Multiscale Computational Fluid DynamicsâPopulation Balance Model

Article pubs.acs.org/IECR

Multiscale Computational Fluid Dynamics−Population Balance Model Coupled System of Atom Transfer Radical Suspension Polymerization in Stirred Tank Reactors Le Xie and Zheng-Hong Luo* Department of Chemical Engineering, School of Chemistry and Chemical Engineering, State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, Shanghai 200240, P. R. China ABSTRACT: Partially water-soluble monomers achieve phase equilibrium between aqueous and dispersed phases, and the interphase mass transfer plays a significant role in suspension polymerization. This work developed a computational fluid dynamics model to simulate the liquid−liquid suspension polymerization process. Complex multiphase flow behaviors and multiscale polymer properties were simultaneously simulated using the combination of an Eulerian−Eulerian two-phase flow model, an atom transfer radical polymerization kinetics model, a population balance model (PBM), and an interphase transfer model. The simulation results for monomer conversion, molecular weight (Mn), polydispersity index (PDI), and Sauter mean diameter (d32) agreed well with the experimental data, thereby validating the soundness of the multiscale model. The proposed model was then employed to predict key variables of the polymerization system. The coupled model can benefit the optimization and scale-up of suspension polymerization reactors with such multiscale characteristics.

1. INTRODUCTION Suspension polymerization is among the most important industrial processes that is commonly employed in the production of many commercial resins, such as polystyrene (PS) and polyvinyl chloride (PVC) as well as poly(methyl methacrylate) (PMMA).1−4 This process is usually implemented in batch stirred reactors, where suspending agents and mechanical agitation combine a mixture of reactants into oil droplets with diameters from μm scale to mm scale.5−8 Living/ controlled polymerization, such as atom transfer radical polymerization (ATRP), benefits the preparation of polymers with narrow molecular weight distribution.9,10 Suspension polymerization helps avoid the two main problems in the polymerization process, namely, intensive heat release and rapid increase in viscosity.11 Combining suspension polymerization with ATRP offers an important technology in the polymerization industry. Suspension polymerization is a liquid−liquid two phase system that includes one aqueous phase and one dispersed phase. This highly complicated process involves multiphase flow coupled with polymerization kinetics, thermodynamic equilibrium in the monomer partitioning process between two phases, and particle dynamics (i.e., particle growth, breakage, and coalescence). This process also involves a multiscale phenomenon (Figure 1), which includes the microscale of the molecular weight of the polymer, the mesoscale of mass transfer, polymerization and particle dynamics of polymer particles, and the macroscale of mixing between two phases.12 Therefore, the operating conditions have a significant role in the suspension polymerization process by determining the © XXXX American Chemical Society

Figure 1. Multiscale phenomenon in suspension polymerization.

distributions of macroscopic flow fields (i.e., velocity, temperature, and concentration), which affect microscopic flow fields (i.e., molecular weight and polydispersity index [PDI]) through polymerization.13,14 Therefore, the multiphase flow characteristics and multiscale properties of the suspension polymerization must be understood to ensure suspension stability and Received: Revised: Accepted: Published: A

January 11, 2017 April 7, 2017 April 10, 2017 April 10, 2017 DOI: 10.1021/acs.iecr.7b00147 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Figure 2. Schematic of submodel coupling in the multiscale model.

Many open works on multiscale properties have been conducted in the suspension polymerization field. Kalfas et al.5 put forward a general two-phase suspension polymerization model that considered PSD evolution. However, the polymerization kinetics was simulated independently from the particle kinetic model. Kiparissides and co-workers23,25,30 developed a two-compartment PBM that established a relationship between turbulence nonhomogeneity and PSD. In addition, they calculated the breakage and coalescence rates based on the local isotropy of the turbulent field which was a common assumption for all RANS-based simulations.20 Similarly, Kotoulas and Kiparissides31 described the dynamic evolution of PSD based on a generalized PBM using the average turbulent energy dissipation rate as a model parameter. The simulated results were consistent with the experimental data. Nogueira et al.20 conducted a multiphase CFD simulation of suspension polymerization reactors. They developed a CFD− PBM coupled system to simulate droplet breakage and coalescence distributions. However, given that the effect of polymerization was ignored, the complete flow fields in suspension polymerization reactors remained unclear. Recently, our group numerically simulated the multiphase flow characteristics and multiscale properties in various polymerization reactors.32−36 Specifically, we built a coupled model comprising the CFD model, PBM, and polymerization moment equations.32,33 However, we mostly focused on the macroscale gas− solid flow field instead of the liquid−liquid system. Wang et al.34,35 conducted several simulations to study the dynamic behavior of the atom transfer radical copolymerization (ATRcoP) in a continuous stirred-tank reactor (CSTR). However, they only considered ideal reactors. Although previous studies have proposed a comprehensive model to simulate the batch suspension ATRP of MMA while considering the polymerization and particle dynamics, this model ignores the flow fields.36

productivity. A better understanding of this process is also critical to suspension polymerization reactor design and amplification. As a valuable engineering tool, computational fluid dynamics (CFD) simulation has been widely employed to predict fluid flow in reactors with a multiscale process. Coupling with the population balance model (PBM), the CFD−PBM is extended to solve particle size distribution (PSD) in a multiphase dispersed system.15−19 Many studies have examined the applications of the CFD models in a liquid−liquid multiphase flow system in stirred tank reactors. These models have also been extended to simulate the suspension polymerization systems.20−25 However, developing a comprehensive model to describe multiscale fields, such as the polymerization reaction rate, number-average molecular weight (Mn), PDI, PSD, phase holdup distribution, and velocity vectors, remains a challenging task. Kiparissides et al.26 had modeled the dynamic behavior of suspension polymerization reactors, and their simulation results agreed well with the experimental data in terms of the time evolution of reactor pressure, temperature, monomer conversion, and Mn. Zhu et al.27 studied the methyl methacrylate (MMA) suspension ATRP kinetic behavior at different reaction temperatures. Wieme et al.28 conducted simulation studies of both pilot- and industrial-scale reactors for the batch suspension polymerization and solved the mass balances simultaneously with the energy balances. Liu et al.29 studied butyl acrylate suspension polymerization in a coaxial capillary microreactor by conducting experiments and CFD simulations. Their microreactor effectively removed polymerization heat and prepared polymer products with a narrow molecular weight distribution. However, the extant models can only describe a single-scale suspension polymerization characteristic in reactors and yield predictions that quantitatively agree with the experimental data. B

DOI: 10.1021/acs.iecr.7b00147 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research In sum, only a few studies have modeled multiscale flow fields and interphase mass transfer in suspension polymerization reactors. As described above, the multiphase flow, polymerization, and particle dynamics models are significant in understanding suspension polymerization, especially when considering the reactor performance and polymer properties. This study proposed a three-dimensional (3D) CFD−PBM coupled system to simulate liquid−liquid suspension polymerization in a stirred vessel. The coupled model comprised the Eulerian−Eulerian two-fluid model, k−ε model, polymerization kinetics model, PBM, and interphase transfer model to simulate multiscale flow fields and interphase mass transfer of monomer. The quadrature method of moments (QMOM) provided by the commercial CFD code FLUENT was used to solve the population balance equation (PBE) and describe the coalescence and breakage of droplets.37,38 The interphase mass transfer model was also solved together with the multiphase model by user-defined functions. To the best of our knowledge, this study is the first to combine the CFD model, polymerization kinetics model, PBM, and interphase transfer model to simulate suspension polymerization in stirred tanks.

Table 2. Polymerization Kinetics Moment Equations models of the moment equations Moment Definitions ∞

μm =

∑ r m[RMr ·],

∞

λm =

r=1

∞

∑ r m[RM rX ], r=1

ϕm =

∑ r m[RM r] r=1

Moment Equations

rRX = − ka,0[RX ][CX ] + kda,0[R•][CX 2] rCX = − ka,0[RX ][CX ] + kda,0[R•][CX 2] − ka[CX ]λ0 + kda[CX 2]μ0 rCX2 = ka,0[RX ][CX ] − kda,0[R•][CX 2] + ka[CX ]λ0 − kda[CX 2]μ0

r R• = ka,0f [RX ][CX ] − kda,0[R•][CX 2] − k p[R•][M ] − k t0[R•][R•] − k tR [R•]μ0 rM = − k p[R•][M ] − (k p + k tr)μ0 [M ]

rμ = k p[R•][M ] − kda[CX 2]μ0 + ka[CX ]λ0 − (k tc + k td)μ0 2 0

rμ = k p([R•] + μ0 )[M ] + k tr(μ0 − μ1)[M ] + ka[CX ]λ1 − kda[CX 2]μ1 1

− (k tc + k td)μ0 μ1

rμ = k p([R•] + μ0 + 2μ1)[M ] + k tr(μ0 − μ2 )[M ] + ka[CX ]λ 2 2

− kda[CX 2]μ2 − (k tc + k td)μ0 μ2

2. MULTISCALE MODEL DESCRIPTION Figure 2 presents the coupling scheme of the developed multiscale model. The two-fluid model, suspension polymerization mechanism and kinetics model, and PBM, which are all widely applied in simulating suspension systems with a twophase reaction, are presented in Tables 1−4. It should be

rλ0 = kda[CX 2]μ0 − ka[CX ]λ0

rλ1 = kda[CX 2]μ1 − ka[CX ]λ1 rλ2 = kda[CX 2]μ2 − ka[CX ]λ 2

1 k tcμ0 μ0 2 Average Properties of Polymer μ1 + λ1 + ϕ1 μ + λ 2 + ϕ2 Mn = Mm , M w = 2 Mm , μ0 + λ0 + ϕ0 μ1 + λ1 + ϕ1

rϕ = k tr[M ]μ0 + k tdμ0 μ0 + 0

Table 1. Elementary Reactions of Suspension ATRP ka,0

RX + CX XooooY R• + CX 2 kda,0

initiation

PDI = M w /M n

kp

R• + M → RM1•

polymerization. As the polymerization continues, the concentration of monomer in the droplets will decrease from its equilibrium value, leading to the interphase mass transfer of monomers.6 Therefore, an interphase transfer model must be incorporated into the continuity equations, momentum balance equations, and monomer mass balance equations through the interphase transfer rate. The interphase transfer rate is expressed as follows:

ka

RMr X + CX XooY RMr• + CX 2 kda

propagation kp

RMr• + M → RMr•+ 1 chain transfer to monomer

k tr

RMr• + M → RMr + M • k t0

R• + R• ⎯→ ⎯ RR k tR

R• + RMr• ⎯→ ⎯ RMr R termination

RMr•

+

k td RMs• ⎯→ ⎯

Fa → d = k ma([M ]e − [M ])

RMr + RMs

(1) e

where kma is the overall mass transfer coefficient, [M] is the equilibrium concentration, and [M] is the MMA concentration in the dispersed phase. Many studies have reported the effect of liquid−liquid equilibrium in suspension/emulsion polymerization.6,40−43 Specifically, the monomer partitioning and monomer interphase transfer rate significantly affect polymerization kinetics as well as determine the Mn and PDI. Furthermore, the reaction processes become increasingly limited by mass transfer along with the decreasing mass transfer coefficient. To describe monomer partitioning, Guillot44 proposed a thermodynamic treatment that required one to calculate the chemical potential of monomers in each phase. If the chemical potentials are unequal in all phases, then an interphase mass transfer takes place. This work employed the same method to calculate the equilibrium concentration. The free energies of mixing in two phases are computed as follows:

k tc

RMr• + RMs• → RMr + sR

pointed out that the employed breakage and coalescence kernels based on Coulaloglou and Tavlarides’ research are independent of droplet viscosity.39 These approximate kernels are used here to describe the breakage and coalescence of the non-Newtonian and even viscoelastic polymer droplets. In this manuscript, much attention is concentrated on the interphase transfer model because of its important role in mass transfer and reaction kinetics. 2.1. Interphase Transfer Model. Partially water-soluble monomers achieve phase equilibrium between aqueous and dispersed phases in suspension polymerization. We assume that the suspension polymerization system is perfectly mixed and that the phase equilibrium is established at the beginning of C


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Industrial & Engineering Chemistry Research Table 3. Models of Population Balance, Breakage, and Coalescencea Population Balance Equation

∂n(L ; x , t ) ∂ + ∇·[vn⃗ (L ; x , t )] = − [G(L)n(L ; x , t )] + B(L ; x , t ) ∂t ∂L − D(L ; x , t )

B(L ; x , t ) = +

1 2

∫0

L

∫L

∞

β(L|λ)b(λ)υ(λ)n(λ ; x , t ) dλ

c[(L3 − λ 3)1/3 , λ]n(λ ; x , t )n[(L3 − λ 3)1/3 ; x , t ]dλ

D(L ; x , t ) = n(L ; x , t )

∫0

∞

c(λ , L)n(λ ; x , t ) dλ + b(L)n(L ; x , t ) Breakage Kernel

b(L) = C1

⎛ σ(1 + αd)2 ⎞ ε1/3 ⎟ exp⎜⎜− C2 ⎟ (1 + αd) ⎝ ρd L5/3ε 2/3 ⎠

2/3

L

υ(λ) = 2, β(L|λ) =

(2L3 − λ 3)2 ⎞ 4.6 ⎛ exp⎜−4.5 ⎟ 3 λ ⎝ ⎠ λ6

Coalescence Kernel

c(λ , L) = C3(λ + L)2 (λ 2/3 + L2/3)1/2 a

⎡ C4μρε ⎛ λL ⎞4 ⎤ ε1/3 ⎟ ⎥ exp⎢− 2 c c 3 ⎜ 1 + αd ⎣ σ (1 + αd) ⎝ λ + L ⎠ ⎦

Reproduced with permission from ref 39. Copyright 1977 Elsevier.

Table 4. Governing Equations of the Two-Liquid Model models

equations

∂ (αqρq ) + ∇·(αqρq vq⃗ ) = msp , ∂t

mass conversion

q = a, d

∂ (αqρq vq⃗ ) + ∇·(αqρq vq⃗ vq⃗ ) = − αq∇p − ∇·τq − mspvq⃗ − F ⃗ + αqρq g ⃗ ∂t

momentum conversion

⎛ ⎞ 2 τq̿ = − αqμeff,q ⎜∇vq⃗ + ∇vq⃗T − I(∇vq⃗ )⎟ ⎝ ⎠ 3 ∂pq ∂ + τq : ∇vq⃗ − ∇·qq (αqρq hq ) + ∇·(αqρq vqhq ) = − αq ∂t ∂t

energy conversion

n

+

∑ (Q ad + ṁ adhad − ṁ dahda),

qq = − αqκq∇Tq

p=1

⎛ μt,m ⎞ ∂ ∇k ⎟ + Gκ ,m − ρm ε (ρm κ ) + ∇·(ρm kvm⃗ ) = ∇·⎜μ + σk ∂t ⎝ ⎠

k−ε turbulence model

⎛ μt,m ⎞ ε ∂ ∇ε⎟ + (C1εGκ ,m − C2ερm ε) (ρm ε) + ∇·(ρm εvm⃗ ) = ∇·⎜μ + σε ∂t ⎝ ⎠ k

⎧ 24 (1 + 0.15Red 0.687) ⎪ C D = ⎨ Red ⎪ ⎩ 0.44 Red > 1000

drag force model

Red =

Aqueous phase (monomer + water):

Red ≤ 1000

dd|vd⃗ − va⃗ |ρa μa

Dispersed phase (monomer + polymer + stabilizer): (ΔG /RT )i ,d = ln(αi ,d) + (1 − miP)αP,d + χiP αP,d 2 N

(ΔG /RT )i ,a = ln(αi ,a) + (1 − mi W )αW,a + χi W αW,a N

+

∑

αj ,aαW,a(χij + χi W − χj W mij) +

∑ j = 1, j ≠ i

2

χij αj ,a +

N−1

N

∑

∑

j = 1, j ≠ i k = j + 1, k ≠ i

∑ (1 − mij)αj ,a

αj ,aαk ,a(χij + χik − χjk mij)

N

αj ,dαP,d(χij + χiP − χjP mij) +

+

∑ j = 1, j ≠ i

(2)

∑ (1 − mij)αj ,d j=1

N

j=1

N

∑ j = 1, j ≠ i

N

j = 1, j ≠ i

+

+

2

χij αj ,d 2 +

N−1

N

∑

∑

αj ,dαk ,d(χij + χik − χjk mij)

j = 1, j ≠ i k = j + 1, k ≠ i

(3)

If the phase equilibrium is achieved, then we obtain D


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Figure 3. Solution procedure for the multiscale model.

Figure 4. Comparison of the effect of number of cells (Case 1: 260k; Case 2: 520k; Case 3: 1020k; Case 4: 2730k) on (A) monomer conversion and (B) d32.

(ΔG /RT )m,a = (ΔG /RT )m,d

transferred into the Eulerian−Eulerian CFD model. Afterward, the species mass fractions distributions are obtained by solving these CFD models. Consequently, an integrated coupling between polymerization kinetics model and CFD is achieved using the interphase transfer model. The phase volume fraction, velocity, and energy dissipation rate calculated by CFD are also transferred into PBM because they are related to the droplet coalescence and breakage rates. After solving the PBE, the Sauter mean diameter (d32) can be calculated by the lower order moments of PSD and used to modify the drag force in the two-fluid model and to update the flow fields of the dispersed phase (i.e., volume fraction and velocity). Overall, CFD−PBM is developed to describe the evolution of droplet size distribution (DSD). Polymerization also has an interactional and interdependent relationship with particle dynamics. The droplet size calculated by PBM can determine the molecular weight distribution.29 In contrast, when the polymerization proceeds, the droplet properties (i.e., density and viscosity) may significantly vary and subsequently affect the droplet coalescence and breakage rates. In other words, PBM

(4)

since the value of interfacial tension energy is several orders of magnitude smaller than the other terms in eq 3. Therefore, this term is usually neglected in suspension polymerization. The presence of water in the dispersed phase is ignored too.6 2.2. Multiscale Model Coupling Mechanism. During the simulation process, the temperature, pressure, phase holdup, and species mass fraction distributions in a 3D stirred reactor are specified before the iterative calculation. The basic macroscale flow field information in each cell is obtained by solving the continuity and momentum equations. Then, the species mass fraction is transferred into the interphase transfer model, which is solved to obtain the interphase transfer rate and real species mass fraction in two phases. The species mass fraction and temperature are then transferred into the polymerization kinetics moment equations. Therefore, the reaction kinetics equations are solved, and the Mn, PDI, and polymerization rate are obtained. Given that polymerization rate is the source term of the species transport, it is interactively E


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Industrial & Engineering Chemistry Research can be integrated into polymerization kinetics. As described above, the CFD, PBM, and polymerization kinetics model can improve one another in the proposed integrated model. Figure 3 presents this scheme in more detail. The above process was performed iteratively until the physical time satisfied a specific criterion. 2.3. Simulation Details. All simulations were performed in a 3D stirred tank reactor with a 45° pitch blade turbine impeller. Roudsari et al.45 described the reactor structure in detail. To reduce the computational cost, the multiple reference frame (MRF) technique was adopted for the impeller rotation. A grid interface was created to exchange data among different fluid zones. The vessel was filled with water (continuous phase) and oil (disperse phase) and contained MMA (monomer), 1chloro-1-phenylethane (initiator), copper(I) chloride (activator), and copper(II) chloride (devitalizer). The vessel was drawn and meshed using commercial software to create unstructured tetrahedral cells because of the complex reactor configuration. In this work, we conducted grid sensitivity analysis based on four cases including 260k, 520k, 1020k, and 2730k cells, respectively. The time evolution of simulated conversion and d32 were given in Figure 4. It seems that the number of grids has little effect on polymerization kinetics and the monomer conversion almost remained about the same. However, the d32 experiences a significant change with the increase of grid number. One knows that the lower number of unstructured grids is not adequate to simulate the turbulence characteristics. Considering calculation time and accuracy, a total amount of 1 018 598 cells was used in this study to predict the multiscale flow fields. The established 3D multiscale model was solved using the commercial CFD code FLUENT (ANSYS, Inc., USA) solved in a double-precision mode. The second-order upwind method was used for all simulations. If the governing equations are solved under unsteady-state conditions, then the convergence criterion of the model equations is 1 × 10−5. The ATRP kinetics model and particle dynamics model were coupled with the CFD model by user-defined functions written in C programming language. The PBM was solved using QMOM, which required solving four additional transport equations to track the four moments of droplet size distribution (DSD) during the simulation when the source terms were specified. The predictions of the multiscale model depend on several parameters, including physical property parameters, polymerization kinetic parameters, particle kinetic parameters, and boundary conditions. Tables 5 and 6 present additional details about these parameters. Given that an isothermal transport model was used in this work, the heat capacities, heat conductivity coefficient, heat transfer coefficient, and diffusion coefficients were all constant.49 All CFD simulations were executed on a 2.6 GHz Intel 2 CPU (64 cores) with 16 GB RAM. When the CFD, polymerization kinetics model, interphase mass transfer model, and PBM are solved together, it needs about 15 days to simulate the physical time of 1000 s.

Table 5. Main Parameters for the Multiscale Model descriptions

values

refs

Polymerization Kinetics Parameters ka, ka,0 (L/mol·s) 1.2853 kd, kd,0 (L/mol·s) 1.2597 × 106 kin kp (L/mol·s) 106.427 exp [−22360/(RT)] ktr (L/mol·s) 8.93 × 10−4 exp [−2240/(RT)] kt (L/mol·s) 9.8 × 107 exp [−701/(RT)] ktd/ktc 2483 exp [−4073/(RT)] ktR, kt0 (L/mol·s) 2.0 × 109 Particle Dynamics Parameters C1 0.86 C2 4.1 C3 0.04 C4 (m−2) 1.0 × 1010 Other Parameters χmw 5.1 χmp 0.50 mmw 0.34 μm (Pa·s) 0.000425 ρm (kg/m3) 866 σ (N/m) 0.019

34 46 5 47

48

40 this work this work this work

Table 6. Simulated Conditions descriptions

values

MMA (mol/L) 1-PECl (mol/L) CuCl (mol/L) initial dispersed phase holdup operating pressure (Pa) drag law wall boundary condition convergence criteria time step (s)

9.36 0.047 0.047 0.2 101 325 Schiller-Naumann no slip for fluid 1 × 10−5 1 × 10−3

model to study the effect of the interphase transfer model on atom transfer radical suspension polymerization. 3.1. Model Validation. 3.1.1. Kinetic Model Validation. To study the ATRP suspension polymerization kinetic behavior, Zhu et al.27 performed a suspension ATRP experiment under the following conditions: [MMA]0/[1PECl]0/[CuCl]0 = 200:1:1, [MMA] = 9.36 mol/L, MMA/ water = 1:4 (v/v), and T = 363 K. Herein, the simulations were conducted under identical conditions to compare the prediction data with the experimental results in terms of MMA conversion, Mn, and PDI. Tables 5 and 6 describe the remaining simulation conditions. As can be seen in Figure 5, the predicted MMA conversion and Mn profile along the polymerization time imply a very good agreement with the experimental data. Furthermore, the Mn increases linearly with reaction time, thereby indicating that the suspension ATRP follows a controlling trend. The time evolution of PDI can also be used to explain the living/controlled suspension polymerization (see Figure 5C). 3.1.2. Dispersion Model Validation. The droplet size distribution is another important issue in suspension polymerization. It is necessary to validate the predicted droplet size in order to ensure the accuracy of the kernels used in this study. However, there was little experimental data of droplet size in ATRP suspension polymerization. Jahanzad et al.50 had studied the Sauter mean diameter of MMA-water dispersion experi-

3. RESULTS AND DISCUSSION A CFD−PBM coupled system describing a multiscale phenomenon was proposed to obtain the entire flow field information. The first subsection verifies the coupled model using experimental data from open literature. The second subsection describes the multiscale flow fields based on the proposed model. The third subsection applies the coupled F


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Figure 5. Comparisons between simulation results (CFD) and experimental data (Exp.) for the batch suspension ATRP: (A) monomer conversion, (B) Mn, and (C) PDI. Reproduced with permission from ref 27. Copyright 2004 Elsevier.

in terms of velocity, phase holdup, breakage source, coalescence source, and DSD, all of which are shown in Figures 7−9 under

ments which are almost identical to our system under different operating conditions, and an empirical correlation was proposed as d32 = 0.022(1 + 3.55αd)We−0.60 DI

(5)

In this work, the simulation cases are conducted under identical conditions (i.e., reactor structure, agitation speed, phase holdup) to compare the prediction data with the experimental results of droplet size. When the agitation speed was 300 rpm and monomer holdup was 0.4, as shown in Figure 6, the d32 reached an equilibrium state (about 85 μm) quickly at

Figure 6. Comparisons between simulation results (CFD) and experimental data (Exp.) for d32 in the water-MMA dispersion system: N = 300 rpm; ϕd = 0.4. Reproduced from ref 50. Copyright 2005 American Chemical Society.

Figure 7. Steady multiscale flow fields of (A) velocity, (B) dispersed phase holdup, (C) breakage rate, and (D) coalescence rate.

a mixing frequency of 300 rpm when the initial dispersed phase distribution is considered homogeneous with a volume fraction value of 0.2. Additionally, the stirred tank reactor was so small that the flow fields would reach steady state in a small amount of time. Therefore, only the steady multiscale flow fields were given here. Figure 7A,B shows the distributions of velocity and phase holdup in the stirred tank reactor, respectively. Despite its large agitation speed, the stirred tank still had some stagnant regions as shown in the velocity distribution profiles. Furthermore, given the density differences between the aqueous and dispersed phases, phase separation was observed at the bottom and top of the stirred tank reactor, thereby presenting a disadvantage for the suspension polymerization in which monomers were dispersed into small droplets under the

the beginning of the polymerization, and the d32 predictions matched well with the open data given by Jahanzad et al.50 In conclusion, the simulation results in Figures 5 and 6 indicate that the model predictions agree well with the experimental data; therefore, the proposed model is then applied to simulate the roles of the key variables (see Section 3.3 “Model Application”). Other-scale flow fields were obtained at the same time. Section 3.2 describes these results in detail. 3.2. Multiscale Flow Field Characterization. The distributions of flow fields based on CFD are also important for the suspension properties of stirred tank reactor. Roudsari et al.45 used a stirred tank reactor that was characterized in this study using CFD−PBM techniques without considering the polymerization. This section reports the multiscale flow fields G


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Industrial & Engineering Chemistry Research influence of suspending agents and mechanical agitation. Additionally, as could be seen, the flow was not yet fully turbulent. These simulated results indicated that the reactor did not have a good stirring, and increasing the stirring speed might be one of the most important means to improve reactor mixing performance. The dispersed droplets in the aqueous phase experienced breakage and coalescence and were determined by the hydrodynamic conditions in the surrounding liquid, their physical properties, and the reactor operating conditions. Figure 7C,D shows the contour plots of the breakage and coalescence of droplets, respectively. Given that the breakage rate is mainly affected by energy dissipation rate (ε) in the stirred reactor,19,51,52 Figure 7C shows that the breakage region is limited to a small zone located around the impeller edges with a high ε. Although the breakage region extends to the adjacent reactor wall, this region is smaller than the coalescence region through the tank, and other researchers obtained similar results.19 Moreover, a small breakage region can significantly affect DSD because the breakage rate has a larger order of magnitude than the coalescence rate. Coalescence is more complex than breakage because of the interactions among the droplets and between the droplets and surrounding fluid.53−55 Coalescence frequency is generally determined by both collision frequency and coalescence efficiency that is related to the droplet size, dispersed-phase fraction, and agitation speed.55 Figure 7D shows that the whole stirred tank reactor has a nonzero coalescence rate. Comparing Figure 7B,D, it is obvious that the distribution of coalescence rate depends on the volume fraction of dispersed phase. Droplet coalescence does not affect the dispersion mechanism for very dilute solutions.56 Therefore, a certain mixing in the vessel may be required for coalescence to occur. Consistent with the literature,19 coalescence showed a dependence on energy dissipation rate. In practice, DSD is simultaneously affected by two characteristic processes, namely, droplet breakage and coalescence. The initial DSD with mean particle size of 160 μm is given in Figure 8, and Figure 9 illustrates the evolution of

Figure 9. Time evolution of droplet size at the impeller speed of 300 rpm.

suspension polymerization, the coalescence between monomer droplets or polymer particles must be avoided for the sake of suspension stability. Therefore, this study investigates suspension particle dynamics based on the validated model. 3.3. Model Application. As mentioned above, MMA monomer achieves phase equilibrium between aqueous and dispersed phases in suspension polymerization. The interphase mass transfer plays a significant role in suspension polymerization. From eq 1, the interphase mass transfer rate is controlled by the overall mass transfer coefficient (kma) and concentration difference (gradient) between aqueous phase and dispersed phase. In general, the mass transfer coefficient (km) can be estimated by the well-known Ranz−Marshall correlations as follows: k md32 = 2 + 0.6Sc1/3Re 0.5 (6) D In this section, the role of interphase mass transfer in suspension polymerization will be investigated in detail. 3.3.1. Effect of Agitation Speed. Agitation significantly affects the particle dynamics in suspension polymerization and determines the breakage and coalescence rates in the vessel by means of macroscopic flow fields. The effect of agitation speed on DSD has been reported elsewhere. Furthermore, the agitation speed can also affect interphase mass transfer rate, which is examined using the CFD−PBM coupling system in this section. Figure 10 shows the time evolution of MMA conversion, MMA consumption rate, MMA interphase mass transfer rate, and MMA mass fraction in aqueous phase when MMA suspension polymerizations and dispersions are performed at impeller speeds of 300, 600, and 900 rpm. According to the eq 6, increasing the impeller speed increases the local Reynolds number and subsequently increases the overall mass transfer coefficient. In addition, the Sauter mean diameter will decrease with the increase of agitation speed. Therefore, the droplet will have a larger specific surface area, which in turn increases the mass transfer flux. Generally speaking, the MMA conversion will experience a decrease because of the interphase mass transfer. In this study, however, the interphase mass transfer has little effect on the conversion and monomer consumption rate. This is not surprising as the amount of monomer in the aqueous phase is low compared to the dispersed phased. These results indicate that the MMA conversion and consumption rates are mainly determined by the monomer concentration in Sh =

Figure 8. Initial droplet size distribution profile.

DSD with polymerization time when both droplet breakage and coalescence are considered. As can be seen, although the breakage region is much smaller than the coalescence region, the breakage effect dominates the stirred tank reactor at the impeller speed of 300 rpm. The numerical calculation with both breakup and coalescence source terms decreased d32 from 160 to 50 μm, and many small droplets were generated. In H


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Figure 10. Time evolution of (A) conversion, (B) consumption rate, (C) transfer rate, and (D) mass fraction in aqueous phase at the impeller speed of black ■, 300 rpm; red ◆, 600 rpm; blue ▲, 900 rpm, respectively.

the dispersed phase. In Figure 10C, there are many interesting phenomena that can be used to analyze the relationship between suspension polymerization kinetics behavior and interphase mass transfer. Obviously, the time evolution of MMA mass transfer rate almost can be defined by three characteristic intervals. In the first stage, it has increased a lot and is sensitive with the increased agitation speed. As the reaction evolves, however, the MMA interphase mass transfer rate decreases significantly (the second stage). Finally, it remains basically unchanged independent of the agitation speed (the third stage). These phenomena can be explained by the ATRP mechanism. It is known that many free radicals are generated in the initiation stage, and MMA monomer is quickly consumed by the chain propagation reaction (see Figure 10B). Then, the created concentration gradient will force monomer to transfer from aqueous phase to dispersed phase. Of course, the decreased droplet diameter will further enhance the interphase transfer rate. That is why the mass transfer rate increases with the increase of agitation speed (see Figure 10C). For suspension ATRP, there is a rapid dynamic equilibrium between active chains and dormant chains. When the dynamic equilibrium is developed, the concentration of free radical will be limited to a low level; thus, the polymerization rate decreases dramatically, and monomer conversion increases linearly with polymerization time (see Figure 10A). In the second stage, although the droplet diameter has continued to decrease (see Figure 11), we find a significant decrease in the interphase mass transfer rate, which indicates that the concentration gradient is

Figure 11. Time evolution of d32 at different impeller speeds.

the main determinant of the mass transfer rate. Similar results can be also found in the third stage where the agitation speed has little effect on the interphase mass transfer rate. Additionally, by comparing Figure 10B,C, we can easily find that the MMA consumption rate nearly has a larger magnitude than the interphase mass transfer rate. This helps us to explain why there are few differences in MMA conversion under different agitation speeds. Figure 10D displays the time evolution of MMA mass fraction in aqueous phase. When the agitation speed increases from 300 to 600 rpm, as can be seen, the mass transfer flux increases a lot at the beginning of I


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interphase mass transfer rate decreases obviously. It is well known that the droplet diameter mainly determines the interfacial area of mass transfer. From Figure 13A, it is easy to find that, although the droplets have different initial diameters, the final interphase mass transfer rates have the same value, which is affected by both mass transfer coefficient and concentration gradient. Figure 14 displays the time

polymerization. When the agitation speed further increases to 900 rpm, however, there is no change in the mass transfer flux. Certainly, the reason can be easily found in Figure 10C. In order to investigate the effect of thermodynamic equilibrium on interphase mass transfer rate, Figure 12 displays

Figure 12. Time evolution of equilibrium concentration and MMA concentration in the dispersed phase. Figure 14. Time evolution of d32 at different initial droplet diameters.

the time evolution of equilibrium concentration and MMA concentration in the dispersed phase when the agitation speed is 600 rpm As can be seen, how far the system deviated from thermodynamic equilibrium is related to the polymerization rate, which is determined by the atom transfer radical polymerization mechanism. 3.3.2. Effect of Initial Droplet Size Distribution. As defined in eq 6, it is obvious that the initial droplet size distribution will play an important role in affecting interphase mass transfer. In this section, when the initial dispersed phase volume fraction value is 0.2 and the agitation speed is 300 rpm, three cases with the initial mean diameter of 160, 300, and 600 μm are carried out to study the interphase transfer behavior. Figure 13 shows the time evolutions of interphase mass transfer rate and MMA mass fraction in the aqueous phase. Obviously, the droplet diameter has a function to play in determining interphase mass transfer rate. As shown in Figure 13, when the droplet diameter increases from 160 to 300 μm, the change of the mass transfer rate is not very noticeable. When the droplet diameter increases to 600 μm, however, the

evolution of Sauter mean diameter. The final droplet size is about 50 μm, which is independent of the initial droplet size and also determines the overall mass transfer coefficient. The influence of droplet diameter on the mass flux can be also found in Figure 13B. As discussed above, the monomer mass fraction in the aqueous phase is influenced by polymerization and interphase mass transfer.

4. CONCLUSIONS This work developed a multiscale model comprising the Eulerian−Eulerian two-phase flow model, ATRP kinetics model, thermodynamic equilibrium equation, and PBM to characterize the multiphase flow behavior and multiscale properties of batch suspension ATRP in a 3D stirred tank reactor. The multiscale coupled model was tested by comparing the simulated results with the open data. The multiscale flow fields in a stirred vessel were described on the basis of the validated model. The proposed model was then employed to

Figure 13. Time evolution of (A) transfer rate and (B) mass fraction in aqueous phase at the initial droplet diameter of black ■, 160 μm; red ◆, 300 μm; blue ▲, 600 μm, respectively. J


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kin = initiation rate constant for monomer addition to primary radical, L·mol−1·s−1 kma = the overall mass transfer coefficient, s−1 kp = chain propagation rate constant for monomer addition to radical chain with length r, L·mol−1·s−1 kt0 = combinative termination rate constant for primary radical addition to primary radical, L·mol−1·s−1 ktc = combinative termination rate constant, L·mol−1·s−1 ktd = disproportional termination rate constant, L·mol−1·s−1 ktr = chain transfer rate constant, L·mol−1·s−1 ktR = combinative termination rate constant for primary radical addition to radical chain with length r, L·mol−1·s−1 mmw = the molar volumes ratio of monomer and water [M] = the concentration of monomer in the droplets, mol· L−1 Mm = molecular weight, g·mol−1 Mn = number-average molecular weight, g·mol−1 Mw = weight-average molecular weight, g·mol−1 [M]e = the concentration of monomer in the droplets at equilibrium with the amount of monomer dissolved in the aqueous phase, mol·L−1 n(L; x, t) = length number density function, m−4 R = universal gas constant, J·mol−1·K−1 Re = Reynolds number Sc = Schmidt number Sh = Sherwood number T = temperature, K v = velocity, m/s Vm = the molar volumes of monomer, L·mol−1 We = Weber number

study the effects of the interphase mass transfer in suspension polymerization. The simulation results showed that the coupled model was highly suitable for simulating multiscale flow fields in suspension polymerization in stirred tank reactors. Although the breakage region is limited to a small zone located around the impeller edges, the breakage effect dominates the stirred tank reactor in this study. Agitation speed is an important factor in determining the droplets breakage rate, coalescence rate, and droplets size distribution as well as interphase mass transfer rate. Increasing the stirring speed will decrease the Sauter mean diameter and increase the overall mass transfer coefficient. The initial droplet size is another key parameter that affects the interphase mass transfer of MMA monomer. Furthermore, the mass flux is also related to the monomer concentration gradient between aqueous phase and dispersed phase. In conclusion, this work described the multiscale fields in suspension polymerization in stirred tank reactors. The simulation results can benefit the scale-up of suspension polymerization in these reactors. In practice, the viscosity changes with the reaction and has a bigger influence on breakage in suspension polymerization. It is of importance to develop a more accurate model to describe droplets evolving from Newtonian viscous droplets to non-Newtonian and even viscoelastic droplets. Our group is currently performing a further study based on the established multiscale model for the liquid−liquid two-phase flow in stirred tank reactors.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +86-21-54745602. Fax: +8621-54745602.

Greek Letters

ORCID

Zheng-Hong Luo: 0000-0001-9011-6020 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the National Natural Science Foundation of China (Nos. U1462101 and 21625603) and the Center for High Performance Computing, Shanghai Jiao Tong University for supporting this work.

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NOMENCLATURE b(λ) = breakage rate, 1/s B(L; x, t) = the birth rate of droplets diameter (L), m/s c(λ, L) = coalescence rate, 1/s C1,2,3 = empirical constant C4 = empirical constant, m−2 d32 = the Sauter diameter, m D(L; x, t) = the death rate of droplets diameter (L), m/s f = initiator efficiency G(L) = the growth rate of droplets diameter (L), m/s ΔG = the Gibbs free energy change, J ka = activation rate constant for dormant chain, L·mol−1·s−1 ka,0 = activation rate constant for initiator, L·mol−1·s−1 kb, kc = model parameters in PBM kd = activation rate constant for radical chain with length r, L· mol−1·s−1 kd,0 = deactivation rate constant for primary radical, L·mol−1· s−1

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ρ = density, kg·m−3 μ = dynamic viscosity, Pa·s μm = mth-order moment of propagating radical λ = droplet size, m λm = mth-order moment of dormant chains ϕm = mth-order moment of dead chains α = volume fraction β = daughter droplet distribution χij = Flory-Huggins interaction parameter of species i and species j σ = interfacial tension, N·m−1 τ = shear stress, N/m2 κ = thermal conductivity, W·m−1·K−1 ε = turbulence energy dissipation rate, m2·s−3

REFERENCES

(1) Vivaldo-Lima, E.; Wood, P. E.; Hamielec, A. E.; Penlidis, A. An Updated Review on Suspension Polymerization. Ind. Eng. Chem. Res. 1997, 36, 939−965. (2) Ni, X.; Johnstone, J. C.; Symes, K. C.; Grey, B. D.; Bennett, D. C. Suspension Polymerization of Acrylamide in an Oscillatory Baffled Reactor: From Drops to Particles. AIChE J. 2001, 47, 1746−1757. (3) Márquez, E. F.; Lagos, L. L. Mathematical Modeling of the Porosity of Suspension Poly(vinyl chloride). AIChE J. 2004, 50, 3184− 3194. (4) Vonka, M.; Storti, G.; Soos, M. Viscosity and Drop Size Evolution during Suspension Polymerization. AIChE J. 2016, 62, 4229−4239. (5) Kalfas, G.; Yuan, H.; Ray, W. H. Modeling and Experimental Studies of Aqueous Suspension Polymerization Processes. 2. Experiments in Batch Reactors. Ind. Eng. Chem. Res. 1993, 32, 1831−1838.

K


Article


in Suspension Polymer Reactors. Comput. Chem. Eng. 1998, 22, S315− S322. (26) Kiparissides, C.; Daskalakis, G.; Achilias, D. S.; Sidiropoulou, E. Dynamic Simulation of Industrial Poly(Vinyl Chloride) Batch Suspension Polymerization Reactors. Ind. Eng. Chem. Res. 1997, 36, 1253−1267. (27) Zhu, C. Y.; Sun, F.; Zhang, M.; Jin, J. Atom Transfer Radical Suspension Polymerization of Methyl Methacrylate Catalyzed by CuCl/bpy. Polymer 2004, 45, 1141−1146. (28) Wieme, J.; De Roo, T.; Marin, G. B.; Heynderickx, G. J. Simulation of Pilot- and Industrial-scale Vinyl Chloride Batch Suspension Polymerization Reactors. Ind. Eng. Chem. Res. 2007, 46, 1179−1196. (29) Liu, Z.; Lu, Y.; Yang, B.; Luo, G. Controllable Preparation of Poly(butyl acrylate) by Suspension Polymerization in a Coaxial Capillary Microreactor. Ind. Eng. Chem. Res. 2011, 50, 11853−11862. (30) Maggioris, D.; Goulas, A.; Alexopoulos, A. H.; Chatzi, E. G.; Kiparissides, C. Prediction of Particle Size Distribution in Suspension Polymerization Reactors: Effect of Turbulence Nonhomogeneity. Chem. Eng. Sci. 2000, 55, 4611−4627. (31) Kotoulas, C.; Kiparissides, C. A Generalized Population Balance Model for the Prediction of Particle Size Distribution in Suspension Polymerization Reactors. Chem. Eng. Sci. 2006, 61, 332−346. (32) Zhu, Y. P.; Chen, G. Q.; Luo, Z. H. Particle Behavior in FBRs: A Comparison of the PBM−CFD, Multi-scale CFD Simulation of Gas− solid Catalytic Propylene Polymerization. Macromol. React. Eng. 2014, 8, 609−621. (33) Zhu, Y. P.; Luo, Z. H.; Xiao, J. Multi-scale Product Property Model of Polypropylene Produced in a FBR: From Chemical Process Engineering to Product Engineering. Comput. Chem. Eng. 2014, 71, 39−51. (34) Wang, W.; Zhou, Y. N.; Luo, Z. H. Modeling of the Atom Transfer Radical Copolymerization Processes of Methyl Methacrylate and 2-(Trimethylsilyl) Ethyl Methacrylate under Batch, Semibatch, and Continuous Feeding: A Chemical Reactor Engineering Viewpoint. Ind. Eng. Chem. Res. 2014, 53, 11873−11883. (35) Wang, W.; Zhou, Y. N.; Shi, L.; Luo, Z. H. Kinetic Modeling of Atom Transfer Radical Copolymerization of Methyl Methacrylate and 2-(Trimethylsilyl) Ethyl Methacrylate in a Train of Continuous Stirred-tank Reactors. Polym. Eng. Sci. 2015, 55, 1030−1038. (36) Xie, L.; Luo, Z. H. Modeling of the Methyl Methacrylate Atom Transfer Radical Suspension Polymerization Process: Polymerization and Particle Kinetics. Macromol. React. Eng. 2016, 10, 479−489. (37) Marchisio, D. L.; Vigil, R. D.; Fox, R. O. Quadrature Method of Moments for Aggregation−breakage Processes. J. Colloid Interface Sci. 2003, 258, 322−334. (38) Petitti, M.; Nasuti, A.; Marchisio, D. L.; Vanni, M.; Baldi, G.; Mancini, N.; Podenzani, F. Bubble Size Distribution Modeling in Stirred Gas−liquid Reactors with QMOM Augmented by a New Correction Algorithm. AIChE J. 2010, 56, 36−53. (39) Coulaloglou, C. A.; Tavlarides, L. L. Description of Interaction Processes in Agitated Liquid-liquid Dispersions. Chem. Eng. Sci. 1977, 32, 1289−1297. (40) Mead, R. N.; Poehlein, G. W. Emulsion Copolymerization of Styrene-methyl Acrylate and Styrene-acrylonitrile in Continuous Stirred-tank Reactors. 1. Ind. Eng. Chem. Res. 1988, 27, 2283−2293. (41) Silva, F. M.; Lima, E. L.; Pinto, J. C. Acrylic Acid/Vinyl Acetate Suspension Copolymerizations. 2. Modeling and Experimental Results. Ind. Eng. Chem. Res. 2004, 43, 7324−7342. (42) Talamini, G.; Visentini, A.; Kerr, J. Bulk and Suspension Polymerization of Vinyl Chloride: The Two-phase Model. Polymer 1998, 39, 1879−1891. (43) Kammona, O.; Pladis, P.; Frantzikinakis, C. E.; Kiparissides, C. A Comprehensive Experimental and Theoretical Investigation of the Styrene/2-Ethylhexyl Acrylate Emulsion Copolymerization. Macromol. Chem. Phys. 2003, 204, 983−999. (44) Guillot, J. Kinetics and Thermodynamic Aspects of Emulsion Copolymerization. Acrylonitrile-styrene Copolymerization. Acta Polym. 1981, 32, 593−600.

(6) Kalfas, G.; Ray, W. H. Modeling and Experimental Studies of Aqueous Suspension Polymerization Processes. 1. Modeling and Simulations. Ind. Eng. Chem. Res. 1993, 32, 1822−1830. (7) Georgiadou, S.; Brooks, B. W. Suspension Polymerisation of Methyl Methacrylate Using Ammonium Polymethacrylate as a Suspending Agent. Chem. Eng. Sci. 2005, 60, 7137−7152. (8) Ni, X.; Zhang, Y.; Mustafa, I. Correlation of Polymer Particle Size with Droplet Size in Suspension Polymerisation of Methyl Methacrylate in a Batch Oscillatory-baffled Reactor. Chem. Eng. Sci. 1999, 54, 841−850. (9) Wang, J. S.; Matyjaszewski, K. Controlled/″Living″ Radical Polymerization. Halogen Atom Transfer Radical Polymerization Promoted by a Cu(I)/Cu(II) Redox Process. Macromolecules 1995, 28, 7901−7910. (10) Kato, M.; Kamigaito, M.; Sawamoto, M.; Higashimura, T. Polymerization of Methyl Methacrylate with the Carbon Tetrachloride/Dichlorotris-(Triphenylphosphine) Ruthenium(II)/Methylaluminum Bis(2,6-di-tert-butylphenoxide) Initiating System: Possibility of Living Radical Polymerization. Macromolecules 1995, 28, 1721− 1723. (11) Chen, Z.; Pauer, W.; Moritz, H. U.; Prüss, J.; Warnecke, H. J. Modeling of the Suspension Polymerization Process Using a Particle Population Balance. Chem. Eng. Technol. 1999, 22, 609−616. (12) Alexopoulos, A. H.; Pladis, P.; Kiparissides, C. Nonhomogeneous Mixing Population Balance Model for the Prediction of Particle Size Distribution in Large Scale Emulsion Polymerization Reactors. Ind. Eng. Chem. Res. 2013, 52, 12285−12296. (13) Xie, L.; Zhu, L. T.; Luo, Z. H. CFD Simulation of Multi-scale Mixing in Anionic Polymerization Tubular Reactors. Chem. Eng. Technol. 2016, 39, 857−864. (14) Xie, L.; Zhu, L. T.; Luo, Z. H.; Jiang, C. W. Multi-scale Modeling of Mixing Behavior in a Three-dimensional Atom Transfer Radical Copolymerization Stirred-tank Reactor. Macromol. React. Eng. 2017, 11, 1600022. (15) Gao, Z.; Li, D.; Buffo, A.; Podgórska, W.; Marchisio, D. L. Simulation of Droplet Breakage in Turbulent Liquid-liquid Dispersions with CFD-PBM: Comparison of Breakage Kernels. Chem. Eng. Sci. 2016, 142, 277−288. (16) Chen, X. Z.; Luo, Z. H.; Yan, W. C.; Lu, Y. H.; Ng, I. S. Threedimensional CFD-PBM Coupled Model of the Temperature Fields in Fluidized-bed Polymerization Reactors. AIChE J. 2011, 57, 3351− 3366. (17) Yan, W. C.; Luo, Z. H.; Lu, Y. H.; Chen, X. D. A CFD-PBMPMLM Integrated Model for the Gas−solid Flow Fields in Fluidized Bed Polymerization Reactors. AIChE J. 2012, 58, 1717−1732. (18) Wang, T. F.; Wang, J. F.; Jin, Y. A CFD−PBM Coupled Model for Gas−liquid Flows. AIChE J. 2006, 52, 125−140. (19) Vonka, M.; Soos, M. Characterization of Liquid-liquid Dispersions with Variable Viscosity by Coupled Computational Fluid Dynamics and Population Balances. AIChE J. 2015, 61, 2403−2414. (20) Nogueira, E. S.; Pinto, J. C.; Vianna, A. S., Jr Analysis of Energy Dissipation in Stirred Suspension Polymerisation Reactors Using Computational Fluid Dynamics. Can. J. Chem. Eng. 2012, 90, 983− 995. (21) Hosseini, S.; Patel, D.; Einmozaffari, F.; Mehrvar, M. Study of Solid−liquid Mixing in Agitated Tanks Through Computational Fluid Dynamics Modeling. Ind. Eng. Chem. Res. 2010, 49, 4426−4435. (22) Cherbański, R.; Milewska, A.; Molga, E. Safety Aspects in Batch Reactors for Styrene Suspension Polymerization. Ind. Eng. Chem. Res. 2007, 46, 5898−5906. (23) Alexopoulos, A. H.; Maggioris, D.; Kiparissides, C. CFD Analysis of Turbulence Non-homogeneity in Mixing Vessels: A Twocompartment Model. Chem. Eng. Sci. 2002, 57, 1735−1752. (24) Vivaldo-Lima, E.; Wood, P. E.; Hamielec, A. E.; Penlidis, A. Calculation of the Particle Size Distribution in Suspension Polymerization Using a Compartment-mixing Model. Can. J. Chem. Eng. 1998, 76, 495−505. (25) Maggioris, D.; Goulas, A.; Alexopoulos, A. H.; Chatzi, E. G.; Kiparissides, C. Use of CFD in Prediction of Particle Size Distribution L


Article

Industrial & Engineering Chemistry Research (45) Roudsari, S. F.; Ein-Mozaffari, F.; Dhib, R. Use of CFD in Modeling MMA Solution Polymerization in a CSTR. Chem. Eng. J. 2013, 219, 429−442. (46) Beuermann, S.; Buback, M.; Davis, T. P.; Gilbert, R. G.; Hutchinson, R. A.; Olaj, O. F.; Russell, G. T.; Schweer, J.; Van Herk, A. M. Critically Evaluated Rate Coefficients for Free-radical Polymerization,2: Propagation Rate Coefficients for Methyl Methacrylate. Macromol. Chem. Phys. 1997, 198, 1545−1560. (47) Zhou, Y. N.; Luo, Z. H. Copper(0)-mediated Reversibledeactivation Radical Polymerization: Kinetics Insight and Experimental Study. Macromolecules 2014, 47, 6218−6229. (48) Azizi, F.; Taweel, A. M. A. Turbulently Flowing Liquid−liquid Dispersions. Part I: Drop Breakage and Coalescence. Chem. Eng. J. 2011, 166, 715−725. (49) Waggoner, R. A.; Blum, F. D.; Macelroy, J. M. D. Dependence of the Solvent Diffusion Coefficient on Concentration in Polymer Solutions. Macromolecules 1993, 26, 6841−6848. (50) Jahanzad, F.; Sajjadi, S.; Brooks, B. W. Comparative Study of Particle Size in Suspension Polymerization and Corresponding Monomer-water Dispersion. Ind. Eng. Chem. Res. 2005, 44, 4112− 4119. (51) Andersson, R.; Andersson, B. On the Breakup of Fluid Particles in Turbulent Flows. AIChE J. 2006, 52, 2020−2030. (52) Liao, Y.; Lucas, D. A Literature Review of Theoretical Models for Drop and Bubble Breakup in Turbulent Dispersions. Chem. Eng. Sci. 2009, 64, 3389−3406. (53) Liao, Y.; Lucas, D. A Literature Review on Mechanisms and Models for the Coalescence Process of Fluid Particles. Chem. Eng. Sci. 2010, 65, 2851−2864. (54) Chesters, A. K. The Modeling of Coalescence Processes in Fluid−liquid Dispersions: A Review of Current Understanding. Chem. Eng. Res. Des. 1991, 69, 259−270. (55) Wright, H.; Ramkrishna, D. Factors Affecting Coalescence Frequency of Droplets in a Stirred Liquid-liquid Dispersion. AIChE J. 1994, 40, 767−776. (56) Chen, H. T.; Middleman, S. Drop Size Distribution in Agitated Liquid-liquid Systems. AIChE J. 1967, 13, 989−995.

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