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Chemical Engineering Science 61 (2006) 332 346

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A generalized population balance model for the prediction of particle sizedistribution in suspension polymerization reactors

Costas Kotoulas, Costas Kiparissides

Department of Chemical Engineering, Aristotle University of Thessaloniki and Chemical Process Engineering Research Institute,

P.O. Box 472 541 24 Thessaloniki, Greece

Received 21 February 2005; received in revised form 1 July 2005; accepted 3 July 2005

Available online 22 August 2005

Abstract

In the present study, a comprehensive population balance model is developed to predict the dynamic evolution of the particle size

distribution in high hold-up (e.g., 40%) non-reactive liquidliquid dispersions and reactive liquid(solid)liquid suspension polymerization

systems. Semiempirical and phenomenological expressions are employed to describe the breakage and coalescence rates of dispersed

monomer droplets 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. The fixed pivot (FPT) numerical method is applied for solving the

population balance equation. The predictive capabilities of the present model are demonstrated by a direct comparison of model predictions

with experimental data on average mean diameter and droplet/particle size distributions for both non-reactive liquidliquid dispersions

and the free-radical suspension polymerization of styrene and VCM monomers.

2005 Elsevier Ltd. All rights reserved.

Keywords: Population balance model; Suspension polymerization; PVC; Polystyrene

1. Introduction

Suspension polymerization is commonly used for pro-

ducing a wide variety of commercially important polymers

(i.e., polystyrene and its copolymers, poly(vinyl chlo-

ride), poly(methyl methacrylate), poly(vinyl acetate)). In

suspension polymerization, the monomer is initially dis-

persed 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 oilphase. The polymerization occurs in the monomer droplets

that are progressively transformed into sticky, viscous

monomerpolymer particles and finally into rigid, spherical

polymer particles of size 50500m (Kiparissides, 1996).

Corresponding author. Tel.: +30 2310 99 6211;fax: +310 231099 6198.

E-mail addresses: [email protected],

[email protected](C. Kiparissides).

0009-2509/$- see front matter 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ces.2005.07.013

The polymer solid content in the fully converted suspension

is typically 3050% w/w.

The suspension polymerization process can in general be

distinguished into two types (Kalfas, 1992): the bead poly-

merization, where the polymer is soluble in its monomer

and smooth spherical particles are produced; and the pow-

der polymerization, where the polymer is insoluble in its

monomer and, thus, precipitates out leading to the forma-

tion of irregular grains or particles. The most important

thermoplastic produced by bead suspension polymerization

is polystyrene (PS). In the presence of volatile hydrocar-bons (C4C6), foamable beads (the so-called expandable

polystyrene, EPS) can be produced. On the other hand,

poly(vinyl chloride) (PVC), which is the second largest ther-

moplastic manufactured in the world, is an example of pow-

der polymerization.

One of the most important issues in suspension poly-

merization process is the control of the final particle size

distribution (PSD) (Yuan et al., 1991). The initial monomer

droplet size distribution (DSD) as well as the final polymer
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C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332346 333

Fig. 1. Schematic representation of drop breakage and coalescence mech-

anism.

PSD in general depend on the type and concentration of

the surface-active agents, the quality of agitation and the

physical properties (e.g., density, viscosity and interfacial

tension) of the continuous and dispersed phases. The tran-

sient droplet/particle size distribution is controlled by two

dynamic processes, namely, the drop/particle breakage and

coalescence rates. 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 sur-

face of a drop. The latter is either increased or decreased by

the turbulent flow field and can be assumed to be negligible

for very dilute dispersions at sufficiently high concentrations

of surface-active agents (Chatzi and Kiparissides, 1992).When drop breakage occurs by viscous shear forces, the

monomer droplet is first elongated into two fluid lumps sep-

arated by a liquid thread (see Fig. 1a). Subsequently, the

deformed monomer droplet breaks into two almost equal-

size drops, corresponding to the fluid lumps, and a series of

smaller droplets corresponding to the liquid thread. This is

known as Thorough breakage. On the other hand, a droplet

suspended in a turbulent flow field is exposed to local pres-

sure and relative velocity fluctuations. For nearly equal den-

sities and viscosities of the two liquid phases, the droplet

surface can start oscillating. When the relative velocity is

close to that required to make a drop marginally unstable,a number of small droplets are stripped out from the initial

one (seeFig. 1b). This situation of breakage is referred to as

erosive one. Erosive breakage is considered to be the dom-

inant mechanism for low-coalescence systems that exhibit

a characteristic bimodality in the PSD (Chatzi and Kiparis-

sides, 1992; Ward and Knudsen, 1967).

Two different mechanisms have been proposed in the lit-

erature to describe the coalescence of two drops in a tur-

bulent flow field. The first one (Shinnar and Church, 1960)

assumes that after the initial collision of two drops, a liquid

film of the continuous phase is being trapped between the

drops that prevents drop coalescence (seeFig. 1c). However,

due to the presence of attractive forces, draining of the liq-

uid film can occur leading to drop coalescence. On the other

hand, if the kinetic energy of the induced drop oscillations

is larger than the energy of adhesion between the drops,

the drop contact is broken before the complete drainage of

the liquid film. The second drop coalescence mechanism

(Howarth, 1964) assumes that immediate coalescence oc-curs when the approach velocity of the colliding drops at

the collision instant exceeds a critical value. In other words,

if the turbulent energy of collision is greater than the total

drop surface energy, the drops will coalesce (seeFig. 1d).

Surface-active agents play a very important role in the

stabilization of liquidliquid dispersions. One of the most

commonly used suspension stabilizers is poly(vinyl acetate)

that has been partially hydrolyzed to poly(vinyl alcohol)

(PVA). By varying the acetate content (i.e., degree of hy-

drolysis), one can alter the hydrophobicity of the PVA and,

thus, the conformation and surface activity of the polymer

chains at the monomer/water interface (Chatzi and Kiparis-

sides, 1994). The solubility of the PVA in water depends on

the overall degree of polymerization (i.e., molecular weight),

the sequence chain length distribution of the vinyl alco-

hol and vinyl acetate in the copolymer, the degree of hy-

drolysis and temperature. Depending on the agitation rate,

the concentration and type of surface-active agent, the av-

erage droplet size can exhibit a U-shape variation with re-

spect to the impeller speed or the degree of hydrolysis of

PVA. This U-type behaviour has been confirmed both ex-

perimentally and theoretically and has been attributed to

the balance of breakage and coalescence rates of monomer

drops.

In regard with the droplet/particle breakage and coales-cence phenomena, the suspension polymerization process

can be divided into three stages (Hamielec and Tobita, 1992;

Maggioris et al., 2000). During the initial low-conversion

(i.e., low-viscosity) stage, drop breakage is the dominant

mechanism. As a result the initial DSD shifts to smaller

sizes. During the second sticky-stage of polymerization, the

drop breakage rate decreases while the drop/particle coales-

cence becomes the dominant mechanism. Thus, the average

particle size starts increasing. In the third stage, 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).For the PS process, Villalobos et al. (1993) reported

that the end of the first stage occurs at approximately 30%

monomer conversion, corresponding to a critical viscosity

of about 0.1 Pa s, while the second stage extends up to 70%

monomer conversion. In the VCM powder polymerization,

at monomer conversions around 1030%, a continuous

polymer network is commonly formed inside the poly-

merizing monomer droplets that significantly reduces the

drop/particle coalescence rate (Kiparissides et al., 1994).

Cebollada et al. (1989)reported that the PSD is essentially

established up to monomer conversions of about 3540%

(i.e., end of the second stage).
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334 C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332 346

The paper is organized as follows. In Section 2, a gen-

eralized population balance model is developed to describe

the dynamic evolution of PSD in batch suspension polymer-

ization reactors. The model takes into account the dynamic

evolution of the physical properties of the continuous and

dispersed phases, in terms of the variation of monomer con-

version and the turbulent intensity characteristics of the flowfield, as well as their relative effects on breakage and coales-

cence mechanisms. In Section 3 of the paper, the fixed pivot

technique (FPT) (Kumar and Ramkrishna, 1996)is applied

for solving the general population balance equation, govern-

ing the PSD developments, in terms of the polymerization

conditions (e.g., monomer to water volume ratio, temper-

ature, type and concentration of stabilizer, impeller energy

input, etc.) and the polymerization kinetic model. An ex-

tensive analysis on the robustness of the numerical method

is carried out in regard with the convergence of the solu-

tion and the conservation of the total mass in the system.

Finally, in Section 4 of the paper, the capabilities of the

present model are demonstrated by a direct comparison of

model predictions with experimental data on average mean

diameter and droplet/particle size distributions for both non-

reactive liquidliquid dispersions and the free-radical sus-

pension polymerization of styrene and VCM monomers.

2. Model developments

To follow the dynamic evolution of PSD in a particulate

process, a population balance approach is commonly em-

ployed. The distribution of the droplets/particles is consid-

ered to be continuous in the volume domain and is usu-ally described by a number density function, n(V,t). Thus,

n(V,t) dVrepresents the number of particles per unit vol-

ume in the differential volume size range (V , V + dV ). Fora dynamic particulate system undergoing simultaneous par-

ticle breakage and coalescence, the rate of change of the

number density function with respect to time and volume is

given by the following non-linear integro-differential popu-

lation balance equation (Kiparissides et al., 2004):

[n(V,t)]t

= Vmax

V

(U,V )u(U)g(U)n(U, t) dU

+ V /2Vmin

k(VU , U )n(VU,t )n(U, t) dUn(V, t)g(V)

n(V, t) Vmax

Vmin

k(V,U)n(U,t) dU. (1)

The first term on the right-hand side (r.h.s.) of Eq. (1) repre-

sents the generation of droplets in the size range (V , V +dV )due to drop breakage. (U , V ) is a daughter drop breakage

function, accounting for the probability that a drop of vol-

ume V is formed via the breakage of a drop of volume U.

The function u(U) denotes the number of droplets formed

by the breakage of a drop of volume U and g(U) is the

breakage rate of drops of volumeU. The second term on the

r.h.s. of Eq. (1) represents the rate of generation of drops in

the size range (V , V +dV ) due to coalescence of two smallerdrops.k(V , U )is the coalescence rate between two drops of

volumeV andU. Finally, the third and fourth terms repre-

sent the drop disappearance rates due to drop breakage and

coalescence, respectively. Eq. (1) will satisfy the followinginitial condition at t=0:n(V, 0)=n0(V ), (2)where n0(V ) is the initial drop size distribution of the dis-

persed phase. In the present study, the initial monomer DSD

was assumed to follow a normal distribution.

2.1. Breakage and coalescence rates

The solution of the population balance equation (Eq. (1))

presupposes the knowledge of the breakage and coalescence

rate functions. In the open literature, several forms ofg(V )andk(V, U)have been proposed to describe the drop break-

age and coalescence rate functions in liquidliquid disper-

sions (Coulaloglou and Tavlarides, 1977; Narsimhan et al.,

1979; Sovova, 1981; Chatzi et al., 1989). According to the

original work of Alvarez et al. (1994) and the proposed

modifications ofMaggioris et al. (2000),the breakage and

coalescence rates can be expressed in terms of the break-

age,b, and collision, c, frequencies and the, respective,

Maxwellian efficiencies,b andc:

g(V)=b(V )eb (V ), (3)

k(V , U )=c(V , U )ec(V,U)

, (4)b and c denote the corresponding ratios of required to

available energy for an event to occur.

In the present study, the breakage of a drop exposed to a

turbulent flow field was supposed to occur as result of en-

ergy transfer from an eddy to a drop having a diameter equal

to the eddy wave length, Dv. Eddy fluctuations with wave-

lengths smaller (larger) than the drop diameter Dv produce

an oscillatory (rigid body) motion of the drop that do not

lead to breakage (Alvarez et al., 1994). The frequency term,

b(V ), was assumed to be equal to the inverse fluctuation

time period, corresponding to the time required for a drop

to reach its mean drop displacement:

b(V )=u(Dv)/Dv, (5)where u(Dv)

2 is the mean square of the relative velocity

between two points separated by a distanceDv, or the mean

square fluctuation velocity of drops of diameter Dv.

For drops in the inertial subrange of turbulence (i.e.,

< Dv L), the energy spectrum will be independent ofthe kinematic viscosity, c, and the mean fluctuation veloc-

ity is solely determined by the rate of energy dissipation

(Hinze, 1959):

u(Dv)2

=kb(s Dv)

2/3, (6)
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wherekb is a model parameter and s is the average energy

dissipation rate for the dispersion.

For droplets in the viscous dissipation range (i.e., Dv< ),

the inertial forces are of the same order of magnitude as

the viscous shear forces and the mean square of the relative

velocity between two points separated by a distance Dv will

be given by (Shinnar, 1961)

u(Dv)2 =kbD2v (s /c). (7)

For high values of the dispersed phase volume fraction, the

damping effect of the dispersed phase on the local turbu-

lent intensity needs to be taken into account.Doulah (1975)

proposed the following cubic equation for the calculation of

the average energy dissipation rate of the liquidliquid dis-

persion,s , in terms of the average energy dissipation rate

of the continuous phase,c, and the kinematic viscosities of

the continuous,c, and liquidliquid dispersion,s :

s /c=(c/s )3

. (8)

Thus, in the presence of a high-volume dispersed phase, the

overall viscosity of the system increases and, therefore, the

energy dissipation rate for the system decreases.

According toAlvarez et al. (1994),for an effective drop

breakage to occur, the drop surface energy and drop vis-

coelastic resistance must be overcome. Considering that the

flow within a drop can be described as one-dimensional

simple-shear flow of a Maxwell fluid, the breakage efficiency

can be expressed as follows:

b=ab(Dv), (9)

where ab is a model parameter and (Dv) is the ratio of

required to available energy for a drop of diameter Dv to

break:

(Dv)=2

Re(1+Re Ve) +CdsWe

, (10)

where Re and We denote the drop Reynolds and Weber

numbers, respectively. The dimensionless quantity Ve ac-

counts for drop viscoelasticity and is a function of the drop

Reynolds number and its physical properties:

Ve=Yo

a exp

1

a

2Re Yo

1+a1a

1a1+a exp

a

ReYo

1

12, (11)

where

Yo=2d

dEdD2v

, =

148Yo. (12)

In the present study, the dispersed-phase elasticity mod-

ulus,Ed, was approximated by the product of the polymer

elasticity modulus, Ep, and the fractional monomer con-

version, (i.e., Ed

=Ep). For highly viscous (Re < 1)

and inelastic dispersions (Yo ), Eq. (11) reduces to

Ve= 112

exp

12

Re

1

. (13)

The scalar quantity Cds in Eq. (10) can be expressed in

terms of the numbers and volumes of daughter and satellite

drops (Chatzi and Kiparissides, 1992):

Cds=Ndar

2/3 +Nda(Ndar+Nsa)2/3

1, r=Vda/Vsa, (14)

where Nda is the number of daughter drops of volume Vdaand Nsa is the number of satellite drops of volume Vsa. In

the present study, the number of daughter drops was set

equal to 2, the volume ratio of daughter to satellite drops,r,

was considered to be constant, while the number of satellite

drops was calculated as a function of the parent drop size

(Chatzi and Kiparissides, 1992):

Nsa=integer(SnsaD1/3u ), (15)

whereSnsais a model parameter estimated from experimen-

tal measurements on DSD or PSD.

Assuming that the daughter and satellite drops are nor-

mally distributed about their respective mean values with

standard deviations ofda and sa, one can derive the fol-

lowing expression for the distribution of drops of volume

V, formed via the breakage of a drop of volume U:

u(U)(U , V )

=Nda 1

da2exp

(VVda)2

22da

+Nsa

1

sa

2exp

(VVsa)

2

22sa

. (16)

It should be noted that the daughter drop number density

function,u(U)(U , V ), should satisfy the following number

and volume conservation equations: U0

u(U)(U , V ) dV=u(U),

U

0

Vu(U)( U , V ) dV

=U. (17)

Accordingly, one can calculate the volumes of daughter and

satellite drops, formed by the breakage of a drop of volume

U, in terms of r, Nda and Nsa (Chatzi and Kiparissides,

1992):

Vda=U

Nda+Nsa/r, Vsa=

U

Ndar+Nsa. (18)

Assuming that the drop collision mechanism in a lo-

cally isotropic flow field is analogous to collisions between

molecules as in the kinetic theory of gases, the collision fre-

quency between two drops with volumes V and Ucan be

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expressed as (Maggioris et al., 2000)

c( V , U )=kc(D2v+D2u)(u(Dv)2 +u(Du)2)1/2. (19)For deformable drops, that is generally the case for low

interfacial tension dispersions or large-size drops, the drop

coalescence efficiency can be expressed as (Coulaloglou and

Tavlarides, 1977)

()c (V , U )=acccs

2

DvDu

Dv+Du

4, (20)

whereac is a model parameter. c, c, and are the vis-

cosity and density of the continuous phase, the interfacial

tension between the dispersed and aqueous phases and the

dispersed phase volume fraction, respectively.

At high monomer conversions, when the polymerizing

monomerpolymer particles behave like rigid spheres, the

coalescence efficiency can be expressed as (Coulaloglou and

Tavlarides, 1977)

(b)c (V , U )=acc

c1/3s (Dv+Du)4/3

. (21)

In general, the monomer drops will behave like de-

formable drops at the beginning of polymerization while,

at high monomer conversions, they will behave like rigid

polymer particles. Thus, the coalescence efficiency over the

whole monomer conversion range can be written as

exp{c(V , U )} =(1) exp{(a)c (V , U )}+ exp{(b)c (V , U )}, (22)

where is the fractional monomer conversion.

2.2. Evaluation of the physical properties

The density of the suspension system, s , can be calcu-

lated as a weighted average of the densities of the dispersed

(d)and continuous(c)phases (Bouyatiotis and Thornton,

1967):

s=d+c(1). (23)The density of the dispersed phase will in turn be a func-

tion of the corresponding densities of the polymer (p)and

monomer(m

) and the extent of monomer conversion, :

d=

p+ 1

m

1. (24)

The viscosity of the liquid(solid)liquid dispersion

was calculated by the following semi-empirical equation

(Vermeulen et al., 1955):

s=c

1

1+ 1.5d

d+c

, (25)

where d and c are the viscosities of the dispersed and

continuous phases, respectively.

For the heterophase suspension polymerization of VCM,

the viscosity of the polymerizing monomer droplets,d, can

be calculated by using the Eulers equation (Krieger, 1972):

d=m1+0.5[n]p

1

p

/cr

2

, (26)

wherep is the volume fraction of the polymer in the dis-

persed phase, given by p=(d/p). cr is the polymervolume fraction corresponding to the critical monomer con-

versionc, at which a 3-D polymer skeleton is formed in-

side the polymerizing monomer drops. When pcr, thedispersed-phase viscosity approaches infinity, indicating the

formation of a rigid structure. Thus, for values ofp larger

than the critical valuecr, the dispersed phase viscosity was

assumed to remain constant.

For the VCM suspension polymerization, the value of

cr was taken to be equal to 0.3, which corresponds to

the monomer conversion at which a continuous polymernetwork is formed inside the polymerizing VCM droplets

(Kiparissides et al., 1994). For the suspension polymeriza-

tion of styrene, the value ofcr was set equal to the 0.7

which corresponds to the monomer conversion at which par-

ticle coalescence stops. Finally, the intrinsic viscosity of the

polymer solution, [n], was calculated by the well-known

MarkHouwinkSakurada (MHS) equation as a function of

the weight average molecular weight of the polymer, Mw:

[] =kMaw. (27)

The viscosity of the continuous phase depends on theconcentration and type of stabilizer that, in turn, affects the

PSD (Cebollada et al., 1989). Okaya (1992) employed the

SchulzBlaschke equation to calculate of the viscosity of

aqueous PVA solutions:

c=w

1+ [PVA]CPVA10.45[PVA]CPVA

, (28)

where c,w, [nPVA] and CPVAare the viscosities of theaqueous PVA solution and pure water, the intrinsic viscosity

and the stabilizer concentration, respectively.

In the open literature, a great number of papers have beenpublished, dealing with the behaviour of polymer molecules

at interfaces. Prigogine and his collaborators (Prigogine and

Marechal, 1952; Defay et al., 1966) presented a remarkably

simple theory on the surface tension of polymer solutions.

Although the Prigogine theory refers specifically to the sur-

face tension of polymer solutions, it is equally applicable to

the prediction of interfacial tension between a polymer solu-

tion and an immiscible liquid or a solid (Siow and Patterson,

1973). In the present study, the model ofSiow and Patterson

(1973)was employed for the calculation of the interfacial

tension between the aqueous and the dispersed phase, . The

change in the interfacial tension with monomer conversion
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was taken into account as in the original work ofMaggioris

et al. (2000).

3. Numerical solution of the population balance

equation

The numerical solution of the PBE commonly requires

the discretization of the particle volume domain into a num-

ber of discrete elements. Accordingly, the unknown num-

ber density function is approximated at a selected number

of discrete points, resulting in a system of stiff, non-linear

differential equations that is subsequently integrated numer-

ically. Several numerical methods have been proposed in the

literature for the solution of the general PBE (Eq. (1)) in

the continuous or its equivalent discrete form (Kiparissides

et al., 2004). In the present study, the FTP ofKumar and

Ramkrishna (1996)was employed for solving the continu-

ous PBE (Eq. (1)).Assuming that the number density function remains con-

stant in the discrete volume interval (Vi to Vi+1), one candefine a particle number distribution, Ni (t ), corresponding

to the i element:

Ni (t )= Vi+1

Vi

n(V,t) dV=ni (V,t)(Vi+1Vi ). (29)

Following the original developments of Kumar and

Ramkrishna (1996), the total volume domain (Vmin to

Vmax) is first divided into a number of elements. The

drop/particle population, Ni (t ), corresponding to the size

range(Vi , Vi+1), is then assigned to a characteristic size xi(also called grid point) as shown inFig. 2. To account for

the formation of new particles of volume V that does not

correspond to the characteristic grid points (xi , xi+1), thefollowing approximation is applied. The particle number

fractionsa(V, xi ) and b(V, xi+1) are assigned to the parti-cle populations at the grid points xi and xi+1, respectively,so that two desired population properties of interest (e.g.,

total number and volume) are exactly preserved:

a(V,xi )f1(xi )+b(V, xi+1)f1(xi+1)=f1(V ), (30)

a(V,xi )f2(xi )

+b(V, xi

+1)f2(xi

+1)

=f2(V ). (31)

By integrating Eq. (1) over the discrete size interval

(Vi , Vi+1) and properly accounting for the respective dropbreakage and coalescence terms, the following set of

Vi-2 Vi-1 Vi Vi+1 Vi+2

xi+1xixi-1xi-2

Fig. 2. Discretization of the particle volume domain.

discretized equations can be derived:

dNi

dt=

Mk=1

nb(xi , xk )g(xk )Nk(t )+jk

j,kxi1 xj+xk xi+1

1 12j,k nc(xi ,V)k(xj, xk )Nj(t)Nk (t )g(xi )Ni (t )Ni (t )

Mk=1

k(xi , xk )Nk(t ), (32)

where nb(xi , xk ) denotes the fraction of drops/particles of

sizexi resulting from the breakage of a drop/particle of size

xk . To preserve the number and volume of particles in the

size range (Vi , Vi+1), nb(xi , xk ) must satisfy the followingequation:

nb(xi , xk )=

xi+1

xi

xi+1Vxi+1xi

(xk ,V)u(xk ) dV

+ xi

xi1

Vxi1xixi1

(xk,V)u(xk ) dV. (33)

The respective expression for nc(xi , V ), accounting for the

fraction of drops/particles of sizeV (=xj+ xk )formed viathe coalescence of two drops/particles of volumes xj and

xk , will be given by

nc(xi , V )=

xi+1Vxi+1xi

xiVxi+1,

Vxi1x

ix

i1xi1Vxi .

(34)

Accordingly, from the calculated values ofNi (t ), one can

easily obtain the values of the average number density func-

tion,ni (V , t ):

ni (V , t )=Ni (t )

(Vi+1Vi ). (35)

It is often desirable to know the number diameter density

function,n(D, t). By noting that n(V, t) dV= n(D, t) dD,one can easily calculate the average number diameter density

function,ni (D,t), in terms ofNi (t ):

Ni (t )= Di+1

Di

n(D, t) dD= ni (Di ,t)(Di+1Di ). (36)

In many cases, experimental measurements on PSD are

given in terms of number or volume fractions, from which

it is not easy to derive the actual particle number dis-

tribution, Ni (t ), or the number volume density function,

n(V,t). Thus, it is better to compare directly the available

experimental measurements on number and volume fraction

distributions with respective simulation results obtained

from the numerical solution of the PBE. Accordingly, one

can define the following number, A(D, t), and volume,

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Table 1

Physical/transport properties and model parameters for VCM/PVC system

m=9471.746(T 273.15) (Kg/m3) (Kiparissides et al., 1997)p=103 exp(0.42963.274104T ) (Kg/m3) (Kiparissides et al., 1997)m=2.1104 106(T 273.15) (Kg/m s)c=10110.4484(T273.15) (Kg/m3) (Kiparissides et al., 1997)

c=0.08 exp(1.5366102

T ) (Kg/m s)[n] = 1.087105 1.67108(T 273.15)M0.851w (m3/Kg) (Polymer Handbook)[s ] =9.13103 +4.317105DP (Okaya, 1992)Ep=2.4109 (Kg/m s2) (Polymer Handbook)r=35, SNsa=110, kb=324, ab=33, kc=3107, ac=2109, ac=1102 (This study)

Table 2

Physical, transport properties and model parameters for styrene/PS system

m=923.60.887(T 273.15) (Kg/m3) (Achilias and Kiparissides, 1992)p=10500.602(T273.15) (Kg/m3) (Achilias and Kiparissides, 1992)m=10528.64(1/T1/276.71)

3(Kg/m s) (Achilias and Kiparissides, 1992)

[n] =1.38105M0.722w (m3/Kg) (Achilias and Kiparissides, 1992)Ep=3.3810

9(Kg/m s

2) (Polymer Handbook)

r=35, SNsa=50, kb=400, ab=33, kc=4107, ac=5109, ac=3103 (This study)

AV(D,t), probability density functions:

A(D, t)= n(D,t)Nt(t )

fNiDi+1Di

,

AV(D,t)=(D3/6)n(D, t)

Vt(t ) fVi

Di+1Di. (37)

A(D, t) dD and AV(D,t) dD represent the number (fNi )

and volume (fV i ) fractions of particles in the size range(D,D+dD), respectively. Nt(t ) and Vt(t ) denote the re-spective total number and volume of particles per unit vol-

ume of the reaction medium. It is apparent that the num-

ber and volume probability density functions will satisfy the

following normalization conditions:

DmaxDmin

A(D, t) dD=1, Dmax

Dmin

AV(D,t) dD=1. (38)

Very often experimental measurements on some average

particle diameter are only available. In general, the average

particle diameter,Dqp, can easily be calculated, in terms ofthe number probability density function, using the following

equation (Chatzi and Kiparissides, 1992):

(Dqp)qp

= Dmax

Dmin

Dq A(D, t) dD

DmaxDmin

DpA(D, t) dD, (39)

where q and p are characteristic exponents that define the

desired average particle diameter (e.g., mean Sauter diam-

eter, D32, average number diameter, D10, average volume

diameter, D30, etc.).

4. Results and discussion

The predictive capabilities of the proposed model were

demonstrated by a direct comparison of model predic-

tions with experimental data on average mean diameter

and droplet/particle size distribution of both non-reactive

liquidliquid dispersions of styrene and VCM in water,

and free-radical suspension polymerization of styrene and

VCM. For polymerization systems, the general popula-tion balance model (see Eq. (1)) was solved together with

the pertinent molecular species differential equations (see

Appendix A) describing the molecular weight develop-

ments (e.g., number and weight average molecular weights)

and the polymerization rate in the heterophase suspension

system. In addition to the above dynamic equations, the

dynamic model included all the necessary algebraic equa-

tions, describing the variation of the kinetic rate constants,

and the physical, transport and thermodynamic proper-

ties of the multi-phase system with respect to the reactor

operating conditions (e.g., temperature, monomer mass,

etc.) and the fractional monomer conversion. Additional

details, regarding the kinetic mechanism (e.g., gel- and

glass-effect), phase equilibrium calculations (e.g., monomer

and initiator partitioning, number of phases in the system,

etc.) can be found in the publications ofKiparissides et al.

(1997), Kotoulas et al. (2003) andKrallis et al. (2004). In

Tables 1and2, the physical, transport and model parame-

ters for the VCM/PVC and styrene/PS systems are reported.

It should be noted that the numerical values of the key

model parameters (i.e., kb, ab, kc, ac) were estimated by

fitting the model predictions to experimental data on DSD

of liquidliquid aqueous dispersions of styrene and VCM.

The system of non-linear ordinary differential equations
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0 200 400 600 800 1000 1200 1400 160010

-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

N E = 30

N E = 50

N E = 80

NE = 100

VolumeProbabilityD

ensityFunction(m

-1)

Particle Diameter (m)

Fig. 3. Effect of the number of volume elements on the calculated volume

probability density function (free-radical suspension polymerization of

styrene).

(Eqs. (32)(34)) together with the necessary kinetic equa-

tions (see Appendix A) were numerically integrated using

the Gear predictorcorrector DE solver.

Validation of the numerical method: The accuracy and

convergence characteristics of the numerical method (FTP)

were first assessed by varying the total number of discretiza-

tion points, the size of the total volume domain and the initial

DSD.Fig. 3shows the effect of the number of equal-size dis-

crete elements (i.e., 30, 50, 80 and 100) on the volume prob-

ability density function for the styrene suspension polymer-

ization. The diameter domain extended from 1 to 2000 m

while the initial DSD followed a Gaussian distribution with

a mean value ofD0=1000m and a standard deviation ofD=100 m. As can be seen, the volume probability den-sity function converges to the same distribution for values

of the number of elements NE80. In the present study,

it was assumed that the numerically calculated distribution

converged to the correct one when the total mass of the dis-

persed phase (i.e., monomer plus polymer), given by the first

moment of particle number distribution,(dN

k=1Vi Ni (t)),differed from the initial monomer mass by less than 23%.

When the upper limit of the total diameter domain, Dmax,

was reduced from 2000 to 1200 m, it was found that the

number of discrete elements, required for the satisfaction

of above mass conservation criterion, was NE

50. Thus, itwas concluded that the numerical solution converged to the

correct distribution when the size of the discrete elements

(i.e., the ratio of the total diameter domain over the num-

ber of elements) was smaller than 25 m. A similar rule was

found to be applicable to the VCM/PVC suspension poly-

merization system.

In liquidliquid dispersions the final DSD is controlled by

the dynamic equilibrium between drop breakage and coales-

cence rates. Thus, for the same operating conditions (e.g.,

input power, dispersed phase volume fraction, temperature,

etc.) the final DSD should be independent of the initial DSD.

Fig. 4 illustrates the effect of the initial DSD on the final

0 200 400 600 800 1000 1200 1400

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

Final Distributions

D = 200 m

D = 700 m

D = 1000 m

Initial Distributions

D = 200 m = 50 m

D = 700 m =100 m

D = 1000 m =100 m

VolumeProbabilityDensityFunction(m

-1)


Fig. 4. Effect of the initial DSD on the calculated volume probability

density function of styrene droplets in water (non-reactive case).

0 50 100 150200

250

300

350

400

450

500

550

600D = 200 m = 50 m

D = 700 m = 100 m

D =1000 m = 100 m

Time (min)

SauterMeanDiameter,

D32

(m)

Fig. 5. Effect of the initial DSD on the dynamic evolution of the Sauter

mean diameter of styrene droplets in water (non-reactive case).

DSD at dynamic equilibrium for the styrenewater disper-

sion system. As can be seen, the calculated final DSD is not

affected by the initial condition. On the other hand, the time

required for the system to attain its final DSD is affected by

the initial DSD condition. Fig. 5, clearly depicts the vari-

ation of the Sauter mean droplet diameter with respect to

time. In all cases, the drop breakage and coalescence ratefunctions were the same. It is apparent that the time required

for the liquidliquid dispersion to reach its dynamic equilib-

rium distribution is larger when the initial DSD had a mean

value of D0=200 m. On the other hand, no significantdifferences in the required times for the system to reach its

dynamic equilibrium were observed when the mean value

of the initial DSD changed from 1000 to 700 m. Notice

that in the former case (i.e., D0= 200m andD= 50m)the drop coalescence mechanism controls the dynamic evo-

lution of DSD, while in the later case (i.e., D0=1000mand D=100 m) the DSD evolution is mainly controlledby the drop breakage mechanism.

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0 200 400 600 800 1000 1200

0.000

0.002

0.004

0.006

0.008

0.010 Initial DSD D

0= 200 m D

0= 700 m D

0=1000 m

D

= 80 m D

= 100 m D

=100 m

Final PSD

(D =1000 m)

(D = 700 m)

(D = 200 m)

VolumeProbabilityDe

nsityFunction(m

-1)


Fig. 6. Effect of the initial DSD on the calculated polystyrene particle

size distribution (suspension polymerization of styrene).

0 50 100 150 200 250 300 350

250

300

350

400

450

500

550

D = 200 m = 80 m

D = 700 m = 100 m

D =1000 m = 100 m

SauterMeanDiameter,

D32

(m)

Time (min)

Fig. 7. Effect of the initial DSD on the dynamic evolution of the Sauter

mean diameter of styrene droplets in water (suspension polymerization

of styrene).

It should be noted that when the polymerization in the

monomer droplets starts before the system has reached its

liquidliquid equilibrium distribution, the final PSD in the

suspension system will not be independent of the initial DSD

condition. In Figs. 6 and 7, the effect of the initial DSDon the final PSD is depicted for the free-radical suspension

polymerization of styrene, assuming that the polymeriza-

tion in the monomer droplets starts at time zero (i.e., be-

fore the liquidliquid dispersion reaches its dynamic equi-

librium point). As can be seen, as the average size of the

initial monomer DSD increases, the final PSD is shifted to

larger sizes. The reason is that drop breakage ceases before

the liquidliquid dispersion has reached its final equilibrium

distribution.

Vinyl Chloride suspension polymerization. The dispersion

of vinyl-chloride monomer (VCM) in aqueous PVA solu-

tions has been studied experimentally byZerfa and Brooks

0 20 40 60 80 100

20

30

40

50

60

MeanDroplet

Diameter(m)

D10

( sim)

D32

( sim)

Time (min)

Fig. 8. Dynamic evolution of the calculated and measured mean

diameter of VCM droplets (non-reactive case: monomer vol-

ume fraction= 0.1; CPVA (72.5% degree of hydrolysis) = 0.02%;temperature=55 C; N=500rpm).

0 30 60 90 120 150

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Volum

eProbabilityDensityFunction(m

-1)


5 min

10 min

30 min

120 min

Experimental

Fig. 9. Dynamic evolution of the calculated distribution of VCM droplets

(non-reactive case: experimental conditions as in Fig. 8; discrete points

represent experimental measurements).

(1996a,b)under different conditions (e.g., monomer hold-

up, agitation speed and type and concentration of stabilizers).

Fig. 8illustrates the dynamic evolution of the number mean

diameter, D10, and the Sauter mean diameter, D32, of VCmonomer droplets in the dispersion. The monomer volume

fraction in the dispersion was 0.1, the temperature was kept

constant at 55 C, the agitation speed was set at 500rpm,while 200 ppm of PVA with a degree of hydrolysis equal to

72.5% were added to the aqueous phase for the stabiliza-

tion of the VCM droplets (Zerfa and Brooks, 1996b). The

continuous lines represent simulation results while the dis-

crete points the experimental measurements. As can be seen,

the droplet size initially reduces (i.e., due to the dominant

drop breakage mechanism) and reaches its final dynamic

equilibrium value, at approximately 30 min. The evolution

of DSD is shown inFig. 9.Initially, the volume probability
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0 30 60 90 120 150 180

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Vo

lume

Pro

ba

bility

Densi

tyFunct

ion

(m

-1)


250 rpm ( sim)

350 rpm ( sim)

500 rpm ( sim)

650 rpm ( sim)

Fig. 10. Calculated and experimentally measured distributions of

VCM droplets in water at different agitation rates (monomer

volume fraction= 0.1; CPVA (72.5% degree of hydrolysis)= 0.03%;temperature=55 C).

density function of VCM is broad. However, as the agitation

continues, it becomes narrower and shifts to smaller sizes.

The predicted steady-state DSD (at 120 min) is in excellent

agreement with the experimentally measured one (black dis-

crete points). It is clear that the proposed model is capable

of predicting satisfactorily the dynamic evolution of VCM

distribution as well as its mean droplet value.

Fig. 10 illustrates the effect of the agitation rate on the

steady-state volume probability density function of VCM

droplets in the aqueous dispersion. All other experimental

conditions were similar to those ofFig. 8, except the con-

centration of the PVA stabilizer, which was 300 ppm (Zerfaand Brooks, 1996a). The discrete points represent the exper-

imental measurements while the continuous lines the model

predictions. As can be seen, as the agitation rate increases

the DSD shifts to smaller sizes and becomes narrower due

to the increased drop breakage rate. In all cases, the model

results are in very close agreement with the experimental

data. An additional comparison study was carried out for a

VCM dispersed volume fraction, of 0.2. It was found that

the VCM droplet distribution shifted to larger sizes as the

monomer hold-up increased. Again, simulation results were

in excellent agreement with experimental measurements on

DSD.Subsequently, experimental measurements on the average

particle size and PSD were compared with model predictions

for the free-radical suspension polymerization of VCM. The

experimental data for the PVC system were provided by

ATOFINA. The experiments were carried out in a 30 L batch

reactor, using 40% v/v VCM in water. The polymerization

temperature was set at 56.5 C while the agitation speedremained constant at 330 rpm.

Fig. 11depicts the variation of the volume mean diam-

eter with respect to polymerization time. The continuous

line represents the simulation results and the discrete points

the experimental measurements. Initially, the mean diameter

0 50 100 150 200 250 300

60

80

100

120

140

160

180

200

Vo

lume

Mean

Dia

meter,

D30

(m

)

Time (min)

Experimental

Simulation

Fig. 11. Dynamic evolution of calculated and experimentally measured vol-

ume mean diameter of PVC particles (reactive case: temperature=56.5 C;dispersed phase volume fraction=0.4; agitation rate=330rpm).

shifts to smaller values due to the dominant drop breakage

mechanism. Subsequently, the drop breakage rate is reduced

while the drop coalescence rate increases because of the in-

creased viscosity of the dispersed phase. Thus, the mean

particle diameter increases until a monomer conversion of

about 75%. After this point, the drop coalescence rate ceases

and the PSD remains almost constant. It is apparent that the

present model predicts very well the dynamic behaviour of

the PSD for the free-radical suspension polymerization of

VCM. InFig. 12, experimental measurements (dash lines)

and simulation results (continuous lines) on PSD are plot-ted at four different conversion levels (i.e., 55%, 65%, 75%

and 83%). As can been seen, the simulation results are in

very good agreement with the experimental measurements.

It should be pointed out that all the simulation results on

VCM suspension polymerization (i.e., for both reactive and

non-reactive cases) were obtained using the same values of

the model parameters (seeTable 1).

Styrene suspension polymerization. The dynamic evolu-

tion of styrene DSD in aqueous dispersions was experimen-

tally studied by Yang et al. (2000). Fig. 13 illustrates the

dynamic evolution of the Sauter mean diameter of styrene

droplets for two different monomer volume fractions. Thetemperature was kept constant at 25 C, the agitation speedwas 350 rpm, while 500 ppm PVA were added to the aque-

ous phase for the stabilization of the dispersion. The PVA

had a degree of hydrolysis equal to 88%, while its molecu-

lar weight varied between 30,000 and 50,000 g/mol. The fi-

nal DSD at dynamic equilibrium was attained after 150 min

of stirring. Apparently, the model predicts fairly well the

dynamic evolution of the Sauter mean diameter of styrene

droplets, as well as the effect of monomer volume fraction.

In Fig. 14, the time evolution of DSD is depicted for the

case of styrene volume fraction of 0.1. It is evident that the

model predictions on DSD are in very good agreement with

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0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Conversion 55%

Experimental

Simulation

Conversion 65%

Experimental

Simulation

0 100 200 300 4000.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Conversion 75 %

Experimental

Simulation

VolumeProbabilityDensityFunction

(m

-1)

0 100 200 300 400

Conversion 83%

Experimental

Simulation


Fig. 12. Predicted and experimentally measured distributions of PVC particles at four different conversion levels: 55%, 65%, 75% and 83% (experimental

conditions asFig. 11).

0 50 100 150 200 250

80

100

120

140

160

SauterMeanDiameter,

D32

(m)

Time (min)

= 0.05 ( sim)

= 0.10 ( sim)

Fig. 13. Dynamic evolution of calculated and experimentally measured

Sauter mean diameter of styrene droplets at two different monomer volume

fractions (non-reactive case: temperature=25 C; agitation rate=350rpm;CPVA (88% degree of hydrolysis)=0.05%).

experimental measurements (discrete points).Fig. 15illus-

trates the effect of the agitation rate on the steady-state DSD

of the styrene droplets. The operating conditions were as in

Fig. 13,while the monomer volume fraction was 0.1. As in

the case of the VCM dispersion, the mean size of styrene

droplets decreases with the agitation rate while the DSD be-

50 100 150 200 250 300

0.000

0.003

0.006

0.009

0.012

0.015

0.018

Vo

lume

Pro

ba

bility

Densi

tyFunct

ion(m

-1)


5 min ( sim)

30 min ( sim)

120 min ( sim)

Fig. 14. Dynamic evolution of calculated and experimentally measured

distributions of styrene droplets in water (non-reactive case: experimental

conditions as Fig. 13; monomer volume fraction=0.1).

comes narrower. In all cases, the simulation results are in

very good agreement with the experimental data that clearly

underlines the predictive capabilities of the present compre-

hensive population balance model.

Finally, the present model was employed to predict the

dynamic evolution of PSD in the free-radical suspension

polymerization of styrene. More specifically, the effects of

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0 50 100 150 200 250 300 350

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

Vo

lume

Pro

ba

bilityD

ensi

tyFunct

ion

(m

-1)


250 rpm ( sim)

450 rpm ( sim)

650 rpm ( sim)

Fig. 15. Effect of agitation rate on the calculated and ex-

perimentally measured distributions of styrene droplets in water

(non-reactive case: temperature=25 C; agitation rate=350 rpm; CPVA(88% degree of hydrolysis)

=0.05%; monomer volume fraction

=0.1).

n-pentane and concentration of stabilizer on the PSD were

investigated for the expandable PS suspension polymer-

ization process. Experimental measurements on PSD were

taken from the work ofVillalobos et al. (1993). The free-

radical styrene suspension polymerization was carried out

in a 1-gal reactor vessel. The dispersed monomer volume

fraction was 0.4. The polymerization took place at 105 C inthe presence of 1,4-bis(terbutyl peroxycarbo) cyclohexane

(TBPCC) bifunctional initiator. The initiator concentration

was 0.01 mol/L-styrene in all the experimental cases, while

tricalcium phosphate (TCP) was used as surface-active agent

at three different concentrations (i.e., 7.5, 5.0 and 3.5 gr/L).

InFig. 16,experimental measurements and simulation re-

sults on PSD are shown for three different addition policies

ofn-pentane into the reactor. More specifically, 7.5% w/w

of n-pentane with respect to the styrene mass was added

200 400 600 800 1000 1200 1400

Experimental

Simulation


200 400 600 800 1000 1200

Experimental

Simulation


0 200 400 600 800 1000 12000.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

VolumeProbabilityDensityFunction(m

-1)


Experimental

Simulation

(a) (b) (c)

Fig. 16. Predicted and experimentally measured distributions of EPS particles for different n-pentane addition policies. (a) 7.5% w/w n-pentane (wrt

styrene) at =0%; (b) 7.5% w/w n-pentane (wrt styrene) at =50% ; (c) in the absence of n-pentane (temperature=105 C; dispersed phase volumefraction

=0.4,

[Io

] =0.01 mol TBPCC/L-styrene;

[TCP

] =7.5 g/L).

to the system at three different conversion levels (i.e., 0, 50%

and 100%). In Fig. 16a, model results (continuous lines)

are compared with experimental data (dash lines) on PSD

for the case ofn-pentane addition at zero monomer conver-

sion. InFig. 16b, the corresponding distributions are illus-

trated for the case ofn-pentane addition at 50% monomer

conversion. The last case (see Fig. 16c) corresponds to theaddition ofn-pentane at the end of polymerization. As can

been seen, for all cases, there is a close agreement between

experimental and simulation results on PSD, indicating the

predictive capabilities of the model for the free-radical sus-

pension polymerization of styrene. It should be noted that,

in the presence of n-pentane, the EPS particles are more

uniform while the PSD becomes narrower.

Villalobos et al. (1993) also investigated experimentally

the effect of the stabilizer concentration on PSD. More

specifically, experiments were carried out at different TCP

concentrations, in the presence of 7.5% w/w n-pentane,

added at 50% monomer conversion. InFig. 17the predicted

and experimental PSDs of the EPS particles are shown for

the three different concentrations of the surface-active agent.

As can be seen, as concentration of the surface-active agent

decreases (i.e., the interfacial tension increases) the PSD be-

comes broader and shifts to larger sizes. Apparently, there is

a very good agreement between calculated and experimental

measurements on the volume probability density function.

5. Conclusions

A comprehensive population balance model coupled with

a system of differential equations governing the conserva-

tion of the various molecular species present in the system

has been developed to describe the dynamic evolution of

the DSD/PSD in free-radical suspension polymerization re-

actors. The fixed pivot technique (FPT) was employed for

solving the PBE. The robustness of the numerical method

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0 500 1000 1500 2000

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

Vo

lume

Pro

ba

bility

Densi

tyFunct

ion

(m

)


7.5 % w/w ( sim)

5.0 % w/w ( sim)

3.5 % w/w ( sim)

Fig. 17. Effect of surface-active concentration on the calculated and

experimentally measured distributions of EPS particles at three different

quantities of surface-active agent (TCP) (experimental conditions as in

Fig. 16).

was examined in regard with its convergence character-

istics and accuracy in terms of the mass conservation of

the monomer, initially loaded into the reactor. The predic-

tive capabilities of the model were demonstrated via the

successful simulation of experimental measurements on

DSD/PSD and the average droplet/particle diameter for both

non-reactive liquidliquid dispersions and the free-radical

suspension polymerization of styrene and VCM.

Notation

A(D,t),Av(D,t) number and volume probability density

functions, 1/m

CPVA concentration of surface-active agent,

Kg/m3

D diameter, m

DP degree of polymerization of the PVA

stabilizer

E elasticity modulus, Kg/m s2

g(V) breakage rate, 1/s

k(V,U) coalescence rate, m3/s

kb, kc model parameters

L macroscale of turbulence, m

Mw weight average molecular weight,

Kg/kmol

n(V,t) number density function, 1/m6

[n] intrinsic viscosity, m3/KgNE number of discrete elements

Ni number of particles having volume

equal to xi per reactor unit volume,

1/m3

Nda, Nsa number of daughter and satellite

droplets per breakage events

r volume ratio of daughter over the satel-

lite drops

Re Reynolds number

Snsa model parameter

t time, s

u(V) number of droplets formed by a break-

age of a droplet of volume V

u(Dv)2 mean square of the relative velocity be-

tween two points separated by a dis-tanceD, m/s

Vda, Vsa volumes of daughter and satellite

drops, m3

V , U , x volumes, m3

We Weber number

Greek letters

ab, ac model parameters

( U , V ) daughter droplets probability function,

1/m3

average energy dissipation rate per unit

mass, m2/s3

microscale of turbulence, m

b, c breakage and coalescence efficiencies

viscosity, Kg/m s

kinematic viscosity, m2/s

density, Kg/m3

interfacial tension, Kg/s2

da, sa standard deviation of the distribution

for daughter and satellite drops

dispersed phase volume fraction

p volume fraction of the polymer in the

dispersed phase

monomer conversionb,c breakage and coalescence frequencies,

1/s

Subscripts

c continuous phase

d dispersed phase

m monomer

p polymer

s suspension system

w water

Acknowledgement

The authors gratefully acknowledge ARCHEMA (ex-

ATOFINA Chemicals) for providing the experimental data

for PVC suspension polymerization.

Appendix A

The free-radical polymerization of vinyl monomers in

general includes the following chain initiation, propagation,

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chain transfer to monomer and bimolecular termination re-

actions (Kiparissides et al., 1997):

decomposition of initiators

Iikd,i 2R, i=1, 2, . . . , N m,

chain initiation

R+M kp P1,chain propagation

Pn+Mkp Pn+1,

chain transfer to monomer

Pn+Mkf m Dn+P1,

termination by combination

Pn+

Pmktc

Dn

+m,

termination by disproportionation

Pn+Pm ktd Dn+Dm,inhibition ofliveradical chains

Pn+Zkz Dn+Z,

whereIi , R, Mand Zdenote the initiator, primary radicals,

monomer and inhibitor molecules, respectively, andPn and

Dn, the corresponding live and dead polymer chains,

having a degree of polymerization n.

In the free-radical polymerization of VCM, the polymer

is insoluble in its monomer, thus, precipitates out to form

a separate phase (i.e., the polymer-rich phase). Thus, the

elementary reactions presented above take place in both the

monomer-rich and polymer-rich phases (Kiparissides et al.,

1997). Additional details, regarding the kinetic modeling

of free-radical polymerization of styrene and VCM (e.g.,

gel- and glass-effect), phase equilibrium calculations (e.g.,

monomer and initiator partitioning, number of phases in the

system, etc.), can be found in the publications ofKiparissides

et al. (1997, 2004)and Kotoulas et al. (2003).

The method of moments is invoked in order to reduce the

infinite system of molar balance equations, required to de-

scribe the molecular weight distribution developments. Ac-cordingly, the average molecular properties of the polymer

(i.e., Mn, Mw) are expressed in terms of the leading mo-

ments of the dead polymer molecular weight distribution.

The moments of the total number chain length (TNCL) dis-

tributions of live radical and dead polymer chains can be

defined as (Krallis et al., 2004)

k=

n=ink Pn, k=

n=i

nk Dn. (A.1)

Accordingly, one can easily derive the corresponding mo-

ment rate functions:

Live polymer moment rate equations

rk=Nmk=1

2fkkdk Ik+kpM

kr=0

k

r

rk

+kfmM(0k )(ktc+ktd)k0kzZk. (A.2)

Deadpolymer moment rate equations

rk=kfmMk+1

2k

jtc

kr=0

k

r

rkr

+ktdk0+kzZk . (A.3)The number- and weight-average molecular weights

can be expressed in terms of the molecular weight of the

monomer, MWm, and the moments of the TNCLDs of live

and dead polymer chains:

Mn=(1+1)(0

+0)

MWm, Mw=(2+2)(1

+1)

MWm. (A.4)

Finally, the total monomer conversion can be calculated by

the following expression, assuming that the long chain hy-

pothesis holds true (i.e., the monomer is mainly consumed

via the propagation reaction):

d

dt=kp

M

M00. (A.5)

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