the mass balance coupled with the fragmentation model (the parameters set-up are in Table 3). The monomer profile through the particle radius is depicted in Figure 6 that shows the monomer transport is not limited by diffusion and the concentration is constant along the radius. This behavior is attributed to the high diffusion coefficients along polymerization. At the beginning of the polymerization, the pores are full of solvent and the diffusion coefficient through this medium is ca. 10-8 m2.s-1, which is three orders of magnitude higher than the respective coefficient in the swollen polymer. Experimental studies38 of slurry pre-polymerization of propylene provide information supporting that mass transfer rate inside the catalyst particle is not limiting.



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Figure 6: Monomer concentration [dimensionless] profile through the catalyst particle in the early moments of polymerization. Parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. As soon as polymer is produced, it starts occupying the pore volume, expelling part of the reaction medium, which creates an advective transport for the monomer. The amount of polymer produced by active sites across the particle radius and the filling factor are shown in Figure 7a,b, respectively. The pore filling is constant along the radius for a given time due to the homogeneous distribution of the active site. The reaction medium velocity corresponding to the advection transport is shown in Figure 7c. The velocity profile along the time evidences a maximum limit related to the decrease of the area available for the medium flow. Along the radial position, the velocity is constant for a given time because of the proportional occupation of pore volume by the nascent polymer. Despite the advective parcel of the transport, the monomer concentration does not decrease along the polymerization, as shown in Figure 6, suggesting that the advection is not a relevant transport mechanism inside the pores when the active sites are well-distributed throughout the catalyst particle.


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23 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

(a)

(b)

(c)

Figure 7: Profiles of (a) the amount of polymer produced by active sites [dimensionless], (b) polymer filling factor [dimensionless], and (c) reaction medium velocity (×106) [m.s-1]. Parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. ACS Paragon Plus Environment


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Figure 8 shows the fragmentation indexes. The delay in the fragmentation of the inner region of the particle in relation to the outer is better evaluated by comparing the differential fragmentation index ΓD (Figure 8b) with the integral fragmentation index ΓI (Figure 8c). Figure 9 joins both indexes in the same graphic, showing that the shell region ca. 0.80 < z ≤ 1 will fragment due to the internal pressure of the respective inner sphere, since ΓI > ΓD for the equivalent time and radial position, as just predicted by the single fragmentation model in Figure 5. This behavior results in the fragmentation of the outer particle region following the integral approach before the fragmentation of the inner particle. The longer time needed to get enough energy to break the inner pore walls is due to the viscoelastic behavior of the polymer, i.e., the compression and the consequent increase of the specific mass. The specific mass of polymer in the fragmentation front is shown in Figure 10 and the polymer in the outer shell is not as compressed as it is in the inner sphere at the fragmentation moment. The results from the different fragmentation mechanisms, i.e., the break of the pore wall due to local increase of the energy stored in the particle skeleton (differential approach) and the break of the outer shell due to the total energy accumulated in an inner sphere (integral approach), allow defining distinct fragmented regions through the catalyst particle. In fact, there is a peripheral fragmented shell and a respective non-fragmented inner sphere in a first moment. This behavior presents similarities to the shrinking core mode of fragmentation.16 As polymerization proceeds, the inner sphere breaks up in an almost instantaneous fashion resembling the continuous bisection mode of fragmentation.16


Page 25 of 46


25 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

(a)

(b)

(c)

Figure 8: Fragmentation profiles: (a) global index [dimensionless], (b) differential index [dimensionless], and (c) integral index [dimensionless]. Parameters are in Table 3. Other units: time [s] and radial position [dimensionless].



Page 26 of 46

26

1.0

Integ. fragmentation index [-]

0.99

0.95

0.8

0.9 0.85

0.6

0.8

0.4

z= 0.7

0.2

0.0

0.0

0.2

0.4

0.6

0.8

1.0


Figure 9: Comparison of the differential with the integral fragmentation index along polymerization. Parameters are in Table 3. The dashed line is the bisectrix. 1.0 0.95

Global fragmentation index [-]

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

0.90

0.8

0.85 z= 0.5 z= 0.8

0.6

0.4

0.2

0.0 930

940

950

960

-3


970

980

Figure 10: Specific mass of polymer along polymerization. Parameters are in Table 3. The generation of fines by the release of small peripheral particles due to the explosion of the particle shell is possible if there is not polymer in the shell to maintain the small pieces together. As noted in Figure 7b, polymer occupies almost the totality of the pore volume available in the shell and, under these conditions, it may be expected that the fragments may stay bound to each other without displacement from the particle. Experimental evidence38 of the outer polymer layer of an uncompleted fragmented catalyst during propylene pre-polymerization corroborates the model results.


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In order to evaluate the restriction of the monomer transport in the fragmentation behavior, the propagation constant of polymerization reaction was set 10-fold higher, as well as the particle diameter was set 2-fold greater than those set in Tables 2 and 3, respectively. The monomer concentration is depicted in Figure 11a. At the beginning of the polymerization, the diffusion occurs mainly through the solvent and the longer path does not affect the monomer profile. However, when pores are full of polymer, the diffusion coefficient decreases and the monomer concentration for the inner part of the particle also decreases, as a result. The aforementioned result contrasts with that in Figure 6, evidencing transport limitations inside the particle. Besides the lower diffusion constant through the polymer, the influence of advection could result in the monomer depletion observed in Figure 11a. In fact, Figure 11b shows the average medium velocity in the pores and, according to the results, accumulation of monomer occurs at the outer regions that show low medium velocities. This result suggests that the advective mechanism presents some level of importance in the monomer transportation when polymer takes place in the pores, at least in these conditions of simulation. The values of the fragmentation indexes related to the results presented in Figure 11 are shown in Figure 12. According to these results, the time needed to fragment the particle is ca. 0.4 s (Figure 12a). The global fragmentation profile shows that particle periphery breaks up earlier than the inner region. The fragmentation seems to occur following the differential approach as denoted by the similarity between global and differential indexes (Figure 12a,b, respectively). The integral fragmentation index is shown in Figure 12c and it is compared with the differential index in Figure 13a also suggesting that the differential approach is preponderant.



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(a)

(b)

Figure 11: Profiles of (a) monomer concentration [dimensionless] and (b) reaction medium velocity (×106) [m.s-1]. Simulation using particle diameter set as 100 µm and kp = 6.7×103 L.mol-1.s-1. Other parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. The aforementioned results are due to the larger generation of polymer in the periphery of the particle, as shown in Figure 13b, which is caused by the higher concentration of monomer in this region. According to the simulation, the generation of fines is not evident, since fragmentation by the integral approach did not occur in this condition.


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29 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

(a)

(b)

(c)

Figure 12: Fragmentation profiles: (a) global index [dimensionless], (b) differential index [dimensionless], and (c) integral index [dimensionless]. Simulation using particle diameter set as 100 µm and kp = 6.7×103 L.mol-1.s-1. Other parameters are in Table 3. Other units: time [s] and radial position [dimensionless].



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30

1.0

(a)


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

0.8

0.6 0.95

0.4

0.90

0.2

0.0

z= 0.80

0.0

0.2

0.4

0.6


0.8

1.0

(b)

Figure 13: (a) Comparison of the differential with the integrated fragmentation index along polymerization and (b) pore filling factor [dimensionless]. Simulation using particle diameter set as 100 µm and kp = 6.7×103 L.mol-1.s-1. Other parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. Active site distribution across the catalyst particle is another issue to be addressed in the analysis of catalyst fragmentation. Figure 14 shows the well- and ill-distributed profiles of active site concentration at the initial condition of polymerization. Intending a comparison with the welldistributed profile of active sites, the ill-distributed cases consider active sites concentrated in the periphery (case 1) and in the inner region (case 2) of the catalyst particle.


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-3

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

Active site conc. at initial condition [mol.m ]

31

6

11

x10

5 ill-distributed 2

4 3 well-distributed

2 1 ill-distributed 1

0

0.0

0.2

0.4

0.6

Radial position [-]

0.8

1.0

Figure 14: Distribution of active site across the catalyst particle. The monomer profile through the particle when simulating actives sites preferentially located in the particle periphery (ill-distributed case 1 in Figure 14) shows a reduction of the concentration at ca. 4 s (Figure 15a). As this behavior is not present when active sites are homogeneously distributed in the catalyst (Figure 6), it suggests some monomer transport restriction through the particle. The average medium velocity is depicted in Figure 15b, in which a higher medium velocity is evident in the particle periphery at the beginning of the polymerization. The aforementioned results are attributed to the active site distribution (case 1), which allows the polymerization to occur primarily on the periphery of the catalyst particle (Figure 15c). The reduction of the monomer concentration at the inner particle is attributed to the restriction of monomer transport through the outer part that is full of polymer. In fact, the monomer diffusivity is reduced because of the change of the medium, i.e., from solvent to swollen polymer. Besides, the advective mechanism is effective in the inner region of the particle after the fragmentation of the shell, i.e., from ca. 4 s to up 7 s (Figure 15b) due to the pore filling. Consequently, the monomer concentration is depleted at the radial position corresponding to high medium velocities (z ~ 0.5).



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32 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

(a)

(a’)

(b)

(b’)

(c)

(c’)

Figure 15: Profiles of (a, a’) monomer concentration [dimensionless], (b, b’) reaction medium velocity (×106) [m.s-1], and (c, c’) pore filling factor [dimensionless] of simulations using ill-distributed active sites (a,b,c- case 1 and a´,b’c’- case 2). Parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. ACS Paragon Plus Environment

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In the case of the active sites are concentrated into the inner particle (ill-distributed case 2 in Figure 14), the simulation shows an almost constant monomer profile through the particle (Figure 15a’). There is only a slight monomer concentration decrease near the particle center, which is attributed to the advective mechanism of transport at this point. Figure 15b’ shows the average velocity profile. At the beginning of polymerization, i.e., before the pore is full of polymer, the average velocity is highest at z ~ 0.5 but, as polymerization proceeds, high velocities are evident in the center (z = 0) and the periphery of the particle. Because of the high velocity in the center, the reaction medium is pushed out to particle peripheral direction decreasing the monomer concentration in the inner part. Nevertheless, the monomer concentration increases as soon as the advective mechanism is not effective and the transport monomer is not limited by diffusion through the outer particle as well. Due to the active site distribution, the polymer fills the inner parcel of the particle before its periphery (Figure 15c). As the peripheral pores are filled with polymer later than the inner pores, a higher diffusion coefficient is possible in the outer region of the particle, allowing a higher flow of monomer to the inner particle. The fragmentation profile is shown in Figure 16a,b,c for ill-distributed case 1. The catalyst particle breaks up early in the particle periphery, as evidenced in Figure 16a. The predominant fragmentation approach is differential because of the similarity between the global and differential fragmentation index profiles (Figure 16a,b, respectively). The integral fragmentation index is shown in Figure 16c and, in spite of the energy accumulated in the inner particle, fragmentation does not occur, as it is also evidenced in Figure 17a.



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(a)

(a’)

(b)

(b’)

(c)

(c’)

Figure 16: Fragmentation profiles: (a, a’) global index [dimensionless], (b, b’) differential index [dimensionless], and (c, c’) integrated index [dimensionless of simulations using ill-distributed active sites (a,b,c- case 1 and a´,b’c’- case 2). Parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. ACS Paragon Plus Environment

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35

1.0

z= 0.80 z= 0.90 z= 0.95

0.8

0.6

0.4

0.2

0.0

0.73

0.8 z= 0.7

0.6

0.4

0.2

0.0

0.0

0.2

0.75 and 0.80 (overlap)

1.0



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

0.4

0.6


0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0


(b)

(a)

Figure 17: Comparison of the differential with the integral fragmentation index. Simulation of the illdistributed active sites (a) case 1 and (b) case 2. The dashed line is the bisectrix. When the active sites are concentrated into the inner particle (ill-distributed case 2 in Figure 14), the predominant fragmentation approach depends on the particle position as depicted in the global fragmentation index (Figure 16a’) in comparison with the differential and the integral fragmentation indexes (Figure 16b’,c’, respectively). The differential approach occurs in the inner particle (ca. 0 < z < 0.73) while the integral approach is dominant in the outer region (ca. z > 0.73), as depicted in Figure 17b. The fragmentation of the outer region occurs when the pores are not full of polymer (Figure 15c’), which is a consequence of the energy accumulation in the respective inner spherical particle. In addition, as Figure 18 shows, the polymer does not fill completely the shell pore volume, in such a way that the fragmented shell is prone to displace some small particles because of the lack of polymer, which serves as a connective medium for the parts.



Page 36 of 46

36

1.0 0.95


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

0.8

0.90

0.85

0.80 0.70

0.6

0.4

0.2 z=0.40

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Pore filling factor [-]

Figure 18: Comparison of the integral fragmentation index with pore filling factor along polymerization. Simulation of the ill-distributed active sites case 2.

Conclusions Some important variables in the polyolefin catalyst synthesis were studied concerning its influence on the particle fragmentation by using a deterministic mathematical model. The model was built using measurable or observable data from the catalyst and the fragmentation was assessed from energy balance following both differential and integral approaches. The model allows different fragmentation behavior to be discriminated as regards the influence of particle size, polymerization rate, and active site distribution. The obtained results are supported by experimental studies available in the literature indicating the deterministic nature of the model and its capabilities of prediction. It is worth mentioning the model was simulated using typical values for the parameters, i.e., without any data fitting. In this way, the proposed model can be a useful tool for development and optimization of catalysts for olefins polymerization, addressing the design in terms of the support nature, particle and pore morphology.


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Notation λ

compressive strength of support [Pa]

Γ

differential fragmentation index [-]

ϕ

interaction parameter [-]

ρ

mass of polymer to pore volume ratio [kg.m-3]

υ

molecular volume of the structural unit [cm3.mol-1]

φ

particle porosity [-]

χ

polymer crystallinity content [-]

τ

pore tortuosity [-]

ρa

specific mass of amorphous polymer [kg.m-3]

φc

critical particle porosity at failure threshold [-]

ρc

specific mass of crystalline polymer [kg.m-3]

ρcat

specific mass of catalyst [kg.m-3]

ϑcat

specific pore volume of catalyst [m3.kg-1]

ΓG

global fragmentation index [-]

∆Hm

polymer fusion heat [J.mol-1]

ΓI

integrated fragmentation index [-]

ρinitial

spatial unconstrained specific mass of polymer [kg.m-3]

(kLa)

volumetric mass transfer coefficient [m.s-1]

ρmax

highest specific mass of polymer [kg.m-3]

λo

pore-free compressive strength of support [Pa]



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38 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

ρpol

specific mass of polymer [kg.m-3]

-3 ρskeleton pore-free specific mass of catalyst [kg.m ]

ηsolv

solvent viscosity [cP]

νX

molecular volume of X at normal boiling temperature [cm3.mol-1]

[H]

hydrogen concentration [mol.m-3]

[I]

Impurities concentration [mol.m-3]

[M]

monomer concentration [mol.m-3]

[Pr]

concentration of growing chain of r polymeric units [mol.m-3]

[S]

active sites concentration [mol.m-3]

[Sd]

inactive sites concentration [mol.m-3]

A

transversal area of pore available for flow [m2]

Deff

effective diffusivity [m2.s-1]

DX

pore diffusion coefficients of X [m2.s-1]

DX-pol

diffusion coefficient of X through swollen polymer [m2.s-1]

DX-solv

diffusion coefficient of X through solvent [m2.s-1]

E

energy stored by the pore wall [J]

E’

pore wall resistance [J]

Ecoh

cohesive energy [J.mol-1]

F

volume fraction of pore occupied polymer [-]

K

bulk modulus of the polymer [Pa]

kd

kinetic constant of bimolecular deactivation [m3.mol-1.s-1]

kp

kinetic constant of propagation [m3.mol-1.s-1]

ktβ

kinetic constant of termination by beta-elimination [s-1]


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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

ktH

kinetic constant of termination by hydrogen m3.mol-1.s-1]

m

randomness of pore [-]

MM

molecular mass of monomer [g.mol-1]

mpol

mass of polymer [kg]

Msolv

molecular mass of solvent [g.mol-1]

P

pressure on the pore wall [Pa]

r

radial position of the particle [m]

RX

reaction rate of X [mol.m-3.s-1]

T

temperature [K]

U

reaction medium velocity [m.s-1]

Vfinal

final polymer volume [m3]

Vinitial

initial polymer volume [m3]

Vpart

particle volume of catalyst [m3]

Vpol

volume of polymer [m3]

Vpore

pore volume of catalyst particle [m3]

Vw

pore wall volume [m3]

Supporting Information The supporting information presents the mathematical treatment for the mass balance according to the method of the moments of the distribution. This information is available free of charge via Internet at http://pubs.acs.org.

References (1) Soares, J. B. P.; McKenna, T. F. L. Polyolefin Reaction Engineering; Wiley-VHC: Weinhein, 2012. ACS Paragon Plus Environment


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List of Figure Captions Figure 1: Definition of spatial domains of the catalyst particle. Figure 2: Influence of pore-free compressive strength on the diff. fragmentation index for φ = 0.45. Figure 3: Influence of particle porosity on the diff. fragmentation index for λo = 109 Pa. Figure 4: Influence of pore distribution on the diff. fragmentation index for φ = 0.65 and λo = 109 Pa. Figure 5: Energy build up within the inner sphere (E) and the corresponding outer shell resistance (E’) for φ = 0.65, λo = 109 Pa, and ρinitial = 940 kg.m-3. Arrow indicates crossover point. Figure 6: Monomer concentration [dimensionless] profile through the catalyst particle in the early moments of polymerization. Parameters are in Table 3. Other units: time [s] and radial position [dimensionless].


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Figure 7: Profiles of (a) the amount of polymer produced by active sites [dimensionless], (b) polymer filling factor [dimensionless], and (c) reaction medium velocity (×106) [m.s-1]. Parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. Figure 8: Fragmentation profiles: (a) global index [dimensionless], (b) differential index [dimensionless], and (c) integral index [dimensionless]. Parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. Figure 9: Comparison of the differential with the integral fragmentation index along polymerization. Parameters are in Table 3. The dashed line is the bisectrix. Figure 10: Specific mass of polymer along polymerization. Parameters are in Table 3. Figure 11: Profiles of (a) monomer concentration [dimensionless] and (b) reaction medium velocity (×106) [m.s-1]. Simulation using particle diameter set as 100 µm and kp = 6.7×103 L.mol-1.s-1. Other parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. Figure 12: Fragmentation profiles: (a) global index [dimensionless], (b) differential index [dimensionless], and (c) integral index [dimensionless]. Simulation using particle diameter set as 100 µm and kp = 6.7×103 L.mol-1.s-1. Other parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. Figure 13: (a) Comparison of the differential with the integrated fragmentation index along polymerization and (b) pore filling factor [dimensionless]. Simulation using particle diameter set as 100 µm and kp = 6.7×103 L.mol-1.s-1. Other parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. Figure 14: Distribution of active site across the catalyst particle.



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Figure 15: Profiles of (a, a’) monomer concentration [dimensionless], (b, b’) reaction medium velocity (×106) [m.s-1], and (c, c’) pore filling factor [dimensionless] of simulations using ill-distributed active sites (a,b,c- case 1 and a´,b’c’- case 2). Parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. Figure 16: Fragmentation profiles: (a, a’) global index [dimensionless], (b, b’) differential index [dimensionless], and (c, c’) integrated index [dimensionless of simulations using ill-distributed active sites (a,b,c- case 1 and a´,b’c’- case 2). Parameters are in Table 3. Other units: time [s] and radial position [dimensionless]. Figure 17: Comparison of the differential with the integral fragmentation index. Simulation of the illdistributed active sites (a) case 1 and (b) case 2. The dashed line is the bisectrix. Figure 18: Comparison of the integral fragmentation index with pore filling factor along polymerization. Simulation of the ill-distributed active sites case 2.

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