Dynamic Evolution of the Particle Size Distribution in Suspension Polymerization Reactors: A Comparative Study on Monte Carlo and Sectional Grid Methods Vassilis Saliakas, Costas Kotoulas, Dimitris Meimaroglou and Costas Kiparissides* Department of Chemical Engineering, Aristotle University of Thessaloniki and Chemical Process Engineering Research Institute, P.O. Box 472, Thessaloniki, Greece 54124
In the present study, an efficient Monte Carlo (MC) algorithm and a fixed pivot technique (FPT) are described for the prediction of the dynamic evolution of the droplet/particle size distribution (DSD/PSD) in both non-reactive liquid–liquid dispersions and reactive liquid(solid)–liquid suspension polymerization systems. Semi-empirical and phenomenological expressions are employed to describe the breakage and coalescence rates of dispersed monomer droplets/particles, in terms of the type and concentration of suspending agent, quality of agitation, and evolution of the physical, thermodynamic and transport properties of the polymerization system. Moreover, the validity of the numerical calculations is first examined via a direct comparison of simulation results obtained by both numerical methods with experimental data on average particle diameter and droplet/particle size distributions for both non-reactive liquid–liquid dispersions and the free-radical suspension polymerization of methyl methacrylate (MMA). Additional comparisons between the MC and the FP numerical methods are carried out under different polymerization conditions. The simulation results reveal that both numerical methods are capable of predicting the mean and the distributed particulate properties of both non-reactive and reactive suspension processes. On d´ecrit dans cet article un algorithme de Monte Carlo (MC) efficace et une technique de pivot fixe (FPT) pour pr´edire l’´evolution dynamique de la distribution de taille de gouttelettes/particules (DSP/PSD) a` la fois dans des dispersions liquide–liquide non r´eactives et des syst`emes de polym´erisation par suspensions liquide (solide)–liquide r´eactifs. On emploie des expressions semi empiriques et ph´enom´enologiques pour d´ecrire les taux de rupture et de coalescence de gouttelettes/particules de monom`eres dispers´es, en fonction du type et de la concentration d’agent de suspension, de la qualit´e de l’agitation et de l’´evolution des propri´et´es physiques, thermodynamiques et de transport du syst`eme de polym´erisation. En outre, la validit´e des calculs num´eriques est d’abord examin´ee par une comparaison directe des r´esultats de simulation obtenus par les deux m´ethodes num´eriques avec les donn´ees exp´erimentales du diam`etre de particule et de la distribution de taille de gouttelettes/particules moyens a` la fois pour les dispersions liquide-liquide non r´eactives et la polym´erisation par suspensions de radicaux libres du m´ethacrylate de m´ethyle (MMA). D’autres comparaisons entre les m´ethodes num´eriques MC et FP sont effectu´ees pour des diff´erentes conditions de polym´erisation. Les r´esultats de simulation r´ev`elent que les deux m´ethodes num´eriques sont capables de pr´edire les propri´et´es particulaires moyennes et distribu´ees des deux proc´ed´es par suspension non r´eactif et r´eactif. Keywords: particulate processes, population balance, suspension polymerization, Monte Carlo, fixed pivot
INTRODUCTION
A
n important property of suspension polymerization processes is the particle size distribution (PSD), which controls key aspects of the process and affects the end-use properties of the product. Suspension polymerization processes are generally characterized by PSDs that can vary in time with respect to the mean particle size as well as to the PSD form (i.e., broadness and/or skewness of the distribution, unimodal
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and/or bimodal character, etc.). The quantitative calculation of the evolution of the PSD presupposes a good knowledge of the
∗ Author to whom correspondence may be addressed. E-mail address:
[email protected] Can. J. Chem. Eng. 86:924–936, 2008 © 2008 Canadian Society for Chemical Engineering DOI 10.1002/cjce.20091
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droplet/particle breakage and coalescence mechanisms. These mechanisms are coupled with the reaction kinetics, thermodynamics and other microscale phenomena, including mass and heat transfer between the different phases present in the system. The time evolution of the PSD is commonly obtained from the solution of a population balance equation (PBE), governing the dynamic behaviour of the dispersed liquid monomer droplets that are being polymerized to solid polymer particles. The numerical solution of the dynamic PBE is a notably difficult problem due to a number of numerical complexities and model uncertainties regarding the particle breakage and coalescence mechanisms and are often poorly understood. It commonly requires the discretization of the particle volume domain into a number of discrete elements and the subsequent numerical solution of the resulting system of stiff, nonlinear differential or algebraic/differential equations (DAEs). In the open literature, a number of numerical methods have been reported on the steady-state and dynamic solutions of the general PBE. These include the full discrete method (Hidy, 1965), the method of classes (Marchal et al., 1988; Chatzi and Kiparissides, 1992), the discretized PBE (DPBE) methods (Batterham et al., 1981; Hounslow et al., 1988), the fixed and moving pivot DPBE methods (Kumar and Ramkrishna, 1996a,b), the high-order DPBE methods (Bleck, 1970; Gelbard and Seinfeld, 1980; Sastry and Gaschignard, 1981; Landgrebe and Pratsinis, 1990), the orthogonal collocation on finite elements (OCFE) (Gelbard and Seinfeld, 1979), the Galerkin method (Nicmanis and Hounslow, 1998) and the wavelet-Galerkin method (Chen et al., 1996). In the reviews of Ramkrishna (1985), Dafniotis (1996), and Kumar and Ramkrishna (1996a), the various numerical methods available for solving the general PBE are described in detail. Moreover, extensive comparative studies have been presented in the publications of Kostoglou and Karabelas (1994, 1995), Nicmanis and Hounslow (1996) and in a recent series of papers by Kiparissides and coworkers (Alexopoulos et al., 2004; Alexopoulos and Kiparissides, 2005; Roussos et al., 2005; Meimaroglou et al., 2006). On the basis of the conclusions of these studies, the DPBE method of Litster et al. (1995), the fixed pivot method of Kumar and Ramkrishna (1996a), the Galerkin and the orthogonal collocation on finite-element methods were found to be the most accurate and stable numerical techniques for the numerical solution of the PBE. The dynamic evolution of the PSD in a particulate process can also be obtained via stochastic Monte Carlo (MC) simulations. Spielman and Levenspiel (1965) were the first to employ an MC approach to study the effect of particle coalescence in a twophase particulate reactive system in well-mixed reactors. Later, Shah et al. (1977) developed a general MC algorithm for time varying particulate processes. In 1981, Ramkrishna (Ramkrishna, 1981) established the precise mathematical connection between population balances and the MC approach. In MC simulations, the dynamic evolution of the PSD is inferred by the properties of a finite number of sampled particles. Based on the method employed for the determination of the sampling time step, MC simulations can be distinguished into time-driven (Domilovskii et al., 1979; Liffman, 1992; Debry et al., 2003) and event-driven ones (Garcia et al., 1987; Smith and Matsoukas, 1998; Tandon and Rosner, 1999; Kruis et al., 2000; Efendiev and Zachariah, 2002). In regard to the total number of simulated particles, MC methods can be further classified into constant number and constant volume MC methods. A more detailed description of the characteristics of the various MC formulations can be found in a number of publications (Maisels et al., 2004; Meimaroglou et al., 2006; Zhao et al., 2007).
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In the present study, two numerical approaches, namely, the fixed pivot technique (FPT) and a stochastic Monte Carlo (MC) algorithm are applied for solving the general PBE, governing the PSD developments in a methyl methacrylate (MMA) free-radical batch suspension polymerization reactor, in terms of the process conditions (i.e., monomer to water volume ratio, temperature, type and concentration of stabilizer, energy input into the system, etc.) and polymerization kinetics. To the best of our knowledge, this is the first time that the two numerical methods (i.e., the FPT and MC) are applied for the calculation of the dynamic evolution of the PSD in a batch suspension polymerization reactor, using a comprehensive model taking into account all the physical and chemical phenomena in the polymerization process. The validity of both numerical methods is examined via a direct comparison of model predictions with experimental measurements on the average particle diameter and the droplet/particle size distributions for both non-reactive and reactive systems.
MODEL DEVELOPMENTS In suspension polymerization, the monomer is commonly dispersed in the continuous aqueous phase by the combined action of surface-active agents (i.e., inorganic or/and water soluble polymers) and agitation. All the reactants (i.e., monomer, initiator(s), etc.) reside in the organic or “oil” phase. The polymerization occurs in the monomer droplets that are progressively transformed into sticky, viscous monomer–polymer particles and, finally, into rigid, spherical polymer particles in the size range of 50–500 m (Kiparissides, 1996). The polymer content, in the fully converted suspension, is typically 30–50% (w/w). Large quantities of commercially important polymers (e.g., polystyrene and its copolymers, poly(vinyl chloride), poly(methyl methacrylate), etc.) are produced by the suspension polymerization process. One of the most important issues in the operation of a suspension polymerization reactor is the control of the final PSD (Yuan et al., 1991). The initial monomer droplet size distribution (DSD) as well as the final PSD largely depend on the type and concentrations of the surface active agents, the quality of agitation and the physical properties (e.g., densities, viscosities, and interfacial tension) of the continuous and dispersed phases. The transient droplet/particle size distribution is controlled by two dynamic processes, namely, the droplet breakage and coalescence mechanisms. The former mainly occurs in regions of high shear stress (i.e., near the agitator blades) or as a result of turbulent velocity and pressure fluctuations along the surface of a droplet. The latter is either increased or decreased by the turbulent flow field and can be assumed to be negligible for dilute dispersions, at sufficiently high concentrations of surface active agents (Chatzi and Kiparissides, 1992). In general, the dynamic evolution of the PSD in a batch suspension polymerization reactor can be calculated from the solution of the governing PBE. The distribution of the droplets/particles is considered to be continuous in the volume domain and is commonly described in terms of the number density function, n(V, t). Thus, n(V, t) dV represents the number of particles per unit volume in the differential volume size range (V, V + dV). For a dynamic particulate system, undergoing simultaneous particle breakage and coalescence, the rate of change of the number density function with respect to time and volume can be described by the following non-linear integro-differential PBE (Kiparissides
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et al., 2004): ∂[n(V, t)] = ∂t
Vmax
ˇ(U, V)u(U)g(U)n(U, t) dU V
V/2
k(V − U, U)n(V − U, t)n(U, t) dU
+ Vmin
(1)
Vmax
k(V, U)n(U, t) dU
−n(V, t)g(V) − n(V, t) Vmin
The first term on the right-hand side (r.h.s.) of Equation (1) represents the generation of droplets in the size range (V, V + dV) due to breakage. ˇ(U, V) is a daughter droplet breakage function, accounting for the probability that a droplet of volume V is formed via the breakage of a droplet of volume U. The function u(U) denotes the number of droplets formed by the breakage of a droplet of volume U and g(U) is the breakage rate of droplets of volume U. The second term on the r.h.s. of Equation (1) represents the rate of generation of droplets in the size range (V, V + dV) due to droplet coalescence. k(V, U) is the coalescence rate between two droplets of volume V and U. Finally, the third and fourth terms represent the droplet disappearance rates due to breakage and coalescence mechanisms, respectively. Equation (1) will satisfy the following initial condition at t = 0: n(V, 0) = n0 (V)
(2)
where n0 (V) is the initial droplet size distribution of the dispersed phase. The numerical solution of the PBE (Eq. 1) presupposes the knowledge of the breakage and coalescence rate functions. In the open literature, several forms of g(V) and k(V, U) have been proposed to describe the droplet breakage and coalescence rate functions in liquid–liquid dispersions (Narsimhan et al., 1979; Sovova, 1981; Chatzi et al., 1989; Coulaloglou and Tavlarides, 1977). More details regarding the selection of the above kernels can be found in Appendix. In regard with the droplet/particle breakage and coalescence phenomena, the suspension polymerization can be divided into three phases (Hamielec and Tobita, 1992; Kotoulas and Kiparissides, 2006, Figure 1). During the initial low-conversion (i.e., low-viscosity) phase, the monomer droplet breakage is the dominant mechanism. As a result, the initial DSD shifts to smaller sizes. During the second sticky-phase of polymerization, the droplet breakage rate decreases while the droplet/particle coalescence becomes the dominant mechanism. Thus, the average particle size increases. In the final phase, the PSD reaches its identification point while the polymer particle size slightly decreases due to shrinkage (i.e., the polymer density is greater than the monomer one). As can be seen in Figure 1, the second phase in the suspension polymerization of MMA coincides with the manifestation of the diffusion-controlled phenomena in the polymerization kinetics. This manifestation is marked by a sharp increase in the polymerization rate, followed by an increase in the monomer conversion and the reaction heat. This high particle polymerization rate can lead to particle overheating (i.e., the individual temperature of a particle, Tp , can exceed the temperature of the continuous phase, Tc ) due to heat-transfer limitations (see Figure 2). Particle overheating can directly affect the droplet/particle size distribution developments through the simultaneously occurring changes in the individual particles’ temperatures and physical
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Figure 1. Dynamic evolution of monomer conversion and mean particle diameter in suspension polymerization of MMA (ϕm = 0.2; I0 = 1 wt% based on monomer; T = 70◦ C; N = 500 rpm).
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Figure 2. Effect of particle diameter on particle overheating. Inset figure: monomer conversion and gel-effect index versus time for the suspension polymerization of MMA (ϕm = 0.2; I0 = 1 wt% based on monomer; T = 70◦ C).
properties (i.e., densities, viscosities, and interfacial tensions). Moreover, particle overheating can considerably increase the particle agglomeration rate (i.e., via the increase of particles’ stickiness). In general, particle overheating (i.e., the temperature difference (Tp − Tc )) can be calculated by the following pseudosteady state equation: (Tp − Tc ) =
(−Hr )Vp Rp hAp
(3)
where (−Hr ) denotes the heat of polymerization. Vp is the volume of the particle and Rp is the particle polymerization rate. Finally, h and Ap are the heat-transfer coefficient and the particle’s external surface area, respectively. In the present study, different correlations (i.e., Ranz and Marshall, 1952, etc.) were considered for calculating the particle’s heat transfer coefficient. During the second sticky-phase of polymerization, the effect of particle overheating on the droplet/particle coalescence rate was taken into account via the dependence of the kc term (see Eq. 36) on the polymerization rate. According to the original work of Achilias and Kiparissides (1992), the onset of the geleffect in the free-radical polymerization of MMA usually occurs in the monomer conversion range of (25–40%) (see the inset in Figure 2). In the region A, the termination rate constant becomes diffusion-controlled since it involves the simultaneous
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Table 1. Physical, transport properties and PSD model parameters for MMA/PMMA system m = 968–1.225 (T − 273.15) (kg/m3 ) p = 1212–84.5 (T − 273.15) (kg/m3 ) −3 m = 10[453.25(1/T −1/254.92)] (kg/m s) 0.64 (m3 /kg) [n] = 9.8 × 10−5 Mw Ep = 3.3 × 1011 (kg/m s2 ) r = 35, SNsa = 30, kb = 40, ab = 41, kc0 = 6 × 10−7 , ac = 1 × 109 , ac = 3 × 103 , km,p = 0.8, lm,p = −3, K = 0.0104, KA = 10.56
Achilias and Kiparissides Achilias and Kiparissides (1992) Achilias and Kiparissides (1992) Achilias and Kiparissides (1992) Achilias and Kiparissides (1992) Brandrup and Immergut (1989) This study
diffusion and reaction of two “live” polymer chains. This results in a decrease of the termination rate constant, ktc , as well as an increase of the gel-effect index, GI (GI = (Rp /Rp0 − 1)). The maximum value of the gel-effect index marks the crossover from region A to region B. In the region B, a decrease in the initiator efficiency takes place, due to the cage-effect, leading to an analogous decrease in the polymerization rate and, ultimately, in the gel-effect index, GI. In accordance to the changes in the gel-effect index, GI, the droplet/particle collision frequency rate constant, kc , was expressed, in the regions A and B, as follows: kc = p1 exp kc 0
t p2
,
t kc = p3 exp − kc0 p4
(4)
where t and kc0 are the time and the initial value of the collision frequency constant, respectively. The parameters p1 , p2 , p3 , and p4 , appearing in the semi-empirical equation (4), were calculated in terms of the starting and ending values of GI in the regions A and B. The variation of the viscosity of the polymerizable monomer phase, d , was calculated using the following non-ideal mixing equation for the monomer/polymer solution (Song et al., 2003): ln d = (1 − p ) ln m + p ln p +[km,p + lm,p (1 − 2p )](1 − p )p ln m,p ln m,p =
ln p − ln m 2
(5) (6)
where p and m denote the viscosity of the polymer and monomer, respectively, ϕp is the polymer volume fraction in the polymerizing monomer droplets and km,p and lm,p are model parameters (Song et al., 2003) (see Table 1). Additional details, regarding the calculation of the densities of the suspension system, s , continuous phase, c , and dispersed phase, d , as well as the viscosity of the continuous phase, c , can be found in the publication of Kotoulas and Kiparissides (2006). The interfacial tension between the aqueous and the monomer/polymer dispersed phase, , is a key parameter governing the suspension stability and, consequently, the droplet/particle size distribution developments. In the present study, the variation of the interfacial tension with monomer conversion was indirectly taken into account through the variation of kc (see Eq. 4). The variation of with respect to the stabilizer (PVA) concentration was calculated by the following equation (Adamson, 1976): = 0 − K
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(7)
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where K and KA are model parameters and 0 denotes the interfacial tension between the pure water phase and the dispersed monomer phase, in the absence of the stabilizer (see Table 1).
THE MONTE CARLO METHOD The stochastic Monte Carlo (MC) method is based on the principle that the dynamic evolution of an extremely large population of particles/droplets, Np (t), (e.g., 1020 ) can be followed by tracking down the relevant events (i.e., coalescence and breakage) occurring in a smaller population of sample particles/droplets, Ns (t), (e.g., 104 ). It is commonly assumed that the effects of the droplet mechanisms on the properties of the sample population are representative of the corresponding effects on the properties of the total droplet population. The MC algorithm employed in the present study is schematically depicted in Figure 3. Initially, the size of the sample population, Vms (0), is defined via the division of the initial monomer volume, Vmp (0), by a factor f. Subsequently, monomer droplets are randomly created to form the sample population in such a way so that the particle array at time zero closely represents the initial particle size distribution and the total volume of the generated droplets is equal to Vms (0). This procedure involves the random assignment of droplet diameters, Di , and respective volumes, Vi (Vi = Di3 /6), to every single droplet in the sample population. It should be noted that for every random procedure described in the present work, a random number generator from a uniform distribution in the range [0,1] was employed. In the present study, the initial droplet-diameter distribution was assumed to follow a normal distribution:
P(Di ) =
1 √ 2
(Di − )2 exp − 2 2
,
i = 1, 2, . . . , Ns
(8)
where the parameters and denote the mean and the standard deviation of the distribution, respectively. The initial sample population was generated according to the approach of Box and Muller (1958). A typical sample population usually contains about 103 –105 droplets in order to ensure an accurate representation of the initial distribution and, at the same time, to minimize the computational requirements. Once the sample population has been formed, the MC algorithm is initiated and the effects of droplet coalescence and breakage mechanisms on the dynamic evolution of the droplet population are stochastically simulated in a series of consecutive variableduration time steps. At the beginning of every time step, two consecutive decisions need to be made on the basis of the calculated droplet/particle mechanism rates, using a random number generator. Initially, the type of event (i.e., coalescence or breakage) that will take place, in the next infinitesimal time interval, is determined and, subsequently, a specific droplet or pair of droplets that will undergo droplet breakage or coalescence, are chosen
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Figure 3. Schematic representation of the MC algorithm.
from the sample population. The occurrence of a droplet/particle coalescence or breakage event is decided in accordance to the following probabilities: Pi =
Ri
2
Ri
i=1 i=2
for droplet breakage for droplet coalescence
(9)
i=1
where Ri denotes the total rate (s−1 ) of each event (i.e., coalescence or breakage) in the sample population. For the selection of a specific droplet/particle (i.e., for breakage) or a pair of droplets/particles (i.e., for coalescence), the fol-
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lowing acceptance–rejection procedure was employed (Garcia et al., 1987):
R∗i ≥ rnj ⇒ Rmax i
g(Vi ) gmax (V)
≥ rnj
k(Vi ,Vj ) kmax (V,U)
for droplet breakage (10)
≥ rnj
for droplet coalescence
where R∗i denotes the rate of droplet breakage (i = 1) or droplet coalescence (i = 2) and rnj is a randomly generated number. Based on extensive numerical simulations (Kotoulas, 2006), it was found that the maximum values of gmax (V) and kmax (V, U) could
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be related to the maximum and minimum values of Vmax and Vmin gmax (V) = g(Vmax ),
kmax (V, U) = k(Vmax , Vmin )
(11)
Finally, the time interval, t, required for the occurrence of a discrete event was given by the reciprocal of the sum of the rates 2 of the two events, i=1 Ri : t =
1 2
(12)
Ri
where Ni (t) is the number of sample droplets/particles in the size range [D(i), D(i + 1)] at time t. Thus, ni (D, t) dD denotes the total number of droplets/particles per unit volume, in the diameter range (D(i), D(i + 1)). Similarly, the respective volume probability diameter density function was calculated by the following equation: Ns
Av,i (D, t) =
j=1 D(i+1)≤Dj
