(264c) Numerical Modelling and Simulation of Semi-Batch Free Radical Polymerization Reactor Operation Using Analytical Solution with AK Gel Model

Numerical Modelling
and Simulation of Semi-Batch Free Radical Polymerization Reactor Operation Using
Analytical Solution with AK Gel Model

Dhiraj Kumar Garg^a,
Christophe A. Serra^b, Yannick Hoarau^c

^aDept. of
Chemical Engineering, Shiv Nadar University, dhiraj.garg@snu.edu.in, ^bICS
(UPR 22 CNRS) & University of Strasbourg, ca.serra@unistra.fr, ^cICUBE
(UMR 7357 CNRS) &  University of Strasbourg,
hoarau@unistra.fr

Introduction

There is a tremendous and growing need of improving
numerical modelling and simulation of various physical and chemical phenomena
to save precious resources like time, material, man-power and money required
for performing costly physical experiments. Polymerization is one of them.

Batch and continuous reactors are the two most common types
of reactors that the industry employs (Levenspiel, 1999). Batch reactor is
transient in nature whereas continuous reactors are generally steady-state
reactors. Batch reactor can undergo the variation of temperature and/or
pressure along with the variation of concentration of chemical species during
transient operation. But there is another category of reactors i.e.
semi-continuous or semi-batch reactor, which becomes important in certain
conditions. Semi-batch operation might be required to control certain aspects
of chemical reactions and/or physical properties of chemical products as in
case of polymerization etc. Thus semi-batch provides certain type of freedom to
control the chemical reaction which otherwise is not available in batch and
continuous reactors.

Modelling and simulation of ideal batch and continuous
reactors is well known. But in real operation, various types of non-ideality
may arise e.g. concentration gradient of various chemical species in bulk in
batch and continuous stirred tank reactor (CSTR) or radially in plug flow
reactor (PFR), temperature gradients etc. The modelling of the chemical process
may itself be difficult or restrictive in nature. This generally happens due to
lack of proper theoretical understanding, lack of adequate computational
resources etc. Free radical polymerization is one such example. Thus a good
modelling with a sound theoretical base is always highly desirable which could
be simulated easily and whose predictions are near to the experimental data. In
absence of such adequate theoretical model, semi-empirical or empirical models
are devised to cater the need. Semi-batch reactor presents various and unique
difficulties not only in terms of physical experiment and prediction of its
output but also in terms of its modelling and simulation.

In this work, we have studied the numerical modelling and
simulation of semi-batch operation of free radical polymerization (FRP) using
analytical solution of FRP with AK gel model (Achilias
and Kiparissides, 1992). The semi-batch operations studied are step
change in reactor temperature and step addition of initiator at a certain time
during reaction.

Work done so far

An analytical solution (AS) was obtained for FRP in our
previous work (Garg et al. 2014a). The mathematical model was based on the
method of moments which was well proven method to predict monomer conversion
(XM), number average molecular weight (MWn) and weight average molecular weight
(MWw) and thus polydispersity index (PDI). The kinetic scheme for FRP included
the elementary steps of initiator dissociation and initiation, propagation,
termination by combination, termination by disproportionation, transfer to
monomer, transfer to solvent and transfer to chain transfer agent (CTA). AS was
obtained for isothermal, homogeneous, variable volume batch reactor for solution
homopolymerization before gel, glass and cage effect. A lot of simplification
in the AS derivation was obtained using a useful assumption of quasi steady
state (QSSA) for the live radical polymer chains production. AS was thoroughly
validated against numerical solution as well as published experimental data for
several monomer-polymer systems with different initiators, solvent fractions
and temperature variations before gel effect sets-in.  

AS was then extended to complete range of conversion using
two gel models (Garg et al 2014b; 2014c) namely ? CCS model (Chiu, Carratt and
Soong, 1983) and AK model. CCS is semi-empirical gel model based on free volume
theory given by Vrentas and Duda (1977). In CCS model, various terms are
clubbed into one term whose value or expression is obtained using curve fitting
against experimental data. On the other hand, AK model retains those physical terms
and whose values are obtained experimentally or theoretically for a given chemical
species. AS was separately integrated successfully with CCS and AK models. The
simulation results thus obtained matched quite well with numerical solution as
well as experimental data for isothermal batch reactor.

A very less experimental data has been obtained for
semi-batch reactor. Researchers have obtained the appropriate curve predicting
the conversion, MWn and MWw using semi-empirical models like CCS models whose
parameters were tuned by best curve fitting methods (Ray et al., 1995; Srinivas
et al., 1996; Dua et al. 1996). Hence the utilization of these models was quite
limited to the operating conditions and specific operations for which they were
obtained. There were two types of semi-batch operations that were carried out ?
one with the step change in temperature at a certain time (Srinivas et al.,
1996) and other with step addition of initiator at a certain time (Dua et al.
1996). The step change in temperature is both sides, i.e. increase as well as
decrease of reactor temperature. Carrying out these operations at different
magnitudes as well as different times and predicting the changes in conversion
and polymer properties is really a difficult task. Previous researchers were
successful in obtaining the curves using the above mentioned model successfully
for one type of task only, i.e. they obtained different model parameters values
for temperature and initiator concentration step change separately. But as
already mentioned, it had little applicability for practical applications.

Current Work

We had applied AS integrated with AK model for modelling and
simulating semi-batch reactor for the same operations. Only two parameters need
to be adjusted to match the experimental data and once fixed, the AS can
predict quite successfully the change in conversion, MWn and MWw for both types
of operations of step change in temperature as well as in initiator. The
preliminary results are presented in Fig. 1 and 2. The blue and green lines
represent the numerical solution for batch reactor in absence of any change in
initiator concentration (as in Fig. 1) or in temperature (as in Fig. 2). The
red line represents the curve by AS using AK model. The red circles represent
the experimental data. It can be seen that AS predictions match quite well with
the experimental data. This is a great simplification as it implies the sound theoretical
basis and validity of the model and thus allows greater applicability to
practical purposes like process control, process optimization etc.

Fig. 1 ? Results for sudden addition of initiator to existing
reacting solution at t = 28 min.

Fig. 2 ? Results for sudden drop of temperature from 70°C to
50°C at t = 25 min.

References

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O (1999) Chemical Reaction Engineering. John Wiley & Sons, New York

2)      Achilias,
D. S.; Kiparissides, C., Development of a General Mathematical Framework for
Modeling Diffusion-Controlled Free-Radical Polymerization Reactions.
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3)      Garg
DK, Serra CA, Hoarau Y, Parida D, Bouquey M, Muller R (2014a) Macromolecules
47(14):4567-4586

4)      Chiu,
W. Y.; Carratt, G. M.; Soong, D. S., A Computer-Model for the Gel Effect in
Free-Radical Polymerization. Macromolecules 1983,
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5)      Garg
DK, Serra CA, Hoarau Y, Parida D, Bouquey M, Muller R (2014b) Macromolecules 47(23):8178-8189.

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