Micromixing effects on the dynamic behavior of continuous free-radical solution polymerization tank reactors

Accepted Manuscript

Micromixing Effects on the Dynamic Behavior of Continuous Free-Radical Solution Polymerization Tank Reactors Bruno F. Oechsler , Pr´ıamo A. Melo , Jose´ Carlos Pinto PII: DOI: Reference:

S0307-904X(16)30534-0 10.1016/j.apm.2016.10.019 APM 11380

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

3 December 2015 24 August 2016 6 October 2016

Please cite this article as: Bruno F. Oechsler , Pr´ıamo A. Melo , Jose´ Carlos Pinto , Micromixing Effects on the Dynamic Behavior of Continuous Free-Radical Solution Polymerization Tank Reactors, Applied Mathematical Modelling (2016), doi: 10.1016/j.apm.2016.10.019

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ACCEPTED MANUSCRIPT Highlights • Micromixing effects on the behavior of free-radical polymerization CSTRs;

• IEM model describes mixing effects for a complex reaction system; • Multiple steady states and oscillations can be predicted under partial segregation;

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• Partially segregated CSTRs may present more complex dynamic behavior

ACCEPTED MANUSCRIPT

Micromixing Effects on the Dynamic Behavior of Continuous Free-Radical Solution Polymerization Tank Reactors

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Bruno F. Oechsler, Príamo A. Melo, and José Carlos Pinto1

Programa de Engenharia Química / COPPE Universidade Federal do Rio de Janeiro Cidade Universitária, CP 68502

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Rio de Janeiro, RJ 21941-972 – Brasil

Abstract – The degree of mixing in polymerization reactions can be influenced by various factors that can also affect the reactor performance. For this reason, a detailed

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micromixing model was implemented to study the effects of micromixing on the dynamic behavior of continuous free-radical solution polymerization tank reactors. The

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reactor model was used to perform the bifurcation analysis of the reacting system, paying special attention to the effect of micromixing parameters on the reactor behavior.

observed

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The bifurcation study showed that multiple steady-states and periodic oscillations can be under

partially segregated

micromixing conditions. Moreover, the

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micromixing model was able to describe the dynamic responses presented by perfectly mixed and completely segregated reactors. These results indicate that this class of

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reactors can exhibit more complex dynamic behavior than shown until now.

Keywords: CSTR Reactor; Micromixing; Bifurcation; Steady State Multiplicity; Periodic Oscillations.

1- Corresponding author: [email protected], Tel:+55-21-39388337, FAX:+5521-39388300

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ACCEPTED MANUSCRIPT 1. Introduction Free radical polymerization reactions conducted in continuous stirred tank reactors (CSTRs) play an important role in industrial polymerization processes, since most freeradical polymerizations are performed in this class of reactors. This explains why significant efforts have been made to develop efficient continuous processes conducted in stirred tanks, as the continuous operation usually benefits the performance of high

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capacity polymerization plants (Hamielec & Tobita, 2005).

In order to successfully design and operate polymerization processes, mathematical models can be very useful as an accurate representation of the physical and chemical phenomena involved in the polymerization system allows for a better understanding of

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the reactor dynamics and control. An efficient mathematical model for a particular polymerization system must take into consideration many fundamental physicochemical aspects of the reaction environment, including a proper description of the chemical reaction kinetics, of mixing effects, of changes of the physico-chemical properties as functions of reaction temperature and composition, among others (Brooks,

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1997).

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One of the main characteristics of solution polymerization reactions is the pronounced increase of the viscosity of the reaction medium with an increase in monomer

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conversion (a significant increase of the system viscosity with the monomer conversion can also be expected in other heterogeneous reactions, such as emulsion

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polymerizations). The increase in viscosity can exert an enormous influence on the reactor performance, affecting the reaction kinetics, the rate of heat removal and the rates of mass and momentum transfer. In particular, the increase in viscosity leads to a

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decrease in the heat transfer coefficient and therefore of the cooling capacity of the reactor (Reichert & Moritz 1989).

Due to the increasing viscosity, the mixing time required to achieve a desired degree of homogeneity inside the reactor can also increase, affecting the micro and macromixing characteristics of the reacting medium. As a result of the increasing viscosity, the reaction medium may become increasingly segregated and the residence time distribution may change due to the possible formation of dead zones and short-circuits.

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ACCEPTED MANUSCRIPT Therefore, monomer conversion, reactor performance, molar mass distribution and the quality of the end products can be affected directly by variations of the medium viscosity (Reichert & Moritz 1989).

Proper micromixing of reactants constitutes an important aspect of polymerizations performed in both homogeneous and heterogeneous systems (Pinto, 1990b). It is

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important to observe that a perfect mixing hypothesis may not be adequate to describe polymerization reactors even when solvents or a dispersion phase are added to the reaction medium. In practice, any polymerization system may present some degree of partial segregation which may vary with the progress of the polymerization due to higher viscosities (in the case of homogeneous systems) or a reduction of the rates of

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particle coalescence (in the case of heterogeneous systems). The behavior of real continuous stirred tank reactors and the properties of the resulting polymer materials probably lay between the extreme cases of maximum micromixing and complete segregation (Reichert & Moritz, 1989).

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Analyses of both stationary and dynamic behavior of free-radical solution polymerization reactors have been widely reported in the literature over the last decades.

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The complex nonlinear dynamic behavior presented by these systems is connected mainly to the high exothermicity of polymerization reactions, which may lead to a

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number of processes instabilities (Papavasiliou & Teymour, 2005; Villa & Ray, 2000). Due to the intrinsic coupling between thermal and kinetic phenomena, when the rate of

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polymerization increases the temperature of the reacting system also increases (assuming that the capacity of heat removal is limited), resulting in an additional increase in the reaction rate. This positive feedback mechanism is eventually halted

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when the monomer concentration falls below a critical value (due to the increasing rates of monomer consumption), leading to a decrease in the reaction rate and a consequent decrease in the reactor temperature. However due to the continuous feed of monomer in a continuous processes the monomer concentration can increase again and lead to a new positive feedback period and the continuous repetition of an oscillatory cycle (Pinto, 1994).

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ACCEPTED MANUSCRIPT Although stability analyses is essential for an assesment of operation safety, operation in unstable regions is not necessarily synonymous with process safety risk. Polymerization reactors can be operated in unstable regions whenever this can allow for economical and safety benefits as long as proper control procedures are developed (Reichert & Moritz 1989). Furthermore, oscillatory operation may allow for production of polymer materials with enhanced quality, when compared to products obtained at steady-state

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conditions (Pinto, 1991).

The systematic use of parametric continuation techniques and bifurcation analysis in polymerization systems is due to the pioneering work of Jaisinghani & Ray (1977), who reported the existence of multiple steady states and Hopf bifurcations in the continuous

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bulk polymerizations of methyl methacrylate and styrene performed under isothermal conditions. Afterwards, the occurrence of distinct patterns of stationary bifurcation (including S-shaped, mushroom and isolated branches of steady states) and sustained periodic oscillations were obtained as functions of the mean reactor residence time for non-isothermal homopolimerizations and copolymerizations performed in bulk and

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solution reactors (Schmidt & Ray, 1981; Hamer et al., 1981). The experimental validation of steady-state multiplicity phenomena and agreement with model predictions

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was originally presented by Schmidt et al. (1984).

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Teymour & Ray (1989) were the first to provide experimental evidences for sustained periodic oscillations in solution polymerizations conducted in stirred tank reactors. In

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subsequent work, the authors showed that experimental data obtained for the solution polymerization of vinyl acetate in a bench scale continuous reactor were in good agreement with dynamic simulations and bifurcation analysis performed with a process

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model (Teymour & Ray, 1992a). More complex dynamic phenomena, including multiplicity of periodic orbits, isolated branches of periodic solutions and perioddoubling cascades leading to chaos, were predicted for polymerizations performed in industrial scale reactors (Teymour & Ray, 1992b, 1991).

Pinto & Ray (1995a,b; 1996) extended the bifurcation analysis to continuous solution copolymerizations of vinyl acetate and methyl methacrylate in the presence and absence of inhibitors. They showed that the bifurcation diagrams can be extremely sensitive to

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ACCEPTED MANUSCRIPT variations in the feed composition. Furthermore, they showed that experimental steadystate and dynamic data were in good agreement with theoretical results provided by dynamic simulations and bifurcation analyses. More complex dynamic behavior was predicted for full-scale copolymerization reactors, although constrained to very small regions of the parameter space; explaining why complex oscillatory behavior has not

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been unequivocally observed experimentally in polymerization systems.

Based on the previous results Freitas Filho et al. (1994) proposed a general mathematical model for bulk polymerization reactors, without defining a priori the specific analyzed reaction system, attempting to identify the generality of steady-state multiplicity phenomena in these reacting systems. The authors showed that free-radical

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bulk polymerization reactors can always present an operating region where as many as five steady states occur. It was also shown that the existence of the strong gel effect can significantly reduce the size the operating range where the five steady-states are expected to be present.

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Melo et al. (2001) extended the previous analysis and showed that self-sustained oscillatory solutions can also be regarded as a generic dynamic characteristic of this

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class of polymerization reactors, regardless of the particular analyzed chemical system. It was shown that oscillatory solutions may occur in ranges of residence times and

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temperatures of commercial interest and that the bifurcation diagrams of periodic orbits are very sensitive to variations in the feed initiator concentration. The increase of the gel

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effect was shown to inhibit the onset of self-sustained oscillatory solutions.

It must be emphasized that all the papers described above assumed that the perfect

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micromixing hypothesis was valid. Consequently the analysis of the influence of micromixing on the dynamic behavior of solution polymerization stirred tank reactor remains open in the literature. Previous studies have investigated the influence of imperfect mixing on the dynamic behavior of polymerization reactors by assuming that the reactors were constituted by compartments (Marini & Georgakis, 1984a,b; Tosun, 1992). These works showed that the bifurcation responses were sensitive to the compartmentalization of the reactor, indicating that the dynamic behavior of the reactor can be affected by mixing effects. However, these works did not use a micromixing

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ACCEPTED MANUSCRIPT model to describe the characteristics of the polymer mixture. Hence it was not possible to provide an analysis of partial segregation and comparison with the extreme limits of perfect mixing and complete segregation.

Pinto (1990b) investigated the effect of complete mass segregation on the dynamic behavior of continuous suspension polymerization reactors. A mathematical model was

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proposed to describe completely segregated particles, subject to perfect macromixing. It was also assumed that the particles did not experience any temperature gradients and that temperatures were uniform throughout the reactor. The mathematical analysis showed that this completely segregated system could not present self-sustained oscillations, in contrast to results obtained when the perfect micromixing assumption is

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made (Pinto, 1990a). This indicates that micromixing effects can exert significant influence on the bifurcation structure of polymerization systems.

Several other studies have used compartmental models to illustrate the influence of imperfect mixing on the stability behavior of polymerization reactors and polymer

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properties. Zhang & Ray (1997) suggested the use of a compartmental model to describe the imperfect mixing at the feed region of a polymerization reactor characterized by fast

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initiator decomposition and low residence time. They showed that imperfect mixing could lead to lower monomer conversions and broader molecular mass distributions

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when compared with the perfect mixing conditions. Villa et al. (1998) used a similar model to describe polyethylene autoclave reactors and observed that imperfect

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macromixing could lead to larger regions of unstable reactor operation. Wells & Ray (2005) studied the effects of macromixing on the stability of adiabatic reactors used for production of low density polyethylene and showed that imperfect mixing of the feed

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stream can expand the region of stable operation in these reactors, due to the low efficiency of the initiator. Taken together, these works indicate that mixing effects can exert a significant influence on the bifurcation structure of polymerization systems.

Based on the previous discussion and to the best of our knowldge, no previous study has addressed the effects of non-ideal mixing on the dynamic behavior of free-radical solution polymerization reactors, describing the mixture of the polymer solution with the aid of a detailed and rigorous micromixing model. As described in the previous

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ACCEPTED MANUSCRIPT paragraphs, prior studies considered perfect micromixing or addressed the mixing problem using compartmentalized models. For this reason, the present work investigates the influence of micromixing effects on the dynamics of free-radical solution polymerization reactors with the aid of a detaile micromixing model. The micromixing model used here is known as the Interaction by Exchange with the Mean (IEM) and was proposed initially by Harada et al. (1962) to describe micromixing effects, i.e. the

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coalesence and redispersion of dispersed droplets on heterogeneous reactions (Harada, 1968). Subsequently Villermaux & Devillon (1972), Villermaux (1986) and Costa & Trevissoi (1972) used the IEM model to describe micromixing effects on homogeneous stirred tank reactors. The IEM model was originally designed for isothermal and steadystate systems. It is extended here to describe dynamic free-radical solution

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polymerization systems in which thermal effects are also considered. The reactor model developed previously by Melo et al. (2001) has been equipped with the IEM micromixing model to allow for micromixing effects. It is shown that these reactors may exhibit multiple steady states (including up to thirteen stationary states and isolas) under conditions of partial segregation. Furthermore, the bifurcation study showed that

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multiple steady-states and periodic oscillations can be observed under partially segregated micromixing conditions, where perfectly mixed and completely segregated

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reactors present trivial dynamic responses.

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2. Reactor Mathematical Model

The IEM model describes micromixing effects in continuous reactors given a well-

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defined Residence Time Distribution (RTD). The model assumes that individual fluid elements, initially segregated in the feed stream are subjected to a mechanism of micromixing characterized by mass and heat transfer with all fluid elements present in

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the reactor (Fox et al., 1990). Figure 1 illustrates the limiting cases of complete segregation and maximum micromixing predicted by the IEM model.

The model comprises a set of population balance equations where fluid elements are fed into the reactor with age equal to zero. As the elements remain inside the reactor, an internal age  can be assigned to individual fluid elements (Fox & Villermaux, 1990). This is equivalent to stating that the system presents a singular feed condition, as discussed by Fox & Villermaux (1990a), with the following properties:

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,   0 f e     0,   0

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(1)

Figure 1: Micromixing limits in continuous chemical reactors: (a) complete segregation, (b) maximum micromixing.

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In this work, the IEM model is used to describe micromixing effects in a polymerization

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solution system performed in continuous stirred tank reactors. Following Freitas Filho et al. (1994) and Melo et al. (2001), a classical free-radical polymerization mechanism,

Kd I   2R*

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including initiation, propagation and termination steps, is adopted:

Initiation Initiation

p Pj  M   Pj 1

Propagation

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K1 R*  M   P1

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K

Ktc Pi  Pj  i  j

Termination by combination

Ktd Pi  Pj  i   j

Termination by disproportionation

It is assumed that each element has the same fluid volume and properties, such as density and specific heat. Balance equations for the average state variables are conveniently obtained by integrating the population balance equations over the internal age distribution (Fox & Villermaux, 1990). In this work, the exponential RTD function

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ACCEPTED MANUSCRIPT is adopted to describe the reactor macromixing behavior. As discussed by Oechsler et al. (2016), heterogeneous reactors, including suspension and emulsion reaction systems, can be simultaneously well mixed in the sense of macromixing (residence time distributions, RTD, can be close to the RTD of stirred tank reactors) and partially segregated in the sense of micromixing (mass and heat transfer among suspended particles can be limited). Although these systems can present the RTD of a perfect

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mixing reactor, the addition of surfactants and suspending agents to avoid the coalescence of the organic particles suspended in the aqueous medium, in order to control the particle size distributions, make these systems partially segregated. When the amount of water and suspending agents are sufficiently high, it is reasonable to assume that the droplets suspended in the aqueous medium do not interact, so that no mass

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transfer takes place among them. Therefore, the resulting set of governing model equations is obtained.

Local balance for initiator:

(2)

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I l  ,t  I l  ,t    +  I ( , I l , T l )  bI  I l  ,t I (t )  t 

where the specific initiation rate is given by:

(3)

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 E  I ( , I l , T l )  K d 0 exp  l d  I l  , t  .  RT  , t  

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Local balance for monomer:

M l  ,t  M l  ,t    +   p ( , M l , Pl , T l )  bm  M l  ,t M (t )  . t 

(4)

Local energy balance:

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ACCEPTED MANUSCRIPT  T l  ,t  T l  ,t     +   p ( , M l , P l , T l )H rV  bhV T l  ,t T (t )  t    

 C pV 

hAT T l  ,t Tc

(5)

 

where the specific rate of polymerization is given by:

(6)

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 E   p ( , M l , Pl , T l )  K p 0 exp  l p  M l  , t  Pl  , t  .  RT  , t  

The micromixing parameters ( bI , bm and bh ) in Equations (2), (4) and (5), respectively, are physically related with the stirring intensity effect on reactor performance. For

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polymerization systems exhibiting low viscosities it is possible to modify the stirring rate without changing the RTD, because the system is typically well described by a perfectly macromixed model. Changing the stirring rate may improve the micromixing level of the system, as stirring may enhance both mass and energy transfer between the

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fluid elements in the reactor. In this way, the dynamic experimental data of conversion and temperature can be used to calculate the micromixing parameters. Assuming that

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the quasi-steady-state assumption is valid for free radicals, the total concentration of growing radicals in the reactor, P l , may be written in terms of local concentration of

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initiator, I l as follows:

 2 fK d I l  , t   P  , t     . K t   12

(7)

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l

The chain termination rate constant may be combined and expressed in terms of and overall rate constant defined as:

Kt  Ktc  Ktd .

(8)

Equations (2), (4) and (5) are defined in the following domains:

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ACCEPTED MANUSCRIPT t>0 and 0    

(9)

and are subject to the following boundary conditions:

 0 =I f

M l  , t  T l  , t 

(10)

 0 =M f

(11)

 0 =T f .

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I l  , t 

(12)

By definition, the average property distribution is obtained from integrating of the property distribution over the system's residence time distribution (RTD). The average

e

0







M (t )   M  , t  l

e

0



T (t )   T  , t  0

l

d



e







 



d

d .

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I (t )   I  , t  l

(13)

(14)

(15)

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

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concentrations and temperature are obtained by the following equations:

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Freitas Filho et al. (1994) and Melo et al. (2001) proposed rewriting the model equations in a form that could lead to more general definition of the model parameters.

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According to the proposed approach, operational and kinetic parameters must be coupled into parametric groups in order to represent arbitrary solution free-radical polymerization systems. Preliminary simulations performed by Melo et al. (2001)

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showed that dynamic effects introduced by a global mass balance did not play any relevant effect on the stability analysis for this system, as also observed for variations of heat capacity of reactants and products. Thus, the global mass balance is not included here explicitly, and it is admitted that the specific mass and the heat capacity are kept constant. In semi-dimensionless form, the model equations are rewritten as follows.

Local balance for initiator:

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ACCEPTED MANUSCRIPT yl  ,  yl  ,    + 1   I ( , y l , T l )  kI  yl  ,  y ( )     

(16)

where the specific initiation rate is rewriten as:

 E2  I ( , y l , T l )  y l  ,   exp  l  D .  T  ,   

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(17)

Local balance for monomer:

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ml  ,  ml  ,    + 1    p ( , y l , ml , T l )  km  ml  , m( )  .    

Local energy balance:

 , Tc

. 

(19)

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 B1G exp  D 

T l 

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T l  ,  T l  ,    + 1    G p ( , y l , ml , T l )  kh T l  , T ( )   

(18)

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The specific rate of polymerization is rewriten as:

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 E1  g 0,5 exp  l  D  A  T  ,   

(20)

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 p ( , y l , ml , T l )  ml  ,    y l  ,   

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where k I , km and k h are dimensionless micromixing parameters. Equations (16), (18) and (19) are defined in the following domains: >0 and 0    1

(21)

and are subject to the following boundary conditions:

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ACCEPTED MANUSCRIPT y l  ,   T l  ,  

l  0 = m  ,    0 =1

(22)

.

(23)

 0 = T f

The average concentrations and temperatures are expressed in an appropriate form to allow for subsequent bifurcation analysis, as follows:

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1 y l  ,   dy ( )  d d  0

(24)

m  ,   d m( )  d d   0 l

1

(25)

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1 T l  ,   dT ( )  d . d  0

(26)





  1  exp   , Il , If



V , q

y

I , If

T v , Tf

hAT  ,  qC p

km   bm ,

kh 

(27)

 bh . C p

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kI   bI ,

Ml , Mf

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M m , Mf

ml 

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yl 

t

 , 

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The new set of dimensionless groups and variables is provided below:

It is important to note that integrals defined by Equations (13)-(15) are defined in a semi-infinite domain, because the upper integration limits tend to infinity. In order to facilitate the computation of these integrals, the integration variable (internal age of fluid elements) is normalized, so that the definite integral remains within the limits 0 and 1, as one can see in Equations (24)-(26).

The new operation parameters are given by:

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D  ln  Kd 0 

(28)

 2 fI f A  ln  K p 0  K d 0 Kt 0 

(29)



(30)

GK d 0

while the physical properties are given by:

H r Cp

E1 

E2 

(31)

0.5Et  0.5Ed  E p

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G 

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B1 

   

R

Ed R

(32) (33)

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where D is a measure of the mean residence time, A is related to the initiator feed concentration, B1 stands for the heat transfer capacity, and G , E1 and E2

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represent the reaction exothermicity, the activation energy of polymerizations kinetics and the activation energy of initiator decomposition, respectively. The ranges of the

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physico-chemical parameters G , E1 , and E2 are obtained from Brandrup &

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Immergut (1989) and are listed in Table 1.

Table 1: Ranges of the physico-chemical parameters (Melo et al., 2001). Unit

Value

E1

K

8,493 – 11,141

E2

K

14,120 – 17,727

G

K

176 – 1,636



Dimensionless

0 – 5.35

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Parameter

The gel effect is a phenomenon caused by the decrease in mobility of polymer chains due to the increased concentration of polymer and increased viscosity of the reaction

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ACCEPTED MANUSCRIPT medium (Odian, 1991). Thus the reaction rates may become limited by diffusion, resulting in the increase of the rate of monomer consumption (because of the increased concentration of free radicals) and the temperature of the system (due to the high exothermicity of the reaction). Typical expressions used to describe the autoaccelaration phenomenon in free-radical polymerizations are usually empirical or based on the Free Volume Theory, as can be seen in O’Neil et al. (1998). The gel effect can be

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calculated with the correlation proposed by Freitas Filho et al. (1994): Kt  Kt' g

(34)

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where:

g  1   xl  . 

(35)

In Equation (34), K t' is the specific termination rate in the absence of the gel effect. The

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parameter  describes the strength of the gel effect. It may depend upon the temperature and the composition of the reactional mixture, but its value is usually

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constrained between well defined limits. The parameters  and x l are the monomer feed fraction and the monomer conversion, respectively. Therefore,  is directly related

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to the amount of solvent in the system, the greater the value of  the lower the solvent

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concentration in the reaction medium.

In this work, the gel effect was excluded from the model analysis because Freitas Filho

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et al. (1994) and Melo et al. (2001) have already shown that the gel effect is not necessary to explain the complex dynamic behavior of the polymerization system studied here. However, for the sake of the complete presentation of the model equations, the term related to the gel effect has been maitained in the model description.

The dynamic behavior of continuous free-radical solution polymerizations reactors is analyzed here in terms of four operational parameters ( D , A , B1 and  ), four physical

16

ACCEPTED MANUSCRIPT parameters ( G , E1 , E2 and  ) and the dimensionless micromixing parameters ( k I , km and k h ).

An important operational parameter for the analysis presented in this work is the parameter D, directly associated with the reactor average residence time (  ) as one may read from Equation (28). One may observe that the parameter D relates operational (  )

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and kinetic (Kd0) parameters. In order to show that the values for parameter D used in the simulations of this work are feasible, Table 2 presents typical values for the kinetic parameter ( K d 0 , as obtained from Brandrup & Immergut (1989)) of some initiators commonly used in free-radical polymerization reactions and assuming a range of

parameter D.

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average residence time from 0.5 to 2.5 hours, thus providing typical values for the

Table 2: Range of parameter D for some common initiators used in free-radical polymerization reactions assuming reactor average residence time in the range 0.5-2.5h.

K d 0 (s-1)

D

3.140×1014

40.8 – 42.5

3.342×1013

38.6 – 40.2

3.927×1013

38.8 – 40.4

1.214×1013

37.6 – 39.2

Toluene (30°C)

2.867×1013

38.5 – 40.1

Styrene (50°C)

1.166×1014

39.9 – 41.5

Solvent

2,2’-Azo-bis-

Di-n-butyl

isobutyronitrile

phthalate (80°C)

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Initiators

Di-n-butyl

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phthalate (117°C) Benzene (50.8°C)

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CE

Benzoyl peroxide

Lauroyl peroxide

Nitrobenzene (30°C)

3. Discretization and Continuation Methods Equations (16)-(26) constitute a set of nonlinear, coupled hyperbolic partial integrodifferential equations. The well-known method of orthogonal collocation on finite elements was used to discretize the independent internal age variable, thus reducing the micromixing model equations to a system of ordinary differential equations.

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ACCEPTED MANUSCRIPT As discussed by Jensen & Ray (1982), the method of orthogonal collocation on finite elements has been quite successfully when applied in problems that exhibit profiles with steep gradients, such as diffusion in catalyst particles with high values of Thiele modulus. The increase in accuracy of this technique, when compared with the global approach, is due to the possibility of including a larger number of discretization points in regions where gradients are more pronounced, avoiding spurious oscillations of

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calculated profiles. For this reason, the method of orthogonal collocation on finite elements (Villadsen & Stewart, 1967) was used to solve the model equations, since the occurrence of the gel effect and the high heat of reaction may lead to very steep dynamic profiles, especially when segregated flow conditions are considered. The Gauss-Radau numerical quadrature (Rice & Do, 1995) was used for calculation of the integrals given

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by Equations (24)-(26). The application of this numerical procedure for the micromixing model results in the following system of dynamical ordinary equations using n+1 collocations points (  k ) for each of the Ne finite elements (i):

(36)

n 1 dmki        ik , j mij     p k , yki  ,mki  Tki     km  mki  m( )  j 0 d

(37)

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n 1 dyki        ik , j y ij    I k , yki  ,Tki     kI  yki   y ( )  j 0 d

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n 1 dTki        ik , jT ji     G p k , yki  ,mki  Tki     kh Tki  T ( )  j 0 d

 B1G exp  D  Tki  , Tc 

(38)

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for k  1, , n  1 and i  1, , Ne

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where:



i k, j

1     A 

Ak , j 

i k

Li

k, j

dl j   d

 

,

,

(39)

(40)

k

18

ACCEPTED MANUSCRIPT 1

Ne

n 1

0

i 0

j 0

y ( )   y , d   Li  H j y ij   , 1

Ne

n 1

0

i 0

j 0

1

Ne

n 1

0

i 0

j 0

(41)

m( )   ml  , d   Li  H j mij   ,

(42)

T ( )   T l  , d   Li  H jT ji   ,

subject to the following continuity conditions:

y0i 1 ( )  yni 1   ,

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(43)

(44)

m0i 1 ( )  mni 1   ,

(45)

, Ne  1.

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T0i 1 ( )  Tni1   , i  1,

(46)

For specific details regarding the implementation of the orthogonal collocation method on finite elements to this problem, we refer to Oechsler (2012) and Oechsler et al.

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(2016). For the sake of comprehension of the equations presented above, the computational implementation of the method initially consisted in the generation of

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orthogonal Legendre polynomials and the subsequent calculation of their roots (  k ) and quadrature weights ( H k ). Then, the matrix for the first-order derivatives of the

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interpolating Lagrange polynomials, Ak , j , is calculated. Finally, the discretized differential equations are obtained by inserting the interpolating Lagrange polynomials

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into the original model equations and making the weighted residuals equal to zero at the collocation points.

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The dynamic reactor model was coupled to the well-known continuation and bifurcation package AUTO (Doedel & Oldeman, 2009), in order to perform a full bifurcation analysis of this system, giving special attention to the effect of the micromixing parameter on the nonlinear behavior of the reactor. AUTO uses the classical pseudo-arc length method (Keller, 1977) in order to carry on continuation along branches of both stationary and periodic solutions. Since the major objectives of this work is to show the influence of the degree of segregation on the dynamical behavior of the system, the monomer micromixing

19

ACCEPTED MANUSCRIPT parameter was selected as the main continuation parameter in most of our bifurcation diagrams. For the sake of simplicity, the three micromixing parameters have been taken to be equal and, when it was desired to analyze only the mass segregation effect, the thermal micromixing was maintained at a sufficiently large value. Following Uppal, Ray & Poore (1976), the reactor mean residence time, here related to the parameter D (see Equation 28), has also been used as a main continuation parameter, because the average

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residence time is a relatively easy variable to manage both for industrial and, especially, at laboratory scale experimental systems. The remaining operation and physicochemical parameters are free continuation parameters for the construction of bifurcation diagrams in higher dimensions. In the present work, for the sake of simplicity the reactor

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bifurcation diagrams are analyzed only in terms of the average reactor temperature.

Eighteen collocation points and three to four finite elements were used in all calculations presented in the next section. In all cases considered here, convergence of the bifurcation diagrams was always obtained under this choice of mesh points. As the number of collocation points can affect the calculated mean concentrations and

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temperature at distinct degrees of micromixing when the numerical mesh is not refined appropriately, dynamic simulations were performed for different degrees of

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micromixing with an increasing number of collocation points, with the objective of guaranteeing the convergence of the numerical results. Similar analyses were performed for bifurcation diagrams, to ensure their convergence. In all cases, this choice of mesh

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points assured convergence with absolute errors bellow 1×10−5.

Occasionally continuation of stationary solutions branches at elevated temperatures presented convergence difficulties, indicating that a larger number of collocation points

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might be necessary or even that numerical problems associated with the pseudo-arc length method continuation used by AUTO occurred, but these solutions are not presented here. In the results presented in the following sections, branches of stable and unstable stationary solutions are represented by continuous and dashed lines, respectively. Hopf bifurcation points are indicated by symbol  and branches of stable and unstable periodic orbits by the symbols ● and ○, respectively.

4. Results

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ACCEPTED MANUSCRIPT Multiple steady states observed in the bifurcation analyses performed for the ideal CSTR model were reported by Melo et al. (2001) and used as benchmark in the present work. As exactly the same diagrams were obtained using the IEM model at the maximum micromixing limit, it was concluded that the IEM model was implemented appropriately. Similarly, oscillatory responses obtained for the perfecttly mixed CSTR and reported by Melo et al. (2001) were also obtained with the IEM model in the limit

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as km   , confirming that the IEM model was implemented appropriately.

It must be emphasized that, with exception of the simulation case presented in Figure 7a, the thermal micromixing parameter has been chosen to be equal to the mass micromixing parameters (all micromixing parameters have been taken as equal). In

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Figure 7a, though, it was admitted maximum thermal micromixing ( kh   ). This condition simulates mass segregated systems, mass segregation is usually expected to be much larger than thermal segregation in polymerization systems (as mass transfer resistance is expected to increase faster with the increase of the system viscosity).

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4.1 Analysis of physical consistency

In order to confirm the physical consistency of our version of the IEM model, dynamic

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simulations are performed between, and including, the limits of complete segregation and perfect micromixing. These were compared with two important benchmarks

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presented in the literature: a completely segregated reactor model (SCSTR), proposed by Danckwerts (1958), and the classical standard model, perfect mixing approach in a

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CSTR. Figure 2 presents dynamical profiles for the mean monomer concentration and temperature at various degrees of micromixing. It is shown that the model predictions

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for km  0 and km   converge to the limiting cases of complete segregation and perfect mixing, thus providing a validation of our the micromixing model. For the sake of simplicity, in the analysis presented here, all dimensionless micromixing parameters have been taken as equal to the values assumed for km , i.e., the micromixing parameter for the monomer mass balance.

21



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PT

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(a)

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ACCEPTED MANUSCRIPT

 (b)

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Figure 2. Dynamic simulations for the average variables at different levels of micromixing: (a) monomer concentration; (b) temperature ( D  39.5 , A  9.5 ,

B1  1020 , E1  9,818K , E2  16, 263K , G  600K ,   0.0 ,   0.20 , n  18 ,

and Ne  3 ).

4.2 Stationary Bifurcations Figure 3 presents bifurcation diagrams obtained for the solution polymerization system in the absense of the gel effect for the perfect mixed CSTR model using the same model

22

ACCEPTED MANUSCRIPT equations reported by Melo et al. (2001). As noticed by Melo et al. (2001), the addition of solvent in bulk polymerization reactions is attractive from an industrial point of view, as it facilitates reactor temperature control, since the solvent acts as a diluent of the polymer solution. Our results corroborate those obtained by Melo et al. (2001), specifically the lower stationary temperatures for lower volume fractions of monomer in the feed. In our model, the adition of solvent in the polymerization rection is performed

(a)

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(b)

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by decreasing the parameter  (which represents the monomer fraction in feed).

T (K)

(c)

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(d)

D

Figure 3: Bifurcation diagram for the solution polymerization system in a well-mixed

PT

CSTR. ( A  9.5 , B1  1020 , E1  9,818K , E2  16, 263K , G  600K ,   0.0 ;

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(a)   0.50 ; (b)   0.40 ; (c)   0.30 ; (d)   0.20 ).

From Figure 3 one can observe that multiple steady states can exist over the middle

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range of the parameter D . Initially, consistency of our micromixing model presented here was evaluated in the limiting case of perfect micromixing. Figure 4b shows the bifurcation diagram using the micromixing parameter as the continuation parameter. The other parameters are described for Case d in Figure 3, with D=39.5. One can see that, for sufficiently large values of km , the same three steady states shown in Figure 3, case d, which are shown in Figure 4a, are observed. As already stated, this result validates the occurence of multiple steady-states through the utilization of the parameter km with the micromixing model presented in the previous section.

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ACCEPTED MANUSCRIPT

In Figure 4b one can see the appearance of aditional limit point bifurcations during the continuation of the lower branches of stationary solutions. Therefore, aditional steady states associated to the lower branch of monomer conversions and temperatures are observed for intermediate values of the micromixing parameter, km . This can be seen more clearly in Figure 5. These results corroborate results presented by Freitas Filho et

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al. (1994) and Pinto & Ray (1995a), who showed that continuous polymerization reactors can show up to five stable stationary states under maximal micromixing conditions. For sufficiently low values of km , a single steady-state solution should be expected, as reported by Pinto (1990b) for completely segregated polymerization systems. Figure 4b and 5 show that the dynamic behavior of partially segregated

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polymerization reactors can be much more complex than the behavior of either perfectly mixed or fully segregated reactors. The results presented in Figures 4b and 5 are also important because the additional stable stationary conditions may find possible industrial applications, as additional stable operation conditions are provided when

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CE

PT

ED

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partial segregation takes place.

km

(a)

24

D

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(b)

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ACCEPTED MANUSCRIPT

Figure 4: (a) Bifurcation diagram for the polymerization system (   0.20 ); (b) Validation of multiple steady states found in (a) for large values Km using a classical perfect

mixing

model

( D  39.5 ,

A  9.5 ,

B1  1020 ,

E1  9,818K ,

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CE

PT

ED

M

E2  16, 263K , G  600K ,   0.0 ,   0.20 , n  18 , and, Ne  3 ).

km Figure 5: Detail of Figure 4 presenting multiplicity of steady states associated to the intermediary degree of micromixing.

25

ACCEPTED MANUSCRIPT One might argue about the numerical convergence of the bifurcation diagram presented in Figures 4b and 5. Figure 6 shows clearly that the size of the discretization mesh used in the solution of the set of partial differential equations directly affects the characteristics of the bifurcation diagrams. In Figure 6, only the intermediate branches of stationary solutions are presented. One can see that additional limit point bifurcations are obtained as the degree of the orthogonal polynomial increases (Figure 6a and 6b).

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However, when the polynomial degree reaches n  18 , the bifurcation diagrams stabilize and remain unchanged for higher values of n , indicating the convergence of the numerical procedure. For this reason, unless stated otherwise, all calculations were

km

(a)

km

(b)

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CE

PT

ED

M

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performed with n  18 .

26

(c)

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km

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ACCEPTED MANUSCRIPT

ED

km

(d)

Figure 6: Convergence analysis to the bifurcation diagram: D  39.5 , A  9.5 ,

PT

B1  1020 , E1  9,818K , E2  16, 263K , G  600K ,   0.0 ,and,   0.20 . (a)

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n=10, Ne=3; (b) n=15, Ne=3; (c) n=18, Ne=3 ; (d) n=20, Ne=3.

Figure 7 shows that, for intermediate values of the parameter D and micromixing

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conditions of partial segregation, additional steady states not predicted by classical models are obtained. As it can be seen in Figure 7b, up to seven steady states can be found under conditions of partial segregation. It must be emphasized that, with exception of the simulation case presented in Figure 7a, the thermal micromixing parameters has been chosen to be equal to the mass micromixing parameter. In Figure 7a, though, it was admitted maximum thermal micromixing ( kh  50 ), while mass micromixing parameters ( ki and km ) were null in this case. This condition simulates mass segregated systems, mass segregation is usually expected to be much larger than

27

ACCEPTED MANUSCRIPT thermal segregation in real polymerization systems (as mass transfer resistance is expected to increase faster with the increase of the system viscosity). In Figure 7b-d, all micromixing parameters were kept fixed on equal values ( kh  ki  km ).

(d)

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(a)

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(a) km = 0.00 (b) km = 2.50 (c) km = 5.00 (d) km = 50.0

(c) (b)

D

Figure 7: Bifurcation diagram to the solution polymerization system: Parameter D effect

B1  1020 ,

E1  9,818K ,

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( A  9.5 ,

E2  16, 263K ,

G  600K ,

  0.0 ,

PT

shown in Figure 11.

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  0.20 , n  18 , and, Ne  4 ). Details regarding the Hopf bifurcation points are

It is interesting to observe that when fluid element populations are allowed to exchange

CE

heat among themselves the reacting system can present multiple steady states. These results support previous results obtained by Pinto (1990b), as multiple steady states were obtained for continuous suspension polymerization systems under complete mass

AC

segregation and perfect thermal micromixing conditions. However, one can also see that for increasing values of the mass micromixing parameter, the bifurcation diagram resembles a typical mushroom-shaped diagram observed under condition of perfect mixing, as expected and shown in Figure 3. This indicates that the system behavior is more sensitive to thermal segregation than to mass segregation, as one might expect based on the very high exothermicity of typical polymerization sytems.

28

ACCEPTED MANUSCRIPT As widely reported in the literature, polymerization reactors are prone to presenting isolated branches of stationary under perfect micromixing, as shown by Jaisinghani and Ray (1977), Schmidt and Ray (1981), Pinto (1990a,b) and Melo et al. (2001). The detachment of isolas (isolated branch of steady-state solutions) has always been related to changes in the upper branch of stationay solutions (higher temperatures). Figure 7c shows, however, the detachment of an isola from modification of the intermediate

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branch of stationary solutions (intermediate temperatures). In Figure 8, this phenomenon is confirmed by performing the bifurcation analysis using the parameter D as the main continuation parameter for various values the micromixing parameter. In particular, the isola obtained in Figure 8b is associated with the intermediate branches of stationary solutions where as many as seven multiple steady states can be observed. This can be

AC

CE

PT

ED

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dynamic analysis of chemical reactors.

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the first time that this kind of behavior is presented in the open literature regarding the

D

(a)

29

D

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ACCEPTED MANUSCRIPT

(b)

( km  6.52 )

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Figure 8: (a) Detachment of stationay isola ( km  5.00 ); (b) Stationary isola ( A  9.5 , B1  1020 , E1  9,818K , E2  16, 263K , G  600K ,

  0.0 ,   0.20 , n  18 , and, Ne  4 ).

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4.2 Hopf Bifurcations

In addition to multiplicity of stationary solutions, Hopf bifurcations are also present on

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the lower branches of steady-state solutions (Figure 9). These bifurcations can be detected over a broad range of values of the parameter D , associated with the mean

PT

residence time, and initiator decay. Given specific decay rates, valid for typical initiators in real systems, this range of the parameter D may be of industrial relevance, as shown

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CE

by Melo et al. (2001).

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ACCEPTED MANUSCRIPT

(b) (c)

(d)

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D

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(a)

Figure 9: Hopf bifurcation to solution polymerization system in well-mixed CSTR ( A  9.5 , B1  1020 , E1  9,818K , E2  16, 263K , G  600K ,   0.0 ; (a)

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  0.20 ; (b)   0.30 ; (c)   0.40 ; (d)   0.50 ).

Figure 10 shows that, starting from the condition of total segregation, a Hopf bifurcation

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point is found at intermediate values of the micromixing parameter. Branches of periodic orbits that start at this particular Hopf bifurcation were then traced. It is observed that the polymerization system does not show oscillatory dynamics in the

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vicinity of total segregation (i.e. all null micromixing parameters), as one might already expect. This confirms results presented by Pinto (1990b). Therefore in order for the

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system to present oscillatory dynamic behavior, the existence of a minimum degree of micromixing is necessary such as the one presented in Figure 10 near the Hopf

AC

bifurcation point. This is the first time that Hopf bifurcations have been observed as a function of the degree of micromixing in solution polymerization reactors.

31

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/298.15

ACCEPTED MANUSCRIPT

km

limit

to

solution

polymerizations

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Figure 10: Numerical verification of oscillatory behavior in maximum micromixing systems

( D  40.1 ,

A  9.5 ,

B1  1020 ,

E1  9,818K , E2  16, 263K , G  600K ,   0.0 ,   0.20 , n  18 , and,

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Ne  4 ).

Figure 11 presents diagrams that are similar to the ones presented in Figure 7, but

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expanded in the region of the Hopf bifurcations. As expected, in the perfect micromixing condition, two Hopf bifurcation points are found, confirming the results displayed in Figure 9. As the degree of segregation is increased (i.e decreasing

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micromixing parameters values) of the system, the range of values of the parameter D over which the system presents oscillatory behavior is decreased. Close to the total

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segregation condition the Hopf bifurcation points coalesce, leading to disappearance of oscillatory solutions. These bifurcation diagrams suggest that the occurrence of limit

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cycles in solution polymerization systems is only possible if some degree of micromixing is present in the polymerization reactor.

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ACCEPTED MANUSCRIPT

(a)kKm 0,0 (a) m == 0.00 (b) Km = 2,5 (b) km = 1.25 (c) Km = 5,0 (c) km = 2.50 (d) kKm== 50,0 (d) 5.00 m

(e) (d) (c) (b)

D

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(a)

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Figure 11: Detailing of Hopf bifurcations associated to the intermediary degree of micromixing presented in Figure 7: Parameter D region of industrial interest.

The continuation of branches of periodic orbits from the Hopf bifurcation points computed in Figure 11 are shown in Figure 12. It is shown that the decrease of the

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micromixing parameter dramatically reduces the complexity of the periodic orbits, as can be confirmed by the disappearance of periodic limit points from the transition from

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Figures 12b to 12a. Figure 11 shows that there is a broad range of values for the parameter D where self-sustained periodic oscillations can be observed under

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intermediate levels of micromixing and that stable and unstable limit cycles may coexist under such conditions. This is an unprecedented result in the dynamical analysis of

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continuous solution polymerization reactors, suggesting that the onset of complex behavior can be investigated more easily with partially segregated polymerization

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systems.

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ACCEPTED MANUSCRIPT

(a) km = 1.25 (b) km = 2.50 (c) km = 5.00 (d) km = 50.0 (d) (c)

(a)

D

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(b)

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Figure 12: Dynamic bifurcations to the solution polymerization system: Parameter D region of industrial interest ( A  9.5 , B1  1020 , E1  9,818K , E2  16, 263K , G  600K ,   0.0 ,   0.20 , n  18 , and, Ne  4 ).

5. Conclusions

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The bifurcation behavior of partially segregated continuous free-radical solution polymerizarion stirred tank reactors was investigated using IEM micromixing model.

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The physical consistency of the micromixing model used here was confirmed for the limits of maximum micromixing and total segregation, regarding the existence of

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multiple steady states and the appearance of periodic oscillations. The bifurcation analyses, for which the micromixing parameter was systematically varied, confirmed the

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existence of multiple steady states and periodic oscillations in the limit of the maximum micromixing and uniqueness of solutions in the limit of complete segregation. In

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addition, limit bifurcation points were observed, revealing that as many as seven stable steady states (and thirteen distinct steady-state solutions) can exist in the vicinity of the intermediate branch of unstable stationary solutions. From the industrial point of view these results are interesting because they illustrate that if due attention is not given to these effects, small perturbations of operating conditions can lead the reactor to a region of low conversion (extinction of polymerization rate). In addition, isolated branches of steady states (isolas) were calculated for the first time as functions of the degree of micromixing and in association with the intermediate branch of unstable stationary solutions.

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ACCEPTED MANUSCRIPT The bifurcation analyses of periodic orbits showed that the emergence of unstable limit cycles in broad ranges of residence times can also be associated with the existence of intermediate degrees of micromixing. These results reinforce that industrial reactors operated under partially segregated conditions may exhibit dynamic instabilities (oscillatory behavior) that cannot be detected by the usual models (where these effects are disregarded). These solutions represent an operational risk, because of the high

Notation

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temperatures that can be reached.

Heat transfer area of the reactor (m2)

A

Parameter relative to the initiator concentration in the feed

Ai , j

Matrix of 1st order differential polynomials of Lagrange

bI

Micromixing mass parameter relative to initiator (1/s)

bm

Micromixing mass parameter relative to monomer (1/s)

bh

Micromixing thermal parameter (J/m3/K/s)

B1

Parameter relative to the thermal exchange capacity

Cp

Heat capacity of the reaction medium (J/mol/K)

D

Parameter relative to the mean residence time

f  , t 

Distribution function of internal ages

f

Initiator efficiency factor

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Parâmetro relativo à medida de exotermicidade da reação (K)

h

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Hi

CE

G

g

ED

M

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AT

Gel parameter Heat transfer coefficient (J/m2/K/s) Numerical quadrature weights

I

Initiator molecule

km

Dimensionless mass micromixing parameter

kh

Dimensionless thermal micromixing parameter

K1

Specific rate of chain initiation step (1/s)

Kd

Specific rate of initiator decomposition step (1/s)

35

ACCEPTED MANUSCRIPT Pre-exponential factor of Arrhenius of initiator decomposition specific rate

Kd 0

(1/s)

Kp

Specific rate of chain propagation step (L/mol/s)

K p0

Pre-exponential factor of Arrhenius of chain propagation specific rate (L/mol/s) Specific rate of termination step in the absence of effect gel (L/mol/s)

K t'

Specific rate of termination step with gel effect (L/mol/s)

K tc

Specific rate of termination by combination step (L/mol/s)

K td

Specific rate of termination by disproportionation step (L/mol/s)

Li

Size of finite element i

M

Monomer molecule

m

Dimensionless monomer concentration

n

Number of collocation points

Ne

Number of finite elements

Pj

Living polimer molecule with chain size j

q

Volumetric flow rate (L/s)

I

Initiator rate (mol/L/s)

p

Polymerization rate (mol/L/s)

R

Universal gas constant (J/mol/K)

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M

ED

PT

R*

Free radical molecule Local temperature (K)

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T l  , t  Tc

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Kt

Cooling jacket temperature (K) Average temperature of reactor (K)

Tf

Temperature reactor feed (K)

t

Time (s)

V

Reactor volume (L)

xl

Local conversion of monomer

y

Dimensionless initiator concentration

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T t 

Greek characters

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Volume fraction of monomer in feed

j

Dead polymer molecule with chain size j

Ed

Activation energy of the initiator decomposition step (J/mol)

E p

Activation energy of chain propagation step (J/mol)

Et

Activation energy for chain termination step (J/mol)

E1

Parameter on the activation energy of polymerization (K)

E2

Parameter on the activation energy of the initiator composition (K)

H r

Reaction entalphy (J/mol)



Internal age (s)



Gel-effect parameter



Dimensionless time



Specific mass of reaction medium (g/L)



Average residence time (s)



Dimensionless internal age

M

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

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Acknowledgements

The authors would like to thank CAPES – Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, CNPq – Conselho Nacional de Desenvolvimento Científico e

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Tecnológico and FAPERJ – Fundação Carlos Chagas Filho de Amparo à Pesquisa do

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Estado do Rio de Janeiro, for providing scholarships and supporting this research.

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