mathematical modeling of emulsion copolymerization reactors: experimental validation and application...

Mathematical Modeling of Emulsion Copolymerization Reactors:Experimental Validation and Application to Complex Systems

Enrique Saldıvar and W. Harmon Ray*

Department of Chemical Engineering, University of WisconsinsMadison, Madison, Wisconsin 53706

In this paper a mathematical model for emulsion copolymerization of several monomers in tankreactors, proposed recently by the authors (Saldıvar et al., 1996), is first validated withexperimental data obtained in our laboratory. Then, a systematic methodology for modelvalidation is presented and illustrated with application to the system methyl methacrylate/styrene. The method of numerical solution of the model is also discussed and shown to be crucialfor the successful application of the model to engineering and fundamental problems. Finally,the generality of the model is demonstrated by applying it to three illustrative problems: (i)simulation of complex dynamics in a flowsheet, (ii) simulation and comparison to experimentaldata for a semicontinuous system with a complex kinetic scheme, and (iii) start-up optimizationin a flowsheet.

Introduction

Emulsion copolymerization processes are of signifi-cant industrial importance. Since these processes usewater instead of solvents, they are also very attractivefrom the environmental point of view. Despite the factthat these processes have been studied for more than50 years, there are still aspects which are not completelyunderstood due to the complexity of these systems.Many models have been proposed in the literature toexplain specific phenomena or tailored to explain spe-cific systems (cf. Storti et al., 1989; Forcada and Asua,1990; Guillot, 1993), but very few of them are of ageneral nature and useful for design and analysis ofemulsion polymerization processes in a comprehensiveway (Hamielec et al., 1987; Dougherty, 1986a,b; Rich-ards et al., 1989). It is not the purpose of this paper tomake a review of previous modeling work published inthe literature; for such a review we refer the reader toSaldıvar et al., 1997. In order to have an effectivereaction engineering approach for the analysis of emul-sion copolymerization systems, it is necessary to developmodels and strategies that overcome the present defi-ciencies in the field. A review of the literature (Saldıvaret al., 1997) revealed three areas of opportunity:1. There is a need for a comprehensive and coherent

framework for modeling of these systems.2. There is a need for implementing model solutions

in a manner that covers all types of operation (batch,semibatch, and CSTR) and the simulation of flowsheetsof CSTR’s.3. The developed models should be systematically

validated by experimentation.One way of satisfying these needs is to build a model

having the following characteristics:(a) For industrial applications it should be able to

model batch, semibatch, single CSTR’s, and CSTR’s inflowsheets.(b) It should be based on a complete free-radical

kinetic scheme including reactions of inhibition, reversepropagation, transfer to monomer, CTA, and polymer,terminal, and internal double-bond polymerization andscission, besides the standard reactions of initiation,propagation, and termination. A complete kinetic de-scription is necessary to realistically study those sameeffects of process variables on the molecular architectureof the polymer.

(c) It should describe the leading moments of theMWD and branching distribution as well as the fullparticle size distribution (PSD). All these are importantvariables that characterize the polymer or latex quality.(d) For practical reasons, it should be applicable to

any number of monomers.A detailed model built along these lines has been

recently presented by Saldıvar et al., 1997, whichintends to cover many of the requirements for engineer-ing applications and includes updated knowledge onthese systems; however, no systematic validation hasbeen presented yet for that model.Systematic validation of models with experiments is

necessary in order to have confidence in their predictivepower; however, a review presented in Saldıvar et al.,1997, concludes that many of the previous studies ofemulsion copolymerization have been focused on veryspecific aspects of these systems; therefore, systematicexperimental programs oriented to the validation of themodels formulated are needed, especially if they areintended for general application to a number of co-polymerization systems. Some guidelines to conductthese studies and which, as a whole, constitute a novelapproach in this problem are as follows:(a) Statistical design of experiments should be used

to explore in a comprehensive way the experimentalspace and to test the model when interactions of severalvariables are present.(b) Experimental measurements should be taken, as

much as possible, for all the relevant measurablequantities that the model can predict, for example, thetime evolution of particle size distributions, total con-version, copolymer composition, and molecular weightdistributions.(c) The systems selected for study should be well

documented in the literature, so their kinetic andphysical parameter values come, as much as possible,from independent studies. Also, only a subset of theexperimental data should be used for the fitting ofunknown or uncertain parameters, while the rest of thedata not used for parameter fitting can be utilized totest the predictive capability of the model.For models intended for general applicability the

previous guidelines should be applied to a number ofcomonomer systems.In this paper the systematic validation of the Saldıvar

et al. model and its solution, implementation, and

1322 Ind. Eng. Chem. Res. 1997, 36, 1322-1336

S0888-5885(96)00464-2 CCC: $14.00 © 1997 American Chemical Society

application to complex engineering systems are ad-dressed. For the validation aspects, the purpose of thispaper is to illustrate the use of the guidelines givenabove by their application to one copolymer system. Wedo not intend here to present a thorough and completevalidation of the model, as this is part of a large researchprogram that will provide material for future papers.Once the model is applied to a number of systems andenriched with the feedback obtained from these studies,the model can be regarded as validated.The organization of the paper is as follows. First, the

methodology for systematic experimental validation ofthe model is illustrated using the system methyl meth-acrylate/styrene (MMA/S) as an example. Then thestructure of the model is summarized, and its numericalsolution is discussed, addressing the problems of con-vergence of the numerical solution and discretizationof the PSD and MWD equations, as well as the specialproblems posed by inserting these equations in flow-sheets. Finally, three examples are presented in whichdiverse systems are studied to analyze complex dynam-ics in a simple flowsheet, semibatch operation withcomplex kinetics, and optimization of the startup of twoCSTR’s to obtain constant composition copolymers.

Experimental System

Experiments have been performed in our laboratoryin batch and CSTR for the methyl methacrylate/styrenesystem. This is a well-documented system for whichmost of the kinetic and physical parameters are avail-able in the literature. The experimental setup is thesame used by Paquet and Ray, 1994, for his tankexperiments. The core of the system is a glass reactorwith a jacket for heating/cooling. Two different vesselsof 500 mL and 1 L can be used, and the system is flexibleso that it can be easily configured for batch or continu-ous operation.The experimental system is presented in Figure 1.The temperature control is achieved by means of a

temperature water bath which is maintained about 1°C above the set point. An on/off temperature controller(Omega CN-9000) receiving a signal from a thermo-couple, operates a solenoid valve to inject cooling waterat 15-20 °C into the jacket when the temperature ofthe reactor exceeds the set point. This system allowsfor precise temperature control ((0.5 °C of reactor setpoint).

Figure 1. Emulsion copolymerization reactor.

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1323

Two systems are needed for continuous feeding of twogroups of chemicals: (i) the aqueous solution of initiatorand emulsifier and (ii) the monomer mixture. Each oneof these feeding systems consists of a reservoir tank,which is maintained with a nitrogen atmosphere, aperistaltic pump, and a rotameter. Emulsion is pumpedout of the reactor through a glass heat exchanger thatcontrols the temperature of the emulsion and is sent toa digital density meter (DMA 40 Paar Scientific Ltd.),jacketed flow tensiometer, and then waste. An oxygensensor has been added for the CSTR experiments toguarantee that the concentration of oxygen in thesystem is sufficiently low (2-3 ppm) before and duringreaction. For batch operation the initiator solution isplaced in a glass reservoir (initiator bomb) on top of thereactor and kept there until the reaction is to be started.A complete description of the experimental setup andits operation can be found in Paquet and Ray, 1994.Materials and Methods. For batch experiments,

monomers, water, and emulsifier were added to thereactor and the initiator bomb was placed on top of thereactor before this was closed. Nitrogen was spargedin the initiator solution and the reactor contents forseveral hours prior to the start of the reaction. Whenthe reactor mixture reached the set-point temperature,the initiator solution was loaded into the reactor andtime zero was marked. Two samples were taken every5 or 10 min for gravimetric and particle size analysis.For the CSTR operation, the monomers, water, and

emulsifier were loaded into the reactor, which then wasclosed. Nitrogen was sparged in the reactor contentsuntil the oxygen concentration in the reactor wasnegligible according to the oxygen sensor (around 2-3ppm). When the reactor reached the set-point temper-ature, time zero was marked and the flow of monomersand the aqueous solution of initiator and emulsifier wasstarted. Detailed operating procedures are given inPaquet, 1993.Styrene monomer from Aldrich (99% purity) (all

percentages in weight) was washed with a NaOHsolution to remove the inhibitor (tert-butylcathecol);methyl methacrylate supplied by Rohm and Haas wasdistilled under vacuum at 29 in. Hg to remove theinhibitor (hydroquinone) and other impurities. Potas-sium persulfate (Aldrich, 99% purity) was used as theinitiator and sodium dodecyl sulfate (BDH Laboratories,99%) as the emulsifier; both were used without furtherpurification. Nitrogen (Liquid Carbonic, 99.999%) wasused for oxygen purging and for keeping an inertatmosphere in the reactor. Deionized water purified bya Millipore filtration system was used in all the reac-tions. The inhibitor to stop the reaction in the sampleswas hydroquinone (Aldrich, 98%). Only in the CSTRreactions was sodium persulfate (Aldrich, 98%) used asthe initiator.Gravimetric analysis was used for measuring conver-

sion. Particle size was determined by dynamic lightscattering, also known as photon correlation spectros-copy (PCS). These analyses were performed off-line ina Malvern 4700c light scattering apparatus. Details ofthe measurement and data analysis can be found inSaldıvar, 1996. The autocorrelation function producedby the apparatus was analyzed by means of the CON-TIN software (Provencher, 1982a,b) including Mie scat-tering factors to recover the moments of the particle sizedistribution. The weight-average diameter was ob-tained from the CONTIN output as the ratio of the first

to the zeroth moment of the recovered particle massdistribution, having as an independent variable thediameter.The composition of the copolymer was determined by

proton nuclear magnetic resonance (NMR) in a WP-200NMR spectrometer at 200 MHz on purified polymersamples.The weight conversion x is defined as follows:

where P(t) and M(t) are the total polymer and totalmonomer, respectively, present in the reactor at time t.The gravimetric calculations of conversion for the CSTRcase, based on the weight of dry solids, required acorrection for initiator and emulsifier. The initiatorconcentration at time t was assumed to follow thewashout equation for a CSTR. In terms of the weightfraction of initiator wI, the equation is

where the superindex 0 and f refer to initial and feedconditions, respectively, and θ is the residence time. Itwas also assumed that the amount of initiator incorpo-rated in the polymer chains is negligible.Batch Experiments. Two groups of batch experi-

ments have been run for the methyl methacrylate/styrene system for the purpose of collecting data formodel validation:(a) A 23 factorial using as variables reaction temper-

ature, initiator concentration, and emulsifier concentra-tion. Table 1 shows the design conditions. In all theexperiments of the design, the initial molar ratio ofmonomers was 50/50 (MMA/S) and the solids contentin the emulsion 20 wt % (based on 100% monomerconversion). A replicate (run 2R) of one of the factorialexperiments (run 2) was run to test the system repro-ducibility.(b) Two experiments at different initial monomer

molar ratios: run 9 (25/75 MMA/S) and run 10 (75/25MMA/S). The rest of the conditions for these experi-ments were the same conditions used for run 4 (50/50MMA/S).In all the experiments 0.800 kg of DI water and 0.200

kg of monomer were used in the 1 L reactor. Anexample recipe is shown in Table 2.

Table 1. Factorial Experimental Design: Batch Styrene/Methyl Methacrylate System

label temp (°C) [I] (mol/m3‚aq) [S] (mol/m3‚aq)

run 1 60 2.77 30run 2 60 2.77 10run 3 60 1.85 30run 4 60 1.85 10run 5 50 2.77 30run 6 50 1.85 10run 7 50 1.85 30run 8 50 2.77 10run 2R 60 2.77 10

Table 2. Example Recipe Run 1: Batch MMA/S

substance mol wt amount (kg)

styrene 104.15 0.102methyl methacrylate 100.12 0.098DI water 18.01 0.800potassium persulfate 270.33 6 × 10-4

sodium dodecyl sulfate 288.38 6.9 × 10-3

x )P(t)

P(t) + M(t)(1)

wI ) wIf + [wI

0 - wIf] exp-t/θ (2)

1324 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

Figure 2 displays the conversion vs time curves forthe replicated experiments showing the reproducibilityof the system.CSTR Experiments. Experiments were run with

the CSTR system for further model validation. Theywere run for more than 10 residence times to make surethat a steady state could be reached. The reactions wererun in the 0.5 L reactor, and the initial conditions werethose given in Table 3 which correspond to the composi-tion of the monomer feed (50/50 MMA/S) and of theemulsifier solution (40 mol/m3‚aq) with a full reactor.Notice that no initiator was initially loaded into thereactor to avoid an early start of the reaction.The conditions of the feed are those given in Table 4.

Mathematical Model

The model has been presented in detail elsewhere(Saldıvar et al., 1997); here only its main featuresnecessary for the discussion are highlighted. The coreof the model is the particle size distribution description.The main assumptions employed for the model are asfollows:(a) The particle size distribution (PSD) is independent

of the molecular weight distribution (MWD). On theother hand, the MWD depends on the PSD.(b) A complete free-radical kinetic scheme based on

the one proposed by Arriola, 1989, has been imple-mented.(c) The partitioning of the monomers in the different

phases (particles, aqueous phase, and monomer drop-lets) is assumed to reach instantaneous thermodynamicequilibrium. Also, the concentration of monomers in theparticles is assumed to be independent of particle size.(d) A pseudohomopolymer approach was used (cf.

Storti et al., 1989).

(e) Perfect mixing of the reactor contents was as-sumed.A population balance equation (PBE) with polymer

mass as the internal coordinate is written for thedistribution function F(m,t) dm, which represents thenumber of particles present per liter of water andhaving a mass of polymer between m and m + dm attime t. This results in the partial differential equation(3) in polymer mass and time for the PSD

where Vw is the volume of water,Q represents mass flowrate, ww is mass fraction of water, Fw is the waterdensity, and the superindex f refers to feed conditions.Its boundary condition is given by

where Vaq is the volume of the aqueous phase, am is thesurface area of a micelle,M is the micelle concentration,the kmmi and kmmR are entry rate coefficients for type iradicals with aqueous phase concentration [Pi]w andinitiator radicals with aqueous phase concentration [R]w,respectively. Only micellar nucleation has been in-cluded in this application by neglecting the contributionof homogeneous nucleation. The initial condition isgiven by

Equivalent equations have been derived by Rawlingsand Ray, 1988, and by Storti et al., 1989.It has been claimed (Lichti et al., 1982) that MWD

effects due to compartmentalization of the growingchains in particles are not correctly accounted for bymodels using only one internal coordinate for growingchains (or “singly distinguished” particles), as in themodeling approach used in this work; they proposeinstead the use of a “doubly distinguished” particlemodel. However, it is not clear if this approach, withits added complexity, is worth the effort. It is recentlybecoming clear that the arguments given by Lichti etal. apply to a limited class of systems: those havingtermination by combination as the dominant mode oftermination and an average number of radicals perparticle close to 0.5 (see Gilbert, 1995). We believe thatby using singly distinguished particles a reasonableapproximation for the calculation of the MWD is ob-tained, given the fact that the model is applied to a verylarge population of particles. Furthermore, the com-plexity added by adopting the approach of Lichti et al.would be considerable, especially when taking intoaccount that the selected model is already a complexone.The average number of radicals in particles is as-

sumed to be a function of the polymer mass and givenas an algebraic relationship in terms of Bessel functionsfrom the classical solution to the Smith-Ewart equationby Stockmayer-O’Toole (O’Toole, 1965). The model iscomplemented by several chemical species balancesperformed over the total volume of the reaction, which

Figure 2. Reproducibility of batch system.

Table 3. Initial Load of Reactor for CSTR MMA/S

substance mol wt amount (kg)

styrene 104.15 0.069methyl methacrylate 100.12 0.066DI water 18.01 0.354sodium persulfate 238.10 0.0sodium dodecyl sulfate 288.38 0.004

Table 4. Feed Conditions for CSTR MMA/S

substance mass flow (kg/s) remarks

styrene 5.50 × 10-5

methyl methacrylate 5.28 × 10-5

DI water 2.83 × 10-4

sodium persulfate 2.70 × 10-6 40 mol/m3

sodium dodecyl sulfate 3.27 × 10-6 40 mol/m3

∂F(m,t) Vw

∂t+∂VwF(m,t)

dmdt

∂m) Ffww

f Qf/Fw - FwwQ/Fw(3)

VwF(m,t)dm

dt |mm+

) FB )

VaqNA(∑i)1

c

amkmmiM[Pi]w + amkmmR

M[R]w) (4)

F(m,t)0) ) F0(m) (5)


result in a set of ODE’s and algebraic equations.Monomer balances and polymer balances for c compo-nents yield 2c ODE’s. Balances for initiator, emulsifier,CTA, inhibitor, water, and the total reaction mass arerepresented by 1 ODE each. Monomer partitioningresults in a nonlinear algebraic system of dimension 3for the partition coefficients approach or dimension 2cfor the Flory-Huggins formulation (cf. Guillot, 1980).The distribution of radical types yields one linear systemof dimension c for each one of the phases: particles andaqueous phase. The aqueous phase radical balanceunder the quasi-steady-state assumption (QSSA) givesonly one nonlinear algebraic equation for the totalaqueous phase radical amount. Other aqueous phasebalances for different polymeric radicals (primary,monomeric, and critical length) under the QSSA aregiven in terms of explicit algebraic relationships. Theemulsifier adsorbed on particles is assumed to follow aLangmuir type adsorption isotherm:

where Sp is the total surface area of particles, SF is thefree emulsifier in the aqueous phase, and Γ∞ and bs areparameters. The value of SF determines the concentra-tion of emulsifier in the aqueous phase [S]w andtherefore the presence or not of micelles through thefollowing equations:

where [S]wcmc is the critical micelle concentration of the

emulsifier, aem is the area of a micelle covered by amolecule of emulsifier, and H is the Heaviside or unitstep function.For MWD a pseudohomopolymer model based on

population balances is written. For live polymer, popu-lation balance equations are derived for the distributionNnl,b(m,t) dm, which represents the number of growing

radicals of length l and branching index b present perliter of water in particles having n live radicals and amass of polymer between m and m + dm at time t.For dead polymer, equations are presented for the

distribution Dnl,b(m,t) dm, which represents the number

of dead polymer chains of length l and branching indexb present per liter of water in particles having n liveradicals and a mass of polymer betweenm andm + dmat time t.The numerical solution of the complete population

balances for the MWD is at present a formidable task,and it is not pursued. Instead, in order to reduce thedimensionality of the problem, only the calculation ofmoments of the MWD is performed.Applying the method of moments to most of the

discrete internal coordinates, the total moments of livepolymer are defined as

and for dead polymer as

Capital letters are used to denote the order of themoments over the corresponding variable. Some of themechanisms give closure problems when moments aretaken over the index n for the live polymer PBE;therefore, partial moments in which this index is keptintact are also defined as:

Applying the above moment definitions to the com-plete PBE’s, partial differential equations (PDE’s) arederived for those moments. The live polymer populationis solved using the partial moments given by eq 11. Inthe resulting PDE, the polymer mass derivative termis neglected based on an order of magnitude analysis(Min, 1976) and then the QSSA is applied, giving thefollowing equation for the live polymer moments:

with boundary condition:

The terms rj on the right-hand side correspond to thedifferent mechanisms that change the classification ofradicals, including all the chemical reactions as well asother physicochemical phenomena such as entry anddesorption of radicals. A complex free-radical kineticscheme has been modeled that includes chain transferto polymer, terminal and internal double-bond poly-merization, scission, inhibition, and reverse propaga-tion, and the traditional free-radical kinetic scheme withinitiation, propagation, termination, and chain transferto monomer and to chain-transfer agent.It has been shown that there are computational

advantages (Arriola, 1989) in solving equations formoments of bulk polymer (live and dead chains) insteadof those directly derived for only dead polymer moments;therefore, equations have been obtained for the mo-ments of bulk polymer by addition of the live and deadcontributions. This results in equations for the mo-ments Ω of bulk polymer defined by

For estimation of polymer properties only the 0thmoments (S ) 0) on the number of radicals (that is, thesummation over particles with all possible number ofradicals) and the 0th moments over the continuousvariable m (summation over all the particle sizes) arerequired. To this end, integral moments are defined:

Sa )SpΓ∞bsSF/Vaq

1 + bsSF/Vaq(6)

M′ )([S]w - [S]w

cmc)NAaemam

(7)

M ) M′H(M′) (8)

µSG,H(m,t) ) ∑

l)0

∞

∑b)0

∞

∑n)0

∞

lGbHnSNnl,b(m,t) (9)

λSG,H(m,t) ) ∑

l)0

∞

∑b)0

∞

∑n)0

∞

lGbHnSDnl,b(m,t) (10)

µnG,H(m,t) ) ∑

l)0

∞

∑b)0

∞

lGbHNnl,b(m,t) (11)

∂µnG,H(m,t) Vw

∂t+

∂dm

dtVwµn

G,H(m,t)

∂m) ∑

j)1

Mech

rj ≈ 0

G, H ) 0, ..., ∞ n ) 1, ..., ∞ (12)

VwµnG,H(m,t) Vw

dm

dt |mm+

)

VaqNA(∑i)1

c

kmmipiw)amM∑

l)0

∞

lF[Pl]w + [R]wδF,0δH,0δn,1

(13)

ΩSG,H ) λS

G,H(m) + µSG,H(m) (14)


The MWD is minimally described by the first threeΛ0G,H moments over chain length (for all possible

branches), represented by the values GH ) 00, GH )10, and GH ) 20. A possible description of the branch-ing distribution can be obtained by the moments GH )01 and GH ) 11 (Graessley, 1968). To include theterminal double-bond polymerization mechanism, anadditional PDE is obtained for the moment:

in which d represents the number of terminal doublebonds per polymer chain.Mathematically the MWDmodel results in five PDE’s

in time and polymer mass for combined moments ofchain length and branching index for the bulk polymer(for fixed zeroth moment on the number of radical index)and an additional PDE for chains with terminal doublebond. Additionally, for each number of radicals (1, 2,..., ∞) there is a set of five PDE’s for combined momentsof chain length and branching index for the live polymerpopulation. This last set is finally reduced to oneconsisting of linear algebraic equations in the livepolymer partial moments when the time and polymermass derivatives are assumed negligible.

Numerical Solution

The main difficulty for the numerical solution of themathematical model is presented by the PDE’s describ-ing the PSD and the moments of bulk polymer. Thesolution of these equations is difficult and has beenreviewed by Rawlings, 1985, who concluded that anefficient numerical solution can be obtained by using asimple version of orthogonal collocation on finite ele-ments on a moving size domain (he used a particle birth-time domain) in which one polynomial is used torepresent each generation of particles produced in thereactor (for oscillatory steady states). This technique(translated to the polymer mass domain) works well torepresent complex dynamics (e.g., limit cycles for con-version in a CSTR); however, it is not accurate enoughto generate detailed profiles of the PSD, as it results innumerical oscillations in the profile, due to the steepfronts that the solution develops. An enhancement ofthe Rawlings technique is necessary for an accuratedescription of the PSD: each generation of particles canbe represented by several polynomials linked by contin-uity conditions. In this work, the enhanced procedurewas applied in two different modes to solve the PDE’swhich constitute the present model. Its essential fea-tures are illustrated by application to the PSD equation.Following the Rawlings approach, the physical do-

main can be divided into K noncontiguous subdomains.For the kth subdomain [mmink(t),mmaxk(t)] a moving massdomain variable uk is defined as

in which mmaxk(t) and mmink(t) are the maximum andminimum polymer mass in particles for the kth sub-domain, respectively.

The distribution function was transformed to facilitatethe calculation of integrals of the distribution functionover the new domain. The function Y(uk,t) is definedsuch that

and defining dimensionless variables:

where M0 and θ are the characteristic micelle concen-tration and characteristic time, the dimensionless ver-sion of the transformed PDE that is obtained is

where g ) dm/dt is the rate of growth of particles ofpolymer mass m. Notice that rigorously the definitionof K noncontiguous domains results in K PDE’s, eachone requiring a proper boundary condition. The physi-cal justification for this representation is that one PDEis associated with each one of the sections (generations)of the PSD. The proper boundary conditions are

The boundary condition for k ) K occurs only at theleftmost subdomain (see Figure 5), the one containingthe micellar size, and it is different from zero only whenmicelles are present. Two additional ODE’s for eachsubdomain are needed for mmaxk(t) and mmink(t):

The PDE’s in the form of eq 22 were discretized usingorthogonal collocation on finite elements on the movingmass domain and using several polynomials (elements)to represent each contiguous section (subdomain) of thedistribution (Carey and Finlayson, 1975). The dis-cretized system of equations plus the complementarybalances and thermodynamic relationships resulted ina differential-algebraic system of equations (DAE)whose solution was implemented, after writing theequations in dimensionless form, in the frame of thesimulation package POLYRED (Ray, 1997). The solu-tion of the DAE system was performed using DDASSL(Petzold, 1985), the integrator implemented in thepackage. The typical simulation time of a batch co-polymerization is 3-4 min in an HP 712/100 work-station. If the molecular weight moments are not

ΛSG,H ) ∫0∞(λSG,H(m) + µS

G,H(m)) dm (15)

ν00,0 ) ∑

n)0

∞

∑l)0

∞

∑b)0

∞

d[Dnl,b + Nn

l,b] (16)

uk )m(t) - mmink

(t)

mmaxk(t) - mmink

(t)0 e uk e 1 (17)

∫Y(uk,t) duk ) ∫F(m,t) dm (18)

F ) Y/M0 (19)

u ) t/θ (20)

∂F∂u |uk + θ

mmaxk- mmink

[g - m′mink -

u(m′maxk - m′mink)]∂F∂uk |t ) θF

m′maxk - m′minkmmaxk

- mmink

+

θ[F f - F ]Rf - θF (uk,t)dgdm

0 e uk e 1 (21)

F (mm,t) ) (mmaxk- mmink

)FB

gM0Vw

k ) K

(22)F (mm,t) ) 0 k ) 1, 2, ..., K - 1 (23)

dmmink

dt) g(mmink

) (24)

dmmaxk

dt) g(mmaxk

) (25)


included in the calculation, the simulation takes 1-2min in that computer.Convergence of the Solution. The numerical

scheme presented gives freedom in the selection of thenumber of elements for each generation of particles andin the location of the breakpoints (mesh). These vari-ables affect two different types of convergence: (i)integral or global which refers to the convergence of thePSD solution measured by quantities that depend onan integral fashion on the PSD, for example, the totalnumber of particles Np or monomer conversion x and(ii) local. Since the collocation procedures minimize theglobal error of approximation, it is possible to obtainglobal convergence on the integral properties of thedistribution using a relatively low number of elements/points even though the detailed distribution is notresolved with high accuracy (Villadsen and Michelsen,1978). Figure 3, which represents the time evolutionof Np for a batch copolymerization case, shows that Npreaches convergence for 80/2 equidistributed elements/points. For these conditions the detailed solution stillwould exhibit numerical oscillations.This is illustrated in Figure 4 in which the time

evolution of the PSD is shown for the numerical solutionusing 270/2 elements/points; the position of the fronthas converged, but even this high-density mesh cannotavoid the spurious oscillations.If one is only interested in the integral properties of

the distribution, one can tolerate some spurious numer-ical oscillations in the detailed profile in order to shortenthe computation time. On the other hand, if the exact

profile is important, as it is for mechanistic studies(Lichti et al., 1983), then one must resort to cumbersomespecial numerical techniques in order to get rid of thespurious oscillations.Finlayson (1992) compares several numerical tech-

niques which can be used to obtain nonoscillatoryprofiles in convection-dominated flows. These systemsrender PDE’s that are mathematically analogous tothose representing the PSD and the moments of theMWD in emulsion polymerization. He concludes thatmost of the successful methods for treating these kindsof problems introduce some form of numerical diffusionto control the growth of unwanted numerical artifacts;however, these methods smear the front, so a balancemust be reached between nonoscillatory solutions andfront smearing. A simple way of applying this conclu-sion in practice is to introduce a small artificial “diffu-sion” or second derivative term in the PDE equation (3)which would result in

where D is an artificial diffusion constant. In order toaccount for the relative size of the convection anddiffusion-like terms, a dimensionless quantity analogousto the Peclet number as Pe ) gm*/D is defined, whereg ) dm/dt andm* is a characteristic polymer mass. Thisrequires the introduction of an additional boundarycondition that does not alter the physical nature of thesolution:

Once the variable transformations defined above wereapplied to eq 26, an excellent approximation to thephysical solution with no significant numerical oscilla-tions was obtained using an equidistant mesh of 220/2elements/points with Pe ) 5× 107. The larger the valueof the Peclet number, the better the approximation tothe true solution but also the stiffer the numericalsolution of the equations; the value used for the Pecletnumber was selected by reaching a compromise betweenthe degree of approximation and the stiffness of thesolution. The comparison of the oscillatory and nonoscil-latory profiles is given in Figure 4. Steep profilesdevelop even before the end of the nucleation period andgrow in steepness at higher reaction times (conversions).Numerical Scheme for Different Types of Op-

eration. In the case of simulation of an emulsionpolymerization in a batch reactor or in a single CSTRwith no feed of particles and with micellar nucleation,the PSD will result, in general, in one or severalnoncontiguous generations which can be discretizedusing the scheme described above withK g 1 (see Figure5). This is the case previously solved by Rawlings, 1985.In the multireactor case, which is the most general

one, the PSD can have any shape, since feed and recycleof particles can occur. For this case we used the movingmass domain with K ) 1 at all times. In order to beable to simulate micellar nucleation at any time, themminK was kept fixed at the micellar nucleation size andonly the maximum polymer mass moves in a waydetermined by the rate of growth of the maximumpolymer mass particle. However, if the polymer mass

Figure 3. Convergence of the collocation procedure for thenumber of particles with a number of elements and two internalpoints.

Figure 4. Comparison of a PSD oscillatory profile with no frontsmearing and a nonoscillatory profile with front smearing. Thecurves from left to right correspond to increasing time for theevolution of the PSD.

∂F(m,t) Vw

∂t+∂(VwF(m,t)

dmdt )

∂m)

DVw∂2F(m,t)

∂2m

+ Ffwwf Qf/Fw - FwwQ/Fw (26)

F(∞,t) ) 0 (27)


in the feed of one reactor surpasses the maximumpolymer mass present in the reactor, the maximummmaxK is increased by a discontinuous jump. An il-lustration of this situation is presented in example 3.

Model Validation

The model was applied to simulate the experimentaldata obtained in our laboratory for the system MMA/S.Figure 6 shows experimental data measured by gravim-etry and simulation results for conversion vs time curvesfor 1/2 of the factorial at 50 °C; in Figure 7 thecorresponding results are shown for 1/2 of the factorialat 60 °C. The effects of initiator and emulsifier initialconcentrations predicted by the model are compared tothe experimental data, showing reasonable agreement.All of the parameters were taken from values reported

in the literature; except for those shown in Table 5. Thecomplete set of parameters used is reported in Saldıvar,1997. Among the parameters fitted were those thatdescribe the area aem of a micelle covered by theemulsifier and the adsorption isotherm of the emulsifier.Morbidelli et al. (1983) point out that the value of aemdepends on the ionic strength of the solution and thatit is erroneous to assume that aem ) aep. Its experi-mental value is generally unavailable for specific sys-tems. The adsorption of emulsifier to particles isreported by the same authors to be reasonably wellrepresented by a Langmuir isotherm, but the param-eters for specific systems are not readily available since

they have to be obtained experimentally for eachsystem. Entry rate coefficients were also fitted sincethey depend heavily on the model employed althoughrecent work (Maxwell et al., 1991) seems to be improv-ing this situation by using more fundamental formula-tions to measure these parameters independently. Also,for the gel effect, available models still lack predictivepower, especially for copolymerization systems; there-fore, two independent parameters were fitted for theglass region (diffusion-limited propagation) of the MMAgel effect correlation. The gel effect equations used werethose given in Table 6 in which Tc and Tk are temper-atures in centigrades and Kelvin degrees, respectively,x is the total weight conversion, and subindexes 1 and2 refer to MMA and styrene, respectively. The indi-vidual gel effect correlations were taken from Schmidtand Ray, 1981, for MMA and from Morbidelli et al.,

Figure 5. Schemes of subdomain allocation for discretization.

Figure 6. Model comparison to data of conversion vs time for 1/2factorial experimental design at 50 °C and 20% solids content forthe MMA/S system.

Figure 7. Model comparison to data of conversion vs time for 1/2factorial experimental design at 60 °C and 20% solids content forthe MMA/S system.

Table 5. Parameter Values Fitted: MMA/S System

parameter value at 50 °C value at 60 °C units

entry rate coefficientskmm1 ) kpm1 4.5 × 10-7 4.5 × 10-7 m/skmm1 ) kpm1 1.5 × 10-7 1.5 × 10-7 m/semulsifier Γ∞ 6.0 × 10-6 3.5 × 10-6 g mol/m2

emulsifier bs 2.0 2.0 m3/g molemulsifier aem 6.0 × 10-20 5.5 × 10-20 m2

gel effect A5 0.032 0.032 Lgel effect A6 1.37 × 10-5 1.37 × 10-5

gel effect A7 350 350 L-1

Table 6. Gel Effect Correlations

Styrenegp2 ) 1gt2 ) exp(S1x + S2x2 + S3x3)

MMAVfi ) A0 + A1(TK - A2)Vfpi ) A0 + A3(TK - A4)Vf ) φp1Vf1 + φp2Vf2 + φpp∑i)1

c ΦpiVfpigp1 ) 1 for Vf > A5

gp1 ) A6 exp(A7Vf) for Vf e A5

Vfc ) A8 - A9Tc

gt1 ) A10 exp(A11Vf - A12Tc) for Vf > Vfc

gt1 ) A13 exp(A14Vf) for Vf e Vfc

Generalgp ) gp1

Φp1gp2Φp2

gt )xgt1gt2ktii ) gtkt011kpii ) gpkp011kt12 ) kt21 )xkt11kt22ktr12 ) ktr21 )xktr11ktr22kpij ) kpii/rij


1983, for styrene. The mixing rule used for averagingthe individual rate constants during the gel effect is ofa semiempirical nature, and it was selected because itprovided a better fitting when compared with otherfunctional forms (e.g., Kalfas and Ray, 1993). Theapproach that we use is a hybrid one between those ofchemically controlled and diffusion-controlled termina-tion, but it can be interpreted (Ma et al., 1993) as a formof diffusion-controlled termination. Ma et al., based onprecise experimental data, argue that the terminationconstant for a copolymer system lies somewhere be-tween the extremes predicted by (i) an average depend-ent on the mean composition of the copolymer chain and(ii) an average dependent on the active terminal unit.Figures 8 and 9 show that the model correctly predicts

the trends experimentally observed by photon correla-tion spectroscopy for weight-average particle diameterfor some of the batch experiments. These data were notused for the parameter fitting procedure.Figure 10 shows the monomer composition effect over

the conversion vs time curves. The trends experimen-tally observed are predicted by the model without anyfurther parameter fitting. Limiting conversion is ap-parent for the run with the highest MMA concentrationas expected due to the higher Tg of the homopolymer ofMMA over that of styrene (cf. Friis and Hamielec, 1976).More recent studies (Adams et al., 1990) conclude thatthe limiting conversion is also affected by a decay ininitiator efficiency with conversion in bulk systems, aneffect that is smaller for heterogeneous systems in whichthe initiator source is in the aqueous phase.

In Figure 11 one of the composition vs conversioncurves predicted by the model is compared to theexperimental one. The agreement shown was obtainedwithout fitting any of the parameters that affect thesecurves, (reactivity ratios, thermodynamic parameters);so, it can be considered that the model was used forthese curves in a predictive way.In Figure 12 the model was used again without any

further parameter fitting to predict the behavior of theMMA/S system for the CSTR runs. It is important tomention that the decomposition rate constant for so-dium persulfate could not be found in the literature andits value was assumed the same as the one for potas-sium persulfate. Nevertheless, the model predictions

Figure 8. Model comparison to data of average particle diameterfor 1/2 factorial experimental design at 50 °C and 20% solids contentfor the MMA/S system. Experimental data were obtained by PCS.

Figure 9. Model comparison to data of average particle diameterfor 1/2 factorial experimental design at 60 °C and 20% solids contentfor the MMA/S system. Experimental data were obtained by PCS.

Figure 10. Model comparison to data of conversion vs timevarying monomer composition at 60 °C and 20% solids content forthe MMA/S system.

Figure 11. Model comparison to data of composition vs conversionfor run R10 at 60 °C and 20% solids content for the MMA/S system.

Figure 12. Model comparison to data of conversion vs time forCSTR runs at 60 °C for the MMA/S system.


were not very sensitive to moderate changes in thisparameter.Figure 13 shows comparisons of model predictions

with data obtained in another laboratory (Forcada andAsua, 1991) for weight-average molecular weight for thesame system at similar conditions (25% solids and 50°C) and varying the monomer composition. The dis-agreement in some of the data is possibly due to theuse of a priori estimates of the cross-transfer to mono-mer rate constants (that we decided not to fit). Byfitting these coefficients Forcada and Asua managed toobtain good agreement between their data and a simplerMWD model.The previous results show that the model is able to

represent in a quantitative way the behavior andpolymer properties of the system MMA/S for a varietyof conditions using minimal parameter fitting. Thesecomparisons and others shown below for ethylene/vinylacetate and for styrene/R-methylstyrene (Saldıvar et al.,1997) give confidence that the model captures correctlythe main features of these complex systems and can beused for design, control, and optimization of emulsioncopolymerization processes.

Application to Complex Systems

Example 1. Complex Dynamics in a Flowsheet.In this example the PSD model is applied to a simpleflowsheet consisting of two CSTR’s connected in series.The system analyzed is again MMA/S, which showssustained oscillations in several variables in someregions of the parameter space or under certain operat-ing conditions (Saldıvar et al., 1997). For this simulationthe parameter values used are essentially the same asemployed in the previous section, except for entry ratecoefficients and adsorption isotherm parameters, whichwere modified in order to favor oscillatory behavior; thecomplete set of values employed can be found in Saldı-var, 1996. Figure 14 shows the oscillations in conver-sion for both reactors and Figure 15 the evolution of thecorresponding total number of particles. The oscilla-tions in conversion are originated by the oscillations inthe number of particles and lag behind these. Theamplitude of the oscillations is larger in the first reactorwhich contains larger concentrations of reactants sinceit receives the fresh feed. The relevance of this exampleis that the discretization procedure used for the PSDcontains sufficient information to convey complex dy-

namics from one unit of the flowsheet to another. 39/2elements/points were used in each reactor to discretizethe PSD.Example 2. Ethylene-Vinyl Acetate System in

a Semibatch Reactor. This is a complex emulsioncopolymerization system for which extensive semibatchexperimentation studies have been recently reported ina series of papers by Scott et al., 1993a,b, 1994a,b. Thesystem is complex since there is monomer in the gasphase, and higher pressures (200-500 psig) are neces-sary to increase the content of ethylene in the polymer.Additionally, complex reactions (transfer to polymer andterminal double-bond polymerization) determine theMWD. Their experiments started with a reactor con-taining only water and emulsifier, and then initiatorand vinyl acetate were added at a constant flow rate. Aconstant pressure of ethylene was kept in the reactor.The monomer and initiator flows were stopped after acertain conversion was reached. From their experi-ments the authors concluded that mass-transfer limita-tions may have occurred in the transport of ethylenefrom the gas phase to the particles. Also, a limitingconversion for VAc was observed.Simulations were run with our model for some of the

experiments reported in Scott et al., 1993b, to verify ifthe model was capable of reproducing the trends ob-served experimentally. For the simulation resultsshown here run 5 of design 2 was selected because itwas performed at high pressure and high agitation rate,both factors contributing to minimal mass-transferlimitations. In the simulations it was assumed that (i)no mass-transfer limitations were present and (ii) all

Figure 13. Model comparison to data of final weight-averagemolecular weight vs monomer composition at 50 °C for the MMA/Ssystem. Experimental data from Forcada and Asua, 1991.

Figure 14. Simulation results showing sustained oscillations inconversion for two CSTR’s in series for the MMA/S system.

Figure 15. Simulation results showing sustained oscillations inthe total number of particles for two CSTR’s in series for theMMA/S system.


the ethylene in the reactor was distributed in theemulsion, with a negligible amount in the gas phase.For monomer partitioning a simple partition coefficientsmodel was used with parameters estimated from lit-erature values of solubilities. The entry rate coefficientswere fitted to approach experimental conversion data.The rest of the parameter values was taken or estimatedfrom literature sources, and the complete set used isreported in Saldıvar, 1996. The rate constants fortransfer to polymer (TTP) and terminal double-bond(TDB) reactions were roughly estimated with the fewdata available for molecular weights in Scott et al.,1993b, as being bounded below 3 × 106 m3/mol‚sec forTTP reactions and 107 m3/mol‚sec for TDB reactions.Higher values could produce divergence in the secondmoment of the MWD due to gelation.Figure 16 shows the comparison of model and experi-

ment for the evolution of conversion. While there is aninflow of reactants, the system reaches a quasi-steadystate because the rate of polymerization equals the rateof addition of monomers as the number of particlesincreases. The dashed line shows the model predictionif the monomer to polymer ratio in the partition coef-ficient expression is kept constant. Only the entry ratecoefficient was fitted to obtain the agreement of themodel with the experimental data shown in the figure.At 120 min the flow of initiator and monomer (VaC) isstopped and the model that uses constant thermody-namic parameters (dashed line) erroneously predictsthat the system reaches total conversion. In order tosimulate with the model the limiting conversion ob-served experimentally, it was necessary to reduce themonomer to polymer ratio parameter near the regionof limiting conversion. It is not clear what the mech-anism is that provokes that change in monomer solubil-ity in the polymer particles if it indeed occurs, but in amore recent paper Scott et al. (1994a) report an exten-sive experimental study of the monomer partitioning inthis system, showing that the ethylene decreases itssolubility in vinyl acetate/polymer mixtures as thepolymer concentration is increased; however, they alsoused an empirical equation to force their model to alimiting conversion.Figure 17 shows the evolution of molecular weight

averages predicted by the model. The figure shows aninitial decay in Mw which can be explained in terms ofa decreasing propagation/entry ratio. The propagationreaction tends to slow down due to decreasing monomerconcentration in the particles and at the same timethere is a buildup of initiator in the reactor due to the

load of initiator until it reaches a steady value. In-creasing initiator concentrations lead to higher radicalconcentrations and therefore higher radical entry rates.Once the quasi-steady state is reached, the ratio propa-gation/entry stays constant and the effect of TTP andTDB reactions takes over producing longer chains, asshown in the figure.Example 3. Methyl Acrylate/Vinyl Acetate Start-

up Control Policy for Two CSTR’s. In this examplethe model is applied to a hypothetical process in whichtwo CSTR’s are connected in series to produce methylacrylate/vinyl acetate (MAVA) copolymer. This systemshows large differences in reactivity ratios (r12 ) 9, r21) 0.1), and therefore it would exhibit a large drift incomposition in a batch polymerization. If a constantcomposition copolymer is to be produced, then poly-merization in a CSTR or train of CSTR’s represents abetter process since in this way a basically constantcomposition copolymer can be produced. However,during startup, there will be a significant drift incomposition until steady state is reached. Furthermore,even at steady state the two units can be producingcopolymer of different compositions, having on theaverage the target composition but being of heteroge-neous composition. Using a model of the system, it ispossible to design optimal policies for the startup andoperation of the process such that the copolymer pro-duced will have constant composition and homogeneity.Let us assume that it is desired to produce a MAVAcopolymer with a 60% content of MA. The process is tooperate with two CSTR’s in series with an intermediatefeed of the faster reacting monomer (MA) (see Figure18). The feed conditions of emulsifier, water, andinitiator for the first reactor are fixed and their valuesgiven in Tables 7 and 8. These conditions were foundby simulation to give the desired composition at steadystate, having as design constraints those contained inTable 9. The initial conditions for all the componentsin the first reactor are also fixed and shown in the

Figure 16. Model comparison to data of conversion vs time forsemibatch EVA copolymerization at 20 °C. Experimental data fromScott et al., 1993b.

Figure 17. Simulation predictions for evolution of molecularweight averages for semibatch EVA copolymerization at 20 °C.

Figure 18. Flowsheet for example 3.


tables. The initial ratio of monomers was calculated tohave the target composition in particles at time zero.The values given apply for a 0.5 L laboratory-scalereactor, but the operating conditions are easily scaledup for industrial-scale operation. All the parametersused in the simulations were estimated from literaturesources, and their values can be found in Saldıvar, 1996.Two startup strategies for producing nearly constant

composition copolymer are compared. The first strategyis an intuitive one and tries to resemble an empiricalapproach in which the process is improved by trial anderror based on experimentation. In this case thesimulations replaced the experiments, and an intuitivesuboptimal profile (ramp) was synthesized for the massflow of the most reactive monomer (MA) in the feed tothe first reactor. The mass flow of the second monomerwas calculated to keep a constant total mass inflow tothe reactor. The complete conditions are shown in Table7. Given the first profile, the input flow of monomer 1to the mixer was found empirically, requiring the driftin composition in the second reactor to be small duringthe transient. The value obtained for the suboptimalflow to the mixer was 8 × 10-6 kg/s. Deviations from

the target composition at the beginning of the reactionare due to polymer produced in the water phase giventhe relatively large and different water solubilities ofthese monomers (5% for MA and 2.5% for VA). Atlonger times the contribution of aqueous phase poly-merization becomes negligible. The transient startupprofiles of MA molar ratio in the copolymer for the firststrategy are shown in Figure 20. There is still somedrift in composition, which is difficult to further reduceusing only intuition and “experimentation”. Still, thisis a considerable improvement over the situation inwhich no control is exerted and the feeds are set to thoseof steady state during all the transient.In the second strategy optimal open-loop policies were

calculated. They yield the optimal flow of MA, giventhe steady-state flow of VA to the first reactor as wellas the optimal flow of MA to the mixer. Optimal initialvalues for MA in both reactors were estimated basedon copolymer produced in particles. The complete setof conditions is given in Table 8. The generation of anoptimal loop policy for obtaining constant compositioncopolymers is easily achieved by running a simulationin which the amount of monomer 1 present in eachreactor at any time is calculated in a backward fashionin two steps as follows: (i) using the copolymer equationand given the target constant composition in copolymer(F1), the necessary composition of the monomer mixturein the particles (f1) is calculated, and then (ii) thethermodynamic partitioning equations are solved inorder to find the amount of monomer 1 present in thereactor which is necessary to yield f1 in particles.The differential equation representing the monomer

1 balance is not solved anymore to define the corre-sponding state; instead, it is modified in order togenerate the optimal flow of monomer. The modifiedequation becomes

where M1 is the amount of monomer 1 in the reactor,QA is the extra inflow of monomer 1 that maintainsconstant composition in the copolymer formed in par-ticles, Q1

f is an inflow of monomer 1 given as a forcingfunction and coming from a previous unit in the flow-sheet, and R1 is the rate of consumption of monomer 1by reactions. When the reactor is the first in theflowsheet, Q1

f is set to zero. The equation is solved forQA, and the derivative dM1/dt is found internally by theintegrator package (DDASSL) which can be used forimplicit systems and supplies the time derivative of the

Figure 19. Suboptimal policies for monomer feed flow to firstreactor for strategy 1, example 3.

Table 7. Feed and Initial Conditions: Strategy 1,Example 3

speciesfeed

CSTR 1 (kg/s)initial valueCSTR 1 (mol)

initial valueCSTR 2 (mol)

Mon 1 see Figure 19 0.23 0.49Mon 2 see Figure 19 1.40 1.14emulsifier 1.59 × 10-6 0.014 0.014initiator 1.31 × 10-6 0 0.005water 1.38 × 10-4 0.35 0.35

Table 8. Feed and Initial Conditions: Strategy 2,Example 3

speciesfeed

CSTR 1 (kg/s)initial valueCSTR 1 (mol)

initial valueCSTR 2 (mol)

Mon 1 see Figure 21 0.23 0.23Mon 2 4.31 × 10-5 1.40 1.40emulsifier 1.59 × 10-6 0.014 0.014initiator 1.31 × 10-6 0 0.005water 1.38 × 10-4 0.35 0.35

Table 9. Design Conditions: Example 3

total feed, kg/s 2 × 10-4

residence time, min 42% solids 30emulsifier sodium dodecyl sulfate, MW ) 288.38initiator potassium persulfate, MW ) 270.33

Figure 20. Evolution of copolymer composition for suboptimalpolicy of strategy 1, example 3.

dM1

dt) QA + Q1

f - R1 -QM1

W(28)


state M1. This scheme can be applied for any numberof reactors in a flowsheet, and it is possible to find theoptimal flow of monomer 1 for each reactor simulta-neously by just performing one simulation.The optimal monomer flows to the first reactor and

mixer are shown in Figure 21, and the results of thesimulation when these flows are fed to the correspond-ing units are shown in Figure 22. The control ofcomposition is excellent and shows only small deviationsfrom the target value at the initial stage due to theaqueous phase polymerization. Final monomer conver-sions at steady state were 28 and 46% for the first andsecond reactors, respectively. It is important to mentionthat optimal open-loop policies for a semicontinuousreactor, using the concepts outlined in this example,were synthesized and experimentally verified (see Saldı-var and Ray, 1997).An interesting aspect of this example is the node

reallocation scheme used to discretize the PSD. In thisexample most of the fresh feed goes into the first unitin the flowsheet, so the rate of growth of particles inthe first reactor tends to be larger than that in thesecond reactor. As a result, the maximum polymermass in the first reactor eventually surpasses themaximum polymer mass in the second reactor, andincoming particles can be unaccounted for in the secondreactor feed unless the grid is reallocated. In thisexample the grid was modified by multiplying themaximum polymer mass in the second reactor by agiven factor and including extra elements/points todiscretize the added interval whenever the front comingin the feed of the second reactor reached the front insidethe second reactor. Figure 23 shows how the realloca-

tion was done a number of times during the simulation.In this example 27/2 elements/points were employed inthe first reactor and initially 11/2 were used for thesecond reactor. At every reallocation of the grid 4/2 newelements/points were added and the maximum polymermass was expanded by a factor of 2.5.

Conclusions

Polymer reaction engineering involves a systematicmethodology for building models of general applicability,which must be thoroughly validated and implementedin a way which can be used for process analysis anddesign. The field benefits from the input from severaldisciplines: polymer science, statistical design of experi-ments, reaction engineering, and numerical methodsamong others. In this paper the systematic experimen-tal validation and practical implementation of a detailedemulsion copolymerization model were addressed.With respect to the truth of postulates used in the

model, it is very difficult, using a model of this kind, toindividually prove or disprove the truth of each one ofthe postulates. Instead, we use commonly acceptedpostulates and hope to represent well the trends, so themodel is useful for engineering applications after datafitting. Since the general trends of a specific systemwere well represented by a general model, the postulatesas a whole are plausible.Although there are still obscure areas in the mecha-

nisms of some of the phenomena occurring in emulsionpolymerization systems, it was illustrated how a com-prehensive model can explain even in a quantitativeway much of the important dynamic behavior andoutput variables needed for the design of polymericproducts and processes, with minimal data fitting. Itwas also illustrated how the practical implementationof the model requires a careful selection of numericaltechniques if the model is to be used to represent thecomplete variety of processes and operation typesemployed in industry.

Notation

aem ) surface area occupied by an emulsifier molecule ona micelle

aep ) surface area occupied by an emulsifier molecule on aparticle

am ) surface area of a micellebs ) parameter in adsorption isotherm for emulsifierd ) number of terminal double bonds in a dead polymermolecule

D ) dead polymer

Figure 21. Optimal profiles for monomer 1 feed flow to CSTR 1and mixer of strategy 2, example 3.

Figure 22. Evolution of copolymer composition for optimal policyof strategy 2, example 3.

Figure 23. Maximum polymer size vs time showing domainreallocation for CSTR 2 for simulations of example 3.


Dnl,b ) distribution function for dead polymer chains oflength l and branching index b in particles having nradicals

D ) numerical diffusionF ) dimensionless PSDF or F(m) ) distribution function for number of particlesof polymer mass m or total number of particles (F)

g ) dm/dt (particle growth rate)gp ) total gel effect factor for propagationgt ) total gel effect factor for terminationgpi ) gel effect factor for propagation component igti ) gel effect factor for termination component iH ) Heaviside functionkmmi ) mass-transfer coefficient for type i radical enteringmicelles

kmmR ) mass-transfer coefficient for initiator radicals enter-ing micelles

K ) number of total subdomains for discretizationm ) polymer massmm ) polymer mass at micellar sizemmin ) minimum polymer massmmax ) maximum polymer massM ) micelle concentrationM0 ) characteristic micelle concentrationNnl,b ) distribution function for growing radicals of lengthl and branching index b in particles having n radicals

NA ) Avogadro’s numberpi ) probability of a radical in particles being of type ipiw ) probability of a radical in the aqueous phase being oftype i

Pwl ) live polymer of length l present in the aqueousphase

PSD ) particle size distributionQA ) extra feed of monomer 1 to a reactor to get constantcomposition copolymer

Q1f ) forcing function of feed of monomer 1 to a reactorcoming from previous units in a flowsheet

ri ) rate of change for mechanism i in population balanceequations

R ) primary radicalsR1 ) rate of consumption of monomer 1 by reactionsS ) emulsifierSa ) emulsifier adsorbed on polymer particlesSd ) emulsifier adsorbed on monomer dropletsSF ) free emulsifier (nonadsorbed)Sp ) total surface of particlest ) timeu ) dimensionless timeu ) dimensionless mass of polymer domainVf ) total free volumeVfc ) critical free volumeVfi ) free volume of component iVw ) volume of water in reactorx ) weight conversionY ) transformed distribution function for number ofparticles (eq 18)

Subindexes and Superindexes

aq ) aqueous phasef ) feedw ) water or aqueous phase

Greek Symbols

Γ∞ ) parameter in adsorption isotherm for emulsifierδa,b ) discrete delta function; )1 when a ) bδ(a) ) continuous delta function; )1 when a ) 0λSG,H ) moments of dead polymer

ΛG,H ) integral over polymer mass moments of bulkpolymer

µSG,H ) moments of live polymer

µnG,H ) partial moments of live polymer

ν00,0 ) moment of dead polymer with TDB’s

θ ) characteristic timeΩG,H ) moments of bulk polymer

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Received for review July 29, 1996Revised manuscript received January 16, 1997

Accepted January 16, 1997X

IE960464Z

X Abstract published in Advance ACS Abstracts, March 1,1997.


mathematical modeling of emulsion copolymerization reactors: experimental validation and application...

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