chapter 5 optimization examiner correction

Chapter FiveChapter FiveChapter FiveChapter Five

OPTIMAL OPERATING STRATEGIES

OF EMULSION

TERPOLYMERISATION PRODUCTS

ABSTRACT .................................................................................................5.1

5.1 INTRODUCTION ...................................................................................5.1

5.2 DYNAMIC OPTIMISATION REVIEW ....................................................5.2 5.2.1 Formulation of Dynamic Optimisation ......................................................5.5

5.2.2 gPROMS Optimisation............................................................................5.11

5.3 OPTIMAL CONTROL OBJECTIVES ..................................................5.13 5.3.1 Particle Size Optimisation .......................................................................5.14

5.3.2 Molecular Weight Distribution Optimisation...........................................5.17

5.3.3 Composition Optimisation.......................................................................5.18

5.4 OPTIMISATION PROCEDURE ...........................................................5.20

5.5 EXPERIMENTAL VALIDATION OF OPTIMISATION PROFILES ......5.21 5.5.1 Optimising Particle Size..........................................................................5.22

5.5.2 Optimising Molecular Weight .................................................................5.26

5.5.3 Optimising Composition .........................................................................5.31

5.6 CONCLUSION.....................................................................................5.32

BIBLIOGRAPHY .......................................................................................5.34

Chapter 5 Optimal Operating Strategies for Emulsion Terpolymerisation Products

5. 1

Optimal Operating Strategies for Emulsion Terpolymerisation Products

ABSTRACT

In this chapter, the dynamic optimisation of emulsion terpolymerisation is presented.

A literature review for the emulsion polymerisation optimisations is firstly addressed.

The theory of process optimisation especially the dynamic one is then explained. The

different problem formulations and solution techniques in gPROMS are discussed.

Off-line dynamic optimisations application to the emulsion terpolymerisation process

of styrene, MMA and MA are investigated for the PSD, MWD and polymer

composition control. The obtained optimal profiles are then validated experimentally.

5.1 INTRODUCTION

Optimisation plays a crucial role in the design of industrial processes, due to the

desire to save valuable resources, meet ambitious production goals, and achieve the

best possible profit margins. The polymer manufacturing and characterisation has

attracted the focus of numerous industrial companies and research institutes. Polymer

manufacturer’s face increasing pressures for production cost reductions and stringent

quality requirements. Many applications of polymer latexes, which are mainly

produced through emulsion polymerisation, require specific physical and mechanical

properties, such as in paints, adhesives and paper coating. These properties are

mainly controlled by their chemical composition, particle size distribution (PSD) and

molecular weight distribution (MWD). Chemical composition of terpolymers for

example determines the glass transition temperature (Tg). The particle size and size


5. 2

distribution affect the dispersion rheology which is critical in achieving the desired

surface finish in coating applications (Anderson and Rapra Technology Limited

(GB), 2003). Also content adhesives require bimodal or multimodal particle size

distributions to maintain a manageable viscosity (Anderson and Rapra Technology

Limited (GB), 2003). MWD has a significant effect on the important end-use

properties of the polymer film such as for adhesion, elasticity, strength, stress-strain

relationships, toughness and solvent resistance(Vicente et al., 2002). Advanced

process optimisation techniques allow the achievement of these requirements as well

as productivity/quality improvements to the chemical process. Therefore, there are

strong incentives to develop strategies to optimise and control the PSD, MWD and

polymer composition.

5.2 DYNAMIC OPTIMISATION REVIEW

Dynamic optimisation is the process of optimising some aspects of process design

and performance taking into account both steady-state and dynamic considerations.

Dynamic optimisation allows operability considerations to be taken into account at

the design stage. This method is applicable to continuous, batch- and semibatch

processes. Dynamic optimisation, in contrast with simulation, allows in formulating

the problem to be solved directly, and then useing the start-of-the-art numerical and

optimisation solvers to determine the optimal values or time profiles of the

optimisation variables.

Several studies have been reported on the optimisation and control of polymerisation

reactors, and these have been recently reviewed by (Cheremisinoff, 1989,

Chakravarthy et al., 1997, Mitra et al., 1998, Kiparissides, 2006). Most of these

studies are on the optimisation and control of lumped properties (Arzamendi et al.,

1992, Kozub and Macgregor, 1992, Crowley et al., 2000, Canu et al., 1994, Saldivar

and Ray, 1997, Sayer et al., 2001b, Vicente et al., 2002, Schoonbrood et al., 1996,

Vicente et al., 2001) and deal with homo- and copolymerisation rather than

terpolymerisation. Table 5.1 which was reported by (Kiparissides, 2006), depicts the


5. 3

most recent publications on monitoring, estimation, optimisation and control of

particulate polymerisation processes.

The major issues regarding the control of emulsion polymerisation reactors was

summarized by (Dimitratos et al., 1994). A hybrid modelling strategy for batch - to -

batch optimisation of PSD for emulsion polymerisation was presented by (Crowley

et al., 2001) and (Doyle et al., 2003). An open loop optimisation study for the

control of the PSD in a semibatch emulsion copolymerisation of vinyl acetate and

butyl acrylate was described by (Immanuel and Doyle, 2002). Semino and Ray

(Semino and Ray, 1995b, Semino and Ray, 1995a) studied the controllability of PSD

in emulsion polymerisation reactors using lumped manipulated variables. They found

that there are appropriate strategies for the control of the PSD through the

manipulation of surfactant, initiator and inhibitor feed concentrations. Models for

emulsion homo- and co-polymerisation were developed by (Zeaiter et al., 2002,

Alhamad et al., 2005) respectively to optimise and control the PSD.

Approaches for the calculation of the optimal monomer addition policies for polymer

composition control in emulsion terpolymerisation are limited (Arzamendi et al.,

1992, Schoonbrood et al., 1996). (Congalidis et al., 1989) presented a feed forward-

feedback control system to regulate the polymer production rate, copolymer

composition, molecular weight and reactor temperature in a simulated solution

copolymerisation of methyl-methacrylate/ vinyl acetate. (Saldivar and Ray, 1997)

presented the control of copolymer composition and averaged molecular weight for

semi-continuous emulsion polymerisation. (Clarke-Pringle and MacGregor, 1998)

developed a batch to batch adjustment strategy for the control of molecular weight

distribution to re-optimise the inputs for the next batch based on end-point

measurement of the MWD.

An iterative dynamic programming (IDP) procedure was applied by (Sayer et al.,

2001b) to optimise the MWD and polymer composition during isothermal

semicontinuous emulsion polymerisation. Bojkov and Luus (Bojkov and Luus,

1995, Bojkov and Luus, 1994a, Bojkov and Luus, 1994b) used intervals of varying

lengths in IDP, and observed that accurate switching times can be obtained, as is

required for time optimal control problems.


5. 4

The optimisation of polymerisation processes may often present conflicting criteria

when multi-objective optimisation for polymerisation reactors is designed. This is the

main motivation of several studies dealing with the use of multi-objective

optimisation for polymerisation reactors (Gupta and Gupta, 1999, Garg and Gupta,

1999, Sareen and Gupta, 1995, Mitra et al., 1998, Mitra et al., 2004, Bhaskar et al.,

2001, Bhaskar et al., 2000, Yee et al., 2003). Bhaskar et al.(Bhaskar et al., 2001,

Bhaskar et al., 2000) pointed out that the multi-objective optimisation of real-life

systems is quite complex and each new application may require the development of

several adaptations of optimisation algorithms to obtain meaningful solutions,

irrespective of which mathematical procedure is used for the purpose.

In this work, the optimisation of semibatch emulsion terpolymerisation reactors was

investigated. The optimisation is performed by IDP procedure through gPROMS

with variable time intervals and is based on our detailed mechanistic model (Srour et

al., 2005) to develop an advanced control strategy for the optimal operation of the

reactor. The major objective of our optimisations is to obtain on a profile of optimal

strategy that would calculate set-point trajectories for the manipulated variables in

order to ensure the production of a polymer with a desired PSD, MWD or terpolymer

composition in the minimum reaction time. The optimisation procedure was applied

to styrene, methyl methacrylate and methyl acrylate emulsion terpolymerisation. Six

variables were used as manipulated variables: styrene monomer feed rate, MMA

monomer feed rate, MA monomer feed rate, surfactant feed rate, initiator feed rate,

and the temperature of the reactor.

The strategy of implementing optimal trajectories obtained from off-line optimisation

problem to corrected process has drawbacks. In the presence of disturbances, such as

changes in the reactor temperature or contamination with inhibitors/impurities, the

optimal trajectories may produce sub-optimal results. In our work, the experimental

validation was carried out in open loop mode with precise reaction procedure and

tuning to minimise the effect of disturbances. Consequently, this work represents an

introduction to model based control and is the first step in the advanced control

hierarchy adopted.


5. 5

Table 5.1 Selective publications on monitoring, estimation, optimisation and control

of polymer quality (Kiparissides, 2006).

Author Test case Comments

Congalidis et al. So: MMA/VAc Control: MWD/CCD

Ellis et al. So: MMA Estimation/control: MWD

Embirucu et al. E/S/C Modelling/estimation/optimisation/control: MW/CCD

Saldivar and Ray E: MMA/Vac Control: MWD/CCD

Yabuki and MacGregor E: SBR Control: MWD

Crowley and Choi So: MMA Optimisation/control: MWD

Crowley and Choi So: MMA Optimisation/control: MWD

Pringle and MacGregor So: St Optimisation/MWD

Crowley and Choi So: St/MMA Optimisation/control: MWD/CCD

Yoo et al. So: St Control: MWD

Sayer et al. E: MMA/BuA Control: MWD/CCD

Santos et al. E: MMA/VAc Modelling/estimation: CCD

Stavropoulos et al. E: St/2-EHA Modelling: CCD

Sayer et al. E: MMA/BuA Optimisation: MWD/CCD

Vicente et al. E: St/BuA Control: MWD/CCD

Chang and Hung So: MMA Optimisation: MWD

Park et al. So: MMA/MA Modelling/estimation: MWD/CCD

Valappil and Georgakis E: St Control: MWD/PSD

Kiparissides et al. S: MMA Optimisation/control: MWD

Catalgil-Giz et al. S: MMA/St Monitoring: MWD/CCD

Chatzidoukas et al. C: Olefin Optimisation: MWD/CCD

Dimitratos et al. E:acryl/vinyl/St Control: MWD/CCD/PSD

Semino and Ray E Control: PSD

Ohmura et al. E: VAc Control: PS

Fevotte et al. E: MMA/VAc Modelling/estimation: PSD

Crowley et al. E: St Control: PSD

Christofides E Control: PSD

Cerrillo and MacGregor E: St Control: PSD

Immanuel et al. E: Vac/BuA Estimation: PSD

Immanu el and Doyle E: VAc/BuA Optimisation/control: PSD

Doyle et al. E: St Control: PSD

Process C: catalytic; E: emulsion; S: suspension; So: solution.

5.2.1 Formulation of Dynamic Optimisation


5. 6

Three main issues need to be addressed in the development of an optimisation/

control policy for a polymer process. Firstly, an understanding of the process

dynamics via mechanistic or data-based modelling and experimentation. Secondly,

the ability to measure and characterize a range of polymer quality variables via on-

line sensors or nonlinear estimation algorithms. Thirdly, the development of

nonlinear model-based predictive controllers with emphasis on achieving superior

performance and constraint handling. Thus, the following steps were used to

optimise our process:

• Development a reliable process model. The key to building a reliable model

is to mathematically describe the chemical and physical phenomena involved

in a process and to derive the necessary material, energy and momentum

balances. These mathematical descriptions involve nonlinear algebraic and

differential equations of a dynamic polymerisation model are strongly related.

• Identify the process variables which can be manipulated and controlled.

• Develop the objective functions. This model involves an equation that

represents the profit made from the sale of products and costs associated with

their production, such as raw materials, operating costs, fixed costs, taxes,

etc. The structure and complexity of the equations for the economic model

and process or plant constraints are very important, since most mathematical

programming procedures take advantage of the mathematical form of these

models. Linear programming is used when all of the equations are linear,

while geometric programming is used when all of the equations are

polynomials.

• Adjust a suitable optimisation procedure. This procedure locates the values of

the independent variables of the process to produce the maximum profit or

minimum cost as measured by the economic model. The procedure also

includes the constraints in materials, process equipment and manpower.

The mathematical model that was used for optimisation purposes is based on the

detailed dynamic model (Srour et al., 2007). Based on this comprehensive model, a


5. 7

multi-layer model-based framework was developed and implemented within an

industrial distributed control system environment, to optimise the PSD, MWD and

terpolymer composition. The expected form of the optimisation algorithm is one that

would calculate set-point trajectories for the manipulated variables, so as to ensure

the production of a polymer with a defined PSD, MWD and terpolymer composition

in the minimum reaction time. The feed rates of different processes, the length of

each interval and temperature profile are the manipulated variables. The off-line

optimal trajectory obtained in this work will be used as a set point within an on-line

multivariable model predictive control scheme to control the PSD, MWD and

polymer composition.

The particle size evolution equations are a set of three coupled partial 2-D integro-

differential equations in radius and time. Also the molecular weight equations are a

set of partial-integro-differential equations in chain length and time. To overcome

this problem, the discretisation technique is used. The particle size evolution

equations and the molecular weight equations are explained and formulated in

chapter three.

The continuous dynamic system is a process model which contains a combination of

differential and algebraic equations of the form:

( ) 0)),(),(),(),( =vtutytxtxf & (5.1)

where x(t) and y(t) are differential and algebraic variables of the model, respectively.

)(tx& is the time derivative of variable x(t), and u(t) is a vector of time-variant control

variables and v is the vector of time-invariant parameters which to be determined by

optimisation. This model description is also subject to the following initial

conditions:

( ) 0)),0(),0(),0(),0( =vuyxxI & (5.2)

The optimisation function (gOPT code in gPROMS) seeks to determine the time

horizon of the process, tf, the time invariant parameters and the dynamic variation of


5. 8

control variables over the whole time horizon, so that the final value of the objective

function ,z, being one of either differential or algebraic variables of the system, to be

maximized or minimised. The mathematical description of the proposed dynamic

optimisation problem (DOP) can be defined as:

)]([],0[),(,,

ftttuvt

tzOptimizeff ∈

(5.3)

This is subject to Eq.s. (5.1) and (5.2) as model description and initial conditions.

When the time variations of the controls (u) and the values of any time invariant

parameters [tk-1, tk] are fixed, the modelling equations together with the initial

conditions completely determine the transient response of the system by performing

a dynamic simulation. The solutions of dynamic optimisation will determine:

• The time horizon (tf).

• The values of the time invariant parameters (v).

• The time variation of the control variables, u(t), over the entire time horizon

],0[ ftt ∈

It is important to note that the maximisation problem is equivalent to the

minimisation problem if the sign of the objective function is reversed; that is,

)]([)]([],0[),(,,],0[),(,,

ftttuvt

ftttuvt

tzMintzMaxffff

−=∈∈ (5.4)

The vector of manipulated inputs u1, u2… um-1 are chosen to have the greatest

influence on the objective function. These variables must be successively adjusted,

during optimisation, to obtain the desired maximum or minimum. Each set of

adjustments to these variables is termed iteration, and in general a number of

iterations are required before an optimum is obtained. In the iterative algorithm a

first estimate of the decision variables must be supplied as a starting point. In the

context of semibatch emulsion terpolymerisation processes as systems studied here,


5. 9

control variable can be PSPI, MWPI, Mn, process conversion and average radius of

the polymer particles, throughout the process, while time invariant parameters can be

feed rates of monomers, initiator, and surfactants ,and reactor temperature.

The optimisation problem often has more than one objective to be maximised or

minimised. The problem then is formulated as a multi-objective optimisation case.

Such problem is dealt with by reformulating the multi-objective problem as a single-

objective case by forming a weighted combination of the different objectives (i.e.

weighted-sum strategy) or else by placing some objectives in the form of constraints

which is called ε-constraint method (Choi and Butala, 1991). The optimal control

problem takes then the following form for the weighted-sum strategy:

...332211)(

+++ JJJMaxix

λλλ i=1,2,…,n (5.5)

Where J1, J2, J3,... are the objectives and λ1, λ2, λ3, ... are the corresponding weights.

Note that, because J is a vector, if any of the components of J are competing, there is

no unique solution to this problem. Instead, the concept of non-inferiority (also

called Pareto optimality) must be used to characterise the objectives. A non-inferior

solution is one in which an improvement in one objective requires a degradation of

another.

In the ε-constraint method, only one of the objectives (mainly the primary one) is

expressed in the cost function while the other objectives take inequality constraints

form.

),....,,( 211)(

nix

xxxJMax

Subject to (5.6)

ini xxxJ ε<),....,,( 21 i=1, 2,…, n

This approach is able to identify a number of non-inferior solutions on a non-convex

boundary that are not obtainable using the weighted sum technique. A problem with


5. 10

this method is, however, a suitable selection of λ to ensure a feasible solution. A

further disadvantage of this approach is that the use of hard constraints is rarely

adequate for expressing true design objectives. There are several types of constraints

which are: Algebraic equality constraints which express the relationships connecting

the decision variables and thus reduces the degree of freedom in the system;

Algebraic inequality constraints which specify the practical operating limits of

certain variables within the process; Differential equality constraints which are

commonly used in chemical processes, where the rate of formation or consumption

of specie is a function of the state variables. Constrained optimisation is very much

more difficult than unconstrained optimisation and a great deal of effort has been

expended to reformulate constrained problems so that constraints are avoided.

For this, our system was subjected to some constraints including the constraints on

the time horizon and decision variables,

maxmin

fff tft ≤≤ (5.7)

],0[),()()( maxmin ftttututu ∈≤≤ (5.8)

maxmin vvv ≤≤ (5.9)

The constraints on the decision variables are stated explicitly since modern

optimization algorithms can handle them very efficiently. This is not generally the

case of other types of constraints. Then we have the end-point constraint variables,

which usually represent certain conditions that the process system must satisfy at the

end of the optimization horizon. For convenience, end-point constraints are divided

into inequality and equality constraints. Although, the latter are a special case of the

former, differentiating them simplifies the definition of some optimization problems.

maxmin )( effef wtww ≤≤ (5.10)

tgt

eefef wtw =)( (5.11)

Due to the transient nature of the emulsion polymerisation process and the inherent

nonlinearities in the system, non-linear constrained dynamic optimisation


5. 11

formulation was used in this work. The optimisation was performed by solving the

comprehensive dynamic models on gPROMS software package (PSE, UK) in the

gOPT dynamic optimisation code. The dynamic formulation of chemical processes in

gPROMS software package is well explained in (PSE, 2004).

The gOPT code utilises the control vector parameterisation algorithm, employing

piecewise control variables. The main attraction of this method is due to its

suitability for solving large-scale dynamic optimisation problems. Five or more equal

horizons were identified for the six manipulating variables. These manipulating

variables remain constant during each horizon and move discretely from one horizon

to the next.

5.2.2 gPROMS Optimisation

The dynamic optimisation problem is set-up in gPROMS using a dynamic model and

a separate optimisation file. This file includes information on the time horizon,

objective function, form of the control variables and any constraints that need to be

imposed on the process.

The solution of dynamic optimisation problems is based on the classical calculus of

variations, the maximum principle of (Pontryagin et al., 1962), and the dynamic

programming of (Bellman and Kalaba, 1965). The most commonly used member of

the triad is the maximum principle as most computational techniques are concerned

with satisfying the maximum principle necessary conditions for an optimum. The

solution of the optimality conditions problem is generally obtained using the quasi-

linearisation approach (Miele, 1975) and through the use of multiple shooting

algorithms such as those proposed by (Bulirsch, 1971, Dixon and Bartholomew-

Biggs, 1981). Other techniques employ a discretisation approach whereby the

optimal control problem is converted to a non-linear problem (NLP) through the

discretisation of all variables. This can be done using the finite difference and

orthogonal collocations methods (Biegler, 1984).


5. 12

The type of control variable to be implemented during the dynamic optimisation in

gPROMS requires specification. The ideal choice generally depends on the

capabilities of the control system from an engineering perspective rather than the

mathematical features of the problem. There are four common control variable

profiles in gPROMS which are: piecewise constant controls where the control

variable remains constant over a period of time before jumping discretely to a

different value for the next interval, piecewise linear controls which follow a linear

path over one time period before jumping discretely to a different linear variation for

the next interval, piecewise linear continuous controls which are the same as

piecewise linear controls, with the added constraint that their values be continuous

over the interval boundaries and polynomial controls which vary smoothly over time.

Dynamic optimisation in gPROMS facilitates piecewise constant and piecewise

linear controls. However, the remaining types of control are easily implemented by

adding additional equations to the model (PSE, 2004). For instance, piecewise

constant controls may often be preferable to other types as they are much easier to

implement and still perform the job efficiently.

The algorithm used in gPROMS, named gOPT code, employs the control vector

parameterisation (CVP) approach coupled with a multi-step backward-difference

method for the integration of the set of differential algebraic equations (Vassiliadis et

al., 1994b, Vassiliadis et al., 1994a, Vassiliadis et al., 1993). When piecewise-

constant approximation over equally spaced time intervals is made for the inputs, the

method is referred to as Control Vector Parameterization (CVP) in the literature

(Srinivasan et al., 2003). The quality of the solution via the CVP approach is strongly

dependent on the parameterization of the control profile. For applying CVP method,

five or more equal horizons were identified for the six manipulating variables. These

manipulating variables remain constant during each horizon and move discretely

from one horizon to the next. The time horizon, over which the process is optimised,

is partitioned into a pre-defined number of stages. The duration of each stage is

divided into a number of control intervals and the continuity of the differential

variables is enforced at the boundaries through simple junction conditions.


5. 13

The solution of the optimisation problem comprises the time horizon, the value of

time invariant parameters and the variation of the control variables over the time

horizon that minimise (or maximise) the objective function, while satisfying the

constraints. gPROMS outputs this data on screen as well as into four files:

� The optimisation report file which contains a summary report on the

optimisation run.

� The detailed results file contains the optimal time profiles of all variables in

the problem.

� The schedule file presents the solution in the form of a gPROMS schedule.

� The gOPT file contains the same information as the schedule file and is

generated at the end of iteration.

The detailed operating procedures of the dynamic optimisation processes within

gPROMS are presented in gPROMS user guide (PSE, 2004).

5.3 OPTIMAL CONTROL OBJECTIVES

The production of emulsion polymers by batch and semibatch processes is of

significant industrial importance. In many cases the amount of material that must be

produced does not justify the use of a continuous process so batch operation is

preferred. However, due to the non-steady nature of batch processes and the different

rates of reaction of the monomers in a recipe, these processes will produce polymers

whose properties change with the progress of the reaction. The instantaneous

terpolymer produced will exhibit a drift in terpolymer composition, particle size and

molecular weight. The end result is that the final terpolymer, which is a blend of all

the terpolymers produced instantaneously, will be heterogeneous. An obvious

remedy to this problem is the use of semibatch operation in which flow rates for

some of the reactants are introduced in order to nullify the drift in composition,

particle size and molecular weight. When the emulsion polymerisation operates,

semibatch process has six manipulating variables while batch has only one which is


5. 14

the temperature. Thus, semibatch has more degree of freedom than that of batch.

Moreover, terpolymer composition control can not be attained in batch process

especially when the process involves monomers with highly different reactivity

ratios. Therefore, semibatch operations are used to optimise and control the end-use

properties of the terpolymer.

Based on the dynamic model, three main different optimal control objectives for

semibatch emulsion polymerisation of styrene, MMA and MA were formulated in

this work and validated experimentally by implementation within a distributed

control system (DCS) environment. These optimal control objectives were set up to

determine the control policy for the control variables (manipulating variables) that

would produce our desired PSD, MWD and polymer composition in a specified

semibatch operation time and monomers amount.

5.3.1 Particle Size Optimisation

The particle size in an emulsion has a significant effect on the conditions of the

polymerisation reaction and the properties of the final product. The particle size of

the latex influences the emulsion viscosity. The viscosity of an emulsion will

increase as the particle size is reduced. Mixtures of particles on the other hand tend

to yield lower viscosities (Fitch et al., 1987, Fitch, 1980). Monodisperse (narrow)

size distribution is required for a glossy finish to latex paints while a polydisperse

(broad) particle size distribution results in a mat finish (Bartsch et al., 1999). The

particle size and size distribution affect the dispersion rheology which is critical in

achieving a smooth surface in coating applications (Anderson and Rapra Technology

Limited (GB), 2003). Adhesives require a broad distribution to maximise their

strength since the different size particles pack closer together, minimising the void

area and thus improving adhesive performance (Fitch, 1980). Also high solids

adhesives applications require bimodal or multimodal particle size distributions to

maintain a manageable viscosity (Anderson and Rapra Technology Limited (GB),

2003). Therefore, two objectives are considered in this study to optimise the

productivity of the reactor with a defined PSD. One objective is to produce


5. 15

monodisperse PSD and the other is to produce a very broad distribution of PSD

(~bimodal).

The framework of population balances is ideally suited to the description of the

complex dynamics of particulate polymerisation processes due to the highly coupled

kinetics, thermodynamics, heat and mass transfer phenomena, taking place in a

heterogeneous process. The numerical solution of the dynamic PBE for a particulate

system, especially for a reactive one, is a notably difficult problem due to both

numerical complexities and model uncertainties regarding the particle nucleation,

growth, aggregation and breakage mechanisms that are often poorly understood.

Usually, the numerical solution of the PBE requires the discretisation of the particle

volume domain into a number of discrete elements that results in a system of stiff,

nonlinear differential or algebraic/differential equations that is solved numerically.

For this reason, the PBE are discretised according to the radius of the particle using

backward finite difference method which has been described in section 3.2.

For achieving the two control objectives of PSD, the particle size polydispersity

index (PSPI), which indicates the spread of the size distribution, is selected as the

objective function. For obtaining monodisperse, PSPI objective function is

minimised while it is maximised to obtain on bimodal PSD. Also this formulation

can control the exact breadth of PSD by setting the PSPI objective function an

appropriate value with specific average radius as a constraint. The particle size

polydispersity index (PSPI) is estimated according to Eq. (5.12). The number

average radius and the mean squared radius are determined by the Eqns. (5.13 &

5.14).

2

2

><

><=

r

rPSPI (5.12)

∑

∑

=

=

⋅

=G

i

G

i

uns

in

inir

r

1

1

)(

)()( (5.13)


5. 16

∑

∑

=

=

⋅

=G

i

G

i

uns

in

inir

r

1

1

2

2

)(

)()( (5.14)

The objective functions to be optimised were defined as:

( )[ ] [ ]ft0, t ; r ; tr,PSPIJmax ∈∀ℜ∈∀= (5.15)

( )[ ] [ ]ft0, t ; r ; tr,PSPIJmin ∈∀ℜ∈∀= (5.16)

The final PSD shape was included in this optimisation in the form of end point

inequality constraints formulated in terms of the final molar concentration density of

particles.

maxmin ),( ntrnn final ≤≤ (5.17)

where nmin and nmax denote the lower and upper limits respectively, and were

specified to match the required distribution.

The system decision variables (manipulating variables) were bounded and explained

in details in section 5.6. In PSPI optimisation, five equal control intervals are

implemented and the time horizon and control horizon interval are subject to the

following bounds:

sts f 250007500 ≤≤ (5.18)

stk 50001500 ≤≤ 5,..,2,1=k (5.19)

where k represent the number of the control horizon interval.

The average particle diameter and the monomer conversion are subjected to the

following end-point constraints:


5. 17

Conversion: 175.0 ≤≤ w (5.20)

Average particle diameter: nmR avg 10075 ≤≤ (5.21)

5.3.2 Molecular Weight Distribution Optimisation

The end-use properties of the produced polymers are directly linked with the

molecular characteristics of the polymer chains. For instance, the proper mixing of

small-size and large-size polymer chains (bimodal MWD) may lead to simultaneous

improvement of both the flow characteristics of the polymer latex and the

mechanical properties of the final polymer film (Chia-Fen et al., 2005). Therefore,

there is a strong incentive to develop strategies to control the complete polymer chain

microstructure (MWD), and not only the molecular weight averages (Mn) (Sayer et

al., 2001a). In this study, we set two different optimised objectives for MWD, one to

produce monodisperse MWD and the other broad MWD. The Molecular weight

polydispersity index (MWPI), which gives an indication of the width of the

distribution is selected as the objective function to control the MWD. For obtaining

monodisperse, MWPI objective function is minimised while it is maximised to

obtain broad MWD. MWPI is defined in Eq. (3.12) which does not indicate that the

distribution is at high or low molecular weight. It mainly indicates the breadth of the

distribution. For this reason, Mn is used as a constraint in the optimisation process in

order to obtain the MWD at specific molecular range. The objective functions to be

optimised were defined as:

( )[ ] [ ]ft0, t ; M ;tM,MWPIJmax ∈∀ℜ∈∀= (5.22)

( )[ ] [ ]ft0, t ; M ;tM,MWPIJmin ∈∀ℜ∈∀= (5.23)

Five equal control intervals are implemented and the time horizon and control

horizon interval are subject to the following bounds:

sts f 250004500 ≤≤ (5.24)


5. 18

st 50001500 1 ≤≤ (5.25)

stk 5000750 ≤≤ 5,4,3,2=k (5.26)

Subjected to process conversion end-point constraint: 18.0 ≤≤ w (5.27)

5.3.3 Composition Optimisation

One of the main objectives in the control of an emulsion polymerisation process is to

maintain the polymer composition at a predetermined level because this variable has

an important role in determining end-use properties of the product, such as glass

transition temperature, particle morphology, and mechanical and chemical resistance.

Intelligent control of the polymer composition is very important when several

monomers with different reactivity ratios are involved in the reaction at a

nonazeotropic monomer composition, which is often the case for industrial systems.

If no control action is taken significants polymer composition drifts occur in batch

process, and this leads to heterogeneity in the polymer properties. Therefore, it is

important to operate the polymerisation reactor in a way such that this desired

polymer composition is obtained. The overall polymer composition Fcomp,i is

determined by integrating the instantaneous polymer composition produced per unit

time. In a terpolymerisation process, constant composition is obtained if the

following two ratios are maintained at the desired values (m & n) throughout the

reaction:

mF

F

comp

comp=

3,

1, (5.28)

nF

F

comp

comp=

3,

2, (5.29)

From the dynamic models, it is noted that the overall polymer composition is a

function of the ratio of the monomers and their reactivity ratios. The reactivity ratios

are mainly affected by the operating temperature. Thus it is sufficient to manipulate

the ratios of the monomers in the reactor in order to keep the composition at a

desired value. Thus, the values of ‘m’ and ‘n’ must be controlled simultaneously in


5. 19

order to guarantee the production of a polymer with the desired composition. This

can be accomplished by manipulating the flow rates of the monomers with ‘m’ and

‘n’ as constraints in a dynamic optimisation scheme to maximise the conversion.

High monomer conversion with desired polymer composition is expected from such

an optimisation. The objective function to be optimised was defined as:

( )[ ] [ ]ft0, t twJmax ∈∀= (5.30)

And the composition of the terpolymer is controlled via the constrained values of ‘m’

and ‘n’ in the following equations.

[ ]maxmin m,m m ∈ (5.31)

[ ]maxmin n,n n ∈ (5.32)

In the case of optimising the terpolymer composition to produce constant

composition at ratios of 5/3/2 for Sty/MMA/MA, the m and n were bounded as

follows:

[ ]2.32,2.83 m ∈ (5.33)

[ ]1.38,1.63 n ∈ (5.34)

The system decision variables (manipulating variables) were bounded and explained

in details in section 5.6. In Composition optimisation, twenty equal control intervals

(not five as for PSPI and MWPI) are implemented because composition control

needs more control action especially when their rates of polymerisation are widely

different, i.e. have different reactivity ratios. The time horizon and control horizon

interval are subject to the following bounds:

sts f 400008000 ≤≤ (5.35)

stk 2000200 ≤≤ 5,..,2,1=k (5.36)


5. 20

5.4 OPTIMISATION PROCEDURE

The objective functions were first defined as mentioned before. For operational and

technical reasons such as the pumps potential limit, the feed rates of monomers,

surfactant and initiator, and temperature were specified within the following upper

and lower bounds:

[ ]0.2g/s 0.0g/s, F lm, ∈ (5.37)

[ ]0.05g/s 0.0g/s, Fs ∈ (5.38)

[ ]0.05g/s 0.0g/s, FI ∈ (5.39)

[ ]K 355 K, 338 T ∈ (5.40)

dt

tdT )( [ ]4,4-∈ (5.41)

The optimisations are for a semibatch operations and so an additional end equality

constraints control the total moles of monomers in the recipe (Nm,T) was added.

[ ]mol mol,4 1.5 N t ∈ (5.42)

An initial time length of 1500 seconds is specified for the first stage of the intervals

which is for seeding stage. The overall horizon is bounded between 6000.0 and

32000 seconds. Note that the feed rates starts after 1500s of the starting point of the

operation even though the feed rates in the feed profiles starts from the starting point

of the reaction. This is because the dynamic model included a 1500s time delay for

all the feed rates. This first interval of the operation is for producing polymers

particles and for this it is called seeding interval.


5. 21

The process equations are solved with a differential algebraic equation solver

(DAES) under the initial guessed conditions. This produces the value of the objective

function, which the optimisation routine then iteratively uses to find the optimal

parameters in the control parameterisation. The sequential method is of the feasible

path type, in which all process equations are feasible during the calculation of the

objective value for the iteration. The piecewise-constant control is set for the flow

rates and temperature in which they remain constant for the first period of time, and

become another value in the second interval of time. This will continue for all the

intervals.

The CVP_SS, a standard mathematical solver for optimisation in gPROMS, is

implemented to solve the model based optimisation problem in this work. CVP SS

can solve optimisation problems with both discrete and continuous decision

variables. For dynamic optimisation problems, CVP_SS is based on a control vector

parameterisation (CVP) approach which assumes that the time-varying control

variables are piecewise constant (or piecewise linear) functions of time over a

specified number of control intervals. The precise values of the controls over each

interval, as well as the duration of the latter, are generally determined by the

optimisation algorithm. The solver implements standard DASOLV and SRQPD

codes. The DASOLV code is to find the solution of the underlying differential

equation (DAE) problem and the computation of its sensitivities. While SRQPD

solver employs a sequential quadratic programming (SQP) method for the solution of

the nonlinear programming (NLP) problem. Both solvers have their own algorithmic

parameters described in the manual of gPROMS (PSE, 2004).

5.5 EXPERIMENTAL VALIDATION OF OPTIMISATION

PROFILES

Semibatch emulsion terpolymerisation reactions of styrene, MMA and MA were

carried out according to the optimal profiles obtained from the optimisation. The

experiential set up is discussed in details in chapter three. The optimal control


5. 22

strategy involved the computation of the monomers feed including styrene and

MMA, surfactant feed, initiator feed and temperature trajectory which would drive

the process to give a terpolymer with a specified properties. The obtained optimal

duration to produce the desired terpolymer was varied between 1.5 and 3 hours

depending on the objectives functions and the constraints.

Three cases were carried out in this work as mentioned above. A standard PID

controller was used for internal and external temperature control, so the temperature

of the reactor was controlled within ±1.5oC to the setting point. A circulator provided

a high (constant) water flow rate through the external reactor jacket, with the water

inside the circulator. The circulator is supplied with a coil for heating and a

refrigeration system for cooling. All the manipulated variables profiles are put into

the Control Builder to be operated automatically.

The experimental results (sections 5.5.1-5.5.3) have good agreement with model

predictions. There is slight difference (≤±5%) between the experimental results and

the model simulations in which the experimental results are still accepted for model

validation. This difference is due to some normal experimental error and reactor

temperature control. The temperature of the model simulation is maintained exactly

the same as its optimal trajectory through the whole reaction, while the temperature

of the reactor that reflecting the experimental results under a PID controller, means

the temperature would oscillate over and under the set point temperature. This

explains that the slight difference is mainly with the PSD and MWD results.

5.5.1 Optimising Particle Size

Particle size is mainly affected by the amount of surfactant, monomers and initiator

involved in the operation. Surfactants are the sources of micelles which change to

new particles through micellar nucleation. Initiators are the sources of oligomers

which generate new particles by homogeneous nucleation or micellar nucleation

when the reactor contains micelles. Temperature has an indirect effect on the

particle size. The strength of its effect depend on the amount of un-reacted


5. 23

monomers presenting in the reaction since temperature increases the rate of

polymerisation which leads to an increase in the growth rate of the particles, hence

narrowing the particle size distribution. Two optimisation operations were carried out

in this section. The first optimisation was carried out to produce bimodal PSD by

maximising the PSPI function (figure 5.1). The second optimisation was for

minimizing the breadth of PSD by minimizing the PSPI function (figure 5.2).

In figure 5.1, the optimisation was aimed to produce bimodal particle size

distribution by maximising the PSPI function. The optimal profiles in figure 5.1

reveal that the monomer feed of Styrene, MMA and MA are at a fixed ratio, since a

constraint was put for their flow rates to be at 5/3/2 ratio respectively. The surfactant

feed started at 0.0014 g/s and decreased to reach about 0.000086g/s at 5513s. This

low feed rate is for stabilising the system since the adsorption of surfactant molecules

increases as the surface area of polymer particle increases. After that the surfactant

flow rate increases significantly to 0.005g/s in order to generate micelles which cause

secondary nucleation to occur through micellar nucleation and hence a new small

stream of polymer particles is generated. This is accompanied with a sharp decrease

in the average size of particles as shown in figure 5.1d.

The amount of oligomeric radicals in the reactor was enough to generate the

secondary nucleation by undertaking entry to the micelles and for this reasons the

flow rate of initiator was zero for the whole reaction. The secondary nucleation

occurred after 177s of increasing the surfactant feed rate which is the time of

generating micelles. The concentration of surfactants was lower than critical micelle

concentration (cmc) and so the surfactant flow rate for a period of 177s was needed

to make the surfactant concentration equal to cmc and then the feed rate generated

micelles which led to micellar nucleation. Thus, bimodal particle size distribution is

obtained as seen in figure 5.1c.


5. 24

0.000

0.002

0.004

0.006

0 2000 4000 6000 8000 10000Time (s)

Fs

(g/s

) &

Fi

(g/s

)

Fs Fi

0.00

0.01

0.02

0.03

0 2000 4000 6000 8000 10000Time (s)

Flo

wra

te (

g/s

)

335

340

345

350

T (

K)

FC FA FB T

(a) (b)

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120 140 160Diameter (nm))

Vo

lum

e F

rac

tio

n

0.0

20.0

40.0

60.0

80.0

100.0

0 2000 4000 6000 8000 10000Time (s)

Dia

me

ter

(nm

)

(c) (d)

0.0

0.2

0.4

0.6

0.8

1.0

0 2000 4000 6000 8000 10000Time (s)

Co

nv

ers

ion

(e)

Figure 5.1 Validation of optimisation for maximising PSPI: (a) & (b) Variable

optimal profiles, (c) PSD, (d) Particle Diameter, (e) Conversion (● experiment; ▬

simulation).

The polymerisation conversion, depicted in figure 5.1e was also used for the

validation of the optimisation process, since a constraint was put to make the

conversion above 80% and also to ensure that the model is able to predict most of

the key polymer properties at different conditions, especially under optimised

conditions. The first section of figure 5.1e [first 1500 s] shows batch reaction with

seeding. The overall conversion dropped at the instant of feeding the monomer. This

is because most of the extant monomers had converted to polymers, and at the

instant of monomer addition, there is more monomer in the reactor than it previously

contained. Thus the overall conversion drops at that instant. Subsequently, the


5. 25

monomers start to propagate and convert to polymers. The decrease in the overall

conversion varies with the feed rate, and is in proportion to the amount added. Thus,

we record a larger drop for a higher flow rate, and a smaller drop for a lower flow

rate.

0.0E+00

2.0E-07

4.0E-07

6.0E-07

8.0E-07

1.0E-06

1.2E-06

1.4E-06

0 4000 8000 12000Time (s)

Fs (

g/s

)

0

0.001

0.002

0.003

0.004

0.005

0.006

Fi

(g/s

)

Fs Fi

0.000

0.008

0.016

0.024

0.032

0 4000 8000 12000Time (s)

Flo

wra

te (

g/s

)

346

348

350

352

T (

K)

FA FB FC T

(a) (b)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140 160Diameter (nm)

Vo

lum

e F

rac

tio

n

0

20

40

60

80

100

120

0 2000 4000 6000 8000 10000 12000Time (s)

Dia

me

ter

(nm

)

(c) (d)

0

0.2

0.4

0.6

0.8

1

0 2000 4000 6000 8000 10000 12000Time (s)

Co

nv

ers

ion

(e)

Figure 5.2 Results for minimising PSPI, (a) & (b) Variable optimal profiles, (c) PSD,

(d) Particle Diameter, (e) Conversion (● experiment; ▬ simulation).

In figure 5.2, narrow particle size distribution was optimised by minimizing the

PSPI. As expected, the feed flow rates of initiator and surfactants were very low in

order to prevent secondary nucleation. The total monomer feed rate was at 0.0001g/s,

which is the lower limit of its constraint, for the first 3000s of the reaction. This

explain the sharp decrease in conversion at 3000s (figure 5.2e) and not at 1500s as


5. 26

shown in figure 5.2e. The growth rate in this interval is very low which is observed

from the slow change in the average size of the particles (figure 5.2d). After that, the

flow rate increased significantly to increase the growth rate of particles and hence

homogenizing the particles size distribution as shown in figure 5.2c. This increase in

the growth rate led to increase the average size of the particles.

5.5.2 Optimising Molecular Weight

In this section, the complete polymer chain microstructure distribution (MWD) and

its molecular weight averages were optimised. The MWD is mainly affected by the

polymerisation temperature and monomer feed rate. The higher the monomer feed

rates, the higher the amount of monomers in the reactor which leads to a higher

propagation rate and thus a higher molecular weight is produced. As the temperature

of polymerisation increases, the coefficients of transfer and termination increase

which leads to a decrease in the degree of polymerisation and thus produces broad

MWD with a low average molecular weight.

The feed rate of initiator and surfactant has a slight effect on the MWD in which

they are indirect proportional to the molecular weights. The reason is that adding

surfactant would cause micellar nucleation, resulting in decreased molecular weight.

Initiator allows for more oligomeric radicals to be formed which increases their rate

of entry and thus increases the termination coefficient with low molecular weights

produced. The agreement between the experimental results and simulations were

good, as can be observed in figures 5.3, 5.4 & 5.5, on optimizing the MWD and Mn.

The conversion and average radius were validated to these processes to prove the

validity of the model to act as a soft sensor under different conditions.

Three optimisation operations were carried out in this section. The first optimisation

was carried out to maximise the breadth of MWD by maximising the MWPI function

(figure 5.3). The second optimisation was for minimising the breadth of MWD by

minimising the MWPI function (figure 5.4). The third optimisation was for

minimising Mn (figure 5.5).


5. 27

In the first optimisation (figure 5.3), the monomers feed rates and reaction

temperature were the only manipulated variables used to produce broad MWD with

high Mn. In figure 5.3b, the temperature of the reaction increases 3oC at the middle of

the reaction intervals in order to increase the conversion since a conversion

constraint (>80%) is set up. But the temperature decreased at the end of the reaction

interval to increase MWPI. Figures 5.3c shows that MWPI decreases sharply during

the first 1500sec and then increases sharply. This is because the first 1500s of the

course of the reaction is a batch process to generate polymer particles (seeding part)

and so the amount of monomers decreases with reaction time and consequently the

rate of polymerisation decreases which led to a sharp decrease in MWPI (figure

5.3b). After that the monomer started to feed and thus MWPI increased sharply

depending on the monomer flow rate.

In the second optimisation (figure 5.4) where MWPI was minimised, monomers feed

rates, initiator feed rate and temperature were the manipulating variables. Figure 5.4a

shows that the initiator has a significant effect in producing polymers with low

molecular weights. This explains the reason for setting initiator feed rate equal to

zero in the first optimisation where MWPI was maximised. This is due to the fact

that as the amount of initiator increases, the rate of termination increases and so low

molecular weights are produced. Thus, the probability of chain growth decreases

resulting in low MWPI. Figure 5.4b shows that the feed rates of styrene and MMA

were zero while the MA monomer feed rate was too high (0.04-0.06g/s).

Styrene and MMA have high polymerisation rate coefficients with low transfer rate

which lead to high molecular weights with high MWPI. This is why the MWPI of all

the polymers produced from styrene and MMA is higher than 2.4. On the other hand,

MA has a high transfer rate coefficient which produces highly branched polymers

with low MWPI and can be reached to 1.01 (Banerjee and Konar, 1986). It can be

also seen from figure 5.4b that the temperature of the operation was high, since

temperature increases the transfer rate coefficients more than polymerisation rate

coefficients. Figure 5.4c shows a simultaneous sharp decrease and then increase to

the MWPI within the first 1800s of the operation. This can be investigated in the

same manner as that for figure 5.4c.


5. 28

0

0.01

0.02

0.03

0.04

0.05

0 1000 2000 3000 4000 5000

Time (s)

Flo

wra

te (

g/s

)

FA FB FC

340

341

342

343

344

345

0 1000 2000 3000 4000 5000

Time (s)

T (

K)

(a) (b)

0

1

2

3

4

0 1000 2000 3000 4000 5000

Time (s)

MW

PI

0.0

0.3

0.6

0.9

1.2

4 6 8log(M)

dw

/dlo

g(M

)

●

●

●

●

●

●

●

●

●

●

(c) (d)

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000 5000

Time (s)

Co

nv

ers

ion

0

20

40

60

80

100

0 1000 2000 3000 4000 5000

Time (s)

Dia

me

ter

(nm

)

(e) (f)

Figure 5.3 Validation of optimisation for maximising MWPI, (a) & (b) Variable

optimal profiles, (c) MWPI, (d) MWD, (e) Conversion, (f) Particle Diameter (●

experiment; ▬ simulation).


5. 29

0.E+00

5.E-05

1.E-04

0 1000 2000 3000 4000 5000Time (s)

Fs

(g

/s)

0.000

0.002

0.004

0.006

0.008

0.010

Fi

(g/s

)

Fs Fi

0.00

0.02

0.04

0.06

0.08

0 1000 2000 3000 4000 5000

Time (s)

Flo

wra

te (

g/s

)

346

348

350

352

T (

K)

FA FB FC T

(a) (b)

0.0

2.0

4.0

0 1000 2000 3000 4000Time (s)

MW

PI (n

m)

0.0

0.3

0.6

0.9

1.2

3.00 4.00 5.00 6.00 7.00

log(M)

dw

/dlo

g(M

)

●

●●

●

●

●

●

●

●

●

●

●

(c) (d)

0.0

0.2

0.4

0.6

0.8

1.0

0 1000 2000 3000 4000Time (s)

Co

nvers

ion

0.0

20.0

40.0

60.0

80.0

0 1000 2000 3000 4000Time (s)

Dia

mete

r (n

m)

(e) (f)

Figure 5.4 Optimisation results for minimizing MWPI, (a) & (b) Variable optimal

profiles, (c) MWPI, (d) MWD, (e) Conversion (f) Particle Diameter (● experiment;

▬ simulation).

In the third optimisation (figure 5.5) where only Mn was minimised with a

conversion constraint (>80%), the optimal profiles, as shown in figures 5.5a and

5.5b, show approximately the same feed rates as that in the second optimisation. The

slight difference is because MWPI was not set as an objective. So the initiator and

MA feed rates were slightly lower than those in the second optimisation, while the

feed rates of styrene and MMA were not zero but very low. Therefore, it can be

concluded that monomer feed rates, initiator feed rate and reaction temperature are

the main manipulation variables for controlling the MWD of the terpolymer. Figure

5.5c shows two small peaks at 1500s and 2500s in the MWPI. These two peaks are


5. 30

due to the stepwise change in the reactor temperature and monomer flow rate. This

also led to the sharp change in the conversion as shown in figure 5.5e.

The MWPI in Figure 5.4 was slightly greater than the MWPI in figure 5.5. It might

be expected that the MWPI in figure 5.4 should be less than that in figure 5.5,

because the objective for figure 5.4 was to minimise MWPI while that for figure 5.5

was to minimise Mn. This slight difference is due to the type of constraints in each

optimisation process. In figure 5.4, the conversion constraint was greater than 85%

while for figure 5.5, it was set greater than 80%. Thus, the optimisation processes of

both figures followed different paths to reach their objectives.

0

0.02

0.04

0.06

0.08

0.1

0 1000 2000 3000 4000

Time (s)

Flo

wra

te (

g/s

)

335

340

345

350

355

T (

K)

Fi FA FB FC T

0

100000

200000

300000

400000

500000

0 1000 2000 3000 4000

Time (s)

Mn

(a) (b)

0.0

1.0

2.0

3.0

4.0

0 1000 2000 3000 4000

Time (s)

MW

PI

0.0

0.3

0.6

0.9

1.2

4 5 6 7 8

log(M)

Vo

lum

e F

rac

tio

n

●

●

●

●

●●●

●

●

●●●

(c) (d)

0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000 4000

Time (s)

Co

nv

ers

ion

0

10

20

30

40

50

60

70

80

0 1000 2000 3000 4000Time (s)

Dia

me

ter

(nm

)

(e) (f)

Figure 5.5 Results for minimizing Mn, (a) optimal profiles, (b) Mn, (c) MWPI, (d)

MWD, (e) Conversion, (f) Particle Diameter (● experiment; ▬ simulation).


5. 31

5.5.3 Optimising Composition

In this case, the terpolymer composition is optimised to produce constant

composition at ratios of 5/3/2 for Sty/MMA/MA respectively with a maximum

conversion (>80%). The composition of the terpolymer was determined using the

Proton Nuclear Magnetic Resonance (1H-NMR) spectroscopy as mentioned in

chapter 4.

Figure 5.6 presents the profiles of the monomer flow rates and the operation

temperature trajectories during the experiment. It also shows the evolution of

cumulative terpolymer composition, conversion and average radius. The terpolymer

composition was controlled by keeping the different monomers concentration ratios

at the required values. In this system, Styrene is the most reactive monomer and MA

is the lowest one. Therefore, the flow feed rate of styrene was the highest flow rate

while MA feed rate was the lowest one as shown in figure 5.6a.

The first 1500s interval of the operation was a batch process for seeding and for this

the terpolymer composition was not controlled in this interval. But the initial amount

of the monomers added to the operation was controlled to get the same terpolymer

composition at the end of the batch interval of the operation. Figure 5.6c shows a

simultaneous sharp decrease and then increase to the conversion within the first

1800s of the operation. This can be investigated in the same manner as that for figure

5.1e.


5. 32

0

0.01

0.02

0.03

0.04

0.05

0 2000 4000 6000 8000Time(s)

Flo

wra

te (

g/s

)340

342

344

346

348

350

352

T(K

)

FA FB FC T

0

0.2

0.4

0.6

0.8

0 2000 4000 6000 8000

Time(s)

Mo

lar

Co

mp

os

itio

n

Sty

MA

MMA

(a) (b)

0

0.2

0.4

0.6

0.8

1

0 2000 4000 6000 8000Time (s)

Co

nv

ers

ion

0

20

40

60

80

100

120

0 2000 4000 6000 8000Time (s)

Dia

me

ter

(c) (d)

Figure 5.6 Results for optimising composition, (a) Variable optimal Profiles, (b)

Mass Composition, (c) Conversion, (d) Particle Diameter (● experiment; ▬

simulation).

5.6 CONCLUSION

In this chapter, a multiobjective dynamic optimisation method for the calculation of

the Optimal control policies for polymer composition, PSD, MWD and Mn in a

semibatch emulsion terpolymerisation is presented. The optimisation procedure is

applied to styrene, Methyl methacrylate and methyl acrylate emulsion

terpolymerisation. The PSPI, MWPI, Mn and composition ratios were used as

objective functions. Six variables were used as manipulated variables, styrene

monomer feed rate, MMA monomer feed rate, MA monomer feed rate surfactant

feed rate, initiator feed rate, and the temperature of the reaction. The mathematical

dynamic model which has been presented in chapter three is used to represent

emulsion polymerisation. The optimisation problem was formulated as a multi-

objective optimisation by reformulating the multi-objective problem as a single-

objective case by placing some objectives in the form of constraints. The gOPT tool


5. 33

of gPROMS was used to perform the dynamic optimisation calculations in which the

control vector parameterisation approach was adopted. The optimisation is carried

out using the SRQPD code which implements a reduced quadratic programming

algorithm. Equality and inequality safety constraints were imposed to the system

operation.

Results show that the optimisation procedure was able to minimise the reaction time

and, simultaneously, obtain a polymer with a desired quality (composition, size

distribution or molecular weight) while taking the safety aspect into account. The

obtained control policies were validated on an experimental stirred tank

polymerisation system and an excellent agreement between them was found. On

maximising PSPI, a bimodal distribution was produced being broader. Temperature

showed to have a strong effect on the molecular weight due to the effect of the

transfer rate coefficient. On maximising the molecular weight, the initiator

concentration was zero at all times showing that any initiator addition could cause

more radicals to be produced, and therefore more transfer. Surfactant feed rate was

also zero at all times when temperature was used as a manipulated variable

concluding that adding surfactant may produce more micelles forming new particles,

and the monomers would then tend to get to the new particles rather than propagating

to produce larger molecular weights.

The results showed that the intelligent control of the polymer composition for several

monomers with different reactivity ratios needs continuous control action and for

this, its time horizon partitioned in twenty short time stages. This work is a first step

in the advanced control hierarchy to the MWD, PSD and terpolymer composition of

the terpolymer product through the manipulation of the reaction temperature and the

monomers, surfactant and initiator feed rates.


5. 34

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Modelling, validation and effect of operating conditions. Chemical

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chapter 5 optimization examiner correction

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