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## Brazilian Journal of Chemical Engineering

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*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol.19 no.1 São Paulo Jan./Mar. 2002

#### http://dx.doi.org/10.1590/S0104-66322002000100007

**A COMPARISON OF DIFFERENT MODELING APPROACHES FOR THE SIMULATION OF THE TRANSIENT AND STEADY-STATE BEHAVIOR OF CONTINUOUS EMULSION POLYMERIZATIONS** **IN PULSED TUBULAR REACTORS**

C.Sayer and R.Giudici

Universidade de São Paulo, Escola Politécnica, Departamento de Engenharia Química,

05508-900 –SP, Phone: (11) 3818-2246, Fax: (11) 3813-2380, São Paulo – SP, Brazil

E-mail: rgiudici@usp.br

(Received: August 1, 2001 ; Accepted: December 6, 2001)

Abstract –Dynamic mathematical models are developed to simulate styrene emulsion polymerization reactions carried out in pulsed tubular reactors. Two different modeling approaches, the tanks-in-series model and the axial dispersion model, are compared. The models developed were validated with experimental data from the literature and used to study the dynamics during transient periods, e.g., the start-up of the reactor and the response to disturbances. The effect of the Peclet number on process variables such as conversion and particle concentration was also verified.: emulsion polymerization, tubular reactors, modeling, axial dispersion, n-CSTRs, styrene.

Keywords

**INTRODUCTION**

In the industrial environment, emulsion polymerizations may be carried out in stirred tank reactors operated in the batch or semi-continuous modes or in continuous reactors. Semi-continuous reactors are used when high production flexibility is required and only relatively small quantities of each polymer are produced. Continuous reactors have economic advantages for production of larger amounts of each product and avoid product differences from batch to batch. Of the different types of continuous reactors the continuous stirred tank reactors (CSTRs) are the most commonly used. On the other hand, continuous tubular reactors present several advantages when compared to CSTRs, such as higher operational flexibility, reduced losses during start-up, grade transitions and shut-down due to the shorter transient periods, and more efficient temperature control due to the higher heat transfer areas (higher area/volume ratios). In addition, for the same final conversion and production rate tubular reactors require lower average residence times (and therefore lower volumes) than CSTRs. Two approaches to preventing the emulsion destabilization and plugging of the tubular reactor were presented in literature: 1) usage of high recycle rates (Rollin et al., 1979; Geddes, 1983, 1989; Lee et al., 1992; Abad et al., 1994, 1995; Araújo et al., 1998, 1999); and 2) introduction of pulses (Hoedemakers, 1990; Meuldijk et al., 1992; Paquet and Ray, 1994a; Mayer et al., 1996; Palma et al., 2001a, 2001b). In the first approach the high recycle rates often lead to perfect mixing and therefore the behavior of the continuous loop reactors is often comparable to that of CSTRs (same length of transient periods). The second approach has the advantage of minimizing sustained oscillations often observed in CSTRs (Greene et al., 1976; Kiparissides et al., 1980; Rawlings and Ray, 1988; Penlidis et al., 1989; Ohmura et al., 1998) or in continuous loop reactors (Araújo et al., 1999). Also, due to the combination of good local agitation with little backmixing, pulsed tubular reactors produce much higher monomer conversions and particle concentrations than single CSTRs. As a possible disadvantage of the pulsed tubular reactors it is important to mention that to our knowledge this kind of reactors are not yet been used in industry, while several industrial emulsion polymerization processes already use continuous loop reactors. This might be attributed to the relatively recent development of this kind of pulsed tubular reactors for emulsion polymerization reactions (Hoedemakers, 1990; Meuldijk et al., 1992; Paquet and Ray, 1994a, 1994b; Mayer et al., 1996; Palma et al., 2001a, 2001b; Sayer and Giudici, 2001).

Besides providing a deeper understanding of the complex mechanisms involved in emulsion polymerization reactions, development of detailed mathematical models is also one of the most powerful tools for optimization of the operational conditions of these reactions.

The aim of this work is to compare two different approaches to the mathematical modeling of emulsion polymerization reactions carried out in pulsed tubular reactors: 1) the dynamic tanks-in-series model and 2) the dynamic axial dispersion model. In the second approach the choice of the most adequate procedure for reducing the partial differential equations and the numerical aspects involved therein is quite challenging (Lee et al., 1999; Botte et al., 2000; Lefévre et al., 2000; Sayer and Giudici, 2001). Therefore, the performances of two different techniques used to solve the axial dispersion model, the method of lines and orthogonal collocation, applied in the spatial direction, are also compared. Polymerization mechanisms addressed by the model include initiation, propagation, transfer to monomer and termination by combination, as well as micelar and homogeneous nucleation mechanisms. The numerical integrator DDASSL (Petzold, 1982) is used to solve the system of differential and algebraic equations in time. Both models were validated with experimental data taken from the literature (Hoedemakers, 1990). The development of dynamic models allowed simulation of important transient reaction periods such as reactor start-up and the effect of perturbations.

**DESCRIPTION OF THE PULSED PACKED TUBULAR REACTOR**

This section intends to provide a brief description of the continuous pulsed tubular reactors. For a more detailed description the interested reader should refer to the original experimental references (Hoedemakers, 1990; Meuldijk et al., 1992). In the pulsed tubular reactors the pulsator is located at the bottom of the reactor (column) and the stroke length (0-14 mm) and pulsation frequency (0 - 3.5s^{-1}), which are independent of the flow rate, can be adjusted in order to vary the axial dispersion. The pulsation provides a combination of good local agitation with little backmixing, reducing radial gradients and preventing emulsion destabilization and reactor clogging. Figure 1 shows a schematic presentation of the continuous pulsed tubular reactor considered in this study. The jacketed column has a length of 500 cm and an internal diameter of 5 cm and is packed with Raschig rings (diameter = 1 cm) resulting in a bed porosity equal to 0.73.

Other experimental emulsion polymerization studies, that were not considered in this work, involve continuous pulsed tubular reactors with no packing (Paquet and Ray, 1994a) or with different kinds of column packing materials like sieved plates (Palma and Giudici, 2001; Palma et al., 2001a, 2001b).

DEVELOPMENT OF THE MODEL

A general mathematical model was developed to describe emulsion polymerization reactions carried out in batch or in continuous reactors. In order to model continuous reactions carried out in tubular reactors, two different approaches were tested: the tanks-in-series model and the axial dispersion model.

Modeling of Emulsion Polymerization

The kinetic mechanism used in this work to represent emulsion polymerizations encompasses the following reactions: initiation, propagation, transfer to monomer and termination by combination, as well as micelar and homogeneous nucleation mechanisms and particle coalescence. The following assumptions are used to derive the model equations

1) Monomer concentrations in polymer particles, monomer droplets and aqueous phase are at thermodynamic equilibrium.

2) The total mass of polymer produced in aqueous phase is negligible.

3) Particle nucleation occurs by the micelar and homogeneous mechanisms.

4) Polymer particles are spherical and monodisperse.

5) The pseudo-steady-state assumption is valid for polymer radicals.

6) Kinetic constants do not depend on chain length.

7) Kinetic constants in the aqueous and polymer phases are the same.

8) The gel effect depends on temperature and polymer fraction in polymer particles.

9) Radicals that enter polymer particles are of length 1.

10) Radicals generated by initiation or chain transfer to monomer have similar reactivities.

11) Reactions are performed at constant temperature.

12) Radial mixing is perfect.

13) Axial velocity is constant.

14) Axial mixing is represented by an axial dispersion coefficient.

Model equations and a detailed description of the model developed are presented in the Appendix.

**Modeling of Tubular Reactors**

Two different approaches to modeling emulsion polymerization reactions carried out in continuous pulsed tubular reactors (PTR) were tested: 1) the tanks-in-series model and 2) the axial dispersion model. The implementation of these two approaches is described in the following sections.

1) The Tanks-in-Series Model

In this first approach the PTR is represented by continuous stirred tank reactors (CSTRs) in series, where each CSTR is described by the model presented in the Appendix and the outlet terms of reactor n are the inlet terms of reactor n+1. In this type of representation the number of reactors (N_{R}) is equal to the Peclet number (Pe) divided by 2 (Hoedemakers, 1990). Pe is computed from the axial dispersion coefficient (De) that is obtained experimentally by tracer studies (Hoedemakers, 1990; Paquet and Ray, 1994a; Palma and Giudici, 2001;** **Ni and Pereira, 2000; Macías-Salinas and Fair, 2000; Benkhelifa et al., 2000). Another equation to compute the number of CSTRs in this approach was proposed by Salmi and Lindfors (1986), apud Lehtonen et al. (1999):

Nevertheless, except for very small Pe values, the values of N_{R} computed by Eq. (1) differ only slightly from Pe/2.

2) The Axial Dispersion Model

In this second approach, which addresses constant axial velocity and axial dispersion, the differential equations that represent the balance equations of a CSTR, presented in the Appendix and written here in vector form (Eq. (2) where x is the vector of state variables), are transformed into second-order partial differential equations, Eq. (3):

The following boundary conditions were used (Danckwerts, 1953):

where z is the dimensionless length, v_{z} is the axial velocity and De is the effective dispersion coefficient in cm^{2}/s. In addition,

where r_{col} and a_{col} are the radius and the length of the PTR in cm, respectively, and f_{void} is the fraction of voids of the PTR (used only in the case of packed reactors).

Two different numerical techniques were tested to solve the second-order partial differential equations: 1) the method of lines and 2) orthogonal collocation.

**a) The Method of Lines (MOL)**

By this procedure the partial differential equations, Eq. (3), are grouped by the finite differences technique; consequently the balance equations are transformed into ordinary differential equations, as follows:

where x_{i} is the vector of state variables at discretization point i. The boundary conditions, Eq. (4) and (5), are transformed into the following algebraic equations:

where ND is the number of internal points and Dz is the length between two discretization points such that

b) Orthogonal Collocation (OC)

By this procedure the partial differential equations, Eq. (3), are grouped by the orthogonal collocation technique; consequently the balance equations are transformed into ordinary differential equations, as follows:

where __X___{i} is the vector of state variables at collocation point i. The boundary conditions, Eq. (4) and (5), are transformed into the following algebraic equations:

where NC is the number of internal collocation points and A and B are the collocation matrices (Villadsen and Michelsen, 1978).

RESULTS AND DISCUSSION

The model developed was implemented in FORTRAN and the system of differential algebraic equations was solved using the DASSL solver (Petzold, 1982). The model was used for the simulation of styrene emulsion polymerization reactions carried out in continuous-pulsed packed tubular reactors. To validate the model developed simulation results were compared with experimental results from Hoedemakers (1990). Experimental conditions, reactor dimensions, kinetic constants and other parameters used in the following simulations are shown in Tables 1 and 2.

The gel effect (x_{gel}), which accounts for the diminution of the termination rate, is computed as proposed by Hui and Hamielec (1972):

where f_{p} is the volumetric fraction of polymer in the polymer phase.

**Model Validation**

Figure 2 compares experimental results, conversion and particle concentration from Hoedemakers (1990) with simulation studies using the tanks-in-series model and the axial dispersion model solved by MOL and by OC. The Peclet number used in these simulations was Pe = 50 (Paquet and Ray, 1994). It can be observed that both models, tanks-in-series and axial dispersion, solved by either MOL or OC, are able to represent quite properly the experimental results.

Effect of the Peclet Number (Pe)

Figure 3 shows the effect of Pe, which is varied by the manipulation of the pulsation frequency and/ or amplitude, on monomer conversion and particle concentration along the tubular reactor. These simulations encompass the whole range of axial dispersions, starting with perfect mixing (very low Peclet numbers, Pe = 0.01) up to plug flow (very high Peclet numbers, Pe > 100). The simulations were performed with the axial dispersion model solved by MOL using 30 internal nodes and by OC using 18 internal collocation points. The tanks-in-series model is not able to represent properly the high axial dispersion region, since the parameter N_{R} can only assume integer values, not allowing to represent the continuous range of axial dispersion. Moreover, the relation N_{R} = Pe/2 is no longer valid for Pe < 10 and no exact nor approximate equivalence between tanks-in-series and axial dispersion models exists for the high dispersion region.

Start-up Dynamics

Figure 4 shows the dynamics of the start-up of a PTR. In these simulations the start-up procedure is to begin the reaction with the reactor filled with water; since this start-up procedure was preferred by Hoedemakers (1990) as it minimizes initial overshoots in conversion and particle concentration. For these start-up simulations, which have rather steep fronts, the results for MOL using 30 internal nodes were more robust than those for the OC technique.

**Effect of Disturbances**

Figures 5 and 6 show, respectively the effects of disturbances on the concentrations of emulsifier (+20% at t = 20 minutes, returning to its original value at t = 70 minutes) and initiator (-20% at t = 20 minutes and +20% at t = 70 minutes) in the feed streams of styrene emulsion polymerization reactions carried out in a PTR. Increasing the emulsifier concentration results in an increase in polymer particle concentration as shown in Figure 5. Decreasing the initiator concentration results in a decrease in polymer particle concentration as shown in Figure 6. It might also be observed that this kind of perturbation in initiator concentration resulted in nonlinear process responses and in slight under- and overshoots in particle concentrations. Comparing the effects of perturbations on the concentrations of emulsifier and initiator, it might be noted that the former has a more significant effect on particle concentration. The emulsifier concentration is directly involved in the number of micelles formed and consequently in the number of polymer particles. On the other hand, radicals formed by the decomposition of the initiator may enter either micelles, thereby increasing the number of particles formed, or an existing polymer particle, in which case the number of polymer particles is not affected. Again the method of lines using 30 internal nodes was more robust than the orthogonal collocation technique in representing these abrupt changes in operational conditions. This same behavior was also observed in other simulations that are not presented here. A general trend in these simulations is the decrease of the numerical stability of the OC with increasing Peclet numbers since higher Peclet numbers lead to rather sharp fronts. In the simulations of emulsion polymerization reactions the stiffness of the system is mainly due to the fast particle nucleation mechanisms that result in quite fast and significant variations in particle number.

In the presented simulations processing time varied from a few minutes to several hours depending strongly of the operational conditions and also of the location of the discretization points, more than of the number of discretization points or of the method of solution (MOL or OC). This behavior is attributed to abrupt changes in the reaction system such as the increase of the average number of radicals per polymer particle due to the gel effect, the disappearance of emulsifier micelles and the consequent ceasing of the micelar particle nucleation mechanism and specially the disappearance of the monomer droplets from the reaction medium. Probably this behavior is further amplified by the iterative computation schemes used by the emulsion polymerization model to compute the monomer concentrations in the different phases and the average number of radicals per polymer particle.

**CONCLUSIONS**

Two different modeling approaches were compared for the dynamic simulation of styrene emulsion polymerization reactions carried out in pulsed tubular reactors: 1) the dynamic tanks-in-series model and 2) the dynamic axial dispersion model. The performances of two different numerical techniques used to solve the axial dispersion model, 1) the method of lines and 2) orthogonal collocation, applied in the spatial direction, were also compared.

Polymerization mechanisms addressed by both modeling approaches include initiation, propagation, transfer to monomer and polymer and termination by combination, as well as micelar and homogeneous nucleation mechanisms.

Both models, the dynamic tanks-in-series model and the dynamic axial dispersion model, were validated with experimental data taken from the literature (Hoedemakers, 1990), showing a good agreement. Nevertheless, the axial dispersion model has the advantage of being able to encompass the whole range of axial dispersions, starting with perfect mixing up to near plug flow, while the tanks-in-series model is not able to represent properly the high axial dispersion region. (Each model uses one parameter to describe axial mixing; however, in the tanks-is-series model this parameter only assumes integer values). In analyzing the influence of the Peclet number, it is verified that with Peclet numbers of around 100 results are already very close to the plug-flow behavior. This type of behavior was also observed by Secchi and Pereira (1998) for high-pressure polyethylene tubular reactors. For simulation of transient reaction periods like reactor start-up or disturbances in the reactant flow rates that may result in rather steep fronts, the MOL has shown more robust results than the OC technique for solution of the axial dispersion model.

ACKNOWLEDGEMENTS

The financial support from FAPESP - Fundação de Amparo à Pesquisa do Estado de São Paulo - and CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico - is gratefully appreciated.

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EMULSION POLYMERIZATION MODEL EQUATIONS

Given the modeling assumptions presented in section 2.1, the population and mass balance equations for a continuous emulsion polymerization system at constant temperature are presented below.

a) Polymer particle number

where F_{Np,e} is the particle inlet rate and q_{s} is the outlet rate in cm^{3}/s, V_{R} is the volume of the reactor, R_{mic} and R_{hom} are the micelar and homogeneous nucleation rates and c_{c} (N_{p})^{2} represents the rate of particle coalescence.

Micelar nucleation

where R_{aq} is the concentration of polymer radicals in aqueous phase, computed by Eq. (A-32), and k^{m}_{abs} is the rate of absorption of radicals from aqueous phase by the micelles, given by

where D_{w}, r_{m} and f^{m}_{abs} represent respectively the diffusion coefficient in aqueous phase, the radius of one micelle and the efficiency of absorption of radicals from aqueous phase by the micelles.

Variable N_{mic} is the number of micelles formed when the emulsifier concentration in aqueous phase, [E]^{aq}, is higher than the critical micelar concentration (CMC) and is given by

where M_{mic} is the number of micelles per volume of aqueous phase, computed by either Eq. (A-16) or Eq. (A-17), depending on the presence or absence of an excess of emulsifier, and v_{p} and v_{aq} are the volumes of polymer and aqueous phases.

Homogeneous nucleation

where c_{hom} is given by

c_{hom}^{0} is the coefficient of homogeneous nucleation and z_{ll} is the course of a critical radical. This latter coefficient is given by

where j_{crit} is the critical length of a radical, which depends on the solubility of the monomer in aqueous phase; k_{p}[M]^{sat}_{aq}^{ }is the propagation rate in aqueous phase during saturation conditions; and a_{p} is the superficial area of one polymer particle swollen with monomer and is given by

where r_{p} is the radius of one polymer particle swollen with monomer and is computed by

Variable R_{jcrit} is the concentration of radicals having critical length j_{crit} in aqueous phase and is computed by Eq.A-31.

b) Initiator (moles)

c) Emulsifier (moles)

In emulsion polymerization systems the emulsifier might be present in three different ways: a) free in aqueous phase b) in the form of micelles in aqueous phase and c) adsorbed on polymer particles and monomer droplets. As monomer droplets have a significantly bigger radius than polymer particles, they have a much lower total superficial area, and therefore, the amount of emulsifier adsorbed on monomer droplets might be neglected. In this case, E is given by

where [E]^{aq} is the concentration of emulsifier in aqueous phase, g is the number of moles of emulsifier per micelle and [E]_{ads} is the concentration of emulsifier adsorbed on polymer particles.

Variable g is computed by

where a_{s} is the superficial area of polymer particles covered by one mole of emulsifier.

Variables [E]^{aq}, [E]_{ads} and M_{mic} must be computed for two different cases

**Case 1 - **The emulsifier concentration is higher than the critical micelar concentration (CMC). In this case, the emulsifier concentration in aqueous phase equals CMC; concentration of emulsifier adsorbed on polymer particles equals the saturation concentration and the excess emulsifier is in the form of micelles. Thus,

and M_{mic} is computed by Eq. (A-12) as follows:

**Case 2** - The emulsifier concentration is lower than the critical micelar concentration (CMC). No micelles are formed; it is assumed that the emulsifier is preferably absorbed on polymer particles, thereby stabilizing them. Thus,

If the emulsifier concentration is sufficient to saturate polymer particles, then

and [E]^{aq} is computed by Eq. (A-12) as follows:

If the emulsifier concentration is not sufficient to saturate polymer particles, then

d) Monomer (moles)

where [M]_{p} is the monomer concentration in polymer particles. Monomer concentrations in the different phases present in emulsion polymerization reactions were computed by the iterative procedure proposed by Omi et al. (1985).

e) Polymer (g)

f) Water (cm^{3})

Computation of the Average Number of Radicals per Polymer Particle

The method proposed by Ugelstad et al. (1967) was used to compute the average number of radicals per polymer particle as follows:

where h and m are the relative absorption/termination and desorption/termination coefficients. These coefficient are defined as

where k_{abs} is the coefficient for the rate of absorption of radicals from aqueous phase by the polymer particles, given by

and R_{aq} is the concentration of radicals in aqueous phase, computed iteratively with the following equations:

where k_{t }R_{aq} and k_{p}[M]_{aq}^{ }are the rates of termination and propagation in aqueous phase, f is the efficiency with which radicals are formed by the decomposition of the initiator and k is the coefficient for the rate of desorption of radicals from polymer particles and is given by

Coefficients k_{0m} and b_{m} are the exit rate of a monomeric radical from a polymer particle and the probability of reaction of a radical in aqueous phase by propagation or termination and are given by

where D_{p} is the diffusion coefficient of a radical in polymer particles in cm^{2}/s.

Finally the following iterative steps are used to compute :

1) Set initial value for ;

2) Compute absorption and termination coefficients;

3) Set initial value for R_{aq};

4) Compute Eq. (A-29) to (A-32) until convergence of R_{aq};

5) Compute Eq. (A-25);

6) Compare computed with the previous value of .Return to item 2 until the error is smaller than the stipulated tolerance.