18th European Symposium on Computer Aided Process Engineering – ESCAPE 18
Bertrand Braunschweig and Xavier Joulia (Editors)
© 2008 Elsevier B.V./Ltd. All rights reserved.
Modeling and simulation of the particle size
distribution for emulsion polymerization in a
tubular reactor
Ala Eldin Bouaswaig,a Wolfgang Mauntz,a Sebastian Engella,
a
Process Control Laboratory, Department of Biochemical and Chemical Engineering,
Technische Universität Dortmund, 44221 Dortmund, Germany,
Email: [email protected]
Abstract
In this paper, a model for the multiphase process of emulsion polymerization in a
tubular reactor is presented. Besides well investigated properties like e.g. conversion.,
the model predicts the particle size distribution of the polymer particles using a
population balance equation. The model consists of a two-dimensional partial
differential equation for the particle size distribution and dynamic balance equations for
the components that are present in the reactor. It also includes a set of algebraic
equations that e.g. describe the monomer distribution among the coexisting phases in
the reactor. Numerically, the use of flux limiters is considered for the growth term in the
PBE to obtain at least second order accuracy and oscillation-free solutions.
Experimental results reported in literature are used to validate the model.
Keywords: Emulsion polymerization, tubular reactor, particle size distribution,
modeling
1. Introduction
Emulsion polymers are usually produced in semi-batch reactors, single continuously
stirred tank reactors (CSTR), multiple CSTRs connected in series or, less frequently, in
batch reactors. In all these cases, the production rate is limited by the cooling capacity
of the reactors, as the heat generated by the polymerization has to be removed to
achieve safe operation.
Tubular reactors have an excellent heat removal capacity due to their large surface area
to volume ratio. Moreover, they are capable of producing large amounts of product with
uniform product quality. However, despite the potential benefits, only few works have
been published in the open literature on the use of tubular reactors for emulsion
polymerization. Paquet and Ray [1] developed a model for emulsion polymerization in a
tubular reactor and validated it with their experimental data [1,2]. In the work of Sayer
and Guidici [3], a dynamic model for styrene emulsion polymerization in a pulsed
tubular reactor was developed. Two different modeling approaches were considered, the
tanks-in-series model and the axial dispersion model. In another work, Sayer et. al. [4]
used a very similar dynamic model to that of Sayer and Guidici [3] for simulating the
emulsion polymerization of vinyl acetate in a pulsed sieved plate column. The
developed model was also used to investigate different start up procedures. Recently,
Marin et. al. [5] developed a model for the emulsion polymerization of styrene in a
tubular reactor with internal-inclined angular baffles. All these models assume the
particles to be monodisperse and they are here referred to as lumped models. Taking
into account the polydispersity of the particles can be done by modeling the particle size
2
A. Bouaswaig et al.
distribution (PSD) by a population balance equation (PBE). As far as the authors are
aware, only Abad et. al. [6] have made use of a PBE with axial dispersion terms to
simulate PSD in a closed loop reactor and no work has been done so far on modeling
the PSD in an open loop tubular reactor which is the aim of the current contribution.
In section 2 of this paper, a lumped axial dispersion model for emulsion polymerization
in a tubular reactor is extended with a PBE to predict the PSD of the produced latex.
The partial differential equations that arise from balancing the quantities in the reactor
are discretized using suitable numerical methods that take into account the highly
convective nature of the flow and the hyperbolic nature of the PBE and this is described
in section 3. In section 4, the model predictions with respect to conversion and PSD are
compared with experimental data reported in literature, and finally, section 5 is devoted
to drawing conclusions and highlighting future research directions.
2. The emulsion polymerization model
The work of Paquet and Ray [1], Sayer et al. [4] and Abad et al. [7] form the basis of
the model. Several assumptions are made:
 The reactor is perfectly mixed in the radial direction. Dispersion is only considered
in axial direction.
 Particles of the same size have the same number of radicals per particle (i.e. the PSD
is modeled by a pseudo-bulk model).
 The fluid density and viscosity are constant.
 Particles are colloidally stable (i.e. the coagulation term is neglected in the PBE).
Based on these assumptions, the PBE for a tubular reactor reads:
F (v, z , t )

F (v, z , t )
 2 F ( v, z , t )
  v(v, z , t ) F (v, z, t )   u z
 Dax
t
v
z
z 2
  (v  v nuc ) nuc .
(1)
F (v, z , t ) is the population density function, v(v, z , t ) represents the growth rate of the
particles with respect to the internal coordinate (v) and  nuc is the rate of particle
generation by nucleation. Following [1], only micellar nucleation is considered, thus:
 nuc  4rm2 k mm N A N m [ R] w
Vw
.
VR
(2)
The coexisting phases in the reactor are assumed to be in thermodynamic equilibrium.
This enables the usage of so-called partition coefficients to determine the monomer
concentration in the droplet phase, water phases and particle phase. In addition, to
determine the overall monomer concentration in the reactor the following monomer
balance is required:
max
[ M ]
[ M ]
 2 [M ]
P
 u z
 Dax
 k p [ M ]  F (v, z, t )n (v, z, t )dv.
t
z
z 2
vmin
v
(3)
The emulsifier is an inert in the reactor and it can be adsorbed on polymer particles,
adsorbed on monomer droplets or be present in the form of micelles. The typical
diameter of the final polymer particle is in the range of 100 nm while the diameters of
Modeling and simulation of the particle size distribution for emulsion polymerization in
a tubular reactor
3
the monomer droplets are in the order of 10 μm. Because monomer droplets possess a
significantly larger diameter, and hence lower specific area, the amount of emulsifier
required to stabilize the monomer droplets is considered negligible. Thus, micelles will
only be present in the reactor if the emulsifier concentration is above the critical micelle
concentration (Ecmc) plus the amount of emulsifier necessary to stabilize the formed
polymer particles. Mathematically this can be written in the following form:

a
 em
N m   4rm2


0,
b

 

a


vmax
 EVR -  E  4V R F (v, z , t )r 2 dv  , a  b
p
 V w  cmc a V w 
.

ep
vmin





otherwise
(4)
Usually a water soluble initiator is used. Assuming that it undergoes a first order
decomposition, the material balance of the initiator is given by:
[ I ]
[ I ]
 2 [I ]
 u z
 Dax
 fk I [ I ].
t
z
z 2
(5)
Since a pseudo-bulk model is used for the PSD, an expression for the average number
of radicals per particle ( n ) is required in Eq. (3). n depends on the rate of radical
generation, concentration of polymer particles, rate of radical entry into particles, rate of
radical desorption from particles and rate of radical termination. These phenomena are
modeled by the Smith-Ewart recursion formula, and the partial fraction expansion
proposed by Ugelstad et.al. [10] is used in this work to calculate n .
For the calculation of n , the concentration of radicals in the water phase is required.
Their balance takes into account initiator decomposition, desorption from particles,
entry into particles, water phase termination and entry into micelles:
[ R] w
[ R] w
 2 [ R] w
 u z
 Dax
 2 fk I [ I ] 
t
z
z 2
vmax
 F (v, z, t )k d (v, z, t )n (v, z, t )dv 
v min
vmax

VR [ R] w

Vw
VR
w
v F (v, z, t )k ap (v, z, t )dv  V w kT [ R]
min

2
(6)
Vw
 4r k mm N A N m [ R]
.
VR
2
m
w
3. Numerical solution
To simulate the set of PDAEs it is transformed to a set of DAEs by discretization.
3.1. Discretizing the spatial derivatives
As reported by Paquet and Ray [1], the Pe numbers that are encountered in emulsion
polymerization tubular reactors are large and this implies that the flow is convection
dominated. The Pe number poses a restriction on the minimum number of discretization
nodes that is required to avoid numerical oscillations [11]. As suggested by Vande
Wouwer et al. [12], upwind finite differencing is considered in this work to avoid
4
A. Bouaswaig et al.
oscillations. A fourth-order (five-point) biased-upwind finite difference method is used
to approximate the first order spatial derivatives, while all second order derivatives are
approximated by a fourth-order (five-point) centered finite difference method.
3.2. Discretizing the growth term in the PBE
Upon discretizing the spatial derivatives in Eq. (1) the following semi-discrete equation
results:
i ( v ,t )



Fi (v, t )
  
   vi (v, t ) Fi (v, t )   i (v, t )
t
v 



(7)
where (i) represents the index of the discretization node in the spatial direction. i(v,t)
includes the nucleation term at node (i) plus the terms that arise from discretizing the
spatial derivatives in Eq. (1) and i(v,t) is the flux of the population density function.
Eq. (7) is a hyperbolic partial differential equation with a source term. Solving this
equation numerically is challenging because low order discretization schemes cause
smearing near discontinuities and steep fronts and high order discretization schemes can
be unstable and produce numerical oscillation in the vicinity of steep fronts.
Furthermore, numerical stability problems may cause convergence to wrong solutions
[13]. The approach used in this work to overcome these difficulties is to describe the
flux (i) by a high order scheme wherever possible and only switch to an oscillationfree low-order scheme when it is inevitable [13]. This can be achieved by using flux
limiters which are auxiliary functions that operate when steep fronts are encountered in
the solution in order to switch to the low-order scheme [13].
Thus, Eq. (7) is discretized with respect to the internal coordinate (v) as follows:
Fi , j

1 
  1   1   i , j
i, j 
v  i , j  2
2 
(8)



  low1   ( si , j ) high1  low1 
i, j
i, j
2
2 
 i, j 2
(9)



  low1   ( si , j 1 ) high1  low1 
i, j
i, j
2
2 
 i, j 2
(10)
t
1
i, j
2
1
i, j
2

where, si,j is a smoothness indicator and it is calculated from the ratios of the
consecutive gradients with respect to the internal coordinate (v):
si , j 
Fi , j  Fi , j 1
Fi , j 1  Fi , j
.
(11)
In Eq. (9,10)  is the flux limiter function and  ,
are the fluxes of the
population density function approximated by a low order and a high order scheme
respectively. For the problem at hand, the low order flux is approximated by the first
order upwind finite difference scheme and the high order flux is approximated by the
low
high
Modeling and simulation of the particle size distribution for emulsion polymerization in
a tubular reactor
5
second order Lax Wendroff scheme [13]. The monotonized central flux limiter function
is chosen for  [14].
3.3. Time discretization
After discretizing the spatial derivatives in all balance equations and the growth term in
the PBE the model reduces to a set of DAEs. To simulate the discretized model an
explicit fourth-order Runge Kutta method is used to march through time to the steady
state solution. The time steps are chosen sufficiently small to guarantee stability.
4. Results and discussion
Paquet and Ray [1,2] reported experimental data for emulsion polymerization of methyl
methacrylate in a tubular reactor. Their experiments are used to validate the model in
this work. All model parameters are taken from literature.
Fig.1 compares the conversion predicted by the model with the experimental data
reported by Paquet and Ray [1]. As can be seen, model prediction of conversion is
good. Figures 2-4 depict the comparison between the experimental PSD and the model
prediction for three different residence times, namely 20, 30 and 40 minutes
respectively. The PSD predicted by the model is narrower than that reported
experimentally. Immanuel et. al. [15] reported similar findings for semi-batch operation
and attributed the mismatch to the nucleation model and to the assumption of the
particles being colloidally stable. Adjusting the rate constant for radical entry into
micelles (kmm) in Eq. (2) and taking into account homogeneous nucleation could
improve model prediction. Part of the available emulsifier would then be used to
stabilize the formed polymer particles and this amount would not be available for
micellar nucleation at the reactor inlet. The formation of larger particles by collision
would release part of the emulsifier and allow for further micellar nucleation at later
stages. The interaction of this improved nucleation model with the coagulation model
would make the PSD prediction broader and more comparable to experimental data.
5. Conclusion and future work
By extending the emulsion polymerization model with a Population Balance Equation
(PBE), the Particle Size Distribution (PSD) of the produced latex can be predicted.
However, modeling the PSD necessitates the use of an internal coordinate and this
significantly increases the size of the discretized model.
Conversion [-]
0.7
0.6
0.5
0.4
0.3
Exp (Pe=253)
Exp (Pe=325)
Exp (Pe=370)
Sim (Pe=370)
Sim (Pe=325)
Sim (Pe=253)
Normalized frequency [%/nm]
1
0.9
0.8
0.2
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Normalized reactor length [-]
Figure 1. Steady state conversion: Model
prediction compared with experimental data
from [1]
1
12
11
10
9
8
7
6
5
4
3
2
1
0
0
Sim.
Exp.
15 30 45 60 75 90 105 120 135 150
Particle diameter [nm]
Figure 2. Steady state PSD (Pe=253): Model
prediction compared with experimental data
from [2]
6
A. Bouaswaig et al.
9
9
8
Sim.
Exp.
7
6
5
4
3
2
1
0
0 15 30 45 60 75 90 105 120 135 150 165
Particle diameter [nm]
Figure 3. Steady state PSD (Pe=325): Model
prediction compared with experimental data
from [2]
Normalized frequency [%/nm]
Normalized frequency [%/nm]
10
8
7
Sim.
Exp.
6
5
4
3
2
1
0
0
15 30 45 60 75 90 105 120 135 150
Particle diameter [nm]
Figure 4. Steady state PSD (Pe=370): Model
prediction compared with experimental data
from [2]
To handle the highly convective flow in emulsion polymerization tubular reactors a
forth order biased upwind differencing method is proposed to discretize the convective
terms. For the growth term in the PBE flux limiters are used to combine an oscillation
free upwind difference method with a higher order method. The developed model
predicts the steady state conversion well; however, additional phenomena have to be
considered to improve the PSD prediction. Neglecting the homogeneous nucleation and
particle coagulation is questionable. Furthermore, since the particle nucleation is
included in the model, the dependence of the monomer concentration in small particles
on the particle size might be significant. This will influence the growth rate of the
particles and the final shape of the PSD. Finally, the flow profile in the reactor will have
an impact on the PSD predictions. The significance of this influence is currently being
studied by our group through the use of a hybrid decoupled CFD-polymerization model
that investigates the effect of the polymerization process on the flow profile in the
tubular reactor and vice versa. Future work will focus on improving the presented model
by reconsidering the assumptions made in this work.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
D.A. Paquet and W.H.Ray, AICHE J, 40 (1994) 88.
D.A. Paquet and W.H.Ray, AICHE J, 40 (1994) 73.
C. Sayer and R.Giudici, Braz. J. Chem. Eng., 19(2002) 89.
C. Sayer, M. Palma and R.Giudici, Ind. Eng. Chem. Res., 41(2002) 1733.
F. L. M. Marin, L. M. F. Lona, M. R. W. Maciel and R. M. Filho, J. Appl. Pol. Sci.,
100(2006) 2572.
C. Abad, J. C. de la Cal and J. M. Asua, Macromol. Symp., 92(1995) 195.
C. Abad, J. C. de la Cal and J. M. Asua, Chem. Eng. Sci., 49(1994) 5025.
H. M. Vale and T. F. McKenna, Prog. Polym. Sci., 30(2005) 1019.
S. Omi, K. Kushibiki, M. Negishi and M. Iso, Zairyou Gijutsu, 3(1985) 426.
J. Ugelstad, P. C. Mörk, and J. O. Aasen, J. Pol. Sci.: Part A: Pol. Chem., 5(1967) 2281.
B. A. Finlayson, Nonlinear analysis in chemical engineering, McGraw Hill, New York,
1980.
A. Vande Wouwer, P. Saucez, and W. E. Schiesser, Ind. Eng. Chem. Res., 43(2004) 3469.
R. J. Le Veque, Numerical methods for conservation laws. Birkhäuser Verlag, Basel,
Switzerland, (1992).
H. Smaoui and B. Radi, Environ. Fluid Mech., 1(2001) 361.
C. D. Immanuel, C. F. Cordeiro, S. Sundaram, E. S. Meadows, T. J. Crowley, and F. J.
Doyle III, Comp. Chem. Eng., 26(2002) 1133.