Download - 1-s2.0-S0009250905006202-main
-
8/12/2019 1-s2.0-S0009250905006202-main
1/15
Chemical Engineering Science 61 (2006) 332 346
www.elsevier.com/locate/ces
A generalized population balance model for the prediction of particle sizedistribution in suspension polymerization reactors
Costas Kotoulas, Costas Kiparissides
Department of Chemical Engineering, Aristotle University of Thessaloniki and Chemical Process Engineering Research Institute,
P.O. Box 472 541 24 Thessaloniki, Greece
Received 21 February 2005; received in revised form 1 July 2005; accepted 3 July 2005
Available online 22 August 2005
Abstract
In the present study, a comprehensive population balance model is developed to predict the dynamic evolution of the particle size
distribution in high hold-up (e.g., 40%) non-reactive liquidliquid dispersions and reactive liquid(solid)liquid suspension polymerization
systems. Semiempirical and phenomenological expressions are employed to describe the breakage and coalescence rates of dispersed
monomer droplets in terms of the type and concentration of suspending agent, quality of agitation, and evolution of the physical,
thermodynamic and transport properties of the polymerization system. The fixed pivot (FPT) numerical method is applied for solving the
population balance equation. The predictive capabilities of the present model are demonstrated by a direct comparison of model predictions
with experimental data on average mean diameter and droplet/particle size distributions for both non-reactive liquidliquid dispersions
and the free-radical suspension polymerization of styrene and VCM monomers.
2005 Elsevier Ltd. All rights reserved.
Keywords: Population balance model; Suspension polymerization; PVC; Polystyrene
1. Introduction
Suspension polymerization is commonly used for pro-
ducing a wide variety of commercially important polymers
(i.e., polystyrene and its copolymers, poly(vinyl chlo-
ride), poly(methyl methacrylate), poly(vinyl acetate)). In
suspension polymerization, the monomer is initially dis-
persed in the continuous aqueous phase by the combined
action of surface-active agents (i.e., inorganic or/and water-
soluble polymers) and agitation. All the reactants (i.e.,
monomer, initiator(s), etc.) reside in the organic or oilphase. The polymerization occurs in the monomer droplets
that are progressively transformed into sticky, viscous
monomerpolymer particles and finally into rigid, spherical
polymer particles of size 50500m (Kiparissides, 1996).
Corresponding author. Tel.: +30 2310 99 6211;fax: +310 231099 6198.
E-mail addresses: [email protected],
[email protected](C. Kiparissides).
0009-2509/$- see front matter 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2005.07.013
The polymer solid content in the fully converted suspension
is typically 3050% w/w.
The suspension polymerization process can in general be
distinguished into two types (Kalfas, 1992): the bead poly-
merization, where the polymer is soluble in its monomer
and smooth spherical particles are produced; and the pow-
der polymerization, where the polymer is insoluble in its
monomer and, thus, precipitates out leading to the forma-
tion of irregular grains or particles. The most important
thermoplastic produced by bead suspension polymerization
is polystyrene (PS). In the presence of volatile hydrocar-bons (C4C6), foamable beads (the so-called expandable
polystyrene, EPS) can be produced. On the other hand,
poly(vinyl chloride) (PVC), which is the second largest ther-
moplastic manufactured in the world, is an example of pow-
der polymerization.
One of the most important issues in suspension poly-
merization process is the control of the final particle size
distribution (PSD) (Yuan et al., 1991). The initial monomer
droplet size distribution (DSD) as well as the final polymer
http://www.elsevier.com/locate/cesmailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://www.elsevier.com/locate/ces -
8/12/2019 1-s2.0-S0009250905006202-main
2/15
C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332346 333
Fig. 1. Schematic representation of drop breakage and coalescence mech-
anism.
PSD in general depend on the type and concentration of
the surface-active agents, the quality of agitation and the
physical properties (e.g., density, viscosity and interfacial
tension) of the continuous and dispersed phases. The tran-
sient droplet/particle size distribution is controlled by two
dynamic processes, namely, the drop/particle breakage and
coalescence rates. The former mainly occurs in regions of
high-shear stress (i.e., near the agitator blades) or as a result
of turbulent velocity and pressure fluctuations along the sur-
face of a drop. The latter is either increased or decreased by
the turbulent flow field and can be assumed to be negligible
for very dilute dispersions at sufficiently high concentrations
of surface-active agents (Chatzi and Kiparissides, 1992).When drop breakage occurs by viscous shear forces, the
monomer droplet is first elongated into two fluid lumps sep-
arated by a liquid thread (see Fig. 1a). Subsequently, the
deformed monomer droplet breaks into two almost equal-
size drops, corresponding to the fluid lumps, and a series of
smaller droplets corresponding to the liquid thread. This is
known as Thorough breakage. On the other hand, a droplet
suspended in a turbulent flow field is exposed to local pres-
sure and relative velocity fluctuations. For nearly equal den-
sities and viscosities of the two liquid phases, the droplet
surface can start oscillating. When the relative velocity is
close to that required to make a drop marginally unstable,a number of small droplets are stripped out from the initial
one (seeFig. 1b). This situation of breakage is referred to as
erosive one. Erosive breakage is considered to be the dom-
inant mechanism for low-coalescence systems that exhibit
a characteristic bimodality in the PSD (Chatzi and Kiparis-
sides, 1992; Ward and Knudsen, 1967).
Two different mechanisms have been proposed in the lit-
erature to describe the coalescence of two drops in a tur-
bulent flow field. The first one (Shinnar and Church, 1960)
assumes that after the initial collision of two drops, a liquid
film of the continuous phase is being trapped between the
drops that prevents drop coalescence (seeFig. 1c). However,
due to the presence of attractive forces, draining of the liq-
uid film can occur leading to drop coalescence. On the other
hand, if the kinetic energy of the induced drop oscillations
is larger than the energy of adhesion between the drops,
the drop contact is broken before the complete drainage of
the liquid film. The second drop coalescence mechanism
(Howarth, 1964) assumes that immediate coalescence oc-curs when the approach velocity of the colliding drops at
the collision instant exceeds a critical value. In other words,
if the turbulent energy of collision is greater than the total
drop surface energy, the drops will coalesce (seeFig. 1d).
Surface-active agents play a very important role in the
stabilization of liquidliquid dispersions. One of the most
commonly used suspension stabilizers is poly(vinyl acetate)
that has been partially hydrolyzed to poly(vinyl alcohol)
(PVA). By varying the acetate content (i.e., degree of hy-
drolysis), one can alter the hydrophobicity of the PVA and,
thus, the conformation and surface activity of the polymer
chains at the monomer/water interface (Chatzi and Kiparis-
sides, 1994). The solubility of the PVA in water depends on
the overall degree of polymerization (i.e., molecular weight),
the sequence chain length distribution of the vinyl alco-
hol and vinyl acetate in the copolymer, the degree of hy-
drolysis and temperature. Depending on the agitation rate,
the concentration and type of surface-active agent, the av-
erage droplet size can exhibit a U-shape variation with re-
spect to the impeller speed or the degree of hydrolysis of
PVA. This U-type behaviour has been confirmed both ex-
perimentally and theoretically and has been attributed to
the balance of breakage and coalescence rates of monomer
drops.
In regard with the droplet/particle breakage and coales-cence phenomena, the suspension polymerization process
can be divided into three stages (Hamielec and Tobita, 1992;
Maggioris et al., 2000). During the initial low-conversion
(i.e., low-viscosity) stage, drop breakage is the dominant
mechanism. As a result the initial DSD shifts to smaller
sizes. During the second sticky-stage of polymerization, the
drop breakage rate decreases while the drop/particle coales-
cence becomes the dominant mechanism. Thus, the average
particle size starts increasing. In the third stage, the PSD
reaches its identification point while the polymer particle
size slightly decreases due to shrinkage (i.e., the polymer
density is greater than the monomer one).For the PS process, Villalobos et al. (1993) reported
that the end of the first stage occurs at approximately 30%
monomer conversion, corresponding to a critical viscosity
of about 0.1 Pa s, while the second stage extends up to 70%
monomer conversion. In the VCM powder polymerization,
at monomer conversions around 1030%, a continuous
polymer network is commonly formed inside the poly-
merizing monomer droplets that significantly reduces the
drop/particle coalescence rate (Kiparissides et al., 1994).
Cebollada et al. (1989)reported that the PSD is essentially
established up to monomer conversions of about 3540%
(i.e., end of the second stage).
http://-/?- -
8/12/2019 1-s2.0-S0009250905006202-main
3/15
334 C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332 346
The paper is organized as follows. In Section 2, a gen-
eralized population balance model is developed to describe
the dynamic evolution of PSD in batch suspension polymer-
ization reactors. The model takes into account the dynamic
evolution of the physical properties of the continuous and
dispersed phases, in terms of the variation of monomer con-
version and the turbulent intensity characteristics of the flowfield, as well as their relative effects on breakage and coales-
cence mechanisms. In Section 3 of the paper, the fixed pivot
technique (FPT) (Kumar and Ramkrishna, 1996)is applied
for solving the general population balance equation, govern-
ing the PSD developments, in terms of the polymerization
conditions (e.g., monomer to water volume ratio, temper-
ature, type and concentration of stabilizer, impeller energy
input, etc.) and the polymerization kinetic model. An ex-
tensive analysis on the robustness of the numerical method
is carried out in regard with the convergence of the solu-
tion and the conservation of the total mass in the system.
Finally, in Section 4 of the paper, the capabilities of the
present model are demonstrated by a direct comparison of
model predictions with experimental data on average mean
diameter and droplet/particle size distributions for both non-
reactive liquidliquid dispersions and the free-radical sus-
pension polymerization of styrene and VCM monomers.
2. Model developments
To follow the dynamic evolution of PSD in a particulate
process, a population balance approach is commonly em-
ployed. The distribution of the droplets/particles is consid-
ered to be continuous in the volume domain and is usu-ally described by a number density function, n(V,t). Thus,
n(V,t) dVrepresents the number of particles per unit vol-
ume in the differential volume size range (V , V + dV ). Fora dynamic particulate system undergoing simultaneous par-
ticle breakage and coalescence, the rate of change of the
number density function with respect to time and volume is
given by the following non-linear integro-differential popu-
lation balance equation (Kiparissides et al., 2004):
[n(V,t)]t
= Vmax
V
(U,V )u(U)g(U)n(U, t) dU
+ V /2Vmin
k(VU , U )n(VU,t )n(U, t) dUn(V, t)g(V)
n(V, t) Vmax
Vmin
k(V,U)n(U,t) dU. (1)
The first term on the right-hand side (r.h.s.) of Eq. (1) repre-
sents the generation of droplets in the size range (V , V +dV )due to drop breakage. (U , V ) is a daughter drop breakage
function, accounting for the probability that a drop of vol-
ume V is formed via the breakage of a drop of volume U.
The function u(U) denotes the number of droplets formed
by the breakage of a drop of volume U and g(U) is the
breakage rate of drops of volumeU. The second term on the
r.h.s. of Eq. (1) represents the rate of generation of drops in
the size range (V , V +dV ) due to coalescence of two smallerdrops.k(V , U )is the coalescence rate between two drops of
volumeV andU. Finally, the third and fourth terms repre-
sent the drop disappearance rates due to drop breakage and
coalescence, respectively. Eq. (1) will satisfy the followinginitial condition at t=0:n(V, 0)=n0(V ), (2)where n0(V ) is the initial drop size distribution of the dis-
persed phase. In the present study, the initial monomer DSD
was assumed to follow a normal distribution.
2.1. Breakage and coalescence rates
The solution of the population balance equation (Eq. (1))
presupposes the knowledge of the breakage and coalescence
rate functions. In the open literature, several forms ofg(V )andk(V, U)have been proposed to describe the drop break-
age and coalescence rate functions in liquidliquid disper-
sions (Coulaloglou and Tavlarides, 1977; Narsimhan et al.,
1979; Sovova, 1981; Chatzi et al., 1989). According to the
original work of Alvarez et al. (1994) and the proposed
modifications ofMaggioris et al. (2000),the breakage and
coalescence rates can be expressed in terms of the break-
age,b, and collision, c, frequencies and the, respective,
Maxwellian efficiencies,b andc:
g(V)=b(V )eb (V ), (3)
k(V , U )=c(V , U )ec(V,U)
, (4)b and c denote the corresponding ratios of required to
available energy for an event to occur.
In the present study, the breakage of a drop exposed to a
turbulent flow field was supposed to occur as result of en-
ergy transfer from an eddy to a drop having a diameter equal
to the eddy wave length, Dv. Eddy fluctuations with wave-
lengths smaller (larger) than the drop diameter Dv produce
an oscillatory (rigid body) motion of the drop that do not
lead to breakage (Alvarez et al., 1994). The frequency term,
b(V ), was assumed to be equal to the inverse fluctuation
time period, corresponding to the time required for a drop
to reach its mean drop displacement:
b(V )=u(Dv)/Dv, (5)where u(Dv)
2 is the mean square of the relative velocity
between two points separated by a distanceDv, or the mean
square fluctuation velocity of drops of diameter Dv.
For drops in the inertial subrange of turbulence (i.e.,
< Dv L), the energy spectrum will be independent ofthe kinematic viscosity, c, and the mean fluctuation veloc-
ity is solely determined by the rate of energy dissipation
(Hinze, 1959):
u(Dv)2
=kb(s Dv)
2/3, (6)
http://-/?- -
8/12/2019 1-s2.0-S0009250905006202-main
4/15
C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332346 335
wherekb is a model parameter and s is the average energy
dissipation rate for the dispersion.
For droplets in the viscous dissipation range (i.e., Dv< ),
the inertial forces are of the same order of magnitude as
the viscous shear forces and the mean square of the relative
velocity between two points separated by a distance Dv will
be given by (Shinnar, 1961)
u(Dv)2 =kbD2v (s /c). (7)
For high values of the dispersed phase volume fraction, the
damping effect of the dispersed phase on the local turbu-
lent intensity needs to be taken into account.Doulah (1975)
proposed the following cubic equation for the calculation of
the average energy dissipation rate of the liquidliquid dis-
persion,s , in terms of the average energy dissipation rate
of the continuous phase,c, and the kinematic viscosities of
the continuous,c, and liquidliquid dispersion,s :
s /c=(c/s )3
. (8)
Thus, in the presence of a high-volume dispersed phase, the
overall viscosity of the system increases and, therefore, the
energy dissipation rate for the system decreases.
According toAlvarez et al. (1994),for an effective drop
breakage to occur, the drop surface energy and drop vis-
coelastic resistance must be overcome. Considering that the
flow within a drop can be described as one-dimensional
simple-shear flow of a Maxwell fluid, the breakage efficiency
can be expressed as follows:
b=ab(Dv), (9)
where ab is a model parameter and (Dv) is the ratio of
required to available energy for a drop of diameter Dv to
break:
(Dv)=2
Re(1+Re Ve) +CdsWe
, (10)
where Re and We denote the drop Reynolds and Weber
numbers, respectively. The dimensionless quantity Ve ac-
counts for drop viscoelasticity and is a function of the drop
Reynolds number and its physical properties:
Ve=Yo
a exp
1
a
2Re Yo
1+a1a
1a1+a exp
a
ReYo
1
12, (11)
where
Yo=2d
dEdD2v
, =
148Yo. (12)
In the present study, the dispersed-phase elasticity mod-
ulus,Ed, was approximated by the product of the polymer
elasticity modulus, Ep, and the fractional monomer con-
version, (i.e., Ed
=Ep). For highly viscous (Re < 1)
and inelastic dispersions (Yo ), Eq. (11) reduces to
Ve= 112
exp
12
Re
1
. (13)
The scalar quantity Cds in Eq. (10) can be expressed in
terms of the numbers and volumes of daughter and satellite
drops (Chatzi and Kiparissides, 1992):
Cds=Ndar
2/3 +Nda(Ndar+Nsa)2/3
1, r=Vda/Vsa, (14)
where Nda is the number of daughter drops of volume Vdaand Nsa is the number of satellite drops of volume Vsa. In
the present study, the number of daughter drops was set
equal to 2, the volume ratio of daughter to satellite drops,r,
was considered to be constant, while the number of satellite
drops was calculated as a function of the parent drop size
(Chatzi and Kiparissides, 1992):
Nsa=integer(SnsaD1/3u ), (15)
whereSnsais a model parameter estimated from experimen-
tal measurements on DSD or PSD.
Assuming that the daughter and satellite drops are nor-
mally distributed about their respective mean values with
standard deviations ofda and sa, one can derive the fol-
lowing expression for the distribution of drops of volume
V, formed via the breakage of a drop of volume U:
u(U)(U , V )
=Nda 1
da2exp
(VVda)2
22da
+Nsa
1
sa
2exp
(VVsa)
2
22sa
. (16)
It should be noted that the daughter drop number density
function,u(U)(U , V ), should satisfy the following number
and volume conservation equations: U0
u(U)(U , V ) dV=u(U),
U
0
Vu(U)( U , V ) dV
=U. (17)
Accordingly, one can calculate the volumes of daughter and
satellite drops, formed by the breakage of a drop of volume
U, in terms of r, Nda and Nsa (Chatzi and Kiparissides,
1992):
Vda=U
Nda+Nsa/r, Vsa=
U
Ndar+Nsa. (18)
Assuming that the drop collision mechanism in a lo-
cally isotropic flow field is analogous to collisions between
molecules as in the kinetic theory of gases, the collision fre-
quency between two drops with volumes V and Ucan be
-
8/12/2019 1-s2.0-S0009250905006202-main
5/15
336 C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332 346
expressed as (Maggioris et al., 2000)
c( V , U )=kc(D2v+D2u)(u(Dv)2 +u(Du)2)1/2. (19)For deformable drops, that is generally the case for low
interfacial tension dispersions or large-size drops, the drop
coalescence efficiency can be expressed as (Coulaloglou and
Tavlarides, 1977)
()c (V , U )=acccs
2
DvDu
Dv+Du
4, (20)
whereac is a model parameter. c, c, and are the vis-
cosity and density of the continuous phase, the interfacial
tension between the dispersed and aqueous phases and the
dispersed phase volume fraction, respectively.
At high monomer conversions, when the polymerizing
monomerpolymer particles behave like rigid spheres, the
coalescence efficiency can be expressed as (Coulaloglou and
Tavlarides, 1977)
(b)c (V , U )=acc
c1/3s (Dv+Du)4/3
. (21)
In general, the monomer drops will behave like de-
formable drops at the beginning of polymerization while,
at high monomer conversions, they will behave like rigid
polymer particles. Thus, the coalescence efficiency over the
whole monomer conversion range can be written as
exp{c(V , U )} =(1) exp{(a)c (V , U )}+ exp{(b)c (V , U )}, (22)
where is the fractional monomer conversion.
2.2. Evaluation of the physical properties
The density of the suspension system, s , can be calcu-
lated as a weighted average of the densities of the dispersed
(d)and continuous(c)phases (Bouyatiotis and Thornton,
1967):
s=d+c(1). (23)The density of the dispersed phase will in turn be a func-
tion of the corresponding densities of the polymer (p)and
monomer(m
) and the extent of monomer conversion, :
d=
p+ 1
m
1. (24)
The viscosity of the liquid(solid)liquid dispersion
was calculated by the following semi-empirical equation
(Vermeulen et al., 1955):
s=c
1
1+ 1.5d
d+c
, (25)
where d and c are the viscosities of the dispersed and
continuous phases, respectively.
For the heterophase suspension polymerization of VCM,
the viscosity of the polymerizing monomer droplets,d, can
be calculated by using the Eulers equation (Krieger, 1972):
d=m1+0.5[n]p
1
p
/cr
2
, (26)
wherep is the volume fraction of the polymer in the dis-
persed phase, given by p=(d/p). cr is the polymervolume fraction corresponding to the critical monomer con-
versionc, at which a 3-D polymer skeleton is formed in-
side the polymerizing monomer drops. When pcr, thedispersed-phase viscosity approaches infinity, indicating the
formation of a rigid structure. Thus, for values ofp larger
than the critical valuecr, the dispersed phase viscosity was
assumed to remain constant.
For the VCM suspension polymerization, the value of
cr was taken to be equal to 0.3, which corresponds to
the monomer conversion at which a continuous polymernetwork is formed inside the polymerizing VCM droplets
(Kiparissides et al., 1994). For the suspension polymeriza-
tion of styrene, the value ofcr was set equal to the 0.7
which corresponds to the monomer conversion at which par-
ticle coalescence stops. Finally, the intrinsic viscosity of the
polymer solution, [n], was calculated by the well-known
MarkHouwinkSakurada (MHS) equation as a function of
the weight average molecular weight of the polymer, Mw:
[] =kMaw. (27)
The viscosity of the continuous phase depends on theconcentration and type of stabilizer that, in turn, affects the
PSD (Cebollada et al., 1989). Okaya (1992) employed the
SchulzBlaschke equation to calculate of the viscosity of
aqueous PVA solutions:
c=w
1+ [PVA]CPVA10.45[PVA]CPVA
, (28)
where c,w, [nPVA] and CPVAare the viscosities of theaqueous PVA solution and pure water, the intrinsic viscosity
and the stabilizer concentration, respectively.
In the open literature, a great number of papers have beenpublished, dealing with the behaviour of polymer molecules
at interfaces. Prigogine and his collaborators (Prigogine and
Marechal, 1952; Defay et al., 1966) presented a remarkably
simple theory on the surface tension of polymer solutions.
Although the Prigogine theory refers specifically to the sur-
face tension of polymer solutions, it is equally applicable to
the prediction of interfacial tension between a polymer solu-
tion and an immiscible liquid or a solid (Siow and Patterson,
1973). In the present study, the model ofSiow and Patterson
(1973)was employed for the calculation of the interfacial
tension between the aqueous and the dispersed phase, . The
change in the interfacial tension with monomer conversion
http://-/?- -
8/12/2019 1-s2.0-S0009250905006202-main
6/15
C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332346 337
was taken into account as in the original work ofMaggioris
et al. (2000).
3. Numerical solution of the population balance
equation
The numerical solution of the PBE commonly requires
the discretization of the particle volume domain into a num-
ber of discrete elements. Accordingly, the unknown num-
ber density function is approximated at a selected number
of discrete points, resulting in a system of stiff, non-linear
differential equations that is subsequently integrated numer-
ically. Several numerical methods have been proposed in the
literature for the solution of the general PBE (Eq. (1)) in
the continuous or its equivalent discrete form (Kiparissides
et al., 2004). In the present study, the FTP ofKumar and
Ramkrishna (1996)was employed for solving the continu-
ous PBE (Eq. (1)).Assuming that the number density function remains con-
stant in the discrete volume interval (Vi to Vi+1), one candefine a particle number distribution, Ni (t ), corresponding
to the i element:
Ni (t )= Vi+1
Vi
n(V,t) dV=ni (V,t)(Vi+1Vi ). (29)
Following the original developments of Kumar and
Ramkrishna (1996), the total volume domain (Vmin to
Vmax) is first divided into a number of elements. The
drop/particle population, Ni (t ), corresponding to the size
range(Vi , Vi+1), is then assigned to a characteristic size xi(also called grid point) as shown inFig. 2. To account for
the formation of new particles of volume V that does not
correspond to the characteristic grid points (xi , xi+1), thefollowing approximation is applied. The particle number
fractionsa(V, xi ) and b(V, xi+1) are assigned to the parti-cle populations at the grid points xi and xi+1, respectively,so that two desired population properties of interest (e.g.,
total number and volume) are exactly preserved:
a(V,xi )f1(xi )+b(V, xi+1)f1(xi+1)=f1(V ), (30)
a(V,xi )f2(xi )
+b(V, xi
+1)f2(xi
+1)
=f2(V ). (31)
By integrating Eq. (1) over the discrete size interval
(Vi , Vi+1) and properly accounting for the respective dropbreakage and coalescence terms, the following set of
Vi-2 Vi-1 Vi Vi+1 Vi+2
xi+1xixi-1xi-2
Fig. 2. Discretization of the particle volume domain.
discretized equations can be derived:
dNi
dt=
Mk=1
nb(xi , xk )g(xk )Nk(t )+jk
j,kxi1 xj+xk xi+1
1 12j,k nc(xi ,V)k(xj, xk )Nj(t)Nk (t )g(xi )Ni (t )Ni (t )
Mk=1
k(xi , xk )Nk(t ), (32)
where nb(xi , xk ) denotes the fraction of drops/particles of
sizexi resulting from the breakage of a drop/particle of size
xk . To preserve the number and volume of particles in the
size range (Vi , Vi+1), nb(xi , xk ) must satisfy the followingequation:
nb(xi , xk )=
xi+1
xi
xi+1Vxi+1xi
(xk ,V)u(xk ) dV
+ xi
xi1
Vxi1xixi1
(xk,V)u(xk ) dV. (33)
The respective expression for nc(xi , V ), accounting for the
fraction of drops/particles of sizeV (=xj+ xk )formed viathe coalescence of two drops/particles of volumes xj and
xk , will be given by
nc(xi , V )=
xi+1Vxi+1xi
xiVxi+1,
Vxi1x
ix
i1xi1Vxi .
(34)
Accordingly, from the calculated values ofNi (t ), one can
easily obtain the values of the average number density func-
tion,ni (V , t ):
ni (V , t )=Ni (t )
(Vi+1Vi ). (35)
It is often desirable to know the number diameter density
function,n(D, t). By noting that n(V, t) dV= n(D, t) dD,one can easily calculate the average number diameter density
function,ni (D,t), in terms ofNi (t ):
Ni (t )= Di+1
Di
n(D, t) dD= ni (Di ,t)(Di+1Di ). (36)
In many cases, experimental measurements on PSD are
given in terms of number or volume fractions, from which
it is not easy to derive the actual particle number dis-
tribution, Ni (t ), or the number volume density function,
n(V,t). Thus, it is better to compare directly the available
experimental measurements on number and volume fraction
distributions with respective simulation results obtained
from the numerical solution of the PBE. Accordingly, one
can define the following number, A(D, t), and volume,
-
8/12/2019 1-s2.0-S0009250905006202-main
7/15
338 C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332 346
Table 1
Physical/transport properties and model parameters for VCM/PVC system
m=9471.746(T 273.15) (Kg/m3) (Kiparissides et al., 1997)p=103 exp(0.42963.274104T ) (Kg/m3) (Kiparissides et al., 1997)m=2.1104 106(T 273.15) (Kg/m s)c=10110.4484(T273.15) (Kg/m3) (Kiparissides et al., 1997)
c=0.08 exp(1.5366102
T ) (Kg/m s)[n] = 1.087105 1.67108(T 273.15)M0.851w (m3/Kg) (Polymer Handbook)[s ] =9.13103 +4.317105DP (Okaya, 1992)Ep=2.4109 (Kg/m s2) (Polymer Handbook)r=35, SNsa=110, kb=324, ab=33, kc=3107, ac=2109, ac=1102 (This study)
Table 2
Physical, transport properties and model parameters for styrene/PS system
m=923.60.887(T 273.15) (Kg/m3) (Achilias and Kiparissides, 1992)p=10500.602(T273.15) (Kg/m3) (Achilias and Kiparissides, 1992)m=10528.64(1/T1/276.71)
3(Kg/m s) (Achilias and Kiparissides, 1992)
[n] =1.38105M0.722w (m3/Kg) (Achilias and Kiparissides, 1992)Ep=3.3810
9(Kg/m s
2) (Polymer Handbook)
r=35, SNsa=50, kb=400, ab=33, kc=4107, ac=5109, ac=3103 (This study)
AV(D,t), probability density functions:
A(D, t)= n(D,t)Nt(t )
fNiDi+1Di
,
AV(D,t)=(D3/6)n(D, t)
Vt(t ) fVi
Di+1Di. (37)
A(D, t) dD and AV(D,t) dD represent the number (fNi )
and volume (fV i ) fractions of particles in the size range(D,D+dD), respectively. Nt(t ) and Vt(t ) denote the re-spective total number and volume of particles per unit vol-
ume of the reaction medium. It is apparent that the num-
ber and volume probability density functions will satisfy the
following normalization conditions:
DmaxDmin
A(D, t) dD=1, Dmax
Dmin
AV(D,t) dD=1. (38)
Very often experimental measurements on some average
particle diameter are only available. In general, the average
particle diameter,Dqp, can easily be calculated, in terms ofthe number probability density function, using the following
equation (Chatzi and Kiparissides, 1992):
(Dqp)qp
= Dmax
Dmin
Dq A(D, t) dD
DmaxDmin
DpA(D, t) dD, (39)
where q and p are characteristic exponents that define the
desired average particle diameter (e.g., mean Sauter diam-
eter, D32, average number diameter, D10, average volume
diameter, D30, etc.).
4. Results and discussion
The predictive capabilities of the proposed model were
demonstrated by a direct comparison of model predic-
tions with experimental data on average mean diameter
and droplet/particle size distribution of both non-reactive
liquidliquid dispersions of styrene and VCM in water,
and free-radical suspension polymerization of styrene and
VCM. For polymerization systems, the general popula-tion balance model (see Eq. (1)) was solved together with
the pertinent molecular species differential equations (see
Appendix A) describing the molecular weight develop-
ments (e.g., number and weight average molecular weights)
and the polymerization rate in the heterophase suspension
system. In addition to the above dynamic equations, the
dynamic model included all the necessary algebraic equa-
tions, describing the variation of the kinetic rate constants,
and the physical, transport and thermodynamic proper-
ties of the multi-phase system with respect to the reactor
operating conditions (e.g., temperature, monomer mass,
etc.) and the fractional monomer conversion. Additional
details, regarding the kinetic mechanism (e.g., gel- and
glass-effect), phase equilibrium calculations (e.g., monomer
and initiator partitioning, number of phases in the system,
etc.) can be found in the publications ofKiparissides et al.
(1997), Kotoulas et al. (2003) andKrallis et al. (2004). In
Tables 1and2, the physical, transport and model parame-
ters for the VCM/PVC and styrene/PS systems are reported.
It should be noted that the numerical values of the key
model parameters (i.e., kb, ab, kc, ac) were estimated by
fitting the model predictions to experimental data on DSD
of liquidliquid aqueous dispersions of styrene and VCM.
The system of non-linear ordinary differential equations
http://-/?- -
8/12/2019 1-s2.0-S0009250905006202-main
8/15
C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332346 339
0 200 400 600 800 1000 1200 1400 160010
-14
10-13
10-12
10-11
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
N E = 30
N E = 50
N E = 80
NE = 100
VolumeProbabilityD
ensityFunction(m
-1)
Particle Diameter (m)
Fig. 3. Effect of the number of volume elements on the calculated volume
probability density function (free-radical suspension polymerization of
styrene).
(Eqs. (32)(34)) together with the necessary kinetic equa-
tions (see Appendix A) were numerically integrated using
the Gear predictorcorrector DE solver.
Validation of the numerical method: The accuracy and
convergence characteristics of the numerical method (FTP)
were first assessed by varying the total number of discretiza-
tion points, the size of the total volume domain and the initial
DSD.Fig. 3shows the effect of the number of equal-size dis-
crete elements (i.e., 30, 50, 80 and 100) on the volume prob-
ability density function for the styrene suspension polymer-
ization. The diameter domain extended from 1 to 2000 m
while the initial DSD followed a Gaussian distribution with
a mean value ofD0=1000m and a standard deviation ofD=100 m. As can be seen, the volume probability den-sity function converges to the same distribution for values
of the number of elements NE80. In the present study,
it was assumed that the numerically calculated distribution
converged to the correct one when the total mass of the dis-
persed phase (i.e., monomer plus polymer), given by the first
moment of particle number distribution,(dN
k=1Vi Ni (t)),differed from the initial monomer mass by less than 23%.
When the upper limit of the total diameter domain, Dmax,
was reduced from 2000 to 1200 m, it was found that the
number of discrete elements, required for the satisfaction
of above mass conservation criterion, was NE
50. Thus, itwas concluded that the numerical solution converged to the
correct distribution when the size of the discrete elements
(i.e., the ratio of the total diameter domain over the num-
ber of elements) was smaller than 25 m. A similar rule was
found to be applicable to the VCM/PVC suspension poly-
merization system.
In liquidliquid dispersions the final DSD is controlled by
the dynamic equilibrium between drop breakage and coales-
cence rates. Thus, for the same operating conditions (e.g.,
input power, dispersed phase volume fraction, temperature,
etc.) the final DSD should be independent of the initial DSD.
Fig. 4 illustrates the effect of the initial DSD on the final
0 200 400 600 800 1000 1200 1400
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
Final Distributions
D = 200 m
D = 700 m
D = 1000 m
Initial Distributions
D = 200 m = 50 m
D = 700 m =100 m
D = 1000 m =100 m
VolumeProbabilityDensityFunction(m
-1)
Particle Diameter (m)
Fig. 4. Effect of the initial DSD on the calculated volume probability
density function of styrene droplets in water (non-reactive case).
0 50 100 150200
250
300
350
400
450
500
550
600D = 200 m = 50 m
D = 700 m = 100 m
D =1000 m = 100 m
Time (min)
SauterMeanDiameter,
D32
(m)
Fig. 5. Effect of the initial DSD on the dynamic evolution of the Sauter
mean diameter of styrene droplets in water (non-reactive case).
DSD at dynamic equilibrium for the styrenewater disper-
sion system. As can be seen, the calculated final DSD is not
affected by the initial condition. On the other hand, the time
required for the system to attain its final DSD is affected by
the initial DSD condition. Fig. 5, clearly depicts the vari-
ation of the Sauter mean droplet diameter with respect to
time. In all cases, the drop breakage and coalescence ratefunctions were the same. It is apparent that the time required
for the liquidliquid dispersion to reach its dynamic equilib-
rium distribution is larger when the initial DSD had a mean
value of D0=200 m. On the other hand, no significantdifferences in the required times for the system to reach its
dynamic equilibrium were observed when the mean value
of the initial DSD changed from 1000 to 700 m. Notice
that in the former case (i.e., D0= 200m andD= 50m)the drop coalescence mechanism controls the dynamic evo-
lution of DSD, while in the later case (i.e., D0=1000mand D=100 m) the DSD evolution is mainly controlledby the drop breakage mechanism.
-
8/12/2019 1-s2.0-S0009250905006202-main
9/15
340 C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332 346
0 200 400 600 800 1000 1200
0.000
0.002
0.004
0.006
0.008
0.010 Initial DSD D
0= 200 m D
0= 700 m D
0=1000 m
D
= 80 m D
= 100 m D
=100 m
Final PSD
(D =1000 m)
(D = 700 m)
(D = 200 m)
VolumeProbabilityDe
nsityFunction(m
-1)
Particle Diameter (m)
Fig. 6. Effect of the initial DSD on the calculated polystyrene particle
size distribution (suspension polymerization of styrene).
0 50 100 150 200 250 300 350
250
300
350
400
450
500
550
D = 200 m = 80 m
D = 700 m = 100 m
D =1000 m = 100 m
SauterMeanDiameter,
D32
(m)
Time (min)
Fig. 7. Effect of the initial DSD on the dynamic evolution of the Sauter
mean diameter of styrene droplets in water (suspension polymerization
of styrene).
It should be noted that when the polymerization in the
monomer droplets starts before the system has reached its
liquidliquid equilibrium distribution, the final PSD in the
suspension system will not be independent of the initial DSD
condition. In Figs. 6 and 7, the effect of the initial DSDon the final PSD is depicted for the free-radical suspension
polymerization of styrene, assuming that the polymeriza-
tion in the monomer droplets starts at time zero (i.e., be-
fore the liquidliquid dispersion reaches its dynamic equi-
librium point). As can be seen, as the average size of the
initial monomer DSD increases, the final PSD is shifted to
larger sizes. The reason is that drop breakage ceases before
the liquidliquid dispersion has reached its final equilibrium
distribution.
Vinyl Chloride suspension polymerization. The dispersion
of vinyl-chloride monomer (VCM) in aqueous PVA solu-
tions has been studied experimentally byZerfa and Brooks
0 20 40 60 80 100
20
30
40
50
60
MeanDroplet
Diameter(m)
D10
( sim)
D32
( sim)
Time (min)
Fig. 8. Dynamic evolution of the calculated and measured mean
diameter of VCM droplets (non-reactive case: monomer vol-
ume fraction= 0.1; CPVA (72.5% degree of hydrolysis) = 0.02%;temperature=55 C; N=500rpm).
0 30 60 90 120 150
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Volum
eProbabilityDensityFunction(m
-1)
Particle Diameter (m)
5 min
10 min
30 min
120 min
Experimental
Fig. 9. Dynamic evolution of the calculated distribution of VCM droplets
(non-reactive case: experimental conditions as in Fig. 8; discrete points
represent experimental measurements).
(1996a,b)under different conditions (e.g., monomer hold-
up, agitation speed and type and concentration of stabilizers).
Fig. 8illustrates the dynamic evolution of the number mean
diameter, D10, and the Sauter mean diameter, D32, of VCmonomer droplets in the dispersion. The monomer volume
fraction in the dispersion was 0.1, the temperature was kept
constant at 55 C, the agitation speed was set at 500rpm,while 200 ppm of PVA with a degree of hydrolysis equal to
72.5% were added to the aqueous phase for the stabiliza-
tion of the VCM droplets (Zerfa and Brooks, 1996b). The
continuous lines represent simulation results while the dis-
crete points the experimental measurements. As can be seen,
the droplet size initially reduces (i.e., due to the dominant
drop breakage mechanism) and reaches its final dynamic
equilibrium value, at approximately 30 min. The evolution
of DSD is shown inFig. 9.Initially, the volume probability
http://-/?- -
8/12/2019 1-s2.0-S0009250905006202-main
10/15
C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332346 341
0 30 60 90 120 150 180
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Vo
lume
Pro
ba
bility
Densi
tyFunct
ion
(m
-1)
Particle Diameter (m)
250 rpm ( sim)
350 rpm ( sim)
500 rpm ( sim)
650 rpm ( sim)
Fig. 10. Calculated and experimentally measured distributions of
VCM droplets in water at different agitation rates (monomer
volume fraction= 0.1; CPVA (72.5% degree of hydrolysis)= 0.03%;temperature=55 C).
density function of VCM is broad. However, as the agitation
continues, it becomes narrower and shifts to smaller sizes.
The predicted steady-state DSD (at 120 min) is in excellent
agreement with the experimentally measured one (black dis-
crete points). It is clear that the proposed model is capable
of predicting satisfactorily the dynamic evolution of VCM
distribution as well as its mean droplet value.
Fig. 10 illustrates the effect of the agitation rate on the
steady-state volume probability density function of VCM
droplets in the aqueous dispersion. All other experimental
conditions were similar to those ofFig. 8, except the con-
centration of the PVA stabilizer, which was 300 ppm (Zerfaand Brooks, 1996a). The discrete points represent the exper-
imental measurements while the continuous lines the model
predictions. As can be seen, as the agitation rate increases
the DSD shifts to smaller sizes and becomes narrower due
to the increased drop breakage rate. In all cases, the model
results are in very close agreement with the experimental
data. An additional comparison study was carried out for a
VCM dispersed volume fraction, of 0.2. It was found that
the VCM droplet distribution shifted to larger sizes as the
monomer hold-up increased. Again, simulation results were
in excellent agreement with experimental measurements on
DSD.Subsequently, experimental measurements on the average
particle size and PSD were compared with model predictions
for the free-radical suspension polymerization of VCM. The
experimental data for the PVC system were provided by
ATOFINA. The experiments were carried out in a 30 L batch
reactor, using 40% v/v VCM in water. The polymerization
temperature was set at 56.5 C while the agitation speedremained constant at 330 rpm.
Fig. 11depicts the variation of the volume mean diam-
eter with respect to polymerization time. The continuous
line represents the simulation results and the discrete points
the experimental measurements. Initially, the mean diameter
0 50 100 150 200 250 300
60
80
100
120
140
160
180
200
Vo
lume
Mean
Dia
meter,
D30
(m
)
Time (min)
Experimental
Simulation
Fig. 11. Dynamic evolution of calculated and experimentally measured vol-
ume mean diameter of PVC particles (reactive case: temperature=56.5 C;dispersed phase volume fraction=0.4; agitation rate=330rpm).
shifts to smaller values due to the dominant drop breakage
mechanism. Subsequently, the drop breakage rate is reduced
while the drop coalescence rate increases because of the in-
creased viscosity of the dispersed phase. Thus, the mean
particle diameter increases until a monomer conversion of
about 75%. After this point, the drop coalescence rate ceases
and the PSD remains almost constant. It is apparent that the
present model predicts very well the dynamic behaviour of
the PSD for the free-radical suspension polymerization of
VCM. InFig. 12, experimental measurements (dash lines)
and simulation results (continuous lines) on PSD are plot-ted at four different conversion levels (i.e., 55%, 65%, 75%
and 83%). As can been seen, the simulation results are in
very good agreement with the experimental measurements.
It should be pointed out that all the simulation results on
VCM suspension polymerization (i.e., for both reactive and
non-reactive cases) were obtained using the same values of
the model parameters (seeTable 1).
Styrene suspension polymerization. The dynamic evolu-
tion of styrene DSD in aqueous dispersions was experimen-
tally studied by Yang et al. (2000). Fig. 13 illustrates the
dynamic evolution of the Sauter mean diameter of styrene
droplets for two different monomer volume fractions. Thetemperature was kept constant at 25 C, the agitation speedwas 350 rpm, while 500 ppm PVA were added to the aque-
ous phase for the stabilization of the dispersion. The PVA
had a degree of hydrolysis equal to 88%, while its molecu-
lar weight varied between 30,000 and 50,000 g/mol. The fi-
nal DSD at dynamic equilibrium was attained after 150 min
of stirring. Apparently, the model predicts fairly well the
dynamic evolution of the Sauter mean diameter of styrene
droplets, as well as the effect of monomer volume fraction.
In Fig. 14, the time evolution of DSD is depicted for the
case of styrene volume fraction of 0.1. It is evident that the
model predictions on DSD are in very good agreement with
-
8/12/2019 1-s2.0-S0009250905006202-main
11/15
342 C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332 346
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
Conversion 55%
Experimental
Simulation
Conversion 65%
Experimental
Simulation
0 100 200 300 4000.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
Conversion 75 %
Experimental
Simulation
VolumeProbabilityDensityFunction
(m
-1)
0 100 200 300 400
Conversion 83%
Experimental
Simulation
Particle Diameter (m)
Fig. 12. Predicted and experimentally measured distributions of PVC particles at four different conversion levels: 55%, 65%, 75% and 83% (experimental
conditions asFig. 11).
0 50 100 150 200 250
80
100
120
140
160
SauterMeanDiameter,
D32
(m)
Time (min)
= 0.05 ( sim)
= 0.10 ( sim)
Fig. 13. Dynamic evolution of calculated and experimentally measured
Sauter mean diameter of styrene droplets at two different monomer volume
fractions (non-reactive case: temperature=25 C; agitation rate=350rpm;CPVA (88% degree of hydrolysis)=0.05%).
experimental measurements (discrete points).Fig. 15illus-
trates the effect of the agitation rate on the steady-state DSD
of the styrene droplets. The operating conditions were as in
Fig. 13,while the monomer volume fraction was 0.1. As in
the case of the VCM dispersion, the mean size of styrene
droplets decreases with the agitation rate while the DSD be-
50 100 150 200 250 300
0.000
0.003
0.006
0.009
0.012
0.015
0.018
Vo
lume
Pro
ba
bility
Densi
tyFunct
ion(m
-1)
Particle Diameter (m)
5 min ( sim)
30 min ( sim)
120 min ( sim)
Fig. 14. Dynamic evolution of calculated and experimentally measured
distributions of styrene droplets in water (non-reactive case: experimental
conditions as Fig. 13; monomer volume fraction=0.1).
comes narrower. In all cases, the simulation results are in
very good agreement with the experimental data that clearly
underlines the predictive capabilities of the present compre-
hensive population balance model.
Finally, the present model was employed to predict the
dynamic evolution of PSD in the free-radical suspension
polymerization of styrene. More specifically, the effects of
-
8/12/2019 1-s2.0-S0009250905006202-main
12/15
C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332346 343
0 50 100 150 200 250 300 350
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
Vo
lume
Pro
ba
bilityD
ensi
tyFunct
ion
(m
-1)
Particle Diameter (m)
250 rpm ( sim)
450 rpm ( sim)
650 rpm ( sim)
Fig. 15. Effect of agitation rate on the calculated and ex-
perimentally measured distributions of styrene droplets in water
(non-reactive case: temperature=25 C; agitation rate=350 rpm; CPVA(88% degree of hydrolysis)
=0.05%; monomer volume fraction
=0.1).
n-pentane and concentration of stabilizer on the PSD were
investigated for the expandable PS suspension polymer-
ization process. Experimental measurements on PSD were
taken from the work ofVillalobos et al. (1993). The free-
radical styrene suspension polymerization was carried out
in a 1-gal reactor vessel. The dispersed monomer volume
fraction was 0.4. The polymerization took place at 105 C inthe presence of 1,4-bis(terbutyl peroxycarbo) cyclohexane
(TBPCC) bifunctional initiator. The initiator concentration
was 0.01 mol/L-styrene in all the experimental cases, while
tricalcium phosphate (TCP) was used as surface-active agent
at three different concentrations (i.e., 7.5, 5.0 and 3.5 gr/L).
InFig. 16,experimental measurements and simulation re-
sults on PSD are shown for three different addition policies
ofn-pentane into the reactor. More specifically, 7.5% w/w
of n-pentane with respect to the styrene mass was added
200 400 600 800 1000 1200 1400
Experimental
Simulation
Particle Diameter (m)
200 400 600 800 1000 1200
Experimental
Simulation
Particle Diameter (m)
0 200 400 600 800 1000 12000.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
VolumeProbabilityDensityFunction(m
-1)
Particle Diameter (m)
Experimental
Simulation
(a) (b) (c)
Fig. 16. Predicted and experimentally measured distributions of EPS particles for different n-pentane addition policies. (a) 7.5% w/w n-pentane (wrt
styrene) at =0%; (b) 7.5% w/w n-pentane (wrt styrene) at =50% ; (c) in the absence of n-pentane (temperature=105 C; dispersed phase volumefraction
=0.4,
[Io
] =0.01 mol TBPCC/L-styrene;
[TCP
] =7.5 g/L).
to the system at three different conversion levels (i.e., 0, 50%
and 100%). In Fig. 16a, model results (continuous lines)
are compared with experimental data (dash lines) on PSD
for the case ofn-pentane addition at zero monomer conver-
sion. InFig. 16b, the corresponding distributions are illus-
trated for the case ofn-pentane addition at 50% monomer
conversion. The last case (see Fig. 16c) corresponds to theaddition ofn-pentane at the end of polymerization. As can
been seen, for all cases, there is a close agreement between
experimental and simulation results on PSD, indicating the
predictive capabilities of the model for the free-radical sus-
pension polymerization of styrene. It should be noted that,
in the presence of n-pentane, the EPS particles are more
uniform while the PSD becomes narrower.
Villalobos et al. (1993) also investigated experimentally
the effect of the stabilizer concentration on PSD. More
specifically, experiments were carried out at different TCP
concentrations, in the presence of 7.5% w/w n-pentane,
added at 50% monomer conversion. InFig. 17the predicted
and experimental PSDs of the EPS particles are shown for
the three different concentrations of the surface-active agent.
As can be seen, as concentration of the surface-active agent
decreases (i.e., the interfacial tension increases) the PSD be-
comes broader and shifts to larger sizes. Apparently, there is
a very good agreement between calculated and experimental
measurements on the volume probability density function.
5. Conclusions
A comprehensive population balance model coupled with
a system of differential equations governing the conserva-
tion of the various molecular species present in the system
has been developed to describe the dynamic evolution of
the DSD/PSD in free-radical suspension polymerization re-
actors. The fixed pivot technique (FPT) was employed for
solving the PBE. The robustness of the numerical method
-
8/12/2019 1-s2.0-S0009250905006202-main
13/15
344 C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332 346
0 500 1000 1500 2000
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
Vo
lume
Pro
ba
bility
Densi
tyFunct
ion
(m
)
Particle Diameter (m)
7.5 % w/w ( sim)
5.0 % w/w ( sim)
3.5 % w/w ( sim)
Fig. 17. Effect of surface-active concentration on the calculated and
experimentally measured distributions of EPS particles at three different
quantities of surface-active agent (TCP) (experimental conditions as in
Fig. 16).
was examined in regard with its convergence character-
istics and accuracy in terms of the mass conservation of
the monomer, initially loaded into the reactor. The predic-
tive capabilities of the model were demonstrated via the
successful simulation of experimental measurements on
DSD/PSD and the average droplet/particle diameter for both
non-reactive liquidliquid dispersions and the free-radical
suspension polymerization of styrene and VCM.
Notation
A(D,t),Av(D,t) number and volume probability density
functions, 1/m
CPVA concentration of surface-active agent,
Kg/m3
D diameter, m
DP degree of polymerization of the PVA
stabilizer
E elasticity modulus, Kg/m s2
g(V) breakage rate, 1/s
k(V,U) coalescence rate, m3/s
kb, kc model parameters
L macroscale of turbulence, m
Mw weight average molecular weight,
Kg/kmol
n(V,t) number density function, 1/m6
[n] intrinsic viscosity, m3/KgNE number of discrete elements
Ni number of particles having volume
equal to xi per reactor unit volume,
1/m3
Nda, Nsa number of daughter and satellite
droplets per breakage events
r volume ratio of daughter over the satel-
lite drops
Re Reynolds number
Snsa model parameter
t time, s
u(V) number of droplets formed by a break-
age of a droplet of volume V
u(Dv)2 mean square of the relative velocity be-
tween two points separated by a dis-tanceD, m/s
Vda, Vsa volumes of daughter and satellite
drops, m3
V , U , x volumes, m3
We Weber number
Greek letters
ab, ac model parameters
( U , V ) daughter droplets probability function,
1/m3
average energy dissipation rate per unit
mass, m2/s3
microscale of turbulence, m
b, c breakage and coalescence efficiencies
viscosity, Kg/m s
kinematic viscosity, m2/s
density, Kg/m3
interfacial tension, Kg/s2
da, sa standard deviation of the distribution
for daughter and satellite drops
dispersed phase volume fraction
p volume fraction of the polymer in the
dispersed phase
monomer conversionb,c breakage and coalescence frequencies,
1/s
Subscripts
c continuous phase
d dispersed phase
m monomer
p polymer
s suspension system
w water
Acknowledgement
The authors gratefully acknowledge ARCHEMA (ex-
ATOFINA Chemicals) for providing the experimental data
for PVC suspension polymerization.
Appendix A
The free-radical polymerization of vinyl monomers in
general includes the following chain initiation, propagation,
-
8/12/2019 1-s2.0-S0009250905006202-main
14/15
C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332346 345
chain transfer to monomer and bimolecular termination re-
actions (Kiparissides et al., 1997):
decomposition of initiators
Iikd,i 2R, i=1, 2, . . . , N m,
chain initiation
R+M kp P1,chain propagation
Pn+Mkp Pn+1,
chain transfer to monomer
Pn+Mkf m Dn+P1,
termination by combination
Pn+
Pmktc
Dn
+m,
termination by disproportionation
Pn+Pm ktd Dn+Dm,inhibition ofliveradical chains
Pn+Zkz Dn+Z,
whereIi , R, Mand Zdenote the initiator, primary radicals,
monomer and inhibitor molecules, respectively, andPn and
Dn, the corresponding live and dead polymer chains,
having a degree of polymerization n.
In the free-radical polymerization of VCM, the polymer
is insoluble in its monomer, thus, precipitates out to form
a separate phase (i.e., the polymer-rich phase). Thus, the
elementary reactions presented above take place in both the
monomer-rich and polymer-rich phases (Kiparissides et al.,
1997). Additional details, regarding the kinetic modeling
of free-radical polymerization of styrene and VCM (e.g.,
gel- and glass-effect), phase equilibrium calculations (e.g.,
monomer and initiator partitioning, number of phases in the
system, etc.), can be found in the publications ofKiparissides
et al. (1997, 2004)and Kotoulas et al. (2003).
The method of moments is invoked in order to reduce the
infinite system of molar balance equations, required to de-
scribe the molecular weight distribution developments. Ac-cordingly, the average molecular properties of the polymer
(i.e., Mn, Mw) are expressed in terms of the leading mo-
ments of the dead polymer molecular weight distribution.
The moments of the total number chain length (TNCL) dis-
tributions of live radical and dead polymer chains can be
defined as (Krallis et al., 2004)
k=
n=ink Pn, k=
n=i
nk Dn. (A.1)
Accordingly, one can easily derive the corresponding mo-
ment rate functions:
Live polymer moment rate equations
rk=Nmk=1
2fkkdk Ik+kpM
kr=0
k
r
rk
+kfmM(0k )(ktc+ktd)k0kzZk. (A.2)
Deadpolymer moment rate equations
rk=kfmMk+1
2k
jtc
kr=0
k
r
rkr
+ktdk0+kzZk . (A.3)The number- and weight-average molecular weights
can be expressed in terms of the molecular weight of the
monomer, MWm, and the moments of the TNCLDs of live
and dead polymer chains:
Mn=(1+1)(0
+0)
MWm, Mw=(2+2)(1
+1)
MWm. (A.4)
Finally, the total monomer conversion can be calculated by
the following expression, assuming that the long chain hy-
pothesis holds true (i.e., the monomer is mainly consumed
via the propagation reaction):
d
dt=kp
M
M00. (A.5)
References
Achilias, D.S., Kiparissides, C., 1992. Development of a general
mathematical framework for modeling diffusion-controlled free-radical
polymerization reactions. Macromolecules 25 (14), 37393750.Alvarez, J., Alvarez, J., Hernandez, M., 1994. A population balance
approach for the description of particle size distribution in suspension
polymerization reactors. Chemical Engineering Science 49, 99113.
Bouyatiotis, B.A., Thornton, J.D., 1967. Liquidliquid extraction studies
in stirred tanks. Part I. Droplet size and hold-up measurements in a
seven-inch diameter baffled vessel. Institution of Chemical Engineers
(London) Symposium Series 26, 4350.
Cebollada, A.F., Schmidt, M.J., Farber, J.N., Cariati, N.J., Valles, E.M.,
1989. Suspension polymerization of vinyl chloride. I. Influence of
viscosity of suspension medium on resin properties. Journal of Applied
Polymer Science 37, 145166.
Chatzi, E.G., Kiparissides, C., 1992. Dynamic simulation of bimodal
drop size distributions in low-coalescence batch dispersion systems.
Chemical Engineering Science 47, 445456.
Chatzi, E.G., Kiparissides, C., 1994. Drop size distributions in highholdup fraction suspension polymerization reactors: effect of the degree
of hydrolysis of PVA stabilizer. Chemical Engineering Science 49,
50395052.
Chatzi, E.G., Gavrielides, A.D., Kiparissides, C., 1989. Generalized model
for prediction of the steady-state drop size distribution in batch stirred
vessel. Industrial Engineering Chemistry Research 28, 17041711.
Coulaloglou, C.A., Tavlarides, L.L., 1977. Description of interaction
processes in agitated liquidliquid dispersions. Chemical Engineering
Science 32, 12891297.
Defay, R., Prigogine, I., Bellemans, A., Everett, D.H., 1966. Surface
Tension and Adsorption. Wiley, New York.
Doulah, M.S., 1975. On the effect of holdup on drop sizes in liquidliquid
dispersions. Industrial Engineering Chemistry Fundamentals 14,
137138.
http://-/?- -
8/12/2019 1-s2.0-S0009250905006202-main
15/15
346 C. Kotoulas, C. Kiparissides / Chemical Engineering Science 61 (2006) 332 346
Hamielec, A.E., Tobita, H., 1992. Polymerization processes. Ullmanns
Encyclopedia of Industrial Chemistry, vol. A21. VCH Publishers,
New York, pp. 305428.
Hinze, J.O., 1959. Turbulence. McGraw-Hill, New York.
Howarth, W.J., 1964. Coalescence of drops in a turbulent flow field.
Chemical Engineering Science 19, 3338.
Kalfas, G.A., 1992. Experimental studies and mathematical modeling of
aqueous suspension polymerization reactors. Ph.D. Thesis, Universityof WisconsinMadison, USA.
Kiparissides, C., 1996. Polymerization reactor modeling: a review of recent
developments and future directions. Chemical Engineering Science 51,
16371659.
Kiparissides, C., Achilias, D.S., Chatzi, E., 1994. Dynamic simulation
of primary particle-size distribution in vinyl chloride polymerization.
Journal of Applied Polymer Science 54, 14231438.
Kiparissides, C., Daskalakis, G., Achilias, D.S., Sidiropoulou, E., 1997.
Dynamic simulation of industrial poly(vinyl chloride) batch suspension
polymerization reactors. Industrial Engineering Chemistry Research
36, 12531267.
Kiparissides, C., Alexopoulos, A., Roussos, A., Dompazis, G., Kotoulas,
C., 2004. Population balance modelling of particulate polymerization
processes. Industrial Engineering Chemistry Research 43, 72907302.
Kotoulas, C., Krallis, A., Pladis, P., Kiparissides, C., 2003. Acomprehensive kinetic model for the combined chemical and thermal
polymerization of styrene up to high conversions. Macromolecular
Chemistry Physics 204, 13061314.
Krallis, A., Kotoulas, C., Papadopoulos, S., Kiparissides, C., Bousquet, J.,
Bonardi, C., 2004. A comprehensive kinetic model for the free-radical
polymerization of vinyl chloride in the presence of monofunctional
and bifunctional initiators. Industrial Engineering Chemistry Research
43, 63826399.
Krieger, I.M., 1972. Rheology of monodispersed lattices. Advances in
Colloid and Interface Science 3, 111127.
Kumar, S., Ramkrishna, D., 1996. On the solution of population balance
equations by discretizationI. A fixed pivot technique. Chemical
Engineering Science 51, 13111332.
Maggioris, D., Goulas, A., Alexopoulos, A.H., Chatzi, E.G., Kiparissides,
C., 2000. Prediction of particle size distribution in suspensionpolymerization reactors: effect of turbulence nonhomogeneity.
Chemical Engineering Science 55, 46114627.
Narsimhan, G., Gupta, G., Ramkrishna, D., 1979. A model for translational
breakage probability of droplets in agitated lean liquidliquid
dispersions. Chemical Engineering Science 34, 257265.
Okaya, T., 1992. General properties of polyvinyl alcohol in relation to its
applications. In: Finch, C.A. (Ed.), Polyvinyl Alcohol Developments.
Wiley, New York, pp. 130.
Prigogine, I., Marechal, J., 1952. The influence of differences in molecular
size on the surface tension of solutions. Journal of Colloid Science 7,122127.
Shinnar, R., 1961. On the behavior of liquid dispersions in mixing vessels.
Journal of Fluid Mechanics 10, 259277.
Shinnar, R., Church, J.M., 1960. Predicting particle size in agitated
dispersions. Industrial Engineering Chemistry Research 35, 253256.
Siow, K.S., Patterson, D., 1973. Surface thermodynamics of polymer
solutions. Journal of Physical Chemistry 77 (3), 356368.
Sovova, H., 1981. Breakage and coalescence of drops in a batch
stirred vessel. II. Comparison of model and experiments. Chemical
Engineering Science 36, 15671573.
Vermeulen, T., Williams, G.M., Langlois, G.E., 1955. Interfacial area in
liquidliquid and gasliquid agitation. Chemical Engineering Progress
51, 85F95F.
Villalobos, M.A., Hamielec, A.E., Wood, P.E., 1993. Bulk and suspension
polymerization of styrene in the presence ofn-pentane. An evaluation ofmonofunctional and bifunctional initiation. Journal of Applied Polymer
Science 50, 327343.
Ward, J.P., Knudsen, J.G., 1967. Turbulent flow of unstable liquidliquid
dispersions: drop sizes velocity distributions. A.I.Ch.E. Journal 13,
356371.
Yang, B., Takahashi, K., Takeishi, M., 2000. Styrene drop size and size
distribution in an aqueous solution of poly(vinyl alcohol). Industrial
Engineering Chemistry Research 39, 20852090.
Yuan, H.G., Kalfas, G., Ray, W.H., 1991. Suspension polymerization.
JMSReviews in Macromolecular Chemical Physics C31, 215299.
Zerfa, M., Brooks, B.W., 1996a. Prediction of vinyl chloride drop sizes
in stabilized liquidliquid agitated dispersion. Chemical Engineering
Science 51 (12), 32233233.
Zerfa, M., Brooks, B.W., 1996b. Vinyl chloride dispersion with relation
to suspension polymerization. Chemical Engineering Science 51 (14),35913611.