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Computers and Chemical Engineering 31 (2007) 552–564

odel based operation of emulsion polymerization reactors with evaporativecooling: Application to vinyl acetate homopolymerization�

Sachin Arora a,∗, Ralf Gesthuisen b, Sebastian Engell a

a Department of Biochemical and Chemical Engineering, Universitat Dortmund, D-44221 Dortmund, Germanyb Innovene Deutschland GmbH, D-50769 Koln, Germany

Received 14 December 2005; received in revised form 24 July 2006; accepted 25 July 2006Available online 25 September 2006

bstract

This work aims at maximizing the productivity of emulsion homopolymerization processes. A dynamic model of emulsion polymerizationrocesses is extended by the inclusion of vaporization from the liquid phases in the reactor to the gaseous phase. The multi-component gas–liquidass transfer phenomenon is described by the Maxwell–Stefan diffusion equations, which are solved by a special algorithm. A novel operation

trategy is developed for running a reactor optimally with respect to batch time. This strategy is applied first to an industrial scale reactor, which isun without using evaporative cooling. Then, based on the extended model, controlled vaporization is included by which additional heat is removed

rom the reaction system. This makes it possible to extend the restrictions imposed by the limited heat removal of the cooling jacket considerably.imulation results are presented for the homopolymerization of vinyl acetate in an industrial scale reactor operated in semi-batch mode. The resultshow that a significant amount of heat can be removed by evaporative cooling thus leading to higher productivity.

2006 Elsevier Ltd. All rights reserved.

al; Ti

ri1thc(ALiwi

eywords: Emulsion polymerization; Evaporative cooling; Limited heat remov

. Introduction

Emulsion polymerization is an important industrial processhat is used for the production of a wide range of polymer mate-ials, e.g. paints, adhesives, coatings and binders. One of thedvantages of this process is the possibility to obtain polymersf high molecular weight at high polymerization rates.

From an industrial perspective, one of the major objectivesn the operation of emulsion polymerization processes is thatf faster and safer operation with consistent polymer productuality. The polymerization reaction proceeds as a classicalouble bond addition reaction initiated via a free-radical mech-

nism. The reaction is very fast and highly exothermic. Thisuts a constraint on the speed of the reaction, due to the lim-ted heat removal capacity of the cooling jacket. With industrial

� A preliminary version of this paper appeared in the proceedings of theuropean Symposium of Computer Aided Process Engineering (ESCAPE 15),arcelona, Spain (29-05: 1 June 2005), Paper No. PO-059.∗ Corresponding author.

E-mail addresses: sachin.arora@uni-dortmund.de,.arora@bci.uni-dortmund.de (S. Arora).

m

iHhmp

tr

098-1354/$ – see front matter © 2006 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2006.07.011

me optimal operation; Temperature control

eactors becoming as big as 200 m3, this limitation is of increas-ng importance in comparison to lab scale reactors of a few00 ml in volume, as thus are often used in adiabatic calorime-ry for runaway experiments (Klostermann et al., 1998). Theigh viscosities of the latexes produced in many industrial pro-esses aggravate the heat-removal problems. It has been shownBuruaga, Armitage, Leiza, & Asua, 1997; Buruaga, Echevarria,rmitage, Leiza, & Asua, 1997; Vicente, BenAmor, Gugliotta,eiza, & Asua, 2001; Vicente, Leiza, & Asua, 2003) that there

s a significant reduction in the overall heat transfer coefficientith the increase in solid content of the medium, especially for

ndustrial size reactors where problems of fouling and non-idealixing are encountered frequently.Increasing the heat transfer area by introducing cooling coils

nside the reactor are often discussed in conventional reactors.owever, the usage of such cooling coils affects the systemydrodynamics (Dimitratos, Elicabe, & Georgakis, 1994). Thisay induce coagulation and cause additional fouling because of

oor mixing in the dead spots created around the coil.Maximizing production by reducing the batch time under

he restriction of limited cooling capacity in industrial scaleeactors for emulsion polymerization processes was discussed

S. Arora et al. / Computers and Chemical Engineering 31 (2007) 552–564 553

Nomenclature

A area of heat transfer between reactor and jacket[A] inverse matrix of effective diffusivity matrixAs surface area of the reactorAi effective area of interface for phase i[B] inverse of general effective diffusivity matrixCpj specific heat of component jct overall concentration in the gas phaseDp diffusion coefficient for radicals in the polymer

phaseDw diffusion coefficient for radicals in the water

phaseDij Maxwell–Stefan diffusivity coefficient for com-

ponents ijf efficiency factor in initiator dissociationf i

a effectiveness factor for the interfacial area�H� enthalpy of reaction�Hv enthalpy of vaporizationHex rate of energy removal by heat exchange to the

jacketHfc rate of cooling by cold feedHg rate of energy generation by reactionHrecc rate of cooling by cold recycle streamHv rate of energy removal by vaporization[I] initiator concentrationJi diffusion molar flux of component ik rate coefficient for radical exit from a single par-

ticlekfm rate constant for chain transfer to monomerkI initiator rate constantkp propagation rate constantkt rate constant for terminationktd rate constant for disproportionationK control valve resistanceKI integral constant of PI controllerKP proportional constant of PI controllerl film thickness[M]i concentration of monomer in phase iMn number average molecular weightMw weight average molecular weightnj moles of component jn average number of radicals per particlenj molar flowrate of component j(N) column vector of Nj from a particular phaseNA Avogadro’s numberNj molar flux of component j from a particular phaseNp total number of particlesNT total molar fluxNi

j molar flux of component j from phase iPI polydispersity index (ratio of Mw by Mn)Pr reactor pressureq parameter in the calculation of n

[R]w overall concentration of radicals in the waterphase

� rate of reactionT reactor temperature�TLMTD log mean temperature difference of reactor with

jacketU overall heat transfer coefficient between reactor

and jacketVpol volume of polymer formedVi total volume of phase iV i

j volume of component j in phase i

(x)i column vector of molar fractions in liquid phasei

(y) column of mole fractions in gas phase

Greek lettersα probability of propagationδ interphaseφs relative solid contentΦ mass transfer ratesλ0 concentration of zeroth moment for dead chainsλ1 concentration of first moment for dead chainsλ2 concentration of second moment for dead chainsμ0 concentration of zeroth moment for growing

chainsσ average rate of radical entry in a single particle correction factor

Subscriptsend final value at conversion 1in input to the reactorini initial value at time zeron nitrogen (used with diffusion coefficients)Nom nominal valueo outletrec recycle stream to the reactorSP setpoint0 bulk phase

Superscriptsd droplet phasep polymer phasew water phase

eitioEnabcp

z adjustable parameter

arlier, e.g. by Vicente et al. (2003). They estimated the decreasen the heat transfer coefficient as a function of conversion andhen, based on this decrease, changed the reaction speed accord-ngly. They also used reaction calorimetry for the estimationf heat transfer coefficient. Recently, Gesthuisen, Kramer, andngell (2004) proposed a hierarchical control scheme for run-ing the emulsion polymerization reactors time optimally. In thispproach the decrease of the heat transfer coefficient is estimated

y extended Kalman filter and then the setpoint for the monomeroncentration in the particle phase is adjusted. The approachroposed in this paper for time optimal operation is similar to

5 emica

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oVmrrrbbRsma6cewepw

motlbtt(tba(

oaclsmgbalbthtt

2

2

ptTidTfimkw

f

are same;(ii) the kinetic rate-constants do not depend on chain length;

(iii) the pseudo-steady state hypothesis is valid for radicals;

Table 1Reaction kinetics

Reaction Explanation

Ikd−→2R◦, R◦ + M

ki−→R1 Initiation (decomposition and consumption)

Rn + Mkp−→Rn+1 Propagation

Rn + Mkfm−→Pn + R1 Chain transfer to monomer

Rn + Pm

kfp−→Pn + Rm Chain transfer to polymer

Rn + Rmktc−→Pn+m Termination by combination

Rn + Pm

ktd−→Pn + Pm Termination by disproportionation

54 S. Arora et al. / Computers and Ch

he idea of maintaining constant isothermal conditions. How-ver, it is applied in a slightly different manner. The differences that no estimation of the decrease in the heat transfer coeffi-ient is required. In order to maintain isothermal conditions, theeedback loop with the monomer feed as the manipulated vari-ble acts on the temperature setpoint that in turn modifies theonomer feed rate and hence the reaction speed. The process

peration strategy is discussed in detail Section 3.Evaporative cooling in solution free radical polymerization

f styrene in a continuous reactor was discussed first by Villa,an Horn, and Ray (1999), where the vaporization of solvent andonomer were considered as an attractive mechanism for heat

emoval. Multiple steady states and oscillatory behavior wereeported based on the effect of the condenser dynamics in theecycle stream. The effect of evaporative cooling in the semi-atch operation of emulsion polymerization reactors for possi-le improvements in productivity have not yet been discussed.ecently, Wang, Bito, Sudol, and El-Aasser (2001) reported

ignificant vaporization of the reactant media in their experi-ents with styrene emulsion polymerization during the early

nd preparatory stages of the process. They reported that around–8% of the monomer was lost during the degassing period. Thisreates a need to analyze the gas–liquid mass transfer phenom-na, which has so far not been considered for such systems andhich could play a major role in process improvements. Mod-

ling of vaporization phenomena in emulsion polymerizationrocesses was first discussed by Arora and Gesthusien (2004)ithout relating it to the new process operation strategy.This contribution specifically deals with the emulsion poly-

erization of vinyl acetate taking into consideration the affectf vaporization of the different species from the liquid phase tohe gaseous phase. The reaction is carried out at temperaturesess than the boiling temperature of the liquid in order to avoidubbling (or foaming). The vaporization fluxes are computed byhe product of the mass transfer coefficients and the concentra-ion gradients in the two phases. The mass transfer coefficientswhich are functions of concentrations) are estimated by solvinghe multicomponent Maxwell–Stefan diffusion equations andy the application of film theory. An algorithm based on thepproaches of Burghardt (1983, 1984) and Taylor and Krishna1993) has been developed for this purpose.

The aim of this paper is to show the advantage of using evap-rative cooling in emulsion polymerization process by usingmodel based operation strategy. This is particularly benefi-

ial for large scale industrial reactors which have comparativelyow heat transfer area and thus reaction rates have to be keptlow. The organization of the paper is as follows: firstly, theodeling approach for emulsion polymerization process and for

as–liquid mass transfer will be presented. This will be followedy the suggested process operation strategy. Simulation resultsre then presented to support the theoretical approach. The simu-ations show a significant improvement in heat removal capacityecause of evaporative cooling. Finally, it will be suggested how

he approaches developed here can be extended to other fast andighly exothermic processes (esp. involving highly viscous sys-ems), where the overall heat transfer coefficient is low and leadso inefficient cooling by the reactor jacket.

l Engineering 31 (2007) 552–564

. Model development

.1. Emulsion polymerization model

Emulsion polymerization is a multiphase process involvingrimarily three liquid phases (the monomer or droplet phase,he water or aqueous phase, and the oil or particle phase).he reaction starts in the water phase where the initiator dis-

ntegrates to form free radicals, which then combine with theissolved monomer in the aqueous phase to form oligomers.hese oligomers either precipitate or are trapped by emulsi-er micelles to form particles. The polymerization reaction thenostly takes place in the particle phase. The complete reaction

inetics is described in Table 1 and the process diagram forhich the model is developed is shown in Fig. 1.The model used here in our investigations is based on the

ollowing assumptions:

(i) the kinetic rate-constants in the aqueous and polymer phases

Fig. 1. Process diagram for emulsion homopolymerization.

emica

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Tanttt

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(

S. Arora et al. / Computers and Ch

iv) radicals generated by initiation or chain transfer to monomerpresent similar reactivities, and

v) the absolute number of particles Np is constant (e.g. by theusage of seed).

The mass balance equations for the liquid phase can now beescribed as:

d

dtnI = nIin − kI[I]wV w (1)

d

dtnM = nMin = � −

∑i=d,w,p

NiMAi + nMrec (2)

dnW

dt= nWin − Nw

WAw + nWrec (3)

d

dtV

ppol = �MwM

ρpol(4)

here

˙ = kpnNp[M]p

NA+ kp[R]wV w[M]w (5)

ere, kI is taken as a overall initiation reaction rate constant.his combines the two rate constants kd and ki (i.e. decompo-ition and consumption, respectively), as shown in Table 1. n

enotes average number of radicals per particle, Np denotes theotal number of particles per unit volume, [R]w denotes the totalumber of radicals in the water phase and NA denotes Avo-adro’s number. The detailed model for the calculation of n isescribed in Appendix A. Np is considered as a constant parame-er in the model as the emulsion polymerization seed is taken forescribing the initial conditions. [R]w is described by Eq. (6).he superscript p and w denote the polymer phase and waterhase, respectively.

∑Ni

jAi denotes the total amount of vapor-

zation of species j from phase i. The subscript ‘in’ denotes theeed to the system and the subscript ‘rec

′denotes the recycle

o the reactor coming from the condenser. The concentration ofonomer in the particle phase [M]p is obtained by solving phase

istribution equations. The equations describing the phase distri-ution are given in detail in Appendix B. An iterative algorithmicrocedure is applied to solve for the current value of volumes ofhe different components in different phases. � denotes the totaleaction rate in the reactor (i.e. in particle and water phase). Vpolenotes the total volume of polymer generated by the reaction.his can then also be used to estimate the total conversion (i.e.

atio of monomer reacted to total amount in recipe). The con-entration of the total number of radicals in the water phase cane described as:

d[R]w

dt= 0 = 2fkI[I] + knNp

NAV w − ka[R]wNp

NAV w − kwt [R]w2

(6)

he left hand side of the radical balance equation is considereds zero due to the pseudo steady state assumption. Since, Eq. (6)

eeds the value of n for the computation of [R]w and n needshe value of [R]w for its computation (see Eqs. (30) and (33)),he solution is obtained by an algorithmic procedure, owing tohe implicit nature of the non-linear algebraic equations. More

tti

l Engineering 31 (2007) 552–564 555

etails about the calculation of n are given in Appendix A. Theeight average molecular weight is calculated by the methodf moments for the live and dead polymer chains (zeroth, firstnd second). These moments are constructed by applying theenerating functions approach as described in Ray (1972). Thequations describing molecular moments and molecular weightverages are described in Appendix C.

The gas phase component mole balance equations are nowerived considering the vaporization from the liquid phase andhe exit of gases from the vapor phase to the condenser:

d

dtn

gM =

∑i=d,w,p

NiMAi − y

g0M

no (7)

d

dtn

gW = Nw

WAw − yg0W

no (8)

d

dtn

gN2

= ngN2

− yg0N2

no (9)

g0j

=n

g0j∑jn

g0j

, where j = W, M, N2 (10)

ere, the effective interfacial area Ai for each phase over whichaporization takes place is assumed to be proportional to anffectiveness factor f i

a, which is equal to the ratio of volume ofhe particular phase to the total volume. This effectiveness factors then multiplied with the total surface area for calculating theffective interfacial area

if iaAs = V i∑

iViAs, where i = d, p, w (11)

The outflow of the gases from the reactor to the condenser isalculated based on the pressure difference and the valve resis-ance as:

˙o = K(Pr − PSP) (12)

Here, the overall molar flowrate of gases going out of theeactor equals no, which is calculated using the pressure differ-nce between the pressure of the reactor and the outside pressure,hich is regulated by a controller. K is the valve resistance coef-cient of the control valve regulating the flowrate. The totalondenser is assumed to be an ideal plug flow reactor. Thisauses the time delay in the recycle feed as compared with thenstant when it enters the condenser. This time delay is taken toave a constant value of 5 min here. It depends on the length ofhe piping in the condensation system. The energy balance forhe liquid phase in the semi batch reactor is written as:

d

dt

(∑njCpjT

)=(∑

njCpjTj

)in

+ �(−�H�)

−∑i,j

NijA

i�Hvj − UA�TLMTD (13)

∑ )

njCpjT represents the total amount of energy in the reac-

or, � the reaction rate, �H� the enthalpy of reaction and �Hvhe enthalpy of vaporization. It should be pointed out that in thendustrial operations, the amount of cold utility is limited, and

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warngteti(a

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A

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Ξ

Φ

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Φ

Tatlflopacfl

56 S. Arora et al. / Computers and Ch

he process is run at the maximum of the cooling flow (Buruaga,rmitage, et al., 1997; Buruaga, Echevarria, et al., 1997; Vicente

t al., 2003). This causes the jacket outlet temperature to belose to the reactor operating temperature. The temperature dif-erence �T between the reactor and the jacket is then calculateds (�T)LMTD, where LMTD denotes log mean temperature dif-erence. Based on the reaction calorimetric estimations (Kramer

Gesthuisen, 2005; Kramer, Gesthuisen, Niggemann, & Asua,005) have suggested to model industrial jackets as plug floweactors rather than continuous stirred tank reactors. However,f the heat transfer is calculated based on the log mean temper-ture difference, then the results thus obtained are similar to theesults as obtained by modeling of a jacket as a plug flow reac-or. Since similar reaction conditions are chosen as in Vicentet al. (2003), the same relationship of the overall heat transferoefficient with solids content is assumed as follows:

= Uini + (Uend − Uini)φzs (14)

ere, Uini denotes the initial overall heat transfer coefficient andend denotes the final overall heat transfer coefficient. φs denotes

elative solid content with respect to the recipe chosen and z isn experimental fit parameter.

.2. Evaporative cooling model

The extension of the emulsion polymerization model iserformed by the inclusion of the gas phase. The dynamicsf the gas–liquid mass transfer is considered by estimatinghe vaporization fluxes from the three liquid phases (droplet,ater and particle). The vaporization fluxes are describedsing a generalized Maxwell–Stefan approach. The general-zed Maxwell–Stefan equation for multicomponent systems ofspecies can be described as:

J) = −ct[B]−1(∇y) (15)

here (J) defines the diffusional flux row column of length n − 1.here are only n − 1 independent equations because of the con-traint that the sum of all diffusional fluxes must be zero. Here, ctepresents the overall concentration in gas phase and (�x) as theolumn of gradients of mole fractions. The n − 1 square matrixB] is described as:

ii = yi

Din+

n∑k=1,i �=k

yk

Din, where i = 1, . . . , n − 1 (16)

Bij = −yi

(1

Dij

− 1

Din

), where

j = 1, . . . , n − 1 and j �= i (17)

˜ij defines Maxwell–Stefan diffusivity constant for species

− j. The overall flux for species i is then calculated by usingn additional independent equation often referred to as the boot-

trap equation. The overall flux (after using a suitable bootstrapelation) can then be computed as:

i = Ji + xiNT (18)

ActS

l Engineering 31 (2007) 552–564

Since, in our case one of the species involved is nitrogen,hich is considered to be stagnant, the condition that the over-

ll flux of nitrogen is zero supplies the necessary bootstrapelation. The reason for using this bootstrap condition is thatitrogen is considered to be present only in the gas phase at theiven temperature and pressure. The physical interpretation ofhis bootstrap condition is that whatever concentration gradientxists across the interface film (which drives nitrogen towardshe liquid phase from the gas phase), an equal and opposite driv-ng force would then exist as well because of the convective fluxdue to the vaporized gases) making the overall flux of nitrogens zero. The vaporization fluxes can then be defined as:

N) = ct

l[Aav]−1Ξ(y0 − y�) (19)

ere, (N) implies a single column vector of vaporization fluxes.he subscripts δ, 0 denote film interface and bulk conditions,

espectively. [A] denotes the inverse diffusivity matrix and isomputed as:

ii =n∑

k=1,i�=k

yk

Dik

, where i = 1, . . . , n − 1 (20)

ij = − yi

Dij

, where j = 1, . . . , n − 1 and j �= i (21)

ere, ct denotes the overall concentration in the vapor phasef the film, l the film thickness and Ξ the non-linear bootstrapelation (that makes the flux of nitrogen equal to zero), which isfunction of the film and bulk conditions. It is calculated basedn the approaches by Burghardt (1983, 1984) and Taylor andrishna (1993), as:

= 1

2Φ

exp Φ + 1

exp Φ − 1(22)

=n−1∑

1

Nk

ctDkn/ l(23)

In view of the assumptions underlying the film model, Φ isonstant which is calculated as:

= lnyn�

yn0(24)

he above equations are explicit in nature and require no iter-tions if the interface compositions are known. It is assumedhat there exists thermodynamic vapor–liquid equilibrium at theiquid side of the interface. Thus, determination of gas inter-ace composition, which satisfies the two conditions (vaporiquid equilibrium and flux of inert component being zero) basedn the existing conditions of temperature, pressure and com-ositions in the bulk phases, is the key to the solution. Anlgorithm was developed (Table 2) that uses an iterative pro-edure to find the interface composition and thus the overalluxes. The vapor–liquid equilibrium is calculated initially by

ntonie equation and then corrected by using activity coeffi-

ients. The activity coefficient for each phase is calculated byhe perturbed chain statistical associative fluid theory (Gross &adowski, 2001).

S. Arora et al. / Computers and Chemical Engineering 31 (2007) 552–564 557

Table 2Algorithm for the calculation of the molar fluxes

Given: (x0), (y0), P, [D]Step 1: Assume x� = x0

Step 2: Guess the interfacial temperatureStep 3: Compute y�

Step 4: Compute yavg

Step 5: Compute [A]Step 6: Calculate Φ by Eq. (24)Step 7: Calculate Ξ by Eq. (22)Step 8: Estimate (N) by Eq. (19)Step 9: Recalculate Φ by Eq. (23)Step 10: Recalculate y� by Eq. (24)Step 11: Check for convergence of y�. If y� have converged, then calculate

the molar fluxes by Eq. (19), else return to Step 2

Aa

3

3

stttatg

cctticmmiFlin Fig. 3.

Ftg

lgorithm for the calculation of molar fluxes when the interface compositionsre unknown.

. Process operation strategy

.1. Extracting heat via evaporative cooling

In order to have high evaporation, the following strategy isuggested. The basic idea is to regulate the concentration ofhe vaporizing species in the gas phase to a setpoint value. Ifhis set point value is below the thermodynamic equilibrium,hen vaporization will take place. The farther the setpoint is

way from the equilibrium point the greater is the vaporiza-ion flux. This property can be exploited here using the nitro-en flow as a manipulated variable for altering the gas phase

K

ig. 3. Comparison of different results based on changes in the setpoint of the molehe setpoint is 0.06, dash-doted lines show results when the setpoint is 0.12 and dasheas phase for the three cases, (B) flowrate of N2 required in the three cases, (C) rate of

Fig. 2. Feedback control loops.

ompositions. Two independent decoupled feedback loops areonstructed to implement the above idea. The first loop controlshe pressure in the reactor by regulating the outflow, whereashe second loop maintains a desired mole fraction of watern the gas phase by regulating the inflow of nitrogen. Bothontrollers are PI-controllers (refer Eqs. (25) and (26)). Sinceost of the vaporized species is water (because monomer reactsuch faster than it can vaporize), the mole fraction of water

n the gas phase is taken as the setpoint in the first loop (referig. 2). The choice of a favorable setpoint is based on simu-

ation studies. The comparison of different setpoints is shown

= KNom + KP1(P − PSP) + KI1

∫ t

0(P − PSP) dt (25)

fraction of water in the gas phase. In all figures, solid lines show results whend lines show results when the setpoint is 0.21. (A) Mole fraction of water in theenergy removal in three cases and (D) total energy removed in the three cases.

5 emical Engineering 31 (2007) 552–564

n

3

paiehoee

H

HddcbnitrtbtTmc

n

HrHhin

bmcc2ft

4

fVci

Table 3Recipe used in the simulations

Variable Value

Temperature (◦C) 68Pressure (atm) 1Water (kg) 6000Vinyl acetate (kg) 3168.3SS

uwTshr

obrinasrrtftoevtdtbOmi(aHiiswlh

tuv

58 S. Arora et al. / Computers and Ch

˙N2 = nN2Nom+ KP2 (yg

W − ygWSP

) + KI2

∫ t

0(xg

W − xgWSP

) dt

(26)

.2. Time optimal operation

Running the process by utilizing the resources in the bestossible way leads to an online optimization problem. However,simpler approach that ensures close proximity to the optimum

s described here. If a reactor is to be run isothermally, then thenergy balance equation (Eq. (13)) implies that the amount ofeat generated should be equal to the amount of heat taken outf the system. This implies that the heat generation rate must bequal to the total heat removal rate as shown in the followingquation:

˙ g = Hex + Hfc + Hrecc + Hv (27)

ere, Hg denotes the rate of heat generation by reaction, Hexenotes the rate of cooling by heat exchange to the jacket, Hfcenotes the rate of cooling by cold feed, Hrecc denotes the rate ofooling by the recycle stream and Hv denotes the rate of coolingy vaporization. Since the emulsion polymerization reaction isot limited by the reaction rate, the speed of reaction can bencreased until the cooling capacity of the system, i.e. sum ofhe terms of the right hand side. If the sum of the terms on theight hand side is not equal to the left hand side then it will lead toemperature variation from isothermal condition. This can thene regulated by altering the monomer feed as it directly is relatedo the reaction rate and hence to the rate of heat generation.hus the feedback controller equation for the monomer as theanipulated variable and temperature as the controlled variable

an be written as:

˙M = nMNom + KP3(T − TSP)KI3

∫ t

0(T − TSP) dt (28)

ere, nMNom can be made time varying such that it approximatelyeconstructs in Eq. (27) (for e.g. nMNom = (Hex + Hv)/�H�).owever, in our simulations was no significant effect seen byaving a time varying nominal feed rate of monomer. It onlyncreased the computational effort and therefore a constant value˙MNom was used in all simulations. The advantage of using feed-ack loop based on temperature measurement is that it can beore easily and robustly measured as compared to the other pro-

ess variables like total monomer holdup, which could also behosen as a basis for time optimal operation (Gesthuisen et al.,004). With regards to the control of temperature by monomereed, it is assumed that there is no existence of droplet phase dueo improper mixing effects.

. Results and discussion

A seeded emulsion polymerization reaction was simulated

or an industrial scale reactor (10 m3), in semi-batch mode.inyl acetate boils at 72 ◦C so the reactor temperature was keptlose to that value but never exceeding it. In order to show themplementation of the proposed operation strategy, three sim-

oFff

odium persulfate (kg) 14.6odium dodecycl sulfate (kg) 29.2

lation runs are provided for comparisons. The same recipeas used in all the three runs, which is depicted in Table 3.ables 4 and 5 describe the parameters and reaction rate con-tants used in the simulation studies. The model for emulsionomopolymerization was validated based upon the experimentalesults of Penlidis, Macgregor, and Hamielec (1985).

In order to determine a favorable setpoint for the mole fractionf water in the gas phase, three simulation runs were performedy changing the setpoints. To focus only on the vaporizationate, the monomer feed was kept constant and the temperaturen the reactor was controlled by the jacket inlet temperature andot by the monomer feed (i.e. the runs were not conducted intime optimal manner). The results of the simulation runs are

hown in Fig. 3. Here, the solid (black) lines show the simulationesults for the setpoint equal to 0.06, dashed (blue) lines showesults for setpoint equal to 0.12 and dash-doted (red) lines showhe results for the setpoint at 0.21. Fig. 3(A) shows the moleraction of water in the gas phase for the three cases, (B) showshe flowrate of N2 required in the three cases, (C) shows the ratef energy removal in the three cases and (D) shows the totalnergy removed in the three cases. It can be seen that higheraporization is obtained and thus more energy is removed fromhe reactor, if the mole fraction of water in the gas phase isecreased (see Fig. 3(C) and (D)). This is an expected result sincehe vaporization rate is proportional to the concentration gradientetween the equilibrium value and bulk phase concentration.n the other hand, the amount of nitrogen required in order toaintain the constant mole fraction of water in the gas phase

ncreased significantly with the decrease of the mole fractionsee Fig. 3(B)). This is due to the constant pressure conditionnd constant mole fraction condition imposed on the controllers.owever, the heat removal rate due to vaporization does not

ncrease proportional to the usage of nitrogen as can be seenn Fig. 3(C) and (D). It is noticeable that even by reducing theetpoint to a very low value of 0.06, not significantly more energyas removed by vaporization. Heat removal by vaporization is

imited and a large amount of vaporization might lead to a veryigh nitrogen consumption making the process less profitable.

A simulation run was performed without evaporative coolingo test the feedback loop for the monomer feed as the manip-lated variable based on reactor temperature as the controlledariable for time optimal operation. Simulation results for timeptimal operation without evaporative cooling are shown in

ig. 4. The log mean temperature difference was fixed to 15 ◦Cor representing the limited heat removal of the jacket. It wasound that the complete run with the maximum jacket cooling

S. Arora et al. / Computers and Chemical Engineering 31 (2007) 552–564 559

Table 4Table of parameters

Symbol Description Value Units Reference

MwM Molecular weight (VAc) 86.09 g/mol Brandrup and Immergut (1989)ρM Density (VAc) 0.8722 g/cc Barudio, Fevotte, and McKenna (1999)CpM Heat capacity (VAc) 100.05 J/mol K Reid, Prausnitz, and Poling (1987)ρpol Density (polymer) 1.0074 g/cc Barudio et al. (1999)Cppol Heat capacity (polymer) 0.35014 cal/g K Brandrup and Immergut (1989)AM Antonie coefficient (VAc) 4.3403 – Gmehling and Onken (1977)BM Antonie coefficient (VAc) 1299.069 – Gmehling and Onken (1977)CM Antonie coefficient (VAc) −46.183 – Gmehling and Onken (1977)BPM Boiling point (VAc) 345.7 K Gmehling and Onken (1977)Aw Antonie coefficient (water) 6.2096 – Gmehling and Onken (1977)Bw Antonie coefficient (water) 2354.731 – Gmehling and Onken (1977)Cw Antonie coefficient (water) 7.559 – Gmehling and Onken (1977)BPw Boiling point (water) 373.15 K Gmehling and Onken (1977)Kd

M Partition coefficient (droplet) 34.7 – Gardon (1968)K

pM Partition coefficient (polymer) 29.5 – Gugliotta, Arzamendi, and Asua (1995)

DM−N2 Diffusion coefficient (gas) 8.5 × 10−6 m2/s Taylor and Krishna (1993)DM−w Diffusion coefficient (gas) 20.4 × 10−6 m2/s Taylor and Krishna (1993)Dw−N2 Diffusion coefficient (gas) 20.4 × 10−6 m2/s Taylor and Krishna (1993)l Interfacial thickness 0.001 m Taylor and Krishna (1993)Dw Diffusion coefficient (water) l.l × 10−5 cm2/s Araujo and Giudici (2003)Dp Diffusion coefficient (polymer) 1.1 × 10−6 cm2/s Araujo and Giudici (2003)�H� Enthalpy (reaction) 89.5 kJ/mol Brandrup and Immergut (1989)�HvM Enthalpy (vaporization) 37.2 kJ/mol Reid et al. (1987)�HwM Enthalpy (vaporization) 45 kJ/mol Reid et al. (1987)zUU

rrtiaFhbto

todTclpg

ovfirswo

FipFs

F

TT

S

kkkkk

Adjustable parameter 3

ini Heat transfer coefficient 490

end Heat transfer coefficient 250

equired around 360 min, see Fig. 4. Fig. 4(A) shows the feedate of monomer which varies over time due to the decrease ofhe heat transfer coefficient. The conversion versus time graphs shown in Fig. 4(B). It can be seen that the control strategypplied based on the temperature as the controlled variable (seeig. 4(D)) works well. Further, it can be seen that the overalleat generation is equal to the overall heat removal as describedy Eq. (27). Thus it can be concluded from this simulation thatime optimal operation is feasible for the isothermal case basedn the temperature measurement only.

Now the results for the simulation run for the same indus-rial reactor as before (10 m3) are presented for time optimalperation with evaporative cooling. A log mean temperatureifference of 15 ◦C was assumed in the previous simulation.he setpoint of the mole fraction of water in the gas phase washosen as 0.15. This value was used based on the results of simu-

ation runs shown in Fig. 3. The results of the simulation run arerovided in Figs. 5–7. The feed rate of monomer is decreasedradually by the controller equation (Eq. (28)) with the reduction

pft

able 5able of kinetic constants

ymbol Rate constant Value

I Initiator 2.6 × 1017 exp(−3.3 × 104

p Propagation 6.14 × 1010 exp(−6.3 × 10

fm Chain transfer (monomer) 2.43 × 10−4 × kp

fp Chain transfer (polymer) 2.36 × 10−4 × kp

t Termination 4.643 × 109 exp(−2.8 × 10

– Vicente et al. (2003)W/m2 K Vicente et al. (2003)W/m2 K Vicente et al. (2003)

f the heat transfer coefficient as shown in Fig. 5(A). The con-ersion versus time graph is shown in Fig. 5(B). It can be seenrom the conversion graph that the complete process can be runn 260 min, which is 100 min less than the previous simulationun for time optimal operation without evaporative cooling. Thishows an increase of productivity by almost 27%. Moleculareight averages and polydispersity index based on the momentsf dead polymer chain are plotted in Fig. 5(C) and (D).

The evolution of the feed rate of nitrogen is shown inig. 6(A). As the reaction proceeds, a smaller amount of nitrogen

s required to maintain the same mole fraction of water in the gashase (see Fig. 7(A)). The overall usage of nitrogen is shown inig. 6(B). The setpoints and the actual controlled variables arehown in Fig. 6(C) and (D).

The vaporization rates of water and monomer are shown inig. 7(A). A high amount of vaporization of water is seen com-

ared to the monomer. This is due to the fact that very littleree monomer is available during the complete process run. Ashe concentration of monomer in the reactor is increasing over

Units Reference

/1.98T) 1/s Penlidis (1986)3/1.98T) cm3/s McKenna, Graillat, and Guillot (1995)

cm3/s Chatterjee, Park, and Graessley (1977)cm3/s Chatterjee et al. (1977)

3/1.98T) cm3/s McKenna et al. (1995)

560 S. Arora et al. / Computers and Chemical Engineering 31 (2007) 552–564

Fig. 4. Time optimal operation of a 10 m3 industrial reactor without evaporative cooling. (A) Feed rate vs. time, (B) conversion vs. time, (C) rate of heat generation(solid) vs. time, rate of heat exchange (dashed) vs. time and rate of cooling by feed (dashed-dotted) vs. time and (D) setpoint temperature (solid) and the reactortemperature (dashed) vs. time

Fig. 5. Time optimal operation of a 10 m3 industrial reactor with evaporative cooling. (A) Feed rate vs. time, (B) conversion vs. time, (C) weight average molecularweight (solid) vs. time and the number average molecular weight (dashed) vs. time and (D) polydispersity index vs. time.

S. Arora et al. / Computers and Chemical Engineering 31 (2007) 552–564 561

Fig. 6. Time optimal operation of 10 m3 industrial reactor with evaporative cooling. (A) N2 feed rate vs. time, (B) total N2 used vs. time, (C) mole fraction of waterin the gas phase (solid) vs. time and the setpoint (dashed) vs. time and (D) setpoint temperature (solid) and the reactor temperature (dashed) vs. time.

Fig. 7. Time optimal operation of a 10 m3 industrial reactor with evaporative cooling. (A) Vaporization rate of water (solid) vs. time on the left y-axis and thevaporization rate of monomer (dashed) vs. time on the right y-axis, (B) total amount of water vaporized (solid) vs. time on the left y-axis and the total monomervaporized (dashed) vs. time on the right y-axis, (C) rate of heat generation (dotted) vs. time, the rate of heat exchange (dashed) vs. time, the rate of vaporizationcooling (solid) vs. time and the rate of cooling by feed and recycle (dash-dotted) vs. time and (D) the total heat generation (dotted) vs. time, the total heat exchange(dashed) vs. time, the total vaporization cooling (solid) vs. time and the total cooling by feed and recycle (dash-dotted) vs. time.

5 emica

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62 S. Arora et al. / Computers and Ch

he batch run so does its vaporization rate. This is because thenitiator is only added at the beginning of the reaction and theate of radical generation is gradually decreasing. However, theeaction rate is still high and it is not the bottleneck as comparedo the heat removal. The reason for not adding initiator duringhe reaction is that, similar conditions are used in this simulations in Vicente et al. (2003). The rates of heat generation and theeat removal by jacket cooling and vaporization are shown inig. 7(C). The decrease in the cooling rate by the jacket due to

he reduction in the overall heat transfer coefficient retards theonomer feed rate and thus the reaction speed. But, due to the

dditional evaporative cooling, the reaction rate is still main-ained at a high value. Overall 26.5% of the total heat generateds taken out by vaporization phenomenon.

. Summary and conclusions

The emulsion polymerization process offers great challengesith respect to industrial operation, optimization and control.ince the reaction is highly exothermic, the process operation

s often restricted by the heat removal constraint. One of theajor issues is to run the process at safe conditions but at higher

eaction rates. Evaporative cooling can be used for this purpose.model has been developed here that describes the effects of

aporization on the process conditions. The exact estimation ofhe molar fluxes requires the knowledge of the mass transferoefficients and the interfacial composition, area and tempera-ure. The approach used in this work is based on the approachesf Burghardt (1983, 1984) and Taylor and Krishna (1993). Sinceurghardt’s solution can be applied only when the interfacend bulk compositions are known, the vapor–liquid equilibriumalculations together with diffusion equations are solved by anterative algorithmic procedure.

Simulation results based on parameters from the literaturere provided based on which a suitable operational setpoint washosen. Further, an operational strategy for maximizing produc-ion rate is described that shows a significant improvement overhe existing processes. As much as 27% of batch reduction times obtained by using the proposed operation strategy. This couldead to an additional batch of 10 m3 of product per day, which is auge gain in the emulsion polymerization process industry. Theimulation results clearly indicate the advantages of using evap-rative cooling in addition to the usual jacket cooling, which istrongly restricted in highly viscous systems and in big reactors.

cknowledgement

The financial support by the ‘Graduate School of Productionngineering and Logistics’ at Universitat Dortmund, Germany

s gratefully acknowledged.

ppendix A. Average number of radicals per particle

The complete solution for all stages (i.e. stages I–III) haseen described rigorously by Ugelstad and Hansen (1976). Thispproach uses modified Basel’s function based on the ratesf absorption, desorption, termination and diffusion of radi-

∑∑

l Engineering 31 (2007) 552–564

als. The approach developed by Li and Brooks (1993) is anpproximate solution of the original problem. The solution thusbtained has been proved to converge to the solutions given bytockmayer (1957), O’Toole (1965), and Ugelstad and Hansen1976). The advantage of the approach lies in its simplicity ofse. This approach is also be used in our work. It can be describeds follows:

¯ = 2σ

k + q(29)

here n denotes the average number of radicals per particle,denotes the average rate of radical entry into single particle

s−1), k denotes the rate coefficient for radical exit from particles−1) and c denotes the rate coefficient for radical terminations−1). If diffusion is considered as the mechanism of radicalntry Urquiola et al. (1991), then σ can be computed as:

= ka[R]w (30)

here,

a = 4πDwrpNAFp (31)

ere, Dwi stands for the diffusion coefficient for radicals in thequeous phase, rp stands for the radius of the monomer swollenarticle and Fp as an adjustable parameter. The rate coefficientor radical desorption is calculated as follows:

d = kfm[M]pK0

K0β + kp[M]p (32)

here,

= kp[M]p + kt[R]w

kp[M]p + kt[R]w + ka(Np/NAV w)(33)

0 = 12(Dw/KpMd2

p )

1 + 2(Dw/KpMDp)

(34)

ere, dp stands for the diameter of the monomer swollen particlend Dp for the diffusion coefficient of radicals in the polymerhase. K

pM is the partition coefficient for radicals between the

ater and the polymer phase as described by Eq. (40). Thearameter q as described by Eq. (29) is computed as:

=√

k2 + 4σfc (35)

here,

= 2(2σ + k)

2σ + k + c(36)

ppendix B. Phase distribution

The following equations describe the phase distribution ofhe monomers in the emulsion polymerization systems:

pM + V d

M + V wM = VM (37)

VpM + Vpol = V p (38)

V wM + V w

W = V w (39)

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d

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w

K

K

K

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M

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A

B

B

B

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B

B

C

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G

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S. Arora et al. / Computers and Ch

VpM/V p

V wM/V w = K

pM (40)

V dM/V d

V wM/V w = Kd

M (41)

dM = V d (42)

An algorithm was developed similar to the algorithm used byraujo and Giudici (2003), to compute the unknown variables,hich is described below:

Step 1. Take initial guesses for Vp, Vd and Vw.Step 2. Solve the set of linear equations and compute the newvalues.Step 3. Take the computed values as new guesses and go tostep 2 until the values converge.

ppendix C. Molecular weight averages

Average molecular weight is calculated using the moments ofhe live and dead chains. The moments are based on the generat-ng functions approach, as described by Ray (1972). Consideringhe overall reaction in particle as well as in water phase, theotal concentration of zeroth moment for growing chains can beescribed as:

0 = (nNP/NA) + [R]wV w

V l (43)

Differential equations for moment of dead chains can then beescribed as:

d

dtλ0 = (kfm[M]p + kfpλ0 + ktdμ0)αμ0 − kfpμ0(λ0

− (1 − α)2αμ0) + 1

2ktμ

20 (44)

d

dtλ1 = μ0

1 − α((kfm[M]p + kfpλ0 + ktμ0)α(2 − α) + ktμ0)

− kfpμ20(1 − α(1 − α)2) (45)

d

dtλ2 = μ0

(1 − α)2 (2α(kfm[M]p + kfpλ0 + ktμ0)

+ ktμ0(2α + 1)) − 2kfpμ20

(1 − α(1 − α)3

1 − α

)+ d

dtλ1

(46)

ere α is called probability of propagation and is defined as:

= Kp

Kp + Kf + 2Kt(47)

here Kf, Kt and Kp are defined as:

kfm[M]pnNP kfpλ0nNP

f =

NA+ kfm[R]wV w[M]w +

NA(48)

t = kt(nNP/NA)2

V p + kt([R]w)2V w (49)

K

l Engineering 31 (2007) 552–564 563

p = kpnNP[M]p

NA+ kp[R]wV w[M]w (50)

Now, the number average molecular weight, the weight aver-ge molecular weight and the polydispersity index can be cal-ulated as:

¯n = MwM

λ1

λ0(51)

¯ w = MwM

λ2

λ1(52)

I = Mw

Mn

(53)

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