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Brazilian Journal of Chemical Engineering
Print version ISSN 0104-6632On-line version ISSN 1678-4383
Braz. J. Chem. Eng. vol.17 n.4-7 São Paulo Dec. 2000
http://dx.doi.org/10.1590/S0104-66322000000400004
MATHEMATICAL MODELING OF POLYSTYRENE PARTICLE SIZE DISTRIBUTION PRODUCED BY SUSPENSION POLYMERIZATION
R.A. F. Machado^{1,2}, J.C.Pinto^{1*}, P.H.H.Araújo^{3} and A.Bolzan^{2 }^{1} Programa de Engenharia Química / COPPE / Universidade Federal do Rio de Janeiro,
Cidade Universitária, C.P. 68502, CEP 21945-970, Rio de Janeiro, RJ - Brazil
Fax: (021) 590-7135, Phone: (021) 590-2241
Email: pinto@peq.coppe.ufrj.br
^{2} Departamento de Engenharia Química e de Alimentos / CTC / Universidade Federal de
Santa Catarina, C.P. 476, CEP 88010-970, Florianópolis, SC - Brazil
^{3} Departamento de Engenharia Química / Escola Politécnica / Universidade de
São Paulo, CEP 05508-900, São Paulo - SP, Brazil
(Received: October 19, 1999 ; Accepted: April 6, 2000)
Abstract - Particle size distribution (PSD) of polystyrene particles produced by suspension polymerization is of fundamental importance in determining suspension stability and product quality attributes. Within a population balance framework, a model is proposed for suspension polymerization reactors to describe the evolution of the PSD. The model includes description of breakage and coalescence rates in terms of reaction kinetics and rheology of the dispersed phase. The model is validated with experimental data of styrene suspension polymerization.
Keywords: Suspension polymerization, particle size distribution, population balance, coalescence, breakage.
INTRODUCTION
Suspension polymerization is an important industrial process usually carried out in a stirred tank reactor. A number of important commercial resins are manufactured by suspension polymerization, including poly(vinyl chloride) and copolymers, styrene resins (ABS, EPS, GPPS, HIPS, SAN), poly(methyl methacrylate) and copolymers, and poly(vinyl acetate). In a typical suspension polymerization system, one or more relatively water-insoluble monomers are dispersed in water (the continuous phase) by a combination of strong stirring and the use of small amounts of suspending agents (stabilizers). Polymerization is started with the addition of an initiator that is soluble in the organic phase (ex. azo-compounds or peroxides). The monomer droplets are slowly converted from a highly mobile liquid state through a sticky stage to hard solid polymer particles. At this point, also known as the Particle Identification Point (PIP), the viscosity is large enough for particle breakage and coalescence to cease (Bishop, 1971; Lontra, 1990; Yuan et al., 1991; Carafilakis, 1993). To prevent reaction runaway during the sticky stage that would lead to disastrous coalescence process, stabilizers are added to the reactor. The stabilizers hinder the coalescence of the monomer droplets first, and later stabilize the polymer beads which tendency to agglomerate may become critical when the polymerization has advanced to the point where the polymer beads become sticky. Other very important project parameters to prevent reaction runaway are: type, position, and ratio between the diameter of the impeller and the reactor, and the agitation frequency.
Three different stages may be distinguished during the evolution of particle size distribution in suspension polymerization. In the first stage a liquid-liquid dispersion is formed. Monomer is dispersed in small drops by strong stirring, and the drops break due to the interaction with the turbulent flux. The coalescence is prevented due to the use of suspending agents. In the second stage, the breakage rate diminishes and the coalescence rate of drops (polymer particles) increases as the viscosity of the particles increases with conversion. The breakage of particles ceases when the viscous forces inside the particles become higher than the turbulent forces generated by the impeller. In the sticky stage, when the action of the stabilizers is not enough, the coalescence of particles increases. In the last stage, particles become rigid (PIP) and the PSD remains the same until the end of the reaction. The viscosity of the particles at this stage is large enough to guarantee that particle breakage and coalescence cease. The final product is obtained in the form of non-uniform spherical particles. The product PSD affects important product quality attributes, for instance, bead impregnability and morphology after expansion, processability, insulation capability and mechanical resistance (Alvarez et al., 1994).
According to Vivaldo-Lima et al. (1997), the most important issue in the practical operation of suspension polymerization is the control of the final PSD. The size of the particles will depend on the monomer type and concentration, the viscosity change of the dispersed phase with time, the type and concentration of stabilizer, and the agitation conditions in the reactor. The development of a mathematical model able to reproduce the behavior of the PSD during suspension polymerization is a very useful tool to determine the optimum reaction conditions to obtain a PSD that is more adequate for the final application. The model can also help to understand the mechanisms involved on breakage and coalescence of drops (polymer particles) during the reaction.
Despite the importance on modeling the evolution of the PSD in suspension polymerization, only few publications have appeared about this subject (Mikos et al., 1986; Kiparissides et al., 1994; Alvarez et al., 1991 and 1994). Alvarez et al. (1991, 1994) were the only ones to consider the possibility of breakage and coalescence, and to include of rheology effects upon breakage and coalescence rate distribution. However, the final PSD calculated by the model was not compared with experimental data.
In this work, a mathematical model able to describe the evolution of PSD of the particles produced in batch-suspension polymerization is presented. A population balance model is used to represent the PSD evolution. The resulting set of integro-differential equations is solved using numerical integration and an adaptive orthogonal collocation method (Pinto & Biscaia, 1996). The model is validated with experimental data of batch styrene suspension polymerization, showing good agreement.
EXPERIMENTAL
The experimental unit consists of: a jacketed glass tank reactor made of boron silicate (FGG Equipamentos Científicos) with internal volume of 1 liter; a thermocryostatic bath (Microquímica); a stirrer (Eberle) that works in the range between 0 and 7000 rpm with a frequency controller (Model Micromaster, Siemens); a three-bladed propeller; a thermocouple model J (Ecil); a digital tachometer (model TD2004-C, Takotron). The agitation frequency is observed with a digital indicator, and can also be observed in-line with a microcomputer (IBM PC), equipped with an acquisition data card AD/DA (Analog-Digital/Digital-Analog, Data Translation) and software for real time processing developed by our laboratory staff. Reaction temperature is observed with a digital indicator of the thermocryostatic bath, and through an acquisition data system. Temperature control is realized by the thermocryostatic bath with an error of ± 0,5^{o}C. Agitation frequency is maintained at the desired value within a precision of ± 5 rpm, and can be changed remotely with the microcomputer. Figure 1 shows the scheme of the experimental unit.
The suspending agent used in all experiments was poly(vinyl pyrrolidone), PVP K-90 (Sigma-Aldrich). The initiator was benzoyl peroxide (BPO) with p.a. quality (Sigma-Aldrich). Distilled water was used as continuous media. Styrene was provided by Nitriflex S.A.. All reactants were used as received, including the monomer that contains p-terc-butylcatecol as inhibitor.
The water volume used in the experiments was of 0.406 liter to 0.130 liter of monomer, corresponding to a volume ratio of 1.0:0.34. This condition was kept in all reactions since it is normally considered as a design parameter. The reaction temperature was 84^{o}C in all experiments. Experimental procedure consists of charging the reactor with distilled water, followed by the addition of stabilizer (PVP K90 reaching a concentration of 2.0g / liter of water) and monomer at the defined agitation frequency. When the reaction medium temperature reaches 84^{o}C, the initiator, benzoyl peroxide (5.0 grams) is added.
The initial size distribution of dispersed styrene drops is not known, due to available equipment limitations. For this reason, it is only possible to know the PSD for conversions above 70%, when the particles have reached the PIP. The high viscosity of the particles at this point allows sampling to analyze the PSD with SEM (scanning electronic microscopy). The deviations observed in the number average particle size of two experiments performed at the same conditions was of 2%.
MATHEMATICAL MODEL
In general, the kinetics of batch suspension polymerization via free radical of relatively water-insoluble monomers, as styrene, are similar to those presented by bulk polymerizations. The process taking place in the suspension droplets can be simply considered to be a bulk polymerization in a small scale. For this reason, each monomer droplet can be considered as a mini bulk reactor, and the global behavior of the reactor is the sum of the behavior of all droplets (Mano et al., 1985; Alvarez et al., 1994; Kalfas & Ray, 1993). Reaction kinetics is not affected by the evolution of PSD, although the reaction kinetics affect the PSD, as the viscosity of the particles (polymer phase) increases very much with conversion, changing the rates of coalescence and breakage of the particles. For this reason, to calculate the conversion, it is not necessary to know the PSD. However, to calculate the evolution of the PSD, it is necessary to know the evolution of the conversion.
In a suspension where there is no mass transfer or nucleation, the population balance equations that describe PSD evolution in a batch stirred tank are described by:
(1) |
where the first and the second terms of the right-hand side of Equation (1) account for the disappearance and appearance of a particle of mass m due to coalescence, respectively. The third and fourth terms of the right-hand side account for the appearance and disappearance of a particle of mass m due to breakage, respectively.
In Equation (1) N_{P} is the total number of particles, K_{C}(m,m’) is the coalescence rate constant, b (m,m’) is the daughter droplet probability, l (m) is the average number of daughter droplets generated by the breakage of a droplet of mass m, g (m) is the
breakage probability, and f(m) is the population density of the PSD, given by:
(2) |
where n(m) is the number of particles of mass m.
If Equation (1) is integrated to include all particles of all sizes, Equation (3) can be obtained for the total number of particles:
(3) |
where the first term of the right hand side of Equation (3) account for particles that disappear after coalescence; the second, for particles that appear due to breakage of a particle.
Analytical solution of Equations (1) and (3) can only be obtained in specific cases, Ramkrishna (1985), Dabral et al. (1996), when very simple functions are used to describe coalescence or breakage rates. For the numerical solution of this framework, many strategies have been implemented: Mikos et al. (1986), Alvarez et al. (1994), Wright & Ramkrishna (1992), Kumar & Ramkrishna (1996.a,b), Das (1996), Chen et al. (1996), Spicer & Pratsinis (1996), Hill & Ng (1996), Alvarez & Alvarez (1989), Pinto & Biscaia (1996), Araújo (1999).
It is possible to observe that there is no consensus in literature about the determination of functions to describe particle breakage and coalescence kinetics with the existence of a significant number of quite different models. These models usually present equations with many parameters that are normally very difficult to obtain or to estimate. For this reason, semi empirical equations with simple correlations are very adequate for specific operation conditions. The same procedure is valid to estimate the minimum and the maximum stable particle size as a function of conversion.
The coalescence model used in this work considers that particles below a minimum size coagulate with any particle and that particles above the minimum size do not coagulate with other particles with size above this minimum size. According to the DLVO Theory (Chen & Kuo, 1996), the coalescence rate can be described as:
(4) |
where, r_{i} and r_{j} are the radius of the particles (considered as spheres) that are coagulating, N_{A} is the Avogrado’s number, T is the temperature of the reaction media, W is a term that accounts for the electrostatic interactions between the particles, C_{1} is the coalescence constant.
The breakage rate used in this work is given by:
(5) |
where m_{max} is the mass of a particle with the maximum stable size and C_{2} is the breakage constant.
For binary breakage (l (m) = 2), the daughter droplet probability is assumed to be (Mikos et al., 1986):
(6) |
where s_{f} is the standard deviation of a normal distribution, and is chosen such that the daughter droplet distribution lies essentially within the range from 0 to v. It can be assumed that:
(7) |
where c is a proportionality constant that assumes the value of 0.4 for binary breakage.
Experimental data obtained by Machado (1997) was used to incorporate the influence of viscous effects in the breakage and coalescence rates. The author observed that agitation frequency and monomer concentration do not affect reaction kinetics, so it was possible to use experimental data (conversion) to obtain a master curve of the time evolution of conversion. Knowing the evolution of conversion and the weight-average molecular weight (Mw), that was the same for all reactions (45^{.}10^{3}), it is possible to obtain the evolution of the viscosity for the system polystyrene-styrene with Equation (8) proposed by Spencer & Williams (Chen, 1994):
(8) |
where h is the viscosity of the polymer particle in poise, T is the particle temperature (that is the same as reactor temperature) in Kelvin, x is the conversion and Mw is the weight-average molecular weight.
With Equation (8) it is possible to couple reaction kinetics to PSD model, considering that there is no mass transfer between dispersed and continuous phase and there is no phase separation during the reaction. These assumptions are very reasonable for styrene suspension homopolymerization.
An initial PSD was assumed for the monomer drops as there is no way to know the real PSD of the monomer drops. Initial PSD is affected by reactor and impeller geometry, impeller speed, emulsifier concentration, fraction of the dispersed phase, density of the dispersed and continuous phase, superficial tension and reaction temperature. It was assumed that the initial average size was of 300 mm and the PSD followed a normal distribution given by Equation (9):
(9) |
where is the average mass of the particles, s is the standard deviation and m is the particle mass.
Due to particle breakage and coalescence, the initial distribution is not stable. Coalescence is prevented at the beginning of the reaction by the addition of stabilizers and breakage occurs until all particles stay below the maximum stable size. However, as reaction proceeds, particle viscosity increases drastically, reducing the particle breakage rate as the viscous forces counterpart turbulent forces.
During viscosity increase there is an intermediary stage when particles become extremely sticky and the action of the stabilizers is not enough to prevent coalescence. At this stage occurs the coalescence of polymer particles with a size below the minimum stable size. The viscosity ranges at which breakage and coalescence occur are assumed model parameters.
To solve the system of integro-differential equations of the population balance framework, an adaptive orthogonal collocation technique as proposed by Pinto & Biscaia (1996) was used. To calculate PSD, 8 collocation points were used, resulting in nine integro-differential equations. The integrals were solved using a numerical method and the integration range was from 0 to 500 mm in relation to particle diameter.
MODEL RESULTS
The following simulations considered that breakage and coalescence occur from a conversion of 0 to 15% and from 10 to 56%, respectively. These values correspond to particle viscosity changes during the reaction, as calculated by Equation (8). The maximum stable particle diameter was taken as 120 mm and the minimum stable particle diameter was considered as a function of particle viscosity, as viscosity increases around 4 to 5 orders of magnitude from 10 to 56% of conversion. The initial minimum stable diameter (dp_{min}(0)) was assumed to be equal to 80 mm, and the following equation was used to calculate the evolution of the minimum stable diameter (dp_{min}) with viscosity (h ):
(10) |
The model was validated with experimental data of two reactions (A and B) with different agitation frequencies. In reaction A agitation frequency was kept constant during the whole reaction at 800 rpm. To obtain experimental PSD data, at least 200 particles per sample were counted. Above this value, the average diameter and the standard deviation did not change significantly. In Table 1 a good agreement is shown between simulated and experimental final average particle diameters and final standard deviations of reaction A.
Figure 2 shows the evolution of conversion, average particle diameter, standard deviation and particle number. At the beginning of the reaction, the breakage rate is very strong as the particles present a very low viscosity (monomer droplets), therefore particle number increases and the average diameter decreases. With viscosity increasing with conversion, the breakage rate diminishes along the reaction until it ceases completely. At this point, the reaction presents a maximum in particle number. At the sticky stage coalescence rate is rather significant and therefore the particle number diminishes. However, when reaction reaches the particle identification point (PIP), particle coalescence ceases completely and the PSD remains the same until the end of the reaction. Average particle diameter decays slowly after 60% of conversion due to the volume contraction of polymer particles.
Final PSD expressed as number fractions computed by the model and obtained experimentally presented a good agreement as can be observed in Figure 3.a, where the experimental PSD is in the form of histogram and the model result is represented by a continuous line. Figure 3.b presents the PSD evolution in volume fraction. Particles larger than the maximum stable diameter start breaking, resulting in a bimodal PSD with an increasing population of small particles. This behavior proceeds until particle viscosity increases and breakage ceases at 15% of conversion (80 minutes of reaction). At 10% of conversion the sticky stage starts and particles begin to coagulate. Therefore after the end of particle breakage, the PSD is displaced to higher sizes until the end of reaction at 300 minutes.
Reaction B is carried out at the same condition as reaction A, although after 60 minutes of reaction, the agitation frequency, w, was increased from 800 rpm to 1000 rpm. To incorporate agitation frequency effect on breakage and coalescence rates in the PSD model the respective constants were multiplied by a factor that accounts for the increase of these rates as the number of collisions between particles (coalescence) and the turbulent forces (breakage) increase with the agitation frequency. The factor is different for each phenomenon, as the influence of agitation frequency is not the same for breakage and coalescence. The modified constants are represented as:
(11) |
(12) |
where w is agitation frequency in rpm.
Analogously, it is assumed that the minimum stable diameter (dp_{min}) increases with agitation frequency with the following rate:
(13) |
As reaction kinetics did not change with the increase in agitation frequency, the ranges of conversion where breakage and coalescence occur did not change either, as they only depend on particle viscosity.
In Table 2 simulated and experimental final average particle diameters and final standard deviations of reaction B are compared. Model results were very close to experimental data.
In Figure 4 the evolution of the average particle diameter and of the standard deviation in reaction B are observed. This reaction presented a stronger coalescence rate when compared to reaction A that resulted in an increase in the final average particle diameter.
Also under this new condition the simulated final PSD in number fraction presented a good agreement with the experimental PSD as can be observed in Figure 5, where the experimental PSD is in the form of histogram and the model result as a continuous line.
DISCUSSION AND CONCLUSIONS
The mathematical model developed in this work is able to compute the evolution of the particle size distributions (PSD) and simulation results were compared with final experimental PSD of batch styrene suspension polymerizations. Adaptive orthogonal collocation technique was shown to be adequate to solve the integro-differential equations of the population balance.
The use of detailed expressions to calculate breakage and coalescence rates, results in significant errors due to the difficulty to obtain precise values for several parameters and most of them, are valid only for a low holdup and no changes in drop rheology. Therefore, these expressions are not applicable to industrial suspension polymerization systems. The use of simple expressions to calculate the breakage and coalescence rates has shown to be very adequate.
Simulation results show that rheology presents a significant effect on breakage and coalescence rates and, consequently, on the final PSD. However, more data is still necessary in order to calculate the effect of agitation speed, stabilizer concentration and organic phase fraction in the reactor, upon the final PSD. This would lead to better correlations for breakage and coalescence rates, increasing the range of operation conditions where the model is applicable.
ACKNOWLEDGMENTS
The authors thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) for providing scholarships.
NOMENCLATURE
Variables
c | Proportionality constant |
C_{1} | Coalescence constant |
C_{2} | Breakage constant |
dp_{min} | Minimum stable particle diameter |
f(m) | Population density of particles of mass m |
K_{c} | Coalescence rate constant |
m | Particle mass |
m_{max} | Mass of the maximum stable particle |
Mw | Weight-average molecular weight |
N_{A} | Avogadro’s number |
Np | Total number of polymer particle |
n(m) | number of particles with mass m |
r_{i} | Particle radius |
T | Reaction temperature |
W | Term that accounts for the electrostatic interactions between the particles |
x | Conversion |
Greek Letters
b (m`,m) | Daughter droplet probability |
g (m`) | Breakage probability |
l (m) | Average number of daughter droplet |
h | Particle viscosity |
s | Standard deviation |
w | Agitation frequency |
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*To whom correspondence should be addressed