A Practical, Systematic Approach for the Scaling‐Up and Modeling of Industrial Copolymerization Reactors

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A Practical, Systematic Approach for the Scaling‐Up

and Modeling of Industrial Copolymerization ReactorsCarlos Guerrero‐Sánchez

a b , Enrique Saldívar

a , Martín Hernández

a c & Arturo Jiménez

b

a CID R&D , Desc Chemical Sector (formerly GIRSA) , Av. de los Sauces 87 MZA 6, Lerma,

52000, Méxicob Instituto Tecnológico de Celaya , Av. Tecnológico y García Cubas S/N, 38010, Celaya,

Méxicoc Resirene SA de CV, Km 15.5 Carr. Puebla‐Tlaxcala, Sto. Toribio Xicotzinco, Tlaxcala, Méx

Published online: 15 Feb 2007.

To cite this article: Carlos Guerrero‐Sánchez , Enrique Saldívar , Martín Hernández & Arturo Jiménez (2003) A Practical,Systematic Approach for the Scaling‐Up and Modeling of Industrial Copolymerization Reactors, Polymer Reaction Engineerin11:3, 457-506, DOI: 10.1081/PRE-120024422

To link to this article: http://dx.doi.org/10.1081/PRE-120024422

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POLYMER REACTION ENGINEERING

Vol. 11, No. 3, pp. 457–506, 2003

A Practical, Systematic Approach for theScaling-Up and Modeling of Industrial

Copolymerization Reactors

Carlos Guerrero-Sanchez,1,2 Enrique Saldıvar,1,*

Martın Hernandez,1,# and Arturo Jimenez2

1CID R&D, Desc Chemical Sector (formerly GIRSA), Lerma, Mexico2

Instituto Tecnologico de Celaya, Celaya, Mexico

ABSTRACT

A systematic methodology for the modeling and engineering analysis of

industrial copolymerization reactors is presented. The methodology,

especially suited for the scaling-up from laboratory experiments topilot plant and industrial reactor level, consists of gradually building

models of more complexity in a modular way as more information is

obtained from experimental data and/or theoretical considerations. In

the first stage, simple models for copolymer composition are written

based on the Mayo – Lewis copolymerization equation and empirical

*Correspondence: Enrique Saldıvar, CID R&D, Desc Chemical Sector (formerly

GIRSA), Av. de los Sauces 87 MZA 6, Lerma 52000, Mexico; E-mail: esaldiva@

mail.girsa.com.mx.#Current address: Martın Hernandez, Resirene SA de CV, Km 15.5 Carr. Puebla-

Tlaxcala, Sto. Toribio Xicotzinco, Tlaxcala, Mexico.

457

DOI: 10.1081/PRE-120024422 1054-3414 (Print); 1532-2408 (Online)

Copyright D 2003 by Marcel Dekker, Inc. www.dekker.com



copolymerization rate data for different reactor configurations (batch and

CSTR) and reactor operations (steady state and some dynamic transients

for the CSTR case). This set of models, which use minimal or no data

fitting, is shown to be highly predictive. In a second stage, as kinetic

information is obtained in the form of an expression for the copolymer-

ization rate, either empirical or mechanistic, the models can be gradually

expanded to include a full non-linear analysis of steady state multi-

plicities and other interesting phenomena, which can have an impact on

the practical operation of the reactor. Also, as a complementary tool for

the modeling of copolymerization reactors, a new model for the gel

effect in polymerization, based on analogies with the familiar diffusion

controlled reactions in heterogeneous catalytic reactors, is outlined and

used. The methodology is illustrated with examples drawn from

industrial reactors in bulk and emulsion, including some industrialreactor data.

K e y W o rd s : C o p o l ym e r i za t i o n; M a t h em a t i ca l m o d e li n g

copolymerization processes; Industrial polymerization reactors; Gel

effect.

INTRODUCTION

Since 1944 several theories have been developed in order to explain the

homo and copolymerization processes (see for example, Bonta et al. (1975);

Mayo and Lewis, (1944), among others); however, the study of

copolymerization reactions and kinetics has been largely dominated in thepast by chemists and physical chemists trying to understand the basic

mechanisms underlying these phenomena. They concentrated earlier efforts

on the region of very low conversions in which the composition of the

monomer mixture in the system is almost constant. As a result of these

efforts, for the last half century there has been important progress in the

understanding of basic mechanisms and in the development of models for

specific aspects of copolymerization systems (for example the ultimate and,

more recently, the penultimate model for copolymer composition), which

may be useful for engineers in process design, operation, control or

optimization. However, many of the aspects that the engineer has to face

when he/she is designing a new copolymerization reactor, or dealing with a

new copolymer system, have not been tackled in a systematic way by the

studies available in the literature. For example, the use of the simple

Mayo– Lewis equation in the frame of different reactor operation types,

which can provide very important information a priori and has proved

highly predictive when conversion data are available, has never received

458 Guerrero-Sanchez et al.



systematic attention in the specialized literature for polymerization

engineers. Also, the fact that the reaction rate for most of the

copolymerization systems, either in bulk, solution or emulsion, cannot be

predicted by any of the existing models, specially when it involves

diffusion controlled reactions, poses an important problem for the engineer

who wants to design a process for a copolymerization system. On the other

hand, many tools for polymerization reaction and reactor analysis have been

developed and are spread throughout the literature of polymer chemistry

and polymer reaction engineering, but have not been put together in a

systematic and accessible framework. In front of this situation, the engineer

that faces the scaling-up of a new copolymerization process from laboratory

experiments to pilot plant and commercial level usually proceeds in a very

empirical and tortuous way, but still he/she is not willing to embark in whatis often perceived as a too complicated, time-consuming and long-term

modeling effort. The general impression is that existing models are either

too simple and confined to rather too low conversions to be of any practical

use, or that no simple predictive models are available for real-life systems

exhibiting gel effect or other complexities.

One solution to this situation is the use of an existing simulation

package such as POLYRED (Ray, 1996), POLYMAT (Gao and Penlidis,

1996, 1998) or Predici, but its use may be limited to those systems in

which all the relevant kinetic constants are known, which is often not the

case. The aim of this work is to make a small contribution to the problem

of modeling real industrial copolymerization reactors providing another

more practical approach that consists of taking some of the existing tools

(laboratory experiments, general reactor models and the copolymerizationequation) and putting them together in a systematic and practical way for

the benefit of the practicing engineer, so he/she can build models of

increasing level of complexity as more experimental and/or scientific data

from the laboratory, the pilot plant or scientific publications dealing with

the specific system of interest, are collected. The methodology is based on

our own industrial experience with a number of copolymerization systems.

The toolbox to be presented here consists of a blend of modeling and

empirical procedures which we have found to be of fairly simple and

practical application, with the final goal of extracting the maximum amount

of reliable information from both approaches: first principles modeling and

empirical data. In this first paper we concentrate on the simplest models,

those mostly related to copolymer composition. Different ways to exploit

well known equations (such as the Mayo–Lewis equation) when applied to

different reactor configurations are offered, as well as systematic ways of

incorporating empirical and more fundamental reaction rate expressions into

the reactor analysis in order to enrich the power of the material and energy

Industrial Copolymerization Reactors 459



reactor balance equations. Although the set of models presented here is not

applicable to all possible situations found in practice, it is hoped that the

ideas used here can be easily extended to other specific situations. In this

first paper the emphasis is also on the chemical aspects of the reactions, so

most of the physical and transport effects related to reactor geometry and

agitation are lumped into empirical reaction rate equations. In future papers,

ongoing work related to other complementary aspects of scaling-up and

copolymerization systems (e.g. transport restrictions, molecular weight

distribution) will be incorporated into this framework. Examples taken from

the industrial practice and containing industrial experimental data are

offered to illustrate the discussed techniques.

The organization of the paper is as follows. First the general meth-

odology, based on practices commonly found in industry, is described.Next, a set of models of increasing level of detail and suited for different

situations is presented. Here, batch and CSTR models are described, along

with assumptions used in their formulation and recommendations for their

application. For the CSTR case, models are offered for the steady state and

transient operations. In the following section, a new model for the gel effect

is presented; this model is based on analogies with heterogeneous diffusion-

controlled reactions in catalytic pellets, so it can be easily grasped by the

general chemical engineer with no specialization in polymers. This model is

introduced in the frame of traditional (‘‘full’’) material balances, formulated

when ultimate model copolymerization kinetics are applied. Finally, in the

last section, the use of some of the presented models is illustrated with

examples taken from our industrial practice. In one of the examples an

innovative graphical way of presenting the effect of reactor configuration(batch or tubular, single or series of CSRT’s) on copolymer composition is

introduced. In this work the emphasis is on composition and kinetics;

therefore, no explicit expressions are given for energy balances. If needed,

they can be readily written for each of the levels of the material balance

models discussed here.

Notice that for the CSTR case, models with slightly different

formulations are offered for the steady state and transient operations; this

arises from the convenience of using different sets of variables better suited

to the way of measuring experimental data.

GENERAL METHODOLOGY FOR

MODELING AND SCALING-UP

Despite the fact that the terminal model, on which the Mayo–Lewis

equation is based, cannot predict at the same time the copolymer




composition and the propagation rate in a copolymerization reaction (see

for example, Coote and Davis (1999)) it is convenient to take advantage of

its capability to predict the copolymer composition given that empirical

data on rate of copolymerization or related data are available, which is

many times the case during process development in industry.

It is well known (c.f. Canegallo et al. (1993), Saldı var and Ray (1997))

that, using the integrated form of the Mayo–Lewis equation, composition in

terms of conversion can be predicted for most copolymerization systems. In

consequence, if one has access to data on conversion evolution with time,

the composition can also be predicted as a function of time. In many

industrial cases when a new copolymer system or a new copolymer

composition for an established system is going to be scaled-up, empirical

kinetic data at the laboratory level or bench are collected at the newmonomer compositions, different temperatures and/or initiator concentra-

tions. This information is usually put in the form of an empirical or semi-

empirical rate equation with the aim of using it for design or control of the

new operation. The proposal here is to use this information in standard

material and energy balances for the type of reactor at hand, taking also full

advantage of the predictive power of the Mayo–Lewis equation. The

guidelines for the use of the models presented in this work can be

summarized as follows:

1. Collect kinetic data (conversion– time relationships) at the labora-

tory level for the copolymerization system at hand, under varying

and controlled conditions of temperature, monomer compositions

and initiator concentrations. This is usually the first step during theinvestigation of a new copolymerization system. It is worth men-

tioning that this step has become recently simplified by the use of

combinatorial equipment that allows the speeding up of the

experimentation. At this step, regardless of the kind of laboratory

reactor used, one has to be careful on what effects reasonably scale-

up regardless of the pattern of mixing of the reactor. This is usually

not a problem for copolymer composition, assuming that reasonable

homogeneity is achieved with mixing (see Zhang and Ray (1997)

for a discussion of this issue). In the simplest cases, historical data

from plant for similar systems can be used in order to estimate the

conversion for a given time or residence time.

2. Depending on the stage of the investigation and the level of detail

of the information collected, use this information in specific forms

of the material and energy balances for the reactor configuration to

be used. The first level of the resulting models provides only

copolymer composition based on a given conversion and requires




only the reactivity ratios, which are available in the literature or

are easily measured for many copolymerization systems. This type

of model is free of the rest of the kinetic constants.

3. At the next level of detail it is expected to count with an empirical

polymerization rate expression that depends possibly on temper-

ature, initiator concentration, monomer composition, etc., of the

general type R p (T , I , M mi). Typically, one can use empirical

models obtained through multiple regression analysis or neural

networks. At this stage, more advanced models can be used.

4. If more information is available on the copolymerization system at

hand (for example, reliable homopolymerization kinetic rate con-

stants or copolymerization gel-effect expressions) enrich the models

of the previous step with standard copolymerization kinetics.

For each one of the steps, one can select the proper model from the set

presented in the following section. The above guidelines are useful in the

scaling-up of a totally new process or during the adaptation of an existing

reactor configuration to a new copolymerization system.

MATHEMATICAL MODELS

Since the Mayo–Lewis equation appeared (Mayo and Lewis, 1944), it

has been used for predicting the copolymer composition for many

copolymerization reaction systems. However, for many years, a more

recent model (penultimate effect) and discussion about Mayo– Lewisequation validity have emerged (Bonta et al., 1975; Coote and Davis,

1999). Despite the fact that the penultimate effect model has stronger

theoretical basis than the Mayo–Lewis equation, it has found many

limitations for its use in real or industrial copolymerization reaction

systems due to the lack of estimated data for their parameters. On the

other hand, the Mayo–Lewis equation has been used to predict the

instantaneous copolymer composition in batch reactors and to study the

composition drift phenomena and its related reactivity ratios and it has

proven predictive for most copolymerization systems, except for those

showing reversibility. However, so far, and with notable exceptions

(Arzamendi and Asua, 1989; Canegallo et al., 1993; Dube et al., 1997;

Saldıvar and Ray, 1997), the Mayo–Lewis equation’s capability to predict

the behavior of copolymer systems has not been exploited to its full extent

for engineering applications. To our knowledge only Ray’s group has

included a model of this sort in his simulation package (Ray, 1996), but

the use of this concept is not widespread and is markedly under-exploited




by practitioner engineers in industry. In this section the models for a

copolymerization batch reactor and CSTR are developed, starting from

those based on the simple Mayo– Lewis copolymerization equation, and

having some of them conversion as independent variable. The simplest

models use only the appropriate reactivity ratios of the pair of monomers

to predict the copolymer composition in a CSTR or in several CSTR’s

connected in series. From there, more sophisticated models are built based

on the simple ones as more knowledge is available on the copolymeri-

zation system at hand.

In this section all the mathematical models for copolymerization reactors

with specific characteristics are shown. All the models can be used for bulk

and solution processes. The models can also be used for suspension and

emulsion polymerizations provided that the concentrations used are those inthe reaction locus. In aqueous heterogeneous systems these concentrations

are simply calculated when the monomers have very low solubility in the

aqueous phase, otherwise, they have to be calculated with coupled equations

representing the partitioning of monomers in the different phases present.

Also, it is worth noting that these models are valid not only for

traditional free radical copolymerizations, but also for the relatively new

quasi-living free radical copolymerizations or true living anionic copolym-

erizations (with the proper reactivity ratios and for chains sufficiently long).

In the last two cases one has to be careful with the interpretation of the

average composition of the copolymer, since in a true living system the

composition drift occurs inside each copolymer chain, giving rise to gra-

dient or even block copolymers in extreme cases.

Batch Reactor Model with Conversion

as Independent Variable

For a copolymerization batch reactor the molar conversion X * can be

defined as in Eq. 1, where (from here on) M mi denotes the molar concen-

tration of monomer i and the extra sub-index 0 denotes initial conditions.

X * ¼ M m1;O þ M m2;O M m1 M m2

M m1;O þ M m2;O

ð1Þ

The monomer material balances can be written with conversion as in-

dependent variable (see Appendix 1) resulting in Eqs. 2 and 3:

dM m1

dX *¼

M m1;O þ M m2;O

1 þ M m2

M m1

M m1 þ r 2 M m2

M m2 þ r 1 M m1

ð2Þ




dM m2

dX *¼

M m1;O þ M m2;O

1 þ M m1

M m2

M m2 þ r 1 M m1

M m1 þ r 2 M m2

ð3Þ

The r i denote the copolymer reactivity ratios. Integration of Eqs. 2 and 3

using any standard ordinary differential equations solver with M m1= M m1,O

and M m2= M m2,O at X *=0, provides prediction of molar copolymer

composition F mi* with molar conversion through Eq. 4.

F m1* ¼ Pm1

Pm1 þ Pm2

Pmi ¼ M mi;O M i ð4Þ

Pmi denotes the molar concentration of units of monomer i in the

copolymer. This model can be used in predictive way for estimation of

copolymer composition with conversion by knowing the reactivity ratios.

The translation of integration results of this model from molar to weight

conversion is trivial.

Steady-State CSTR Model with Conversion as

Independent Variable

A brief outline of the derivation of this model is given in Appendix 1.

In order to describe the copolymerization in a CSTR when the output

conversion is known a priori, the following definitions are useful:

PRm1 ¼ 1

1 þ 2 M m2

r 1 M m1

þr 2 M 2m2

r 1 M 2m1

þ 1

2 þr 1 M m1

M m2

þr 2 M m2

M m1

ð5Þ

PRm2 ¼ 1

1 þ 2 M m1

r 2 M m2

þr 1 M 2m1

r 2 M 2m2

þ 1

2 þr 1 M m1

M m2

þr 2 M m2

M m1

ð6Þ

The monomer concentrations in Eqs. 5 and 6 are those inside the reactor at

the steady state. Density of reaction mixture is given by Eq. 7,

r ¼ M m1W m1 þ M m2W m2 þ Pm1W m1 þ Pm2W m2 ð7Þ

where W mi is the molecular weight of monomer i. Monomer and monomer

units in polymer concentrations correspond to those in the reactor at steady




state. The molar conversion increment per reactor D X *, can be defined as in

Eq. 8,

D X * ¼ polymer output polymer input

monomer input

¼

Pm1

y þ

Pm2

y

Fx pm1

VW m1

Fx pm2

VW m2

Fxm1

VW m1

Fxm2

VW m2

ð8Þ

where F is the total mass flow at the reactor inlet; V and y are reactor

volume and residence time respectively, and xmi, x pmi are the mass fractionof monomer i and monomer i units in polymer, respectively, at the reactor

inlet. The weight conversion increment D X is given by Eq. 9

D X ¼ Pm1W m1 þ Pm2W m2 rð x pm1 þ x pm2Þ

rð xm1 þ xm2Þ ð9Þ

The monomer material balances result in Eqs. 10 and 11

PRm1 þ r xm1

W m1

M m1

ð1 D X *Þ

D X *ð M m1 þ M m2Þ ¼ 0 ð10Þ

PRm2 þ r

xm2

W m2 M m2 ð1 D X *Þ

D X *ð M m1 þ M m2Þ ¼ 0 ð11Þ

Solution of non-linear algebraic Eqs. 9– 11, coupled with explicit

expressions 7 and 12 – 15, provides copolymer composition and monomer

composition at steady state in a CSTR if D X (weight) and the inlet weight

fractions are known. The non-linear system is solved for unknowns M m1,

M m2 and D X *. Pm1 and Pm2 are then given explicitly by:

Pm1

Pm2

¼ AIB ð12Þ

where

A ¼a11 ¼ 1 x pm1 a12 ¼ x pm1

W m2

W m1

a21 ¼ x pm2

W m1

W m2

a22 ¼ 1 x pm2

0B@

1CA ð13Þ




B ¼

b11 ¼ YPRm1 þ x pm1

W m1

½ M m1W m1 þ M m2W m2

b21 ¼ YPRm2 þ x pm2

W m2

½ M m1W m1 þ M m2W m2

0B@

1CA ð14Þ

where Y (which represents the product of R p and y) is given in terms of

state variables as:

Y ¼ ð M m1 þ M m2ÞD X *

1 D X * ð15Þ

These equations are useful for situations in which the output conversion of

a CSRT (or each CSTR in a train) can be reasonably estimated a priori.Typical examples include 1) the addition of some amount of a new co-

monomer to an existing process (the maximum amount depends on how

different the reaction rate is expected to be; usually a co-monomer of the

same chemical family is added, allowing large compositions of the second

monomer without a significant effect on the reaction rate), 2) small

variations in inlet composition to manipulate the composition drift in a train

of CSTR’s, 3) any other case in which the manipulated variables of the

reactor can be adjusted to reach a desired conversion (e.g. by the use of a

control system), regardless of the composition feed.

Steady-State CSTR Model Given

Empirical Expression for R p

If an empirical or theoretical expression is available for predicting the

rate of copolymerization, the monomer material balances for a CSTR in

steady state are transformed from those in Eqs. 10 and 11 to the form given

in Eqs. 16 and 17.

PRm1 þ Fxm1

R pVW m1

M m1

y R p

¼ 0 ð16Þ

PRm2 þ Fxm2

R pVW m2

M m2

y R p

¼ 0 ð17Þ

Solution of the non-linear algebraic system (Eqs. 16 and 17) provides M m1

and M m2; Pm1 and Pm2 can be calculated by Eqs. 12–14 with Eq. 18 instead

of Eq. 15.

Y ¼ R py ð18Þ




Steady-State CSTR Model Given

Theoretical Expression for R p

If information is available in the form of kinetic rate constants, as in

Eqs. 19 and 20, a theoretical R p can be inserted into the previous model.

R p ¼ M m1* ðk p11 M m1 þ k p12 M m2Þ þ M m2

* ðk p21 M m1 þ k p22 M m2Þ ð19Þ

M m1* þ M m2

* ¼

ffiffiffiffiffi Ri

k T

r ð20Þ

where M mi* is the molar concentration of polymeric radicals of type i, the

kpij is the propagation rate constant of type i radical with monomer j, Ri isthe radical generation rate and k T is the rate constant of termination.

Dynamics of Copolymerization Composition in CSTR’s

Given Exit Conversion or an Empirical Expression for R p

In this section a simple model for the dynamics of composition in a

copolymerization CSTR is developed, assuming that the output conversion

is known. This model uses only the appropriate reactivity ratios of the pair

of monomers to predict the copolymer composition in a CSTR or in several

CSTR’s connected in series. When the conversion value of a copolymer-

ization CSTR (see Figure 1) is known or can be estimated a priori (see

Steady-State CSTR Model with Conversion as Independent Variable), thematerial balances of the monomers and the monomer units incorporated in

copolymer chains can be written as in Eqs. 21–24:

d ð M j xm1; jÞ

dt ¼ F j1 xm1; j1 F jD X jCm1; j F j xm1; j ð21Þ

d ð M j xm2; jÞ

dt ¼ F j1 xm2; j1 F jD X jCm2; j F j xm2; j ð22Þ

d ð M j x pm1; jÞ

dt ¼ F jD X jCm1; j F j x pm1; j ð23Þ

d ð M j x pm2; jÞ

dt ¼ F jD X jCm2; j F j x pm2; j ð24Þ

where xs, j means the mass fraction of species s contained in reactor R j; the

symbols D X j and M j designate the weight conversion increment per reactor




and the mass of reaction mixture, respectively; in reactor R j, F j stands for

the outlet and inlet mass flow to reactor j. Subscripts refer to the different

chemical species, stream number and reactor. For example, x pm1, j,represents the mass fraction of polymer made of units of monomer 1 in

stream F j (which comes out from reactor R j).

Analyzing Eq. 21, for example, one may notice that the first term on

the right hand side represents the amount of monomer 1 in the feed stream

to reactor, the second term is the consumption of monomer 1 due to the

copolymerization reaction and the third term refers to the amount of

monomer 1 in the outlet stream of reactor. The symbol C is defined in

Eqs. 25 and 26, and stands for the mass fraction of monomer 1 in the

copolymer formed in reactor R j according to the Mayo–Lewis equation. In

Eq. 25, r i designates the reactivity ratio of monomer i, f i* is the molar

fraction of monomer i in the mixture of monomers, and C*i the molar

fraction of monomer i incorporated into the copolymer.

Cm1* ¼

r 1 f m1* 2

þ f m1* f m2

*

r 1 f m1* 2

þ 2 f m1* f m2

* þ r 2 f m2* 2

ð25Þ

Figure 1. Conceptual scheme of a copolymerization reaction carried out in a CSTR.




Eq. 25 represents the Mayo–Lewis equation in molar basis. In order to use

it in the proposed model it is necessary to express it into mass basis using

Eq. 26, where W is the molecular weight of the corresponding monomer.

Cm1 ¼ Cm1

* W m1

Cm1* W m1 þ Cm2

* W m2

ð26Þ

where Ci is the weight fraction of monomer i.

In order to solve the set of Eqs. 21–24, one first expands the derivative

of the products on the left hand side of the equations. For example, for

Eq. 21 this would result in an equation of the form:

M jd ð xm1; jÞ

dt þ xm1; j

d ð M jÞ

dt ¼ F j1 xm1; j1 F jD X jCm1; j F j xm1; j

ð27Þ

which could be solved by knowing the way in which the total mass in the

reactor changes with time and having proper initial conditions. However, in

many practical cases, the following simplifying assumptions are applicable

and facilitate the solution of the model:

. The reaction mass of R j reactor, M j, is constant at any time.

. The reactor R j is well-stirred (CSTR).

. The inlet and outlet mass flows of reactor R j are constant and the

same; there is no mass accumulation in reactor.

In this way, the M j are known at all times and can be taken out of the

derivative, so the proper initial conditions become:

xs; j ¼ xs; j0 ð28Þ

for s=(m1, m2, pm1, pm2). This model is useful for dynamic transitions of

composition at conditions similar to those specified for the model of

Steady-State CSTR Model with Conversion as Independent Variable.

In the case that an empirical expression, R p (T , I , xmi), for the rate of

copolymerization R p is available, then Eqs. 21 –24 are transformed into

Eqs. 29–32.

d ð M j xm1; jÞ

dt ¼ F j1 xm1;

j1 Rp jW m1C

m1;

j

* V F j xm1

;

j ð29Þ

d ð M j xm2; jÞ

dt ¼ F j1 xm2; j1 Rp jW m2Cm2; j

* V F j xm2; j ð30Þ




d ð M j x pm1; jÞ

dt ¼ Rp jW m1Cm1; j

* V F j x pm1; j ð31Þ

d ð M j x pm2; jÞ

dt ¼ Rp jW m2Cm2; j

* V F j x pm2; j ð32Þ

General Kinetic Model with New Approach for

Modeling the Gel Effect

In this section it is assumed that the traditional ‘‘full’’ material balance

will be used; that is, that all the relevant kinetic base parameters (at

conversion approaching zero) are available. In these cases, however, one isoften faced with the difficulty of incorporating reliable descriptions of the

gel effect for prediction of the rate of copolymerization. Here, a new

approach for modeling the gel effect is introduced; this approach is based

on analogies with the diffusion-controlled reactions in heterogeneous

catalytic reactors. In Application Examples the use of this approach is

illustrated using experimental data reported by Garcia-Rubio et al. (1985)

for the copolymerization system styrene–acrylonitrile.

For years a large number of mathematical models have appeared in

order to predict free radical polymerization systems at intermediate and

high conversions, where the gel effect phenomena emerges. Reviews of

these models are carried out by Mita and Horie (1987), Achilias and

Kiparissides (1988), and Tefera et al. (1997a). Among the most important

correlation parameters and concepts used to formulate these models onefinds: conversion, viscosity, polymer fraction, chain entanglement and

theories such as reptation, diffusion-controlled reactions, and free volume

theory. The model proposed in this section makes use of the diffusion-

controlled reaction concept, but using a new approach based on an analogy

with the internal transport process in catalytic pellets found in heteroge-

neous catalytic reactions.

In order to describe the proposed model, a classical terminal model

kinetic scheme and the material balance equations for copolymerization

systems used by Hamer et al. (1981) will be invoked and adapted (see

Appendix 2).

In their paper Hamer et al. (1981) define an empirical factor related

with the termination rate constant for modeling the gel effect. The empirical

factor, gtii=k tii / k tiio, is defined as the ratio of the termination rate constant

observed during the polymerization, k tii, and the termination rate constant

when the conversion is close to zero, k tiio. For the value of the cross-

termination rate constant, k t 12, they chose the geometric mean between the




two homopolymerization termination rate constants multiplied by the factor

f, that can take the value of unity for diffusion-controlled copolymerization

reactions according to North (1963). On the other hand, the model proposed

here is similar to that of Hamer et al. (1981) except for the definition used

for the empirical factor gt to model the gel effect. Instead of gt ,

effectiveness factors, Z p and Zt , are defined in this work to account for

the gel effect in the propagation and termination reactions respectively.

Next, expressions for the effectiveness factors are developed and finally the

complete kinetic model for copolymerization is shown.

In an heterogeneous catalytic reaction (where the reactants are

consumed inside a porous catalyst), two transport processes are known to

occur: the external and the internal transport processes. For the case of the

external transport process in a catalytic particle, the mass transfer of reactants from the bulk fluid to the particle surface is ruled by

concentration gradients of reactants and by the mass transfer coefficient,

k m (see for example Smith (1970)). However, in heterogeneous catalytic

reactions, the catalyst is generally supported and located on the walls inside

porous particles; therefore, the reactants have to diffuse through the pores

of the particle in order to reach the catalyst and participate in a chemical

reaction. This is the main concept behind the analogy for the proposed

model of the polymerization. At a certain point when a bulk polymerization

reaction is taking place, the polymer chains are long enough and their

concentration large enough for entanglements to occur. At that point some

reference volume element of entangled chains can be conceived as a porous

particle. Around and inside the reference volume there will be monomer

molecules and propagating and terminating free radicals.If one looks at existing gel effect models for high conversion

polymerization (Achilias and Kiparissides, 1992; Chiu et al., 1983; Hoppe

and Renken, 1998; Russell et al., 1988; Sharma and Soane, 1988), under the

light of this analogy, their modeling of mass transfer and diffusion

phenomena can be more easily identified with the idea of the external

transport process in the catalytic pellet. Here, we turn our attention to the

analogy of polymerization gel effect with internal resistances in the

catalytic pellet as we consider that the diffusion paths experimented by the

monomer or polymer chains are similar to the paths experimented by the

reactants in the pores of a catalytic pellet.

For propagation reactions at intermediate or high conversion, the

polymerization system is similar to the heterogeneous catalytic system in

the sense that the monomer molecules (reactants) have to diffuse along the

‘‘holes’’ left after polymer chain entanglements have occurred, and have to

reach a free radical in order to propagate the active chain (‘‘catalytic site’’).

Generally, in catalytic reactions the catalyst is not modified when a






Fs ¼ r s

3

ffiffiffiffiffiffiffiffik r p

De

s ð35Þ

Using these definitions, one can get an expression for the evaluation of the

actual rate for the whole catalytic particle, Rc, in terms of the rate evaluated

at the outer surface conditions, Rs, as shown in Eq. 36.

Rc ¼ Z Rs ¼ 1

Fs

1

tanh 3Fs

1

3Fs

kC s ð36Þ

According to Eq. 35, Fs will be small when the particle is small, the

diffusion process is fast or the chemical reaction is very slow. However,

for F

s values greater than 5, a good approximation for Eq. 34 is given byEq. 37.

Z ¼ 1

Fs

ð37Þ

For high values of Fs, the diffusion of reactants through the particle has an

important effect on the rate of reaction, such that the diffusion of reactants

towards the interior of the particle is slow.

The previous analysis is applicable only to specific conditions of

reaction for which an analytical solution is available. Different conditions

of reaction or particle geometries lead to problems without analytical

solution; for this reason other authors have given approximate solutions for

more complex reaction systems (Aris, 1975; Bischoff, 1965). In order to

complete the analogy between the heterogeneous catalytic model andpolymeric systems, the previous analysis will be applied to the propagation

and termination reactions of a polymer reaction system. The assumption

that the propagation and termination reactions are pseudo-first order

reactions will be used as an approximation in order to simplify the analysis.

In propagation reactions the monomer molecules will be considered as

reactants and the free radicals as the catalyst in the particle. In termination

reactions one of the two free radicals will have the role of reactant and the

other one that of the supported catalyst.

The proposed analogy applied to the propagation and termination

reactions of a polymerization (or copolymerization) system can be sum-

marized in Eqs. 38 and 39. In these equations and according to Eq. 36, the

product of the effectiveness factor and the kinetic constant measured at zero

conversion (k po or k to) for each reaction is grouped into an effective kinetic

constant (k p or k t ). At first sight, one can see that at the beginning of the

polymerization Z p or Zt are close to one, the mass transfer through the

mixture of reaction does not have effect on the rate of polymerization and




the chemical process controls the rate. However, at intermediate or high

conversion, when the chains of the polymer begin to entangle, Z p or Zt can

be reduced even orders of magnitude, so the effective kinetic constants

significantly modify their values.

k p ¼ k poZ p ð38Þ

k t ¼ k toZt ð39Þ

In order to take this analogy to a practical level one can calculate the Thiele

modulus either in an empirical way or using Eq. 35 in terms of the polymer

system parameters. The latest approach is more desirable but it is beyond

the scope of this work. Also, considering the level of approximations

used so far, the difficulty of finding reliable diffusion coefficients and

the practical focus of this work, it was decided here to relate the Thiele

modulus with experimental variables more easily measured such as con-

version, viscosity or molecular weight. Given the experimental informa-

tion at hand, an empirical expression in terms of conversion is used in

this work.

It was also decided to use the expression in Eq. 34 for the calculation of

the effectiveness factor in termination reactions due to the fact that mass

transfer limitations for polymeric free radicals start to be considerable at

intermediate conversion (low values of Thiele modulus). On the other hand, for

the calculation of the effectiveness factor in propagation reactions, Eq. 37 will

be used, as this expression is valid for high values of the Thiele modulus when

the diffusion process for the monomer molecules is severely hindered.According to the previous discussion, Eqs. 40 and 41 are the proposed

empirical expressions for modeling the Thiele modulus for propagation and

termination reactions, respectively, in terms of conversion. The resulting

functional forms were found as best fit for experimental data for the

copolymerization system styrene– acrylonitrile. Although the factor A for

modeling the Thiele modulus in propagation reactions is empirical, it can

be easily related with the limiting conversion of the polymerization system.

F p ¼ 1

tanhð A X *Þ ð40Þ

Ft ¼ expð BX *Þ ð41Þ

A summary for the model of this section is shown in Eqs. 42–61. The

result is an extension of Hamer et al. (1981) model including the proposed

approach for modeling the gel effect. Eq. 48 includes an expression to

calculate the variation of the initiator efficiency with conversion; this




expression was modified from the semi-empirical expression reported in

Tefera et al. (1997b), where the free volume theory was used.

dM m1

dt ¼ k p11 M m1 M m1

* k p21 M m1 M m2* ð42Þ

dM m2

dt ¼ k p12 M m1

* M m2 k p22 M m2 M m2* ð43Þ

dI

dt ¼ k d I ð44Þ

M m2* ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi2 f ik d I

k t 11

k p21 M m1

k p12 M m2

!2

þ2k t 12

k p21 M m1

k p12 M m2

þ k t 22

v uuuuutð45Þ

M m1* ¼

k p21 M m1

k p12 M m2

M m2* ð46Þ

X * ¼ M m1o þ M m2o M m1 M m2

M m1o þ M m2o

ð47Þ

f i ¼ 2 f io

1 þ expð X *Þ ð48Þ

Z p ¼ 1

F p

ð49Þ

F p ¼ 1

tanhð A X *Þ ð50Þ

k p11 ¼ k p11oZ p ð51Þ

k p22 ¼ k p22oZ p ð52Þ






APPLICATION EXAMPLES

In this section, several applications for the mathematical models

developed in the previous sections are shown. It was decided not to

illustrate step by step the methodology proposed in General Methodology

for Modeling and Scaling-Up; instead, the most relevant aspects of different

applications were chosen. Three of them were directly taken or arose from

real industrial problems and only one is based on data taken from the

literature. First, a set of plots to determine the copolymer composition in

several configurations of copolymerization CSTR’s are generated with a

steady state CSTR model (see Steady-State CSTR Model with Conversion

as Independent Variable). Next, the application of the dynamic model of a

copolymerization CSTR (see Dynamics of Copolymerization Compositionin CSTR’s Given Exit Conversion or an Empirical Expression for R p) and

comparison of model predictions to experimental data from an industrial

copolymerization process are discussed. In the third example, the same

dynamic model is applied to simulate an industrial copolymerization

process in emulsion where several CSTR’s are connected in series. Finally,

the illustration of the general kinetic model with a new approach for

modeling of the gel effect (see General Kinetic Model with New Approach

for Modeling the Gel Effect) is discussed, using experimental data taken

from the literature. Also, a bifurcation analysis of this model applied to a

CSTR is carried out.

Example 1. Copolymer Composition Prediction in

Different Reactor Configurations

In this section a set of plots for determining the copolymer composition in

several configurations of copolymerization reactors are presented. The kind of

plots presented here are useful for the design of continuous copolymerization

processes when the output conversion can reasonably be predicted a priori and

the reactivity ratios are known, especially if one is interested in the variation of

composition when the reactor configuration is changed. Due to the com-

position drift in copolymerization, these charts are useful in design since they

predict the ‘‘jumps’’ in composition in each reactor of a series of CSTR’s,

which often give rise to heterogeneous products. The plots were generated

solving the model proposed in Steady-State CSTR Model with Conversion as

Independent Variable (steady state) where the weight conversion increment

and monomer composition of the feed stream are known.

Figures 2 and 3 represent two typical cases for pairs of monomers

where r 1<1, r 2<1 (styrene– acrylonitrile) and r 1>1, r 2<1 (styrene– vinyl

acetate), respectively. In these plots, the output copolymer composition in a




single CSTR is given in terms of monomer composition in the feed stream

at different conversions. Two limiting conditions are included: when

conversion is 1 (line of 45, mass fraction of monomer in the feed stream is

the same mass fraction of the monomer incorporated to copolymer) and

when conversion tends to 0 (this case corresponds to the instantaneous

composition of a copolymer synthesized in a batch reactor given directly by

the Mayo– Lewis equation). Figure 4 exhibits the same information of

Figure 3, but with different coordinates.

Figures 5 and 6 show the instantaneous copolymer composition for

every single CSTR of a series of CSTR’s. For this analysis the system

styrene– vinyl acetate was chosen with a feed stream composition of 0.4

mass fraction of styrene. In this case the number of reactors was varied and

Figure 2. Copolymer composition vs. monomer composition in the feed stream of a

single CSTR for the system styrene (1)–acrylonitrile (2) for different output con-versions. Case r 1< 1, r 2<1 (r 1=0.36, r 2=0.078; Garcia-Rubio et al., 1985).




the total or global conversion ( X =1) was divided by the number of reactors

in every set, for example, for the case where the number of CSTR’s was 20,

the corresponding conversion for each reactor of this set was 0.05, therefore

the global conversion is always 1. As in previous analyses, it is interesting

to mention the limiting conditions in this case: when the conversion in one

reactor is close to 1 (case for one CSTR), the instantaneous copolymer

composition is the same as the monomer composition of the feed stream.

On the other hand, when the conversion in every single reactor tends to 0

(case tending to an infinite number of CSTR’s connected in series, e.g.

1000 CSTR’s with X = 0.001 each), the behavior is similar to that of a plug

flow reactor (PFR), which also tends to the Mayo–Lewis equation

behavior. For better visualization, Figure 5 shows the first 10 reactors for

every set, while Figure 6 shows the cases when the number of CSTR’s

Figure 3. Copolymer composition vs. monomer composition in the feed stream of a

single CSTR for the system styrene (1)– vinyl acetate (2) for different output

conversions. Case r 1>1, r 2<1 (r 1=55, r 2=0.01; Hamer et al., 1981).




connected in series tends to infinite ( R20, R100 and R1000). This kind of

analysis finds application in the quantification of the heterogeneity of the

produced copolymer in continuous processes.

Figure 7 corresponds to the behavior of a system of alternating nature

(styrene– maleic anhydride) with nearly ideal values for alternation:

r 1 = 0.001, r 2 = 0.001. The plot shows the output composition from a CSTR

for different output conversions and for different feed stream compositions.

Notice that the value of the output composition in all cases remains almost

flat at 0.5 until the value of the output weight conversion nearly doubles the

value of the feed stream composition. After that conversion the copolymer

composition for monomer 1 (styrene), richer in the feed, rises rapidly with

conversion. This is due to the fact that a unit of each one of the monomers

alternates until one of them is exhausted (also notice that the molecular

weights of both monomers are similar, 104 and 98). A curve is also

Figure 4. Copolymer composition vs. conversion of a single CSTR for the system

styrene (1) – vinyl acetate (2) for different styrene feed compositions ( fst ). Case r 1

>1,

r 2<1 (r 1=55, r 2=0.01; Hamer et al., 1981).




included in order to show the high sensitivity of this system to the values of

the reactivity ratios used. Notice that when values of r 1 = 0.02, r 2 = 0.01

(Brandrup et al., 1999) are used, the behavior deviates significantly from

the ideal one, but one has to be careful with values reported in the lit-

erature, as they show high dispersion.

Example 2. Application of Dynamic Model of a

Copolymerization CSTR

Many industrial copolymerization processes, and their corresponding

products, are sensitive to composition drift, which is well described by the

Mayo–Lewis equation. In order to illustrate the application of the model

described in Dynamics of Copolymerization Composition in CSTR’s Given

Exit Conversion or an Empirical Expression for R p, the prediction of

copolymer composition in a dynamic transition between steady states in an

industrial bulk copolymerization process was considered. Predicted values

Figure 5. Instantaneous copolymer composition vs. number of CSTR’s connected in

series for the system styrene (1)–vinyl acetate (2) with 0.4 mass fraction of monomer

1 in the feed stream.




were compared to industrial experimental data of copolymer compositionobtained by 1H Nuclear Magnetic Resonance.

The experiment started departing from a steady state in which the

monomer composition of the feed stream was known, and at time zero a step

change of composition in the feed stream with a second monomer was

imposed. During the transition, samples were taken at different times and the

copolymer composition measured. The transition was also simulated with the

model to determine how long it would take to reach the new steady state, as

some additional capacitances (tanks) were present in the process. All the

assumptions made for the development of the model were approximately met,

and the difference of monomer composition between the liquid phase and the

vapor phase was ignored. Due to proprietary reasons some details of the

experiment cannot be revealed; therefore, the process diagram has been

simplified and the raw materials will be named only as monomer 1 and

monomer 2, and their reactivity ratios as r 1 and r 2, respectively.

A simplified scheme of the bulk copolymerization process is shown in

Figure 8. The experimental step of monomer composition in the feed stream

Figure 6. Instantaneous copolymer composition vs. number of CSTR’s connected in

series for the system styrene (1)–vinyl acetate (2) with 0.4 mass fraction of monomer

1 in the feed stream.




was from 0– 100% to 10–90% (monomer 1– monomer 2, weight basis).Table 1 shows the process parameters used in the model and their values.

Table 2 shows the conditions defining the initial steady state. The reactivity

ratios used were r 1 = 0.53 and r 2 = 0.58. According to the scheme of Figure

8, one can formulate Eqs. 62–71, which are derived from the basic model

of Dynamics of Copolymerization Composition in CSTR’s Given Exit

Conversion or an Empirical Expression for R p adapted for the specific

configuration of the process.

Eqs. 62 and 63 describe the material balances in reactor 1 ( R1) for

monomer 1 and 2 respectively.

d ð M R1 xm1;1Þ

dt

¼ F o xm1;o þ F 3 xm1;3 F 1 xm1;1 F 1D X 1Cm1;1 ð62Þ

d ð M R1 xm2;1Þ

dt ¼ F o xm2;o þ F 3 xm2;3 F 1 xm2;1 F 1D X 1Cm2;1 ð63Þ

Figure 7. Copolymer composition vs. conversion for styrene (1) – maleic anhydride

(2) alternating copolymerization system. A,B curves calculated with r 1 = r 2 = 0.001,

curve C with r 1 = 0.02, r 2 = 0.01. f 2 is the feed weight fraction of (2).




Eqs. 64 and 65 describe the material balances for monomer 1 and monomer

2, respectively, in reactor 2 ( R2).

d ð M R2 xm1;2Þ

dt ¼ F 1 xm1;1 F 2 xm1;2 F 2D X 2Cm1;2 ð64Þ

d ð M R2 xm2;2Þ

dt ¼ F 1 xm2;1 F 2 xm2;2 F 2D X 2Cm2;2 ð65Þ

Downstream reactor 2 ( R2), a monomer recovery tank ( MR, unit 3) is

located in order to separate non-reacted monomer from the copolymer and

return it to the process. The material balances for the monomers in this

flash tank are Eqs. 66 and 67. They are trivially derived canceling out the

reaction term from the basic model. It is assumed that all the monomer is

recovered, therefore F 2 xm1,2=F 3 xm1,3 and F 2 xm2,2=F 3 xm2,3.

d ð M MR xm1;3Þ

dt ¼ F 2 xm1;2 F 3 xm1;3 ð66Þ

d ð M MR xm2;3Þdt

¼ F 2 xm2;2 F 3 xm2;3 ð67Þ

Finally, material balances for the produced copolymer in each reactor are

necessary. These ones are represented by Eqs. 68–71, which are written in

Figure 8. Conceptual scheme of industrial bulk copolymerization process analyzed,

example 2.




Table 1. Parameters of process used in the model and

their values, example 2.

Parameter Value Unit

xm1,o 0.1 –

xm2,o 0.9 –

M R1 15000 Kg.

M R2 5500 Kg.

M MR 1200 Kg.

D X 1 0.3 –

D X 2 0.5 –

F o 4800 Kg.hr.1

F 1 6000 Kg.hr.1

F 2 6000 Kg.hr.1

F 3 1200 Kg.hr.1

F 4 4800 Kg.hr.1

Table 2. Values at initial steadystate, example 2.

Variable Value

xm2,o 1

xm1,o 0

xm2,3 1

x pm2,4 1

xm2,1 0.7

xm2,2 0.2

xm1,3 0

x pm1,4 0

xm1,1 0

xm1,2 0

x pm2,1 0.3 x pm2,2 0.8

x pm1,1 0

x pm1,2 0






A natural application of this model is to reduce the transition periods

by using different feed-forcing policies. Figure 11 shows the comparison

between the dynamic response of the copolymerization process to a step

input, and the response when the input for the transition is started by a

sudden drop from 100% to 85% of monomer 2 followed by a ramp of

increasing monomer 2 from 85% up to 90% in a linear fashion for 5 hours.

In this last case it is possible to reduce the transition period by 40%.

Example 3. Application of Dynamic Model to Another

Continuous Copolymerization System

In this section, the model proposed in Dynamics of Copolymerization

Composition in CSTR’s Given Exit Conversion or an Empirical Expression

for R p is applied for the simulation of an industrial copolymerization

process where an elastomer is produced by emulsion polymerization. Again,

the dynamic transition for a step change in the feed stream monomer

Figure 9. Dynamic transition for the industrial copolymerization process described

in Figure 8. A step of monomer composition in feed stream from 0–100% to 10–90%

(monomer 1–monomer 2) was applied to the process at time zero.




composition is simulated. Figure 12 shows a conceptual diagram of theprocess that includes a series of CSTR’s.

For this process, Eqs. 21– 24 are applicable, considering only the

copolymer feed term, which uses the cumulative composition of the

previous reactor, and making j=1. . .20.

The simulated transition corresponds to a step change of monomer

composition in the feed stream from 29–71% to 26.5–73.5% (monomer 1–

monomer 2, weight basis), applied to the process at an arbitrary time zero.

The values of the reactivity ratios used were r 1=0.60 and r 2=1.35. The

residence time in each reactor is 15 minutes and the weight conversion

increment in every single reactor is 0.05, such that the total conversion

obtained at the end of the process is 1. These conditions are similar, but not

identical, to those used in an existing industrial process. Exact details are

not revealed due to proprietary reasons. All assumptions described in

Steady-State CSTR Model with Conversion as Independent Variable are

met to a good approximation (including the almost null effect of the small

change in the feed stream monomer composition on the copolymerization

Figure 10. Dynamic transition of the instantaneous, cumulative and global

copolymer composition in the industrial copolymerization process described in

Figure 8.






Conversion as Independent Variable on a mass basis. Once the initial

conditions for every reactor of the process were known, the simulation for

the step perturbation in the feed stream was carried out. Figure 13 shows

the simulation results (only some reactors are shown). As can be seen,

reactor 20 takes about 9 hours for reaching the new steady state. Since

several CSTR’s are connected in series, the process approaches the

behavior of a tubular reactor with dispersion. In plug flow reactors the input

perturbation takes one residence time to be reflected in the output stream.

On the other hand, in a CSTR the change in the output occurs from the

very beginning of the perturbation, but the process takes around three

residence times to reach the steady state. The case illustrated in this section

lies somewhere between these two extreme cases; for this reason, for

reactor one the step change starts to be detected from the beginning of thesimulation, but at the final reactor the change starts to be reflected after

Figure 13. Dynamic transition for the industrial copolymerization process described

in Figure 12. A step of monomer composition in feed stream from 29–71% to 26.5–

73.5% (mon 1–mon 2) was applied to the process at time zero.




around 200 minutes (less than one residence time) and the settling time for

the whole process is around two residence times.

Example 4. Illustration of the General Kinetic Model

with New Approach for Modeling the Gel Effect

In order to illustrate the use of the kinetic model of Genetic Kinetic Model

with New Approach for Modeling the Gel Effect, the model was fitted to

experimental data for batch copolymerization of styrene (m1) with acry-

lonitrile (m2) taken from the work of Garcia-Rubio et al. (1985), in which

variations of initiator, temperature and ratio of monomers were studied.

Table 3 shows the values used for the model parameters. Figures 14– 17

Table 3. Parameter values used for model testing (copolymerization system

styrene (1)–Acrylonitrile (2)), example 4.

Param. Unit Value Reference

k d s1 1.581015 exp (30.8/ RT ) Jaisinghani and

Ray (1977)

k p11o l mol1 s1 1.051107 exp (3557/ T ) Husain and

Hamielec (1978)

k p22o l mol1 s

1 1.047108 exp (3663/ T ) Brandrup and

Immergut (1975)

r 1 – 0.36 Garcia-Rubio et al.(1985)

r 2 – 0.078 Garcia-Rubio et al.

(1985)

k t 11o l mol1 s1 1.26109 exp (1.68/ RT ) Jaisinghani and Ray

(1977)

k t 22o l mol1 s

1 782106 Brandrup et al. (1999)

f io – 0.4 Odian (1990)

rm1 g l1

9240.918(T 273.15) Tefera et al. (1995)

rm2 g l1 806 Brandrup and Immergut

(1975)

W m1 g mol1 104.14 Brandrup and Immergut

(1975)

W m2 g mol1 53.06 Brandrup and Immergut

(1975) R Kcal mol

1K 1

1.987103

Brandrup and Immergut

(1975)




compare the experimental data with the model simulation. The values of

parameters A and B were fitted in each case; for parameter A the values

used were the limiting conversions taken directly from the experimental

data. It is worth highlighting that the model uses only two fitting

parameters to predict the experimental data in a wide range of initial

monomer composition.

According to the results shown in Figures 14– 17, one can conclude

that the empirical parameter A increases its value with increments of the

temperature of reaction, and must reach the value of one when the

temperature of reaction is greater than the glass transition temperature.

Also, A stays constant when the concentration of initiator changes. For the

case of B, this parameter shows an inverse relationship with temperature,

which is consistent with the fact that a high temperature corresponds to a

low B, low Ft and effectiveness factor close to one. Physically, this means

Figure 14. Comparison between model predictions and experimental data for the

system styrene –acrylonitrile. Initial styrene composition in % mol, 0.01 molar

concentration for initiator AIBN, temperature 40C. Values for empirical parameters

are A =0.9 and B =14.752.




that at high temperature, diffusion limitations are less important because the

reaction system is less viscous; therefore the monomer molecules and

macromolecules of polymer can diffuse more easily. Also, B shows an

inverse relationship with initiator concentration. Again, diffusion limitations

are less important at high concentration of initiator because at these

conditions the chains of polymer exhibit lower molecular weight, reducing

the viscosity of the reaction media.

When more detailed kinetic information is available, such as the mo-

del considered here, one can make a more complex analysis of the reactor

behavior. Here, the kinetic model proposed in General Kinetic Model with

New Approach for Modeling the Gel Effect and fitted to experimental

data for the system styrene– acrylonitrile is applied to a copolymerization

CSTR in order to carry out a bifurcation analysis of the reactor. The

bifurcation analysis is based on the methodology proposed by Hamer et al.


system styrene– acrylonitrile. Initial styrene composition in % mol, 0.05 molar


are A =0.9 and B =11.808.




(1981) for the isothermal case. Eqs. 72–74 show the material balances for

the reactants.

V dM m1

dt ¼ ð M m1 f M m1Þq M m1ðk p11P* þ k p21Q*ÞV ð72Þ

V dM m2

dt ¼ ð M m2 f M m2Þq M m2ðk p12P* þ k p22Q*ÞV ð73Þ

V dI

dt ¼ ð I f I Þq k d IV ð74Þ

For the bifurcation analysis, the dimensionless groups reported in Hamer

et al. (1981) were used, and the software XPP-AUTO (Ermentrout) was

employed for its solution. The results for the analysis using the parameters

from Figure 15 case, but applied to a CSTR, are shown in Figure 18,


system styrene –acrylonitrile. Initial styrene composition in % mol, 0.01 molar


are A =0.9 and B =10.372.




where the dimensionless parameters Da and x1 are defined (as in Hamer et

al. (1981)) by Eqs. 75 and 76, respectively.

Da ¼ yk p22

ffiffiffiffiffiffiffiffiffik d I f

k t 22o

r ð75Þ

x1 ¼ M m1 f M m1

M m1 f

ð76Þ

According to Figure 18, there are three possible steady states for a

determined value of Da. Given that Da is proportional to the reactor

residence time, it is possible to reach higher conversion with the same

residence time (same size of reactor or same flow), which is desirable for

economic reasons.


system styrene– acrylonitrile. Initial styrene composition in % mol, 0.05 molar


are A =0.9 and B =9.441.




Figure 19 shows diagrams generated from the bifurcation analysis

under different operating conditions. In all cases the abscissa is the

Damkohler number and the ordinate is the output conversion. The diagrams

in the upper half (A, B, C, D, E and F) were calculated at 40 C temperature

and the lower half at 60C. Rows 1 and 3 (A, B, C, G, H, I) correspond to

low initiator concentration in the feed: 0.01 mol L1 and rows 2 and 4 (D,

E, F, J, K, L) to high initiator concentration (0.05 mol L1). Finally, mol

fraction of styrene in the feed ( xm1) was changed by columns. Diagrams in

the first (left) column (A, D, G, J) correspond to xm1=0.5, diagrams in the

middle column (B, E, H and K) to xm1=0.7 and those in the right hand side

column (C, F, I, L) to xm1=0.9. All the diagrams are calculated at

isothermal conditions; therefore, the multiplicities must be due to the gel

effect. For isothermal conditions to be achieved, it is assumed that the

reactor system has enough capacity for heat removal for all the conditions

analyzed. Of course, this may not be true for a specific real reactor, but it is

Figure 18. Bifurcation analysis of a copolymerization CSTR for conditions of

Figure 16, case of 50% mol of initial styrene composition.




interesting to study pure kinetic effects on reactor multiplicities. From the

diagrams one can make the following observations:

. In all cases there is a steady state multiplicity of 3, with 2 stable steady-

state regions (darker line) and an unstable steady-state region.

Figure 19. Maps from bifurcation analysis of a copolymerization CSTR for

conditions of Figure 16, case of 50% mol of initial styrene composition.




. A temperature increase tends to remove the multiplicities, apparently

because the larger reaction rates brought about by the higher

temperature make the steady state more easily ‘‘jump’’ towards higher

conversions. The temperature increase also shifts the multiplicities

towards larger values of the Damkohler number.

. An increase in the feed initiator concentration has a qualitatively

similar effect to that of a temperature increase.

. Higher styrene compositions in the feed also tend to remove the

multiplicities and shift them towards higher values of the Damkohler

number. The loss of multiplicities at higher styrene compositions

seems to be related to less violent reactions linked to a ‘‘softer’’ gel

effect that occurs when styrene composition is increased. Simulations

in which a solvent is included (not shown) confirm that a less violentreaction diminishes the presence of multiplicities (multiplicities

disappear at even low solvent contents).

FINAL REMARKS

A framework for building models of increasing level of detail/com-

plexity, suitable for design and scaling-up of some classes of industrial copo-

lymerization reactors, has been presented. The models discussed are focused

on copolymer composition and copolymerization rate; other aspects, such as

molecular weight distribution and transport issues, are not considered in these

models or are lumped into empirical parameters. The models presented are not

useful for all possible situations, but they have proven useful for several casesfound in industry. The approach presented here is practical, but using as much

information of scientific nature as possible. In the same practical spirit, a new

approach for modeling the gel effect phenomena, based on an analogy with

internal resistances to diffusion in a catalytic pellet, has been introduced.

Additional work is needed to test this analogy further, but its main value

resides on the use of a concept already familiar to the chemical engineer.

Hopefully this work will contribute to a more extended use of mathematical

modeling tools in the copolymerization reactors area.

APPENDIX 1

Outline of Derivation for Batch Reactor Model with

Conversion as Independent Variable

Material balances for monomers in a batch reactor with terminal model

kinetics are:

dM m1

dt ¼ PRm1 ðA1-1Þ




dM m2

dt ¼ PRm2 ðA1-2Þ

where

PRm1 ¼ k p11 M m1* M m1 þ k p21 M m2

* M m1 ðA1-3Þ

PRm2 ¼ k p12 M m1* M m2 þ k p22 M m2

* M m2 ðA1-4Þ

Taking the time derivative of the molar conversion Eq. 1 for a batch reactor

we get:

dX *

dt ¼ 1

M m1;O þ M m2;O

dM m1

dt þdM m2

dt

ðA1-5Þ

Using the chain rule for derivatives and the inverse function theorem, as

well as Eqs. A1-1 and A1-5, it is possible to write:

dM m1

dX * ¼

1

dX *dt

dM m1

dt

¼ ð M m1;O þ M m2;OÞ

1

1 þdM m2

dM m1

0BB@

1CCA

ðA1-6Þ

Again, the term dM m2 / dM m1 can be calculated by using the chain rule for

Eqs. A1-1 and A1-2. The calculation of this term followed by the

application of the quasi-steady state assumption for radical types (Eq. A1-7),

and the use of the standard definition for reactivity ratios r 1 and r 2 results in

Eq. 2. A similar derivation can be used in order to obtain Eq. 3.

M m1* ¼

k 21 M m1 M m2*

k 12 M m2

ðA1-7Þ

Outline of Derivation for Steady-State CSTR Model with

Conversion as Independent Variable

Material balances for monomers in a CSTR with terminal model

kinetics can be written as follows:

dM m1

dt ¼ PRm1 þ

Fxm1

VW m1

M m1

y ðA1-8Þ

dM m2

dt ¼ PRm2 þ

Fxm2

VW m2

M m2

y ðA1-9Þ




The total rate of polymerization, R p, is given by:

R p ¼ M m1* ðk p11 M m1 þ k p12 M m2Þ þ M m2

* ðk p21 M m1 þ k p22 M m2Þ

¼ PRm1 þ PRm2 ðA1-10Þ

Making the derivatives zero (steady state) in Eqs. A1-8 and A1-9, dividing

them by R p and using the QSSA for radical types (see Eq. A1-7) one gets

Eqs. A1-11 and A1-12:

PRm1 þ Fxm1

R pVW m1

M m1

R py ¼ 0 ðA1-11Þ

PRm2 þ Fxm2

R pVW m2

M m2

R py ¼ 0 ðA1-12Þ

Equations for balances of polymer in the form of monomer units in

polymer, in steady state, are:

dPm1

dt ¼ PRm1 þ

Fx pm1

VW m1

Pm1

y ¼ 0 ðA1-13Þ

dPm2

dt ¼ PRm2 þ

Fx pm2

VW m2

Pm2

y ¼ 0 ðA1-14Þ

Using the last two equations one can get expressions for Pm1 / y and Pm2 / y.

Replacing these expressions in the definition of molar conversion (Eq. 8),re-writing the denominator of Eq. 8 in terms of the steady state equations

for monomer balances and rearranging, one gets:

D x* ¼ R p

R p þ M m1

y þ

M m2

y

ðA1-15Þ

From Eq. A1-15 one can write:

R p ¼ ð M m1 þ M m2ÞD x*

yð1 D x*Þ ðA1-16Þ

Using expression A1-16 in the monomer balances A1-11 and A1-12 and

noting that F / V =r / y, one may get Eqs. 10 and 11. Dividing the steady state

form of polymer balances (Eqs. A1-13 and A1-14) by R p, replacing F / V

by r / y, writing r in terms of Eq. 7 and solving for Pm1 and Pm2, one gets

Eqs. 12–15.




APPENDIX 2

Equations of Hamer et al. (1981) Model

The main chemical reactions are:

I !k d

2 R*

R* þ M m1 !k i1

M m1*

R* þ M m2 !k i2

M m2*

Initiation

M m1* þ M m1 !k p11

M m1*

M m1* þ M m2 !

k p12

M m2*

M m2* þ M m1 !

k p21

M m1*

M m2* þ M m2 !

k p22

M m2*

Propagation

M m1* þ M m1

* !k t 11

Inactive Polymer

M m1* þ M m2

* !k t 12

Inactive Polymer

M m2* þ M m2

* !k t 22

Inactive Polymer

Termination

The corresponding material balances for the monomer species are obtained

using the long chain assumption (Ray, 1972), resulting in Eqs. A2-1 andA2-2

dM m1


* k p21 M m1 M m2* ðA2-1Þ

dM m2


* k p22 M m2 M m2* ðA2-2Þ

Using the quasi-steady state assumption and making a material balance for

the free radical species, it is possible to obtain Eqs. A2-3 and A2-4:

k p12 M m2 M m1* ¼ k p21 M m1 M m2

* ðA2-3Þ

M m1* þ M m2

* ¼ 2 f ik d I

k t 11 þ 2k t 12 þ k t 22

12

ðA2-4Þ




Solving simultaneously Eqs. A2-3 and A2-4, explicit expressions can be

obtained for the free radical species as in Eqs. A2-5 and A2-6.

M m1* ¼

k p21 M m1

k p12 M m2

M m2* ðA2-5Þ

M m2* ¼

2 f ik d I

k t 11

k p21 M m1

k p12 M m2

2

þ2k t 12

k p21 M m1

k p12 M m2

þ k t 22

8>>><>>>:

9>>>=>>>;

12

ðA2-6Þ

NOMENCLATURE

A empirical factor for modeling the Thiele modulus in propagation

reactions

B empirical factor for modeling the Thiele modulus in termination

reactions

C s molar concentration evaluated at outer surface conditions

CSTR continuous stirred tank reactor

Da Damkohler number

De effective diffusion coefficient

F mass flow

f mass fraction of a monomer in the monomer mixture f * molar fraction of a monomer in the monomer mixture

F * molar fraction of monomer units of some type in the copolymer

f i initiator efficiency

gt empirical factor for modeling gel effect in termination reactions

I molar concentration of initiator

k first order rate kinetic constant

k rate kinetic constant

k d dissociation rate constant for the initiator

k m mass transfer coefficient

k p propagation rate constant of a specific propagation reaction

k po propagation rate constant of a specific propagation reaction at zero

conversion

k p average propagation rate constant of a specific propagation reaction

k t average termination rate constant of a specific propagation reaction

k T global termination rate constant for termination reactions

k t termination rate constant of a specific termination reaction




k to termination rate constant of a specific propagation reaction at zero

conversion

M mass inside reactor

M molar concentration of a specific monomer

M mj* molar free radical concentration of a growing polymer chain with

terminal unit of monomer type j

P molar concentration of polymer formed from specific monomer

units

P* free radical molar concentration of a growing polymer chain with

terminal unit of monomer type 1

PFR plug flow reactor

PR polymerization rate of a specific monomer

q volumetric flowr 1,2 copolymerization reactivity ratio of monomer type 1 or type 2,

respectively

R reactor

R* free radical molar concentration generated from decomposition of

initiator

Rc actual rate for the whole catalytic particle

Ri rate for initiation reactions

R p rate of copolymerization

r s radius of a spherical catalytic particle

Rs rate evaluated at outer surface conditions

t time

T Temperature

V reactor volumeW molecular weight of a specific monomer

X weight conversion

x weight fraction of a specific chemical entity

X * molar conversion

Subscripts

f feed stream

j reactor number or stream number

l number of units of monomer type 1 incorporated to the copolymer

m1 monomer type 1

m2 monomer type 2

mi monomer type i or polymer from type i monomer

n number of units of monomer type 2 incorporated in the

copolymer

o initial conditions or conditions at time or conversion zero

pm1 polymer from monomer type 1

pm2 polymer from monomer type 2




r number of units of monomer type 1 incorporated in the copolymer

s number of units of monomer type 2 incorporated in the copolymer

Greek Letters

r p density for a catalytic particle

Z effectiveness factor

Z p effectiveness factor for propagation reactions

Zt effectiveness factor for termination reactions

C mass fraction of monomer incorporated into copolymer

D X * molar conversion increment per reactor

F p Thiele modulus for propagation reactions

Fs Thiele modulus for spherical particle

Ft Thiele modulus for termination reactions

r density or density of reaction mixturef factor to calculate the cross termination rate constant

C* molar fraction of monomer incorporated into copolymer

y residence time of corresponding reactor

D X weight conversion increment per reactor

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Received October 15, 2002

Accepted May 5, 2003


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A Practical, Systematic Approach for the Scaling‐Up and Modeling of Industrial Copolymerization Reactors