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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
Contents lists available at SciVerse ScienceDirect
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Chemical Engineering Science
0009-2509/$- see front matter& 2012 Elsevier Ltd. All rights reserved.
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: alireza.hosseini@bci.tu-dortmund.de (A. Hosseini).
Chemical Engineering Science 88 (2013) 108120
http://www.elsevier.com/locate/ceshttp://www.elsevier.com/locate/ceshttp://dx.doi.org/10.1016/j.ces.2012.11.021mailto:alireza.hosseini@bci.tu-dortmund.dehttp://dx.doi.org/10.1016/j.ces.2012.11.021http://dx.doi.org/10.1016/j.ces.2012.11.021mailto:alireza.hosseini@bci.tu-dortmund.dehttp://dx.doi.org/10.1016/j.ces.2012.11.021http://dx.doi.org/10.1016/j.ces.2012.11.021http://dx.doi.org/10.1016/j.ces.2012.11.021http://www.elsevier.com/locate/ceshttp://www.elsevier.com/locate/ces8/10/2019 Novel Approaches to Improve the Particle Size Distribution Prediction of a Classical Emulsion Polymerization Model
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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.
A. Hosseini et al. / Chemical Engineering Science 88 (2013) 108 120110
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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
ffiffiffiffiffiffiffi2D
p : 7
Taking expectation gives
dxrdt
@xr
@r Gr,tD @
2xr@r2
@xr
@r
dWtdt
ffiffiffiffiffiffiffi2D
p : 8
By definition of the Wiener process
dWtdt
ffiffiffiffiffiffiffi2D
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
A. Hosseini et al. / Chemical Engineering Science 88 (2013) 108120 111
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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.
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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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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]
Densityfunction[nm-1]
Densityfunction[nm-1]
Densityfunction[nm-1]
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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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]
Densityfunction[nm-1]
t= 6600 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]
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.
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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]
Densityfunction[nm-1]
t= 6600 sec
Error= 2.7105
0 20 40 60 80 100 120 1400
0.05
0.1
Radius [nm]
Densityfunction[nm-1]
t= 7800 sec
Error= 2.3104
0 20 40 60 80 100 120 140
0
0.05
0.1
Radius [nm]
Densityfunction[nm-1]
t= 10800 sec
Error= 3.88104
0 20 40 60 80 100 120 140
0
0.05
0.1
Radius [nm]
Densityfunction[nm-1]
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]
Densityfunction[nm-1]
Densityfunction[nm-1]
Densityfunction[nm-1]
Densityfunction[nm-1]
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.
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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.
References
Abad, C., De la Cal, J.C., Asua, J.M., 1994. Emulsion copolymerization in continuousloop reactors. Chem. Eng. Sci. 49, 50255037.
Alhamad, B., Romagnoli, J.A., Gomes, V.G., 2005. Advanced modelling and optimaloperating strategy in emulsion copolymerization: application to styrene/MMAsystem. Chem. Eng. Sci. 60, 27952813.
Araujo, P.H., De-la-Cal, J.C., Asua, J.M., Pinto, J.C., 2001. Modeling particle sizedistribution (PSD) in emulsion copolymerization reactions in a continuousloop reactor. Macromol. Theory Simul. 10, 769779.
Beers, K.J., 2006. Numerical Methods for Chemical Engineering: Applications inMATLAB. Cambridge University Press.
Borges, R., Carmona, M., Costa, B., Don, W., 2008. An improved weightedessentially non-oscillatory scheme for hyperbolic conservation laws. J. Com-put. Phys. 227, 31913211.
Bouaswaig, A.E., Engell, S., 2010a. Comparison of high resolution schemes forsolving population balances. Ind. Eng. Chem. Res. 49, 59115924.
Bouaswaig, A.E., Engell, S., 2010b. An inverse problem approach to extract thegrowth kernel in particulate processes. In: Proceedings of the 9th InternationalSymposium on Dynamics and Control of Process Systems (DYCOPS 2010),
Leuven, Belgium, July 57 .Bouaswaig, A.E., 2011. Simulation, control, and inverse problems in particulate
processes. Ph.D. Dissertation, Schriftenreihe des Lehrstuhls fuer Anlagen-steuerungstechnik der Universitaet Dortmund, Shaker Verlag.
Caceres, Manuel O., Diaz, Mario C., Pullin, Jorge A., 1987. Stochastic processes incosmology. Phys. Lett. A 123 (7), 329335.
Coen, E.M., Gilbert, R.G., Morrison, B.R., Leube, H., Peach, S., 1998. Modelingparticle size distributions and secondary particle formation in emulsionpolymerisation. Polymer 39 (26), 70997112.
Crowley, T.J., Meadows, E.S., Kostoulas, E., Doyle III, F.J., 2000. Control of particlesize distribution described by a population balance model of semibatchemulsion polymerization. J. Process Control 10, 419432.
Dokucu, M., Park, M., Doyle III, F.J., 2008. A Reduced-order methodologies forfeedback control of particle size distribution in semi-batch emulsion copoly-merization. Chem. Eng. Sci. 63, 12301245.
Dokucu, M., Doyle III, F.J., 2008. Batch-to-batch control of characteristic points onthe PSD in experimental emulsion polymerization. Am. Inist. Chem. Eng. J. 54,31713178.
Ermak, D., Buckholz, H., 1980. Numerical integration of the Langevin equation:
Monte Marlo simulation. J. Comput. Phys. 35, 169182.Ferguson, C.J., Russell, G.T., Gilbert, R.G., 2002. Modelling secondary particle
formation in emulsion polymerisation: application to making core-shellmorphologies. Polymer 43, 45574570.
Fokker, A.D., 1914. Die mittlere Energie rotierender elektrischerdipole im Strah-lungsfeld. Ann. Phys. 348 (4. Folge 43), 810820.
Forcolin, S., Marconi, A.M., Ghielmi, A., Butte, A., Storti, G., Morbidelli, M., 1999.Coagulation phenomena in emulsion polymerization of vinyl chloride. PlasticsRubber Composit. 28, 0915.
Gilbert, R.G., 1995. Emulsion polymerization: A Mechanistic Approach. AcademicPress, San Diego.
Grosso, M., Galan, O., Baratti, R., Romagnoli, J.A., 2010. A stochastic formulation forthe description of the crystal size distribution in antisolvent crystallizationprocesses. Am. Inist. Chem. Eng. J. 56, 20772087.
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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025
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100125
0
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Time[m
in]Radius[nm]
Gro
wth[nms1]
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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