Modelling and Estimation of Polycondensation Processes

Pratima Ramkhelawan

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Graduate Department of Chernical Engineering & Applied Chemistry

O Copyright by Pratima Ramkhelawan 2000

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Modelling and Estimation of Polycondensation Processes, Master of Applied Science,

2000, Pratima Ramkhelawan, Department of Chernicd Engineering and Applied

Chemistry, University of Toronto

Abstract

A polyester made from adipic acid, isophthalic acid and 2-rnethyl-1.3-propanediol was

studied in this project. There were two main objectives: the first was to develop a first

principles process model in a way that it could provide useful data for the purposes of

modelling, state estimation and process control: the second objective was CO develop a

state estimator that can be used on-Iine, to predict and update the process model as the

reaction proceeds.

A first principles model was developed and kinetic parameters were estimated using the

expcrimental data. The experimental data was also used to empirically correlate variables

such as number and weight average moIecular weight and viscosity to the outputs of the

mode 1.

A state estimator (extended Kalman filter, EKF) was developed for the system studied.

The EKF uses the process model and available online measurements to provide online

optimal estimates of the states and outputs of the model.

Acknowledgements

1 would like to sincerely thank Dr. Alex Penlidis for al1 his help and guidance throughout

this thesis. 1 would also like to thank Dr. Will Cluett for his advice and support.

1 am grateful to Adrian Thompson for al1 his help whenever 1 needed it.

I would also like to thank the following people at [CI for their help throughout this project:

Guy Stella

Charlie DeBrosse

Sam Rostami

1 would also like to acknowledge ICI Paints Canada, OGS and NSERC for their financial

support.

.. A bstract ~ ~ ~ ~ ~ ~ ~ . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ e g ~ ~ ~ ~ ~ ~ g ~ g m ~ ~ ~ m ~ ~ ~ e ~ ~ ~ ~ ~ ~ o ~ g ~ g g ~ g ~ g ~ ~ ~ ~ ~ I e ~ ~ ~ e e g ~ g g ~ ~ g g ~ ~ ~ ~ ~ ~ g ~ ~ e ~ ~ o ~ ~ e ~ ~ ~ g e ~ ~ ~ œ ~ ~ ~ ~ ~ œ o ~ ~ œ ~ œ ~ ~ ~ ~ o ~ ~ ~ ~ o ~ ~ o ~ II

... Acknowledgements ........................................................................................... .Oo...........me...lll List of Figures ...................................................................................................................... vi

... List of Tables .................................................................................................................. v i i i

1 Introduction .................................................................................................................... 1

1 - 1 Motivation ........................................................................................................ 1

7 ......................................................................................................... 1 -2 Objectives - ...................................................................................... 1 -3 Description of System 2

.......................................................... 1.3.1 Polycondensation Polymerization 2

...................................................................................... 1 -3 -2 Mode l Structure - 3

.................................................................................. 1 .3.3 System of Interest 4

.......................................................................................... 1 -4 Overview o f Thesis 5

................................. 2 Literature Review ...... .................................................................... 6

............................................................................................ 2.1 System of Interest 6

2.2 Modelling ......................................................................................................... 7

.................................................................................. 2.2.1 General Modelling 7

2.2.2 Nylon Modelling ..................................................................................... 9

2.2.3 PET Modelling ..................................................................................... 12

.................................................................... 2.2.4 Solid State Polymerization 14

.................................................................................. 2.3 Estimation and Control 16

3 Mode1 Development ...................................................................................................... 18

.................................................................................................... 3.1 Bac kground 18

3.2 First Principles Approach ............................................................................. 18

................................................................................. 3 .2.1 Mode1 Description 19

......................................................................... 3.2.2 A Benchmark Exarnple 24

.............................................. 3 .2.3 MPD. Adipic Acid. Isophthalic Acid 2 9

.......................................................................................................... 3 -3 Dry Add -31

1 Experimental Results ................................................................................................ 34

........................................................................................ 4.1 Experimental Senip 34

..................................................... 4.2 Experimental Procedure ............... ...... -35

4.3 Experimental Data ......................................................................................... 38

Pacameter Estimation .................................................................................................. 47

................................................................ 5.1 Estimation Procedure and Results 4 7

........................................................................................ 5.2 Confidence Bounds 55

S tate Estimation Results .............................................................................................. 61

..................................................... ............................................ 6.1 Introduction ... 61

6.2 Kalman Filtering ............................... .. ........................................................ 62

............................................................................ 6.3 Extended Kalman Filtering 64

..................................................................................... 6.4 Simulation Examples 68

............................................................................................. 6.4.1 Effect of R 69

6.4.2 Effect of Q ........................................................................................... 74

6.4.3 Effect of G ............................................................................................. 74

....................................................................................................... Application Resulb 78

7.1 Tuning the Estimator ..................................................................................... 78

7.1 - 1 Tracking Experimental Data ................................................................. 79

........................ 72Combining Estimation Results with Empirical Correlations 87

.............................................................................. 7.3 Process-Mode1 Mismatch 89

............................................................................. 7.3.1 Estimating the States 89

......................................................... 7.3 -2 Confidence in Mode1 Parameters 92

.................... Conclusions and Recommendations ................................................. 95

8.1 Conclusions .................................................................................................... 95

.......................................................................................... 8.2 Recommendations 96

Bibliography ........................................................................................................................ 98 Appendix A ....................................................................................................................... 105

List of Figures

................................................ Figure 3.1 Typical weight fraction distribution plot 20

Figure 3 -2 DPn vs . Time at 1 66OC: Experimental (e). Mode1 ( - ) .................... ... . 27

Figure 3.3 Mode1 predictions of DPn at 166°C ( -) and 702OC ( .. ); ................. 28

Figure 3.4 Conversion vs . Time at 1 6 6 O ~ ; Experimental (m) . Mode1 ( -) ............ 29

Figure 4.1 Flow Diagram of Industrial Process .....................................~................ 3 5

Figure 4.2 Viscosity vs Acid Number; Exp#l(t). Exp#2 (=). Exp#3 (A). Exp#4 ........................................................................................... (x). Exp#5 (O) -37

Figure 4.3 Viscosity (centipoise) vs Acid Number; Al1 Data (t) with exponential trendline ................................................................................................... 40

Figure 4.4 Batch Temperature ( O C ) vs . Time; Exp#l ( 6 ) . Exp#2 (m) . E x P 3 ( A ) . .............................................................................. Exp#4 (x). Exp#5 (O) -41

Figure 4.5 Acid Number vs Time; Exp#l (e). Exp#2 (i). Exp#3 (A). Exp#4 (x) . .................................................................................................. (0) -42

Figure 4.6 Conversion vs Time; Exp#I (+). Exp#2 (m). Exp#3 (A). Ex* (x). Exp#5 (4 .................................................................................................. 43

Figure 4.7 Mn . MW vs Acid Number; Mn (O) with trendline . MW (=) with trendline ................................................................................................................. 44

Figure 4.8 Mn. MW vs Viscosity; Mn (.). MW (t) ................................................... 45

Figure 4.9 Mn, MW vs Time; Mn (a). MW (.) .......................................................... 46

Figure 5.1 Conversion vs Time; Exp#l (*) . Exp#2 (I) . Exp#3 (A) . Exp#4 (.x). Exp#5 (m) .................................................................................................. 48

Figure 5.2 Conversion vs Time: Mode1 ( - ) . Experimental (O) ............................. 53

Figure 5.3 Conversion vs Time; Mode1 ( -). Experimental (e) ............................. 54

Figure 5 -4 Joint Confidence Regions for the Parame ter Estimates ........................ - 5 9

................................................................... Figure 6.1 Kalman filter block diagram 62

Figure 6.2 Simulated Experimental Data (a); Model alone (-); Filter ( .. ); ....... -71

......................................... Figure 6.3 Mode1 states; Mode1 alone (-); Filter ( .. ); 71

Figure 6.4 [RI increased; Simulated Experimental Data (.); Model alone (-); Filter ( .. ); ......................................................... -72

Figure 6.5 Mode1 States as [RI is increased; Mode1 alone (-); Filter ( .. ); .......... 72

Figure 6.6 ER] decreased; Simulated Experimental Data (m); Model alone (-); .............................................................................................. .. Filter ( ); 73

Figure 6.7 Mode1 states as [RI is decreased; Mode1 alone (-); Filter ( .. ); ......... 73

Figure 6.8 [QJ increased; Simulated Experimental Data (a); Model alone ( ) ; .. .....-..............*......................................................................... Filter ( ); 75

Figure 6.9 Mode1 states as [QI is increased; Mode1 alone (-); Filter ( .. ); ......... 75

Figure 6.10 [QI decreased; Simulated Experimental Data (.); Model done(-); .. ........--.-................................................................................... Filter( ); 76

Figure 6.1 1 Mode1 states as [QI is decreased; Mode1 alone (-): Filter ( .. ); ....... 76

Figure 6 . I 2 Effect of changing [G]; Simulated Experimental Data (m): Mode 1 alone (-); Filter with G = 1 (O); Filter with G= GI ( O ) ...................... 77

Figure 7.1 Conversion vs Time; Exp#I (.), Exp#2 (.), Exp#3 (A) . Exp#4 (x). ExpM (4 .................................................................................................. 80

Figure 7 -2 Experimen ta1 Data (m); Filter ( .. ) ................................................... 81

Figure 7.3 Experiment # 1 ; Experimental Data (a); Filter (-) ................................. 82

Figure 7.4 Experiment # 1 ; Model alone (-); Filter ( - . ) ...................................... 82

Figure 7.5 Experirnent #2; Experimental Data (a); Filter ( - ); ............................. 83

Figure 7.6 Experiment #2; Model alone (-); Filter ( .. ) ... ................................... 83

Figure 7.7 Experiment #3; Experimental Data (m); Filter (-); ............................... 84

Figure 7.8 Experiment #3; Mode1 alone (-); FiIter ( .. ) ...................................... 54

Figure 7.9 Experiment #4; Experimental Data (m); -Filter (-); ................... .... ........ 85

Figure 7-10 Experiment #4; Model alone (-); Filter ( .. ) .................................... 85

Figure 7.1 1 Experiment #5; Experimental Data (m); Filter (-); ............................. 86

.. Figure 7.12 Experiment #5; Mode1 alone (-); Filter ( ) .................................... 86

Figure 7.13 Mode1 (a); Correlation ( .. ); ................... ... .................................. 88

Figure 7.14 Simulated Experimental Data (m); Filter ( - . ) ..................................... 90

Figure 7.15 Tme Process ( - ); Nominal Mode1 ( - - ); Filter ( - - ); ............... 91

Figure 7-16 Simulated Experimental Data (a); Filter ( - . ) ..................................... 93

Figure 7.1 7 True Process ( - ); Nominal Model ( - - ); Filter ( - - ); ............... 93

Figure 7.1 8 Simulated Experimental Data (m); Filter ( - - ) .................................... -94

Figure 7.19 True Process ( - ); Nominal Mode1 ( - - ); Filter ( - - ); ............... 94

vii

.......................... ................ Table 1.1 Ingredients of polyester of studied .. A

Table 3.1 Polyrner Chain Definition ........................................................................ 21

Table 3.2 Copolymer Moment Definition ................................................................ 21

Table 3.3 Vector Operations Defined for Generalized Modeling ........................... 21

Table 3.4 Polycondensation Reactions ..................................................................... 21

Table 4.1 Size of experiments .................................................................................. 38

Tabte 4.2 Typical Experimental Data - Experiment # I Data Set .......................... .. 39

.................................................................... Table 5.1 Optimal parameter estimates 51

.......... ............................................................. Table 7.1 EKF Tuning Parameters ,., 81

Table 8.1 Typical Experimental Data - Experiment #2 Data Set .......................... 105

Table 8.2 Typical Experimental Data - Experiment #3 Data Set .......................... 106

Table 8.3 Typical Experimental Data - Experiment #4 Data Set .......................... 107

Table 8.3 Typical Experimental Data - Experiment #5 Data Set .......................... 108

viii

1 Introduction

Motivation

A common problem that plagues polymerization processes is the lack of on-line sensors. It

is desirable to track variables like molecular weights and conversion as a polymerization

proceeds. Measurernents like temperature and pressure are readily available; however,

molecular weights, conversion, as well as rnany other quality variables. are typically not

available. Polymer sarnples are usually taken and analyzed offiine: this may be done

during the course of the reaction or after the fact. In either case, large time delays are

associated with the offline analysis. As a result, operators generdly run polymerizations by

trying to achieve a preset temperature profile that has historïcally proven successful.

The process models that exist in the literature are typically a set of non-linear coupled

equations. The kinetic parameters associated with these models are not well known and

difficult to estirnate. The absence of sensors and inability to estimate such model

parameters make the design and control of polymer processes a challenging task.

A process model that will reasonably predict the behavior of the process under changing

conditions and in the presence of disturbances can prove to be a very useful tool. A model

can provide estimates for the key variables needed to assess the course of the reaction. In

addition to identifying kinetic model parameters. the variables or States in the model need

to be correhted with desired end use properties of the polymer in order to be useful.

1.2 Objectives

This thesis attempts to address some of the shortcornings outiined in the motivation. The

primary objective was to develop a process model for a polycondensation system using a

first principles approach. The goal was to construct a process model in such a way that it

could provide usefid data for the purposes of modelling, state estimation and process

control.

A secondary objective was to address the issue of the lack of online sensors. The objective

here was to use the identified process model to develop a state estimator. A state estimator

can be used on-line, to predict and update the model as the polymerization proceeds. It can

also be extended to incorporate offline mezsurements.

1.3 Description of System

1.3.1 Polycondensation Polymerization

Polymerization processes can be divided into two main types: chain-growth (or addition)

polymerization and step-growth (or polycondensation) polymerization. Polycondensation

processes differ from c h a h growth potymerization in that the growth of the polymer

molecule can occur from the reaction of any two molecules, not just by monomer addition

to a growing radical chain. The growth process is typically much slower than addition

polymerization. resulting in a slow increase in molecular weight. The polymerization

reaction frequentl y produces a low molecu t ar weight by-product (usuall y water). Common

examples of polycondensation products include nylon 6, nylon 6.6 and PET (polyethylene

terepthalate).

A polycondensation reaction c m be represented schematically as:

P m + P , < k P >P,,+,+W

Equation 1 . 1

rvhere Pi represents the growing pol ymer chain of length i

W represents the byproduct

kp represents the rate constant for the reaction

Polycondensation processes are frequently equilibrium limited and thus necessitate the

removal of the condensation byproduct in order to achieve desired conversions. As the

reaction proceeds and viscosity increases, this goal becomes increasingly challenging. The

lack of on-line measurements makes it even more difficult to ascertain what is happening in

the process. These factors make polycondensation processes very challenging to design

and control. The need for a good process mode1 that will reasonably predict the behavior

of the process under changing conditions is of paramount importance.

These processes are highly exothermic and therefore have a high rate of heat release

throughout the reaction. The viscosity of the polymer can typically increase by orders of

magnitude as the reaction proceeds. In the advanced stages of polymerization, these factors

contribute to many problems including heat transfer, diffusion limitations and byproduct

removal. The combination of these effects makes polycondensation processes very non-

1 i n e ~ .

1.3.2 Male l Structure

Polymerization systems are often modelled by sets of non-linear differential equations.

The equations can be represented in a state space form as:

"Y(!) = . g ( X . ~ l )

Equation 1 .S

rulzere .r( t ) is a vector of n States

u( t ) is a vector of r manipulated inputs

y( t ) is a vector of m outputs

This type of mode1 structure is extremely useful as it lends itself to a muttivariable control

strategy.

1.3.3 System of Interest

The polyester studied in this project is made from adipic acid, isophthalic acid m d 2-

methyl- l,3-propanediol. The chernical formulas of each compound are shown in Table 1.1.

Very little is found in the open literature about ihis particular recipe. Two articles that

examine some of the physical properties of this system (SulIivan and Cooper, 1995;

Duncan et al.. 1990) are discussed in Section 2.1. The polyester resin is typically produced

in a batchwise manner and used in applications such as pop bottle caps.

/ Material 1 Molar M a s (g/mol) 1 Chemical Formula

1 Adipic Acid I 146.14 1 HOOC(CHr)dCOOH

lsophthalic Acid

Table 1.1 Ingredients of polyester of studied

166.13 C6b(COOH)2

1.4 Overview of Thesis

Chapter 2 presents a literature review of the general field of polycondensation relating to

modelling, estimation and control. Chapter 3 provides background on the model

development for the system of interest. It discusses first principles vs. black box modelling

and provides details of the modelling approach taken in this thesis. It also provides insight

into a Monte-Carlo simulation package, DryAdd. which was investigated. Chapter 4

provides a description of the experimental setup, experimental procedure and presents the

experimental results. Chapter 5 discusses the identification work that was performed to

estimate the kinetic parameters for the model devcloped in Chapter 3. Section 5.2 also

discusses confidence intervals associated with the parameter estimates. In Chapter 6 an

extended Kalman filter (state estimator) is developed for the system studied. Chapter 7

il lustrates some of the potential applications of the mode].

2 Literature Review

Numerous papers in the open literature relate to polymerization processes in general;

however, only a small subset of these publications focus on polycondensation or step-

growth type polymerization. The majority of the polymerization literature relates to

addition (or chah growth) polymerization. It is the intent of this chapter to review the

literature of the past decade or so that in some sense relates to polycondensation. Le..

presents a polycondensation mode1 or example. This review attempts to cover the general

field of polycondensation as it relates to process modelling and process control.

2.1 System of Interest

Two articles (Sullivan and Cooper, 1995; Duncan et al., 1990) specifically relate to this

project as they both discuss polyesters made from 2-methyl-1.3-propanediol (MPD), adipic

and isophthalic acid. In Duncan et al. (1990). MPD is substituted for neopentyl glycol

(NPG) and isophthatic acid partially replaces adipic acid in the manufacture of a polyester

resin used as a coating. NPG is one of the most common polyol monorners used for this

application but has ri slow reaction time. cornes in a solid form and tends to form a slightly

opaque coating. The advantages of MPD include a Iiquid form. significantly shorter

cooking times and improved flexibility when compared with the NPG coating. If aslightly

harder. less flexible film is desired, isophthalic acid c m partially replace adipic acid. The

substitution of MPD and isophthatic acid does not appear to affect the adhesive properties

of the film and no other adverse changes are observed. Sullivan and Cooper (1995)

compare polyester coating resins made with NPG and MPD. The focus of this paper is

polyester weatherability under conventional accelerated tests; it also examines polyester

decomposition mechanisms. The experiments show that the MDP formulation maintains its

gloss just as well as the NPG formukaion; the differences observed are marginal. They

d s o show that as the isophthalic acid content is increased the resin maintains its gloss for a

longer duration.

There were no models or kinetic parameters found in the literature for the system of

interest.

The need for a process model that will reasonably predict the behaviour of the process

under ctianging conditions is of paramount importance. Modelling of various

polycondensation systems have been widely studied in the literature. The models that exist

are typically a set of non-linear coupled differential and algebraic equations. The kinetic

parameters associated with these models, in many cases. are not well known and difficult to

estimate. There is also the added challenge of relating variables in the model to desired,

end-use properties of the polymer.

2.2.1 General Modelling

Jacobsen and Ray ( l992a) provide a generalized modelling technique for polycondensation

processes. They present a general kinetic scheme that c m be applied to many condensation

processes. The models presented are valid for both homopolymerization and

copolyrnerization, and what they coin Type 1, ii or III nonlinear kinetics (as detemined by

the type of monomer). The modelled rate equations are developed using the method of

moments. A set of general reaction mechanisms is presented which represents the standard

reactions. Also presented are mechanisms that would include other side reactions, like ring

opening reactions. For a particular polymer, a subset of the general set of equations

availabIe would be used. Severai examples of models are presented and compared with

literature data. These include models for PET, nylons, polyurethanes, epoxy resins as well

as other systems. The paper also discusses the modelling implications of batch vs. flow

reactors as well as some mass transfer considerations. This paper stands out from other

papers cited in that it presents a generdized approach that can be applied to a host of

systems. as opposed to examining a specific system(s).

Jacobsen and Ray (1992b) examine the design of polycondensation processes. Many

pclycondensation processes cm be divided into three stages: a pre-polyrnerization,

polymerization and finishing stage. The paper examines the problems that are inherent in

each stage and discusses tools that are useful for design in each stage. The use of the design

tools is illustrated with nylon and polyethylene terephthalate examples.

Ravindranath and Mashelkar (1986) examine the influence of reversible and interchange

reactions on the molecular weignt distribution of a condensation polymer produced in a

CSTR. The results show that in the presence of reversible and interchange reactions. the

polydisperity decreases, and at high conversions the polydisperity approaches that of a

batch reactor. The analysis illustrates the usefulness of a CSTR for preparing condensation

polymers having a low polydisperity index.

Diaranieh ( 199 1 ) looks at the viscosity buildup in large-scale reactors. Specifically. the

viscosity buildup of a sulfonated melamine-formaldehyde resin is studied. The effects of

pH, temperature and reaction time on viscosity buildup are examined. He also examines

scale-up. reactor geometry, agitator geometry and agitator tip speed on the viscosity

buildup. It is concluded that pH is the most important variabIe for controlling viscosity.

fol lowed by agitator geometry.

Park (1988) proposes a new mode1 for the chain length dependence of the reaction rate

constant on the rnolecular weight distribution and average molecular weights in linear

condensation polymerization.

A few textbooks have also been published that cover the area of polymer modelling and

polymer reaction engineering (Platzer, 1975; Dotson et al., 1996; Gupta and Kumar, 1987).

Gupta and Kumar's (1987) book is specific to step growth polymerization. The first half of

the book discusses principles of step-growth polymerization, including kinetic modelling,

equal reactivity assurnption, m a s transfer considerations, optimization and control. The

second half discusses in more detail exarnples of many industrially important step-prowth

polymers, like polyesters, PET, nylon and potyurethanes.

2.2.2 Nylon Modelling

There are an increasing number of papers on nylon 6 and nylon 6,6 because of its

commercial importance. In the production of nylon, some undesired cyclic side products

are formed. referred to as cyclic oligomers, which tend to create problems during polymer

processing. Therefore, a useful mode1 should try to incorporate the formation of the side

products. Unfortunately, none of the kinetic models in the literature can predict the total

concentration of the side products. Many try to predict their concentration indirectly. There

has a1so been a lot of optimization work done with these polymers in an attempt to

minimize undesirable side products and reaction time.

Steppan et al. ( 1987) use a simple activity-based solution model to develop therrnodynamic

correlations for the description of the rate and equilibrium behaviour of non-ideal

polymerization of nylon. Steppan et al. ( 199 1 ) develop a simplified degradation model for

nylon 6.6 polymerization. Thermal degradation c m have a major impact on the quality of

nylon 6.6 polymer. This model attempts to improve upon previous models by incorporating

side reactions that are significant at high temperatures for nylon 6.6.

Gupta and Tjahjadi (1987) simulate a commonly used energy efficient tubular nylon 6

reactor under steady state conditions. The effects of various parameters and operating

variables on molecuiar weight are studied, including feed compositim, temperature.

flowrate, heat transfer coefficients and reactor dimensions.

Wajge and Gupta (1994) and Sareen and Gupta (1995) have worked on multiobjective

dynamic optirnization for nylon 6 polymerizations. in scalar objective functions, weighting

factors are associated with each objective. Typically the objective function to be optimized

is the weighted average of severd individual objectives (e.g., minimize experimental time,

minimize formation of side products!. However, in many cases, optimal solutions may be

rnisleading. The choice of the weighting factors is not a trivial choice and could be an

arduous task if a large number of weighting factors is required. The objective function

begins to lose physical meaning and becomes very subjective.

Recent trends in 'vector optimization' are now being applied to polymer engineering. The

objective function is a vector of individual objective functions. This approach is called

multiobjective decision making or optimization and has it roots in management science.

Wajge and Gupta (1994) is the first study to be applied to an industrial reactor. This

technique has been used to optimize a copoiymerization reactor as well as other

engineering applications; this is the first time (Wajge and Gupta. 1994) multifunctional

optimization has been applied to a nylon 6 reactor. The most important drawback to

conventional optimization methods is the possibility of losing optimal solutions irrespective

of the weighting factors.

Mipp and Ray (1995) present a dynamic model for melt polycondensation reactions in

tubular reactors. The model ailows for axial dispersion and is capable of simulating

removal of low molecular weight by-products from the reaction mixture into an adjacent

vapor phase. General features of the model include a genenc kinetic framework that

permits modeling of numerous polymeric systems and the ability to readily configure

multistage systems. Simulation results are shown for two sample systems. the

polymerization of nylon 6 and nylon 6,6. The model is suitable for analyzing startup and

grade transition operations as well as for steady-state design. The model is also a useful

tool for control system design.

The modeling of nylon 6 produces a set of non-linear ODE's. There is a need to optimize

the model to determine the best operating conditions. Therefore. an efficient numerical

method for solving these models is necessary. Haswani et al. (1995) use a semi-analytical

solut ion to sol ve the non-linear ODE's descri bing the three main reactions (pol yaddi tion,

polycondensation and ring opening) involved in nylon 6 production. Their technique is

easily implemented and is said to be faster and more efficient than previous solutions.

Kumar and Gupta (1997) have attempted to improve upon the existing models by

incorporating into the kinetic scheme a rnodel for Iow order cyclic oligomers. The mode1

also incorporates mass baiance. moment and appropriate closure equations. They have used

regression techniques to fit data and estimate rate and equilibrium constants. They claim to

be able to predict the individual concentration of some of the low-order oligomers as wel

as the total cyclic oligomer concentration under various reaction conditions.

Aatmeeyata and Gupta ( 1998) model and optimize an industrial semi-batch nylon 6 reactor.

The model accounts for important phenomena including heat transfer. vaporization and

pol ymerization.

Giudici et al. ( 1997) describe transient experiments in an industrial twin-screw extruder

reactor used for the finishing stage of nylon 6,6 polymerization. Transient experiments

were desisned to obtain information from the extruder such as degree of filling and average

residence time. A mode1 was developed from the experimental results. Giudici et al. (1998)

develop a mathematical model for the finishing stage of nylon 6,6 polycondensation. The

model is compared with experimental data and shows good agreement with the data after

optimal fitting of the rate constants.

Mallon and Ray ( 1 998a) develop a model for nylon 6 and nylon 6,6. The kinetic parameters

for polycondensation can vary unexpectedly with reaction conditions. Many empirical

correlations have been developed to describe this observed behaviour: however, many of

these correlations are very compIex and do not hold when other conditions are varïed. In

this article a model is presented that will handle a wide range of water concentrations and

temperature variations as welI as interchange reactions and ring oligomer formation.

Kalfaî (1998) proposes a simplified mechanism for the depolymerization of nylon 6 and

nylon 6,6. This work has applications in treating post-consumer waste for the recycle of

post-consumer nylon mixtures back to monomers.

2.2.3 PET Modelling

Polyethylene terephthalate (PET) is the most commercidly produced polyester and as a

result many studies have been published on various modelling aspects of PET.

In the 1980's Kumar and Gupta performed extensive work on modelling of PET. Kumar et

al. ( 1 984) develop an optimal temperature profile for the transesterification step of PET to

maximize conversion and minimize side products. Kumar et al. (1984) model a thin film

PET reactor. The design variables included film thickness, surface area. concentration.

exposure time, residence time and temperature. Kumar e t al. (1984), and Kumar and Mista

( 1986) develop a general kinetic mcdel accounting for the intermolecular reaction in step-

growth polymerization of multifunctional monomers. Kumar and Sainath (1987) develop

an optimal temperature profile for the polycondensation stage of PET using a control vector

iteration technique. Et was also determined that the optimal reactor pressure would be the

lowest possible pressure for the entire reaction. which is important for flashing off volatile

compoi-ients.

Yamada and coworkers have also done a great deal of modelling work related to PET. In

Yamada et al. (1985), Part I of a mathematical model is developed for the continuous

esterification of terephthalic acid (TPA) and ethylene glycol (EG). The key parameter in

this model is the weight fraction of the liquid phase. This allows more precise estimation

of the concentration of each component and can be used to predict oligomer and distiliate

properties. In Part II (Yamada et al., 1986) of the model development, the reaction rate

constants of the proposed model are estimated by applying the Simplex method to

experimental plant data. Good agreement between experimental data and model predictions

is observed.

Another paper (Yamada and Imamura, 1988) studies the effect of a potassium titanium

oxyoxalate catalyst on the esterification process between TPA and EG; they conclude that

during continuous production of PET, potassium titanium oxyoxalate is an effective

catalyst for both the polycondensation and esterification reactions. This catalyst accelerates

the main reactions and has little effect on undesirable side reactions. Yamada and Imamura

( 1989) develop a simulation model for PET that can simultaneousiy express al1 oligomer

properties and concentrations of vapour phase compositions.

Yamada ( 1 992a,b) develops a new matheinaticül model for a continuous recycle

esterification process as well as a semi-continuous recycle process. These models are useful

for optimization and design applications. i.e., they can be used to predict many useful

concentrations (of carboxyl and hydroxyl end groups), number average properties.

concentrations of diethylene glycol, ethylene glycol and water in the vapour phase. melting

points and degree of esterification.

Yamada (1996) examines the effect of reaction variables on the continuous recycle

production of PET. The influence of reaction pressure, recycle ratio and residcnce time on

the oligorner characteristics is investigated. Some of the major conclusions were: ( 1 ) when

increasing the reaction pressure, the main reactions progress more easily but the side

rcactions are favoured over the main ones. and (2) the reaction pressure primarily affects

the formation of diethylene glycol forrned in the first reactor.

Martin and Choi (1991) examine the transient behaviour of a continuous melt

polycondensation reactor for the finishing stage of PET. The effects of reactor operating

variables such as polymerization pressure, temperature, residence time, feed prepolymer

molecular weight and ethylene glycol flow rate have been examined with simulations.

Laubriet et al. (199 1) propose a new approach to the modelling of a continuous PET

finishing polycondensation reactor. The model consists of a polymer melt phase and a

vapour phase and no distinction between the film phase and bulk melt phase is made. This

model can be used for any reactor geometry, which is important since reactor geometries

Vary and can be complex. A kinetic rnodel is incorporated into the steady state reactor

model for the prediction of various function end group concentrations and side product

formations.

Yoon and Park (1994) develop a mathematicai mode1 for a final stage PET reactor.

Detailed side reactions as well as diffusional removal of small molecules through the film

are considered. The mode1 explains the effects of film exposure time, pressure, initial

TFAEG temperature and catalysts on the degree of polymerization.

Cheong and Choi (1996) propose a dynamic multicompartment rnodel for a continous flow

rotating disk reactor for the finishing stage of PET. The effects of rector design and

operating variables on molecuiar weight and ethylene glycol removal are investigated via

model simulations. A detailed analysis of ethylene glycol removal rate from two phases is

presented.

2.2.4 Solid State Polymerization

Solid state polymerization (SSP) takes place in the solid phase and is a powerful way to

produce high molecular weight polymers. tt is difficult to produce high molecular weight

polymer in melt polycondensation because at higher temperatures. degradation reactions

increase faster than chain building reactions. One also encounters high viscosity that leads

to mixing, heat and m a s transfer limitations. In SSP the polymerization is carried out

below the melting point but above the glass transition temperature and this allows the end

groups to have sufficient mobility. At these lower temperatures, the rate of the degradation

reactions decreases dramatically and in many cases can be assumed negligible. However,

since the temperature is lower, the reaction proceeds at a slower rate and the total reaction

time is longer than melt polymerization. Many papers have been published in the area of

solid-state polymerization.

Ravindranath and Mashelkar (1990) analyze the previous SSP models and attempt to

overcome some of their limitations. These authors developed a model considering both

diffusion and generation of ethylene glycol during the course of the reaction. They also

analyzed the limiting cases of SSP in kinetically or diffusion controlled regimes and were

able to predict in a semi-quantitative way. the influence of particle shape, particle size and

temperatgre on the polycondensation process.

Devotta and Mashelkar (1993) develop a model for the SSP of PET that incorporates the

effect of crystallinity and the influence of the carrier gas on the degree of polymerization.

They suggest that carbon dioxide can be used as a carrier gas, instead of nitrogen: this will

result in higher molecular weight polymer and reduce the formation of side products.

Kulkarni and Gupta (1994) develop an improved mathematical model for SSP of nylon 6.

The paper studies the effects of changing operation conditions on SSP, e.g., size and degree

of crystailinity of polymer particles and water concentration.

Zhi-Lian et al. (1995) propose a numerical method to solve the SSP process equations, to

analyze the mechanism of SSP. Their results suggest that for the industrial SSP of PET. the

overall reaction rate in a single pellet is simulated by diffusion and reaction rate jointly

controIling the process.

Gao et al. ( 1997) propose a semi-analytical model to analyze the mechanism for SSP of

PET. The overall reaction rate can be simulated by a model that is jointly controlled by

diffusion and reaction rates. Wu et al. (1997) develop a comprehensive model of SSP by

anal yzing simi larities and di fferences between solid-state and melt polymerization. The

degradation and other side reactions neglected in the earlier models for SSP were included

in this model. MalIon and Ray (1998b) aIso develop a model to handle the reactions in

polymers undergoing polycondensation reactions in the solid state. The model inchdes

equations for previously neglected effects including variable crystallinity and gas phase

mass transfer effects.

2.3 Estimation and Control

There has been little work published on state estimation and control of industrial

polycondensation processes. This may be an indication that many of the industrial

polycondensation pot ymerization process are implemented without a formal feedback

control strategy; instead, histoh-d time, temperature and pressure profiles are followed.

Many survey and review papers have been published for the estimation and control of

polymer reactors. Some of these inciude Embirucu et al. (1996). Penlidis (1994), Chien

and Penlidis (1990), Elicabe and Meira (1988) and MacGregor et al. (1984). The most

recent work that reviews the literature of control of polymerization reactors is Tanaka

( 1997). Tanaka has reviewed papers related to optimization, modelling in polymer reactor

control, online monitoring, batch and semi-batch reactor control and continuous reactor

control.

There have been only a few estimation and control papers that examine a polycondensation

process:

Tobita and Ohtani (1992) examine controlling the molecular weight distribution for

irreversible step-growth polymerization. They investigate the effect of intermediate

monomer feed in batch and continuous plug flow reactors.

Robertson et al. (1995) examine the control of a nylon 6,6 batch reactor. Typical control

strategies employ the use of pre-detennined temperature and pressure trajectories for

con trol ler setpoints. These nominal trajectories corne from historical good mns in the

absence of disturbances. This paper examines the effect of disturbances on the desired

polymer properties. A P D cascaded controller is used to track the pre-determined

setpoints. They conclude that while the addition of feedback improves the ability to

cornpensate for some disturbances (like temperature and heat transfer coefficients), the

system is still extremely sensitive to changes in the feed conditions (like initial water

concentration).

Appelhaus and Engell (1996) design and implement a non-linear observer for the

polymerization of PET. The observer is able to determine important concentrations in the

polymer as well as the overall mass transfer coefficient. The knowledge of the latter

parameter (which was unknown before) offers new possibilities for improved process

control.

There has been some recent estimation work chat cites chain growth polymerization

examples, and hence can also be applied to polycondensation systems:

Kozub and ~MacGregor ( 1992) evaluate nonlinear state estimation for application to semi-

batch polymerization problems. The semi-batch emulsion copolymerization of

styrenehtadiene rubber is used as a case study.

Semino et al. (1996) examine the issue of parameter estimation using an extended Kalman

filter. A continuous rnethylmethacrylate system is used as a case study. The effect of errors

on both updated and non-updated parameters is analyzed.

Karjala et al. (1997) present results from an on-line. real time implernentation of an

extended Kalman filter for an industrial polyethylene reactor. The extended Kalman filter

provides real time estimates of the melt tlow index of the polymer. When imbedded within

a inultivariable control scheme, the state estimation has the potential to significantly

improve process performance by reducing the arnount of offgrade material produced during

grade changes.

Mutha et al. (1997) develop a model-based estimation and control strategy with

applications to polymer reactors. A methylmethacrylate system is used as a case study. The

paper presents an experimental application of both state estimation and nonlinear model-

based predictive control.

Mode1 Development

3.1 Background

One of the main objectives of this work was to develop a first-principles description of the

process. A first principles model uses knowledge based on the underlying physical and

chemical principles goveming the process to develop a mathematical representation.

Statistical, Monte-Carlo, modelling, is another technique that can be applied to

polymerization processes. In the Monte-Carlo approach to polymer modelling, one would

specify what elements of the system are allowed to react and the associated probability of a

successful reaction. This stochastic simulation is usuaily mn many times to determine that

the result is statistically significant.

Both of the Monte-Carlo and first principles approaches are powerful and important

techniques. Often Monte-Carlo techniques are employed because the system of interest is

difficult to model using a first principles approach. The focus of this work is to develop a

first principles model, however, a Monte-Carlo approach was also investigated. Our

i ndustrial partner has developed a Monte-Carto mode1 ling software package that was made

available for this work.

3.2 First Principles Approach

The modelling approach presented in Jacobsen and Ray ( l992a) was adopted for this work.

Jacobsen and Ray ( 1992a) provide a generdized modelling technique for polycondensation

processes. This paper (Jacobsen and Ray, 1992a) stands out from other papers cited in

Section 2.2 in that it presents a generalized approach that can be applied to a host of

systems. as opposed to examining a specific system(s). The models presented are valid for

al1 types (homopolymerization, copolymerization) of polycondensation processes. A set of

general reactions is presented which represents the standard reactions. Also presented are

other side reactions, Iike ring opening reactions. For a particular polymer, a subset of the

genenl set of equations available wouid be used. Several examples of rnodels are

presented and compared with experimental data published in the literature. The

completeness of the paper and the ease of use of the models made it a very attractive

choice.

3.2.1 Mode1 Description

The mode1 equations are developed using the method of moments. From a kinetic

mechanism, a moment expression c m be derived. A polymer sampIe is made up of a

distribution of many polymer molecules, P,. of varying lengths (n). This distribution

(molecular weight or chain length distribution) is often presented by plotting the mass or

mole fraction of each chain length versus the chah length, Figure 3.1.

A rnolecular weight distribution can be equivalently represented by it moments. An infinite

series of moments completety describes the molecular weight distribution. In many cases it

is not necessary to caiculate the entire infinite series. Typically, only the first few moments

rire calculrited because they are sufficient to describe the most important characteristics of

the distribution.

Sarnple Weight Fraction 0.05

I 1

i

1 0.00 O 10 20 30 40 50 60 70 80 i n (Chain Length)

Fisure 3.1 Typical weight fraction distribution plot

For a simple system, the i th moment is defined by:

Equation 3.1

where [P,, ] is the concentration of the polymer chains of length n.

Some of the lower order moments have a physical meaning. For example. the zeroeth

moment is equivaIent to the concentration of a11 polymer chains. The zeroeth and first

moments have a direct physical interpretation.

According to Jacobsen and Ray (1992a), polymer moments are defined in the following

tables. Table 3.1 shows the definition of a polymer chain; TabIe 3.2 shows how a polymer

moment is defined; Table 3.3 shows the notation for a vector and also the structure of the

unit vector used: Table 3.4 shows typical polycondensation type reactions that may be

observed in a given system.

Table 3.1 Polymer Chain Definition

- Pn.o.b - a polymer chain with

ni - monomer units of type i

a, - A end groups of type j

bk - B end groups of type k

where n,a,b are vectors - - - - -

Table 3.2 Copolymer Moment Definition

where the foilowing moments are calcuIaied:

di th moment for the i th monomer

e, th moment for the i th A end group

f; th moment for the i th B end group

where d,e f are vectors

Table 3.3 Vector Operations Defined for General ized Modeling

The i th element of vector x

d(i) = O for i# j

: where O(i) is a unit vector

Table 3.4 Polycondensation Reactions

Ring Addition Pn1.o.b + Rn tf Pm+n.u.b-~(i)+6(j)

Deactivation Pm,,,b + Pm.a-6(i).b

To develop the modelling equations the following steps are completed:

1 . Wt-ite down the kinetic mechanism.

In Jacobsen and Ray (1992a) a table of possible reactions is presented; a subset of

theses reactions is shown in Table 3.4. For a specific polymer systern, seitct al1

reactions that apply.

2. Write out a population/mass balance for each comportent involved in the mechanism.

The general rate expression for the population balance for fn,ri,t>, is given by:

Equation 3.2

3. From the population balance, the method of moments may be used to derive the general

moment rate equations:

Equation 3.3

ivhrre N,, number of A end group types in vector a

Nb number of B end group types in vector b

k.. Arrhenius rate constant

The result will be a set of nonlinear differential rate equations that c m be used to solve for

any specific moment. As the number of moments that are solved for increases, the amount

of information that can be obtained from the model aiso increases; however, it is important

to note that if higher moments are desired, the number of equations that must be solved also

increases exponentiall y. Typicall y, one will see models that calculate second moments as

the highest moment.

Only the rate expressions given for the kinetic mechanisms are required for modelling the

reactions in a batch reactor. If it is desired to model another type of reactor. the fiow effects

of the reactor must be included in the modelling.

The rate constants, k,i, are key parameters required to adequately model the system. These

parameters are, in general. not well known and not available in the literature. The rate

constant is an Arrhenius expression:

Equation 3.4

where A pre-exponential factor

E, activation energy

R universai gas constant

T temperature in degrees Kelvin

3.2.2 A Benchmark Example

A polyester system that is rnodelled in Jacobsen and Ray (199%) was chosen for the

purpose of benchmarking. The example selected is a polyester that is made with diethylene

glycol and adipic acid. The 2-component (adipic acid, ethylene glycol) system was chosen

as a benchmark pnmarily because it was hypothesized that it would behave in a similar

manner to the 3-component system (MPD. adipic acid, isophthalic acid) examined in this

work (Section 2.1). Several other researchers have also studied this adipic acid, ethylene

glycol system (Lin and Hsieh. 1977; Lin and Yu. 1978(a,b); Gupta and Kumar, 1987).

From Table 3.4 the relevant reaction mechanisin is the Polycondensation Type 1 reaction:

rt.1zere Monorner A = adipic acid - HOOC(CH~)JCOOH

Monurner B = dieth_t.lerre giycol - CdHlo03

Cl, = water

The moment rate equations. Equation 3.3. reduce to the following set of equations:

Equation 3.5

h = Uf' moment (total concentration of poiymer chains) [;].o.o

h = I"' monomer A niornent (concentration of A un ifs. including those in polyner) [;].o.o

h = I" monomer B moment (concentration of B mirs . including those in pofymer)

h[".l.o = f" A end moments ((active A end grmp

O

h = f" B end moments (active B end group

concentration)

concentration)

Conversion is a measure of how much a particular functional

analogous to a percentage yield. Conversion can be calculated on

on a particular functional group. The expression for conversion. p.

Equation 3.6

group has reacted: it is

an overall basis or based

is given by:

where A is the functional group of interest

[ A ] =concentr(ationofA

[ A,, ] = initial concentration of A

The number average chain length, DPn, is a measure of the ratio of the total number of

molecules at time t to the total number of molecules at time (r=O). The expression for DPn

is given by:

- DPn =- - - 1 --

Equation 3.7

Figure 3.2 shows experimental data and model predictions for the adipic acid, diethylene

glycol system. The system was simulated using Matlab and the model predictions (dashed

line in Figure 3.2) coincide with those published in Jacobsen and Ray (1992a). The

experimental data is from Flory (1939); the experiment was mn at two different

temperatures. Jacobsen and Ray (1992a) use their model to simulate DPn versus time. The

iwatlab mode1 simulated the results from Jacobsen and Ray and curves for two

temperatures are shown in Figure 3.3.

Benchmaking Jacobsen and Ray 1992a 40.

O' O 1 2 3 4 5 6 7 8 9 10

lime (seconds) 104

Figure 3.3 Mode1 predictions of DPn at 166°C ( - ) and 202°C ( - - ):

Figure 3.4 shows model predictions and experirnental data for conversion versus time at

166 OC. The model follows the experirnental data quite well. Conversion data is very

useful to an operator since it is a direct rneasure of how far the polymerization has

advanced.

Benchrnarking Jacobsen and Ray 1 ! I

Time (seconds) x l o4

Figure 3.4 Conversion vs. Time at 1 6 6 ~ ~ : Experimental (a), Mode1 ( - )

3.2.3 MPD, Adipic Acid, IsophthaUc Acid

The 2-component benchmarking system was expanded to a 3-component system in order to

mode1 the 2-methyl- l,3-propanediol, adipic acid, isophthalic acid system. It was also

expanded to incorporate a non-isothermai temperature profile. In the benchmarking case

there was one possible reaction type, an alcohol group reacting with an acid group. In the 3-

component case, there are two types of possible reactions: an alcohol group reacting with

an acid group from the adipic acid, and an alcohol group reacting with an acid group from

the isophthalic acid.

From Table 3.4 the relevant reaction is the Polycondensation I type reaction:

P,>i.a.b + 'n.e. t, Pm+n.a+e-b(i).b+ +Cij

Equation 3.8

whet-e Monamer A = adipic acid - HOOC(CH&COOH

Morzorner B = isophthalic acid - C&i.dCOOH)_i

Monomer C = 2-metizyl- 1,3-propanediol - HOCH2CH(CH3)CH20H

Cti = ivater

The moment equations were derived for the zeroeth and first moments. using the procedure

outlined in Section 3.2.1. The moment rate equations, Equation 3.3, reduce to the following

set of equations:

Equation 3.9

where

x 1 = )ro,o.o = O'h nzornent (total concentration of polymer clznins)

- 2 = hsc 1 1.o.o = 1" A end moments (active A end groicp ccocentration)

~3 = hs~z>,o,o = 1" B end moments (active B end group concentration)

~4 = ?q3 = 1" C end mornenrs (active C end group concentration)

~5 = b.3 1.0 = 1'' monomer A moment (conc. of A ~cnits, inclicding those in p o b e r )

= )ro.apl.o = 1" monomer B moment (conc. of B icnits. inclriding those in polyner)

= 1 = 1." monorner C moment (conc. of C units, including those in polyner)

kl = rcrte corzstant for the reaction between adipic acid arid MPD

kl = rate constant for the reaction between isophthalic acid and MPD

In order to simulate the system, the rate constants must be estimated. Chapter 5 discusses

how these parameters were estimated for this model. In the model defined nbove (Equation

3.9). there are seven states: however, since X j to x,- are constant. the model can be reduced

to four states. The model inputs are the initial concentrations of the monomers and the

temperature profile. The mode1 outputs are conversion and DPn. These variables are

calculated in a manner similar to the benchmarking case.

3.3 DryAdd

DryAdd is a simulation package that has been developed by our industrial partner. This

package can be used to simulate various types of free-radical and condensation type

polymerizations. DryAdd is based upon Monte Carlo simulations, which employ random

number generation to simulate the random nature of a real process. It is a purely statistica1

attempt to model a process. It is a huge bookkeeping task but ha.s the advantage of being

able to deai with very cornplex processes and a large numbers of variables.

For a particular simulation, a monomer molecule will be chosen at random from the

reaction vessel and will be ailowed to react according to the reactions that have been pre-

specified. The monomer may or may not react. After a successful reaction. it will be

accounted for and retumed to the reaction vessel for further reaction and another randorn

monomer unit will be selected for possible reaction; if it does not react, it will be put back

into the reaction vessel. A running tdly of what has and has not reacted, including

connectivities, branching, network formation, etc. is kept.

The latest version of DryAdd has the ability to mode1 free-radical and condensation

polymerizations. It can account for reversible reactions. growth in pre and post-gel regions,

cross-linking and network formation. The user can specify temperature as well as feed

profiles. After a simulation is run, graphical output is available. Figures and tables

showing the number and weight average molecular weight versus temperature and time as

well as sequence data. number of unreacted sites/monomer and other pieces of information

are available. Multiple simulations of a systern can be run. and the results will be averaged

and standard deviations of al1 variables will be calcuIated. The user must specify a

minimum amount of information in order to run a simulation. One must specify:

1. The type of material present. Le., functionai group(s) on the molecule

2. The amount of each type of material used, i-e.. mass

3. The molecular weight of each type of material

4. How each type of materiai reacts, i-e., one must have some notion of the reaction

mechanism

These four items are the bare minimum required. However, there are more items that the

user could specify. These include kinetic data (i.e., rate constants) or relative reaction

rates. The current default weights al1 relative reaction rates to 100%, Le. if no rate

information is specified, al1 reactions are assumed to react at the same rate. In this instance

it is not practicai to simply use the default value. Additionally, the user has the option to

specify a feed schedulc, a temperature schedule and a removal schedule. There are also

many more advanced features available that are not discussed here, such as pre-polymer

handling.

In both DryAdd and a first principles approach. kinetic information is necessary. As

previously mentioned, there is no kinetic information in the literature for the system of

interest. Therefore, to use either approach, rate constants must first be estimated.

DryAdd was examined as a simulation package, but not used further in this work.

4 Experimental Results

4.1 Experimental Setup

Shown in Figure 4.1 is a simplified flow diagram of the process. The polymer is produced

in a batchwise manner in a 100-gallon tank as the main reactor. This tank is instrurnented

with a viscosity sensor, a thennocouple and a video camera. The main reactor is jricketed

which facilitates temperature control. The tank is connected to a partial condenser.

Condensate from the partial condenser can either be returned to the reaction vessel o r

coilected in a tank on a load cell. The load ce11 is used to meawre the weight of water

(condensate) that is removed from the vessel during the reaction.

During the reaction, viscosity and temperature are measured online. The online viscosity

measurements, for this system, were not especialiy useful; the reaction is carried out ac

approxirnaely 2 4 0 ~ ~ and at that temperature the viscosity of the polymer does not change

appreciably. Offline measurements of viscosity and acid number are made after the initial

stages of the experiment. Unlike the online viscosity, the offline viscosity measurements at

30°c are very usehi1 and change appreciably throughout the reaction. All of the viscosity

data presented in this work is measured at 30°c.

One source of error in this experiment is MPD losses in the early part of the reaction. The

iMPD monomer is quite volatile and if the temperature goes too high, MPD losses c m

occur. MPD losses in the early part of the experiment will directly affect the ability to

achieve high conversions o r low acid numbers. If during the course of the reaction the acid

number is not dropping to the desired value, additional MPD is usually added. MPD losses

c m be caused by poor temperature control during the initial stages. This is not an issue

during the latter part of the reaction because there is no unreacted MPD monorner in the

vessel. Other sources of error include contamination of the reaction vessel, weighing errors.

and the assumed purity of the monomers.

Al1 experiments were run at a pilot plant facility of our industrial partner.

Total Condenser

100 Sampling Port Gallon * - -

Load CeIl I

Figure 4. I Flow Diagram of Industriol Process

4.2 Experimental Procedure

The polyrner is made in a batchwise process. Monomer is added to the reaction vessel and a

reaction temperature of approximately 240°C is set as a target. During the initial stages of

the experiment, the reactor operatcs with partial reflux. Condensate passes through the

partial condenser, through a full condenser and into a holding tank. The condensate, which

is primarily water, is weighed and removed. The removal of water is necessary in order to

achieve high conversion and drive this equilibnum reaction to the products. During the

latter stages of the experiment, the partial condenser is shut off and any remaining

condensate wilI pass through the total condenser and coilect in the holding tank.

When the partial condenser is shut off, at the midpoint of the experiment, a light aromatic

solvent is added to the reaction vessel. The solvent does not take part in the condensation

reaction but forms an azeotrope with water, lowering the boiling point of water in the

mixture. and thereby making it more volatile. The solvent is then separated from the water

and recycled back to the reaction vessel.

Once the temperature of the reaction mixture reaches approximately 240°C, polymer

samples are taken from the reactor at irregular intervais and analyzed in the laboratory for

acid number and viscosity. Acid number is directly proportional to conversion. The offline

measurements of acid number and viscosity are typically available 15-30 minutes after the

sample has been taken. Acid number and viscosity are the key variables that are used to

track the course of the reaction. Figure 4.2 shows a plot of viscosity versus acid number for

al1 five experiments. Typically, the customer requires a polymer with a viscosity between

300-400 centipoise (at 30°C) and an acid number less than 1.

Relevant Definitions

Acid number is a measure of the concentration of carboxyl end groups in a polymer and is

another way to track conversion. To measure acid number a polymer sample is taken.

cooled to room temperature, and titrated with a base (potassium hydroxide) to its

equivalence point. Acid number is defined as the weight in milligrarns of KOH required to

neutralize 1 g of polymer:

,..N Acid #= 5.6 1-

Equation 4.1

rihere x = grams of polymer

N = concentration of base

y = cm' of base required for neutralization

In polymer systems, there are different ways to define an average molecular weight. Two

functional definitions are referred to as number average rnolecular weight (Mn) and weighi

average molecular weight (MW). These are very useful polyrner properties to know and in

many cases the customer desires a polymer of a specific Mn and MW range. They are

defined as:

i

Equation 4.2

Equation 4.3

wlzere N; is the nimber of polymer chahs of lengtlr i and ntulecirlar rwighr Mi

Altematively. Mn can aIso be calculated by multiplying the average molecular weight of

the repeat unit by the number average chain length. DPn (calculated in Equation 3.7).

Viscosity vs. Acid Number

1000 :

10 1 1

45 40 35 30 25 20 15 10 5 O

Acid Number

Figure 4.2 Viscosity vs Acid Number: Exp#l (e), Exp#2 (i), Exp#3 (*), Exp#4 (*),Exp#5 (a)

4.3 Experimental Data

The procedure outlined in the previous section was run on five separate occasions by the

industrial partner, and therefore five sets of experirnental data were available for this work-

Four experiments were run at the pilot plant facility in the 100-gallon reactor; the fifth

experiment was a full production mn (approximately 15000 lbs of materiais), run at a

manufacturing faciIity. It should be noted that during one of the experiments (Experiment

#4). the author was there to observe how the experiment was run.

Shown in Table 4.1 are the batch sizes for each experirnental run. There are three

experiments that are approximately the same size (700-735 Ibs.) and two larger mns (1000

ibs. and 15000 Ibs.). The combination of repeated experiments and varying batch sizes

provide some insight on the reproducibility of experimental data as well as on how the

system is affected by scde-up.

1 Experiment # l ~ a t c h Size 1 1 Pilot Plant - 700 Ibs.

2 Pilot Plant - 700 Ibs.

/ Pilot Plant - 1000 lbs. 1

Table 4.1 Size of experiments

4 l

5

Shown in Table 4.2 is a typical data set that is obtained from each experiment.

Temperature, time, acid number, viscosity and the weight of condensate removed are

recorded during the course of the experiment. Temperature is measured online, whereas

acid number and viscosity are measured offline. Experiments #1-3 and 5 are representative

sets of data that were made available. For experiment #4, there are additional variables that

have been measured on and offline. Specificdly, temperature measurements were measured

more frequently (at 3 minute intervals); the inlet and outlet water temperature to the

condenser was measured; weight average molecular weight, number average molecular

Pilot Plant - 735 Ibs. - Full Data Set

Full Production Run - 15000 Ibs.

weight and polydispersity were determined from the polymer samples after the experiment

was completed via GPC (gel penneation chromatography). The experimental data tables for

experiments #2-5 are in Appendix A.

Expriment #l

I~aterial 1 Lbs. 1

Time (min) Temp. (C) Acid # ' Viscosity H20 Off

(centipoise) (Ibs.)

O 142.2

Table 4.2 Typical Experimental Data - Experiment # I Data Set

6 1

118

178

203 237.8 19 122 97.0

213 I 1 , I I

240.0

163.9 55.8

195.6 1 230.6

8 1.8

95.2

The experimental results will be presented in a graphical form in the following pages. The

variables plotted (e-g.. conversion. acid number) have been defined in previous sections.

Viscosity vs. Acid Number

45 40 35 30 25 20 15 10 5 O Acid Number

Figure 4.3 Viscosity (centipoise) vs Acid Number: All Data (+) with exponential trendline

As discussed in Section 4.2, the operators tracked the course of the reaction exclusively by

acid number and viscosity measurements. The variables seem to have an exponential

relationship. which is observed in Figure 4.3. Viscosity is plotted versus acid number: the

data from al1 experiments are combined and an exponential (linear on a log scale) trendline

is plotted.

The temperature profile for each experiment is shown in Figure 4.4. The desired reaction

temperature is approximatel y 2 4 0 ~ ~ for al1 batches.

Batch Temp vs. Time 260 ,

1

120 . 100 -

O 100 200 300 400 500 600 700 800 900 1000

Tirne (minutes)

Figure 4.4 Batch Temperature (OC) vs. Time: Exp#l (e). Exp#2 (m). Exp#3 (A). Exp#l (XI.

Exp#5 (a)

I t is important to note that the tirne axis in Figure 4-4-Figure 4.6 has some degree of error or

subjectivity associated with it. From the experimental data it is assumed that the actual

reucfion began shortly after al1 the reactants were added to the reaction vessel . The full

production batch (Experiment #5) in most plots seems to be i a g g i ~ g behind the other four

experimen ts; however. the reaction start time may have k e n underestimated. Since it

would take longer for a larger batch to reach the desired reaction temperature, it i s plausible

that the assumed reaction start tirne has k e n estimated to be too early. If this i s the case.

the plots associated with Experiment #5 should be shifted to the left somewhat. It is also

plausible that the reaction time has been adequately estimated and scaling up the

experiment simply increaes the duration.

Figure 3.5 shows acid number versus time. The desired acid number is less thm 1. A11

experiments except for Expriment #4 achieved the specification. In Experiment #4 it was

postulated, in retrospect, that some temperature control problems that occurred in the initial

stages of the reaction caused MPD losses and as a result the desired acid number could not

be reached.

Another way to look at the acid number versus time is to convert acid number to

conversion. Figure 4.6 shows conversion venus time data for al! experiments. It should be

noted that polymer samples were not taken until after the initial stages of the experiment.

Unfortunately, as a resuIt. there is no low conversion data to observe.

- -

Acid Nurnber vs. Time

45

Time (minutes)

Figure 4.5 ~ c i d Number vs Time; Exp# 1 (*), Exp#î (i), Exp#3 (A), Exp#4 (x), Exp#5 (a)

Conversion vs. Tirne

O 100 200 300 400 500 600 700 800 900 1000 Time (minutes)

Figure 4.6 Conversion vs Time; Exp#l (e), Exp#2 (m), Exp#3 ( h ) , Exp#4 (r). Exp#S (.)

The next few figures contain data from Experiment #4 alone. As previously mentioned.

there was more analysis perforrned during this experiment and as a result there are other

variables that can be examined. The samples that were analyzed during the reaction for acid

number and viscosity were also analyzed (after the expriment was completed) for

rnolecular weight averages, Mn and MW.

Often when niaking a polymer, Mn or MW are variables that a customer looks at. instead or

specifying a conversion, a Mn and/or MW is specified. These properties are not typically

measured online and in most cases are only measured for the final polymer sample after the

reaction is cornpleted. The white box mode1 that is developed in this work does not

calculate MW from a first principles approach but can be used to approximate Mn.

The first principles mode1 was combined with empirical correlations from the experimental

data in an attempt to predict useful variables (like MW) that are otherwise difficult to

estirnate using a first principles approach or measure online.

Mn, MW vs Acid Number

35 30 25 20 15 10 5 O

Acid Number

Figure 4.7 Mn. MW vs Acid Nurnber; Mn (.) with trendline, MW (i) with trendline

Figure 4.7 shows the relationship between Mn. MW and acid number. Since acid number is

measured during the course of the reaction, this chart c m be used to estimate Mn or MW

when an acid number measurement is available. An exponential trend correlates acid

number with Mn and MW.

Similarly, Figure 4.8 shows the relationship between viscosity and Mn and MW. A trend

line is not drawn in this case but such a figure could nevertheless be used to estimate Mn or

MW from a viscosity measurement.

Viscosity vs. Mn, MW 1 O000

Viscosity (cpoise)

Figure 4.8 Mn. MW vs Viscosity; Mn (D), MW ( O )

Figure 4.9 shows Mn and MW as function of time. Mn and MW should. in general. foliow

the same trend. In Figure 4.9 it appears that MW changes very sharply around 400 minutes

and Mn does not. This observation is likely due to experimental error. In sorne polymer

sarnpIes duplicate analysis was performed; the PvIn duplicate samples varied by as much as

+ i 50, and the MW samples varied by as much as -00. Within this region of experirnental

error, Mn and MW do follow approximately the same trend.

Since al1 five experiments follow approximately the same trends (Figure 4.3, Figure 3.4,

Figure 4.5. Figure 4.6), Figure 4.9 could also be used to estimate Mn or MW as the

experiment progresses in time.

Mn, MW vs. Time

O O 100 200 300 400 500 600 700 800 900 1000

Time (minutes)

Figure 4.9 Mn, MW vs Time; Mn (i), MW (.)

5 Parameter Estimation

As discussed in Section 3.2.3, the kinetic parameters must be estimated in order to model

this system. There were no kinetic parameters found in the literature. Thus, experiments

were performed, as discussed in Chapter 4, to obtain data for estimation of these key

parameters. In order to estimate the kinetic parameters, the outputs of the model have to be

correlated with the experirnental data. The experimental data contain measurements of acid

number, viscosity and temperature. Recail that acid nurnber can be converted to

conversion. The model can calculate conversion or number average chain length as its

outputs. Therefore. the conversion (or acid number) data was used to estimate the model

parameters.

5.1 Estimation Procedure and Results

The acid number measurements from al1 five sets of experimental data were converted to

conversion (Figure 5.1 shown below, which is a repeat of Figure 4.6)- and used to estimate

the kinetic parameters.

Rccall that the rate constant, k, is an Arrhenius expression of the form:

Two

acid

rate constants

group and a

must be determined: one that describes the reaction between an adipic

MPD group, k l l , and one that describes the reaction between an

isophthalic acid goup and a MPD group, k2,, For each rate constant, k, the parameters Eu

and A must be identified. Therefore, a total of four parameters need to be estimated to

descri be the overall polycondensation process.

I Conversion vs. Erne

O 200 400 600 800 1 O00 Time

F i g u ~ 5.1 Conversion vs Time: Exp#l (e), Exp#2 (.). Exp#3 (A), Exp#4 (x). Exp#5 (a)

The parameter estimation problem was formulated in a non-linear Ieast squares fashion and

was solved using Nelder-Mead simplex optimization (Nelder and Mead. 1965: Lagarias et

al.. 1998). Yamada et al. (1986) used the same optimization technique to identify rate

constants for another polycondensation system. The performance index was selected to be

the squared error, ( y - )', between the experirnental conversion data ( jP ) and the mode1

prediction ( ).

Initially, the estimation was performed by taking one set of experimental data (and the

associated temperature profile) and estimating a set of kinetic parameters. This was done

for each of the five experiments. During the estimation it was observed that the 'optimal'

parameters were sensitive to initial guesses. For example. the optimization would return

different parameter estimates that had approximately the same value of the performance

index. These observations of non-unique solutions are quite common when trying to

ident i fy rate constants for pol ymerization (nonlinear) systems, with highl y correlated

parameters (A and &).

Next. additional factors were considered in order to allow selection of a 'single' solution

'best' suited to the problem at hand. Since the parameter estimates were sensitive to initial

guesses, the best possible initial guess was desired. The four-dimensional parameter

estimation problem was reduced to a two-dimensional problem. Instead of three-

components (two acids and MPD), the system was treated as a two-component system. The

rate constant for isophthalic acid was constrained to be equivalent to that of adipic acid:

therefore, only two parameters needed to be estimated. Since the amount of isophthalic acid

in the system is very small compared to adipic acid. it is reasonable to assume that the

parameters estimated using the reduced two-component simulation would be close to the

'tme' value of the parameters describing the reaction between adipic acid groups and MPD

groups and thus provide a reasonabte initial estimate for the three-component system.

These types of assumptions are commonly made in order to make the estimation probkm

tractable. In order to validate this type of assumption many more experiments would have

to be performed.

The two-component simulation was mn in the same rnanner that the three-component was:

i.e,. one set of experimental data was used at a time. The results were similar to the three-

component simulation in that a non-unique set of parameters was estimated. However, the

parameter estimates were very close together: i.e., the optimal parameter estirnates from

each data set were, in many cases. approximately the same point. whereas in the four-

dimensional estimation, an average could not be chosen among the optimal estimates.

Additionally, in the two-dimensional estimation, the parameter estimates were much less

sensitive to initial gesses.

The three-component (four-dimensional) optimization was then re-run with the new initial

estimates for the adipic acid parameters (based on the two-component. two-dimensional

optimization results). A unique solution was not obtained but the results were more

consistent in that the sets of 'optimal' estimates were much closer together. Next. the

optimization was rerun with al1 five experiments combined. In the previous set of runs, a

set of parameters was estimated for each individual set of experimental data. With al1 the

experiments combined, a single set of parameters could be estimated to best describe the

overall results. As in the previous cases, non-unique sets of parameters were obtained that

yielded approximately the sarne performance indices.

A method had to be devised to reduce the number of solutions and choose a final pararneter

vector. Therefore, in addition to the performance index. the nom of the parameter

covariance matnx was examined, Equation 5.10. The covariance matrix was constructed in

order to generate confidence bounds and joint confidence regions. which will be elaborated

on in the next section. The norm of the covariance matrix was calculated for al1 parameter

estimates whose performance indices were similar. Parameter vectors with smaller norms

were given preference because the nom of the covariance matrix was related to the size of

the confidence bounds.

The norm of the covariance matrix proved to be a very useful tool to choose between what

appeared to be equivalent parameter vectors. For seemingly equivalent estimates. the nom

indicator could vary by orders of magnitude and made choosing the optimal parameter

quite easy. This can be understood because a pararneter vector with a srnall performance

index for one set of experimental data, may not be optimal for ail five experiments. The

calculation of the norm of the covariance matrix was based on al1 experimental data. and

not on a single data set.

It should be noted that, whether ri parameter estimate has been obtained using the data in

only one experiment or in al1 five experiments. the final nom-based selection of the

parameter estimate (Equation 5.10) is based on the combined covariance matrix (using al1

the experimental data), which is defined in the next section.

It was hypothesized that the parameter vector estimated when al1 five experiments were

combined. would be the best to describe this system. However. when the norms of the

covariance matrix were calculated frorn these estimates, they were much greater than the

noms of the parameter estimates from the individual systems. When al1 five experiments

were combined, the error ( y - 5 )' from two of the experiments dominated the estimation.

Therefore, the 'optimal' estimate did not accurately reflect al1 five experiments.

The overdl optimal parameter estimate was chosen based on ri combination of the

performance index and the n o m of its covariance matrix. The optimal parameter estimate

is shown in Table 5.1. Recall that El and Al are the kinetic pararneters associated with

adipic acid and MPD and E2 and At are the kinetic pararneters associated with isophthalic

acid and MPD.

Figure 5.2 compares the model predictions for conversion with the experimental vaIues for

al1 five experiments. The model fits for experiments 1, 3 and 5 appear to follow the

experimental data quite well. The model for experiments 2 and 4 does not fit as nicely as

the other three experiments. In experiment 2, the first conversion data point seems to

suggest that the reaction start time may have been estimated too early and as a result the

data points should be shifted to the left. Similarly in experiment 4, it appears that the

reaction start time may have been estimated too late and as result the data points should be

shifted to the right. Recail that it was discussed in Section 4.3 that there was some

subjectivity (or guessing) involved in determining when the reaction actually started. The

model predictions from experiments 2 and 4 suggest that the reaction start time may have

been overesti mated and underestimated, respective1 y.

Parameter

Estimate

The shape of the model prediction is also a function of temperature. The temperature

profiles for the five experirnents (Figure 4.4), are simiIar but not identicdly the same. In

Figure 5.2, experiment 4 appears to have initial dynamics that are dissimilar from the other

four experiments. It should be noted that there is much more temperature data available in

experiment 4 than in the other four experiments; the temperature was measured online

every three minutes and this was incorporated into the simulation. In the other four

experiments, temperature data is only available at irregular intervals, particularly when

samples were taken. Therefore, the initial dynamics reflected in experiment 4 (Figure 5.2),

Table 5.1 Optimal parameter estimates

E [caVmol]

13 1 0 0

Er [caVmol]

13800

Al [kg/mo12-hr ] A. [kgL/molL-hr] 6

180 55 !

are likely a better representation than the initial dynamics modelled in the other four

experiments since it incorporates a more complete temperature profile. Figure 5.3 is

identical to Figure 5.2; the axes have simply been adjusted to a different scale, for a more

detailed picture.

One should also consider that the amount of error or noise associated with the experimental

data is unknown; if, for example, the tme value of the first point in experiment 2 was

sli_ohtly higher, the mode1 would appear to fit the data very well. This point (first point in

experiment 2) in particular appears that it may be inconsistent because the conversion

changes quite dramatically over a very short period of time; such a jump is not observed in

any other experiment, and it may be simply experimental error.

As discussed in the experimental procedure (Section 4.3), offline sarnpling did not occur

until after the initial stages of the experiment and as a result there is no low conversion

data. If lower conversion data can be obtained, some of these issues could be resolved and

c o d d possibly resuit in a unique set of parameters when the optirnization is perforrned.

O -' O 1 2 3 4 5 6

Trne (Seconds) to4

Figure 5.2 Conversion vs Time; Model ( - ), Experimental (e)

Time (Seconds) 104

Figure 5.3 Conversion vs Time; Mode1 ( - ), Expenmental (.)

5.2 Confidence Bounds

In this section, a linear approximation of the nonlinear system is used to generrrte 95%

maqinal confidence bounds and 95% joint confidence regions (Bates and Watts. 1998) for

the parameters estimated in the previous section.

In the linear case, given the model:

Y = X J + Z

Equation 5.1

rvizere Y is rhe response vector

X is the regressor (derivative) nlntrix

fl is the mode1 parorneter vector

Z is n izonnaliy distributed, zero mean white noise sequence

it foIlows that:

Var[a = E[ZZ~] = a' 1

Equation 5.2

@ is the least squares estirnate such that

Equation 5.3

is minimized

The least squares estimator has the following properties:

1 . Given that is a linear function of Y and 2, and that Z is assumed to be normally

distributed, then is normally distributed

-. U D l = p

3. vartfi] = d (xTx)-' A

4. A P-dimensional, 1-a joint confidence region for P is the ellipsoid

( / 3 - p ) T ~ T ~ ~ - f i ) 5 P s 2 F ( P . iV-P: a)

Equation 5.4 .

7 s ( D ) CVitere S- = -

N - P

(IV-PI is the rirtmber of degrees of freedom

In the non-linear case. you are given the model:

Y = R X . e) + Z

Equation 5.5

This model is of exactly the sarne form as the linear case except that the expected response

(Y) is a non-linear function of the parameters. 8.

A

The 1 -a joint confidence region for 0 is the ellipsoid:

^ T T ( 6 - 6 ) v v t e - e ) g s 2 ~ ~ N - P . ~ )

Equation 5.6

icllere V is the (N-rP) derivarive matrir obrairzed by lirzenrizing the madel

Equation 5.7

crnd V is npproxirnated by a finite difference

Equation 5.8

whert. S is rhe percenr pertcïrbation

N i s the nttmber of data points

P is the rzrrrnber o f parameters

The variable V is the gradient of the model and reflects how the model behaves if the

parameters are perturbed. V was calculated by numerical differentiation. The parameters.

ê . were perturbed in a positive and negative direction by a small amount (typically O. 1 -

1%). and the output of the model, Y (in our case. conversion) was calculated at each of the

rxperiniental data points. This finite difference was used to approximate a derivative matnx

V. Once V was determined, the covariance matrix. confidence bounds and joint confidence

regions could be estimated as in the linear case.

Equation 5.6 can be rewritten as

Equation 5.9

cov, ê ) =s (v'v) -' Equation 5.10

Equation 5.9 can rearranged and solved for 6' which defines an ellipsoid centered on the

parameter point estimate.

S ince 8 is assurnecf to be nomally distributed, then any linear transformation of ê would

also be normaily distributed:

Equation 5.1 1

A

A linear transformation of the parameter vector H ~ a l l o w s the isolation of any two

parameters. 6, and Oj with a new transformed covariance rnatnx ( H C O V ( @ ) H ~ ) . With this

transformation, the joint confidence region for any two parameters can be plotted in two-

dimensional space.

Shown in Figure 5.4 are the joint confidence regions for the parameter estimates. The

vertical and horizontal Iines are 95% marginal confidence bands of the individual

parameter estimate. The ellipses are 95% joint confidence regions, as defined by Equation

5.9.

It is important to remember that these joint confidence regions are Zinear approximations to

an underlying nonlinear system. The extent to which these approximate regions adequately

represent the regions of reasonable parameter values is determined by the adequacy of the

linear approximation to the expectation fùnction. Linear approximation regions con be

extremely misleading (Bates and Watts, 1998).

Figure 5.4 Joint Confidence Regions for the Parameter Estimates

Given that the joint confidence regions shown in Figure 5.4 are only approximations and

can be misleading, the results should be interpreted in a qualitative manner. From this

figure, one may not be able to say with 95% certainty that the parameter estimates lie in the

regions shown but can get a sense of how dependent the mode1 parameters are on each

other.

During the parameter estimation it W ~ S observed that the performance index was very

sensitive to changes in El and E-. This cm be understood since El and Er appear in the

exponential part of the rate constant, and hence small changes in the parameter will result

in large changes in the rate constant. AI and A7 do not affect the output to the extent that El

and E2 do, and therefore are more difficult to estimate with the sarne certainty or

confidence. From Figure 5.4, El and El appear highly correlated and have a very srnall joint

confidence region when compared with a11 other joint confidence region ?airs. This

suggests that these parameter estimates are more accurate than the others, but also suggests

that. given the high degree of correlation, perhrips both parameters were not necessary to be

estimated separately: i-e., maybe one could simply have been calculated based on the

estimated value of the other. This may, however, be a coincidence specific to the test

reaction/reactants chosen and therefore for generality. it is best to leave the parameters as

separate entities. Additionafly, since there is very little isophthalic acid in the system. it is

reasonable to suspect that identifying A- and El may be very difficult.

6 State Estimation Results

6.1 Introduction

In the previous chapters a first principles model wm developed and the kinetic parameters

of the model were estimated using experimental data. From the available data. the best

possible mode1 was constructed. Nonetheless, models are only approximations of the

average behaviour of the system; no model is perfect. Model parameters are estirnuteci to

minimize the error between the expenmental data and the predictions based on the fina1

model. Experimental data sets are often incomplete, as was the case in this work.

Disturbances and noise are always present and these c o m p t the experimental data.

A model alone will not reflect the effect of disturbances and noise on the outputs and states

of the true system. Another problem commonly faced when mnning an experiment is that.

in many cases. the states of the system are unknown and unmeasurable. For example. in

this work. the states of the model are directly proportional to the concentrations of the

monomers in the reactor. The output that is being rneasured during the experiment is

conversion (or acid number). Since the states are not being measured. they cannot be

tracked during the experiment, even though knowledge of the states could be very useful.

A model can be coupled with an estimator that can attempt to address the [imitations of

using the model alone. An estimator combines both the model and the measurements

during an experiment. An estimator can compensate for noise and disturbances that may be

present when mnning an experiment. An estimator can also be used to infer the states of

the system from the measured outputs, even though the states may be unmeasurable.

Another term for an estimator is afilrer. There are many examples in engineering where

filtering is necessary. Radio communications signals are often compted with noise; a good

filtering algorithm can remove the noise from electromagnetic signals while still retaining

the useful information. T h e Kalman filter is the estimation algorithm used in this work-

The filter is very powerful in severai respects: it supports estimations of past, present, and

even future states. It is an optimal recursive least squares estimator. Kaiman filtering

(Kalman. 1960) has k e n applied in areas as diverse as aerospace, marine navigation,

nuclear power plant instrumentation, dernographic modelling, manufacturing, and many

others. Kalman filtering is also used extensivdy in electricai engineering applications.

The question addressed by the Kalman filter is this: Given our knowledge of the behaviour

of the system, and given our measurements. what is the best estimate of states and outputs

of the true system? We know how the system behaves according to the process model, and

have measurements of the outputs, so how can we determine the best estimate of the true

states? Surely we can do better than just take each measurement at its face value.

especially if we suspect that we have measurement noise. Figure 6.1 is a block diagram of

the Kalrnan filter and shows schematicaily how the filter works.

6.2 Kalman Filtering

The Kalrnan filter addresses the general problern of trying to estimate the state of a process

t hat is governed by the linear stochastic differential equation:

Mode1 Equation

X( t ) = F(t) s(t) + G(t) w ( t )

w 0) - M O , Q( r )) Equation 6.1

Measurement Equation

Equation 6.2

The KaIman filter is forrnulated as follows:

Assume that the process noise w (t) is white Gaussian noise with a covariance matrix Q(t).

Further assume that the measurement noise u ( r ) is white Gaussian noise with a covariance

niatrix R(r). and that it is not correlated with the process noise. The estimation algorithm is

formulated such that the following statistical conditions hoid:

1 . The state estimate is equal to the expected vaiue of the state

i-e.. E ( X ( Z )) = X( t )

2 . The estimate of the state minimizes the expected value of the square of the estimation

error: i.e.. min { E ( [ ~ ( t ) -i([)][~(t) -.?(t ) lT) }

The Kalman filter updates the estimate of the model's states and outputs given:

1 . the a priori estimate of the state at time t . z(t- ) . 2. the curent measurement, y( t )

3. an estimate of the covariance of the noise associated with the measurements. R(r)

4. an estimate of the covariance of the noise associated with the states. Q(t)

The Kalman filter implernentation equations are:

63

Impiementation Eauations

? ( t ) = H ( t ) i ( t )

Equation 6.3

where P(t) is the error covariance matrix and K(z) is the Kalman gain matnx.

The matrix K(t) (in Equation 6.3) is calculated by the filter to be the gain that minimizes the

error covariance P(t) . Looking at K(t) we see that as the measurernent error covariance.

R(r). approaches zero, the gain K(t) weighs the error between the actual measurement and

estimated measurement more heavily. Conversely, as the error covariance estimate

approaches zero, the gain ut) weighs the error between the actual measurement and

estimated measurement less heavil y.

In other words. as the confidence in the measurements increases, the estimator relies

heavily on the measurements. and, as the confidence in the measurements decreases, the

estimator relies more on the underlying process model. Therefore, a balance is achieved by

weighting the confidence in the state estimates with the confidence in the measurements,

using the associated covariance matrices, R(t) and Q(r).

6.3 Extended Kalman Filtering

As discussed in the previous section. the Kalman filter is an optimal way to estimate the

states and outputs of a process, given the process measurements and covariance matrices

for the states and measurements. The development of the filter assumes that the process

model is linear. In the case of a non-linear model, some type of linearization must be done

in order to apply the filter. In one approach, the process model can be linearized about the

current estimate of the state. over each sampling interval; in this approach the error

associated with the linearization is very small since it is done over a relatively srnall region.

Altematively, a non-linear model could first be linearized about some steady state and then

combined with the Kalman filter; in this approach, the error associated with the

linearization could be large, depending on how non-linear the system is. The latter

approach is. cornputationally, simpler since the model is only linearized once. and not

online, at every sampling instant. The former approach, although more computationally

intensive is more robust since it only assumes linearity over the sampling intend. Such a

Kalman filter that linearizes about the estimate of the state is referred to as an extertded

Kcllnrarr filter (EKF).

Using a Taylor Series expansion, the model can be linearized around the current estimate

using the partial derivatives of the process and measurement functions to compute

estimates even in the case of non-linear relationships.

In the nonlinear case. the process is assumed to be governed by the non-linear stochastic

differen tial equations:

Mode1 Equation

X( t ) = f(-u(r), t ) + G ( t ) w ( r )

(Nt, - N (O. Q( t ),

Equation 6.4

~Measurement Equation

' ( t ) = Ir(x(t), t ) + v ( t )

v 0) - N (O, R ( N

Equation 6.5

Recall that the model equations are:

The measurement equation is:

Equation 6.7

itdzere y is conversion

The implementation of the extended Kalman filter differs from the linear case in that it uses

the linearized partial derivatives. F(I) and H(t ) , which are updated continuously.

ictiere F(r) is the Jacobian matrix of partial derivatives of f(r) with respect to x.

Equation 6.8

m d H(r) is the Jacobian matrix of partial derivatives of M t ) with respect to x.

In the system studied in this work. f(r) is a non-linear function of the states (Section 3.2.3).

However. h(t). is a linear function of the states. Therefore. in the implementation of the

extended Kalman filter, f(t) is linearized to obtain F(t) at every integration step, whereas

h(r) is Iinear and needs not to be linearized.

An underlying assumption of the Kalmm filter is that the states are obsewuble. A linear

dynamic system model is observable if and only if its states are uniquely determinable from

the rnodels inputs and outputs. The observability of a system depends only on H ( t ) and

F(y): a row or column of zeros in F(t) will make the system non-observable. It the case of

the linearized model presented in Equation 6.8. the first column is a column of zeros. and

thercforee, .rl is not observable; the model is therefore modified (eliminating, xl, to make

it observable) for use with the EKF and includes only 3 states:

Equation 6.10

idrere

X, = i > . o , ~ = 1"' A end moments (active A end group concentrarion); iviriclr is eqriivnlent

to 2-r the concen~ration of ndipic acid

,o,o = ISr B end moments (active B end group concentrution): whicli is

quivalent to 2-r the concentration of isophtiralic acid

0.0 = 1"' C end moments (active C end group concentration) : ivhicir is

eqriivcrlent to 2.r the concentration of MPD

The exclusion of the zeroeth moment does not prevent the calculation of any

polymer variables that have been previously mentioned.

6.4 Simulation Examples

In this section. the use of the extended Kalman filter is dernonstrated. The mode[ that was

developed in the previous chapters was used to generate experimental data. Zero mean.

normally distributed white noise was added to the model's output to simulate the presence

of noise. RecaIl that the model's output is conversion and the states of the model, -Y,, .Q and

XJ, are directly proportional to the concentrations of adipic acid, isophthalic acid and MPD,

respectively. In al1 figures, the underlying process model is shown as a solid line; the

simulated experimental data that the filter uses are shown as dots; and the estimated output

from the EKF is shown as a dashed line.

In Figure 6.2 normally distributed white noise with a variance of 0.01 is added to the model

output to simulate experimental data. The tuning parameters of the EKF are Q. R and G. Q

is the covariance mauix associated with the states; R is the covariance matrix associated

with the outputs: G is a weighting matrix that describes the cross-correlation of noise of the

states. Unless othenvise stated, it is assurned that there is no cross-correlation of noise in

the states and G is equal to the identity matrïx. Since there is only one output in this model,

R is the estimated variance of the output. Q is a 3x3 diagonal matrix of the f o m :

In the following exarnples it is assumed that the individual variances of the states are equal,

1.e. q l = q-, = q3.

Although the underlying model equations of the EKF are irnplemented in a continuous

fishion, it is not practical to assume that measurements are available continuously. A three-

minute time span is chosen as a sampling interval. The EKF is forrnulated in such a way

that between measurements, the filter assumes that the 1 s t available measurement is the

current measurement. In practice, this is what is done: it is better to use the last known

measurement as the current measurement, until the next measurement becomes available,

instead of using no measurernent at al1 (Grewal and Andrews, 1993).

Figure 6.2 demonstrates the ability of the filter to estimate the trrie process output in the

presence of a noisy signal. Figure 6.3 shows the states of the nominal model and the

estimated states from the EKF. Recall that the states of the model are not measured and the

EKF must infer the tme states from the measurements and the model.

6.4.1 Effectof R

R is the estimated variance of the output. As R is increased, the confidence in the

measurement decreases and the EKF will weigh its optimal estimate more heavily on the

model. Similarly. as R is decreased, the confidence in the measurement increases and the

filter will rely to a greater extent on the measurements to construct the estimate.

Figure 6.4 and Figure 6.5 show the effect on the filter as R is increased. The experirnental

data and estimates for Q and G are the sarne as in Figure 6.2. From Figure 6.4 it is evident

that the filter is relying more on the underiying estimated states and to a lesser extent on the

measurernents when compared to Figure 6.2. Figure 6.5 shows the model states and the

estimated states. When compared with Figure 6.3, the states estimates are rnuch smoother

and closer to the model alone because R has been increased.

Figure 6.6 and Figure 6.7 show the effect on the filter as R is decreased. When compared to

any of the previous figures, it is evident that the filter is attempting to follow the

measurements to a greater extent.

-0.2 O 0.5 1 1.5 2 2.5 3

Time (s) 104

Figure 6.2 Simulated Experirnentd Data (a); Model alone (-): Filter ( - - );

Figure 6.3 Model States; Model alone (-); Filter ( - - );

Time ( s ) 104

Figure 6.4 [RI increased; Simulated Experimental Data (w); Model aione (-): Filter ( - - ):

Figure 6.5 Model States as [RI is increased; Model done (-); Filter ( - - );

-0.2 O O. 5 1 1.5 2 2.5 3

Time (s) l o 4

Figure 6.6 [RI decreased; Simulated Experimental Data (.); Model alone (-); Filter ( - - );

Figure 6.7 Model States as [RI is decreased; Model alone (-); Filter ( - - );

73

6.4.2 Effect of Q

Q is the estimated variance of the states. As Q is increased, the confidence in the estimated

states decreases and the EKF will weigh its optimal estimate more heavily on the

measurements. Similarly. as Q is decreased, the confidence in the estimated state increases

and the filter will rely more on the model when constructing an estimate. In this

application of the EKF, Q is a 3x3 diagonal matnx. In the examples presented in this

section, al1 of the diagonal elements are set equal. This assumes that the estimated variance

of each state is approximately equal.

Figure 6.8 and Figure 6.9 show the effect on the filter as -) is increased. Increasing Q has a

simiIar effect as decreasing R. The filter relies less on the estimated states (model) and

more on the measurements. Similarly, Figure 6.10 and Figure 6.1 1 show the effect on the

filter as Q is decreased. Decreasing Q has a similar effect as increasing R. The filter relies

more on the estimated states and less on the measurements.

6.4.3 Effect of G

In the previous simulations. matrix G had been set equal to the identity matrix which

assumes that there is no correlation between the noise of the states. In this application,

there is no reason to assume that there is cross-corretation with the state noise. However,

for demonstrative purposes. assume that there is some phenornenon that causes one to

believe that the noise elements associated with the two acids in the system are correlated. In

this case. G could be forrnulated to reflect this as:

Figure 6.12 shows the effect of changing G from the identity matrix to GI . The differences

in the optimal estimates between G = 1 and G = G f are very subtle but do reflect the effect

of the cross correlation. The differences in the estimated states (not shown) are also very

subtle.

T h e (s)

Figure 6.8 [QI increased; Simulated Experimental Data (m): Model alone (-): Fiiter ( - - ):

\ '. '._. . '\. .

5 .- K..

O O O . 5 1 1 . 5 2 2 5 3

Tim e ( 2 ) x I O 4

Figure 6.9 Model states as [QI is increased; Model alone (-); Filter ( - - );

75

-0.2 O 0.5 1 1.5 2 2.5 3

Time (s) 104

Figure 6.1 O [QI decreased; Simulated Experimental Data ( e ) : Model done(-): Filter( - - ):

O ' O 0 . 5 1 1 . 5 2 2 . 5 3

T i m e (s) x I O 4

Figure 6.1 1 Model states as [QI is decreased; Model alone (-); Filter ( - - );

76

Time (s)

Figure 6.12 Effect of changing [G]: Simulated Experimerital Data (e); Mode1 alone (-):

Filter with G = 1 (O): Filter with G= G I ( O )

With al1 the simulation results of this chapter, it is established that the filter implementation

equations and its responses are correct.

7 Application Results

This c hapter illustrates potential applications of the mode1 and estimator developed in the

previous chapters. First, the estimator matrices Q and R are tuned to best reflect the

experirnental data. Next, the tuned estimator is used to illustrate the EKF's benefits and

practical applications of the work done in this thesis.

7.1 Tuning the Estimator

In the previous chapter it was assumed that Q was a diagonal matrix with al1 the diagonal

elements equal. The assumption of al1 the elements being equal is not necessarily the best

assumption and it is, in fact, more prcictical to assume that the variance of each state is not

equal. There are many reasons why the estimated error (or variance) in one state could be

greater or less than in another. For example, in this systern. there is only a small amount of

isophthalic acid added to the system. If there are any weighing errors or impurities in any

of the monomers. this type of error would be relatively more significant when attempting to

estimate the concentration of isophthalic acid. The MPD is a relatively volatile component

in the reaction mixture. If. during the initial stages of the reaction, some MPD is lost to

evsiporation, the experiment will not reach desired conversion levels. During the parameter

estimation process, it was deemed that there was greater confidence in the kinetic

paranieters associated with adipic acid and less confidence with the kinetic parameters

associated with isophthalic acid. Additionally, there was greater overall confidence

associated with the kinetic parameters (El and Ez) in the exponential part of the rate

constant and Iess confidence with Al and A,, the pre-exponential factors. Given al1 of these

sources of error and uncertainty, it is felt that the variance of each state is not necessarily

equal.

Since the rate constant associated with isophthatlic acid has the most uncertainty and given

that isophthatic acid is present only in very small amounts, it is believed that - 2 ha! the

most variance.

Since the possibly also exists that MPD could be lost by evaporation in the initial stages of

the reaction. it is believed that XJ has a variance greater than xl but less than -Q. Therefore,

when tuning the estimator, Q is chosen such that:

7.1.1 Tracking Experimental Data

The estimator was tuned to best reflect the experimental data, balancing the confidence in

the mode1 with the confidence in the rneasurements. Before the EKF can be used with the

experimental data. a few additional issues must be resolved. First. the measurements are

sampled at irregularly spaced intervals. Secondly, there are no measurements available

during the initial stages of the reaction.

In the simulations of the EKF shown in the previous chapter, measurements were available

at every sampling interval; this is not always the case with experimental data. In order to

adapt the EKF to use the experimental measurements, a three-minute sampling interval was

chosen. When the filter was applied, it was formulated such that whenever a measurement

was not available, it was assumed that the current measurerner-t was the I a s t available

measurement. This assumption is reasonable if the time between measurements is not too

long. For example, in Figure 7.1 (which is a repeat of Figure 4.6), the time between the

start of the reaction and the first measurement is mo long. The estimator will try to use the

mode1 but the estimate will eventually become biased because it has some Ievel of

confidence in the measurement (Figure 7.2). When the next measurement is finaily used,

the estimator tries to follow the measurements but it has k e n badly biased in the period

where no new measurements were available and as a result. the estimates are poor. Since

al1 of the experimental data sets do not have measurements rivailable until sorne relatively

long time after the start of the reaction, the estimator must be modified to accommodate

this. The estimator is modified such that until the first measurement is available, the

estimator relies on the process model alone; after the first measurement is available. the

estimator assumes the current measurement is the last available measiirement. The final

tuning pararneters of Q and R used that best refiect the data and the confidence in the

process model are shown in Table 7.1.

Figure 7.3 - Figure 7.12 show how the estimator copes with the experimental data using the

tuning parameters in Table 7.1. The pararneters were obtained by trial and error.

Conversion vs. Tirne

Figure 7.1 Conversion vs Time; Exp#l (.), Exp#2 (.), Exp#3 (A). Ex* (r). Exp#5 (a)

Data From Expriment #3

O O O. 5 1 1.5 2 2.5 3 3.5

Tirne (s) x 104

Figure 7.2 Expenmentd Data (.): Filter ( - - )

Table 7.1 EKF Tuning Parameters

Experirnent #1

O O 0.5 1 1.5 2 2.5 3

Time (s) X I O '

Figure 7.3 Experiment #1: Experimental Data (e): Filter (-)

Figure 7.3 Experiment # 1 ; Mode1 alone (-); Filter ( - - )

82

Experiment #2

Time (s)

Figure 7.5 Experiment #2: Experimental Data (.); Filter ( - );

. Tim e (s) x 1 0 .

Figure 7.6 Experiment #2; Model alone (-); Filter ( - - )

83

Expenment #3 -- --d----

/-----

O - O 0.5 1 1 .S 2 2.5 3 3.5

Erne (s) x IO

Figure 7.7 Experiment #3: Experimental Data (.): FiIter (-);

Figure 7.8 Experiment #3; Model alone (-); Filter ( - - )

84

Time (s)

Figure 7.9 Experiment #4: Experimental Data (.): Filter (-);

O O 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5

T i m e ( 5 ) x I O 4

Figure 7.10 Experiment fi; Mode1 alone (-): Filter ( - - )

85

0.1 - 1

0 : O 1 2 3 4 5 6

Tirne (s) x 10

Figure 7.1 1 Experiment #5: Experïmental Data (.); Filter (-):

f l---- -- _- . ---. - - - - - - - - O

O 1 2 3 4 5 6 Tim e (s) x 1 0 '

Figure 7.12 Experiment #5; Model alone (-); Filter ( - - )

86

7.2 Combining Estimation Results with Empirical Correlations

The model's outputs can be combined with empirical correlations to estimate other usefrtl

polyrner properties. During an experiment, operators track the course of the reaction using

viscosity and acid number (Figure 4.3). The mode1 developed calculates conversion which

can be converted to acid number. Acid number has been empirically correlated with

viscosity (Figure 4.3). number average molecular weight (Mn), and weight average

molecular weight (Figure 4.7). Mn c m also be calculated from the model by multiplying

DPn (calcuiated in Equation 3.7) with the average molecular weight of the repeat unit. In

this approximate calculation it is assumed that the repeat unit is an adipic acid-iMPD unit

(since there is only a small amount of isophthalic acid in the system). Therefore. as the

mode! is combined with the EKF to predict the outputs and the States. it can also be

combined with other empirical correlations to estimate acid number, viscosity. number and

weight average molecular weight.

Viscosity and acid number are measured offiine during the course of the reaction. There are

no online measurements of number and weight average molecular weight: these variabIes

c m be measured from the sarnples taken, after the reaction is completed. Although Mn and

MW are not measured during an experiment, knowledge of theses variables could be very

useful for both the operators and the customer. Often the customer requires a final product

within a certain Mn and MW range. Since Mn and MW are not typically measured. the

operators associate Mn and MW with a viscosity and acid number range. Having an

estimate of acid nurnber and viscosity as well as number and weight average molecular

weight during the entire experiment, gives the operators more information to work with.

Shown in Figure 7.13 is an example of online estimates of acid number, viscosity. number

average molecular weight and weight average molecular weight, Acid number and Mn crin

be calculated directly from the process model and are shown as a dotted line: Mn, MW and

viscosity can be calculated from empirical correlations (Figure 4.3. Figure 4.7) and are

shown as a dashed line. Therefore, in addition to estimating the states and outputs of the

process. the process rnodel can be used to estirnate the variables shown in Figure 7.13.

Acid Number 300 r

Viscosity

F ipre 7 .13 Model (a); Correlation ( - - );

This section illustrates how the EKF copes with process-mode1 mismatch. If the true

system is somewhrit different from the identified rnodef, the model alone will not accurately

calculate the trrre states. If the EKF c m identify the tnre states in the face of mismatch.

greater confidence can be placed in the results of the estimacor dunng an experiment. This

wiII be illustrated using simulations. To sirnulate process-mode1 mismatch. the model

parameters are petturbed and the perturbed model, which represents the [rue process. is

used to generate the measured outputs. y. The identified model is combined with the EKF

to estimate the states of the true system. If the EKF can cope with the mismatch. the

est imated states will follow the trrie states.

7.3.1 Estimating the States

The knowledge of the MPD concentration can be very useful dunng an experiment. As

previously mentioned. MPD is a retatively volatile component and can be lost during the

initial stages of the reaction. If this happens. the desired conversion levels will not be

achieved and the operators will observe that acid nurnber is not decreasing to the desired

value: eventually, the operators will add more MPD to the system. Ul timately, the polymer

will reach the desired conversion, but corrections to the system can add several hours to the

experimental duration. If an online estimate of not only conversion but also of the

concentration of MPD is available throughout the entire reaction, corrections can be made

much earlier in the experiment. thus avoiding lengthening the experirnental duration.

Recall that the states of the model are proportional to the concentrations of the monomers.

Therefore. one can track the concentration of the monomers with the EKF. To simulate

loss of MPD, measurernents have been simulated with the initial concentration of MPD

being lowered by 10%. This can reflect MPD losses during the initial stages of the

reaction. If the EKF can cope with such a disturbance. the estimator should be able to track

the true states (based on the 1090 change) of the nominal model.

Figure 7.14 shows the measurements and the estimated measurements from the EKF.

Figure 7.15 shows the states of the true model, the states of the nominal model alone and

the estirnated states. Frorn Figure 7.15. it is evident that the tuned EKF, using the Q and R

values previously deterrnined, improves upon the model alone is tracking the trrre process

in the face of disturbances to the MPD concentration. (RecalI that Q is the estimated

variance of the states, and R is the estirnated variance of the measurements.) Therefore, if a

few measurements are taken during the initial stages of the reaction, the EKF can be used

to improve tracking the true concentrations of the monomers, and if there is a deviation

from the desired concentrations. the operators wiil have a chance to correct for i t earl ier on-

Alcohol Concentration -1 Ooh 1

o i O O. 5 1 1.5 2 2.5 3

Tirne (s) x 104

Figure 7.14 Simulated Experimental Data (a); Filter ( - - )

Figure 7.1 5 Tme Process ( - ); Nominal Mode1 ( - - ); Filter ( - - );

7.3.2 Confidence in Mode1 Parameters

Another area of uncertainty in the nominal model is the estimated model parameters. Four

parameters were estimated and joint confidence intervals were constmcted (Chapter 5).

During the parameter estimation process. it was deemed that there was greater confidence

in the kinetic parameters associated with adipic acid and less confidence with the kinetic

parameters associated with isophthalic acid. Additionally, there was greater overall

confidence associated with the kinetic parameters Ei and Ez in the exponential part of the

rate constant and less confidence with AI and A?, the pre-exponential factors.

Process-mode1 mismatch c m also be simulated by perturbing the estimated parameters. In

Figure 7.16 and Figure 7.17. the kinetic panmeters associated with isophthalic acid are

perturbed. The pre-exponential constant is increased by 30% and the activation energy is

decreased by 30%. Figure 7.17 illustrates how the estimated states of EKF track very well

the (rue states. Similarly, in Figure 7.18 and Figure 7.19, the pre-exponential constants

were increased by 50% and 30% for ridipic acid and isophthalic acid. respectively. Figure

7.19 shows that the estimated states eventually follow the true states satisfactorily again.

Kinetic Parameters [E l A l €2-30% A 2 + 3 0 % ]

o. O 0.5 1 1.5 2 2.5 3

Time (s) 104

Figure 7.16 Simulated Experimental Data (a); Fiiter ( - - )

O 0 5 1 1 5 2 2 5 3 1 3 r t o 4

1 0 y

\, 1 '-.. - - -.-

5 - '--. . ----% .- ------ -------- '- ----.--- --- *---- --- -7 -

0 ' O O . 5 1 1 5 2 2 5 3

TI^ e ( s ) r 1 0 4

Figure 7.17 True Process ( - ); Nominal Mode1 ( - - ); Filter ( - - );

93

Kinetic P arameters [E 1 A 1 +50% E2 A2+30°/~] 1 -

izI.iL~œ------ &a &&O

0.9 i de*g

f 0.8 - 9

w w

0.7 - .u 0,

0' . ' 0.6 ' O d

O .- a,' 2 2 0.5 - 9' C 0, '

a' 0.4 : *

0 0.3 - .

0- O O . 5 1 1.5 2 2.5 3

Time (s) x l o d

Figure 7.18 Simulated Experimental Data (.); Filter ( - - )

Tim e (s) x I O *

Figure 7.19 Tme Process ( - ); Nominal Model ( - - ); Filter ( - - );

94

8 Conclusions and Recommendations

8.1 Conclusions

The contributions in this thesis include developing a process rnodel and combinins the

model with an extended Kalman filter for a polyester system made from adipic acid.

isophthalic acid and 2-methyl- l,3-propanediol.

Several experiments were perfonned by an industrial partner at a pilot plant facility to

generate experimental data. A first principles model w u developed and kinetic parameters

were estimated using the experimental data. The identified model has inputs of initial

concentrations of ingredients and a temperature profile. The model has outputs of

conversion and number average chain length. The experimental data were atso used to

empirically correlate variables like number and weight average molecular weight and

viscosity to the outputs of the model.

An extended Kalman filter was developed to facilitate online estimation of the model's

states and outputs. The extended Kalman filter uses the process model and online

measurements to provide online optimal estimates of the states and outputs of the model.

The extended Kalman f i l ter was combined with empirical correlations to predict additional

poiymer properties like viscosity, number average molecular weight and weight average

molecuiar weight. Tuning parameters were selected for the EKF to best track the

experimental process data.

8.2 Recommendations

The following are recommendations for future work:

Tuke more frequent rneasctrernents drtring the initial stages of the renction. The EKF is

dependent on process measurements. As the number of measurements increases, the

filter's ability to adequately estirnate the true states also increases. In the five

experiments performed, sampling of the polymer began after the initial stages of the

experiment. On average, there were no sarnples taken during the first 2-3 hours: after

this time. measurements were taken at irregularly spaced time intervals until

completion. If samples were taken during the first hour o r two. the information

obtained would not have k e n particularly useful to an operator. However, with respect

to the EKF, measurements taken early on are vital and could provide v e n useful

information to an operator. AdditionaIly. if low conversion data were obtained. the

parameter estimation could be re-run and a better set of rate constants (parameters)

could be obtained.

Modifiirlg the EFK to rtse delryrd process ntensrtrernertts. When a polymer sample is

taken from the reaction vessel, it typically takes 15-30 minutes before laboratory

analysis is available. Given the duration of the experiment. this is a reasonably small

amount of delay. However, the EKF could be reforrnulated to incorporate delayed

measurements, as illustrated in Mutha et al. (1997).

Encorrrcrge ortr irzdccstrinl partner tu rtse the EKF in un ornine fbshiorz dtririg the

esperirrient. It is uncertain whether our industrial partner would go as far as to

implement the EKF in an online fashion. However, it is more feasible that they could

use the EKF in a semi-online fashion, where it is re-run offline every time a

rneasurernent becomes available. In this approach, the operators will have many more

tools to guide them on how the polyrnerization is proceeding. Specificaily, tracking the

MPD concentration couid be very beneficial in minimizing the experimental duration.

This will eventually pave the way for future online implementation.

3. Develop a control srraregy bnsed on the extended Kalnran filrer and nrodel. Currently,

there is no formal control strategy being implemented when the experiment is being

run. Now that the EKF can estimate the concentrations of the monomers and the acid

number during polymerization, the EKF can naturally be combined with a control

strategy that is based not only on the process measurements but also on the States of the

model (model predictive non-lineaï- control, as in Mutha et al,. 1 997).

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Appendix A

Expriment #2 1

cllaterial 1 Lbs. 1 292.W

\di pic

Time (min)

0 60 120 170 240 260 270

sophthalic 1 41.39 1

Temp. (C )

141.7 1 60-0 195.6 220.0 225.6 233.3 238.9

Acid #

1

Table 8.1 Typical Experimental Data - Experiment #2 Data Set

Viscosi ty (centipoise)

Conversion H20 Off (Ibs.)

55.8 53.8

94.6

! 100.4 (

-.I

[COOH] molkg

Expriment #3

' ~ a t e r i a c MPD

Acid # Time (min)

O 60

.

Lbs. 4 16.82

Adipic Iso~hthalic

-- -- - - - - -

Table 8.2 Typical Experimental Data - Experiment #3 Data Set

Temp. (C)

14 1.7 140.6

Viscosity (centipoise)

58 1.54 59.07

Solvent Total

[COOH] m o l k g

H-O Off (Ibs.)

55.4

99.9 1 999.34

Conversion

I I Expriment #4

Time (min)

O 5

i ~ d i ~ i c 1 4 : a l Isonhthalic

Material MPD

Temp. (C)

139.4 138.3

Lbs. 306.36

Table 8.3 Typical Expetimentai Data - Experiment #4 Data Set

Solvent Total

Acid #

73 -43 734.30

Viscosity (centipoise)

Conversion H20 Off (lbs.)

[COOH] mollkg

I I Experiment #5 1

1 (centipoise)

Table 8.4 Typical Experimental Data - Experiment #5 Data Set

Conversion H20 Off (1 bs. )

i

[COOH] molkg