Novel Approaches to Improve the Particle Size Distribution Prediction of a Classical Emulsion Polymerization Model

8/10/2019 Novel Approaches to Improve the Particle Size Distribution Prediction of a Classical Emulsion Polymerization Model

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Novel approaches to improve the particle size distribution predictionof a classical emulsion polymerization model

Alireza Hosseini a,n, Ala Eldin Bouaswaig b, Sebastian Engell a

a Process Dynamics and Operations Group, Technische Universitat Dortmund, Germanyb BASF SE, Ludwigshafen, Germany

H I G H L I G H T S

c

We proposed two novel approaches to improve the PSD prediction of the classical emulsion polymerization PBE models.cAddition of a particle size dependent stochastic term to the growth kernel is investigated in the first approach.

c In second approach, the growth kernel is extracted from the evolution of the experimental PSDs.

c Two proposed approaches are effective in overcoming the limitations of the original PBE models of emulsion polymerization.

a r t i c l e i n f o

Article history:

Received 29 March 2012

Received in revised form

14 August 2012

Accepted 14 November 2012Available online 30 November 2012

Keywords:

Emulsion polymerization

Particle size distribution

Population balance equation

Stochastic growth

FokkerPlanck equation

Inverse problems

a b s t r a c t

A recent investigation on the homopolymerization of styrene (Hosseini et al., 2012a) showed that the

classical population balance models are incapable of predicting the evolution of the breadth of the

experimental particle size distributions correctly when a high resolution discretization method is used

to suppress the numerical errors. Also by re-tuning the model parameters the model predictions did not

fit the experimental results which points to a structural inadequacy of the conventional deterministic

growth models in describing the experimentally observed broadening phenomenon. Two novel

approaches are suggested in this work to improve the predictions. In the first approach, a possibly

size dependent stochastic term is added to the deterministic growth kernel to account for theinhomogeneities of the growth process. The probability distribution of the resulting stochastic

differential equation evolves over time based on the FokkerPlanck equation. The parameters of the

(possibly size dependent) dispersion term of the FokkerPlanck equation are used as tuning parameters

to fit the model to the experimental results. In the second approach, the growth kernel is extracted

from the characteristics of the transient experimental particle size distributions. The extracted growth

kernel is described in terms of the states of the system which affect the growth phenomenon. The

advantages and disadvantages of both approaches are highlighted.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Emulsion polymerization is a complex heterogenous process

that is used for producing a wide rage of products in an

environmentally friendly manner due to the utilization of the

water as a solvent in the process. Adhesives, paints, synthetic

rubbers, binders and coatings are some products which are

produced by emulsion polymerization. The end-use properties

of these products (latexes) such as viscosity, film forming, drying

time and adhesion are highly correlated with the particle size

distribution (PSD) of the latex, therefore the control of the PSD is

critical in emulsion polymerization processes and developing a

model to predict the PSD is highly desirable.

The modeling of the particle size distribution is a special case of

population balance equation (PBE) modeling. Several PBE models

for different types of emulsion polymerization processes have been

developed during the last decades (Abad et al., 1994;Coen et al.,

1998;Immanuel et al., 2002;Min and Ray, 1978;Melis et al., 1998;

Paquet and Ray, 1994;Rawlings and Ray, 1988;Saldivar and Ray,

1997; Saldivar et al., 1998; Zeaiter et al., 2002). The complete

model of the process is composed of the PBE and the balance

equations for the states of the continuous phase of the system.

In general, it is not possible to cover all aspects of the real

processes in mathematical models. Researchers in the field of

particulate processes have considered several strategies to deal

with the discrepancies which are observed when validating the

PBE models against experiments.

Mallikarjunan et al. (2010)attempted to fit the PSDs that were

predicted by the PBE model of emulsion copolymerization to the

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Chemical Engineering Science

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http://dx.doi.org/10.1016/j.ces.2012.11.021

n Corresponding author. Tel.:49 231 7553419; fax:49 231 7555129.E-mail address: [email protected] (A. Hosseini).

Chemical Engineering Science 88 (2013) 108120
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experimental results by introducing some correction factors.

Some of the factors were used to correct the original parameters

of the system while the others were used to correct the delayed

nucleation of the second generation of the particles or to correct

the growth rate of the large particles by multiplying the growth

kernel by a monotonically decreasing particle size dependent

function. In their work, the issue of the conservation of the mass

when the growth kernel is multiplied by another function was not

discussed, however any change in the growth rate has to becompensated by modifying the reaction rate.

Data-driven models are another option when the predictions

of the first principle models deviate considerably from the

observed reality. In cases where models for the individual kernels

of the PBE model are either unavailable or unreliable, researchers

used the inputoutput data to construct data-driven models

(Dokucu and Doyle, 2008; Dokucu et al., 2008). Due to the

inflexibility of black-box models, other intermediate approaches

where a black-box model is integrated within the body of a semi-

rigorous deterministic model (gray-box models) have been also

proposed. These approaches maintain the main structure of the

fundamental model (i.e. the PBE model) and aim at developing

models for the individual kernels by applying inverse problem

techniques (i.e. extraction of the individual kernels from the

measured data), e.g. as discussed in Bouaswaig and Engell

(2010b). Such intermediate models partly sacrifice the rigorous-

ness of the model but simultaneously maintain, to some extent,

the global nature of the model. Inverse problem approaches to the

modeling of the particulate processes have been applied for

extracting separable growth kernels in the presence of nucleation

(Mahoney et al., 2000,2002;Ramkrishna, 2000), the coagulation

kernel (Ramkrishna, 2000; Wright and Ramkrishna, 1992), the

breakage kernel (Ramkrishna, 2000), and for non-separable

growth kernels in the absence and presence of coagulation

(Bouaswaig and Engell, 2010b).

Recently, Hosseini et al. (2012a) showed that the standard

deterministic PBE models of emulsion polymerizations do not

predict the broadening of the PSDs that was observed experimen-

tally for the growth dominated semi-batch emulsion polymeriza-tion of styrene correctly. This insufficiency is hidden when a

significant numerical diffusion occurs due to the use of unsuitable

discretization methods, but is evident if a suitable discretization

method such as the modified weighted essentially non-oscillatory

method (WENO35-Z) of Bouaswaig and Engell (2010a) is used.

Investigations of the potential sources of the observed discrepan-

cies showed that most likely the reason for the mismatch is a

structural inadequacy of the model, the exclusion of the stochastic

nature of the growth phenomenon. To overcome this problem, it

was proposed to augment the model by a stochastic term with a

dispersion coefficient that is added to the deterministic growth

kernel to account for the stochastic nature of the growth process.

The solution of the resulting stochastic differential equation

evolves over time based on the FokkerPlanck equation (FPE).The FPE was used instead of the original PBE, in combination with

the other differential-algebraic equations of the standard model, to

simulate the evolution of the PSDs. The dispersion coefficient of the

FPE was used as a tuning parameter to fit the PSDs predicted by the

model to the experimental results.

In this paper, two novel approaches to improve the methods

fromHosseini et al. (2012a) and Bouaswaig and Engell (2010b)in

order to overcome the limitations of the growth dominated PBE

models of emulsion polymerization in the prediction of the experi-

mental PSDs are proposed and investigated experimentally. The

first approach is an extension of our recent work regarding the

stochastic modification of the growth kernel (Hosseini et al., 2012a).

It is proposed here to make the stochastic term which is added to

the growth kernel to account for the inhomogeneity of the growth

phenomenon dependent on the particle size. The motivation for

this is that otherwise the transient PSDs tend towards pseudo-

Gaussian shapes when they evolve over time. This might not be

suitable for applications in which not only the mean size and the

standard deviation of the PSDs but also the shape of the PSDs (eg.

skewness) should be captured. Therefore, dispersion coefficients

with linear, quadratic and exponential dependencies on the particle

size are tested in order to explore the flexibility of the suggested

approach in capturing the shape of the distributions. The secondapproach is an extension of previous work regarding the extraction

of the growth kernel from transient experimental PSDs (Bouaswaig

and Engell, 2010b). To make the extracted growth kernel generally

applicable, it is proposed here to define the evolution of the growth

kernel in terms of a new state of the system instead of the time

which was used in Bouaswaig and Engell (2010b).

The structure of this paper is as follows: after a glance at the

pseudo-bulk PBE model of emulsion polymerization and its

limitations in describing the experimental results in Section 2,

the theoretical framework of our suggested solutions for the

observed inadequacies is presented in Section 3. The experimen-

tal part of the work is presented inSection 4the result of which is

then used for the discussions of the Section 5. In Section 5, the

two methods which were presented in Section 3are applied to

the pseudo-bulk PBE model of emulsion polymerization of styr-

ene. The suggested approaches are then compared in Section 6

and finally conclusions are presented in Section 7.

2. The standard PBE model of emulsion homopolymerization

The dynamic evolution of the PSD is usually described by a PBE

(Min and Ray, 1974; Penlidis et al., 1986). Different modeling

assumptions influence the form of the PBE, among which the

types of polymerization (homo/copolymerization) and the

kinetics are the most important ones (Vale and McKenna, 2005).

In this work, the pseudo-bulk model is used (Abad et al., 1994;

Araujo et al., 2001; Forcolin et al., 1999; Herrera-Ordonez andOlayo, 2000; Immanuel et al., 2002; Kiparissides et al., 2002;

Saldivar and Ray, 1997; Sood, 2004). In this approach, it is

assumed that the number of radicals can be averaged over the

particles of the same size. The PBE of an emulsion polymerization

process can be expressed as follows:

@nr,t@t

@nr,tGr,t@r

Rnuc: Rcoag:, 1

where r is the radius of the particles, n is the population density

function, G is the growth rate, and the first and second term on

the right hand side stand for the nucleation and coagulation rates,

respectively.

The growth kernel of the model is computed as follows

(Alhamad et al., 2005;Crowley et al., 2000;Ferguson et al., 2002;Immanuel et al., 2002; Saldivar et al., 1998; Vale and McKenna,

2005;Zeaiter et al., 2002):

Gr,t kpMwt4pr2rpNA

nr,tMpt, 2

where kp is the propagation rate constant, Mwt is the monomer

molecular weight, rp is the polymer density, NA in the Avagadronumber,nr,t is the average number of radicals per particle andMp is the monomer concentration in the particle phase. The bestcontrol of the polydispersity of the PSD is possible in seeded

process where nucleation and coagulation are suppressed. There-

fore, the focus of this paper is on the growth dominated PBE and

the experiments were designed such that the growth phenomenon

is dominant. The validity of the assumptions of nucleation and

A. Hosseini et al. / Chemical Engineering Science 88 (2013) 108120 109


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coagulation not being significant was shown in Hosseini et al.

(2012a).

The discretized form of the PBE in combination with the

dynamic balances of the average number of radicals per particle,

the monomer, the initiator, the emulsifier, water, the radicals in

the water phase, the volume of the liquid inside the reactor and

the equations to calculate the partitioning of the monomer and

emulsifier constitute a set of differential algebraic equations

(DAE), the integration of which provides the dynamic evolutionof the PSD and the other states of the system (refer to Hosseini

et al., 2012afor more details).

As it was discussed inHosseini et al. (2012a), the simulation of

the experiments using the WENO35-Z scheme (Borges et al.,

2008; Bouaswaig and Engell, 2010a) showed a good agreement

regarding the conversion of the monomer and the evolution of the

mean diameter. However, the simulated PSDs are narrower than

the experimental ones; using the zero-one PBE model (Coen et al.,

1998; Zeaiter et al., 2002) delivered the same results (Hosseini

et al., 2012a). A thorough investigation of the potential sources

of the observed discrepancy such as the quality of the PSD

measurement, the occurrence of nucleation and coagulation in

the experiments, insufficient mixing, the implementation of the

model, the growth kernel, the model parameters showed that

due to the structural inadequacy in describing the growth

phenomenon, the classical PBE models, even after parameter

adaptation, are incapable of predicting the results of the

performed experiments. In the next section, the theoretical

frameworks of two different approaches to these model limita-

tions are presented. The first approach extends the stochastic

formulation of the PBE model with a dispersion term as proposed

in Hosseini et al. (2012a) whereas in the second approach the

growth kernel is extracted from experimental data. Both

approaches are general and can be used to reduce or to avoid

similar discrepancies between the prediction of the models and

the experimental results in other particulate processes.

3. Solutions to overcome the model inadequacies

3.1. Stochastic formulation of the particle size distribution

The PBE is derived from first principles by applying the

Reynolds transport theorem. In the formulation of this equation,

it is assumed that the state of the population changes through a

deterministic model and the stochastic nature of the process is

neglected (Ramkrishna, 2000). In the deterministic formulation of

the PBE, the density function (nr,t) is assumed to be a smoothand differentiable function with respect to time and the internal

and external coordinates. This means that the number density is

an average quantity (Ramkrishna, 2000) hence, the PBE delivers

average information about the transient PSDs of the populations.

If the fluctuations of the particle state around the mean value arelarge, there will be a stochastic distribution of the growth rate

between particles of the same size. This can result in stochastic

broadening of the PSD which is the commonly used deterministic

PBE models that provide average information about the popula-

tion are incapable of describing.

In emulsion polymerization the stochastic broadening may

result from the distribution of the number of radicals over the

particles of the same size. In the zero-one kinetics, the particles

with two or more radicals are neglected. In theory, the applic-

ability of the zero-one model is limited to small particle sizes in

which the occurrence of zero-one kinetic is likely and the

rate of radical termination inside the particles is much greater

than the rate of radical absorption (Saidel and Katz, 1969). Even

though the contribution of the particles with more than one

radical might not be significant in the kinetic behavior of the

system, their effect on the PSD is not necessarily negligible

because the existence of particles with more than one radical

leads to broader PSDs than what is predicted by zero-one kinetic

model (Vale and McKenna, 2005). In the pseudo-bulk kinetic

approach the compartmentalization is neglected and it is

assumed that the number of radicals can be averaged over

particles of the same size. In general this assumption is valid

when the average number of radicals per particle is high (Valeand McKenna, 2005). If this condition is not met then the particles

which contain less radicals than the average number of radicals

lag behind those particle which have more radicals than the

average number of radicals; this leads to a phenomenon which is

called stochastic broadening (Gilbert, 1995) and it is not con-

sidered in pseudo-bulk models.

Researchers have used different approaches to overcome the

deficiency of the population balance equation in describing the

size dispersion resulted from the growth fluctuations. Randolph

and White (1977) tried to model the size dispersion in the

crystallization process by using an effective number dispersion

term in the population balance model for the process of crystal-

lization of the sugar crystals. The same issue was discussed in

Middleton and Brock (1976) for modeling of the PSD in the

aerosol systems. Randolph and Larson (1988) also discussed the

addition of an dispersion term to the population balance equation

when the fluctuations of the particles around the mean size is

pronounced based on the observed evidences reported in Human

et al. (1982). However, Randolph and Larson (1988) did not

provide a theoretical explanation to motivate the addition of the

dispersion term to the population balance equation.

Matsoukas and Lin (2006)investigated the use of the Fokker

Planck equation (FPE) for modeling the particle growth by

monomer conversion. They studied the behavior of the standard

deviation of the transient distributions of the size of the particles

when a growth rate of power-law type is considered. They

showed that the first and second moments of the distribution

resulting from the FPE are exact while the PBE cannot exactly

track the second moment of the distribution.Grosso et al. (2010)used the FPE for modeling the crystal size

distribution (CSD) in an anti-solvent crystallization process.

A simple logistic function with two adjustable parameters was

considered as the growth kernel that was augmented by a

stochastic part to account for the fluctuations of the growth rate

and unknown dynamics. The same authors investigated the effect

of different diffusive terms on the performance of the FPE in

predicting the shape of the CSD in Grosso et al. (2011).

Niemann et al. (2006)studied the precipitation of nanoparti-

cles in miniemulsion droplets of water-in-oil type to produce

particles with a precise and controlled PSD. They compared the

Monte Carlo stochastic simulation of the process with the

classical PBE modeling approach. They concluded that even

though the computation time of the PBE approach is shorter thanthe Monte Carlo method, the latter provides a better agreement

with experimental data especially in terms of the prediction of

the PSD.

Haseltine et al. (2005)used the stochastic master equation to

model the stochastic behavior of a crystallization process.

They used the Monte Carlo method to simulate the system.

The simulations showed that the results of the FPE approximation

of the master equation is in good agreement with the Monte Carlo

simulation of the original discrete stochastic master equation.

The main disadvantage of direct simulation of the stochastic

master equation is the computational effort which increases by

increasing the number of particles that are simulated, therefore

the FPE approximation can be the method of choice for the

stochastic simulation of particulate systems.



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The FPE was originally derived by Fokker (1914) and Planck

(1917) to describe the temporal evolution of the probability

distribution of the particles under the Brownian motion. Later

on, FPE found its application in many different areas such as

plasma analysis (Park and Petrosian, 1996), cosmological studies

(Caceres et al., 1987), population genetics and cell replication

(Katsuhiko and Kunihiko, 2006) and more recently in particulate

processes (Grosso et al., 2011;Hosseini et al., 2012a).

The FPE can be derived based on the procedure provided inRamkrishna (2000). It is assumed that the state of an individual

particle is determined by a stochastic differential equation ofIto

type as follows :

dr

dt Gr,t

ffiffiffiffiffiffiffi2D

p dWtdt

, 3

where ris the size of the particle, G is the deterministic growth

rate,D is the dispersion coefficient and dWtis the increments of a

Wiener (white noise) process. Assuming that x is a property

which is associated with the particle size r, then this property

can be written for the population as

Xt Z rmaxrmin

xrnr,tdr: 4

The trajectory of this average property evolves over time based on

the following ordinary differential equation:

dXtdt

Z rmaxrmin

xr @nr,t@t

dr: 5

This rate equals the net birthdeath rate (Br,t) of the particlesplus the average change due to the stochastic behavior of the

particles in the particle space as follows:

dXtdt

Z rmaxrmin

xrBr,tdr dxrtdt

: 6

Considering Eq. (3) and using Itos lemma (the trajectory of an

arbitrary function fyt, where the quantity y(t) obeys the Itosstochastic differential equation dy

t=dt

ay,t

by,t

dW

t=dt

,

obeys the Itos formula dfy=dt ay,tf0y12b2y,tf00yby,t

f0ydWt=dt) gives

dxrdt

@xr@r

Gr,tD @2xr@r2

@xr@r

dWtdt


p : 7

Taking expectation gives

dxrdt

@xr

@r Gr,tD @

2xr@r2

@xr

@r

dWtdt


p : 8

By definition of the Wiener process

dWtdt


p 0, 9

therefore,

dxrdt

@xr

@r Gr,tD @

2xr@r2

Z rmaxrmin

@xr@r

Gr,tD @2xr@r2

nr,tdr: 10

Substituting Eq.(10)in Eq.(6)gives

dXtdt

Z rmaxrmin

xr @nr,t@t

dr

Z rmaxrmin

xrBr,t @xr@r

Gr,tD @2xr@r2

nr,t

dr: 11

The integrated terms that result from integrating the terms within

the square bracket by parts in Eq.(11)vanish due to the boundary

conditions (n1,t n1,t 0), therefore

Z rmaxrmin

xr @@t

nr,tBr,t @@r

Gr,tnr,t@2

@r2Dnr,t

dr 0:

12Since x(r) is stochastic, the expression within the square bracket

should be zero, therefore

@

@tnr,t @

@rGr,tnr,t D @

2

@r2nr,tBr,t: 13

Eq.(13)is the FPE and replaces the original PBE in combinationwith the other differential-algebraic equations of the model to

simulate the evolution of the PSD.

Hosseini et al. (2012a)used the FPE with a constant dispersion

term (Eq. (13)) to describe the growth dominated emulsion

polymerization process in the absence of birth and death of the

particles (Br,t 0). Eq. (13) with Br,t 0 is similar to thegrowth dominated PBE apart from the diffusive term on the right

hand side of the equation which arises from considering the

fluctuation of particles. In the above mentioned work, the disper-

sion coefficient of FPE was considered as a tuning parameter to fit

the model prediction to the experimental data. By using the FPE

with a constant dispersion coefficient, the transient PSDs tend

toward pseudo-Gaussian shapes which is not always suitable.

Therefore in this work it is proposed to use dispersioncoefficients which depend on the particle size to overcome this

limitation. When the dispersion coefficient itself is a function of

the particle size, the FPE results as follows (Beers, 2006):

@

@tnr,t @

@rGr,tnr,t @

@r Dr @nr,t

@r

Br,t: 14

Using the chain rule:

@

@r

@

@rDrnr,t @

@r Dr @nr,t

@r

@

@r nr,t dDr

dr

: 15

By substituting@=@rDr @nr,t@r from Eq. (15) in Eq. (14), thenew form of the FPE is obtained:

@

@tnr,t

@

@r G

r,t

dDr

dr nr,t

@2

@r2D

rnr,t

Br,t

16

and the corresponding Langevin equation is (Beers, 2006)

dr

dt Gr,t dDr

dr

ffiffiffiffiffiffiffiffiffiffiffiffi2Dr

p dWtdt

: 17

3.2. Extraction of the growth kernel

Mahoney et al. (2000,2002)investigated the extraction of the

growth kernel from available PSD measurements. The key

assumption that is made in this approach is that the deterministic

growth kernel is separable, i.e.

Gr,t drdt

GrGt: 18

In general, the growth kernel Gr,t describes how the character-istics of the solution of the PBE evolve over time. The character-

istics are the paths along which the individual particles evolve

such that r(t) satisfies dr=dtGr,t with the initial conditionrt0 r0 (Mahoney et al., 2002). The characteristics can beextracted from the available transient PSD measurements and

on the basis of this knowledge, the kernel that describes the

growth process can be obtained. As explained in Mahoney et al.

(2002), the quantity nr,tGr is invariant along the characteris-tics. This assumption only holds if no coagulation takes place.

This implies that the original method as described in Mahoney

et al. (2002)cannot be used to extract the growth kernel if growth

and coagulation take place simultaneously. Furthermore, in many

particulate processes, the growth kernel is not separable.

The growth kernel of emulsion polymerization model belongs to

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class of nonseparable kernels. In Eq.(2), the explicit expression fornr,t developed byLi and Brooks (1993)is used,dnr,t

dt rrkdrnr,tfrCrn2, 19

in which r is the overall absorption rate of radicals, kd is thedesorption rate coefficient, fr 4rr2kdr=2rrkdrCr and Cr ktNA=vs. Due to the nonlinear dependency of non both particle size and time, the growth kernel G

r,t

in

emulsion polymerization is nonseparable.

InBouaswaig and Engell (2010b), a novel and general approach

that can be used for extracting the growth kernel was introduced.

This approach is applicable when the assumption of separable

growth rate does not hold and when growth and coagulation are

taking place simultaneously. The simplified form of the algorithm

(without coagulation) proposed inBouaswaig and Engell (2010b)is

described here. This algorithm is more general than the algorithm

ofMahoney et al. (2002)since it can be applied without having to

impose conditions on the structure of the growth kernel. However,

the core concept of both techniques is the same; i.e. the character-

istics are extracted from the experimental data and the growth

kernel is computed from this information. This approach is based

on the fixed pivot on a moving grid method (FPMG) ofKumar and

Ramkrishna (1997). The main idea in FPMG is to define a numberof bins and discretize the original PBE (Eq. (1)) in terms of these

bins. The evolution of the boundaries of the bin i is determined by

the growth kernel. Hence, the discretized PBE for a pure growth

process is given by

dNitdt

0, 20

dridt

Gri,t, 21

where Eq. (21) describes the evolution of the characteristics of the PBE

andNi(t) represents the total number of particles in the bin iat timet:

Ni

t

Z ri 1

ri

n

r,t

dr:

22

For a pure growth problem, the total number of particles contained in

the moving bin remains constant. Making use of these facts, the

proposed approach for extracting the growth kernel for a pure growth

problem proceeds as follows (adapted from Bouaswaig and Engell,

2010b):

Step 1. Discretize the known distribution att t0 intoMbins.Step 2. Calculate Ni for each bin using Eq.(22).Step 3. At every instant of time t, at which a measurement of

the number density function nr,t is available, movethe boundaries of the bins such that Eq. (20)holds for

each bin.Step 4. Train a perceptron neural networks (PNN) or use any

other nonlinear black-box model to approximate thefunction f1which is defined as follows:

rit f1rit0,t i 1,2,. . . ,k, 23

Grit0,t dridt

df1rit0,tdt

i 1,2,. . . ,k,24

wherek is the number of discretization points. The function f1in

Eq.(23) maps the initial location of the bin boundaries obtained

in Step 1 and the time instances at which the measurements are

available,t, to the location of the boundaries obtained in Step 3.

That is, starting from an initial grid r0, Eq. (23) describes the

evolution of the characteristics of the PBE. Eq. (24) gives the

extracted growth kernel. Since it is desirable to compute Gr,t

instead of Gr0,t an additional step is added to the aboveprocedure:

Step 5. Train a PNN or use any other nonlinear black-box

model to approximate the functionf2which is defined

as follows:

rit 0 f2rit,t i 1,2,. . . ,k: 25

By substituting rit 0 from Eq. (25)in to Eq. (24), the growthkernel is obtained. For the case of simultaneous growth and

coagulation refer to Bouaswaig and Engell (2010b). It is worth

mentioning that for a pure growth problem, Step 5 can be

skipped. The nonlinear mapping described by Eq.(23)is sufficient

for describing the evolution of the PSD.

InBouaswaig and Engell (2010b), this algorithm was shown to

be efficient in the extraction of nonseparable growth kernels even

in the presence of the coagulation. However, the dependency of

the growth kernel on the lumped states is only implicitly defined

via its dependency on time and this limits its value. For an

extracted growth kernel to be useful, it should be described in

terms of the states of the system that affect the growth kernel and

evolve over time. This was recognized in the work of Mahoney

et al. (2002)in which the separable growth kernel for a precipita-

tion system was extracted as a function of the supersaturation of

the medium and the particle size.

In case of emulsion polymerization, from Eq.(2), the variables

that have an impact on the growth kernel for the emulsion

polymerization problem are kp,Mpt and nr,t. Among thesevariable only n is distributed over the particle size. f1 in the

extraction algorithm is a nonlinear function that describes the

evolution of the characteristics and is a function of the integral of

the growth kernel. Therefore it make sense to introduce an

artificial state w to replace time as an independent variable inthe algorithm of extracting growth kernel. This artificial state is

defined as follows:

dwtdt kptMwt4prpNA navtMpt: 26

It is suggested to take all variables on the right hand side of

Eq.(26) from the simulation of a lumped model of the emulsion

polymerization process (Kramer, 2005). The lumped model con-

sists of the equations mentioned inSection 2apart from the PBE

as it is assumed that all the particles are of the same size. As it

was shown inBouaswaig (2011), the prediction of the conversion

using the lumped model is in very good accord with the results of

the distributed model indicating the feasibility of the suggestion.

By adding Eq. (26)to the lumped part of emulsion polymer-

ization, the trajectory of the artificial state wt can be obtained.wt, together with the initial location of the characteristics(ri

t

0

), are used as inputs to the neural network that approx-

imates f1 in the extraction algorithm. As mentioned before, thisnonlinear mapping, together with a lumped emulsion polymer-

ization model, is sufficient for describing the evolution of the PSD

if only the pure growth problem is considered. The growth kernel

itself can be obtained straightforwardly following the proposed

algorithm.

4. Experiments

As it was mentioned before, the PSD in emulsion polymeriza-

tion can be controlled best in a seeded process where nucleation

and coagulation are suppressed properly. Furthermore, the goal of

our work is to provide a reliable model to predict the evolution of

the PSD in emulsion polymerization which is not only capable of

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predicting the final PSD but also provides a good prediction of the

transient PSDs between the seed and the final PSD. Therefore,

several PSD measurements are needed during the process to

accomplish this purpose. To accomplish the above mentioned

issues, several experiments were carried out in our experimental

facilities by modifying the original recipe of Rajabi-Hamane

(2007) to obtain a recipe which assures that coagulation and

nucleation are not significant in the process and to obtain the

transient PSDs which are needed for model validation.

The absence of significant coagulation or nucleation in the

processes that were conducted using the modified recipes was

checked by monitoring the number of the particles (Hosseini

et al., 2012a).The experiments, using styrene as monomer, were performed in

a 1 l reactor. The reactor is equipped with a marine type stirrer

(Fig. 1) and a glass condenser. For heating and cooling purposes,

the setup is connected to an external thermostat through which

the cooling medium (water) is circulated. For the data acquisition

and operation of the process an ABB control system with Freelance

software is used. The reactor is schematically illustrated in Fig. 2.

The seed was produced based on the recipe which is given in

Table 1, in batch mode. Initially, all ingredients except the initiator

were poured in to the reactor and the temperature was set to 65 1C.

The emulsion was stirred for approximately 1 h with an impeller

speed of 400 rpm to reach the desired temperature and to remove

almost all traces of dissolved oxygen under nitrogen atmosphere in

the reactor. After 2 h from the addition of the initiator, the reactor

was cooled down and the seed was collected from the reactor.

This seed was then used in the semi-batch experiments (according

to the recipes given inTable 1). Since the purpose of this work is to

investigate a pure growth process, three different recipes which

could assure the pure growth condition (E1, E2 and E3) were used

for the comparison with the model predictions. In semi-batch

mode, the emulsifier, seed, deionized water and a small amount of

monomer to swell the particles of seed were poured in the reactor

at the beginning and left for approximately 1 h to reach the desired

reaction temperature of 65 1C under nitrogen atmosphere. Then

the initiator was added and the feeding of the monomer with

constant flow rate (flow rates and the amount of monomer to be

fed in each batch are provided inTable 2) was started. The PSD and

the solid content were measured at the predetermined points of

time during the batch. For that purpose the samples were takenfrom the reactor by using a pipet. A dynamic light scatteringFig. 1. The marine type impeller used in the 1 l reactor.

B 201

P 201

B 301

P 501

P 401

HKS

R 101

M

P 301

B 501

B 401

TIR

TIR

SIR

TIR

TIR

TIR

TIR

NIR

TIR

603

601

101

102

105

104

101

101

602

LA

FIR

FIR

FIR

FIR

501

201

301

401

LIR

LIR

LIR

LIR

WIR

WIR

WIR

WIR

TIR

TIR

TIR

TIR

201

301

401

501

301

301

401

501

201

301

401

501

Fig. 2. P&ID of the 1 l reactor available at TU Dortmund.



7/13

instrument from Brookhaven (NanoDLS) and a moisture analyzer

instrument from Mettler-Toledo (HB43-S) were used for measuring

the PSD and the solid content. The samples collected from the

reactor were treated with hydroquinone to stop the reaction

immediately.

To measure the solid content of the samples, a glass fiber filter

is located in an aluminum sample pan. The aluminum sample pan

with the filter inside is put into a structure which can then

directly be placed in the moisture analyzer device which is

equipped with a halogen heating system. Using a circular halogen

lamp in this device ensures the quick and uniform heating and

drying of the samples. After weighting of the filter, sample pan

and the structure which is done automatically by the instrument,

several droplets of the sample are dropped on the filter and by

closing the cap of the instrument the measurements commences.

After several minutes (210 min which depends on the accuracy

which is needed) the sample is dried and the solid content is

reported by the instrument.

As mentioned above dynamic light scattering is used to

measure the PSD in this work. This technique is a relatively fast

technique (26 min per sample) for determining the PSD.

Dynamic Light Scattering can be used for measuring the size of

the particles typically in the sub-micron region where it measures

the fluctuation of laser beam intensity due to the Brownian

motion of the particles after passing through the liquid sample,

and relates this to the size of the particles. The larger the particle,the slower is the Brownian motion, and hence the fluctuation of

the intensity is smoother. InHosseini et al. (2012a), the accuracy

of the results of this dynamic light scattering device was checked

by comparison with another measurement technique with higher

resolution but slower (11.5 h per sample) such as scanning

electron microscopy (SEM).

5. Results and discussion

As mentioned before, in this work only the growth dominated

PBE is considered. The experiments were designed such that the

growth phenomenon is dominant by using a sufficient amount of

emulsifier. Monitoring the number of particles during the experi-ments confirmed these assumptions (Hosseini et al., 2012a).

Hosseini et al. (2012a) showed that the simulation of experi-

ments using the WENO35-Z scheme to discretize the PBE is in

good accord with the experimental data in terms of the monomer

conversion and the mean size of the particles while the PSDs

predicted by model are narrower than the experimental PSDs.

Here, the two approaches described inSection 3are applied to

improve the accuracy of the PBE models of emulsion polymeriza-

tion in predicting the experimental PSDs.

5.1. Stochastic approach in emulsion polymerization of styrene

Different dependencies of the dispersion coefficient (linear,

quadratic and exponential) on the particle size are considered to

increase the flexibility of the approach proposed byHosseini et al.

(2012a) in capturing the shape of the experimental PSDs.

The different forms of the dispersion coefficient are monotoni-

cally increasing and are give by

Dr a rr0

, 27

Dr a rr0

b r

r0

2, 28

Dr a expbr, 29where r0 is the minimum radius. The parameters of these

functions (a and b) were adapted to fit the PSDs predicted bythe model to the experimental data. The parameters are esti-

mated using least square technique in which the objective

function is the sum of the squares of the differences between

the experimental and simulated PSDs. The considered objectivefunction is

JXmi0

nnti,rnti,r:nnti,rnti,rT, 30

in whichm is the number of measurements and nnti,ris the thevector of experimental PSD at time ti. The parameters of the

functions D(r) were estimated for experiment E1, using the

glbDirect global solver from the TOMLAB optimization package.

The resulting parameters for linear, quadratic and exponential

dispersion functions are given inTable 2.Fig. 3demonstrates the

dispersion functions that resulted by using the adapted para-

meters.Fig. 4shows the comparison of the transient PSDs that are

predicted by model with the results of experiment E1 when

instead of the PBE (Eq.(1)), the FPE (Eq. (16)) is used consideringconstant (D 1:132 1018 dm2=sec, fromHosseini et al., 2012a),linear, quadratic and exponential dispersion coefficients.

The results of the model, using the estimated parameters for

experiment E1, are compared with the experimental results of E2

and E3. InFigs. 5 and 6, there is a very good accord between the

experiments and the model predictions. In comparison to the

incorporation of the constant dispersion coefficient used in

Hosseini et al. (2012a), linear, quadratic and exponential disper-

sion functions deliver better results among which the exponential

dispersion function is the most suitable for this system.

Due to the implicit change in the deterministic growth rate,

the reaction rate must be adapted to assure the conservation of

mass. Moreover, one has to track the relative magnitude of the

stochastic and of the deterministic part of the growth kernel to

Table 1

Recipes of the POLYDYN experiments.

Experiment Seed E1 E2 E3 Ingredient Provider

Seed (g) 50.86 50.01 51.38 Polystyrene TU Dortmund

Monomer (g) 100 5.18 5.06 5.03 Styrene MERCK

Initiator (g) 0.3423 1.26 1.29 1.26 Ammonium peroxydisulfate MERCK

Emulsifier (g) 12.112 1.03 1.16 1.07 Sodium dodecyl sulfate MERCK

Water (g) 800 435.46 410.59 431 Deionized water TU Dortmund

Flow rate (g/min) 0.832 0.585 0.75 Monomer to be fed (g) 168.37 158 132

Table 2

Parameters.

Dispersion function Parameters

Linear a 1:24 1019 dm2 s1Quadratic a 1:4 1021 dm2 s1, b 2:18 1020 dm2 s1Exponential a 1:44 1019 dm2 s1, b 0:041 dm1

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8/13

assure that the fluctuations do not lead to negative values of the

growth rate within the range of particle sizes considered in

this work.

The overall monomer consumption rate is given as

ripkpZ rmaxrmin

Mprnr,tnr,tdr: 31

By multiplying Eq.(16)by r3 and some parameters and integrat-

ing both sides of the equation over the whole range of particlesizes, one obtains the effective reaction rate of the system when

the FPE is used:

r0ip4prpNA

3Mwt

Z rmaxrmin

r3 @@r

Gr,tdDrdr

nr,t

@2Drnr,t

@r2

dr: 32

This reaction rate expression should be used instead of Eq. (31) to

ensure the conservation of mass. As one can see in Fig. 7 the trajectory

of the polymer mass using the FPE and considering Eq. (32)exactly

matches the results of the original PBE model (using Eqs. (1) and (31)).

There is also a good agreement with the experimental results.

To monitor the relative magnitude of the stochastic and determi-

nistic growth kernels, the Langevin equation can be solved directly for

many particles using the Monte Carlo method and averaging (Ermak

and Buckholz, 1980). As an alternative to this computationally

intensive approach, one can use the FPE which describes the evolu-

tion of the probability of the particles having a specific growth rate in

the interval of [r,

rdr] (Hosseini et al., 2012b). For that purpose, onecan extract the stochastic growth kernel from the characteristics ofthe transient PSDs obtained from the simulation of the FPE using the

method described inSection 3.2. Using this approach, the stochastic

growth rate averaged along the characteristics can be compared with

the deterministic one. To reduce the error which arises from taking

the derivative of the output of a neural network, in step 4 of the

algorithm of Section 3.2, instead of neural networks polynomials

were fit to the characteristics and the derivatives of these polynomials

were calculated in order to determine the growth kernels along the

characteristics. Polynomials of sixth order were found to be in a good

fit for the extracted characteristics (Fig. 8). By taking the derivatives of

the characteristic functions with respect to time, the growth rate of

the stochastic process along each characteristic curve is obtained.

As one can see inFig. 9,the comparison of the average stochastic and

deterministic growth kernels along the extracted characteristics

shows that the average stochastic growth rate, at the upper boundary

of the particle sizes, is nowhere greater by more than 24% and, at the

lower boundary of the particle sizes, is nowhere smaller by more than

10% of the deterministic growth kernel. So, the consistency of our

approach was shown.

5.2. Extraction of a state dependent growth kernel in emulsion

polymerization of styrene

The PNN that is used to approximate f1 in the algorithm of

Section 3.2for the case of the pure growth of a polystyrene seed

consists of one hidden layer. The hidden layer contains three

neurons with sigmoid transfer functions and the output layer

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

t=5400 sec

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

Densityfunction[nm-1]

t=7800 sec

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]




t=9600 sec

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

t=16800 sec

Fig. 4. Evolution of the PSD, FokkerPlanck model with constant, linear, quadratic and exponential dispersion coefficient vs. experiment E1 (dashed). The dispersion

coefficient was fitted to the data. Error PSDsimulationPSDExperiment PSDsimulationPSDExperimentT.

0 0.2 0.4 0.6 0.8 1 1.2

x 106

0

0.5

1

1.5

2

x 1017

Radius [dm]

Dispersioncoefficient[dm

2.se

c1]

LinearQuadratic

Exponential

Fig. 3. Comparison of the dispersion functions, the parameters were fitted to the

results of experiment E1.

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9/13

contains one neuron with a linear transfer function. Since there

are two inputs to the PNN, the total parameters that have to be

determined by training the PNN are fifteen (five biases and ten

weights). The artificial statew is used instead of time.From the selected PSD measurement of the experiment E1, the

evolution of the characteristics was extracted as described in Step 3 of

the algorithm presented in Section 3.2. This information is used to

train the neural network. The information obtained from the char-

acteristics of the selected PSD measurements of the experiments E2

was used for validation. The trained PNN is used to predict the

evolution of the PSD starting from the seed for experiment E3

according to the algorithm ofSection 3.2. The results of the proposed

model to the experiments E2 and E3 are illustrated inFigs. 10 and 11.

Clearly the transient PSDs are captured well indicating that the

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

Density

function[nm-1]

Density

function[nm-1]

t= 6000 sec

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

t= 7200 sec

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

t= 9600 sec

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

Dens

ityfunction[nm-1]

Dens

ityfunction[nm-1]

t=12000 sec

Fig. 6. Evolution of the PSD, FokkerPlanck model with constant, linear, quadratic and exponential dispersion coefficient vs. experiment E3. The dispersion coefficient

was fitted to the data inFig. 4. Error PSDsimulationPSDExperiment PSDsimulationPSDExperimentT.

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]


t= 6600 sec

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]


t= 7800 sec

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

D

ensityfunction[nm-1]

t= 10800 sec

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

D

ensityfunction[nm-1]

t= 16200 sec

Fig. 5. Evolution of the PSD, FokkerPlanck model with constant, linear, quadratic and exponential dispersion coefficient vs. experiment E2 (dashed). The dispersion

coefficient was fitted to the data in Fig. 4. Error PSDsimulationPSDExperiment PSDsimulationPSDExperimentT.



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designed nonlinear mapping function f1 can predict the evolution

of the characteristics accurately. The quadratic error (PSDmodelPSDExperimnet:PSDmodelPSDExperimnetT) between the PSDs predictedby the model and the experimental PSDs are also provided in the

above mentioned figures.

6. Comparison of the two approaches

In this work, two different novel approaches to the modeling of

the PSD in emulsion polymerization were proposed. Both

approaches were shown to be efficient to overcome the limitations

of the classical PBE model of emulsion polymerization in predicting

the experimental PSDs. The pros and cons of these approaches can

be summarized as follows:

Growth kernel extraction Theoretically, any shape of the distribution can be approximated. For the pure growth problem, the solution of the model is fast,

due to the incorporation of the lumped model instead of the

distributed model. The approach is effective also if the growth kernel is non-separable and = or the growth and coagulation take place

simultaneously provided a model of coagulation is available

Limited extrapolation capacity in comparison to a rigorousmodel.

Online correction of the extracted kernel is not simple, if notimpossible.

Stochastic correction of the growth kernel There is a physical justification behind the approach. It is rigorous and flexible. If needed, online estimation of the dispersion coefficients is

possible. The shape of the dispersion function is not physically justified.

The use for bimodal PSDs is unclear.

Based on the results of Section 5, the extraction approach is

more accurate which is not a surprise as it is more flexible.

The FPE prediction is not good in the early stages of the process

but gets better towards the end. The growth rates of the two

different approaches proposed in Sections 3.1 and 3.2 of the

paper, are compared inFig. 12for experiment E1. To obtain these

growth rates, the algorithm described in Section 3.2 was used.

To compute the growth rate of the first approach (the Fokker

Planck equation) first, the characteristics of the solution are

obtained from the evolution of the PSD obtained from the

simulation of the experiment E1 by the FokkerPlanck equation

and usingstep 3of algorithm ofSection 3.2. Then, the growth rate

of the FokkerPlanck equation, along the characteristics, is

obtained by taking the derivative of the extracted characteristics

with respect to time. This growth kernel is then compared with

the growth kernel obtained from the evolution of the character-

istics of the PSD of the experiment E1 in Section 5.2. As one can

see in Fig. 12, the growth kernels extracted from the two

approaches are in good agreement.

The choice of the appropriate method to overcome the inade-

quacies of the original models of emulsion polymerization in the

prediction of the PSD depends on the application at hand.

For example, if the model is applied for the online monitoring

or the control of the PSD and there exists a transient discrepancy

between the model predictions and the experimental results, the

extraction approach cannot be used for further online model

correction, but the stochastic correction may be the method of

choice since there exists the possibility of adapting the model tothe observed discrepancies by estimating the parameters of the

dispersion functions online. In other applications one might need

a higher accuracy in the prediction of the PSD and the extraction

method serves this purpose within its domain of validation.

The approaches presented in this work are general and can be

used for other particulate processes.

7. Conclusion

Hosseini et al. (2012a) showed that the inadequacy of the

classical PBE models of emulsion polymerization in predicting the

experimental PSDs cannot be surmounted by simply re-tuning

the parameters of the PBE models of emulsion polymerization

0 5000 10000 150000

0.5

1

1.5

2

Time [sec]

Radius[dm]

Characteristic Curves

Char.

Fitted Poly.

Char.

Fitted Poly.

Char.

Fitted Poly.

106

Fig. 8. Fitted polynomials of sixth order to the characteristic curves, using

FokkerPlanck equation model, simulation of experiment E3.

Fig. 9. Relative difference between the deterministic and the extracted stochastic

growth rates.

0 1 2 3 4 50

50

100

150

Time [hr]

M

assofpolymer[g]

Model

Modified Model

Experiment

Fig. 7. Prediction of the mass of polystyrene, using the FokkerPlanck equation

with the modified reaction rate (dashed) and using the deterministic population

balance model with the standard reaction rate (solid) vs. experiment E1.



11/13

within reasonable bounds. In this work, two different novel

approaches to overcome the inadequacy of the classical PBE

models of emulsion polymerization in describing the experimen-

tal PSDs are presented and investigated.

In the first approach, a stochastic term with a particle size

dependent dispersion coefficient is added to the growth kernel in

order to account for the stochastic nature of the growth phenom-

enon which is neglected in the original model. The probability

distribution of the resulting stochastic differential equation

evolves over time based on the FPE. The dispersion coefficients

of the FPE were assumed to have a linear, quadratic or

exponential dependency on the particle size. By using this new

approach, one can capture the shape of the transient PSDs better

than with a constant dispersion coefficient (Hosseini et al.,

2012a).

The second approach provides a general method to extract a

semi-deterministic growth kernel from the trajectories of the

characteristic curves of the transient experimental PSD. This

approach is useful for extracting (separable or nonseparable)

growth kernels in the presence or in the absence of coagulation.

In this paper, for a pure growth problem the extracted growth

kernel is described in terms of the state of the system that affects

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]


t= 6600 sec

Error= 2.7105

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]


t= 7800 sec

Error= 2.3104

0 20 40 60 80 100 120 140

0

0.05

0.1

Radius [nm]


t= 10800 sec

Error= 3.88104

0 20 40 60 80 100 120 140

0

0.05

0.1

Radius [nm]


t= 16200 sec

Error= 8.4105

Fig. 10. Evolution of the PSD, model with extracted growth kernel (solid) vs. experiment E2 (dashed). Error PSDsimulationPSDExperiment PSDsimulationPSDExperimentT.

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

t= 6000 sec

Error= 1.086104

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

t= 7200 sec

Error= 6.92104

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]





t= 9600 sec

Error= 1.61104

0 20 40 60 80 100 120 1400

0.05

0.1

Radius [nm]

t=12000 sec

Error= 1.0139104

Fig. 11. Evolution of the PSD, model with extracted growth kernel (solid) vs. experiment E3 (dashed). Error PSDsimulationPSDExperiment PSDsimulationPSDExperimentT.



12/13

the growth phenomenon instead of in terms of time which was

used in Bouaswaig and Engell (2010b). This effective state is

computed from the simulation of the lumped model of emulsion

polymerization.

Both proposed approaches were found to be effective in

overcoming the limitations of the original PBE models of emulsion

polymerization to describe experimental results. Our current

research is focused on using the FPE model for controlling the

PSD in emulsion polymerization.

Nomenclature

Br,t net rate of birth-death for FokkerPlanckequation, mol.l1 dm1 s1

D(r) dispersion coefficient of the FokkerPlanck

equation, dm2 s1

G growth rate of polymer particles, nm s1

Gt time dependent part of a separable growth kernel, s1Gr size dependent part of a separable growth kernel, dmkd desorption rate coefficient, s

1

kp propagation rate constant, l mol1 s1

kt termination rate constant, l mol1 s1

MP monomer concentration in the particle phase,mol l1

Mwt molecular weight of monomer, g mol1

n population density function, mol l1 dm1

n average number of radicals per particle

navet average number of radicals calculated from thelumped model of emulsion polymerization

NA avogadros number, mol1

Ni concentration of the particles within bini, mol l1

r particle radius, dm

t time, s

vs volume of monomer swollen particle, dm3

Rnuc: nucleation rate, mol l1 dm1 s1

Rcoag: coagulation rate, mol l1 dm1 s1

x(r) a property which is associated with particle sizer

X a property which is associated with the whole

population of the particles

Greek letters

r absorption rate, s1

rp polymer density, g l1

rm monomer density, g l1

wt artificial state introduced in inverse growthproblem, l

Abbreviations

DAE differential algebraic equation

DLS dynamic light scattering

FMPG fix pivot on amoving grid method

FPE FokkerPlanck equation

PSD particle size distributionPBE population balance equation

WENO35-Zmodified weighted essentially nonconciliatory

scheme (order 3,5)

Acknowledgments

The financial support of the Deutscher Akademischer Austausch

Dienst(DAAD) is gratefully acknowledged.

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Grosso, M., Cogoni, G., Baratti, R., Romagnoli, J.A., 2011. Stochastic approach for theprediction of PSD in crystallization processes: Formulation and comparativeassessment of different stochastic models. Ind. Eng. Chem. Res. 50 (4),21332143.

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Extracted stochastic growth kernel from the FPE

Extracted growth kernel from the experiment E1

Fig. 12. The stochastic growth kernel extracted from the simulation of experiment

E1 using FokkerPlanck equation with exponential dispersion function vs. growth

kernel extracted from the experiment E1.



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Novel Approaches to Improve the Particle Size Distribution Prediction of a Classical Emulsion Polymerization Model