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Mathematical Modeling,
Optimization, and Quality
Control of High-Pressure
Ethylene Polymerization
Reactors
a
a
Costas Kiparissides , George Verros & John F.
Macgregor
b
a
Department of Chemical Engineering, Chemical
Process Engineering Research Institute Aristotle
University of Thessaloniki, P.O. Box 472,
Thessaloniki, 54006, Greece
b
Department of Chemical Engineering,
McMaster University Hamilton, Ontario, L8S 4L7,
Canada
Published online: 23 Sep 2006.
To cite this article: Costas Kiparissides , George Verros & John F. Macgregor
(1993): Mathematical Modeling, Optimization, and Quality Control of HighPressure Ethylene Polymerization Reactors, Journal of Macromolecular Science,
Part C, 33:4, 437-527
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J.M.S-REV.
MACROMOL. CHEM. PHYS., C33(4), 437-527 (1993)
Mathematical Modeling,
Optimization, and Quality
Control of High-pressure
Ethylene Polymerization
Reactors
'
COSTAS KIPARISSIDES and GEORGE VERROS
Department of Chemical Engineering
Chemical Process Engineering Research Institute
Aristotle University of Thessaloniki
P.O. Box 472, Thessaloniki 54006, Greece
JOHN F. MacGREGOR
Department of Chemical Engineering
McMaster University
Hamilton, Ontario L8S 4L7, Canada
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
438
1.1. High-pressure LDPE Process Technology. . . . . . . . . . . . . . 440
1.2. LDPE Reactor Modeling: Literature Review. . . . . . . . . . . 441
2. REACTION KINETICS AND RATE FUNCTIONS.. . . . . . . . .
2.1. Kinetics of Ethylene Polymerization. . . . . . . . . . . . . . . . . . .
2.2. Polymerization Rate Functions. . . . . . . . . . . . . . . . . . . . . . .
2.3. Kinetics of Ethylene Copolymerization. . . . . . . . . . . . . . . .
2.4. Copolymerization Rate Functions. . . . . . . . . . . . . . . . . . . . .
~
'To whom correspondence should be addressed.
437
Copyright @ 1993 by Marcel Dekker, Inc.
443
443
451
457
459
KIPARISSIDES, VERROS, AND MacGREGOR
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438
3. THERMODYNAMIC, PHYSICAL, AND TRANSPORT
PROPERTIES. . .
..........................
3.1. Ethylene Th
nd Transport Properties
3.2. Physical and Thermodynamic Properties of LDPE.. . . . . .
3.3. Thermodynamic and Transport Properties of Comonomers and
Solvents.
.......................
3.4. Calculation of the Thermodynamic and Transport Pr
of the Reaction Mixture.. . . . . . . . . . . . . . . . . . . . . .
465
465
47 1
472
475
4. MATHEMATICAL MODELING OF HIGH-PRESSURE LDPE
REACTORS
...........................
479
4. I . The Modeling of Tubular Reacto
. . . . . . . . . . . . . . 480
4.2. Comprehensive Tubular Reactor
. . . . . . . . . . . . . . . . 483
487
4.3. Simulation Results on Tubular Reactors
493
4.4. The Modeling of Vessel Reactors. . . . . . . . . . . . . . . . . . . . .
4.5. Comprehensive Vessel Reactor Model
. . . . 499
4.6. Simulation Results on Vessel Reactors. . . . . . . . . . . . . . . . . 501
5. SENSITIVITY ANALYSIS, OPTIMIZATION, AND
QUALITY CONTROL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1. Sensitivity Analysis. .
................
5.2. Optimization of LDPE Reactors.. . . . . . . . . . . . . . . . . . . . .
5.3. Multivariate Statistical Quality Control
...........
503
507
5 10
515
.........................
52 1
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
522
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
524
6. CONCLUSIONS.. .
1.
INTRODUCTION
Polyethylene (PE) is the most widespread polymer and also the most studied
by macromolecular scientists. In 1990, polyethylene world production was
estimated at approximately 25 x lo6 tonnes per year: 65% of this was lowdensity, made in high-pressure reactors, and 35 % was high-density
homopolymer and linear low-density polyethylene produced in low-pressure
reactors.
Density and degree of branching are the most important physical and
molecular characteristics of PE, respectively. PE of density ranging from 0.91
to 0.925 g/cm3 is classified as low-density polyethylene (LDPE). Mediumdensity polyethylene (MDPE) has a density in the range of 0.926 to 0.94
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
TUBULARTECHNOLOGY
439
VESSEL TECHNOLOGY
FIG. 1. Schematic representation of LDPE molecular structure.
g/cm3, and high-density polyethylene (HDPE) has a density in the range of
0.941 to 0.965 g/cm3. The density of PE is determined by the degree of shortchain branching (SCB). The lower the degree of SCB, the higher the density
of PE. Figure 1 shows schematically the chain structures of the various
polyethyleneproducts [ 11. It is interesting to note that the branching type (long
or short), functionality, shape, and the degree of branching distribution (DBD)
are strongly related to the polymerization process and reactor operating
conditions employed. Typical branching frequencies in LDPE are 10-40 SCB
and 0.3-3 LCB per one thousand backbone carbon atoms, respectively.
PE is commercially produced by both free-radical (high pressure) and ionic
(low pressure) addition ethylene polymerization processes. The free-radical
high-pressurepolymerization processes essentially employ two types of reactors:
tubular and stirred autoclave. Ethylene free-radical polymerization is conducted
at very high pressures (1000-3500 atm) and high temperatures (140-330 “C)
in the presence of free-radical initiators such as azo compounds, peroxides,
or oxygen. Under the reaction conditions employed in high-pressure processes,
LDPE is produced as a result of short-chain branching formation.
Low-pressure ionic ethylene polymerization processes have been developed
more recently for the production of MDPE, HDPE and “linear low-density
polyethylene,” LLDPE. Ionic ethylene polymerization is carried out at relatively
low pressures (8-80 atm) and temperatures less than 150°C using a transition
metal catalyst of the Ziegler-Natta or Phillips type. Developments in transition
metal catalyzed ethylene polymerization have been described in a review paper
by Choi and Ray [ 2 ] .Today, low-pressure polyethylene is produced by three
polymerization processes: 1) solution process, 2 ) suspension (liquid slurry)
440
KIPARISSIDES, VERROS, AND MacGREGOR
process, and 3) gas-phase process. LLDPE with a wide range of densities
(0.88-0.95 g/cm3) is produced in low-pressure polyethylene reactors by
regulating the amount of an a-olefin comonomer.
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1.l. High-pressure LDPE Process Technology
A high-pressure process includes three units: 1) the compression unit, 2)
the reactor@),and 3) the product separation system [3]. A tubular LDPE reactor
consists of a spiral-wrappedmetallic pipe with a large length-to-diameter ratio.
The total length of the reactor ranges from 500 to 1500 m while its internal
diameter does not exceed 60 111111. The heat of reaction is partially removed
through the reactor wall by a heat transfer fluid which flows through the reactor
jacket. Only approximately one-half of the heat of reaction is usually removed
through the reactor wall. This results in a nonisotherrnal reactor operation.
In relation to the heat requirements of the process, the reactor can be divided
into a number of zones, including a preheating zone, the reaction zones, and
the cooling zones. The conversion achievable with this technology ranges
between 20 and 35% per pass. The polymer produced in these reactors can
have a density ranging from 0.915 to 0.93 g/cm3 and a melt flow index
varying in the range of 0.1 to 150 g/10 minutes. A schematic diagram of a
two-zone LDPE tubular reactor is shown in Fig. 2. A commercial reactor line
may consist of 3-5 reaction zones and several cooling zones. The reactor usually
includes a number of monomer, initiator, and chain-transfer agent side-feed
points. The temperature and flow rate of each coolant stream entering a
reaction/cooling zone is used to control the temperature profile in the reactor.
Ethylene, a free-radical initiator system, and solvent@)are injected at the reactor
inlet. Additional amounts of ethylene, initiators, and chain transfer agents may
be fed along the reactor length.
A vessel reactor is a constantly stirred autoclave which operates under
controlled temperature and pressure conditions [3]. These reactors are usually
long vessels with length-to-diameter ratios as high as 20 to 1. In some cases
they are well agitated with a high degree of directional flow imposed, depending
on the product to be produced. The reactor may be subdivided into multiple
reaction zones. In this case it is called a “multizone vessel.” Reaction conditions
(i.e., temperature, pressure, initiator concentration, etc.) can be adjusted
separately in each zone to give polymers of a wide molecular-weight range
[3]. LDPE resins produced in vessel reactors are more hazy than those produced
in tubular reactors. However, LDPE resins produced in autoclave reactors are
more suitable for extrusion coating and molding applications.
A schematic diagram of a typical autoclave reactor is shown in Fig. 3. The
reactor is separated into three zones and is provided with a vertical stirrer shaft.
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
at
I
ZONE 1
ZONE 2
441
+
FIG. 2. Schematic representation of a two-zone high-pressure tubular LDPE reactor:
(1) Reactor feed, (2) quenching stream, (3) and (5) coolant inlet, (4)and (6) coolant
outlet, and (7) initiator feed.
A low temperature initiator is fed to the first zone which is well agitated, and
a uniform temperature is maintained in the zone. In the second zone an
intermediate temperature initiator is fed. In this zone the end-to-end mixing
is reduced and a temperature gradient is established. Finally, in the third zone
a still higher temperature initiator is injected. This zone is well mixed to establish
and control the reactor exit temperature.
1.2.
LDPE Reactor Modeling: Literature Review
In the past 20 years, several mathematical models have been developed for
high-pressure LDPE reactors with varying degrees of complexity. Table 1
summarizes the main publications on the modeling of LDPE reactors.
As can be seen from Table 1, a large number of computer models have been
published in the open literature. These models provide a sound basis for the
mathematical description of commercial high-pressure LDPE tubular and vessel
reactors. However, it should be pointed out that careful consideration should
be given to the modeling assumptions in relation to a commercial process. In
particular, emphasis should be placed on the following aspects of a computer
model:
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442
KIPARISSIDES, VERROS, AND MacGREGOR
I
FIG. 3. Schematic representation of a three-zone high-pressure vessel LDPE reactor:
(1) Reactor feed, (3) quenching stream, and (2), (4), and (5) initiator feed.
1. Physical state of the reaction mixture (one-phase versus two-phase
system)
2. Kinetic mechanism and the selection (estimation) of the values of the
kinetic rate constants
3. Reactor flow conditions and mixing effects
4. Variation of the physical properties of the reaction mixture
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
443
A steady-state computer model consists of a set of nonlinear differential
equations (tubular reactors) or algebraic equations (vessel reactors) describing
the conservation of various molecular species, total mass, energy, and
momentum in the reactor. The model equations are usually coupled with a set
of algebraic equations describing the variation of kinetic, physical, and transport
parameters with respect to reactor operating conditions.
A sufficiently comprehensive model should permit calculation of monomer
conversion, initiator consumption, reaction temperature, the moments of radical
and polymer size distributions, the degree of long-chain and short-chain
branching, and the number of unsaturated double bonds in polymer chains as
affected by initiator concentration, temperature, pressure, concentration of chain
transfer agent, heat-transfer coefficient, and other design and operating variables
of the process.
In Section 2 of this review paper we deal in detail with free-radical ethylene
homopolymerization and copolymerization kinetics. The dependence of the
physical, thermodynamic, and transport properties of the reaction mixture on
reactor operating conditions (i.e., temperature, pressure, and composition) must
be known in any comprehensive modeling study. In addition to the variation
of these properties, appropriate expressions are needed for the calculation of
the overall heat transfer coefficient and friction factor in LDPE tubular reactors.
These topics are discussed in Section 3 of the paper. In Section 4, a unified
mathematical framework is developed for modeling tubular and vessel LDPE
reactors. Simulation results are presented to demonstrate the ability of these
models to predict molecular weight and other structural properties of PE in
high-pressure reactors. Finally, in Section 5 we examine the optimization,
sensitivity, and statistical quality control of high-pressure LDPE reactors.
2.
REACTION KINETICS AND RATE FUNCTIONS
2.1.
Kinetics of Ethylene Polymerization
The industrial importance of the high-pressure ethylene free-radical
polymerization process has led to very extensive studies of the kinetic
mechanism of the polymerization. A large number of papers, books, and patents
have been published on this subject: Ehrlich and Pittilo [34], Ehrlich and
Mortimer (351, Luft [36], Marano and Jenkins [37], Yamamoto and Sugimoto
[38], Goto et al. [13], Luft et al. [39, 401, Ogo [41], Beasly [42].
The free-radical ethylene polymerization mechanism includes the following
elementary reactions.
PFR
6. Lee and Marano [ l l , 121
PFR
PFR
PFR
Vessel
Vessel
PFR
PFR
8. Donati et al. [14, 151
9. Hwu and Foster [16]
10. Hollar and Ehrlich [I71
11. Marini and Georgakis [18, 191
12. Feucht et al. [20]
13. Gupta et al. [21]
14. Kiparissides and Mavridis [22, 231
PFRivessel
PFR
5. Chen et al. [lo]
7 . Goto et al. [13]
Vessel
Vessel
PFR
PFR/vessel
~
Reactor type
2 . Van der Molen and van Heerden [7]
3. Mercx et al. [8]
4. Agrawal and Han [9]
1. Thies and Schoenemann [4-61
References
Summary and comments
An excellent series of papers. Experimental and theoretical
results on x, T, M,, M,, LCB, SCB, and DB
Kinetics and initiator efficiency
Effects of residence time distribution on initiator productivity
Effect of axial mixing on the reactor performance. Prediction
of x, T, M,,, and M ,
Use of double moments to predict x , T, M,,, M, and LCB.
Variation of physical properties with reaction conditions
Prediction of molecular properties (i.e., M,, M,). Sensitivity
analysis of reactor performance with respect to operating
conditions
Computer model for vessel and tubular LDPE reactors. Comparison of experimental and theoretical values of x, ME,
M,, LCB, SCB, and DB
Effects of fluid pulsed motion on axial mixing, pressure drop,
and heat transfer
Prediction of reactor fouling using time-series analysis
Investigation of residual reaction in cooling zones. Prediction
of runaway conditions in LDPE reactors
Investigation of mixing phenomena in LDPE vessel reactors.
Prediction of initiator productivity and polymer quality
A detailed mathematical model on autoclave reactors. Prediction of molecular properties of LDPE
A comprehensive model on an LDPE tubular reactor. The effect of multiple intermediate feeds is investigated
Sensitivity analysis of product quality and reactor performance
with respect to operating conditions. The optimization of
tubular LDPE reactors is examined in the second publication
TABLE 1
High-pressure LDPE Reactor Models
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0
G)
8rn
(u
iz
0
z
b
-$
<
z
rn
P
v)
rn
r;
W
D
z
G
u,
ir
P
Vessel
PFR
PFR
PFR
PFR
PFR
PFR
Vessel
PFR
PFR
15. Villermaux et al. [24]
16. Yoon and Rhee [25]
17. Shirodkar and Tsien [26]
18. Brandolin et al. [27]
19. Azevedo and Howell [28]
20. Tilger and Luft [29]
21. Zabisky et al. [30]
22. Chan et al. [31]
23. Kiparissides et al. [32]
24. Verros et al. [33]
The shrinking aggregate and the IEM models are applied to
high-pressure vessel reactors to account for partial aggregation of initiator feed stream
The plug flow model includes the axial dispersion term. An optimal temperature policy which maximizes the exit monomer
conversion is determined
A computer model is developed to study the polymerization of
ethylene in a one- or two-zone tubular reactor. The sensitivity of product molecular properties to various process
variables is also investigated
A mathematical model for ethylene polymerization in a
multizone tubular reactor is proposed. The model allows
good prediction of x, M,,, M,, and LCB for different reactor
configurations
A second-order model is developed for high-pressure LDPE
tubular reactors including mass and thermal diffusion effects
A two-dimensional dynamic model is developed for a highpressure LDPE reactor. Variation of the physical properties
along the reaction coordinate is also considered
A copolymerization model for tubular reactors is proposed. The
model is used to simulate the operation of commercial
reactors
A copolymerization model for vessel reactors is developed.
Two-phase kinetics and gel formation from crosslinking reactions are taken into account. The model is used to simulate
the operation of commercial reactors
A comprehensive mathematical model is developed for the
homoploymerization of ethylene in a two-zone tubular reactor
with intermediate feed
A mathematical model based on double moments is employed to
calculate the molecular weight and compositional changes for
the copolymerization of ethylene in a two-zone tubular LDPE
reactor
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I]
g
8
P
R
20
RD
z
5
5
N
5a
P<
'c1
z
m
r
<
rn
-I
I
a
m
C
0,
u)
m
KIPARISSIDES, VERROS, AND MacGREGOR
446
1. Initiation (peroxides, azo compounds, or oxygen):
O2
- 2R';
kd02
+ MI
I
kd
2R'
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2. Chain initiation reactions:
R'
-
+ MI
kI 1
Rl
3. Propagation:
R,
kP
--Rx+l
+ Ml
4. Termination by combination:
Rx
ktc
+ Ry
Dy+x
5. Termination by disproportionation:
R,
-D, + D,
+ Ry
krd
6 . Chain transfer to monomer:
R,
+ MI
ktm
D,
+ RI
7. Chain transfer to solvent or chain transfer agent:
R,+S
kts
D,+R'
8. Chain transfer to polymer (intermolecular transfer):
R,
+ D,
ktP
Dx
+ Ry
9. Intramolecular transfer (backbiting):
R,
kb
R,
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
447
10. Scission of radicals:
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11. Retardation by impurities (or oxygen):
R,
+ impurities (0,)
kr
D,
12. Decomposition of ethylene:
-
2C2H4
C2H4
kdec
kdec
2C
2C
+ 2CH4 + heat
+ 2H2 + heat
where the symbols R, and D, denote “live” radicals and “dead” polymer
chains of chain length x , respectively.
2.1.1. Initiation
The initiation process in free-radical ethylene polymerization is much like
other vinyl polymerizations when common free-radical generators such as
peroxides and azo compounds are used to initiate the polymerization.
Buback [43] studied the thermal initiaton of ethylene. His experimentalresults
on pure ethylene, carried out at temperatures of 180 to 250°C and pressures
up to 2500 atm, showed that a very slow thermally initiated reaction to high
molecular weight PE could be established. The actual mechanism is not known
but it can be expressed as an overall third-order reaction:
In general, the rate of thermal initiation will be lower than the corresponding
rate obtained by chemical initiation.
Oxygen has been a traditional initiator for the high-pressure PE process.
However, the mechanism by which oxygen initiates the formation of radicals
does not appear to be well understood. In general, oxygen initiation is considered
as a multistep process where at low temperatures the rate-controlling step is
a reaction of oxygen with ethylene to form peroxides. The peroxides formed
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448
KIPARISSIDES, VERROS, AND MacGREGOR
can subsequently generate normal chain radicals which initiate the
polymerization. At high temperatures the radical chain initiation reaction
becomes the rate-controlling step.
The kinetics of oxygen-initiated polymerization of ethylene at high pressures
up to 2200 atm and temperatures between 60 and 250 “C was investigated by
Tatsukami et al. [44]. They found that above temperatures of 19O”C, no
induction period exists in the polymerization. The rate equations for oxygen
and monomer consumption were derived by considering a retardation by oxygen
reaction in addition to initiation, propagation, and termination reactions.
2.1.2. Short-Chain Branching
Intramolecular chain transfer produces short-chain branches by Roedel’s
[45] “backbiting” mechanism, according to which the growing radical curls
back on its own chain, occasionally transferring the radical to the third or fifth
carbon from the growing end.
The formation of short-chain branches in PE has received considerable
attention. Willbourn [46] used infrared and mass spectroscopic analysis of model
compounds and found that LDPE contained ethyl and butyl short-chain branches
at a ratio of 2 : 1 in favor of the ethyl branches. Similar studies have been reported
by Dorman et al. [47], Randall [48], Bovey et al. [49], and Cudby and Bunn
[50]. All investigators agree that the principal type of short branching in LDPE
is n-butyl and ethyl, with possibly n-amyl and n-hexyl in smaller proportions.
Ethyl branches are also believed to be present and could be accounted for by
a second backbiting reaction of the branched polymer radical formed during
the first backbiting reaction.
Short-chain branching is well known to be particularly critical in its effects
on the morphology and solid-state properties of semicrystalline PE. LDPE
molecules can contain 10-40 SCB per thousand carbon atoms. Short-chain
branching controls the density of LDPE and its crystalline melting point. The
effects of synthesis conditions on the SCB of LDPE and its crystalline melting
point have been investigated experimentally by Luft et al. [51]. They reported
that SCB increases with increasing temperature and decreases with increasing
pressure. On the other hand, density and crystalline melting point decrease with
increasing SCB.
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
2.1.3.
449
Lang-Chain Branching
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Long-chain branches in LDPE are formed by an intermolecularchain transfer
reaction. Long-chain branching (LCB) probably arises from abstraction by a
growing radical of a hydrogen atom from the backbone of a polymer chain,
followed by monomer addition to the new radical site.
Long-chain branching has been identified experimentally in LDPE, and it
is mainly responsible for the broad MWD and its rheological behavior (i.e.,
solution viscosity, viscoelastic properties) (Mullikin and Mortimer [52] ; Small
[53,54]). In measuring LCB, such methods as size exclusion chromatography
(SEC), viscosity measurements, and C-13 NMR have been utilized. In
particular, SEC coupled with automatic viscometry or low-angle laser light
scattering (LALLS) measurementsappears to be the most suitablemethod. Since
1953 a great deal of work has been directed toward the estimation of LCB in
LDPE. The more important studies on LCB in LDPE have been summarized
by Yamamoto [%I. Luft et al. [5 11 reported the effects of synthesis conditions
on LCB. In general, LCB increases with increasing temperature and decreases
with increasing pressure.
2.1.4.
Formation of Unsaturated Structures
In general, for the formation of the vinyl groups (-CH=CH2) the
following elementary reactions can be considered: 1) termination by disproportionation, 2 ) chain transfer to monomer, 3) &scission of sec-radicals.
However, we can assume that the rate of formation of vinyl groups by &scission
reactions will be higher than the rates of formation by termination and transfer
to monomer reactions.
Note that when an a-olefin such as propylene is used as a solvent, vinyl groups
can also be formed by a transfer to solvent reaction.
450
KIPARISSIDES, VERROS, AND MacGREGOR
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Similarly, the formation of vinylidene groups (>C=CH2) can be explained
by the scission reaction of tertiary radicals:
The formation mechanism of the trans-vinylene groups (-CH=CH-)
has
not been sufficiently clarified. Holmstrom and Sorvic [56]considered that the
reactions 1) p-scission of sec-radicals that branch at the a-position, 2) ally1
migration of vinyl groups, and 3) disproportionation of see-radicals explain
the formation of (-CH=CH-)
groups.
The trans-vinylidene content in LDPE is lower than that of the other two
unsaturated bonds. The total unsaturation per lo3 CH2 of any sample is
obtained by summing the contents of (-CH=CH2), (-CH=CH-),
and
(>C=CH2) determined by IR analysis. The total unsaturation content per lo3
carbon atoms in LDPE is usually less than 0.5.
It is unclear how important P-scission is to the determination of the molecular
weight distribution under usual polymerization conditions. The LCB and /?scission reactions compete, one building up molecular weight and the other
narrowing the high molecular weight tail.
2.1.5. Other Reactions
Control of molecular weight necessitates control of the amount of any material
that acts as a chain-transfer agent (CTA). In the commercial production of LDPE,
hydrocarbons, alcohols, ketones, and esters are usually employed as chain-transfer
agents. Note that the addition of small amounts of an inhibitor can have marked
effects on the free-radical polymerization of ethylene. For example, acetylene,
in amounts between 1.5 and 2.5 mol%, completely stops the polymerization.
Ethylene is known to undergo a highly exothermic decomposition at high
temperatures and pressures. It has been established that decompositionreactions
lead to the formation of carbon, hydrogen, and methane (Beady [42]). The
decomposition of ethylene is exothermic with an energy of activation of about
125 kJ/mol. Therefore, once initiated, it proceeds rapidly, consuming ethylene
and causing large temperature and pressure increases. Decomposition of
ethylene may be caused by hot spots in the reactor.
2. I.6. Kinetic Rate Constants
The dependence of the rate constants upon temperature and pressure is given by
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
45 1
where AE,AV, P, T, and R are the activation energy (cal/mol), the activation
volume (cm3/mol), pressure (atm), the absolute temperature (K), and the ideal
gas constant, respectively. Notice that a negative activation volume implies
that the corresponding rate constant increases with pressure. Typical values
of kinetic rate constants related to ethylene polymerization are listed in Table
2. It should be noted that the decomposition rate constants of peroxides will
also have activation volumes associated with them.
Although a great number of papers have been published on the modeling
of LDPE reactors, a consistent set of rate constants has not been established
in the open literature. This may be attributed to the complexity of the reaction
mechanism, the large number of kinetic parameters to be identified
experimentally, and the wide range of experimental conditions over which the
kinetic parameters are estimated. It should be pointed out that under normal
experimental conditions the absolute values of kp and kr cannot be obtained.
Therefore, while most investigators agree on the value of the kp/kp5
parameter, the reported values for kp and k, show a large variation. This means
that one of the two parameters must be estimated by another independent
method. Indeed, Takahashi and Ehrlich [57] and Luft et al. [39] obtained
absolute estimates of propagation (kp)and termination (kJ rate constants using
the rotating sector method. The problem of estimation of kinetic rate constant
is also discussed in Section 4.3 of this review.
Detailed kinetic information on high-pressure polymerization of ethylene
is given in the articles of Ehrlich and Mortimer [35],Luft and coworkers [39,
401, Goto et al. [13], and Lorenzini et al. [58, 591. The most complete set of
reaction constants has been reported by Goto et al. [13]. The reported values
were estimated from experimental measurements on monomer conversion,
number- and weight-average molecular weights, amount of unsaturated double
bonds, and total methyl content per lo3 carbon atoms. The measurements were
obtained from an autoclave reactor operated under typical industrial conditions.
2.2.
Polymerization Rate Functions
To describe the conservation of various molecular species present in a
reactor, we need to know their corresponding net production rates. The
expressions for these rate functions can be obtained by combining the various
elementary reactions describing the generation and consumption of initiator(s),
monomer(s), solvents, and “dead” and “live” macromolecules. Let r, and
r,* denote the net rate of production of “dead” and “live” polymer chains
Agrawal et al. [9]
Chen et al. [lo]
Lee and Marano [ll, 121
Goto et al. [13]
Donati et al. [14, 151
Feucht et al. [20]
Gupta et al. [21]
Shirodkar and Tsien [26]
Brandolin et al. [27]
2.2 x loLo
1.6 x lo9
1.075 X lo9
8.33 x 10'
3.1 X 10'
9.7 x los
1.6 x lo9
2.8 X 10'
3.0 X 10'
7800 + 0.5P
709 1
7099.5 - 0.556P
10520 - 0.447P
6164 - 0.6P
8880
7091
7769 - 0.52P
5245
1.25 X 10'
2.95 x 10'
5.887 x lo7
1.56 X 10'
3.1 x lo4
4.8 x 10'
2.95 x lo7
5.8 x lo7
1.0 x lo6
kldl
(L/gmol/s)
EP
(caligmol)
(Llgmolis)
krdO
+
-
-
-
-
0
720
0.121P
(cal/gmol)
Erd
Termination by disproportionation
loo0 0.244P
2400
298.05 - 0.3398P 3.246 X 10'
3000 0.3148P
750
720
9.7 x 10'
2400
1.3 x 10'
298 + 0.0243P
3950
+
E*c
(cal/gmol)
Termination by combination
kpo
(L/gmol/s
Propagation
Kinetic Constants Related to Free-Radical Polymerization of Ethylene
TABLE 2
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14080 + 0.1065P
4.86 x 10'
1.7 X lo6
9.0 x lo5
7.5 X lo6
4.4 x lo6
Goto et al. [13]
Donati et al. [14, 151
Feucht et al. [20]
Gupta et al. 1211
Shirodkar and Tsien [26]
Brandolin et al. [27]
8492 - 0.038P
9500
9000
4680
-
9Ooo
7704.11 - 0.484P
9 x 105
4.116 X lo5
Agrawal et al. [9]
Chen et al. [lo]
Lee and Marano [l 1, 121
-
EP
(cal/gmol)
kPQ
(L/gmol/s)
Chain transfer to polymer
5.823
X
lo5
(L/gmol/s)
knd
-
11050 - 0.484P
-
-
(cal/gmol)
Em
Chain transfer to monomer
6.445 X lo6
3.41 x 10'
3.306 X lo7
kI3a
(L/gmol/s)
-
-
9400
(continued)
10032 0.484P
12820 0.4722P
E*
(cal/gmol)
Chain transfer to solvent (n-hexane)
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4.6 X lo6
2.95 X lo8
1.3 x lo9
1.56 x lo9
-
5500
9417
9935
-
13030 - 0.569P
-
-
7.3
X
lo6
2.36 X lo7
11315
-
b
-
-
-
-
a
1.61 X 10'
kB*,O
(s -3
15760 0.5473P
EB"
(cal/gmol)
p-Scission of tert-radicals
-
14530 - 0.447P
-
4'
(cal/gmol)
2.72 x 10" 2oooO
6 -9
kB,O
P-Scission of sec-radicals
"kB.= 2.315 x 1022exp(-33576/RT)/{8.51 X 1010exp(-13576/Rl) + 5.821 x 101'exp(-14665/RT)}.
bkB. = 1.583 X 1Ouexp(-34665/RT)/{8.51 X 1010exp(-13576/RT)+ 5.821 x lO"exp(-14665/RT)}.
D o ~ teti al. [14, 151
Feucht et al. [201
Gupta et al. [21]
Shir0dka.rand Tsien [26]
Brandolin et al. [27]
Agrawal et al. [9]
Chen et al. [lo]
LeeandMarano[ll, 121
Goto et al. [13]
Eb
(cal/gmol)
km
(s-9
Intramolecular chain transfer
TABLE 2. Continued
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
455
with a degree of polymerization x , respectively. Based on the kinetic mechanism
of free-radical polymerization of ethylene described in the previous section,
the following general composite rate functions for rx and r,” can be derived:
2Jkdcd
+
(k,~,
+
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i= 1
+ k,C,[R(x
- 1) - R(x)]
m
m
x= 1
L
x=2
i=l
x=l
’
It should be noted here that, for modeling purposes, it is not practical to solve
the resulting infinite system of differential equations describing the conservation
of macromolecular species in the reactor. As a result, one has to resort to modeling techniques such as the method of moments (MM), the instantaneous property
method (IPM), and the property moment method (PMM) to obtain information
on the polymer quality. In recent articles by Konstadinidis et al. [60]and Achilias
and Kiparissides [611, these modeling methods are reviewed in detail. The method
of moments is based on the statistical representation of the molecular properties
of interest (e.g., M,,, M,) in terms of the leading moments of the respective
distributions (Arriola [62]). Accordingly, the leading moments of the total number
chain length distributions (TNCLDs) of “live” and “dead” polymer chains are
defined as
m
m
x=l
x=2
KIPARISSIDES, VERROS, AND MacGREGOR
456
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where R(x) and D(x) denote, respectively, the concentrations of “live” and
“dead” polymer chains of length x. Accordingly, one can define the
corresponding moment rate functions of the total number chain length
distributions of “dead” and “live” polymer chains by multiplying each term
of Eqs. ( 2 ) - ( 3 ) by x n and summing the resulting expressions over the total
variation of x:
n
i=O
+
To break down the dependence of the n-moment rate function on the (n
1) moment, Lee and Marano [l 1, 121 noticed that the sums of the moment
(r,) 1} and { ( rh)2 ( r p ) 2 }were only dependent on
rate functions { ( rA)l
the zeroth and first moment of the live radical distribution. By assuming p1
= A,
p I and p2 = A2
p2, they were able to express the reaction rates
for b,XI,po, (Al + p l ) , and (A,
p2) in a closed form. Following Lee and
Marano’s approach, Eq. (6) can be further simplified to
+
+
+
+
‘
1=0
+
i=l
/
c
n
+ ( 1 / 2 ) k,,
i=O
(?)&Anpi,
n = 1, 2
(7)
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
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2.3.
457
Kinetics of Ethylene Copolymerization
At high pressures and temperatures, ethylene will undergo free-radical
copolymerization in the presence of a comonomer such as vinyl acetate, methyl
acrylate, ethyl acrylate, acrylic acid, and methacrylic acid. The reactivity ratios
of ethylene with various comonomers are given in Table 3 (Beady [42]). Note
that both reactivity ratios of ethylenehinyl acetate (EVA) are approximately
equal to 1 (rl = r2 = 1). This means that EVA with constant composition
can easily be produced in either vessel or tubular reactors.
Comonomers can also promote transfer to monomer reactions, thus reducing
the molecular weight of the polymer. When a-alkenes are employed, their
transfer activity combined with a much lower propagation rate tend to limit
the amount of comonomer that can be incorporated into the copolymer.
A fairly general kinetic mechanism describing the free-radical copolymerization includes the following elementary reactions.
1. Initiation (by peroxides or azo compounds):
I
kd
2R
2. Chain initiation reactions:
R'
+ M,
klj
R$-j,j-];
j = 1, 2
3 . Propagation reactions:
Ri,q
+ Mj
kpij
Ri+2-j,q+j-l;
i = I , 2 and j = 1, 2
4. Chain transfer to monomer reactions:
5 . Chain transfer to solvent (chain transfer agent) reactions:
6. Chain transfer to polymer:
KIPARISSIDES, VERROS, AND MacGREGOR
458
TABLE 3
Reactivity Ratios of Ethylene with Various Comonomers
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Comonomer
Propylene
Butene- 1
Vinyl acetate
Acrylic acid
Methacrylic acid
Methyl acrylate
Ethyl acrylate
RS.4
+ DXJ
-
kpl2 lkpl 1
kp2 1 4 7 2 2
3.1 f 0.2
3.4 f 0.3
1.07 0.06
0.02
0.1
0.05
0.04
0.77 f 0.05
0.86 f 0.02
1.09 f 0.02
4
6
8
15
.
ktpij
+ Dp,q;
RJX,,
Pressure
(arm)
Temperature
("C)
1030-1720
1030-1720
1010
1180-2070
140-226
1380
2070
130-152
180
i = 1, 2 a n d j = 1, 2
7. Termination by disproportionation:
Rk,q
+ Ri,,
ktdij
Dp,q
+ D,,,;
i = 1, 2 and j = 1, 2
8. Termination by combination:
Rb.q
+ Ri,,
k,
-
Dp+x,q+y; i = 1, 2 a n d j = 1, 2
9. Intramolecular transfer (short-chain branching):
Ri,q
- Rk,q or R{,q;
kbi
i = 1, 2
10. &Scission of see- and tert-radicals:
R6,q
- DP7q + R';
kpi
i = 1, 2
In the above mechanism the subscript i stands for the ethylene (i = 1) and
the comonomer (i = 2), and the superscripts refer to the ultimate monomer
unit in the polymer chain. The above mechanism is sufficiently general and
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
459
describes most high-pressure ethylene copolymerizations. Besides initiation and
propagation reactions, it includes termination by both combination and
disproportionation, molecular weight control by transfer to monomer and
modifier, long-chain branch formation by transfer to polymer, short-chain
branch formation by intramolecular transfer, and double bond formation by
&scission. In the above kinetic mechanism it is assumed that no depropagation
reactions occur and the penultimate effect is negligible.
2.4.
Copolymerization Rate Functions
To identify a copolymer chain, we introduce a general notation Gp,qwhich
denotes the concentration of “live” or “dead” polymer chains having p units
of monomer 1 (M,) and q units of monomer 2 (M2) in a polymer chain. It
should be noted that the ultimate monomer unit in a “live” copolymer chain
can be of either the MI or M2 type. As a result, two different symbols, P and
Q, are introduced to identify the live copolymer chains ending in an M1 or
an M2 monomer unit, respectively.
Let rGbe the polymerization rate of various species present in the reaction
mixture [i.e., initiator(s), monomer(s), solvent(s), “live” polymer chains of
type P or Q and “dead” polymer chains, D]. These rate functions can be
obtained by combining the rates of the various elementary reactions describing
the generation and consumption of “live” and “dead” copolymer chains based
on the general kinetic mechanism of ethylene copolymerization described above.
For simplification, we choose to work with the bivariate number chain length
distributions (NCLDs) of the polymer chain populations, P@,q), Q(p,q) , and
D(p,q). Accordingly, we write the following generalized expressions describing
the net rates of appearance/disappearanceof individual molecular species [33,
611.
Initiator consumption rates:
Primary radical formation rate:
Monomer(s) consumption rate (propagation rate) :
KIPARISSIDES, VERROS, AND MacGREGOR
460
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Net formation rate of “live” polymer chains:
- ktp2iQ@jq)
m
m
p=l
q=l
C C qD@,q)
Net formation rate of “dead” polymer chains:
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
46 1
and Poo and Qoo denote the concentrations of “live” polymer chains of type
“p,, and ‘Q,” respectively:
om
p=l
q=l
p=l
q=l
Based on the above definitions of rate functions and the fundamental reactor
design equation for a plug-flow reactor (PFR) or a continuous stirred tank reactor
(CSTR), one can derive a low-order system of molar balance equations using
the method of moments. This system of differential equations can be solved
numerically to obtain desired information on molecular weight and
compositional developments in a high-pressure copolymer reactor.
The leading moments of the bivariate number chain length distributions of
“live” and “dead” macromolecules can be defined as (Arriola [62], Achilias
and Kiparissides [61])
m
w
p=l
q=l
m
m
m
m
p=l
q=l
Accordingly, one can obtain the corresponding rate functions for the moments
of the bivariate number chain length distributions of “dead” and “live” polymer
KIPARISSIDES, VERROS, AND MacGREGOR
462
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chains by multiplying each term of Eqs. (12)-(14) by the term pmqnand
summing the resulting expressions over the total variation of p and q:
i=O
k=O
m
n
j=O
k=O
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
463
It should be pointed out that when transfers to polymer reactions are included
in the kinetic mechanism, the n-order polymer moment equations will depend
on the (n 1)-order moments. This is due to the fact that the polymerization
rate function for the transfer to polymer reaction depends on the total degree
of polymerization, x . To break down the dependence of the moment equations
on higher order moments several closure methods have been proposed [ 11,
30,631. The closure method of Hulburt and Katz [63] has been used in several
model developments. This technique assumes that the molecular weight
distribution can be represented by a truncated (after the first term) series of
Laguerre polynomials by using a gamma distributionweighting function, chosen
so that the coefficients of the second and third terms are zero. By assuming
that the first three terms of the Laguerre polynomials are sufficient for
representing the molecular weight distribution, Hulburt and Katz derived the
following approximation for the third moment, p 3 :
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+
P3 =
P2
- (2P2PO
PlPO
- P:)
Assuming that the molecular weight distribution of polyethylene produced
in high-pressure tubular reactors follows a log-normal distribution, Zabisky
et al. [30] proposed an alternative approximation for the third moment, 1.13:
However, a comparison of model predictions with experimental results [30]
showed that only the geometric mean of Eqs. (24)-(25) was in satisfactory
agreement with the experimental data.
Lee and Marano [ l l , 121 proposed an alternative way to break down the
dependenceof the “dead” polymer moment equationson higher order moments.
By adding Eqs. (21) and (22) to the corresponding moment rate function of
“dead” polymer chains, Eq. (23), one can obtain the following expression
for rPmnwhich is independent of the higher order moments:
KIPARISSIDES, VERROS, AND MacGREGOR
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464
n
m
n
k=O
j=O
k=O
m
n
j=O
k=O
The other rates of interest will be given by the following expressions.
Long-chain branching formation rate:
rLcB = ktpiiGopio
+
k t p i 2 Go p 0 1
+
ktp2ih%p10
+ ktp22~%p01
(27)
Short-chain branching formation rate:
rSCB
= kblh& -k kb2h%
(28)
Rate of @-scissionof sec-radicals:
rp, = kp,lh&
+ kp,J$
(29)
Rate of &scission of rerr-radicals:
rot, = kp-,X&,
+ k,.,h&
(30)
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
THERMODYNAMIC, PHYSICAL, AND TRANSPORT
PROPERTIES
3.
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465
The dependence of thermodynamic and transport properties of the reaction
mixture (i.e., density, specific heat, viscosity, thermal conductivity)on pressure,
temperature, and composition must be known in any comprehensive modeling
study. Furthermore, the values of the overall heat transfer coefficient and the
friction factor must be computed at each point along the reaction length. In
what follows, a detailed discussion on the calculation of the thermodynamic
and transport properties of the reaction mixture is presented.
Ethylene Thermodynamic and Transport Properties
3.1.
3.1.1.
SpeciJic Volume of Ethylene
According to Benzler and Koch [64], the reduced volume of the ethylene
gas phase can be related to the reduced temperature (T, = T/TJ and pressure
(P, = P / P J by
P, = u
+ b(Tr - 1 ) + c ( T , - 1 ) 2
(31)
The coefficients a , b, and c are functions of the reduced density ( p , = p / p c )
and will be given by: 1) for p r < 1:
a = 1 - (1
+ O.445pr)(p, -
1)4
+ 1 . 4 4 8 ~ ~0.603~;)
-6.55~; + 2.077p:(4.31
p,)
b = 3.555(1
c =
2) for p r
-
> 1:
a = 1
+ p;(p,
- 1)4[1.331 - O.692(pr
1)
b = 6.55 i- P ; ( p , - 1)[7.4 - 2.8(pr - 1)
+ 1.282(pr - 1 ) 2 - O.312(pr - 1)31
c = 16.65 + 3 0 . 2 2 ~-~ 15.01~; + 1.6~:
+ 0.126(pr
KIPARISSIDES, VERROS, AND MacGREGOR
466
The above equations can be solved numerically by a Newton-Raphson
routine to calculate the value of reduced density for given values of T, and Pr.
In Fig. 4 the specific volume of ethylene is plotted with respect to temperature
at different pressures.
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3.1.2. Specific Enthalpy of Ethylene
Benzler and Koch [64] proposed the following equation for the estimation
of the specific enthalpy of ethylene:
+ %P r- 2 ( 1 -
+ ) j G Pr
d p r
where Ho is a constant reference enthalpy and C , represents the isochoric heat
capacity of an ideal gas, J/(kmol.K).
c,
=A
+ B exp ( - C I T D )
-R
(39)
whereA = 3.925E+04, B = 1.155E+05, C = 1.234E+03, D = 1.0977,
and R = 8.314E+03; J/(kmol-K). The expressions for a, band c will be given
by Eqs. (32)-(37). Finally, Pc and V, denote the critical pressure and critical
specific volume of ethylene, respectively. In Fig. 5 the specific enthalpy of
ethylene is plotted against temperature at different pressures.
3.1.3. Spec$c Heat Capacity of Ethylene
The isobaric and isochoric heat capacities of ethylene can be calculated by
Eqs. (40)-(41) (Benzler and Koch [64]):
The partial derivatives of VE with respect to temperature and pressure are
given by
( a v E / a p ) T 1= ( P ~ / V , ) [ U+~ b f ( T r - I )
+ C ~ T ,-
1 ) 2 /r ~ ] (42)
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
0.0024
467
1
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0 0022 -
0 0020
-
00018-
-
00016-
0.0014
1
3000
n
!
Temperature (K)
FIG. 4. Specific volume of ethylene versus temperature.
where a ’ , b ’ ,and c ’ denote the first derivatives of a, b, and c (see Eqs. 32-37)
and the partial derivative ( d P / d T ) will be equal to
In Fig. 6 the specific heat of ethylene is plotted against temperature at different
pressures.
3. I . 4. Ethylene Thermal Conductivity
The thermal conductivity of ethylene at high pressures can be calculated
from the Stiel and Thodos correlation (see Ref. 65) based on the corresponding-
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KIPARISSIDES, VERROS, AND MacGREGOR
200
-
100
-
200
2000
a m
2500
n
3000
300
n
400
500
600
700
Temperature (K)
FIG. 5. Specific enthalpy of ethylene versus temperature (reference conditions: 2000
atm, 300 K).
states principle. From data on 20 nonpolar substances, Stiel and Thodos
established the following analytical approximations:
( A - Ao)I'zs = (14.0 x 10-8)(e0.535pr- 1);
where: A = dense gas thermal conductivity
A' = low-pressure gas thermal conductivity
r
= ~:/6&fl12p-2/3
c
pr
< 0.5
(45)
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
-
087
Y
\
ul
-.
7
m
v
V
al
-
469
2000 atm
3000
n
c
al
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7
2
07-
4
W
e
0
-
n
d
0
m
a
m
u
0.6 -
c)
m
al
I
-0
e
0
al
a
UJ
-
05
I
I
1
I
T e m p e r a t u r e (K)
FIG. 6. Specific heat of ethylene versus temperature.
The low-pressure value of thermal conductivity, A', can be expressed by
Xo = 10-6(14.52Tr
- 5.14)2'3 ( c p l r ) , cal/cm.s.K
(48)
In Fig. 7 the thermal conductivity of ethylene as calculated by the Stiel-Thodos
correlation is plotted with respect to temperature at 2000,2500, and 3000 atm.
3. I . 5 Ethylene Viscosity
The viscosity of ethylene at high pressures can be calculated from the Stiel
and Thodos correlation reported in the textbook on R e Properties of Gases
andfiquids by Reid, Prausnitz, and Sherwood [65]. Stiel and Thodos established
the following correlation for nonpolar gases:
KIPARISSIDES, VERROS, AND MacGREGOR
4 500e-4 -
L
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4 000e-4 -
\
3 500e-4-
3 000e-4
2 500e-4
-
--t
2000 atm
2500
n
3000
n
2 000e-4
FIG. 7. Thermal conductivity of ethylene versus temperture.
[(q
- 7')E
where: q
'71
4
+
l]0.25= 1.023
+ 0 . 2 3 3 6 4 ~+~ 0.58533~;
-
0.40758~: +O.O9332p;
(49)
= dense gas viscosity
= low-pressure gas viscosity
=~ ~ / 6 ~ - 1 / 2 p ~ - 2 / 3
M = molecular weight
The low-pressure gas viscosity (q') for nonpolar gases can be calculated by
= 4.61Tr -2.04e-0.449Tr
+ 1.94e-4.058Tr+ 0.1
(50)
Note that the above correlation will be valid for values of p r in the range 0.1
< p r < 3. In Fig. 8 the viscosity of ethylene is plotted with respect to
temperature for different pressures.
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
--O-
471
2000 atm
.
e
-
0)
m
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0
2 00e-3
a
v
-
n
u
m
c
100e-3-
-na
S
d
W
I
0 OOe+O200
300
400
500
600
700
T e m p e r a t u r e (K)
FIG. 8. Ethylene viscosity versus temperature.
3.2.
Physical and Thermodynamic Properties of LDPE
The density of polymer can be calculated from Eq. (51) [66]:
pp
+
= (9.61 x 1 0 - ~ 7.0 x ~ o - ~-T5 . 3 x I O - ~ P ) - '
(51)
Bogdanovic et al. [67] calculated the thermodynamic properties of different
grades of polyethylene (i.e., linear and branched) using the Tait state equation:
where Vpo and V, represent the specific polymer volume at atmospheric
pressure and pressure P , respectively. C ( = 0.985) is an empirical constant
for the polymers considered in the study [67]. Bogdanovic et al. [67] found
that the parameter B can be expressed as
KIPARISSIDES, VERROS, AND MacGREGOR
472
B = bo exp (-b,T)
(53)
where the numerical values of bl and bo are given in Table 4. The specific
volume of polymer at atmospheric pressure (Vpo)is adequately represented by
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VPo = Cexp
@In
(54)
where the value of ul is also reported in Table 4.
To derive the other thermodynamic properties (i.e., internal energy,
enthalpy, and entropy), the approach of Maloney and Prausnitz [68] can be
followed. In Figs. 9 and 10, calculated values of specific volume and specific
enthalpy of PE, respectively, are plotted as a function of temperature at different
pressures.
The specific heat capacity of polyethylene is given by
Chen et al. [lo] gave the following approximation for the specific heat capacity
of polyethylene:
c,, = 1.041
+ 8.3
x 10-4T;
ca1ig.K
(56)
Finally, the thermal conductivity of LDPE can be assumed to remain constant
[4.0 x l o p 4 cal/(cm-s.K)] according to the work of Eiermann [69].
3.3. Thermodynamic and Transport Properties of Comonomers
and Solvents
For the prediction of the thermodynamic properties of comonomers and
solvents, a generalized thermodynamic correlation based on Pitzer's threeparameter corresponding states was employed.
Lee and Kesler (see Ref. 65) developed a method of representing analytically
the various thermodynamic properties based on Pitzer 's three-parameter
corresponding states principle. The properties include: densities, enthalpy
departures, entropy departures, fugacity coefficients, isobaric and isochoric
heat capacity departures, and the second virial coefficients.
To facilitate the analytical representation of the thermodynamic properties,
the compressibility factor of the fluid is expressed in terms of the compressibility
factor of a simple fluid z(O)and the compressibility factor of a reference fluid
z(') as follows:
Linear PE
Branched PE
Ultrahigh MW linear PE
179.04
179.45
170.53
b, (Mpa)
X
4.661
4.699
4.292
6 , (IiK)
lo’
0.9172
0.9399
0.8992
8.50
Vm (m’/kg) x
7.80
7.34
a , (l/K) x lo4
Numerical Values of the Constants in Eqs. (52)-(54)
TABLE 4
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5
N
5a
?
4
in
z
rn
TI
m
KIPARISSIDES, VERROS, AND MacGREGOR
474
000130
--O-
h
\
m
W
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n
0
-C
.
d
v
000120-
2000alm
2500
3000
x
3)
J
e
0
a
-:
>
-0
0
000110-
L
0
m
n
(0
000100
200
300
400
500
600
700
Temperature (K)
FIG. 9. Specific volume of LDPE versus temperature.
where w is the acentric factor of the fluid. By using a modified Benedict-WebbRubin equation of state, the compressibility factors (z ( r ) , z ( O ) ) are expressed
by the following equations:
30
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
200
300
400
Temperature
500
600
475
700
(KI
FIG. 10. Specific enthalpy of LDPE versus temperature (reference conditions: 2000
atm, 300 K).
The values of the b, c, d, /3, and y constants are given in Table 5. From &.
(58), one can easily derive expressions for the thermodynamicdeparture functions
of enthalpy, isochoric heat capacity, and isobaric heat capacity. Thus,for the calculation of a thermodynamic quantity at a given temperatureand pressure, one needs
to know the critical properties T,, P,, and o of the fluid of interest. The thermal
conductivity and viscosity of the various comonomers and chain-transfer agents
can be calculated by using expressions similar to those reported in Subsection 3.1.
3.4.
Calculation of the Thermodynamic and Transport
Properties of the Reaction Mixture
The specific volume, specific enthalpy, and specific heat as well as the
thermal conductivity of the reaction mixture can be calculated by a simple
addition rule.
KIPARISSIDES, VERROS, AND MacGREGOR
476
TABLE 5
Constants in the Lee-Kessler Equation of State [65]
Constant
b,
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4
b3
b4
CI
c2
c3
c4
d, x lo4
d2 x lo4
P
Y
c
Simple fluid
Reference fluid
0.1181193
0.265728
0.154790
0.030323
0.0236744
0.0186984
0
0.042724
0.155488
0.623689
0.65392
0.060 167
0.20266579
0.33151 1
0.027655
0.203488
0.03 13385
0.0503618
0.016901
0.04 1577
0.48736
0.0740336
1.226
0.03754
N
P, =
WjPi
i= I
where P,,, is the property of the mixture, wi is the weight fraction of the i
component, and PI denotes the corresponding property of the i component.
3.4.1. Solution Viscosity of Ethylene-LDPE
For the calculation of the solution viscosity of ethylene-LDPE, the work
of Ehrlich and Woodbrey [70] seems to provide the best available experimental
(semiempirical) approach. Ehrlich and Woodbrey measured the relative
viscosity of moderate concentrated solutions of LDPE in ethane experimentally
and reported the following correlation:
where
is the relative viscosity of the solution (ql,/qO,ethane)
is the viscosity of pure ethane
q s is the viscosity of the ethane-LDPE solution
[q] is the intrinsic viscosity of LDPE measured in p-xylene at
105 "C
[Cj represents the concentration of polymer in g/dL
qr,ethane
qO,ethane
HIGH-PRESSURE ETHYLENE POLYMERlZATlON REACTORS
477
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Ehrlich and Woodbrey found that the relative viscosity of LDPE in ethylene,
qr,eth,was related to the relative viscosity of LDPE in ethane, qr,ethane, by the
following simple equation:
Furthermore, they found that the relative viscosity of LDPE in ethane
depends on the temperature and concentration of LDPE:
qr,ethane
- qr,ethane,l5O0C exp
According to the work of Ehrlich and Woodbrey, the intrinsic viscosity [ q ]
of LDPE in p-xylene can be expressed in terms of the viscosity-average
molecular weight, M,, using the well-known Mark-Houwink correlation:
[q] =
2.347
X
10-3@556
(66)
Substituting Eqs. (64)-(66) into Eq. (63), we obtain the following
semiempirical correlation for the relative viscosity of LDPE in ethylene:
In qr,eth = 2.00
+ 0.912
x 10-3@556[Cj +
Assuming that M, can be approximated by the number-average molecular weight
of polymer (28pl/h), and the mass concentration of polymer can be expressed
in terms of the first moment of MWD ( p , ) , one can write Eq. (67) as follows:
E,
=
-500
+ 5 6 0 ~ ~ ; cal/gmol
(69)
KIPARISSIDES, VERROS, AND MacGREGOR
470
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It is important to point out that Ehrlich and Woodbrey carried out their
experiments at conditions similar to the industrial operating ones (i.e.,
temperature, 150-270 "C; pressure, 1400-2000 atm; LDPE concentration,
50-200 g/L). It is interesting to note that Chen et al. [lo] reported an expression
similar to Eq. (68) for the calculation of the relative solution viscosity of LDPE
in ethylene:
In Fig. 11 the solution viscosity of the reaction mixture as predicted by Eqs.
(68) and (70) is plotted with respect to the temperature.
3.4.2. Calculation of the Overall Heat Transfer Coeficient
The calculation of the overall heat-transfer coefficient may be based either
on the inside or outside diameter of the tube at the discretion of the designer.
Accordingly, the inside overall heat transfer coefficient, Ui, can be expressed
as
For the calculation of the inside film heat transfer coefficient, hi, typical
correlations for Newtonian fluids can be used. Thus, for fully developed
turbulent flow, hi can be obtained from the following empirical relation:
Nu
=
0.027Re0~8Pr0~33(q,/~~,,,)o~14
(72)
where Nu (= hiDi/X,) is the Nusselt number
Re (= psuDi/r,) is the Reynolds number
Pr (= cpsq-s/Xs)is the Prandtl number
All properties are evaluated at bulk temperature conditions of the fluid, except
the solution viscosity v,, which is evaluated at the wall temperature.
3.4.3.
Calculation of the Friction Factor
As discussed in Section 2.1 of this review, the kinetic rate constants vary with
pressure. It is therefore necessary to know the pressure at each point of the reactor. For the calculation of pressure drop in the reactor, the Fanning friction factor
must be known. The friction factor can be calculated by the following equations:
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
100
!
-
---e
01
In
\
7
0
a
v
-
-
eq ( 6 8 )
eq ( 7 0 )
10:
3
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479
.
a
In
0
0
In
7
>
S
0
c
4
3
1:
0
cn
W
a
0
1
1
100
200
I
I
300
400
T e m p e r a t u r e (‘C)
FIG. 11. LDPE solution viscosity versus temperature (pressure 2000 atm, C,,,, =
10 gIdL, M, = 45,000).
f = 16/Re;
Re < 2100
(1/fl1’’ = 4.0 log (Ref”’) - 0.4;
4.
(73)
2.1
X
lo3 < Re < 5
X
lo6
(74)
MATHEMATICAL MODELING OF HIGH-PRESSURE LDPE
REACTORS
If detailed information on the molecular structure of the polymer is required,
the reaction steps outlined in Sections 2.1 and 2.3 must be considered. The
need to include a reaction step in a kinetic model will depend on the final use
of the reactor model and the required information on the molecular properties
KIPARISSIDES, VERROS, AND MacGREGOR
480
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of the polymer. Needless to say, the number of kinetic parameters to be
determined experimentally increases as the complexity of the reaction
mechanism increases.
To model a high-pressure LDPE multizonc reactor, the following balance
equations have to be established:
1.
11.
...
111.
iv .
V.
vi.
vii.
...
v111.
ix.
A total mass balance for the reaction mixture
Molar balances for initiators, monomers, solvents (CTA)
Molar balances for “live” radicals
Molar balances for “dead” polymer
Molar balances for SCB and LCB
An energy balance for the reaction mixture
An energy balance for the cooling (heating) fluid
A momentum balance
Energy balances at the quenching points
In the following sections, the most common assumptions related to the modeling
of high-pressure tubular LDPE reactors are examined as well as the design
equations describing the operation of these reactors.
4.1.
The Modeling of Tubular Reactors
In Table I , a number of computer models for high-pressure LDPE tubular
reactors is listed. The most common modeling and computational assumptions
made in these models are
1. One phase flow
2. Plug flow conditions and absence of axial mixing
3. Stationary flow conditions and constant fluid velocity
4. Constant reactor pressure
5 . Constant wall temperature
6 . Quasi-steady-state approximation (QSSA) for radicals
7. Constant initiator efficiency
8. Rate constants are independent of viscosity
9. Negligible heat of reaction due to chain initiation, termination, and
transfer reactions
10. Constant physical properties
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
481
As can be seen, the number of assumptions employed in a computer
model of a high-pressure LDPE tubular reactor can vary considerably. A
model description can include as many details of the real reactor as present
knowledge of the system allows. However, it should be pointed out that
even the most detailed model description will largely depend on a number
of kinetic and physical parameters which are not always possible to measure
precisely.
A model builder should keep in mind that one must often compromise model
detail and complexity with available information and final use of the model.
Subsequently, an attempt is made to examine the validity of each of the modeling
assumptions outlined above.
4.1.1. Reactor Flow Conditions (assumptions I , 2, and 3)
One common assumption made by most investigators refers to the physical
state of the reaction mixture. Under many industrial operating conditions
a single ethylene-polyethylene (E-PE) phase exists. Figure 12 (Buback [43])
shows that an E-PE system is homogeneous in an extended pressure and
temperature region above 1500 bar and 150 "C, respectively. The critical
curve (CC) for the E-PE system separates the homogeneous region from
the two-phase region. Note that the precise form of the CC depends on the
molecular weight distribution. The melting pressure curve (MPC) of PE limits
the homogeneous region to lower temperatures. If separation of the polymer
phase from the ethylene phase occurs, a two-phase reactor model must be
considered. The process conditions which determine the existence of oneor two-phase system are discussed in the following references: Ehrlich 1711,
Bonner et al. [72], Harmony et al. [73], Bogdanovic et al. [74], and Walsh
and Dee 1751.
In tubular reactors the pressure is lowered periodically for a short time by
means of a control valve so that the corresponding increase in the velocity
tears away any polymeric deposits on the tube wall. Some investigators have
argued that this pulsed motion inside an LDPE reactor can be described by
a plug flow with axial mixing model. However, more recent studies (Donati
et al. 1141, Yoon and Rhee [25], and Tilger and Luft [29]) have shown that
the effect of axial mixing on polymerization is minor and can be neglected.
Under typical industrial flow conditions of very high Reynolds numbers, an
ideal plug flow model can be used to describe the reactor flow behavior.
Thies [6] investigated the influence of unsteady-state flow conditions on
temperature, conversion, and molecular properties. He reported that ethylene
KIPARISSIDES, VERROS, AND MacGREGOR
482
300 n
0
250 -
--
CCCUPE)
0
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U
m 200 L
=
4
a
L
m
a
E
l-
15010050
MPC
I
I
I
0
FIG. 12. Phase diagram for ethylene-polyethylene (Buback [43]).
conversion and product properties are not affected significantly by the process
dynamics, and he concluded that the stationary hypothesis for tubular LDPE
is justified.
Many investigators assume a constant flow velocity along the tube length.
However, velocity changes as the density of the reaction mixture, which depends
on pressure, temperature, and composition, varies along the tube length.
Therefore, the additional effect of variable velocity has to be considered in
a comprehensive reactor model.
4. I . 2. Reactor Operating Conditions (assumptions 4 and 5)
Most of the models published in the open literature do not take into account
the variation of pressure along the reactor. This variation is not negligible
(200-300 atm). It is well known that pressure has a profound effect on the
polymerization rate and molecular properties of LDPE. Therefore, in a
comprehensive reactor model, it will be important to calculate the pressure
profile along the reactor length.
Although the assumption of constant wall temperature is not well justified,
it has been extensively used by many investigators. The change of the wall
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
483
temperature can be modeled by including in the reactor model the appropriate
energy balances for the cooling (heating) fluid in the jacketed reactor.
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4. I. 3. Kinetic Assumptions (assumptions 6, 7, 8, and 9)
The molecular weight moment equations can be considerably simplified by
application of the quasi-steady-stateapproximation (QSSA) for ‘‘live” radicals.
The QSSA is accurate under many operating conditions and should be used
whenever possible. It should be noted that our computer simulations [23] have
shown that the errors resulting from the application of the QSSA in the computed
values of conversion, M,, and M,, are not significant.
In free-radical polymerization, the bimolecular termination step can become
diffusion-controlledat high monomer conversion. This process causes a lower
than expected rate of termination and a much higher rate of polymerization
than expected, especially at high monomer conversions. This diffusioncontrolled termination is caused by a decrease in the mobility of the polymer
chains due to the increase of the viscosity of the reaction medium. Goto et al.
[131 suggested that the termination rate was controlled by the segmental diffusion
of polymer chains, and they calculated a diffusion-controlledtermination rate
constant by considering the effect of excluded volume.
With respect to initiator efficiency, most investigators assume a constant
value. However, initiator efficiency can significantly vary with polymerization
temperature as van der Molen and van Heerden [7], Goto et al. [131, and Luft
et al. [76] pointed out. One way to model the variable initiator efficiency is
to include a side reaction leading to the deactivation of generated initiator
primary radicals in addition to the main primary radical production mechanism.
Notice that the heats of reaction of all but propagation reactions are usually
neglected.
4. I.4. Constant Physical Properties (assumption 10)
In general, this assumption will not be valid and the dependence of the
physical properties on the polymerization conditions must be taken into account.
This is discussed in detail in Section 3 of this review.
4.2.
Comprehensive Tubular Reactor Model
To describe the conservation of individualpolymer chains in a high-pressure
plug flow ethylene copolymerization reaction, an infinite set of partial
differential equations of the general form
KIPARISSIDES, VERROS, AND MacGREGOR
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484
is required. G denotes the various species present in the reaction mixture (i.e.,
initiator(s), monomer(s), solvents(s), “live” polymer chains of type P or Q
and “dead” polymer chains D). The indices p and q can vary from 1 to a very
high degree of polymerization (i.e., lo4). At steady-state the first term of Eq.
(75) is dropped. Application of the general design Eq. (75) to the various
molecular species present in the reactor results in a prohibitively large set of
differential equations which must be solved numerically to obtain information
on the desired molecular properties of polyethylene. In an attempt to reduce
the high dimensionality of the numerical problem, several mathematical
techniques have been developed to recast the “infinite” set of differential
equations into a lower order system which can be easily solved. These techniques
have been reviewed by Tirrell et al. [77] and include the use of generating
functions (Ray [78]), z-transforms (Mills [79]), and the continuous variable
approximation (Coyle et al. [80], Ellis et al. [81], Gonzalez-Romero and
Rodriguez [82], and Baillagou and Soong [83]). In the present paper a
comprehensivemathematical model for a multizone tubular reactor is developed
based on the general copolymerization mechanism described in Sections 2.3
and 2.4 and assumptions 1 , 2 , 7 , 8, and 9. Accordingly, the steady-state mass,
molar, energy, and momentum balances are written as follows.
Total mass balance:
(76)
Initiatior(s) molar balances:
d(Cd,u)/dx = -rdi;
i = 1, 2, ..., Ni
(77)
i = I, 2
(78)
Monomer(s) molar balance:
d(C,p)/dx
= -rpi;
Solvent(s) (CTA) molar balances:
(79)
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
Moment balances for ‘‘live” radicals:
d(X;,u)/du = r!mn
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d(X$,u)/& = rQ
hmn
Moment balances for “dead” polymer:
d(pmnU)/&
=
rpmn
Balance on long-chain branching:
~(CLCBU) =
~LCB
Balance on short-chain branching:
d ( CSCBU
= rSCB
Balance on double bonds:
d ( CVNLU) =
~VNL
d ( CVNDU)
/& =
~ V N D
Energy balance for fluid in the reactor jacket:
dT,/& = - TDU( T - T,) / (m,~,,)
Energy balance for the reaction mixture:
d(Tpc,u)/A = (-AHr,T)trpl
+
+
+
( - A H r , ~ f 2 r p 2 4U(T,.
( d In p-’ld In T ) , ( d P / & )
Momentum balance:
dP/& = -4fpu2/2D
At the quenching points, the temperature of the reaction mixture is lowered
by the addition of fresh ethylene (Fig. 2 ) . At these points, the following mass,
KIPARISSIDES, VERROS, AND MacGREGOR
486
molar, and energy balances must be established in order to calculate the initial
conditions at the inlet of the following reaction zone:
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Total mass balance:
where is the mass flow rate, Q is the volumetric flow rate, and p is the density
of the corresponding stream.
Component mass balances:
where wi is the weight fraction of the i component and Ciis the concentration
of the i species (Cmip Csi, Cdi, Pmn, CSCB,CLCB).
Energy balance:
h3H3 = mlH1
+ m2Hz
(92)
where His the specific enthalpy associated with a stream. Finally, the cumulative
monomer conversion, ycum,is calculated by
and Fm0,2
are the molar flows of monomer at the inlet of the first
where Fmo,l
and second reaction zone, respectively, and ym,land ym,2are the corresponding
fractional monomer conversions.
The molecular weight averages, the SCB and LCB per 1000 carbon atoms,
as well as the cumulative copolymer composition can be expressed in terms
of the moments of the bivariate NCLD of “dead” polymer chains as follows:
Number-average molecular weight:
Mtl = (MWlPlO
+ MW2POIYPCL00
(94)
Weight-average molecular weight:
(95)
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
487
Long-chain branches per 1000 carbon atoms:
(LCB/1000C)
=
+
5oOc~c~/(p10 poi)
(96)
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Short-chain branches per 1000 carbon atoms:
(SCB/lOOOC) = 5oOcs~~/(p10
+ pol)
(97)
Cumulative copolymer composition:
cc
=
~ 1 0 / ( ~ 1+0 Pol)
(98)
Vinylene double bonds:
(VNL/1000C) = 500[VNL]/(pol
+ pie)
Vinylidene double bonds:
(VND/1000C)
4.3.
=
SOO[VND]/(pol
+ plo)
Simulation Results on Tubular Reactors
One of the most difficult problems in simulating the operation of industrial
high-pressure LDPE reactors is the selection of appropriate values of the various
kinetic rate constants. In spite of the great number of papers published on the
modeling of LDPE reactors (see Table l), a consistent set of rate constants
has not yet been established in the open literature. Many investigators have
proposed full sets of kinetic rate constants obtained by fitting model predictions
to industrial or pilot-plant operating data (Shirodkar and Tsien [26], Brandolin
et al. [27]). However, by assuming that 1) the long-chain hypothesis (LCH)
and the quasi-steady-stateapproximation (QSSA) are valid and 2) the velocity
gradient is negligible, one can show (Verros et al. [33]) that the reactor model
equations will not depend on the absolute values of kinetic rate constants but
on the ratios “eg, defined as
”
KIPARISSIDES, VERROS, AND MacGREGOR
488
TABLE 6
Numerical Values of the c j Parameters"
k0
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(L/gmol/s
or s-')
2500
10-~
3.11
6.45 X
0.218
0.169
1.034
8
AE
(cal/gmol)
AV
(cm3/gmol)
7594
-26.2
-
-
3560
3560
2300
4010
5241
25 10
~
Kiparissides et al.
Odian 1841
Goto et al. [13]
Present work
Goto et al. [13]
24.1
24.1
0.2
0
2.9
3.8
~~
Ref.
>>
>,
1.
~~~
ac = k o e - ( A E + P A V ) / R T ,
'Tubular reactor.
Yessel reactor.
As a result, one cannot obtain independent estimates of the absolute values
of the individual kinetic rate constants from simple industrial reactor data. In
Table 6 the values of the kinetic parameters used in the simulation of ethylene
polymerization are reported. For simplicity, the ratio of ktd/k,c was taken as
equal to 1.
For our simulation studies, the reactor geometry shown in Fig. 2 was
employed. The reactor consists of two reaction zones. Each zone has a total
length of 150 rn and the internal tube diameter is 3.8 cm. The mass flow rate
of ethylene at the inlet of the first zone and at the side quenching point is 11.O
and 4.1 kg/s, respectively. The initiator mixture contains equal amounts per
weight of tert-butyl peroctoate (TBPO), tert-butyl perpivalate (TBPP), and tertbutyl perbenzoate (TBPB). The kinetic constants for the initiators are given
in Table 7. For reasons of simplicity it was assumed that the initiator
concentration at the inlet of each reaction zone was the same. In all simulations
runs the temperature of the reaction mixture at the inlet of each reaction zone
was set at 192 "C. A fouling factor equal to 0.24 cal/cm2/K was assumed for
both reaction zones. The coolant flow was considered to be 50 kgls and its
outlet temperature was set at 127 "C in each zone. The nominal reactor inlet
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
489
TABLE 7
Kinetic Constants for Initiators [85]"
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KO
TBPO
TBPB
TBPP
ak
d
(s-9
AE
(cal/gmol)
AV
(cm3/gmol)
5.75 x 10"
2.89 x 1017
7.95 x 1013
26,112
38,440
28,024
6.11
5.02
3.45
k -(At+PAV)lRT,
- Oe
-
320
W
270
v
0)
L
d
3
m
i
al
a
5
t
220
-
Cdo = 2.E-5 Kgrnol/rn3
Dimensionless A x i a l L e n g t h
FIG. 13. Effect of the initial initiator concentrationon the reactor temperature profile.
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KIPARISSIDES, VERROS, AND MacGREGOR
01
-
Cdo = 2 E-5 Kgmol/m3
I
CdO =
00
00
02
04
06
4E-5
08
>>
10
Dimensionless A x i a l Length
FIG. 14. Effect of the initial initiator concentration on the ethylene cumulative
conversion.
pressure was 2900 atm and the TBPO nominal concentration was 4.E-5
kgmol/m3. The reactor model equations together with the appropriate algebraic
equations describing the variation of thermodynamic, physical, and transport
properties of the reaction medium were numerically solved to calculate the
monomer and initiator concentrations, temperature and pressure profiles, as well
as the variation of molecular properties of LDPE (i.e., M,,,
M,, LCB, SCB, etc.)
along the reactor length. A modified Adams-Moulton routine was employed for
the numerical integration of the stiff differential equations of the model.
Figures 13-2 1 illustrate some representative model results obtained for
different initial molar concentrations of TBPO as a function of the dimensionless
axial length (x/L).The molar concentrations of the other two initiators are
calculated in terms of the molar concentration of TBPO, assuming that the weight
fraction of each initiator is equal to 1/3.
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
491
70000
-t-
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60000
coo = 2.E-5 Kgmol/rn3
Cdo = 4.E-5
D
Coo = 6.E-5
n
50000
40000
30000
20000
0.0
0.2
0.4
0.6
0.8
1 .o
Dlmensionless A x i a l Length
FIG. 15. Effect of the initial initiator concentration on the number-averagemolecular
weight.
Figure 13 shows that as the initial concentration of TBPO increases, the
peak temperature increases while its position shifts to the inlet of the reaction
zone. An increase in the initial initiator concentration results in an increase
of monomer conversion (Fig. 14), weight-average molecular weight (M,)
(Fig. 16), LCB (Fig. 17), and SCB (Fig. 18). Note that the number-average
molecular weight (M,)
decreases with initial initiator concentration (Fig. 15)
since a higher initiator concentration results in the formation of a larger number
of polymer chains. Similarly, the polydispersity index ( = M,,,/M,) increases
from 9 to 13 as the initial initiator concentration of TBPO changes from 2.0
kgmol/m3.
x lop5to 6.0 x
The initiator concentration also has a significant effect on the long-chain
branching and short-chainbranching of LDPE. Both types of branching increase
with initiator concentration (Figs. 17 and 18). The calculated values of LCB
(= l.0/103 C) and SCB (= 25/103 C) are in good agreement with
KIPARISSIDES, VERROS, AND MacGREGOR
500000
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400000
I
1
f--
300000
200000
100000
--3-
0
Cdo = 6E-5
n
I
Dimensionless A x i a l Length
FIG. 16. Effect of the initial initiator concentration on the weight-average molecular
weight.
experimental results on molecular structure analysis of LDPE produced in
industrial reactors [26].
Figure 19 illustrates the variation of the Reynolds number in the two-zone
reactor of Fig. 2. It is interesting to note that the Reynolds number exhibits
a significant decrease from lo6 to 6 X lo4 along the reactor length. Under
these flow conditions the plug flow assumption is fully justified. Figure 20
illustrates the variation of the inside film heat transfer coefficient with respect
to the reactor length. Note that as the initiator concentration increases, the inside
film heat transfer coefficient decreases due to the higher solution viscosity of
the reaction mixture caused by an increase of monomer conversion.
It should be noted here that the ethylenehinyl acetate copolymer composition
remains almost constant along the reactor length (Fig. 21). Actually, simulation
results show that the weight copolymer composition is independentof the peak
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
493
14
1.2-
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1.0-
08-
0.6-
"1
t
,
,
.
,
00
00
02
04
.
7
06
COO = 2.E-5 Kgmol/m3
Cdo = 4.E-5
1)
C:=6€[5
b,
08
,
10
Dimensionless A x i a l L e n g t h
FIG. 17. Effect of the initial initiator concentration on long-chain branching.
temperatures (Fig. 13) and equal to the weight fraction of vinyl acetate in the
feed stream. This behavior can be explained by the fact that the values of the
reactivity ratios (kp121kpland kp21/kp22)are approximately equal to 1.
4.4.
The Modeling of Vessel Reactors
Vessel reactors operate at temperatures and pressures similar to those of tubular
reactors, but their operation is more like that of a continuous stirred tank reactor.
The vessel reactor may have a length to diameter ratio as high as 20 to 1. Mixing
is provided by a shaft running down the center of the vessel with several impeller
blades of various types. The heat transfer through the wall is small, so the reaction
is essentially adiabatic. The inflow of monomer and initiator at several points down
the reactor (Fig. 3) provides satisfactory temperature control although the
temperature may vary down the length of the reactor.
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26
24
22
20
18
-
--t-
--t-
Cdo
2 E - 5 Kgmol/m3
CdO = 4 E - 5
x
CdO = 6.E-5
n
16
0.0
0.2
0.4
0.6
I
I
0.8
1 .o
Dimensionless A x i a l Length
FIG. 18. Effect of the initial initiator concentration on short-chain branching
In Table 1 a number of computer models for high-pressure LDPE vessel
reactors are presented. The most common modeling and computational
assumptions made in these models are:
1.
2.
3.
4.
5.
6.
7.
One-phase versus two-phase model
Ideal versus nonideal mixing conditions
Constant initiator efficiency
Constant reactor pressure and temperature
Quasi-steady-state approximation for ‘‘live’’ radicals
Rate constants are independent of viscosity
Negligible heat effects due to chain initiation, termination, and
transfer reactions
8. Constant physical properties
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
495
lo6
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-t-
Cdo = 2 E - 5 Kgrnol/m3
CdO = 4E-5
z
lo5
lo4
1031
0.0
0.2
0.4
0.6
0.8
1 .o
Dimensionless A x i a l Length
FIG. 19. Effect of the initial initiator concentration on the variation of the Reynolds
number in the reactor.
We have already discussed assumptions 5-8 in relation to the modeling of
tubular reactors. In the following subsections, emphasis will be placed on
assumptions 1-3.
4.4.1. Two-PhaseModel (assumption 1)
Under certain operating conditions (see Fig. 12), the polymer may precipitate
out from the monomer phase, forming a two-phase system. As a result,
polymerization occurs in two phases, namely, a monomer-rich phase and a
polymer-rich one. Reactions involving growing macromolecules (i.e.,
termination, transfer to polymer) may become diffusion-controlled in the
polymer-rich phase. Therefore, PE produced under heterogeneous kinetics can
differ significantly from the polymer produced in a one-phase system. As
KIPARISSIDES, VERROS, AND MacGREGOR
-
12000
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10000
t
CdO = 2E-5 Kgrnol/m3
I
Cdo = 4.E-5
u
aooo
6000
4000
2000
0
0.0
0.2
0.4
0.6
0.8
1 .o
Dimensionless Axial Length
FIG. 20. Effect of the initial initiator concentration on the inside film heat transfer
coefficient.
discussed before, if separation of the polymer phase from the ethylene phase
occurs, a two-phase reactor model must be considered. Such a model should
take into account the partition of all species (i.e., monomers, initiators, polymer)
in the two-phase system as a function of reactor operating conditions.
4.4.2. f i e Effect of Mixing on Reactor Performance (assumption 2)
Two main features characterize the operation of vessel reactors: 1) the very
high power input per unit volume required to maintain good mixing conditions
in the reaction zone, and 2) the absence of appreciable heat exchange, so that
the reactor can be considered practically adiabatic.
Several authors have shown that the degree of mixing affects the behavior of
high-pressure LDPE vessel reactors (Mercx et al. [8], Marini and Georgakis [18,
191, Villermaux et al. [24], Donati et al. [15]). The importance of end-toend mixing
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
0.0
0.2
0.4
--.--C
2%VA
--Q-
5%VA
-f-
0%VA
0.6
0.0
497
1 .o
Dimensionless A x i a l Length
FIG. 21. Effect of the initial vinyl acetate concentration on the copolymer weight
composition.
was stressed in a patent by Christl and Roedel[86]. The critical feature of this
invention was to make a uniform product by providing constant conditions for
polymerization. To obtain proper end-to-end circulation and to maintain
temperature differences between any two points in the autoclave below 5 "C,
Christl and Roedel defined a mixing number as
where NC is the number of circuits, G, is the end-to-end circulation mass rate,
and Gf is the inlet mass rate of monomer. They reported that for NC values
greater than 100, sufficient end-to-end mixing was achieved in the vessel.
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498
o l
100
I
I
I
I
150
200
250
300
Polymerization Temperature ("C)
FIG. 22. Effect of the initiator type on initiator consumption: (I) tert-butyl peroctoate,
(11) di-tert-butyl peroxide, (111) tert-butyl perpivalate, and (IV) diisononanoyl peroxide
(residence time, 65 seconds; pressure, 1700 bar; C, = 40 mol ppm).
4.4.3. Initiator Productivity {assumptions 2 and 3)
The effectiveness of organic peroxides as initiators in the polymerization
of high-pressure polyethylene has been investigated by van der Molen and van
Heerden [7], Luft et al. [76], and Goto et al. [13]. Luft and his coworkers camed
out a large number of experiments with 10 different peroxides at various
temperatures (e.g., 110-300°C). It was found that the organic peroxide
consumption per kilogram of polyethylene produced exhibited a minimum with
respect to the reaction temperature. Luft et al. [76] investigated the effects of
polymerization temperature, pressure, initiator concentration, mean residence
time of the reactor, and stirring rate on the initiator productivity.
The initiator consumption with respect to polymerization temperature for
various types of initiators is plotted in Fig. 22 (Luft et al. [76]). Apart from
the level of minimum consumption and the corresponding temperature, the
different forms of the consumption curves should be considered. Peroxides with
a broad consumption curve, such as tert-butyl perpivalate, have a constant
consumption over a wide temperature range. Therefore, they are particularly
suitable for polymerization in tubular reactors where the temperature changes
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
499
considerably along the reaction tube. In addition, the shape of the curve affects
the operating behavior of the reactor and particularly the temperature profile.
That is, if the operating conditions of the reactor lie below the minimum
temperature on a steep curve branch, then a slight reduction in the reactor
temperature causes a rapid increase in the initiator consumption, and this then
leads to a further decrease of the reactor temperature without any external
intervention. Conversely, if the temperature increases, the initiator consumption
drops drastically, and the temperature then rises to the optimal minimum
temperature. This process bears a certain similarity to the extinction-ignition
processes observed in exothermic CSTRs and catalytic reactions.
Several investigators have tried to explain these important phenomena
observed in continuous LDPE reactors. Marini and Georgakis [18, 191
considered an imperfectly mixed reactor model to show that increases in the
initiator consumption with polymerization temperature are due to mixing
limitations at the initiator feed. Goto et al. [13] introduced a variable initiator
efficiency by assuming, in addition to the main radical initiation mechanism,
a competitive side reaction which consumes initiator without producing any
active free radicals. Finally, Villermaux et al. [24] considered two different
partial segregation models, namely the shrinking aggregate model and the
interaction by exchange with the mean model, to describe the phenomena related
with the variable initiator productivity in commercial reactors.
4.5.
Comprehensive Vessel Reactor Model
To describe the dynamic operation of a high-pressure vessel ethylene
polymerization reactor, assuming one-phase flow, a set of nonlinear differential
equations of the general form
is required, where Fa and F, denote the corresponding inlet and outlet molar
flow rates of various molecular species (i.e., monomer, solvent, initiator,
moments of “live” and “dead” polymer chains, etc.). At steady-state, the
left-hand side term of Eq. (106) is dropped. Accordingly, the steady-state molar,
mass and energy balances are written as follows.
Total mass balance:
mo = m
(107)
KIPARISSIDES, VERROS, AND MacGREGOR
500
Initiator molar balances:
QoCdjo- QCdi = rdi;
i = 1, 2, ..., Ni
(108)
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Monomer molar balance:
Q0Cmio - QC,, = rpi;
i = 1, 2
(109)
Solvent (CTA) molar balances:
QoCsj0- QC,; = r,,;
i = 1, 2 , ..., N,
Moment balances for “live” radicals:
Moment balances for “dead” polymer:
QPmn
=
rpmn
Balance on long-chain branching:
Balance on short-chain branching:
Balance on double bonds:
Energy balance for the reaction mixture:
(110)
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
501
All reaction rates appearing in the above balance equations are given by
Eqs. (5)-(7) for the homopolymerization case and by Eqs. (21)-(30) for the
copolymerization case.
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4.6.
Simulation Results on Vessel Reactors
For our simulation studies, a single isothermal stirred tank reactor was
considered. As discussed above, the initiator consumption rate does not remain
constant with the polymerization temperature (see Fig. 22). To model this
variable initiator consumption rate, Goto et al. [13] assumed a competitive side
reaction consuming initiator radicals in addition to the free radical formation
reaction.
experimental
8-
theoretical
6-
4-
2-
.
...
01
0
100
I
I
150
200
I
250
300
350
Polymerization Temperature ("Cl
FIG. 23. Effect of the polymerizationtemperature on initiator consumption. (Initiator,
DTBP; experimental conditions as in Fig. 22.)
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502
Typical values of the kinetic rate constants kd and k, are reported in the original
paper of Goto et al. [ 131. In the present work, a di-tert-butyl peroxide (DTBP)
was used as initiator.
Figure 23 illustrates the effect of temperature on initiator consumption as
calculated by the present model (continuous curve). The discrete points represent
experimental measurements obtained by Luft et al. [76]. Note that initiator
consumption decreases with temperature until it reaches a minimum value. After
that point, initiator consumption increases with temperature.
100
150
200
250
300
350
P o l y m e r i z a t i o n Temperature ('CI
J?IG. 24. Effect of the polymerization temperature on ethylene conversion. (Initiator,
DTBP; residence time, 40 seconds; Ca = 40 mol ppm; pressure, 1700 atm.)
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100000 '
0
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80000 -
60000
-
40000
-
20000
-
&
-
experimental
theocellcal
0 ,
P o l y m e r i z a t i o n Temperature ('C)
FIG. 25. Effect of the polymerization temperature on the number-average molecular
weight. (Experimental conditions as in Fig. 24.)
Figures 24-28 depict the effect of polymerization temperature on monomer
conversion (Fig. 24), number-average molecular weight (Fig. 25),
polydispersity index (Fig. 26), and short- and long-chain branching (Figs. 27
and 28). It can be seen that monomer conversion, SCB, and LCB initially
increase with temperature until they reach a maximum value, after which they
decrease with temperature. This behavior can be explained by the fact that higher
temperatures favor the competitive side reaction which reduces the effective
number of primary radicals and thus the initiation of new polymer chains. It
is important to point out that the present simulation results are in satisfactory
agreement with the experimental data reported by Luft et al. [51, 881
5.
SENSITIVITY ANALYSIS, OPTIMIZATION,
AND QUALITY CONTROL
In the LDPE industry there is considerable economic incentive to develop
real-time optimal policies that will increase production and improve the PE
KIPARISSIDES, VERROS, AND MacGREGOR
504
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6
100
150
200
250
300
350
P o l y m e r i z a t i o n T e m p e r a t u r e ("C)
FIG. 26. Effect of the polymerization temperature on the polydispersity. (Experimental
conditions as in Fig. 24.)
quality. One of the most important problems in the operation of PE reactors is
the selection of optimal operating conditions that maximize the reactor productivity
at the desired product quality. It is well known that the structure of polymer
molecules can be described in terms of the following molecular properties:
Molecular weight distribution, MWD
Number-average molecular weight, M,,
Weight-average molecular weight, M,
Degree of branching distribution, DBD
Long-chain branching, LCB
Short-chain branching, SCB
Copolymer composition distribution, CCD
Number of vinyl and vinylidene groups
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
505
P o l y m e r i z a t i o n T e m p e r a t u r e ("C]
FIG. 27. Effect of the polymerization temperature on the short-chain branching.
(Experimental conditions as in Fig. 24.)
A number of PE end-use properties such as density, melt index, impact
strength, rigidity, tensile strength, etc. can be related through some semiempirical relationships to the molecular structure of polymers. In fact, several
quantitative correlations have been published in the open literature relating enduse properties to the molecular structure of PE. On the other hand, for a given
reactor configuration, the molecular properties will strongly depend on the
operating conditions. Therefore, the production of polymers with specified enduse properties will be closely related to the optimal control of molecular
properties during production.
A general approach to the optimization of polymerization reactors will
encompass four aspects: reactor modeling, optimal design, on-line measurements,
and optimal control. The modeling of high-pressure LDPE reactors has been
discussed in previous sections of this review. Problems related with on-line
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0.3
theoretical
-
02-
01-
0.0
1
I
I
I
FIG. 28. Effect of the polymerization temperature on the long-chain branching.
(Experimental conditions as in Fig. 24.)
measurements and control of polymerization reactors were recently reviewed
by MacGregor et al. [89] and Chien and Penlidis [90].
In the present review we focus our attention on the optimal design problem.
In particular, in Subsection 5.1 we analyze the sensitivity of molecular properties
of LDPE produced in a tubular reactor with respect to variations in the nominal
operating conditions. Subsection 5.2 deals with the optimization of high-pressure
LDPE reactors. Finally, in Subsection 5.3 we discuss the application of
multivariate statistical methods to high-pressure LDPE reactors. Statistical
methods are used in the development of models from process data. These models
are employed for prediction of molecular properties and monitoring the process
performance.
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5.1
507
Sensitivity Analysis
To optimize the operation of an existing LDPE reactor or to design an optimal
reactor system, it is necessary to know the sensitivity of system responses to
deviations of system operating parameters from their nominal values
(Kiparissides and Mavridis [22]). It is well known that polymerization reactors
are very sensitive to changes in operating conditions and kinetic parameters.
Most reactor instabilities and upsets occur because of poor mixing of the initiator
stream and “gel” formation due to polymer holdup at the wall of tubular
reactors. Therefore, it is very important to know the underlying relationships
between the system outputs and the model parameters.
The high-pressure polymerization of ethylene in a tubular reactor can be
described in terms of the set of differential equations (Eqs. 76-89). This can
be written in vector form as:
where y is a vector of dimensionless system variables
and the vector p includes the design, operating, and kinetic parameters of the
reactor model, e.g.,
A means of assessing the effect of parameter uncertainties on the model
output variables is the sensitivity coefficient. The sensitivity coefficient, pg,
is defined as the rate of change of the value of the output variable yi with
respect to perturbations in the parameter pP
‘pij
= ayi/apj;
i = 1, 2,
..., Ny and j
= 1, 2,
..., Np
(122)
The sensitivity coefficient, Eq. (122), indicates the magnitude and direction
of change of the output variable yi caused by deviations of the parameter p j
from its nominal value. By differentiation of Eq. (122), one can obtain the
corresponding sensitivity differential equations in terms of the model nonlinear
functions f(y,p):
KIPARISSIDES, VERROS, AND MacGREGOR
508
d'pkj
=
dr
afk
i=l
j = 1, 2,
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pii
+ -;ah
k = 1, 2, ..., N,and
apj
aYj
..., Np
The initial conditions of these Eqs. (123) are
Cpij(0) =
0 if pj is not an initial condition (i.e., U , D)
6 , if p j is an initial condition (i.e., C,, Cso)
(124)
The total number of sensitivity equations will be equal to the number of
state variables times the number of model parameters (N, x N J . The variation
of the sensitivity coefficients along the reactor is determined from the numerical
integration of the sensitivity Eqs. (123) and the model Eqs. (76)-(89). It should
be noted that the sensitivity equations are extremely stiff. Thus, extra care must
be taken in integrating these equations. Accordingly, a multistep predictor
corrector method suitable for stiff differential equations should be used.
Typical results of the sensitivity analysis for a single zone high-pressure
LDPE tubular reactor are shown in Figs. 29-31 (Kiparissides and Mavridis
1221). In these figures the normalized sensitivity coefficients
are plotted as a function of reactor length. The reason for plotting the normalized
sensitivity coefficient, piJ, instead of (oii is to show the relative influence of
various parameters on the state variables. Note that for a positive sensitivity
coefficient, a positive change of the corresponding parameter from its nominal
value will result in an increase of the output variable. On the other hand, a
negative sensitivity coefficient indicates that a positive change in the parameter
will cause a decrease in the output variable. The effect of perturbations in the
overall heat transfer coefficient U on the system responses is shown in Fig.
29. It can be seen that the system responses are very sensitive near the hot-spot
temperature. It is interesting to note that the sensitivity coefficients for M,,,
M,, and T change sign along the reactor length. This means that, depending
on the location in the reactor, a positive variation in U might cause either a
decrease or an increase in these variables. The final values of the normalized
sensitivity at the reactor exit can be obtained from Fig. 29 for x/L = 1. These
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
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40
2.0
0.0
-2.0
0.0
0.2
0.4
0.6
0.8
1 .o
Dimensionless A x i a l Dtstance
FIG. 29. Effect of perturbations in the heat-transfercoefficient on system responses.
values show that as the heat transfer coefficient increases, the exit monomer
conversion, M,,, M,, and PD increase while the temperature of the reaction
fluid at the reactor exit decreases. This implies that conversion improvements
due to heat transfer can be made. However, the degree of improvement of
monomer conversion will depend on the initiator system used for the
polymerization. These results of the sensitivity analysis are in good agreement
with earlier results reported by Lee and Marano [12], Chen et al. [lo], and
Gupta et al. [21]
The effect of perturbations in the reactor tube diameter D on the system
responses is shown in Fig. 30. It can be seen that perturbations in D have exactly
the opposite effect on the output variables to that obtained by deviations in the
overall heat transfer coefficient U . This is explained by the fact that both
parameters affect the rate of heat transfer. Thus, an increase in D will bring
about a decrease in the exit monomer conversion, M,,, and M,, and will cause
KIPARISSIDES, VERROS, AND MacGREGOR
510
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31
-5
0.0
0.2
0.4
0.6
0.8
1.o
Dimensionless Axial Distance
FIG. 30. Effect of perturbations in the tube diameter, D , on system responses.
an increase in the exit temperature of the reaction fluid. The results of Fig.
30 indicate that a small reactor diameter will improve the heat transfer
characteristics of the process and will increase the final monomer conversion.
Finally, the effect of solvent perturbations on the system variables is shown
in Fig. 3 1. It is clear from these results that solvent perturbations have no effect
on monomer conversion, initiator conversion, or reaction temperature. On the
other hand, solvent variations do affect the number-average and weight-average
molecular weights. In fact, a positive variation in C,, will cause a decrease
in the output variables M,, and M,,,.
5.2.
Optimization of LDPE Reactors
The diverse uses and forms of polymer products require different physical,
chemical, and mechanical properties that are difficult to express in a single
objective function. Fortunately, it is often possible to correlate the properties
of a polymer product with some relatively easy measured quantities, i.e., M,,
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-0.50
0.0
0.2
0.4
0.6
0.8
511
1.0
Dimensionless Axial Distance
FIG. 31. Effect of perturbations in the initial solvent concentration, CSo,on system
responses.
and PD. Because the MWD of a polymer for a given reactor configuration
is a function of the reaction variables, namely, concentrations of initiators,
solvents, pressure, temperature, etc., the final properties of a polymer will
depend on the precise control of these variables.
With respect to the more specific problem of optimization of high-pressure
LDPE reactors, very little work has been published in the open literature
although a number of LDPE plants are equipped with process computers. In
an interesting paper by Marini and Georgakis [181, the modeling, optimization,
and control of a vessel LDPE reactor are discussed.
Yoon and Rhee [25]applied the maximum principle to determine an optimal
temperature policy that would maximize the exit monomer conversion in an
LDPE tubular reactor. However, no product quality specifications were
considered. Mavridis and Kiparissides [23] developed an optimization procedure
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512
KIPARISSIDES, VERROS, AND MacGREGOR
for selecting the optimal wall temperature and the initiator and chain transfer
agent concentrations in a fixed-size high-pressure PE tubular reactor in order
to maximize the reactor productivity at the desired product quality. Finally,
Brandolin et al. [91] examined the optimization problem for fixed-size LDPE
tubular reactors through the maximization of production by using several
operational policies. The usefulness of some of these policies, as well as the
feasibility of their practical implementation, was also examined.
5.2.1. Classijkation of Control Variables
For a given PE reactor configuration, the polymer production and the
molecular properties of polymer chains will depend upon a large number of
control variables. These can be classified into
1. Initiator variables: namely type of initiator, number of initiators, number
and location of initiator feed points, initiator feed rates.
2 . Solvents (chain-transfer) variables: namely, type and number of chaintransfer agents, number and location of feed points, chain-transfer feed
rates.
3. Heat transfer variables: feed temperature, number and length of
heating-cooling zones in LDPE tubular reactors, inlet temperature of
the cooling-heating fluid in each zone or jacket, the heat transfer
coefficient in each zone.
4. Reaction variables: that is, polymerization temperature, pressure, degree
of mixing, monomer feed concentration and rate, number and location
of monomer side feed points.
As can be seen, the number of possible control variables is very large.
Obviously, the final selection of the control variables will be related to the
specific design of a process, its operational modes, the sensitivity of the
particular process to manipulations of the control variables, and the optimization
objective function.
5.2.2. Dejinition of the Objective Function
Many optimization problems in polymerization have been studied by defining
a single scalar objective function which combines all identifiable performance
measures with weighting factors chosen a priori. However, the combination
of several terms in a single objective function is not always easy in the sense
that some controlled variables may react in opposite directions to manipulations
of a control variable. An objective function reflects the optimization objectives
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of a problem. In general, these objectives will fall into the following categories:
economic objectives, safety, environmental constraints, product specifications.
Let us assume that we desire to operate a high-pressure LDPE tubular reactor
at a maximum production rate to obtain a polymer with desired molecular
properties (i.e., Mnd, Pod). These two optimization objectives can be
combined into a single objective function of the following form:
J = w,y,
+ w,
(-
Mn - 1),
Mnd
+ w3(E
- l),
pDd
Subsequently, we must select the nominal values of the weighting factors wl,
w,, and w3.It is obvious that there are practically no limitations on the exact
form of the objective function. The above performance index is an implicit
function of the control variables. Therefore, the general static optimization
problem becomes one of maximizing (minimizing) the objective function subject
to model equations describing the quantitative relationships between state and
control variables.
Mavridis and Kiparissides [23] considered the optimization of a tubular
reactor.The tubular reactor was divided into three heat-transfer zones of
specified length. The preheating zone had a length of 0.4L. The other two zones,
the reaction and the cooling zones, had lengths of 0.1L and 0.515,respectively.
The wall temperature in each zone was unknown. The reaction feed consisted
of ethylene, two initiators, and a chain-transfer agent. The decision variables
of the problem were six:
1.
2.
3.
4.
5.
6.
The reactor wall temperature in the first zone, T,,
The reactor wall temperature in the second zone, Tw2
The reactor wall temperature in the third zone, Tw3
The initial concentration of the first initiator (Cd,o)
The initial concentration of the second initiator (Cao)
The ratio of the chain-transfer agent feed rate to the monomer feed
rate, S
The objective function was written as in Eq. (126). The values of the
weighting factors were w , = 1 and w2 = w3 = 100. The desired M d at the
reactor exit was set to 25,000, and the final polydispersity was assumed to vary
in the range of 3 to 7. They used a random search optimization method [92]
and the mathematical model of the process to determine the optimal control
variables that maximized the objective function (Eq. 126). They found that,
in principle, one can obtain a polymer with specified molecular weight
characteristics by proper manipulation of the control variables.
KIPARISSIDES, VERROS, AND MacGREGOR
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514
W'
OI
2
c
0.0
0.2
0.4
0.6
0.8
1.o
DIMENSIONLESS LENGTH
FIG. 32. Optimal temperature profiles as a function of final polydispersity index.
The optimal temperature, initiator, monomer, and number-averagemolecular
weight profiles are plotted in Figs. 32, 33, 34, and 35, respectively. In Fig.
32 the temperature profiles (2) and (3) exhibit two peak temperatures that
correspond to the total consumption of the first and second initiator (Fig. 33).
On the other hand, the temperature profile (1) shows a single temperature peak
since only the first initiator is competely consumed. In this case the conversion
of the second initiator at the reactor exit is limited to 2.84%.
In Fig. 34 the conversion profiles are plotted for different values of POd.
Note that the final conversion values are very close to those obtained in industrial
reactors. At the end of the preheating zone, the monomer conversion does
sharply increase to a point at which the first initiator is completely consumed.
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
0.0
0.2
0.4
0.6
0.8
515
1.0
DIMENSIONLESS LENGTH
FIG. 33. Optimal initiator conversion profiles as a function of final polydispersity index.
After this point the monomer conversion continues increasing as a result of the
decomposition of the second initiator. A final conversion value is obtained before
the reaction exit due to the complete consumption of initiators . The number-average
molecular weight profiles are plotted in Fig. 35. Note that M, continuously
decreases up to a final value that corresponds to the desired value of M,.
The procedure developed by Mavridis and Kiparissides 1231 is fairly general
and provides a systematic way for selecting the optimal initiator, solvent, and
heat-transfer conditions that maximize monomer conversion at a given product
quality in a fixed sized reactor.
5.3.
Multivariate Statistical Quality Control
In high-pressure tubular and vessel reactors it is extremely difficult to
measure on-line such fundamental product properties as Mn,M,, LCB, SCB,
KIPARISSIDES, VERROS, AND MacGREGOR
1
1
1
I
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0.2
02
0.1
0 .c
00
0.2
0.4
06
0.8
10
DIMENSIONLESS LENGTH
FIG. 34. Optimal monomer conversion profiles as a function of final polydispersity
index.
etc. These molecular product properties, which are closely related to the melt
index and density of PE, are usually measured infrequently and off-line.
However, many indirect on-line process measurements of the temperature
profile and the initiator and solvent flow rates are made on a frequent basis.
In this section we look at the use of multivariable statistical methods which
utilize the available process data to build models which are capable of predicting
fundamental molecular properties, monitoring the process performance,
detecting process operating problems, and diagnosing assignable causes for
these problems. More details can be found in Skagerberg et al. [93].
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
I
I
I
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M,d = 25,000
I
517
+
!'2 PD=3
POX5
)3:PD=7
-
-
-
DIMENSIONLESS LENGTH
FIG. 35. Optimal number-average molecular weight profiles as a function of final
polydispersity index.
5.3.1. Inferential Models for Polymer Properties
Inferential or predictive models for the product properties Y = (Mn,M,,
LCB, SCB, ..., ) T in a high-pressure LDPE reactor would be of great value.
Nonlinear dynamic models developed from fundamental material and energy
balances could be used to develop a nonlinear state estimator such as the cxtended
Kalman filter to obtain estimates of unmeasured molecular properties from online measurements of temperature, initiator and solvent flow rates, etc. Although
this approach is an optimal one, it is very time-consuming and would still require
KIPARISSIDES, VERROS, AND MacGREGOR
518
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the estimation of some model and filter parameters to match plant data. An
easier and more practical approach is to build empirical models directly from
the process operating data. The general form of these models would be
for the ith property variable. The process variables used in the model can be
the temperature profile measurements (T, in each section of a tubular reactor,
the coolant temperature (T,) in each zone, and the feed rates of monomer (Fm),
solvent ( F s ) ,and initiators (FJ to each zone. The process data (X) contains
the information necessary to provide good predictions of all molecular properties
(Y) of the polyethylene. The major difficulties associated with building empirical
process models arise from the ill-conditioned nature of the process data and
from the way in which the process data are collected (Kresta et al. [94]). To
overcome this problem, Skagerberg et al. [93] used multivariate partial least
squares or projection to latent structures (PLS) methods to develop inferential
models for an LDPE tubular reactor.
PLS is a multivariable regression method based on projecting the information
in high dimensional spaces (X, Y) down into low dimensional subspaces defined
by a small number of latent vectors (ti, t2, ..., t,). Such latent variables are
linear combinations of original variables and express the dominant patterns in
the data. These new latent vectors summarize all the important information
contained in the original data sets. The scaled matrices of process (X) and
product property (Y) measurements are then modeled in PLS by means of a
series of principal components like bilinear decompositions according to Eqs.
(128) and (129):
n
X =
tip:
+E
i= I
n
Y =
C uiqT+ F
i= 1
where ti and ui are the latent vectors in X and Y, and pi and qi are the loading
vectors that express the contribution from each X and Y variable toward defining
the new latent variables ti and uir respectively. E and F are matrices of
residuals. The final PLS prediction models can always be expressed in the form
of the original variables as
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
519
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where the vector of regression coefficients is given by
P, Q , and W are the corresponding loading and weight matrices calculated by
the PLS algorithm [93].
Skagerberg et al. [93] investigated the application of PLS to LDPE reactors
by simulating a single zone tubular reactor using the model of Kiparissides and
Mavridis [22,23]. Process measurements considered to be available were the reactor wall temperature, 20 temperatures along the reactor, and the chain-transfer
agent feed rate. In spite of unmeasured variations in the initiator feed rates, the
wall heat transfer coefficient, and the substantial measurement errors in the product properties (Y), good predictions of all polymer properties were obtained (Figs.
36 and 37). The PLS loading vectors p and q also provided considerable insight
into which groupings of process variables were important in predicting the various
properties.
5.3.2. Process Monitoring via Multivariate SPC Charts
Statistical process control charts such as the Shewhart or CUSUM chart are
well-established statistical procedures for monitoring and detecting problems with
stable univariate processes. These charts are usually applied to the final product
property measurements and do not utilize the large volumes of process data that
are avadable on a much more frequent basis.
Kresta et al. [94] proposed the use of multivariate control charts in the low
dunension space defined by the dominant latent vectors (ti, i = 1, 2, 3) calculated
from a PLS analysis of the process data. These latent variables are determined
from the frequently available process measurements, but contain the information
relevant to inferring the finalproduct properties. If an LDPE process is operating
in a stable and acceptable fashion, new “observations” of the latent variables should
continue to fall inside a target region defined by past periods of good operation,
and the squared prediction errors (SPE) of new process measurements should remain small.
For the LDPE tubular reactor study described by Skagerberg et al. [93], a threedimensional SPC chart ( t l , tz, SPE) is shown in Fig. 38. Under normal operation the process observations are clustered in a well defined region close to the
(tl-tz)plane with small values of SPE. When problems occur, such as 1) a
gradual fouling of the reactor tube wall, 2) a change in the level of impurities
or in the initiator efficiency, or 3) coolant overheating, these affect the process
temperature profile and result in large changes in the latent vectors and the SPE.
KIPARISSIDES, VERROS, AND MacGREGOR
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520
Calculated (Mw)' x lo6
FIG. 36. Observed versus calculated values of the reciprocal weight-average molecular
weight, M,.
/'
36
U
w
t
32
VI
Y
30
28
28
NVrn
30
32
34
36
Calculated SCB
FIG. 37. Observed versus calculated values of short-chain branching, SCB.
HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
52 1
I
I
I
g ?
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I
I
ad
I
I
I
FIG. 38. Multivariate monitoring chart for the LDPE reactor. The points within the
reference set represent normal operating conditions ( o ). The following abnormal events
are depicted: 1) fouling ( 0 ), 2) coolant overheating ( a ) , and 3) impurities ( 4).
In each case an alarm is provided by the multivariate chart (Fig. 38). Although
the charts do not provide direct information on the source of the problems,
diagnosis of possible causes is usually possible through further analysis of the
PLS model.
6.
CONCLUSIONS
This paper reviews recent modeling, optimization, and statistical quality
control developments for high-pressure LDPE tubular and vessel reactors. A
unified mathematical framework is developed for modeling high-pressure vessel
and tubular LDPE reactors. This methodology is based on the most general
kinetic mechanisms for ethylene homo- and copolymerization. Advanced
thermophysical correlations describing the variation of physical and transport
properties of the reaction mixture are presented. To predict the molecular and
structural developments of LDPE in high-pressure reactor, the method of
moments is employed. Application of QSSA and the LCH to the moment
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522
KIPARISSIDES, VERROS, AND MacGREGOR
equations of “live” polymer chains shows that ethylene copolymerization
lunetics will depend on the values of cij ( = ku/kpi)parameters. This means
that one cannot obtain individual estimates of the kinetic rate constants by fitting
model predictions to lunetic data obtained from an industrial reactor. Model
predictions for tubular and vessel reactors are in good agreement with reported
experimental data on conversion and molecular properties of LDPE obtained
in laboratory and in industrial reactors [26].
To analyze the effects of design, operating, and kinetic parameters on the
reactor performance, a systematic procedure based on sensitivity analysis is
presented. The main aspects of optimizing the performance of LDPE reactors
are also discussed. Finally, the use of multivariate statistical process control
methods for prediction of polymer properties and monitoring the process
performance is briefly reviewed.
It should be noted that the models and procedures developed in this work
are both realistic and general and can lead to a more systematic design and
operation of industrial LDPE reactors.
NOMENCLATURE
E”
f
f
concentration (gmol/L)
specific heat capacity (cal/g/K)
ideal specific heat capacity of ethylene (cal/g/K)
inside tube diameter (cm)
“dead” polymer chains having p ethylene and q comonomer
monomer units
activation energy for viscous flow (cal/gmol)
Fanning friction factor (dimensionless)
initiator efficiency (dimensionless)
end-to-end circulation mass rate (g/s)
inlet mass rate of monomer (g/s)
ethylene specific enthalpy (l/g)
inside film heat transfer coefficient (cal/cm2/s)
rate constant for intramolecular transfer (l/s)
rate constant for p-scission of sec radicals (1/s)
rate constant for @scission of tert radicals (l/s)
rate constant for dissociation of initiator ( U s )
propagation rate constant (L/gmol/s)
termination by combination rate constant (L/gmol/s)
termination by disproportionation rate constant (L/gmol/s)
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HIGH-PRESSURE ETHYLENE POLYMERIZATION REACTORS
523
rate constant for chain transfer to monomer (L/gmol/s)
rate constant for chain transfer to polymer (L/gmol/s)
rate constant for chain transfer to solvent (L/gmol/s)
reactor length (m)
mass flow (g/s)
viscosity-average molecular weight
number of circuits
Nusselt number
reactor operating pressure (am)
critical pressure of ethylene (atm)
Prandtl number
reduced pressure of ethylene (dimensionless)
volumetric flow rate (m3/s)
universal gas constant (cal/gmol/K)
Reynolds number
“live” polymer chains having p ethylene and q comonomer
monomer units
temperature (K)
critical temperature of ethylene (K)
reduced temperature of ethylene (dimensionless)
fluid velocity (m/s)
reactor volume (m3)
LDPE specific volume (L/g)
ethylene specific volume (L/g)
axial distance
weight fraction
cumulative monomer conversion
fractional conversion of the j component
Greek Letters
activation energy (callgmol)
propagation reaction enthalpy (cal/gmol)
activation volume (cm3/gmol)
characteristic kinetic ratio defined by Eqs. (101)-(104)
intrinsic viscosity (dL/g)
ethylene viscosity (g/m/s)
ethylene-polyethylene reduced solution viscosity (dimensionless)
524
%
17SW
x
L n
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Pmn
P
Pc
‘pij
KIPARISSIDES, VERROS, AND MacGREGOR
ethylene-polyethylene solution viscosity (g/m/s)
viscosity at the wall temperature (g/m/s)
thermal conductivity (cal/cm/s/K)
moments of the bivariate NCLD of “live” copolymer chains
(gmo1/m3)
moments of the bivariate NCLD of “dead” copolymer chains
(gmol/m3)
reaction mixture density (kg/m3)
critical density of ethylene (kg/m3)
sensitivity coefficient
Subscripts
0
C
di
LCB
m
SCB
S
si
VNL
VND
conditions at the inlet of the reaction zone
coolant
“i” initiator
long-chain branching
monomer(s)
short-chain branching
solution
“i” solvent
vinyl groups
vinylidene groups
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