A generalized population balance model for the prediction of particle size distribution in...

Chemical Engineering Science 61 (2006) 332–346www.elsevier.com/locate/ces

A generalized population balancemodel 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 2005Available online 22 August 2005

Abstract

In the present study, a comprehensive population balance model is developed to predict the dynamic evolution of the particle sizedistribution in high hold-up (e.g., 40%) non-reactive liquid–liquid dispersions and reactive liquid(solid)–liquid suspension polymerizationsystems. Semiempirical and phenomenological expressions are employed to describe the breakage and coalescence rates of dispersedmonomer 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 thepopulation balance equation. The predictive capabilities of the present model are demonstrated by a direct comparison of model predictionswith experimental data on average mean diameter and droplet/particle size distributions for both non-reactive liquid–liquid dispersionsand 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)). Insuspension polymerization, the monomer is initially dis-persed in the continuous aqueous phase by the combinedaction of surface-active agents (i.e., inorganic or/and water-soluble polymers) and agitation. All the reactants (i.e.,monomer, initiator(s), etc.) reside in the organic or “oil”phase. The polymerization occurs in the monomer dropletsthat are progressively transformed into sticky, viscousmonomer–polymer particles and finally into rigid, sphericalpolymer particles of size 50–500�m (Kiparissides, 1996).

∗ Corresponding author. Tel.: +302310996211;fax: +3102310996198.E-mail addresses:cypress@cperi.certh.gr,

cypress@alexandros.cperi.certh.gr(C. Kiparissides).

The polymer solid content in the fully converted suspensionis typically 30–50% 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 monomerand smooth spherical particles are produced; and the pow-der polymerization, where the polymer is insoluble in itsmonomer and, thus, precipitates out leading to the forma-tion of irregular grains or particles. The most importantthermoplastic produced by bead suspension polymerizationis polystyrene (PS). In the presence of volatile hydrocar-bons (C4–C6), foamable beads (the so-called expandablepolystyrene, 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 sizedistribution (PSD) (Yuan et al., 1991). The initial monomerdroplet size distribution (DSD) as well as the final polymer

C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332–346 333

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

PSD in general depend on the type and concentration ofthe surface-active agents, the quality of agitation and thephysical properties (e.g., density, viscosity and interfacialtension) of the continuous and dispersed phases. The tran-sient droplet/particle size distribution is controlled by twodynamic processes, namely, the drop/particle breakage andcoalescence rates. The former mainly occurs in regions ofhigh-shear stress (i.e., near the agitator blades) or as a resultof turbulent velocity and pressure fluctuations along the sur-face of a drop. The latter is either increased or decreased bythe turbulent flow field and can be assumed to be negligiblefor very dilute dispersions at sufficiently high concentrationsof 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 (seeFig. 1a). Subsequently, thedeformed monomer droplet breaks into two almost equal-size drops, corresponding to the fluid lumps, and a series ofsmaller droplets corresponding to the liquid thread. This isknown as Thorough breakage. On the other hand, a dropletsuspended 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 dropletsurface can start oscillating. When the relative velocity isclose to that required to make a drop marginally unstable,a number of small droplets are stripped out from the initialone (seeFig. 1b). This situation of breakage is referred to aserosive one. Erosive breakage is considered to be the dom-inant mechanism for low-coalescence systems that exhibita 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 liquidfilm of the continuous phase is being trapped between thedrops 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 otherhand, if the kinetic energy of the induced drop oscillationsis larger than the energy of adhesion between the drops,the drop contact is broken before the complete drainage ofthe liquid film. The second drop coalescence mechanism(Howarth, 1964) assumes that immediate coalescence oc-curs when the approach velocity of the colliding drops atthe collision instant exceeds a critical value. In other words,if the turbulent energy of collision is greater than the totaldrop surface energy, the drops will coalesce (seeFig. 1d).Surface-active agents play a very important role in the

stabilization of liquid–liquid dispersions. One of the mostcommonly 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 polymerchains at the monomer/water interface (Chatzi and Kiparis-sides, 1994). The solubility of the PVA in water depends onthe 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 ofPVA. This U-type behaviour has been confirmed both ex-perimentally and theoretically and has been attributed tothe balance of breakage and coalescence rates of monomerdrops.In regard with the droplet/particle breakage and coales-

cence phenomena, the suspension polymerization processcan 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 dominantmechanism. As a result the initial DSD shifts to smallersizes. During the second sticky-stage of polymerization, thedrop breakage rate decreases while the drop/particle coales-cence becomes the dominant mechanism. Thus, the averageparticle size starts increasing. In the third stage, the PSDreaches its identification point while the polymer particlesize slightly decreases due to shrinkage (i.e., the polymerdensity 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 viscosityof about 0.1Pa s, while the second stage extends up to 70%monomer conversion. In the VCM powder polymerization,at monomer conversions around 10–30%, a continuouspolymer network is commonly formed inside the poly-merizing monomer droplets that significantly reduces thedrop/particle coalescence rate (Kiparissides et al., 1994).Cebollada et al. (1989)reported that the PSD is essentiallyestablished up to monomer conversions of about 35–40%(i.e., end of the second stage).

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 describethe dynamic evolution of PSD in batch suspension polymer-ization reactors. The model takes into account the dynamicevolution of the physical properties of the continuous anddispersed 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 pivottechnique (FPT) (Kumar and Ramkrishna, 1996) is appliedfor solving the general population balance equation, govern-ing the PSD developments, in terms of the polymerizationconditions (e.g., monomer to water volume ratio, temper-ature, type and concentration of stabilizer, impeller energyinput, etc.) and the polymerization kinetic model. An ex-tensive analysis on the robustness of the numerical methodis 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 thepresent model are demonstrated by a direct comparison ofmodel predictions with experimental data on average meandiameter and droplet/particle size distributions for both non-reactive liquid–liquid 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 particulateprocess, 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)dV represents 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 thenumber density function with respect to time and volume isgiven by the following non-linear integro-differential popu-lation balance equation (Kiparissides et al., 2004):

�[n(V, t)]�t

=∫ Vmax

�(U, V )u(U)g(U)n(U, t)dU

+∫ V /2

k(V − U, U)

× n(V − U, t)n(U, t)dU − n(V, t)g(V )

− n(V, t)

∫ Vmax

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 breakagefunction, accounting for the probability that a drop of vol-umeV is formed via the breakage of a drop of volumeU .The functionu(U) denotes the number of droplets formedby the breakage of a drop of volumeU and g(U) is the

breakage rate of drops of volumeU . The second term on ther.h.s. of Eq. (1) represents the rate of generation of drops inthe size range(V , V +dV ) due to coalescence of two smallerdrops.k(V, U) is the coalescence rate between two drops ofvolumeV andU . Finally, the third and fourth terms repre-sent the drop disappearance rates due to drop breakage andcoalescence, respectively. Eq. (1) will satisfy the followinginitial condition att = 0:

n(V,0) = n0(V ), (2)

wheren0(V ) is the initial drop size distribution of the dis-persed phase. In the present study, the initial monomer DSDwas 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 coalescencerate 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 liquid–liquid disper-sions (Coulaloglou and Tavlarides, 1977; Narsimhan et al.,1979; Sovova, 1981; Chatzi et al., 1989). According to theoriginal work of Alvarez et al. (1994)and the proposedmodifications ofMaggioris et al. (2000), the breakage andcoalescence rates can be expressed in terms of the break-age,�b, and collision,�c, frequencies and the, respective,Maxwellian efficiencies,�b and�c:

g(V ) = �b(V )e−�b(V ), (3)

k(V, U) = �c(V , U)e−�c(V ,U), (4)

�b and �c denote the corresponding ratios of required toavailable 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 equalto the eddy wave length,Dv. Eddy fluctuations with wave-lengths smaller (larger) than the drop diameterDv producean oscillatory (rigid body) motion of the drop that do notlead to breakage (Alvarez et al., 1994). The frequency term,�b(V ), was assumed to be equal to the inverse fluctuationtime period, corresponding to the time required for a dropto reach its mean drop displacement:

�b(V ) = u(Dv)/Dv, (5)

whereu(Dv)2 is the mean square of the relative velocitybetween two points separated by a distanceDv, or the meansquare fluctuation velocity of drops of diameterDv.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(�sDv)2/3, (6)

wherekb is a model parameter and�s is the average energydissipation rate for the dispersion.For droplets in the viscous dissipation range (i.e.,Dv < �),

the inertial forces are of the same order of magnitude asthe viscous shear forces and the mean square of the relativevelocity between two points separated by a distanceDv willbe 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 ofthe average energy dissipation rate of the liquid–liquid dis-persion,�s , in terms of the average energy dissipation rateof the continuous phase,�c, and the kinematic viscosities ofthe continuous,�c, and liquid–liquid dispersion,�s :

�s/�c = (�c/�s)3. (8)

Thus, in the presence of a high-volume dispersed phase, theoverall viscosity of the system increases and, therefore, theenergy 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 theflow within a drop can be described as one-dimensionalsimple-shear flow of aMaxwell fluid, the breakage efficiencycan be expressed as follows:

�b = ab�(Dv), (9)

whereab is a model parameter and�(Dv) is the ratio ofrequired to available energy for a drop of diameterDv tobreak:

�(Dv) = 2

Re(1+ ReVe)+ CdsWe

, (10)

whereRe andWe denote the drop Reynolds and Webernumbers, respectively. The dimensionless quantity Ve ac-counts for drop viscoelasticity and is a function of the dropReynolds number and its physical properties:

Ve= Yo

(− 1− a

2Re Yo

×[1+ a

1− a− 1− a

1+ aexp

(− a

)]− 1

12, (11)

Yo = �2ddEdD2

, =√1− 48Yo. (12)

In the present study, the dispersed-phase elasticity mod-ulus,Ed , was approximated by the product of the polymerelasticity 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

(− 12

)− 1

). (13)

The scalar quantityCds in Eq. (10) can be expressed interms of the numbers and volumes of daughter and satellitedrops (Chatzi and Kiparissides, 1992):

Cds = Ndar2/3 + Nda

(Ndar + Nsa)2/3

− 1, r = Vda/Vsa, (14)

whereNda is the number of daughter drops of volumeVdaandNsa is the number of satellite drops of volumeVsa. Inthe present study, the number of daughter drops was setequal to 2, the volume ratio of daughter to satellite drops,r,was considered to be constant, while the number of satellitedrops was calculated as a function of the parent drop size(Chatzi and Kiparissides, 1992):

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

whereSnsa is 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 withstandard deviations of�da and�sa, one can derive the fol-lowing expression for the distribution of drops of volumeV , formed via the breakage of a drop of volumeU:

u(U)�(U, V )

�da√2

(− (V − Vda)

2�2da

�sa√2

(− (V − Vsa)

2�2sa

)}. (16)

It should be noted that the daughter drop number densityfunction,u(U)�(U, V ), should satisfy the following numberand volume conservation equations:∫ U

0u(U)�(U, V )dV = u(U),∫ U

0V u(U)�(U, V )dV = U . (17)

Accordingly, one can calculate the volumes of daughter andsatellite drops, formed by the breakage of a drop of volumeU, 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 betweenmolecules as in the kinetic theory of gases, the collision fre-quency between two drops with volumesV andU can be

expressed as (Maggioris et al., 2000)

�c(V , U) = kc(D2v + D2

u)(u(Dv)2 + u(Du)2)1/2. (19)

For deformable drops, that is generally the case for lowinterfacial tension dispersions or large-size drops, the dropcoalescence efficiency can be expressed as (Coulaloglou andTavlarides, 1977)

�()c (V , U) = ac

�cc�s

Dv + Du

]4, (20)

whereac is a model parameter.�c, c, � and� are the vis-cosity and density of the continuous phase, the interfacialtension between the dispersed and aqueous phases and thedispersed phase volume fraction, respectively.At high monomer conversions, when the polymerizing

monomer–polymer particles behave like rigid spheres, thecoalescence efficiency can be expressed as (Coulaloglou andTavlarides, 1977)

�(b)c (V , U) = a′

c�1/3s (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 rigidpolymer particles. Thus, the coalescence efficiency over thewhole 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) andmonomer(m) and the extent of monomer conversion,�:

+ 1− �

. (24)

The viscosity of the liquid(solid)–liquid dispersionwas calculated by the following semi-empirical equation(Vermeulen et al., 1955):

�s = �c

1− �

(1+ 1.5�d�

�d + �c

), (25)

where�d and �c are the viscosities of the dispersed andcontinuous phases, respectively.

For the heterophase suspension polymerization of VCM,the viscosity of the polymerizing monomer droplets,�d , canbe calculated by using the Eulers equation (Krieger, 1972):

�d = �m

(1+ 0.5[n]�p

1− �p/�cr

)2, (26)

where�p 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-version�c, at which a 3-D polymer skeleton is formed in-side the polymerizing monomer drops. When�p → �cr, thedispersed-phase viscosity approaches infinity, indicating theformation of a rigid structure. Thus, for values of�p largerthan the critical value�cr, the dispersed phase viscosity wasassumed to remain constant.For the VCM suspension polymerization, the value of

�cr was taken to be equal to 0.3, which corresponds tothe 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 of�cr was set equal to the 0.7which corresponds to the monomer conversion at which par-ticle coalescence stops. Finally, the intrinsic viscosity of thepolymer solution, [n], was calculated by the well-knownMark–Houwink–Sakurada (MHS) equation as a function ofthe 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 thePSD (Cebollada et al., 1989). Okaya (1992)employed theSchulz–Blaschke equation to calculate of the viscosity ofaqueous PVA solutions:

�c = �w

(1+ [�PVA]CPVA

1− 0.45[�PVA]CPVA

), (28)

where�c, �w, [nPVA] andCPVA are the viscosities of theaqueous PVA solution and pure water, the intrinsic viscosityand the stabilizer concentration, respectively.In the open literature, a great number of papers have been

published, dealing with the behaviour of polymer moleculesat interfaces. Prigogine and his collaborators (Prigogine andMarechal, 1952; Defay et al., 1966) presented a remarkablysimple 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 tothe 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 interfacialtension between the aqueous and the dispersed phase,�. Thechange in the interfacial tension with monomer conversion

was taken into account as in the original work ofMaggioriset al. (2000).

3. Numerical solution of the population balanceequation

The numerical solution of the PBE commonly requiresthe 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 numberof discrete points, resulting in a system of stiff, non-lineardifferential equations that is subsequently integrated numer-ically. Several numerical methods have been proposed in theliterature for the solution of the general PBE (Eq. (1)) inthe continuous or its equivalent discrete form (Kiparissideset al., 2004). In the present study, the FTP ofKumar andRamkrishna (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), correspondingto the ‘i’ element:

Ni(t) =∫ Vi+1

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

Following the original developments ofKumar andRamkrishna (1996), the total volume domain (Vmin toVmax) is first divided into a number of elements. Thedrop/particle population,Ni(t), corresponding to the sizerange(Vi, Vi+1), is then assigned to a characteristic sizexi

(also called grid point) as shown inFig. 2. To account forthe formation of new particles of volumeV that does notcorrespond to the characteristic grid points(xi, xi+1), thefollowing approximation is applied. The particle numberfractionsa(V, xi) andb(V, xi+1) are assigned to the parti-cle populations at the grid pointsxi andxi+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+1 xi xi-1 xi-2

Fig. 2. Discretization of the particle volume domain.

discretized equations can be derived:

M∑k=1

nb(xi, xk)g(xk)Nk(t) +j �k∑

j,kxi−1� xj +xk � xi+1

×(1− 1

2�j,k

)nc(xi, V )k(xj , xk)Nj (t)Nk(t)

− g(xi)Ni(t) − Ni(t)

M∑k=1

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

wherenb(xi, xk) denotes the fraction of drops/particles ofsizexi resulting from the breakage of a drop/particle of sizexk. To preserve the number and volume of particles in thesize range(Vi, Vi+1), nb(xi, xk) must satisfy the followingequation:

nb(xi, xk) =∫ xi+1

xi+1 − V

xi+1 − xi

�(xk, V )u(xk)dV

+∫ xi

xi−1

V − xi−1

xi − xi−1�(xk, V )u(xk)dV . (33)

The respective expression fornc(xi, V ), accounting for thefraction of drops/particles of sizeV (=xj + xk) formed viathe coalescence of two drops/particles of volumesxj andxk, will be given by

nc(xi, V ) =

xi+1 − V

xi+1 − xi

xi �V �xi+1,

V − xi−1

xi − xi−1xi−1�V �xi.

Accordingly, from the calculated values ofNi(t), one caneasily obtain the values of the average number density func-tion, n̄i (V , t):

ni(V , t) = Ni(t)

(Vi+1 − Vi). (35)

It is often desirable to know the number diameter densityfunction,n(D, t). By noting thatn(V, t)dV = n(D, t)dD,one can easily calculate the average number diameter densityfunction,ni(D, t), in terms ofNi(t):

Ni(t) =∫ Di+1

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

In many cases, experimental measurements on PSD aregiven in terms of number or volume fractions, from whichit 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 availableexperimental measurements on number and volume fractiondistributions with respective simulation results obtainedfrom the numerical solution of the PBE. Accordingly, onecan define the following number,A(D, t), and volume,

Table 1Physical/transport properties and model parameters for VCM/PVC system

m = 947− 1.746(T − 273.15) (Kg/m3) (Kiparissides et al., 1997)p = 103 exp(0.4296− 3.274× 10−4T ) (Kg/m3) (Kiparissides et al., 1997)�m = 2.1× 10−4 − 10−6(T − 273.15) (Kg/ms)c = 1011− 0.4484(T − 273.15) (Kg/m3) (Kiparissides et al., 1997)�c = 0.08 exp(−1.5366× 10−2T ) (Kg/ms)[n] = �1.087× 10−5 − 1.67× 10−8(T − 273.15)�M0.851

w (m3/Kg) (Polymer Handbook)[�s ] = 9.13× 10−3 + 4.317× 10−5DP (Okaya, 1992)Ep = 2.4× 109 (Kg/ms2) (Polymer Handbook)r = 35, SNsa= 110, kb = 324, ab = 33, kc = 3× 10−7, ac = 2× 109, a′

c = 1× 102 (This study)

Table 2Physical, transport properties and model parameters for styrene/PS system

m = 923.6− 0.887(T − 273.15) (Kg/m3) (Achilias and Kiparissides, 1992)p = 1050− 0.602(T − 273.15) (Kg/m3) (Achilias and Kiparissides, 1992)

�m = 10528.64(1/T −1/276.71)−3(Kg/ms) (Achilias and Kiparissides, 1992)

[n] = 1.38× 10−5M0.722w (m3/Kg) (Achilias and Kiparissides, 1992)

Ep = 3.38× 109 (Kg/ms2) (Polymer Handbook)r = 35, SNsa= 50, kb = 400, ab = 33, kc = 4× 10−7, ac = 5× 109, a′

c = 3× 103 (This study)

AV (D, t), probability density functions:

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

Nt (t)≈ fNi

Di+1 − Di

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

Vt (t)≈ fVi

Di+1 − Di

. (37)

A(D, t)dD andAV (D, t)dD represent the number(fNi)

and volume(fV i) fractions of particles in the size range(D, D + dD), respectively.Nt(t) andVt (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 thefollowing normalization conditions:

∫ Dmax

A(D, t)dD = 1,

∫ Dmax

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

Very often experimental measurements on some averageparticle diameter are only available. In general, the averageparticle diameter,Dqp, can easily be calculated, in terms ofthe number probability density function, using the followingequation (Chatzi and Kiparissides, 1992):

(Dqp)q−p

=∫ Dmax

DqA(D, t)dD

/∫ Dmax

DpA(D, t)dD, (39)

whereq andp are characteristic exponents that define thedesired average particle diameter (e.g., mean Sauter diam-eter,D32, average number diameter,D10, average volumediameter,D30, etc.).

4. Results and discussion

The predictive capabilities of the proposed model weredemonstrated by a direct comparison of model predic-tions with experimental data on average mean diameterand droplet/particle size distribution of both non-reactiveliquid–liquid dispersions of styrene and VCM in water,and free-radical suspension polymerization of styrene andVCM. For polymerization systems, the general popula-tion balance model (see Eq. (1)) was solved together withthe pertinent molecular species differential equations (seeAppendix A) describing the molecular weight develop-ments (e.g., number and weight average molecular weights)and the polymerization rate in the heterophase suspensionsystem. In addition to the above dynamic equations, thedynamic 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 reactoroperating conditions (e.g., temperature, monomer mass,etc.) and the fractional monomer conversion. Additionaldetails, regarding the kinetic mechanism (e.g., gel- andglass-effect), phase equilibrium calculations (e.g., monomerand 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). InTables 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 keymodel parameters (i.e.,kb, ab, kc, ac) were estimated byfitting the model predictions to experimental data on DSDof liquid–liquid aqueous dispersions of styrene and VCM.The system of non-linear ordinary differential equations

0 200 400 600 800 1000 1200 1400 160010-14

NE = 30 NE = 50 NE = 80 NE = 100

Particle Diameter (µm)

Fig. 3. Effect of the number of volume elements on the calculated volumeprobability density function (free-radical suspension polymerization ofstyrene).

(Eqs. (32)–(34)) together with the necessary kinetic equa-tions (see Appendix A) were numerically integrated usingthe Gear predictor–corrector 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 initialDSD.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�mwhile the initial DSD followed a Gaussian distribution witha mean value ofD0 = 1000�m and a standard deviation of�D = 100�m. As can be seen, the volume probability den-sity function converges to the same distribution for valuesof the number of elementsNE�80. In the present study,it was assumed that the numerically calculated distributionconverged to the correct one when the total mass of the dis-persed phase (i.e., monomer plus polymer), given by the firstmoment of particle number distribution,(d

∑Nk=1ViNi(t)),

differed from the initial monomer mass by less than 2–3%.When the upper limit of the total diameter domain,Dmax,

was reduced from 2000 to 1200�m, it was found that thenumber of discrete elements, required for the satisfactionof above mass conservation criterion, wasNE�50. Thus, itwas concluded that the numerical solution converged to thecorrect 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 wasfound to be applicable to the VCM/PVC suspension poly-merization system.In liquid–liquid 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 14000.000

Final DistributionsD = 200 µmD = 700 µmD = 1000 µm

Initial Distributions D = 200 µmσ = 50 µm D = 700 µmσ =100 µm D = 1000 µmσ =100 µm

Fig. 4. Effect of the initial DSD on the calculated volume probabilitydensity function of styrene droplets in water (non-reactive case).

0 50 100 150200

600 D = 200 µm σ = 50 µm D = 700 µm σ = 100 µm D =1000 µm σ = 100 µm

Time (min)

Fig. 5. Effect of the initial DSD on the dynamic evolution of the Sautermean diameter of styrene droplets in water (non-reactive case).

DSD at dynamic equilibrium for the styrene–water disper-sion system. As can be seen, the calculated final DSD is notaffected by the initial condition. On the other hand, the timerequired for the system to attain its final DSD is affected bythe initial DSD condition.Fig. 5, clearly depicts the vari-ation of the Sauter mean droplet diameter with respect totime. In all cases, the drop breakage and coalescence ratefunctions were the same. It is apparent that the time requiredfor the liquid–liquid dispersion to reach its dynamic equilib-rium distribution is larger when the initial DSD had a meanvalue ofD0 = 200�m. On the other hand, no significantdifferences in the required times for the system to reach itsdynamic equilibrium were observed when the mean valueof the initial DSD changed from 1000 to 700�m. Noticethat in the former case (i.e.,D0= 200�m and�D = 50�m)the drop coalescence mechanism controls the dynamic evo-lution of DSD, while in the later case (i.e.,D0 = 1000�mand�D = 100�m) the DSD evolution is mainly controlledby the drop breakage mechanism.

0 200 400 600 800 1000 12000.000

0.010 Initial DSD D0= 200 µm D0= 700 µm D0=1000 µm

σD= 80 µm σD= 100 µm σD=100 µm

Final PSD(D =1000 µm)(D = 700 µm)(D = 200 µm)

Fig. 6. Effect of the initial DSD on the calculated polystyrene particlesize distribution (suspension polymerization of styrene).

0 50 100 150 200 250 300 350

D = 200 µm σ = 80 µm D = 700 µm σ = 100 µm D =1000 µm σ = 100 µm

Time (min)

Fig. 7. Effect of the initial DSD on the dynamic evolution of the Sautermean diameter of styrene droplets in water (suspension polymerizationof styrene).

It should be noted that when the polymerization in themonomer droplets starts before the system has reached itsliquid–liquid equilibrium distribution, the final PSD in thesuspension systemwill not be independent of the initial DSDcondition. In Figs. 6 and 7, the effect of the initial DSDon the final PSD is depicted for the free-radical suspensionpolymerization of styrene, assuming that the polymeriza-tion in the monomer droplets starts at time zero (i.e., be-fore the liquid–liquid dispersion reaches its dynamic equi-librium point). As can be seen, as the average size of theinitial monomer DSD increases, the final PSD is shifted tolarger sizes. The reason is that drop breakage ceases beforethe liquid–liquid dispersion has reached its final equilibriumdistribution.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 10020

D10 ( sim) D32 ( sim)

Time (min)

Fig. 8. Dynamic evolution of the calculated and measured meandiameter of VCM droplets (non-reactive case: monomer vol-ume fraction= 0.1; CPVA (72.5% degree of hydrolysis) = 0.02%;temperature= 55◦C; N = 500 rpm).

0 30 60 90 120 1500.00

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 inFig. 8; discrete pointsrepresent experimental measurements).

(1996a,b)under different conditions (e.g., monomer hold-up, agitation speed and type and concentration of stabilizers).Fig. 8 illustrates the dynamic evolution of the number meandiameter,D10, and the Sauter mean diameter,D32, of VCmonomer droplets in the dispersion. The monomer volumefraction in the dispersion was 0.1, the temperature was keptconstant at 55◦C, the agitation speed was set at 500 rpm,while 200ppm of PVA with a degree of hydrolysis equal to72.5% were added to the aqueous phase for the stabiliza-tion of the VCM droplets (Zerfa and Brooks, 1996b). Thecontinuous 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 dominantdrop breakage mechanism) and reaches its final dynamicequilibrium value, at approximately 30min. The evolutionof DSD is shown inFig. 9. Initially, the volume probability

0 30 60 90 120 150 1800.00

250 rpm ( sim) 350 rpm ( sim) 500 rpm ( sim)650 rpm ( sim)

Fig. 10. Calculated and experimentally measured distributions ofVCM droplets in water at different agitation rates (monomervolume fraction= 0.1; CPVA (72.5% degree of hydrolysis) = 0.03%;temperature= 55◦C).

density function of VCM is broad. However, as the agitationcontinues, it becomes narrower and shifts to smaller sizes.The predicted steady-state DSD (at 120min) is in excellentagreement with the experimentally measured one (black dis-crete points). It is clear that the proposed model is capableof predicting satisfactorily the dynamic evolution of VCMdistribution 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 VCMdroplets in the aqueous dispersion. All other experimentalconditions were similar to those ofFig. 8, except the con-centration of the PVA stabilizer, which was 300ppm (Zerfaand Brooks, 1996a). The discrete points represent the exper-imental measurements while the continuous lines the modelpredictions. As can be seen, as the agitation rate increasesthe DSD shifts to smaller sizes and becomes narrower dueto the increased drop breakage rate. In all cases, the modelresults are in very close agreement with the experimentaldata. An additional comparison study was carried out for aVCM dispersed volume fraction, of 0.2. It was found thatthe VCM droplet distribution shifted to larger sizes as themonomer hold-up increased. Again, simulation results werein excellent agreement with experimental measurements onDSD.Subsequently, experimental measurements on the average

particle size and PSDwere compared with model predictionsfor the free-radical suspension polymerization of VCM. Theexperimental data for the PVC system were provided byATOFINA. The experiments were carried out in a 30L batchreactor, using 40% v/v VCM in water. The polymerizationtemperature was set at 56.5◦C while the agitation speedremained constant at 330 rpm.Fig. 11 depicts the variation of the volume mean diam-

eter with respect to polymerization time. The continuousline represents the simulation results and the discrete pointsthe experimental measurements. Initially, the mean diameter

0 50 100 150 200 250 30060

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= 330 rpm).

shifts to smaller values due to the dominant drop breakagemechanism. Subsequently, the drop breakage rate is reducedwhile the drop coalescence rate increases because of the in-creased viscosity of the dispersed phase. Thus, the meanparticle diameter increases until a monomer conversion ofabout 75%. After this point, the drop coalescence rate ceasesand the PSD remains almost constant. It is apparent that thepresent model predicts very well the dynamic behaviour ofthe PSD for the free-radical suspension polymerization ofVCM. In Fig. 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 invery good agreement with the experimental measurements.It should be pointed out that all the simulation results onVCM suspension polymerization (i.e., for both reactive andnon-reactive cases) were obtained using the same values ofthe model parameters (seeTable 1).Styrene suspension polymerization. The dynamic evolu-

tion of styrene DSD in aqueous dispersions was experimen-tally studied byYang et al. (2000). Fig. 13 illustrates thedynamic evolution of the Sauter mean diameter of styrenedroplets for two different monomer volume fractions. Thetemperature was kept constant at 25◦C, the agitation speedwas 350 rpm, while 500ppm PVA were added to the aque-ous phase for the stabilization of the dispersion. The PVAhad a degree of hydrolysis equal to 88%, while its molecu-lar weight varied between 30,000 and 50,000g/mol. The fi-nal DSD at dynamic equilibrium was attained after 150minof stirring. Apparently, the model predicts fairly well thedynamic evolution of the Sauter mean diameter of styrenedroplets, as well as the effect of monomer volume fraction.In Fig. 14, the time evolution of DSD is depicted for thecase of styrene volume fraction of 0.1. It is evident that themodel predictions on DSD are in very good agreement with

0.014Conversion 55%

Conversion 65%

0 100 200 300 4000.000

0.014Conversion 75 %

0 100 200 300 400

Conversion 83%

Fig. 12. Predicted and experimentally measured distributions of PVC particles at four different conversion levels: 55%, 65%, 75% and 83% (experimentalconditions asFig. 11).

0 50 100 150 200 25080

Time (min)

ϕ = 0.05 ( sim)ϕ = 0.10 ( sim)

Fig. 13. Dynamic evolution of calculated and experimentally measuredSauter mean diameter of styrene droplets at two different monomer volumefractions (non-reactive case: temperature=25◦C; agitation rate=350 rpm;CPVA (88% degree of hydrolysis) = 0.05%).

experimental measurements (discrete points).Fig. 15 illus-trates the effect of the agitation rate on the steady-state DSDof the styrene droplets. The operating conditions were as inFig. 13, while the monomer volume fraction was 0.1. As inthe case of the VCM dispersion, the mean size of styrenedroplets decreases with the agitation rate while the DSD be-

50 100 150 200 250 3000.000

5 min ( sim) 30 min ( sim) 120 min ( sim)

Fig. 14. Dynamic evolution of calculated and experimentally measureddistributions of styrene droplets in water (non-reactive case: experimentalconditions asFig. 13; monomer volume fraction= 0.1).

comes narrower. In all cases, the simulation results are invery good agreement with the experimental data that clearlyunderlines 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 suspensionpolymerization of styrene. More specifically, the effects of

0 50 100 150 200 250 300 3500.000

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 wereinvestigated for the expandable PS suspension polymer-ization process. Experimental measurements on PSD weretaken from the work ofVillalobos et al. (1993). The free-radical styrene suspension polymerization was carried outin a 1-gal reactor vessel. The dispersed monomer volumefraction was 0.4. The polymerization took place at 105◦C inthe presence of 1,4-bis(terbutyl peroxycarbo) cyclohexane(TBPCC) bifunctional initiator. The initiator concentrationwas 0.01mol/L-styrene in all the experimental cases, whiletricalcium phosphate (TCP) was used as surface-active agentat three different concentrations (i.e., 7.5, 5.0 and 3.5 gr/L).In Fig. 16, experimental measurements and simulation re-

sults on PSD are shown for three different addition policiesof n-pentane into the reactor. More specifically, 7.5% w/wof n-pentane with respect to the styrene mass was added

200 400 600 800 1000 1200 1400

200 400 600 800 1000 1200

0 200 400 600 800 1000 12000.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

(a) (b) (c)

Fig. 16. Predicted and experimentally measured distributions of EPS particles for differentn-pentane addition policies. (a) 7.5% w/wn-pentane (wrtstyrene) at� = 0%; (b) 7.5% w/wn-pentane (wrt styrene) at� = 50% ; (c) in the absence ofn-pentane (temperature= 105◦C; dispersed phase volumefraction= 0.4, [Io] = 0.01mol TBPCC/L-styrene;[TCP] = 7.5g/L).

to the system at three different conversion levels (i.e., 0, 50%and 100%). InFig. 16a, model results (continuous lines)are compared with experimental data (dash lines) on PSDfor 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% monomerconversion. The last case (seeFig. 16c) corresponds to theaddition ofn-pentane at the end of polymerization. As canbeen seen, for all cases, there is a close agreement betweenexperimental and simulation results on PSD, indicating thepredictive capabilities of the model for the free-radical sus-pension polymerization of styrene. It should be noted that,in the presence ofn-pentane, the EPS particles are moreuniform while the PSD becomes narrower.Villalobos et al. (1993)also investigated experimentally

the effect of the stabilizer concentration on PSD. Morespecifically, experiments were carried out at different TCPconcentrations, in the presence of 7.5% w/wn-pentane,added at 50% monomer conversion. InFig. 17the predictedand experimental PSDs of the EPS particles are shown forthe three different concentrations of the surface-active agent.As can be seen, as concentration of the surface-active agentdecreases (i.e., the interfacial tension increases) the PSD be-comes broader and shifts to larger sizes. Apparently, there isa very good agreement between calculated and experimentalmeasurements on the volume probability density function.

5. Conclusions

A comprehensive population balance model coupled witha system of differential equations governing the conserva-tion of the various molecular species present in the systemhas been developed to describe the dynamic evolution ofthe DSD/PSD in free-radical suspension polymerization re-actors. The fixed pivot technique (FPT) was employed forsolving the PBE. The robustness of the numerical method

0 500 1000 1500 20000.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

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 andexperimentally measured distributions of EPS particles at three differentquantities of surface-active agent (TCP) (experimental conditions as inFig. 16).

was examined in regard with its convergence character-istics and accuracy in terms of the mass conservation ofthe monomer, initially loaded into the reactor. The predic-tive capabilities of the model were demonstrated via thesuccessful simulation of experimental measurements onDSD/PSD and the average droplet/particle diameter for bothnon-reactive liquid–liquid dispersions and the free-radicalsuspension polymerization of styrene and VCM.

Notation

A(D, t), Av(D, t) number and volume probability densityfunctions, 1/m

CPVA concentration of surface-active agent,Kg/m3

D diameter, mDP degree of polymerization of the PVA

stabilizerE elasticity modulus, Kg/ms2

g(V ) breakage rate, 1/sk(V, U) coalescence rate, m3/skb, kc model parametersL macroscale of turbulence, mMw weight average molecular weight,

Kg/kmoln(V, t) number density function, 1/m6

[n] intrinsic viscosity, m3/KgNE number of discrete elementsNi number of particles having volume

equal to xi per reactor unit volume,1/m3

Nda, Nsa number of daughter and satellitedroplets per breakage events

r volume ratio of daughter over the satel-lite drops

Re Reynolds numberSnsa model parametert time, su(V ) number of droplets formed by a break-

age of a droplet of volumeVu(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 satellitedrops, 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 unitmass, m2/s3

� microscale of turbulence, m�b, �c breakage and coalescence efficiencies� viscosity, Kg/ms� kinematic viscosity, m2/s density, Kg/m3

� interfacial tension, Kg/s2

�da, �sa standard deviation of the distributionfor daughter and satellite drops

� dispersed phase volume fraction�p volume fraction of the polymer in the

dispersed phase� monomer conversion�b, �c breakage and coalescence frequencies,

Subscripts

c continuous phase

d dispersed phasem monomerp polymers suspension systemw water

Acknowledgement

The authors gratefully acknowledge ARCHEMA (ex-ATOFINA Chemicals) for providing the experimental datafor PVC suspension polymerization.

Appendix A

The free-radical polymerization of vinyl monomers ingeneral includes the following chain initiation, propagation,

chain transfer to monomer and bimolecular termination re-actions (Kiparissides et al., 1997):decomposition of initiators

kd,i−→2R•, i = 1,2, . . . , Nm,

chain initiation

R• + Mkp−→ 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 + Pmktd−→ Dn + Dm,

inhibition of ‘ live’ radical chains

Pn + Zkz−→ Dn + Z•,

whereIi, R•, M andZ denote the initiator, primary radicals,monomer and inhibitor molecules, respectively, andPn andDn, 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 forma separate phase (i.e., the polymer-rich phase). Thus, theelementary reactions presented above take place in both themonomer-rich and polymer-rich phases (Kiparissides et al.,1997). Additional details, regarding the kinetic modelingof 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 thesystem, etc.), can be found in the publications ofKiparissideset al. (1997, 2004)andKotoulas 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 bedefined as (Krallis et al., 2004)

�k =∞∑

nkPn, �k =∞∑

nkDn. (A.1)

Accordingly, one can easily derive the corresponding mo-ment rate functions:

‘Live’ polymer moment rate equations

r�k=

Nm∑k=1

2fkkdkIk + kpM

)�r − �k

+kfmM(�0−�k)−(ktc+ktd)�k�0−kzZ�k. (A.2)

‘Dead’ polymer moment rate equations

r�k= kfmM�k + 1

k∑r=0

)�r�k−r

+ ktd�k�0 + kzZ�k. (A.3)

The number- and weight-average molecular weightscan be expressed in terms of the molecular weight of themonomer,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 bythe following expression, assuming that the long chain hy-pothesis holds true (i.e., the monomer is mainly consumedvia the propagation reaction):

dt= kp

M0�0. (A.5)

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A generalized population balance model for the prediction of particle size distribution in...

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