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Modelling condensed mode cooling for ethylene polymerization.
Part II. Impact of Induced Condensing Agents on Ethylene
Polymerization in an FBR operating in Super-Dry Mode.
Rita Alves, Muhammad Ahsan Bashir, and Timothy F.L. McKenna
Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b02963 • Publication Date (Web): 25 Oct 2017
Downloaded from http://pubs.acs.org on October 30, 2017
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Modelling condensed mode cooling for ethylene polymerization. Part II. Impact
of Induced Condensing Agents on Ethylene Polymerization in an FBR operating in
Super-Dry Mode.
Rita Alves, Muhammad Ahsan Bashir, Timothy F.L. McKenna*
University de Lyon, CNRS, CPE-Lyon, UCB Lyon-1, Chimie Catalyse Polymères et
Procédés (C2P2), 43 Blvd du 11 Novembre 1918, 69616 Villeurbanne Cedex, France
*timothy.mckenna@univ-lyon1.fr
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Graphic for TOC.
Abstract
The Sanchez-Lacombe Equation of State (SL EoS) was used to estimate the
concentration of ethylene and different induced condensing agents (ICA) in
polyethylene, and the effects of adding an ICA on parameters such as reactor
temperature, production rate and particle size were discussed. A simple CSTR-like
reactor model was validated from production data, and the results reveal that serious
errors will be found in the prediction of the reactor temperature and production rate if
the interaction between ethylene and ICA is not accounted for. It is also shown that
adding an ICA can lead to increased production rates at the cost of decreased catalyst
mileage.
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1. Introduction
Industrial processes for the production of polyethylene (PE) can be divided into
different categories according to the phase in which the polymerization takes place:
solution, slurry, gas-phase processes, with the latter two being more significant in terms
of production volumes. While slurry phase processes are commercially important for a
number of reasons, gas-phase processes are even more widely used due to their
versatility. They can be used to produce resins with a full range of densities, from linear
low density polyethylene (LLDPE) to high density polyethylene (HDPE) in the same
process.1 The only type of reactors used for production of gas-phase PE are Fluidized
Bed Reactors (FBR), since this is the only reactor type that can be used to evacuate
enough heat from the reactor to achieve commercially pertinent rates of
polymerization.1
A diagram of a typical FBR for PE production is shown in Figure 1. The reactor is
essentially an empty cylinder with an expansion zone at the top (to reduce the gas
velocity and help prevent any fine particles from flowing out of the reactor and into the
recycle compressor), and a distributor plate at the bottom. Catalyst (or prepolymerized
catalyst) is fed into the reactor at point slightly above the distributor plate, and the fluids
are typically fed through the bottom of the reactor, usually (but not always) below the
distributor plate. The polymer is removed through a product discharge valve, following
into a series of degassing tanks to separate the unreacted monomer. The gaseous recycle
stream is compressed, cooled and afterwards mixed with fresh monomer, hydrogen and
eventually other compounds, then fed back into the reactor.
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Compressor
Heat Exchanger
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C2H4
Catalyst
H2
Comonomer
ICA
N2
Figure 1. Unipol process for polyethylene production.
As mentioned above, one of the key points in the safe and economical operation of an
FBR to produce PE is heat removal; a typical commercial scale reactor will generate
several 10s of megawatts of energy during a polymer production rate that can surpass
750 kt/year.1 In fact, heat removal is the single most important factor that places an
upper limit on the PE production rate.
It is well-known that most of the heat generated by the polymerization is removed via
the gas phase as it flows over the particles in the bed. However, this is limited by the
maximum flow rate of gas through the bed, and by the temperature of the feed stream.
The feed stream temperature is typically regulated with cooling water, and can be as
low as is economically possible when operating in “dry mode” (i.e. when only ethylene,
vaporized comonomer, hydrogen and nitrogen are present in the feed). Increasing the
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flow rate of gas through the reactor would help, but this limited in scope: if the flow rate
is too low, the bed collapses, and if it is too high, a significant fraction of the particles
will be blown out of the bed and into the recycle stream.2
The only other effective means of improving the heat removal capacity of the reactor is
to alter the physical nature of the feed stream. Compounds such as ethane, propane or
butane can be introduced into the reactor in what can be referred to as “super dry
mode,” leading to two main consequences: i) increased ability to remove heat through a
higher gas phase heat capacity; ii) increased ethylene concentration in the amorphous
polymer phase (co-solubility effect). Note that heavier alkanes can be used as well, but
then the feed stream can only be cooled so far in this case without condensation taking
place.1,2
However, even more heat can be removed when the reactor is operated in what is called
“condensed mode”. In this case the recycle stream is compressed, and then cooled by
passing it through at least one external heat exchanger to a temperature below that of the
dew point of the gas mixture. The resulting stream is then fed into the lower zone of the
reactor in such a way that the liquid is sprayed into the reacting zone, and the droplets of
liquid are vaporized by the heat of reaction. Alkanes such as isomers of butane, pentane
or hexane are most commonly used to this end. In the case of super dry mode, or
condensed mode, the compounds used to help heat removal can be referred to as
induced condensing agents (ICA). Monomers such as 1-butene, or 1-hexene can also be
liquefied and contribute to energy evacuation as well, but since they are reactive, we
will differentiate between liquefied reactants and chemically inert ICA. In the current
paper, we will look at the impact of vaporized ICA on the performance of a FBR
operating in super dry mode. In normal condensing mode, it has been shown that the
liquid droplets evaporate rapidly, and that the clear majority of the powder bed in a
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typical reactor contains only solid particles and a continuous gas phase.3 Thus, using a
super dry mode simulation to investigate the importance of accounting for the
interactions between the different components present is a reasonable first step in
understanding how to represent the thermodynamics of the polymerization.
In a series of recent papers from our research group, experimental work has shown that
adding a chemically inert ICA (isomers of pentane or hexane) has a significant effect on
the observed rate of polymerisation during gas phase polymerization on Ziegler-Natta
catalysts, even in a closed, semi-batch reactor.4,5,6 For example, adding 2 bars of npentane to 7 bars of ethylene provokes an increase of 40% in the average activity as
compared to 7 bars of ethylene alone. This was attributed to the fact that the presence
of heavier alkanes can increase the solubility of lighter compounds such as ethylene,
thus adding n-pentane to the gas phase polymerization of ethylene provokes an increase
in the monomer concentration at the active sites, and thus an increase in the observed
reaction rate. It was also seen that the higher the solubility of an ICA in the polymer is,
the greater its enhancing effect on the solubility of ethylene in the polymer will be.
Furrhermore, the swelling of the amorphous phase of the PE can also change the actual
volume of the particles, and perhaps have an influence on the fluidisation behaviour of
reacting powder.7 It would clearly be interesting to understand what, if any, impact this
co-solubility effect would have on reactor operation.
An earlier paper by Hutchinson and Ray discussed the importance of understanding the
importance of solubility in multicomponent systems, and pointed out that heavier
alkanes can enhance the solubility of lighter components such as ethylene.8 They also
showed that by using the correct ethylene concentration (i.e. that at the active sites
corrected for the cosolubility effect), one finds similar reactivity ratios in slurry and gas
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phase polymerizations. These same authors compared gas and slurry polymerizations,
but did not investigate reactor behaviour under a range of operating conditions.
Since fluidized beds are widely used in many forms, one can find numerous models for
predicting their behaviour in the open literature.9,10,11 Such studies describe fluidized
beds in great detail and provide an extensive list of empirical correlations which may be
used to estimate properties of importance when designing FBR. Studies on the
modelling of FBRs in the specific case of PE production are numerous as well, and
exhibit many levels of complexity. For example, Choi and Ray,12 and Grosso and
Chiovetta13 proposed a 2 phase model of a bubbling FBR which included an emulsion
phase (mixture of particles and vapour) and a bubble phase (vapour only), and were able
to track temperature and concentration gradients in the reactor assuming a constant
bubble size. Other groups extended this analysis to include variable bubble sizes with a
uniform emulsion phase,14 with regard to the temperature and concentration gradients in
the gas phase. And, even more complex models have been developed that divide the
different phases into separate zones in order to obtain a more accurate picture of the
gradients as well as of the particle size distribution in the reactor.15
However, in terms of the impact of adding ICA to the reactor, very few studies have
been published in the open literature. A series of studies looked at the impact of the
addition of ICA on the heat balance around FBRs for PE production, but did not look at
the impact of the ICA on the product itself, nor its impact on the reaction rate.15,16 Other
authors have studied the impact of adding an ICA to the feed stream, and demonstrated
that cooling the feed stream, and including an evaporation term in the heat balance
allows one to increase the reaction rate, and thus obtain higher levels of production.
16,17,18
However, none of the papers published in the field of reactor modelling considers
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the impact that ICA might have on the swelling of the polymer particles, nor on the
observed rate of reaction.
In the current paper, we propose to investigate the impact of adding an ICA on the
overall reactor behaviour. To do so, we will use a simplified approach to model the
reactor model and assume that the residence time distribution of the reactive powder
bed in the FBR is that of a continuous stirred tank reactor. It has been shown elsewhere
that this simplification has a limited impact on the calculation of the final PSD and
conversion in the reactor.19
On the other hand, we will include a more complex
description of the solubilities of different species in the reactor to allow for interactions
between the different species in the polymer phase, as these interactions turn out to be
significant. In the event that co-solubility effects have an impact on reactor operation,
more complex models (that include single particle models) can be developed in the
future.
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2. Modelling Development
Model Description
The development of the mathematical model used here will be divided into different
sections: mass and heat balances, calculation of the particle size distribution, FBR
design equations and thermodynamic model.
The present model is based on the following simplifying assumptions:
•
Reactor is operating at steady-state;
•
The residence time distribution of the FBR is that of an ideal CSTR;
•
We will consider super-dry mode only – i.e. no liquid droplets in the feed,
only vapour phase ICA;
•
Catalyst activation is instantaneous;
•
The rate of polymerization will be modelled using a global propagation
constant (not attempt is made to differentiate between families of active
sites);
•
Catalyst particles are considered spherical;
•
The elutriation of solids is neglected;
•
Gas entrainment by production discharge is neglected;
•
The solid feed to the reactor consists only of fresh catalyst (no
prepolymerization);
•
No breakage or aggregation is considered;
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•
The gaseous outlet of the reactor consists of unreacted ethylene, ICA and
nitrogen;
•
The solid outlet of the reactor consists of a polymer phase, containing the
polymer and catalyst particles, as well as dissolved ethylene and ICA;
•
Heat transfer between the growing particles and the gas phase is by
convection only.
All correlations for bed density, heat transfer coefficients and other reactor properties
are calculated using an average size for catalyst and for polymer particle. It is, of course,
possible to adopt a more detailed modelling approach, such as full scale population
balances. However, since the objective of this work is to explore how important it is to
use accurate thermodynamic models, the additional complication created by using
complex models is not useful at this point.
2.1.
Mass Balances
The general form of the ethylene mass balance is written as follows:
Q, − Q, − R T, P ∙ V ∙ MW − Q, = 0
Where Q, is the ethylene mass flow rate entering the reactor, Q,
(1)
is the ethylene
mass flow rate exiting the reactor, R is the reaction rate at a given temperature and
pressure , V is the catalyst volume in the bed and Q, is the flow rate of ethylene
dissolved in the outlet polymer stream. This last variable can be defined as:
Q, =
C
∙ MW
∙ Q ∙ w ρ (2)
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is the concentration of ethylene in the amorphous polymer phase, Q is the
Where C
PE production rate (as defined in equation (15)) and w is the weight fraction of
amorphous phase in the polymer, which varies with the temperature.20
The general form of the alkane mass balance is given as:
Q !", − Q !", − Q !", = 0
(3)
Where Q !", is the ICA mass flow rate entering the reactor, Q !",
is the ICA mass
flow rate exiting the reactor and Q !", is the flow rate of ICA dissolved in the polymer
phase. The Q !", equation is similar to the ethylene:
Q !", =
C!" ∙ MW !"
∙ Q ∙ w ρ (4)
Where C !" is the concentration of ICA in the amorphous polymer phase.
Since the ICA is a chemically inert compound, and we are not interested in modelling
the molecular weight distribution in this work, the kinetic scheme considered here will
be the homopolymerization of ethylene. Even though the polymerization includes
several well-known steps, a simple expression for the overall reaction is enough to
reflect the impact of the cosolubility effect:21
R = k ∙ C∗ ∙ C
(5)
Where k represents the kinetic rate constant. C ∗ is the active sites concentration on the
catalyst is given by equation (7), presented below: 22
Q ⁄ρ ∙ C'∗ − k ∙ V! ∙ C ∗ −
C∗ = ,
()
*
∙ C∗ = 0
!∗+
(6)
(7)
- ∙*./
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Q is the catalyst mass inlet flow rate, ρ is the catalyst density, τ is the average
residence time and C'∗ is the initial concentration of active sites. It is important to
mention the use of Arrhenius Law to predict the kinetic rate (k and catalyst
deactivation (k constants at the reaction temperature, as described in equations (8)
and (9).
k = k
1234
E
1
1
∙ exp 8 ∙ :
− ?@
R T<= T>
E
1
1
1
k = k 234 ∙ exp 8 ∙ :
− ?@
R T<= T>
(8)
(9)
In Equation (5) C
is the ethylene concentration in the amorphous polymer phase. This
last parameter is of the utmost importance and an accurate estimation is likely needed in
order to predict the polymerisation rate. C
changes with the presence of different
ICAs, comonomers, and with the operating conditions (reactor temperature and
pressure). It is important to point out that at present, no EoS can accurately predict this
parameter à priori in multicomponent systems (when one of the components is a
polymer), which leaves fitting of these models to the experimental data as the only
option.23
A thermodynamic model was implemented to estimate the polymer density, and
ethylene and ICA concentration in the polymer phase. The chosen model was the
Sanchez-Lacombe Equation of State (SL EoS),24,25 a widely applied model in the
polymer industry due to its simplicity and good accuracy. This model is described in
section 2.5.
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2.2.
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Energy Balances
Inside the reactor two temperatures can be observed: The bulk temperature (TA ) and the
solid particles temperature (T> ).
The rate of heat transfer between the growing particles and the continuous phase can be
written as follows:26
h ∙ A ∙ T>−TA = V
,
∙ Rp ∙ D−∆H G
(10)
Rearranging (10),
T> −TA =
d I ∙ R ∙ D−∆H G
Where ∆H
(11)
6 ∙ KKK
d ∙h
L
is the heat of reaction and h represents the convective heat transfer
coefficient.26
The reactor heat balance is written as follows:
∆H − ∆H + ∆HN< = 0
(12)
Reference State:
•
Reference Temperature – Inlet Temperature (T ;
•
Reference Pressure – Reactor working pressure;
•
Ethylene, nitrogen and alkane in gaseous form;
•
Solid catalyst;
•
Semi-crystalline polyethylene.
Assuming this reference state and that the operation occurs in steady state, the heat
balance in equation (12) is reduced to:
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−∆H + ∆HN< = 0
(13)
Replacing the parameters, the equation (13) takes the following form:
KKKKKKK
KKKKKKK
KKKKKKK
TA − T ∙ DQ, ∙ KKKKKK
C , + Q !", ∙ C
, !" + Q OP , ∙ C ,OP + Q ∙ C ,
(14)
+ Q! ∙ KKKKK
C ,! G + Q,<> ∙ ∆H = 0
KKK is the average heat capacity, as described in supplementary material equation
Where C
(S1) for gases and presented in Table 5 and Table 8 for PE and catalyst. It is assumed
that changes in pressure will not affect heat capacity. Q,<> represents the ethylene
flow rate that is consumed as a reactant, which is the same as the PE flow rate
production:
Q,<> = Q = R ∙ V ∙ MW
2.3.
(15)
FBR Design Equations
Correlations used to predict minimum superficial velocity and bed porosity are listed in
Table 1.
Table 1. Fluidization and bed properties correlations.
Re= = 33.7L + 0.0408 ∙ Ar'.W
Minimum fluidization
velocity10
Average gas fraction27
δ = 1 − Y0.466 + 0.534 exp Y
H= = HA ∙ 1 − δ
Height at minimum
fluidization28
Bed voidage for HA ≤H= 29
Bed voidage for H= ≤ HA ≤
^H= + 2 HA − H= `
29
Weight of Solids in bed10
u= − u>
\\
0.413
1 − ε =
1 − ε =
H= ∙ 1 − ε= H
H=
H= ∙ 1 − ε= ∙ HA − H= −
HA ∙ 1 − ε=
2H ∙ HA − H= W = S ∙ HA ∙ 1 − ε ∙ ρ
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Particle Size Distribution
The calculation of the PSD of the polymer produced inside a CSTR is based on the
model proposed by Soares et al.:30
d = d 1 + α t//I
F d =
(16)
3 1 + α t//I ef/*
α d
τ
(17)
Where,
α=
k C
C ∗ MW
ρ (18)
d is the diameter of the polymer particle, d is the diameter of the catalyst particle (m),
t is the reaction time, τ is the average residence time of the reactor and α is a combined
kinetic parameter.
The main limitation of the model is that it assumes that the catalyst particles entering
the reactor are monodispersed. Thus, the catalyst PSD is discretized and all the
correlations are applied to each catalyst particle size. From the particle size distribution
of the polymer, it is also possible to obtain the average particle size of the polymer
phase, 31 which is considered to be particle diameter (KKK
d ) for all calculations.
KKK =
d
2.5.
1
F d ∑
d
(19)
Thermodynamic Model
The SL EoS was implemented with two different approaches to predict the ethylene
concentration in the amorphous polymer phase, and amorphous polymer density: a
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ternary model applied to Ethylene(1)/ICA(2)/Polymer(3) system that accounted for cosolubility effects, and a binary model, applied to the Ethylene(1)/Polymer(2) and
ICA(1)/Polymer(2) systems. Details regarding the EoS parameters and solution strategy
can be found elsewhere.32
Additional assumptions specific to the implementation of the SL-EoS include:
•
The gaseous inlet of the reactor consists only of ethylene, one single ICA and
nitrogen. No experimental data are available for more complex systems;
•
The thermodynamic equilibrium between the polymer and gas phase is attained
instantaneously;
•
Ethylene and ICA solubility in the polymer phase are calculated at the said gas
phase temperature.
In other words, no impact of particle temperature rise on
solubility is included here;
•
Due to its very low solubility, nitrogen content in the polymer phase and its impact
on ethylene solubility are neglected;
•
Due to the inherent inability of the SL EoS to account for the negative impact of
polymer crystallinity on the solubility of penetrants (i.e., ethylene, ICA or
ethylene/ICA mixture) in polyethylene, the binary interaction parameter(s) of the
EoS were fine tuned in a way that the relative error between the experimental
solubility values (of penetrant(s) in polyethylene) and those estimated by SL EoS
was less than 10%. It has been shown in the literature that in order to account for
the effects of polymer crystallinity on penetrant(s) sorption one can use an elastic
constraints model or adjust the binary interaction parameter(s) of the SL EoS in
order to fit the model to the experimental data or use a non-equilibrium version of
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the SL EoS which requires polymer swelling data as an input that is not available
for most of the penetrant(s)/polyethylene systems in the open literature.33,34,35
Binary System - ICA/Polymer
The experimental solubility data of pure ethylene and various pure alkanes in
polyethylene (PE) at different conditions was obtained from the open literature as
shown in Error! Reference source not found.. The SL EoS was fit to these binary
solubility data points in order to obtain the binary interaction parameter (kij) for each
binary system (i.e., penetrant(1)/PE(2)). It should be noted that penetrant refers to the
solute and that the solubilities were estimated in grams per gram of amorphous PE.
Table 2 shows some of the representative kij values obtained for different binary
systems at different temperatures by fitting the SL EoS to the experimental data.
Table 2. Binary interaction parameter(kij) values obtained by fitting the SL EoS to the
experimental solubility data.
Diluent/PE
T (°C)
kij
Ethylene/LLDPE36
70
0.028
Ethylene/LLDPE36
90
0.012
Propane/LLDPE37
70
0.023
iso-Butane/LLDPE38
74
0.025
iso-Butane /LLDPE38
82
0.022
n-Hexane/LLDPE39
70
0.0135
n-Hexane /LLDPE39
80
0.0135
Fitting the SL EoS to the experimental solubility data provides mass fraction of the
penetrant in the polymer phase as well as the swollen polymer phase density. Utilizing
both of these values allows one to estimate the concentration of penetrants inside the
swollen polymer phase.
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1800
CICA,polymer (mol/m3)
1600
1400
1200
1000
800
n-Hexane, T=80ºC
600
iso-Butane, T=80ºC
400
iso-Butane, T=70ºC
200
Propane, T=70ºC
0
0
2
4
6
8
10
ICA Partial Pressure (bar)
12
14
16
Figure 2. Concentration of iso-butane, n-hexane or propane in LLDPE at 70ºC and 80ºC
(ethylene partial pressure of 7 bar) obtained from fitted SL EoS.
900
800
ρ,pol (kg/m3)
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700
600
n-Hexane, T=80ºC
500
iso-Butane, T=80ºC
iso-Butane, T=70ºC
400
Propane, T=70ºC
300
0
2
4
6
ICA Partial Pressure (bar)
8
10
Figure 3. Amorphous polymer density in the presence of propane or iso-butane or nhexane, at 70ºC and 80ºC (ethylene partial pressure of 7 bar) obtained from fitted SL
EoS.
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To simplify the calculations, correlations were developed for the ICA concentration in
the amorphous polymer phase and the amorphous polymer density, as shown by the
fitted lines in Figure 2 and Figure 3, respectively. Equations (20) and (21) show the
mathematical form of these correlations. The coefficients are shown in Table 3.
C!" = A ∙ P !"
(20)
ρ
(21)
h<
= −B ∙ P !" + C
Table 3. Coefficients for the binary correlations of ICA concentration on amorphous
polymer phase (mol.m-3) and polymer density (kg.m-3).
70ºC
80ºC
Units
A
B
C
bar ∙ mol
mI bar f/
kg
mI Propane
Iso-butane
Iso-butane
n-hexane
100.2
165.3
140.1
542.0
9.9
42.1
35.3
25.2
854.0
854.0
854.0
854.0
As the pressure of ethylene is kept constant at 7 bar throughout the reactor simulations
and the co-solubility effect is not accounted for in binary system, the concentration of
ethylene is considered to be constant at 70ºC and 80ºC.
Ternary System – Ethylene/ICA/Polymer
As the experimental data for ternary systems (i.e., ethylene(1)/alkane(2)/PE(3)) is scarce
(rather non-existent) in the open literature, the concentration of ethylene or ICA in
polyethylene in such systems at a given reactor temperature and pressure was estimated
by first fitting the SL EoS to the experimental solubility data of the respective binary
systems (i.e., ethylene(1)/PE(2) or ICA(1)/PE(2)). In the second step, the fitted binary
interaction parameters of the first step were then directly used in the estimation of
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ethylene and ICA concentrations inside the amorphous polymer phase in the
corresponding ternary systems without any further modification. It is important to
mention here that such method of estimation can lead to overestimation of the solutes’
solubility in the polymer phase due to the inherent incapability of the used SL EoS
model (which is not specific to this model but to all the existing EoS models employed
in this field). Details about the calculation of equilibrium solubility and polymer phase
density can be found elsewhere.23,40
Figure 4 shows the solubility of ethylene in ethylene(1)/iso-butane(2)/LLDPE(3)
mixtures with different gas phase compositions by using the binary interaction
parameters obtained by fitting the respective binary experimental solubility data. As the
iso-butane concentration in the gas phase increases, the overall mixture solubility
increases and tends to shift towards the pure iso-butane/LLDPE system. The density of
the amorphous polymer phase decreases with increasing iso-butane content in the gas
phase which is in agreement with the fact that the higher the solubility of a penetrant the
higher is the swelling of the polymer. The same trends were also observed for all the
other systems discussed in this work and for the sake of brevity they are not shown here.
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200
190
180
Cet,polymer (mol/m3)
1
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170
160
n-hexane T=80ºC
150
Iso-butane T=80ºC
140
Iso-butane T=70ºC
130
Propane T=70ºC
120
0
1
2
3
4
5
6
7
8
9
10
11
ICA Partial Pressure (bar)
Figure 4. ICA effect on the concentration of ethylene in the amorphous polymer phase
at 70º C and 80º C (ethylene partial pressure at 7 bar) obtained from the SL EoS in
ternary (ethylene(1)/ICA(2)/PE(3)) systems.
Using the SL EoS based solubility and polymer phase density, the concentration of each
gaseous component in the polymer phase in ternary systems was calculated and the
results are shown in Figure 4. The increase in the ICA partial pressure in the gas phase
leads to the increase of ethylene concentration in the polymer phase. In addition, the
higher the carbon number of the ICA the higher is the ethylene concentration in the
polymer phase at the same conditions. This observation can be attributed to the cosolvent effect of alkanes on the solubility of ethylene, which manifests itself in
multicomponent gases/polymer systems and is well known in the open literature. For
detailed discussion about the co-solvent or co-solubility effects the reader is referred to
see.35,40,36,39
Figure 5 shows the effect of ethylene/ICA mixtures on the density of amorphous PE in
ternary systems estimated by the SL EoS. It can be observed here that the higher the
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carbon number of the ICA the lower is the polymer phase density at given conditions
which agrees with the discussion made above.
835
n-hexane T=80ºC
830
ρpolymer (kg/m3)
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Iso-butane T=80ºC
Iso-butane T=70ºC
825
Propane T=70ºC
820
815
810
805
800
0
1
2
3
4
5
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7
8
9
10
11
ICA Partial Pressure (bar)
Figure 5. ICA effect on the amorphous polymer density at 70ºC and 80ºC (ethylene
partial
pressure
at
7
bar)
obtained
from
the
SL
EoS
in
ternary
(ethylene(1)/ICA(2)/LLDPE(3)) systems.
For the sake of simplification, empirical correlations were again developed for the
and C !") and the polymer
ethylene and ICA concentration in the polymer phase (C
density, as shown by the equations (20)-(22). The parameters are shown in Table 4.
C
= D ∙ P !" + E
(22)
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Table 4. Coefficients for the correlations of ethylene and ICA concentration on polymer
phase (mol.m-3) and polymer density (kg.m-3).
70ºC
80ºC
Units
Propane
Iso-butane
Iso-butane
n-hexane
75.25
237.69
120.38
343.50
kg
mI
1.71
2.87
3.26
9.11
828.97
828.97
825.90
825.90
mol
mI 2.16
4.82
4.84
13.71
152.35
152.35
129.5
129.5
bar ∙ mol
mI A
bar f/
B
C
bar f/
D
E
1400
1200
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1000
800
600
n-Hexane 80ºC - Binary
400
n-hexane 80ºC - Ternary
200
Propane 70ºC - Binary
Propane 70ºC - Ternary
0
0
2
4
6
8
n-Hexane partial pressure (bar)
10
12
14
Figure 6. Comparison of Propane and n-Hexane concentrations in the amorphous
polymer phase for binary and ternary systems at 70ºC and 80ºC (partial pressure of
ethylene of 7 bar in ternary systems).
As seen in Figure 6, the concentration of ICA in the amorphous polymer phase is
expected to be higher when the binary approach is considered. This is an expected
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result, as ethylene is known to act as an anti-solvent for ICA, thus decreasing the
concentration of ICA in amorphous PE. The existence of co-solubility effects (i.e., cosolvent and anti-solvent effects) has been discussed in detail by different authors for
different penetrant(s)/polyolefin systems.35,36,39,40
2.6.
Model Implementation
To solve the reactor equations, the Matlab’s Optimization Toolbox’s fsolve function
was used. This function requires a matrix with the initial guess for every variable, since
all the equations are solved simultaneously. Consequently, an auxiliary and simplified
version of the model was developed in Microsoft Excel® to obtain the required initial
values. The model was solved using the algorithm shown in supplementary material
Figure S2.
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3. Results
3.1.
Model Validation
The model validation was carried out by replicating examples 7A and 7C of the patent
US 6,864,332 B2.2 Example 7A does not include any ICA, so was used to determine a
reference reaction rate. In example 7C a mixture of propane and iso-butane are
introduced in a “super dry” mode industrial production run. The data used in both
examples is shown in Table 5 whereas example specific data is shown in Table 6.
Table 5. Data used for both examples in the validation of the model.
Parameter
Units
Value
Reactor Diameter2 (d )
m
4.75
Reactor Bed Height2 (Hb)
m
13.3
Catalyst type41
-
Ziegler Natta
kg/m3
2300
J/(kg.K)
2000
kgpol/(kgcat.h.bar)
1500
J/(kg.K)
2000
J/mol
-107600
-
0.476
Ethylene Partial Pressure2
bar
7.8
Reactor Abs. Pressure2
bar
22.4
Inlet Temperature2
ºC
35
Catalyst Density41 (ρc)
Catalyst Heat Capacity41 (Cp,c)
2
Catalyst Specific Activity
Polymer Heat Capacity41 (Cp,p)
Heat of Reaction41 (∆H)
Minimum Fluidized Bed Porosity11 (εm.f.)
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Table 6. Data used in the validation of the model.2
Parameter
Units
7A
7C
Reactor Abs. Pressure
bar
22.4
22.4
Inlet Temperature
ºC
35
35
Inlet gas flow rate
kg/s
280
355
Inlet Catalyst flow rate
kg/s
8.20x10-5
2.70x10-4
Ethylene Partial Pressure
bar
7.8
7.8
Propane Partial Pressure
bar
0
4.3
Iso-butane Partial Pressure
bar
0
3.3
Since in example 7C there are two ICA, some minor alterations were made to the
reactor equations. All equations regarding the ICA are still written in the same fashion,
but accounting for two ICA compounds instead of one.
For the ethylene concentration in the polymer (C
), a blunt approximation was made.
Since no data is available to estimate the model parameters for the quaternary system in
question (i.e., propane/iso-butane/ethylene/PE system), we defined a “pseudo” ICA
which is mixture of propane and iso-butane as follows. C
is estimated by interpolating
the data obtained for ternary systems using a weighted average of the slopes for propane
and for butane in equation (22). Strictly speaking, the solubility of one ICA in the
polymer will be influenced by the presence of other ICA in the mixture, but since the
interpolation is only for one composition, it is likely that the error caused by this
approximation will be small.
Table 7 shows the comparison between the results presented in examples 7A and 7C2
and the results obtained in the simulations.
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Table 7. Comparison between the results presented in example 7A and 7C and the
simulation (Sim.) and the corresponding variation (∆).
7A2
Sim. 7A
∆ (%)
7C2
Sim. 7C
∆ (%)
16
16.3
2%
28.9
29.8
3%
Reactor Temperature 88
(ºC)
84
-5%
88
90
2%
Superficial Velocity
(m/s)
0.75
0.75
0%
0.75
0.75
0%
Productivity
(gpolymer/gcatalyst)
53,650
55,365
3%
29,700
30,691
3%
Residence Time (h)
4.6
4.1
-12%
2.5
2.2
-14%
PE Production Rate
(tonne/h)
These preliminary simulations suggest that the model is a decent approximation to the
system, and we can therefore use it to achieve the objectives laid out above. The slight
difference in the polyethylene production rate, productivity and reactor temperature can
be explained by the approximations made above (clearly reasonable in light of the good
agreement), and due to the fact that the solubility values are only available at 70ºC for
the considered system and the reactor in the patent operates at 88ºC. The slight
differences observed in the residence time can be explained by the use of a CSTR
approach for the powder phase.
3.2.
Case Studies
Several simulations were run to evaluate the influence of increasing the ICA partial
pressure in the reactor on productivity, PE production rate and the polymer PSD.
The parameters that were kept constant in all upcoming simulations are presented in
Table 8. Values that are specific to each simulation will be presented when relevant.
Table 8. Thermodynamic and physical properties of the solid phase, reaction parameters
and reactor properties.
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Parameter
Units
Reactor Diameter2 (d)
m
4.75
Reactor Bed Height2 (Hb )
m
13.3
Catalyst type41
-
Ziegler Natta
Initial Catalyst Active Site Concentration41 (C0* )
mol/m3c
0.52
Catalyst Density41 (ρc)
kg/m3
2300
J/(kg.K)
2000
J/(kg.K)
2000
Kinetic rate constant41 (kp,ref)
m3/(mol.s)
180
Catalyst deactivation rate constant41 (kd,ref )
s-1
1 x 10-4
Reaction Activation Energy41(Ea)
J/mol
42000
Catalyst Deactivation Energy41 (Ed)
J/mol
42000
Heat of Reaction41 (∆Hpol)
J/mol
-107600
Minimum Fluidized Bed Porosity11 (εm.f.)
-
Catalyst Size42
µm
Catalyst Heat Capacity41 (Cp,c)
41
Polymer Heat Capacity
(Cp,p)
Value
0.476
45;65; 80
The bed height is kept constant in all simulations. The reactor dimensions were taken as
the same as the ones from patent US 6,864,332 B2.2
The pressure of nitrogen is variable within each simulation to accommodate the increase
of ICA partial pressure whilst keeping the total pressure constant. It was assumed that
nitrogen was insoluble in the polymer phase, and had no impact on the solubility of
ethylene and the ICA considered.43
Since the model is developed for super-dry mode, the maximum partial pressure of the
ICA and the inlet temperature are not independent since there is a need to ensure that
the feed stream temperature is always above its dew point (i.e., the feed stream contains
no liquid droplets). The values of the partial pressure were chosen to ensure there in no
liquid present in the reactor and to keep the bulk temperature within the 70ºC or 80ºC
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due to the correlations developed for the concentration of ethylene and ICA in the
amorphous PE and PE density.
3.2.1. Simulation I – Binary Vs Ternary
Simulation I compares the two different approaches to the thermodynamic modelling
for the prediction of ethylene and ICA concentration in amorphous polymer and
amorphous polymer density: the ternary model and the binary model. This simulation
aims to show the importance of accounting for the co-solubility effect and the influence
of increasing ICA partial pressure on reactor behaviour when the remaining inlet
parameters are kept constant.
All the inlet parameters are kept constant (inlet gas temperature and flow rate, catalyst
inlet flow rate and ethylene partial pressure), changing only the ICA partial pressure.
For this simulation, the values used are summarized in Table 9.
Table 9. Simulation I reactor parameters.
Parameter
Unit
Value
Catalyst inlet flow rate
kg/s
0.0033
ºC
35
mol/s
10000
Reactor Abs. Pressure
bar
20
Ethylene Partial Pressure
bar
7
ICA Partial Pressure
bar
0 to 4
Nitrogen Partial Pressure
bar
13 to 9
Inlet gas temperature
Inlet gas flow rate
2,44
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78
Isobutane, Binary
Propane, Binary
Isobutane, Ternary
Propane, Ternary
75
Bulk Temperaute (ºC)
73
70
68
65
63
60
58
55
0
1
2
3
ICA Partial Pressure (bar)
4
5
Figure 7. Influence of propane or iso-butane on the reactor bulk temperature (ºC), using
two different approaches to the thermodynamic model (binary and ternary).
14
Isobutane, Binary
PE production rate (tonne/h)
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Propane, Binary
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Isobutane, Ternary
Propane, Ternary
12
12
11
11
10
10
0
1
2
3
ICA Partial Pressure (bar)
4
5
Figure 8. Effect of propane or iso-butane on PE production rate (tonne/h), using two
different approaches to the thermodynamic model (binary and ternary).
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Figures 7 and 8 show, first of all, that not accounting for the interaction between the
different species in the system can lead to very different predictions. Regardless of the
thermodynamic model chosen, the PE production rate decreases with the increase of
ICA partial pressure. As the heat capacity of the gas phase increases quickly with ICA
content, it can remove more heat and keep both the gas phase and particle temperatures
lower than in the absences of ICA. The decrease in temperature leads to a decrease in
the reaction rate although it would increase the solubility of ethylene.
However, using a purely binary description of the solubility of ethylene and ICA means
that the enhancement of the ethylene concentration is not taken into account. So the
only effect of the ICA is to lower the predicted reactor temperature. This leads to
predictions of a lower specific rate of polymerization and lower productivity than is
seen for the ternary case.
If one considers the parameters shown in Table 10, it is evident that using the ternary
model for calculating solubility leads to a very different solubility than in the binary
case.
This allows one to see that although iso-butane will increase the ethylene
concentration more than will propane, its higher heat capacity means that we can
evacuate more energy from the reactor. Figure 7 shows that this leads to a lower
temperature with iso-butane than with propane, so the rate constant is lower for the
former. The trade-off means that, under the conditions chosen for the simulation, we
observe that the production rates in the presence of both iso-butane and propane are in
fact similar. A very different conclusion would be reached should one use a binary
solubility model.
These results prove the importance of using an adequate thermodynamic model. It is
clear that the use of the binary approach will underestimate the temperature, PE
production rate and particle size in the reactor, while overestimating the residence time.
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Table 10. Comparison and variation of kinetic parameters k ∙ C ∗ , ethylene
G and reaction rate DR G for systems
concentration in amorphous polymer phase DC
with no ICA and iso-butane in binary/ternary correlations (ICA partial pressure at 4 bar,
ethylene partial at 7 bar).
Parameter
k ∙ C∗
C
Units
No ICA
Iso-butane
(Binary)
∆ (%)
Iso-butane
(Ternary)
∆ (%)
mI ∙ s
molrs
mI 32.2
19.5
-65
22.3
-44
152
152
0
171.6
11
4.9x103
3.0x103
-65
3.8x103
-28
mI mol
mI ∙ s
R
75
73
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71
Solids
69
Bulk
67
65
63
61
59
57
55
0
1
2
3
ICA Partial Pressure (bar)
4
5
Figure 9. Effect of adding iso-butane on the bulk and solids temperature in the gas phase
polymerization of ethylene at partial pressure of 7 bar.
As seen in Figure 9, the solids and bulk temperature are very similar, presenting a ∆T of
0.34ºC. This is an expected result according to Floyd et al26 and McAuley et al45 as the
particles considered in the model are relatively large, thus allowing for significant heat
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evacuation. Also, the reactor operates at a high superficial velocity, which also aids the
heat removal by increasing the convective heat transfer coefficient.
From Table 11 it is possible to conclude that the bed porosity and bed weight are not
significantly affected by the presence of ICA. The bed weight will increase with the
presence of ICA, which is an expected result since the ICA is going to be adsorbed into
amorphous PE which also leads to higher ethylene solubility in the polymer phase
(assuming that all the ethylene is not reacted), adding weight to the bed.
Table 11. Maximum and minimum values for the bed porosity, bed weight, superficial
velocity and residence time for simulation I.
Parameter
Units
Binary
Ternary
Bed Porosity
-
0.71-0.70
0.70-0.69
Bed Weight
tonne
64.0-66.1
65.5-66.7
m/s
0.80-0.74
0.80-0.75
h
5.1-6.4
5.1-5.8
Superficial Velocity
Residence Time
The change in superficial velocity can be attributed to the decrease in the
compressibility factor (z) and to the bulk temperature decrease. The more ICA is added
into the system, the further the gas strays from the ideal behaviour (see Figure S2 of
supplementary material). The decrease in the z decreases the volumetric flow rate
according to Equation (S5), thus decreasing the superficial velocity.
The residence time increases in the binary and ternary approaches. The residence time is
directly linked to the PE production rate, since the bed height is kept constant. As less
polymer is being produced with the increase of ICA (see Figure 8), the average
residence time increases. It is also shown that the increase in residence time is more
accentuated for the binary approach, as less PE is being produced in that case.
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The results shown in this simulation prove the importance of using an adequate
thermodynamic model for the estimation of penetrant(s) solubilities in the polymer
phase and polymer phase densities under industrial conditions. The binary approach
(i.e., not considering co-solubility effects) will underestimate the reactor temperature,
PE production rate and swollen polymer particle size, while overestimating the
residence time.
3.2.1. Simulation II – Constant Bulk Temperature
In this simulation, the reactor temperature is kept constant at 70ºC as well as the bed
height, as mentioned above. Note that while the temperature of 70°C is somewhat
lower than 80-90°C typically seen in industry, it is the temperature for which the
thermodynamic data are available which ensures that the solubility and swelling
predictions are as accurate as possible. To keep the reactor temperature constant, the
catalyst flowrate is varied, since it is not possible to change the reactor gas feed flowrate
due to fluidization constraints. Table 12 summarizes the values used in this simulation.
Table 12. Simulation II reactor parameters.
Parameter
Unit
Value
Inlet Temperature
ºC
35
Bulk Temperature
ºC
70
Inlet gas flow rate
mol/s
10000
Reactor Abs. Pressure
bar
20
Ethylene Partial Pressure
bar
7
ICA Partial Pressure2,44
bar
0 to 7
Nitrogen Partial Pressure
bar
13 to 6
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16
PE Production Rate (tonne/hr)
15
14
13
12
11
10
Propane
9
Isobutane
8
0
1
2
3
4
5
ICA Partial Pressure (bar)
6
7
8
Figure 10. Influence of propane or iso-butane on PE production rate (tonne/hr) in gas
phase ethylene polymerization at a partial pressure of ethylene of 7 bar and 70ºC.
1130
1120
Propane
Productivity (kgpolymer/kgcatalyst)
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1110
Isobutane
1100
1090
1080
1070
1060
1050
1040
1030
0
1
2
3
4
5
ICA Partial Pressure (bar)
6
7
8
Figure 11. Influence of propane or iso-butane on the catalyst productivity in gas phase
ethylene polymerization at a partial pressure of ethylene of 7 bar and 70ºC.
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In this simulation – perhaps more representative of commercial operation than
Simulation I – it can be seen that the presence of ICA increases the PE production rate
when the temperature is kept constant, as shown in Figure 10. This can again be
explained by the co-solubility effect, where the presence of ICA increases the ethylene
concentration in the amorphous polymer phase. Figure 11 however, the productivity, or
mileage of the catalyst decreases at the same time which can be explained by the
increased polymerization rate in the bed. To keep the same bed height, the polymer
needs to be withdrawn at higher rate as the ICA concentration increases. This leads to a
decrease of the average residence time of the powder (c.f. Table 13), and thus a shorter
time for the catalyst to produce polymer, and an increase of catalyst feed rate (c.f. Table
14).
Iso-butane has a more pronounced effect than propane, since the higher PE
production rates are obtained when using this alkane.
810
800
Average polymer particle size (µm)
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Propane
790
Isobutane
780
770
760
750
740
730
720
0
1
2
3
4
5
ICA Partial Pressure (bar)
6
7
8
Figure 12. Influence of propane or iso-butane (at ICA partial pressure of 7 bar, ethylene
partial pressure of 7 bar and 70ºC) on the average swollen particle size of the PE.
As shown in Figure 12, the average swollen particle size will decrease. From Figure 5 it
could be expected that the average particle size would increase, however equations (16)
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and (17) show that the particle size distribution has three central parameters that are
directly affected by the presence of ICA: polymer density, ethylene concentration on
amorphous polymer and residence time. Table 13 shows a comparison of the results for
these parameters.
Table 13. Comparison and variation of polymer density Dρ
G,
ethylene concentration
in polymer amorphous phase
and residence time τ for systems with no ICA,
propane and iso-butane (ICA partial pressure at 4 bar, ethylene partial at 7 bar and
70ºC).
DC
G
Parameter
ρ
C
τ
Units
No ICA
Propane
∆ (%)
Iso-butane
∆ (%)
(kg/m3)
945
941
-0.4
938
-0.7
(mol/m3)
152
167
9.1
186
18.1
(hr)
6.7
5.2
-22.1
3.9
-29.8
Since the average residence time decreases with the increase of ICA in the system, the
particles spend a much shorter time in the reactor (on average) so the particles are
smaller. Once again, this effect is more pronounced for iso-butane, as it is a bigger
molecule than propane and has more significant co-solubility effects due to its higher
solubility in PE than propane.
Table 14. Maximum and minimum values for the bed porosity, bed weight, superficial
velocity and catalyst inlet flow rate for simulation II.
Parameter
Units
Propane
Iso-butane
Bed Porosity
-
0.70
0.70
Bed Weight
tonne
65.8-66.6
65.8-66.1
Superficial Velocity
m/s
0.79-0.74
0.79-0.76
Catalyst inlet flow rate
kg/s
0.0024-0.0039
0.0024-0.0036
As seen in Table 14 the bed porosity remains invariable in this simulation. The bed
weight increases slightly and the superficial velocity decreases for reasons discussed
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above. The catalyst inlet flow rate increases with the increase of ICA to keep the bulk
temperature constant. Since iso-butane has a higher heat capacity, more catalyst needs
to be fed when using this alkane.
4. Conclusions
The importance of correct estimation of thermodynamic properties has been emphasized
by numerous publications in the open literature and to show this for an industrial scale
reactor, a mathematical model was developed in order to analyse the impact of ICA
solubility in polyethylene (with and without ethylene) on the production rates, reactor
behaviour and polymer PSD.
The model has been validated and has shown a good agreement with chosen examples.
The results show that most of the reactor parameters are sensitive to the presence of
ICA (i.e., correct estimation of penetrant(s) concentration(s) inside the amorphous
polymer phase and polymer phase density in multicomponent mixtures). When keeping
all inlet parameters constant apart from the ICA partial pressure (simulation I), a
decrease in the bulk temperature and PE production rate is observed. In simulation I the
importance of accounting for the co-solubility effect was also shown; if this is not the
case, the model underestimates the reactor temperature, PE production rate and particle
size. When the reactor temperature is kept constant (simulation II), the increase of ICA
partial pressure increases PE production rate, but decreases catalyst productivity.
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List of symbols
A – Particles surface area (m2)
C∗ – Concentraction of active sites concentration in the catalyst (mol.m-3)
C'∗ - Initial concentration of active sites (mol.m-3)
C – Concentration of ith component in amorphous PE (mol/m3)
KKK – Component average heat capacity (J.kg-1.K-1)
C
H – bed height (m)
h – Convective heat transfer coefficient (W.m-2.K-1)
k – Catalyst deactivation constant (s-1)
k = – Gas phase thermal conductivity (W.m-1.K-1)
k – Kinetic rate constant (mol.m-3cat.s-1)
MW – Molecular weight (kg.mol -1 )
Nu – Nusselt number (-)
P – Pressure (bar)
P – ith component partial pressure (bar)
Pr – Prandlt number (-)
Q – Flow rate (kg.s-1)
Q( – Inlet gas volumetric flow rate (m3.s-1)
Re – Reynolds number (-)
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R – Reaction Rate (mol.m-3cat.s-1)
S – Reactor cross-section area (m2)
T – Temperature (K)
u> – Gas superficial velocity (m.s-1)
V – Catalyst volume in fluidized bed (m3)
V
,
– Catalyst Particle volume (m3)
w – Amorphous phase mass fraction (-)
W – Weight solids in fluidized bed (kg)
z – Compressibility factor (-)
Greek letters
∆H – Heat of reaction (J/mol)
δ – Average gas fraction (-)
ε – bed porosity (-)
μ – Gas phase Viscosity (Pa.s)
ρ – Density (kg/m3)
τ – Average residence time (s)
Subscripts
b – Bulk (gas) phase
C – Catalyst
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d – dissolved in amorphous PE
Et – Ethylene
ICA – Induced Condensing Agent
in – Entering reactor
mf – Minimum fluidization
out – Exiting reactor
p – Particle
PE - Polyethylene
pol – Polymer
s – Solid phase
5. Associated Content
Supporting information
The gas-phase physical and thermal properties estimation has been elaborated.
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6. References
(1)
Soares, J. B. P.; McKenna, T. F. L. Polyolefin Reaction Engineering; WileyVCH, 2012.
(2)
Bragança, A.; Morschbacker, A.; Rubbo, E.; Miro, C.; Barlem, T.; Mukherjee, A.
Process for the Gas Phase Polymerization and Copolymerization of Olefin
Monomers. US 6864332 B2, 2005.
(3)
Alizadeh, A.; McKenna, T. F. L. Condensed Mode Cooling in Ethylene
Polymerisation: Droplet Evaporation. Macromol. Symp. 2013, 333, 242–247.
(4)
Namkajorn, M.; Alizadeh, A.; Somsook, E.; McKenna, T. F. L. Condensed Mode
Cooling for Ethylene Polymerisation : The Influence of Inert Condensing Agent
on the Polymerisation Rate. Macromol. Chem. Phys. 2014, 215, 873–878.
(5)
Alizadeh, A.; Namkajorn, M.; Somsook, E.; McKenna, T. F. L. Condensed Mode
Cooling for Ethylene Polymerization: Part I. The Effect of Different Induced
Condensing Agents on Polymerization Rate. Macromol. Chem. Phys. 2015, 216,
903–913.
(6)
Alizadeh, A.; Namkajorn, M.; Somsook, E.; McKenna, T. F. L. Condensed Mode
Cooling for Ethylene Polymerization: Part II. From Cosolubility to Comonomer
and Hydrogen Effects. Macromol. Chem. Phys. 2015, 216, 985–995.
(7)
Kanellopoulos, V.; Mouratides, D.; Tsiliopoulou, E.; Kiparissides, C. An
Experimental and Theoretical Investigation into the Diffusion of Olefins in SemiCrystalline Polymers: The Influence of Swelling in Polymer-Penetrant Systems.
Macromol. React. Eng. 2007, 1, 106–118.
(8)
Hutchinson, R. A.; Ray, W. H. Polymerization of Olefins through Heterogeneous
Catalysis. VIII. Monomer Sorption Effects. J. Appl. Polym. Sci. 1990, 41, 51–81.
(9)
Davidson, J.; Harrison, D. Fluidized Particles; Cambridge University Press,
1963.
(10)
Kunii, D.; Levenspiel, O. Fluidization Engineering, 2nd editio.; Reed Publishing
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