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Model Building Methodology for Complex

Reaction Systems

A thesis submitted to the

University of Manchester Institute of Science andTechnology

for the degree of

Doctor of Philosophy

By

Wenling Zhang

under the supervision of

Professor Robin Smith

Department of Process Integration

Manchester, United Kingdom

April, 2004



Declaration

No portion of the work referred to in this thesis has been submitted in support of an

application for another degree or qualification of this or any other university, or any

other institution of learning.

Wenling Zhang

II



Abstract

The complexity of chemical reaction processes and the short market window of some

chemical products mean that detailed model building can often not be justified. With

little knowledge of chemistry, this work aims to provide a new methodology for model

building of chemical reaction systems with minimum experimental measurements, for

the purpose of reactor design and optimisation. Most often reactor designs are scaled

from experimental measurements, especially for the manufacture of fine and speciality

chemicals. Yet, without a model of the reaction system, major opportunities can be

missed in the design and optimisation of the reactor. When models are developed for a

reaction system in the laboratory, they are often inappropriate for reactor design and

optimisation.

In the first part of this thesis, the reaction scheme that best describes the production of a

given chemical and suitable kinetic equations are obtained simultaneously using

optimisation. A hybrid optimisation method is used to deal with this large problem

where more than one model fits the same experimental data within a certain confidence

level. Stochastic optimisation methods provide multiple solutions that are rival models

for model discrimination. An NLP method improves model precision from the

stochastic optimisation in the narrowed search space. A strategy for reaction scheme

construction is used to generate all reactions from the reacting species and to provide

plausible reaction schemes during optimisation. These reaction schemes are screened

simultaneously with kinetic models to fit the most appropriate reaction scheme and

kinetic model from the rival models.

Optimal experiments then need to be designed to discriminate among rival models. The

experimental design exploits the potential for mixing, as well as temperature and

concentration effects to discriminate between models through the reactor superstructure.

The oleic acid epoxidation reaction is usedto demonstrate the methodology.

For refinery heterogeneous catalytic reactions, due to the complex nature of catalysis, a

large number of rival models pose difficulties for model building and discrimination. In

the rest of the thesis, three-level kinetic study method is developed for model building

III



to reduce the model complexity by separating diffusion effects from kinetic equations.

In addition, catalyst characterisation is used to assist model discrimination. There are a

largenumber of

techniquesavailable

toconnect catalyst properties, catalyst activities

with model performance with different capabilities and limitations. However, not all of

thesewill be useful in a given application. A classification of those techniques specified

for hydrodesulphurisation (HDS) processes provides guidance for selecting suitable

techniques to yield the most information with accuracy, speed, and economy.

Furthermore, plausible ways for model discrimination and model improvement for

thiophene and diesel HDS are explored, including operating condition, feedstock and

catalyst effects.

IV



Acknowledgements

Thisthesis

is by farthe most significant accomplishment

inmy

lifeand

itwould

be

impossible without people who supported me and believed in me.

Although I always find it difficult to express myself, I know thanks never be enough to

express my gratitude to my supervisor, Prof. Robin Smith, for his guidance,

encouragement and support inside and outside academia. Whatever I gain is indebted to

him, his contribution and inspiration that put me in the right direction towards

accomplishing my research, and his patience and kindness that always drew me out of

wandering nowhere.

Special thanks to the Department of Process Integration for giving me the opportunity

and the sponsorship to study at U MIST and the chance to meet many people. My

sincere thanks go to all staffs, for their help and support whenever needed. I would like

to thankChris for his

magic thatbrings beautiful

results and thankNan for

always

giving me useful tips.

I am also grateful to all students in the department for creating a friendly atmosphere

and making a great time in which I spent and worked, and all friends for supporting and

helping me through those uneasy and depressing times of being away from families.

Naming all of them will definitely double the volume of this thesis.

I would like to send deep thanks to my parents, parents-in-law and my older brothers,

for their love and support through my life without any doubt about my decision ever.

Last but not least, I give unlimited thanks and love to my dearesthusband, Dr. Zhiqiang

Meng, for his love, encouragement and being with me over last 15 years, especially

companying me in the final year by compromising his own career.

V



Table of Contents

Declaration....................................................................................................................... II

Abstract...........................................................................................................................

II

Acknowledgements..........................................................................................................

V

Table of Contents............................................................................................................

VI

List of Figures .................................................................................................................. X

List of Tables................................................................................................................

XIII

Chapter 1: Introduction.....................................................................................................

1

1.1 Model building challenges for complex reaction systems..................................3

1.2 Researchmotivation and objectives ....................................................................5

1.3 Overview of thesis...............................................................................................

6

Chapter 2: Literature Review............................................................................................

8

2.1 Introduction.........................................................................................................

9

2.2 Model building procedure: step-by-step method ..............................................10

2.2.1 Reaction paths and schemes.....................................................................10

2.2.1.1 Knowledge-based methods ......................................................... 11

2.2.1.2 Logic-centred methods................................................................12

2.2.2 Reaction kinetics......................................................................................

13

2.2.2.1 Traditional approach: white box.................................................

14

2.2.2.2 Tendency model: grey box..........................................................

15

2.2.2.3 Approximate methods: black box...............................................

16

2.2.3 Model reduction ....................................................................................... 172.2.4 Processdevelopment and optimisation ....................................................

19

2.2.4.1 Heuristic methods........................................................................20

2.2.4.2 Geometric technique (Attainable Region)...................................

22

2.2.4.3 Mathematical methods (basedon superstructure) .......................23

2.3 Model building for refinery heterogeneouscatalytic reactions .........................24

VI



2.3.1 Current approaches...................................................................................25

2.3.2 Difficulties and challenges.......................................................................29

2.4 Summary...........................................................................................................

30

Chapter 3: Construction of Reaction Schemes................................................................

32

3.1 Introduction.......................................................................................................

33

3.2 Feasible reaction generation..............................................................................36

3.2.1 Stoichiometric analysis ............................................................................37

3.2.2 Two-stage reaction generation method

....................................................

40

3.2.2.1 Reacting species classification ....................................................42

3.2.2.2 Reaction set-up ............................................................................42

3.2.2.3 Reaction feasibility: linear programming ....................................44

3.2.2.4 Stage II reaction generation .........................................................46

3.2.2.5 Simplification..............................................................................

46

3.3 Reaction scheme construction ...........................................................................47

3.3.1 Incidence matrix .......................................................................................49

3.3.2 Feasibility check procedure......................................................................52

3.4 Illustrative example ...........................................................................................53

3.5 Conclusions.......................................................................................................

59

Chapter 4: Hybrid Optimisation of Rival Models...........................................................

60

4.1 Introduction ....................................................................................................... 61

4.2 Problem Description..........................................................................................

62

4.2.1 Preliminary experimental design..............................................................

63

4.2.2 Data fitting................................................................................................

64

4.2.3 Reaction system modelling ......................................................................65

4.3 Rival Models.....................................................................................................

69

4.3.1 Optimisation framework .......................................................................... 70

4.3.2 Objective function....................................................................................

71

4.3.3 Hybrid optimisation ..................................................................................72

4.3.3.1 Simulate Annealing (SA)............................................................

72

4.3.3.2 Nonlinear Programming (NLP)...................................................

77

4.3.4 Implementation of the optimisation .........................................................79

VII



4.3.4.1 Simulated Annealing framework................................................

79

4.3.4.2 Optimisation moves.....................................................................80

4.3.4.3 Parameterprecision

improvement...............................................

82

4.4 Illustrative cases................................................................................................82

4.4.1 Base case..................................................................................................82

4.4.1.1 Reaction list generation...............................................................84

4.4.1.2 Parameterestimation ...................................................................84

4.4.1.3 Results and analysis ....................................................................85

4.4.2 Oleic acid epoxidation reaction system................................................

87

4.4.2.1 Model building............................................................................

89

4.4.3 Discussion................................................................................................

90

4.5 Conclusions.......................................................................................................

91

Chapter 5: Model Discrimination and Optimal Experimental Design............................

92

5.1 Introduction.......................................................................................................

93

5.2 Model discrimination criteria ............................................................................96

5.3 Optimal experimental design..........................................................................

101

5.3.1 Laboratory reactors ................................................................................102

5.3.2 Operating conditions ..............................................................................103

5.3.3 Reactor superstructure............................................................................104

5.3.4 Simulated Annealing (SA) optimisation ................................................106

5.4 Case Studies .................................................................................................... 107

5.4.1 Base case................................................................................................107

5.4.2 Oleic acid epoxidation ............................................................................109

5.5 Conclusions.....................................................................................................

110

Chapter 6: Model Building for Refinery HeterogeneousCatalytic Reactions..............

111

6.1 Introduction ..................................................................................................... 112

6.2 Heterogeneous catalytic reactions...................................................................114

6.2.1 General features......................................................................................

114

.............................2.2 Hydrodesulphurisation (HDS) processes...................

118

6.3 Model building methodology for HHDS rocesses...........................................122

6.3.1 Catalyst characterisation ........................................................................124

VIII



6.3.2 Catalyst kinetics.....................................................................................

128

6.3.3 Model discrimination.............................................................................

132

6.4 Casestudies.....................................................................................................

133

6.4.1 Hydrodesulphurisation of thiophene......................................................

133

6.4.2 Hydrodesulphurisation (HDS) of diesel........................... .........

137

6.4.2.1 Temperature effects...................................................................141

6.4.2.2 Sulphur compound addition effects...........................................142

6.4.2.3 Catalyst effects ..........................................................................143

6.4.3 Discussion

..............................................................................................

145

6.5 Conclusions.....................................................................................................

146

Chapter 7: Conclusions and Future Work.....................................................................

147

7.1 Conclusions.....................................................................................................

148

7.2 Future work .....................................................................................................152

7.3 Remarks...........................................................................................................

153

Notation.........................................................................................................................

154

References.....................................................................................................................

158

Appendix A...................................................................................................................

172

Appendix B...................................................................................................................

174

Appendix C...................................................................................................................

176

Appendix D...................................................................................................................

177

IX



List of Figures

Fig. 1.1 Typicalchemical processes 2

Fig. 1.2 Overall chemistry of net reaction of R4

Fig. 2.1 Conventional model building methodology 10

Fig. 2.2 Ideal reactors 19

Fig. 2.3 Ideal reactor combinations 20

Fig. 2.4 Industrial reactors 20

Fig. 2.5 Triangular representation 24

Fig. 2.6 Illustration of the complexity of a Co-Mo/A1203 hydroprocessing catalyst 27

Fig. 2.7 The role of microkinetic analysis 28

Fig. 3.1 Reaction model representation 33

Fig. 3.2 Reaction schemeof 2-methyl-6-trifluoromethyl aniline 34

Fig. 3.3 Dimethylbenzene oxidation 39

Fig. 3.4 Reaction generation framework 41

Fig. 3.5 Reaction set-up 43

Fig. 3.6 Changesof reactant and product set 46

Fig. 3.7 Complex monomolecular reaction schemes 48

Fig. 3.8 Reaction scheme examples 58

Fig. 4.1 General model building framework 61

Fig. 4.2 Simulation method framework 69

Fig. 4.3 Optimisation method 70

Fig. 4.4 Data fitting curve 71

Fig. 4.5 Simulated Annealing optimisation framework 78

Fig. 4.6 Optimisation moves in SA 80

Fig. 4.7 Example of state-to-state moves 81

X



Fig. 4.8 Base case 82

Fig. 4.9 Data graph for basecase 84

Fig. 4.10 SA and NLP optimisation results 86

Fig. 4.11 Residual analysis 86

Fig. 4.12 Rival models 87

Fig. 5.1 General model building framework 93

Fig. 5.2 Data fitting and model performance 94

Fig. 5.3 Model discrimination framework 96

Fig. 5.4 Optimal experimental design framework 101

Fig. 5.5 Laboratory reactors for homogeneousreactions 102

Fig. 5.6 Laboratory reactors for heterogeneousreactions 103

Fig. 5.7 Homogeneous experimental reactor superstructure 104

Fig. 5.8 Multiphase experimental reactor superstructure 104

Fig. 5.9 Model performance 108

Fig. 6.1 Operating conditions change with sulphur content target 114

Fig. 6.2 Steps in a heterogeneouscatalytic reaction 115

Fig. 6.3 Method for special distribution of active ingredients 116

Fig. 6.4 Modem refinery 118

Fig. 6.5 Feed and product chromatogram 118

Fig. 6.6 Dimethyldibenzothiophene reaction scheme 119

Fig. 6.7 Relative reaction rate of sulphur compounds 120

Fig. 6.8 Catalyst synthesis procedure 121

Fig. 6.9 Model building framework 124

Fig. 6.10 Catalyst properties 125

Fig. 6.11 Catalytic kinetic model composition 127

XI



Fig. 6.12 Concentration profile 131

Fig. 6.13 Laboratory reactors for heterogeneousreactions 132

Fig. 6.14 Parity plot for Model A 135

Fig. 6.15 Model predictions 136

Fig. 6.16 Optimal operation conditions for model discrimination 136

Fig. 6.17 Pseudo-component for sulphur compounds 137

Fig. 6.18 Boiling curve of diesel fuel 138

Fig. 6.19 Pseudo-component performance of different models 142

Fig. 6.20 Model difference of each pseudo-component addition 142

Fig. 6.21 Catalyst active material distribution 143

Fig. 6.22 Model performance changing with active material location 144

Fig. 6.23 Model performance changing with particle size 145

XII



List of Tables

Table 4.1 Experimental data for basecase 83

Table 4.2 Comparison of optimisation results 85

Table 4.3 A 24-1 ractional factor design 88

Table 4.4 Operating levels for variables of interest 88

Table 4.5 Experimental data from the factorial experiments 88

Table 6.1 Quantity of sulphur in the various distillation fractions 113

Table 6.2 European diesel specifications 113

Table 6.3 Typical commercial hydrotreating catalysts properties 120

Table 6.4 Compilation of techniques for HDS catalysts 127

Table 6.5 Factors of Eq. 6.14 134

Table 6.6 Diesel properties 138

Table 6.7 Kinetic parametersfor Model 1 140

Table 6.8 Kinetic parameters for Model 2 140

XIII



Chapter 1

Chapter 1: Introduction

1.1 Model building challenges for complex reaction systems

1.2 Researchmotivation and objectives

Introduction

1.3 Overview of thesis



Chapter 1 Introduction

The chemical reactor is the core of any chemical process. In a typical chemical process

(Fig. 1.1), the capital and operating costs of chemical reactors may be only 10 - 25% of

the total cost, withseparation units

dominating thesize of cost of the process.

Yetthe

performance of the chemical reactor usually controls the costs and modes of the

operation of the separation units. Thus, improvements in the reactor may have an

enormous impact on upstream and downstream separation processes. However, the

performance of the chemical reactor relies on the accuracy of processmodels.

Recycle

Products

Raw Separation Chemical SeparatioTB

Materials Process Reactor Process mroducts

Fig. 1.1 Typical chemical processes

A wide variety of reactions are complex multi-step processes, particularly those

encountered in processes for fine and speciality chemicals and pharmaceutical

chemicals. Processes for fine and speciality chemicals and pharmaceutical chemicals

often feature products with short product life cycles (Mills & Chaudhari, 1997) that

make the development of a detailed model unattractive, because the company that first

markets the product tends to get 70 % of the total sales (Cussler & Moggridge, 2001).

This creates a disincentive to study the reaction chemistry and kinetics in any detail. In

turn, this often leads to reactor designs being scaled from laboratory experiments

without any model of the chemistry or reaction kinetics. Where a model is developed,

there tends to be a compromise between the desire to spend a lot of time to study the

reaction details and the desire to start early production. Yet, without a model that

reflects the key features of the reaction system, major opportunities can be missed.

Heterogeneous catalytic reactions are more complex thanhomogeneous

reactionsdue to

the combination of diffusion, adsorption, surface reaction, and desorption, which makes

the model building become the most difficult task in the development of a new process.

In general, the procedure of model building for heterogeneouscatalytic reactions might

last up to 5 years, with a large number of experiments carried out.

2




1.1 Model building challenges for complex reaction systems

The design of a chemical reactor, and its optimisation and control require a quantitative

description of reaction system behaviour in the form of equations. Existing methods for

reactor design and optimisation can only be used after a reaction model has been

obtained from laboratory experiments.

In the process of the product development, chemists and chemical engineers are

working at different stages.Chemists aim to find the synthesis routine for the product,

while chemical engineers focus on process design and optimisation.

The conventional approach for model building of reaction systems is generally done

step by step. First, several mechanisms or schemes are proposed according to the

detailed identification of the reaction products and intermediates, together with

physicochemical insights into the reaction mechanism and catalysts. Then, reaction rate

data are obtained to derive the rate law for a specific reaction by appropriate

experimental planning, data collection and analysis. Rival models are screened

according to statistical analysis. Finally, model reduction might be needed to simplify

models for engineering purposes.

Certainly, there are drawbacks using the conventional methodology. Experiments

carried out without simultaneous evaluation are likely to omit exploring the most

important informationessential

forreactor

design and optimisation. Important factors

that affect scale-up of processesmay be ignored by chemists and may cover regions that

are not important. The result is likely to be a process operated under non-optimal

conditions.

For example, an important product R in pharmaceutical industry is produced from raw

materials A and B, as shown in Fig. 1.2.

3




HN OCH3

\S

O- OH

N ýCHZOCONHZ

a-ý

sC

11 1 OCH3H2C-N\

Cý' COOCHZOCH,s

JýCHZCOCI

SO2NH N CHZOCONHZO

COOCH2OCH3 COOCHZOCH,

ABR

Fig. 1.2 Overall chemistry of net reaction of R

The actual reactions are composed of five reaction steps (Paul, 1988):

A+BHR*+HCI

R*+BHR**+S, +HCI

R*+HC1->S1

R**+HCIHR+B

S1-> S2+ HCl

When the process was scaled up from a laboratory reactor to an industrial scale reactor,

it was observed that no R was produced at all. From the analysis, it became clear that

the side product effect was overlooked in the model based on the laboratory reactor.

That is, if the concentration of by-product (HC1) is not controlled, this reaction system

will give no yield of the desired product.

At the early stage of model building, serious errors, such as overlooking some variables,

an incorrect reaction scheme, extrapolation outside the region where the model has been

tested, or unjustified expansion of the number of variables, may cause the reactor design

to be inaccurate or entirely erroneous.

Detailed intrinsic models are critical for the design, optimisation, and control of

chemical reactors. Unfortunately, the use of such models is hindered by several factors,

due to stiffness of the equations, high dimensionality and the large scale of the models.

The numerical solution of such models is computationally demanding, especially when

kinetics are coupled with transport phenomena. On the other hand, for gaining insight

into the reaction system behaviour, key features and components of the system in

detailed models are often disguised.

4




1.2 Research motivation and objectives

Accuracy and efficiency are the two requirements of the desired models for reaction

systems. From the initial stages, it is preferable to direct the experimental design to

extract a model suitable for scale-up using process design techniques. Hence, it is

necessary to find approximate models for these complex reaction systems by the fast

and effective evaluation of experimental data, to shorten development time to market.

Given all stable species involved (as opposed to unstable intermediates) in the system,

without any information about reaction pathways and mechanisms, this work aims to

find suitable reaction schemes and kinetic equations for the purpose of reactor design

and optimisation.

The methodology to be developed here will optimally determine a set of reaction

schemes and associated kinetics for existing reaction systems and to then design

experiments that give maximum information on the choice between models, based on

alternative mechanisms. Model discrimination is emphasised for the earlier experiments

and gradually the emphasis switches to precision of parameter estimation. In this work,

the first consideration will be restricted to homogeneous reaction systems that are

kinetically controlled where mass transfer does not play an important role.

In addition, a systematic methodology for heterogeneous catalytic reactions is also

required. The methodology for this casewillbe developed

usinghydrodesulphurisation

(HDS) processes as one of typical refinery heterogeneouscatalytic reactions. In this

thesis, the procedure of model building for heterogeneouscatalytic reactions including

catalyst characterisation, kinetic studies and model discrimination will be explored also,

where mass transfer cannot be neglected.

In the model building methodology, the participating speciesare used to generatethe set

of all reactions in a two-stage method. Atom-molecule matrix formulation allows all

species to be systematically representedfor reaction system analysis and can be used to

describe the reaction equations. After generating all possible reactions by a two-stage

method, integer linear programming is used to test the stoichiometric feasibility of the

reactions through checking if the massconservation law is satisfied. A reaction scheme

5



1Introduction

construction algorithm is used to provide all feasible reaction schemes for the

optimisation search, while reaction models are obtained through the best fitting of

experimental data.

Since the true mechanism is rarely known, a number of alternatives will lead to a set of

so-called rival models. The experimentation needsto provide evidence in favour of one

model. Further experiments for model discrimination are designed by maximising the

divergence among all rival models. All kinds of operating conditions, including feeding

policies, phase contacting patterns, mixing types, recycles and discharges should be

searched through optimisation. Here an optimisation method is needed again to

guaranteea robust and fast global solution.

The objective of the present work is to develop a new method to link the work of

chemists and chemical engineers to reduce the risk of direct scale-up of chemical

reactors, make full use of experimental information, to save expenseof laboratory and

pilot experiments, and to shorten time from laboratory to market.

1.3 Overview of thesis

The next chapter generally reviews existing methods from all aspectsof model building,

their advantages and disadvantages.It also includes the complexities and difficulties of

model building for heterogeneous catalytic reactions, focusing on the

hydrodesulphurisation (RIDS)processes.

In Chapter 3, with little knowledge of reaction chemistry, feasible reaction schemesare

derived directly from the information of the inlet and outlet components of reaction

systems from the chemical engineering viewpoint. It is guaranteed that all possible

reaction steps and all feasible reaction schemesare generated. Here a new strategy for

reaction scheme construction is discussed in detail with examples, which provide

feasible reaction schemesfrom the raw materials to products. A feasible reaction list in

each stage obtained through a two-stage method is used to construct reaction schemes.

Because more than one model can fit the same experimental data set, it is essential that

all possible combinations of reaction schemes and kinetics are obtained before further

experimental information is available. As a mixed integer and nonlinear programming

6




problem, hybrid optimisation can provide a global optimal solution. Chapter 4 considers

a new methodology to identify suitable reaction schemes and kinetic equations by an

optimisation methodsimultaneously with

theaid of experimental

data. Astochastic

method is designed to obtain a set of solutions close to the optimal region by setting up

reaction scheme moves and kinetic moves to generate alternatives and monitor the

random search.A basecaseand oleic acid epoxidation will be used to illustrate how the

methodology can interpret the experimental data and be used for reactor design and

optimisation.

Further experimental data are necessary for model discrimination, so optimal

experimental design is carried out by an optimisation method. All kinds of laboratory

reactors and operating conditions are embedded in a reactor network superstructure to

be optimised. The same two cases continue to function as examples in Chapter 5, to

demonstrate the whole procedure for model discrimination and model accuracy

improvement.

In Chapter 6, a systematic methodology of model building and model discrimination for

heterogeneous catalytic reactions is explored. Due to the complex nature of catalysis, a

large number of rival models pose difficulties for model building and discrimination. In

the procedure of model building for heterogeneous catalytic reactions, a three-level

kinetic study method is used in this thesis to reduce the model complexity by separating

diffusion effects from kinetic equations. In addition, catalyst characterisation is used to

assist model discrimination. There are a large number of techniques with different

capabilities and limitations available to connect catalyst properties, catalyst activities

with model performance, but not all of these will be useful in a given application. A

classification of those techniques discussed here provides guidance for selecting suitable

techniques to yield the most information with accuracy, speed, and economy. Because

of the importance of the hydrodesulphurisation (HDS) process in the refinery industry,

plausible ways for model discrimination and model improvement for thiophene and

diesel HDS are explored, while operating conditions, feedstock and catalyst effects are

included.

Chapter 7 summarises the work, discusses the limitations of the methodology and

recommends future work.

7



Chapter 2

Chapter 2: Literature Review

2.1 Introduction

2.2 Model building procedure: step-by-stepmethod

2.2.1 Reaction paths and schemes

2.2.1.1 Knowledge-based methods

2.2.1.2 Logic-centred methods

2.2.2 Reaction kinetics

2.2.2.1 Traditional approach: white box

2.2.2.2 Tendency model: grey box

2.2.2.3 Approximate method: black box

2.2.3 Model reduction

2.2.4 Processdevelopment and optimisation

2.2.4.1 Heuristic methods

2.2.4.2 Geometric technique (Attainable Region)

2.2.4.3 Mathematical methods (basedon superstructure)

2.3. Model building forrefinery

heterogeneouscatalytic reactions

2.3.1 Current approaches

2.3.2 Difficulties and challenges

2.4. Summary

Literature review

8



Chapter 2Literature Review

In a chemical process, reactions and reactors are at the centre of the whole process,

influencing the upstream, downstream and other units. So reactor design and

optimisation is central to the field of process synthesis.This posesgreat pressureon the

model accuracy and efficiency and requires systematic methodologies of model

building. Previous work for model building is reviewed in this chapter, including

conventional methods for homogeneous reaction systems. Special considerations for

refinery heterogeneouscatalytic reactionsarealsodiscussed ere.

2.1 Introduction

The reactor design and optimisation can only be carried out after the reaction models,

reaction schemes and kinetic expressions have been obtained by chemists from

laboratory experiments.

Generally, the development chemist often has a strong background in synthetic

chemistry,but

maylack

skillsin

manipulation ofkinetics

and chemical thermodynamicsto benefit from those techniques when needed. The gap between the work of chemists

and chemical engineers becomes the weakest link of process development and scale-up.

As a result, processes will run under non-optimal conditions.

Since the true mechanism is rarely known, there are several mechanisms consistent with

the data. Even if one mechanism remains that is in agreement with all the known facts,

there is no assurance that it is unique, or that new experiments will not provide evidence

to discredit it.

In order to meet the requirement of cutting costs and making profit, detailed study of the

reaction system is not preferable since an enormous number of experiments will be

necessary. On the other hand, a fast decision on getting the best model at this point

might lead to the wrong direction for research. There is always a trade off between the

model accuracy and the model efficiency.

Advantages and disadvantages of conventional methods in the procedure of model

building will be reviewed for homogeneous and heterogeneous catalytic reactions to

spark the inspiration for new methodology.

9




2.2 Model building procedure: step-by-step method

A step-by-step approach is widely used for model building of reactionsystems, as

shown in Fig. 2.1.

Reaction scheme or Parameter Model

mechanism estimation reduction

Fig. 2.1 Conventional model building methodology

As the first step of model building, a reaction mechanism or scheme composes the

fundamental structure of the reaction model. The current research of reaction

mechanisms or schemes focuses on the situation in which reaction schemes

(mechanisms) that satisfy the given specification for the transformation of available raw

materials to desired products are composed through either a fixed, predetermined set of

elementary reactions, or simultaneously generated reactions. This is a problem

encountered quite frequently during research and development of chemical and

biochemical processes.

Reaction rate data are needed to obtain the rate law for a specific reaction by proper

experimental planning, data collection and analysis. From this, reaction rate parameters

(i.e. reaction order, frequency factor and activation energy) are determined and modified

(Froment & Bischoff, 1979). Appropriate statistical analysis is necessary for model

screening.

Model reduction is used to simplify complex reaction systems, such as combustion

system, metabolism process and diesel HDS. Lumping, sensitivity analysis, target factor

analysis and perturbation theory are commonly used methods.

2.2.1 Reaction paths and schemes

There are two approaches to the study of reaction paths and schemes: optimal design of

reaction paths to desired products carried out in advance, and analysis of existing

reaction systems.

10



Chapter 2 Literature Review

Reaction path design is currently used by experienced chemists and engineers.

Knowledge of similar reaction paths, availability of raw materials, named organic

reactions and basic reaction mechanisms are used to guide their research for new

reaction paths. These researchersare remarkably adept when one considers the immense

number of paths, which could lead to a simple organic molecule.

The aim is to find relevant paths in this multitude that could potentially lead to

economic, flexible, and safe processes.These paths could be subjected to more detailed

analysis and experiments to further resolve their potential.

The methods of generating reaction paths are based on structure transformation (atom,

bond, electron, and functional group). Structure transformation can provide many more

reaction options than the reaction pathways of the reaction mechanism.

The current approaches for generating reaction steps are commonly used in organic

synthesis to design reaction paths can be divided into two categories (Nishida et al.,

1981): knowledge-based methods and logic centred methods.

They are commonly used as a part of synthesis design that aims to find novel, feasible

and economic pathways, which is considered as the potential synthesis routine for the

desired chemical product. Significant progress has been made by organic chemists

towards the systematic and automatic generation of reaction paths.

2.2.1.1 Knowledge-based methods

For knowledge-based methods, a databaseof reactions or a databaseof possible reaction

types for a certain molecular structure is defined a priori and stored. A number of

programmes (REACT, SECS, LHASA) have been developed using this concept, with

different representations of molecules and reactions, evaluation methods and search

strategies.

The REACT (Govind & Powers, 1977,1981) programme was the first effort to

automate the generation of chemical reaction paths for the petroleum industry. The

generation of a target molecule by the REACT programme involves searching a

databaseof 200 generalized reactions, the search being driven by the structural features

11




of the parent structure and certain basic strategies. In REACT, a molecule is represented

by a connection table (CT), containing the relevant information: connectivity, atom and

bandtypes. The changesbetween products and reactants are checked by a list of logical

functions, if the changesare feasible, reactions occur.

Knowledge-based methods have the advantagethat reliable reactions are used,but have

the disadvantages that a large database s required and that no new reactions can be

considered.

2.2.1.2 Logic-centred methods

Logic centred methods (Ugi & Gillespie, 1971; Hendrickson, 1971) focus on the

transformations of atom site or bond, generating sets of intermediates that can be

converted to the desired molecule. The advantage is that new reactions, forward and

backward searchesare considered. The large number of possible reactions generated out

of the range of the reality is its disadvantage.

Hendrickson (1971) provides a constructive representation of the chemical molecule in

which molecules are represented as carbon sites and was able to classify all possible

reaction sites and reactions that might interconnect them. For a simple molecule, acrylic

acid, well over 500 paths are generated. This method simply generates reaction steps.

So an evaluation method is an important but difficult step of chemical path synthesis to

prune infeasible reactions.

Another situation often encountered in refinery, combustion, and biochemical reactions:

construction of reactions and mechanisms from elementary reactions, or construction of

pathways to form reactions. Accordingly Al (Mavrovouniotis, 1993,1995) and P-graph

theory (Fan et al., 2002) can be used. They have one common feature: overall and all

plausible elementary reactions aredefined

a priori.

The P-graph method is based on the unique graph-representation in terms of P-graphs, a

set of axioms, and a group of combinatorial algorithms. In the methods, the inclusion or

exclusion of one elementary reaction in the mechanism of concern hinges on the general

combinatorial properties of feasible reaction networks. The decisions are facilitated by

solving a linear programming problem comprising a set of mass balance constraints to

12




determine the existence or absenceof any feasible solution. The search is accelerated

further by exploiting the inferences of preceding decisions, thereby eliminating

redundancy.

To analyse existing reaction systems, selectivity versus conversion plots of

experimental data are general and useful aids to obtain insights into a reaction scheme,

but it is difficult to apply to complex reaction systems. For instance, maximum

concentration versus reaction time plots are a typical feature of consecutive reactions.

On the other hand, the appearance of a maximum might indicate that, rather than the

system involving consecutive reactions, or consecutive reactions as a part of a more

complex reaction scheme, there might be some combinations of reversible and

irreversible slow and fast reactions. In most cases,the reaction pathway or mechanism

is proposed based on the chemist's expertise and experience, not from systematic

methods.

The approaches to automatically design reaction paths are over complicated to apply if

only stable species (as opposed to unstable intermediates) in the system are detectedand

measured. Also, visual plot methods are not suitable for complex reaction systems.So it

is necessary to develop a systematic methodology to find feasible reaction schemes or

existing reaction systems without missing any options.

2.2.2 Reaction kinetics

In order to determine the operational strategy for a reactor, it is necessary to consider

the reaction kinetics, reactor dynamics and operational constraints. Anything related to

reaction kinetics cannot be estimated reliably from theory, and must be determined by

experiments. So experiments are carried out to obtain data for kinetic studies.

The kinetic analysis of chemical reactionsin

complex systemsis

adifficult

problem.

The major complication results from the complex stoichiometry and thermodynamics

and an enormous number of possible kinetic models, combinations of reaction networks

and kinetic equations for each reaction. Furthermore, heterogeneous reactions involve

mass transfer and hence the reaction rate is influenced by factors such as stirred speed,

interfacial area, diffusion coefficients, etc.

13




It may also be necessary to investigate some intermediate reaction steps separately,

particularly these whose response in the complete process is weak. With complex

feedstocks, the investigation may start with a single component, with binary and ternary

mixtures, to study interaction effects before progressively increasing the complexity

(Froment, 1987).

Anything related to reaction kinetics cannot be estimated reliably from theory, and must

be determined by experiments. So experiments are carried out to obtain data for kinetic

studies.

Three main methods to perform kinetic analysis with handling the experimental data:

2.2.2.1 Traditional approach: white box

The traditional approach emphasises he development of a detailed kinetic model based

on molecular kinetics in which any of the elementary reactions in the schemes are

studied. This method is not always possible, since even for simple reactions the

complete mechanism might still not be understood.

A mechanistic state-space representation based on stoichiometric and kinetic

knowledge, and on energy and material balances for the reactor, is the preferred

approach for modelling reactors. The kinetic model describes the effect that the

temperature and the concentrations have on the rate of each reaction.The

reactor model

relates the states (concentrations, temperature and volume) to the inlet streams,reaction

terms, and possible disturbances. Mechanistic models are typically derived from

physico-chemical laws, but they can also contain qualitative information in the form of

expert and/or linguistic knowledge. They are well suited for a wide range of process

operations.

Data regression for parameter estimation is also a difficult task for complex reaction

systems. Generally, it is carried out using the conventional methods of linear and

nonlinear regression. The Levenberg-Marquardt method (also called the Marquardt

method) works very well in practice and has become the standard for nonlinear least-

squares routines. In recent years, Genetic Algorithms (GA) (Moros et al., 1996; Wolf &

Moros, 1997; Park & Froment, 1998) have also been used for large-scale systems,

14




incorporating traditional method to increase calculation accuracy by providing good

starting points.

However, models are difficult and time-consuming to build for industrially relevant

reaction systems. A sensitivity analysis can help evaluate the terms in a model and

retain those that are most relevant to the processing objectives. No realistic model is the

purely mechanistic, as new physical parameters typically need to be estimated on the

basis of process data. As with data-driven models, experimental design techniques are

useful tools for building sound models from a limited amount of data.

2.2.2.2 Tendency model: grey box

This method (Filippi et al., 1986,1989; Fotopoulos et al., 1998) is based on a

phenomenological approach and on the estimation of a small number of kinetic

parameters by matching the model predictions to experimental data. It does not require a

detailed kinetic description, but takes into account general knowledge about the process,

such as mass and energy balances. This option reduces the kinetic scheme to a very

simple one, often lumping chemical species with simple kinetics, and normally using

power functions in the stages of the scheme considered.

The tendency models are developed by using fundamental material and energy balances,

along with an approximate overall reaction network. The model's stoichiometric

coefficients and the kinetic parameters are estimated by least-squares regression of

experimental data. Because the kinetics assumed n the tendency model may not closely

represent the real kinetic network, several models may be neededfor process simulation

and optimisation. Although the proposed model is usually less accuratethan the detailed

mechanism, it may provide important insights into the true kinetics and guide the search

to a more optimal operation.

The first stage of model development is the identification of a simple stoichiometric

network. Necessary information includes experimental initial and final compositions of

the significant reactants for each experiment.

The resulting kinetic network is unique in that the calculation of the stoichiometric

coefficients is user-defined. Parameters, such as the maximum allowable error and the

15




number of reactions, are also selected in applying available heuristic information in the

model. If an appropriate solution is unattainable, the reaction may be divided into stages

with each section representedby a distinct kinetic model.

Identification of the tendency model's Arrhenius constants is performed with the

experimental temperature/concentration profiles. Each reaction's preexponential factor

and activation energy (for non-isothermal processes) are evaluated by minimising the

deviations between the model and experimental compositions over the course of

reaction. The heats of reaction are then estimated by comparing the experimental heat

generation with values calculated using the kinetic network. The final product is a

process model that can be used in simulation and optimisation studies.

The methods applied for calculating the parameters n traditional approach and tendency

model are normally basedupon regression, and have many variations. The most utilised

methods have been the differential method and the integral method.

Although the tendency model approach appears o be of high industrial relevance, it has

received little attention in the academic research community. The real engineering

challenge here consists of generating reliable parameterestimatesby matching, as much

as possible, the prediction error to be minimised in the estimated step to the criterion to

be optimised in the optimisation step.

In tendency models, reaction networks are extracted from the experimental data, so the

number of reaction steps is equal to the number of independent reactions. In most

complex reaction systems this assumption is not true.

2.2.2.3 Approximate methods: black box

This does not take into account of physical properties of actual reaction systems. The

development of mathematical analysis has led to the discovery and study of important

classes of approximation functions, which under certain conditions, have proven to be

the natural means of approximating other, more or less arbitrary functions. These

approximationfunctions include polynomials, trigonometric series, orthogonal

functions, splines, etc. In this case, the adjustable parameters are basically viewed as

vehicles for fitting the data and, in principle, do not reflect physical considerations

16




about the reacting system. Neural network methods have been proposed to model the

dynamics of discontinuous reactors (Galvan et al., 1996; Krothapally & Palanki, 1997).

In this method, the amount of available data is crucial for the accuracy of the model.

Although simple and relatively easy to obtain, black-box models have certain

drawbacks:

1. Even when they exhibit good interpolative capabilities, they are inadequate for

predicting the reactor behaviour outside the experimental domain in which the

data were collected for model building.

2. Black-box models represent a dynamic relationship only between variables that

are manipulated or measured. Unfortunately, key variables usually remain

unmeasured in most reactors.

In brief, for the investigation of chemical reaction kinetics, it is not necessarythat the

detailed mechanism of reaction is studied, and also it is not easy to find approximation

functions to represent experimental data for large-scale complex reaction systems. So

mathematical functions (hyperbolic or power law), based on experimental results, can

represent or interpret the experimental data best in the range of experiments must be

developed.

In particular, for complex reaction systems,the selectedregression method shouldhave

the ability to handle the estimation of a large number of kinetic parameters.

2.2.3 Model reduction

Detailed modelling of complex reaction systems s becoming increasingly important in

the development, analysis, design and control of chemical reaction processes.However,

the use of such complex models is hindered by two obstacles. First, because of their

sheer size and the presence of multiple time scales, these models are difficult to solve.

Second, the models contain a large number of uncertain or unknown kinetic parameters.

Regression to determine the parameters of complex nonlinear models is both difficult

and unreliable, and the sensitivity of simulations to parameter uncertainties cannot be

easily ascertained. Furthermore, for the purpose of gaining insight into the reaction

17




system behaviour, it is usually preferable to obtain simpler models that bring out the key

features and components of the system. For those reasons, model simplification and

order reduction are becoming central problems in the study of complex reaction

systems. The simulation, design and control of a complex reaction system benefit from

the derivation of accurate and reliable reduced models tailored to particular process

modelling tasks (Okino & Mavrovouniotis, 1998).

Model reduction methods used for simplifying complex reaction systems in the last two

decades nclude lumping, sensitivity analysis, target factor analysis, perturbation theory,

invariant manifold method and variable selection techniques (Edwards & Edgar, 2000).

The choice of model reduction methods depends on the accuracy required for the

modelling task, the structure of the reaction system and the extent of available kinetic

information.

Lumping (Weekman, 1979) is a widely used method for model reduction in which the

reaction vector is transformed. When a kinetic scheme is known, discrete lumping is

possible. The difficulty in finding appropriate lumping schemes ncreases dramatically

for large nonlinear reaction networks.

Model reduction through sensitivity analysis (Tilden et al., 1981; Rabitz et al., 1983;

Edelson & Flamm, 1984) is feasible when the complete reaction scheme and its full

solution areknown. In sensitivity analysis, which species hat need to be retained in the

reduced model can be specified, aswell as the desired accuracy. Recently, a further step

in this direction was done by Zhu and Petzold (1999). A nonlocal in time criterion of

closeness of solutions between the full and of the reduced systems of chemical kinetics

is used. It requires not just a closeness of derivatives but a true closeness of the

dynamics.

By identifying the important models that embody the different time scales of the

reaction system, the singular perturbation method (van Breusegem & Bastin, 1991) is

able to perform model simplification in a manner similar to that of sensitivity analysis

without the need for solutions to the full kinetic model (Duchene & Rouchon, 1996).

18




Target factor analysis (TFA) (Bonvin & Rippen, 1990) has been used successfully with

reaction data to determine, the number of independent reactions and the corresponding

stoichiometry without knowledge of reaction kinetics.

The Method of Invariant Manifold (MIM) for model reduction of dissipative systems of

reaction kinetics has been developed during last two decades (Gorban & Karlin, 1992a,

b, 2000,2003). The MIM is based on a formulation of the condition of invariance as an

equation, and its solution by Newton iterations. The systematic use of thermodynamics

structures and of the quasi-chemical representation allows to construct approximations,

which are in concordance with physical restrictions. Dynamic and static post-processing

procedures give the opportunity to estimate the accuracy of obtained approximations,

and to improve this accuracy significantly.

All of these techniques have been applied with some success, but there remain

unanswered problems: which reduced model is the best one, what is the smallest

dimension of the reduced model, and what is the range of application of the reduced

model?

2.2.4 Process development and optimisation

The aim of reactor design and optimisation is to find the reactor network structures and

operating conditions that give maximum performance.

There are 3 types of ideal reactor: batch, plug flow (PFR) and stirred tank (CSTR), see

Time10

Feed

Product

Batch PFR

Fig. 2.2 Ideal reactors

CSTR

uct

Fig. 2.2. Also, combinations of different ideal reactors can be used in the form of series,

19




ý. ýýeries Parallel Series- Parallel

Fig. 2.3 Ideal reactor combinations

00000

00000

00000

00000

00000

00000

Fig. 2.4 Industrial reactors

parallel or series-parallel_ of ideal reactors, as shown in Fig. 2.3. In reality, generally

more complex reactors are used (Fig. 2.4).

Also it is necessary to determine the operating conditions to be used. For example, it is

necessary to determine the feeding policy and temperature profiles as well as flow

contacting types, or equipment sizes.

Furthermore, other information such as physical properties, mass transfer, heat transfer,

chemical thermodynamics, fluid mechanics, and especially chemical kinetics are

required to design and optimise a reactor.

Different reaction systems, homogeneous, heterogeneous or catalytic, have different

characteristics. Accordingly, the design methods should have different features, but in

generally they can be divided into 3 types: heuristics, graphical techniques (Attainable

Region) and mathematical methods, and sometimes combinations of these.

2.2.4.1 Heuristic methods

Levenspiel (1962) gives the most useful rules for homogeneous reactors and can also be

applied directly to heterogeneoussystems. It includes six general rules:

20




Rule 1: For Single Reactions

To obtain the minimum reactor volume, the concentration of a reactant whose reaction

order is n>0 should be kept ashigh aspossible, low for components where n<0.

Rule 2: For Reactants In Series

Consider reactions in series:

A-ýR-ýS-->"""Y-Z

To maximise any intermediate, fluids that have different concentrations of the active

ingredients - reactant or intermediates, must not be mixed.

Rule 3: For Parallel Reactions

Consider the parallel reactions with reaction orders ni :

R nl ...low order

AS n2 ...intermediate

T n3 ...high order

To get the best product distribution:

o Low CA favours the reaction of lowest order

s High CA favours the reaction of highest order

0 If the desired reaction is of intermediate order then some intermediate CA

will give the best product distribution

0For

reactions all ofthe same order, the product distribution is not affected

by the concentration level.

Rule 4: Complex Reactions

Reaction schemes can be analysed by decomposing them into their simple series and

simple parallel components. Therefore, rules for parallel reactions and series reactions

21




can be applied. For example, for the following reactions, where R is the desired product,

the breakdown is as follows:

A+B-- R B->R

B+RýSAýRýS and

BSS

The breakdown means that A and R should be in plug flow, without any recycle, while

B can be introduced as desired at any concentration level, since it will not affect the

product distribution.

Rule 5: Continuous Versus Discontinuous Operations

Any product distribution that can be obtained in continuous steady-state low operations

can be obtained in a non-flow reactor and vice versa.

Rule 6: Effect of Temperature on Product Distribution

High temperature favours the reaction with higher activation energy E, while a low

temperature favours the reaction with smaller E.

According to the above general rules, reactors and operating conditions to achieve

maximum performance can be determined. However, heuristic methods are easy to use

but they often contradict each other, especially for complex reactions, so systematic

methods are necessary.

2.2.4.2 Geometric technique (Attainable Region)

In the early 1960's, Horn (1964) advocated the study of optimal reactor synthesis by

geometric methods. He called it the attainable region, corresponding to a prescribed

feed and kinetics, which means the full set of outcomes achievable by all possible

designs. In the 1980s, Glasser and Hildebrant (1987,1990) were instrumental in

reviving this attainable region idea.

Of special importance are its extreme points, as these determine the region completely,

and the reactor optima are often realised there. Although for isothermal problems the

strategy provided an elegant tool for the graphical interpretation of the synthesis

22




problem, the extension of the method to more complicated mechanisms and non-

isothermal problems unavoidably goes through the proposition of a series of curves and

three-dimensional graphical simulations. An implementation of the attainable region

concept has been presented by Balakrishna and Biegler (1992), who proposed a

targeting approach that endeavours to bypass the difficulties in constructing the

attainable region in high-dimensional spaces.

2.2.4.3 Mathematical methods (based on superstructure)

Jackson (1968) postulated a reactor superstructure made up of plug flow reactors

connected by side streams.Adjoint relations were used to model the effect of flow in the

side steams on the concentration of species at the exit of the reactor network. Achenie

and Biegler (1986,1988) allowed for component splits and postulated a series- parallel

combination of axial dispersion reactors. A reactor network superstructure that includes

CSTRs and PFRs with various interconnections was formulated by Kokossis and

Floudas (1989,1990 & 1991). In the problem formulation, the PFRs were approximated

by a series of equal-sized sub-CSTRs to eliminate the differential equations, and integer

variables were used to represent the existence of reactor units. This approach was

capable of handling arbitrary kinetics for both isothermal and non-isothermal situations.

Mehta and Kokossis (1996) extended this method to multiphase reactor design and

optimisation.

The key advantage of the superstructure-based approaches is that they can determine

simultaneously the objective value and the explicit optimal reactor network

configuration and operating conditions. One of the limitations of the superstructure-

based approaches is that the optimal solution is only as rich as the initial superstructure.

The true optimal solution can be missed ignored if the initial superstructure does not

include all possibility.

So far three different approaches for reactor design and optimisation are addressed,

provided that reaction schemesand kinetics are available.

23




2.3 Model building for refinery heterogeneouscatalytic reactions

Catalysis is used to describe thephenomenon of

increasing therate of a chemical

reaction by a chemical present in the reaction medium (homogeneouscatalysis), or by a

solid surface on which the reaction can occur (heterogeneouscatalysis). A material that

Reaction

Mechanism

Catalyst Catalyst

Synthesis Characterisation

Fig. 2.5 Triangular representationof Catalysis

can cause catalysis is a catalyst. Heterogeneous catalysis is commonly used in the

chemical industry because of the easy separation of products from catalysts than

homogeneous catalysis. Heterogeneous catalysis not involving solid catalysts will not

be in the scope of this work.

Heterogeneous catalysis is more than a subfield of chemical kinetics and includes

catalyst synthesis, catalyst characterisation and reaction mechanism, as shown in

Boudart's triangular representation (Boudart & Djega, 1984), Fig. 2.5.

Catalyst synthesis deals with the composition, structure and texture of catalytic

materials. Characterisation of catalysts provides quantitative information on catalyst

properties that affect the catalyst performance with the aid of chemical and physical

techniques. The reaction mechanism is the reaction information on an atomic level,

whichis

mostlikely

unknown.

When the three aspects are combined together, quantitative and systematic methods to

describe the interrelationships among them become very difficult. For this reason vast

amounts of empirical knowledge exist and await systematic investigation.

24




Catalysis has been applied in the chemical industry for at least 150 years, whereas the

experimental techniques for investigation of catalysis at the atomic level were not

available until 25 years ago. Also, the computational techniques are even younger andhave yet to become routine.

The major processes in the chemical or petroleum refining industry are based on

heterogeneous catalytic reactions, such as, hydrodesulphurisation, methanol synthesis,

methanation, Fischer-Tropsch synthesis, ammonia synthesis, toluene disproportionation,

and ethyl benzene isomerisation etc. Every process has unique features. The model

building methodology here will be developed based on the hydrodesulphurisation

(RIDS) process in order to be more specific.

2.3.1 Current approaches

Hydrodesulphurisation (HDS) processes are used to meet the requirements for high

qualityfuels, in

whichthe

sulphur contentis the

main concern.

In petrochemical applications, such a complex process must be modelled if an optimum

plant design and operating policy are to be realized. In petroleum refining, fuel

specifications become more and more strict, so improving the accuracy of the model

prediction and speeding up the time for model building of new processesand catalysts

become increasingly important for cost reduction of HDS processes.

Industrial reactions are usually complex, involving several simultaneous and

consecutive reactions with complex feedstocks, leading to a variety of products. In

kinetic studies the actual reaction network is frequently reduced to simplify overall

reactions, while the experimental data are interpreted by a power law that is valid only

within a narrow range of operating conditions. More often Hougen-Watson type rate

equations (Hougen & Watson, 1947) are imported to get abetter fit

of experimental

data, due to the increased number of parameters.

The sulphur compounds in crude oil are present largely in the form of thiols, sulphides,

and various thiophenes and thiophene derivatives, which are difficult to detect and to

measure. The complexity of reaction models will increase rapidly with the number of

reacting species going through a complex reaction scheme. Ma et al. (1994) has

25




detected more than 60 kinds of sulphur compounds in a gas oil, all exhibiting very

different reactivities. If these are described by reaction equations individually, the size

of the reaction model will be huge. A commonly used method is to lump the compoundsinto several groups. Each group is represented by one single compound (pseudo-

component) that features similar properties. Moreover, the reaction scheme between

those pseudo-components is composed of several typical reactions regardless of the

actual reaction mechanisms.

Catalytic reactions depend on the interrelationship between chemicals and catalysts.

Hence, properties of the contact area between the phases are important reaction

variables, which need to be accounted for simultaneously in order to model

heterogeneouscatalytic reactions. More complicated approachesare built on the surface

science.

With the aid of chemical and physical methods for catalyst characterisation, bulk,

texture and surface properties can be obtained to provide proofs for the mechanism

assumptions made in kinetic analysis and interrelationship between catalyst properties

and kinetic behaviour. Cobalt-molybdenum oxides supported on y-alumina (CoMo/y-

A1203) and Nickel-molybdenum oxides supported on y-alumina (NiMo/y-A1203) are the

most generally used catalysts for hydrotreatment.

Catalytic activity is related to the presence of sulphides of Co, Mo and Ni elements.

However, the most important role of Co and Ni is to act as promoters, while Mo is the

precursor of the catalyst. Most efforts are focusing on the understanding of the

promoting effects of cobalt and nickel on Mo-based catalysts.

Delmon (1979) proposed the presence of two distinct phasesto explain the synergetic

effect between cobalt and molybdenum. From this, van Parijs and Froment (1986a)

introduced a remote control model to account for the varying concentration of active

sites of the catalyst. It is a further step towards linking catalyst properties to reaction

rate, but it still lacks a solid basis from catalyst characterisation.

26



Chapter2 Literature Review

On the other hand, Topsoe and co-researchers (1981,1986,2001) developed the

CoMoS theory standing on the discovery of a special signal in Mössbauer emission

spectroscopy.The behaviour

can be ascribed to a mixed CoMoS phase, in which cobaltis located on the edges of MoS2 crystallites as shown in Fig. 2.6. However, the detailed

structure of the CoMoS phase s still unclear after many years of efforts.

Co9S8

ri iraýsar-.ra r-

MoS2-Itke

domains

Qs

" Co (Nf)

0 Ma

iCo: AE203

Co 'Z'Co

Co CoCo

Fig. 2.6 Illustration of the complexity of a Co-Mo/A1203 hydroprocessing catalyst

The term microkinetic is used to distinguish between the approach outlined from a

simple description of kinetics, i. e. power-law kinetics or Hougen-Watson rate

expressions, and the surface science approach.

Microkinetic analysis (Dumesic et al., 1993) is an examination of catalytic reactions in

terms of elementary chemical reactions that occur on the catalyst surface and their

relation with each other and with the surface during a catalytic circle, to provide a

reaction mechanism basis for chemical reaction engineering. From another application

point of view, the level of understanding of the catalytic chemistry is very useful for

catalyst development.

In the last decade, microkinetic analysis has turned into a valuable tool for unravelling

reaction mechanisms in heterogeneouscatalysis and for evaluation of kinetic parameters

of the elementary steps. In the microkinetic approach, the kinetic model for a catalytic

27



I


reaction is formulated with kinetic parameterswhich are physical meaningful and, as far

as is possible, can be derived from theoretical calculations or from experimental results.

Fig. 2.7 displays the interplay of the most important data required for microkinetic

analysis (Hinrichsen, 1999).

Single crystalsurfaces (UHV)

MaterialGap

1

>idarly-f TrteAxY1erirnents

irt ri- tOcrc kineli czato analysis

3p ktros- PressureCUPIC

studiesGap

Real catalysts(high pressure)

Fig. 2.7 The role of microkinetic analysis.

In the past 10 years, a new type of laboratory reactor, now popularly called a TAP

(temporal analysis of products) reactor, has been developed that shows promise of being

able to characterize the reaction kinetics at the elementary step level. So both transient

operation and steady-state operation can be carried out to provide experimental data for

microkinetic analysis.

Due to the non-Langmuirian behavior, a coverage-dependence included into the rate

constant of the elementary steps would yield a better agreement between experimental

and calculated data.

Hydrodesulphurisation has been studied by numerous researchers. Most studies have

investigated sulphur compounds dissolved in pure solvents to simulate petroleum

fractions (Daly, 1978; Girgis & Gates, 1991; van Parijs & Froment, 1986b; van Parijs

et al., 1986; Broderick & Gates, 1981; Vanrysselberghe & Froment, 1996; Kilanowski

et al., 1978; Vanrysselberghe et al., 1998). Thiophene is the most studied sulphur

28



Chapter2 Literature Review

compound (Satterfield & Roberts, 1968; Lee & Butt, 1977; Morooka & Hamrin, 1977;

Van Parijs & Froment, 1986b).

2.3.2 Difficulties and challenges

Industrial processes generally involve reactions that consist of several steps and quite

frequently deal with complex feedstocks. Difficulties and challenges arise in the SIDS

model building process accordingly.

Ina

heterogeneouscatalytic reaction, one or more of the reactants

form intermediates

with the catalyst on the surface of the catalyst. These intermediates then take part in

subsequent reactions that result in the final products and the regenerated catalyst.

Understanding of the reaction mechanisms has always been the most difficult task for

model building. This has attracted significant attention, but still lacks prominent

theories. The ability to reliably predict the structure of catalyst or catalyst intermediate

is a very important aspect to the model,design

and catalystimprovement.

The Hougen-Watson approach is generally recommended for expressing rates of

catalytic reaction, which contains the terms that characterise the adsorption of reacting

species rather than power law equations. But it still cannot adequately account for the

interaction of the reacting specieswith the catalyst.

Although the mechanism of HDS with model compounds can be conveniently studied,

they cannot be used effectively to study the interactions between the individual HDS

reactions and the interactions between catalyst preparation, catalyst property, and

catalyst reactivity in the hydroprocessing of real petroleum feedstocks.

Unfortunately, few studies have reported detailed analysis of HDS with real petroleum

feedstocks (e.g.,

light cycle oil) due to the difficulty involved in the interpretation of

experimental results. In most of the real-feedstock studies, data analysis was limited to

considering lumped sulphur (pseudo-component) removal (Ma et al., 1995; Froment et

al., 1994). Moreover the reaction scheme between those pseudo-components is

composed of several typical reactions, regardless of the actual reaction mechanisms.

Very few of these studies dealt with sulphur removal in terms of individual sulphur

29




species (Kabe et al., 1992; Ma et al., 1994,1996), while simple first-order or second-

order power law reaction rate equations were employed.

Catalytic rate-modelling forms an integral part of process development and design, as

well as providing insight into the underlying reaction mechanisms. Complex catalytic

chemistry needs to be explored in terms of the choice of the reaction scheme,

mechanism assumptions and different mixture representations. The effects of different

catalysts are usually unclear with no quantitative correlation available to examine

structure sensitivity. Unfortunately, accurate data fitting cannot provide information for

mechanism discrimination. All of this results in a large number of rival models to be

discriminated between, without any of the assumptions being able to be validated in the

early stages of model development. For example, there are 174 rival models for

Dibenzothiophene HDS (Vanrysselberghe & Froment, 1996) or 15 rival models for

Diesel HDS (Hidalgo, 1999) in the literature.

2.4 Summary

The current methods of model building for complex reaction systems have been

reviewed in this chapter. It can be seen that, although progress has been made in all

aspects of the model building procedure, there is still no systematic methodology to

derive reaction models from detectable speciesbecauseof:

Complex multiple or multi-step reactions

Process chemistry not well understood

Short product market window

" Lack of high reliable computation tools

Theshortcomings of

thestep-by-step

method are summarised asbelow:

" Important information might be missed

" Might lead to inappropriate models

" Measurements carried out under conditions not appropriate for the final

optimised design

30




9 Chemists do not work closely with chemical engineers

0 Might lead to process scale-up failure

0 Detailed models for complex reaction systems are time-consuming to build

Accuracy and efficiency are the two requirements of desired models for the reaction

systems. So it is important to explore all the possibilities and determine important

information to avoid being disguised by minor factors with minimum experimental

measurements. Using process design techniques to instruct the experiment design, to

extract a model suitable for scale-up, is a preferable approach in the model building for

complex reaction systems.

31



Chapter 3 Construction of Reaction Schemes

Chapter 3: Construction of Reaction Schemes

3.1 Introduction

3.2 Feasible reaction generation

3.2.1 Stoichiornetric analysis


3.2.2.1 Reaction species classification

3.2.2.2 Reaction set-up

3.2.2.3 Reaction feasibility: linear programming

3.2.2.4 Stage II reaction generation

3.2.2.5 Simplification

3.3 Reactionscheme construction

3.3.1 Incidence matrix

3.3.2 Feasibility check procedure

3.4 Illustrative example

3.5 Conclusions

32




For the models to describe reaction systems, two aspectsmust be included: the reaction

scheme and associated kinetic models, neither of which can be neglected as shown in

Fig. 3.1. Reactionschemes are considered first, but later the reaction scheme and

kinetics will be optimised simultaneously. In this chapter, a strategy to find appropriate

reaction schemesto represent reaction systems s presentedand discussed.

Model

Reaction scheme: Reaction kinetics:

" Which of the reactions involved " Reaction equation type

" How many reactions involved " Kinetic parameters

Experimental data:

" Reactor type (CSTR, PER, Batch, etc.)

" Raw materials and products

" Operating conditions (T, P, Feeding policy,

etc.)

Fig. 3.1 Reaction model representation

This new strategy, including generating single reaction steps and finding all plausible

reaction schemes, is suitable for complex reaction systems. Feasible reaction generation

is based on a two-stage method. Once all the reacting species are classified into

reactants and products, all possible reactions are set up in a two-stage procedure. Linear

programming is used to check the reaction feasibility.

Construction of reaction schemesprovides all plausible reactions from the combinations

of feasible reactions generated from the two-stage procedure. The algorithm that

searchesall possible reaction step combinations can guaranteethat all plausible reaction

schemesare obtained.

3.1 Introduction

In industrial practice, many reactions of interest have a complex reaction scheme.

Complex reaction systemsgenerally involve more than one reaction step. Using one

33




reaction to describe a reaction system from raw materials to products is in most cases

inadequate for reaction system analysis.

For instance, 2-methyl-6-trifluoromethyl aniline is the intermediate for producing the

widely used speciality agricultural chemical, Herbide (Tremont et al., 1988). It is

converted from 2-methylthiomethyl-6-trifluoromethyl aniline using hydrodesulphuri-

sation. The desired reaction is:

33

F3CJAH2SCH3 H2

H3i

Catalyst

In practice, many other side reactions occur at the sametime, the whole reaction scheme

is given in Fig. 3.2.

NH

3F3C H2 S-CH

H2 FCC

F3C

CH3SH

H2

H2

+

CH3 S

-CH3

Fig. 3.2 Reaction scheme of 2-methyl-6-trifluoromethyl aniline

CH4 + H2S

A reaction scheme is defined to describe a multi-step reaction, containing one or more

reaction steps, which covers all the relationships among the reacting species. The

determination of the reaction scheme is the first step of the model building

methodology, on which kinetics are based. Hence experimental effort is required to

validate the postulated reaction scheme and to obtain the coefficients of reaction

kinetics.

For one existing reaction system, given all stable species involved (as opposed to

unstable intermediates) in the reaction systems, without any information about

34



Chapter 3 Construction of ReactionSchemes

elementary reactions, reaction pathways or reaction mechanisms, this work aims to find

a set of plausible routes to connect raw materials with final products. That is, a reaction

schemeis

composed of routes, any of whichis defined

asone reaction step.

The reaction steps are single stoichiometric reactions instead of elementary reactions,

for example, one single reaction (Doraiswamy, 2001):

CH4 +C'2-> CH3Cl + HCl (3.1)

canbe described by

thefollowing

reaction mechanism:

C12-> 2C1"

Cl "+CH4 -->HCl + CH3 (3.2)

CH3 +C12 -->CH3C1+ Cl

Even though the kinetic equation for every elementary reaction can be written in the

form of power law and the reaction order is the same as the reaction coefficients, free

radicals or active intermediates cannot be detected and measuredeasily. Accordingly,

the reaction kinetics should be derived by the quasi-steady state or rate-determining

step. Furthermore, it is inappropriate for complex reaction systems because the

important insights into reaction systemsmight be disguised by using a complicated and

detailed kinetic model. So, in this thesis, only single stoichiometric reactions are valid

for reaction steps on which reaction kinetics are based. This means using Eq. (3.1)

instead of Eq. (3.2).

Certainly there is more than one reaction scheme that can provide connections between

the same reactants and products. For a complex reaction system, even through the

number of feasible reaction steps s small, the number of combinations of reaction steps

constituting the reaction schemecan be very large. Finding the most appropriate

reaction scheme to describe the reaction system is the aim of model building, but at the

early stage of research, determining which reaction scheme is more appropriate is

impossible without further experimental study. In order to avoid important information

being missed, it is critical to first find all potential reaction schemes,and then validate

them.

35




First of all, the task of the model building methodology is to identify which reactions

and how many reactions can be used to describe the reaction system, and what is the

skeleton of the reaction schemebefore kineticsarecoupled

Candidate reaction scheme construction is done in two stages n this thesis in order to

find all plausible reaction schemes. In the first stage, all possible reaction steps in the

reaction schemes should first be generated. A database of molecular structure

transformations (atom, bond, electron, and functional groups) is good for the study of

reaction mechanisms, but from the standpoint of chemical engineering, stoichiometric

reactions are enough to describe the reaction system, regardless of the reactivity of the

actual molecular structure. So stoichiometry is used to analyse the existing reaction

system.

In the second stage, the reaction scheme is constructed by picking up reaction steps

from generated reactions. In order to guarantee that all plausible reaction schemesare

searched, enumeration and combination methods are used first, then a matrix

transformation procedure checks the feasibility of the reaction schemes.

Oleic acid epoxidation will be used to illustrate how the strategy can find all feasible

reaction schemes through matrix transformations after the two-stage reaction step

generation.

3.2 Feasible reaction generation

In the first stage, the reaction steps should be generated for constructing reactions

schemes. This should cover all possible reactions between stable species.All chemical

change is subject to the law of conservation of mass, including the conservation of the

chemical elements making up the species involved. For any element in a reaction

system, there is a conservation equation stating that the amount of that element is fixed,

no matter how combined or recombined, and regardless of reaction rate or whether

equilibrium is attained.

The conservation of atomic species is commonly expressed in the form of chemical

equations, corresponding to chemical reactions. We refer to the stoichiometric

constraintsexpressed this way as chemical reaction stoichiometry. A simple system is

36




represented by one chemical equation and a complex system by a set of chemical

equations. Determining the number and a feasible set of chemical equations, for a

specified list of species (reactants and products), is the role of chemical reaction

stoichiometry. So the first task of the model development is the identification of simple

stoichiometric reactions one-by-one.

3.2.1 Stoichiometric analysis

When limited information is available from initial experiments on the reaction system,

the best tool for analysing the species involved in the reaction system is chemical

reaction stoichiometry, which can give us basic information about the system.

Knowledge of the stoichiometry ensures that all the reacting species are detected, and

forms a starting point for selection of analytical methods suitable for the study of

reaction kinetics of the reaction system. It also enables the formulation of an initial

hypothesis about the reaction and provides important information regarding the number

of independent reactions, N1nd. n turn, this allows preliminary experiments to be

designed.

Assume there are N reacting species (molecules) in the reaction system, M types of

atoms (or functional groups) consisting of all species, R reaction steps in one reaction

scheme.So chemical equations for a complex system are described as:

AX=o

where A is the atom-molecule matrix,

all a12 ... alN

_

a21 a22 ... a2N

AMxN

aij

aM1

aM2 ... aMN

a,j

is the number of ith atom (functional group) of jth species.

1>0 ith atom exists in jth speciesaý..

=0 otherwise

(3.3)

(3.4)

37




X is the stoichiometric coefficient matrix of one reaction scheme,

x11

X21

NxR

XN1

x12 ... x1R

x22 ... X2R

= Lx1X

Jk

XN2 ... XNR

X2 ... XRI (3.5)

xk is the reaction stoichiometric coefficient column vector for one reaction, xJk

s the

stoichiometric coefficient of jth speciesof kth reaction and is an integer variable.

>0

X1k=<0

=0

for products

for reactants

not involved

For an existing reaction system with little chemical information, the number of reaction

steps R in any candidate reaction scheme is to be determined and varies according to

different schemes. At this moment X remains unknown. The construction of reaction

schemes becomes the construction of matrix X, in which the reaction scheme and

reaction stoichiometric coefficients information are included.

The reaction system analysis starts from the atom-molecule matrix A and useful

information is extracted from the matrix characterisation. The rank of matrix, R, is the

larger number of the independent columns and the independent rows. For any reaction

system, in order to get the reliable rank of an atom-molecule matrix R, the SVD

(Singular Value Decomposition) method is used to calculate the rank (see Appendix A),

because in some cases the atom-molecule matrix might be a singular matrix. So, the

rank of atom-molecule matrix cannot be obtained through the Gaussian elimination

method.

R must be less than or equal to M or N. Thus the number of independent reactions is

gained by (Aris & Mah, 1963):

Njnd <_min(M, N) -R(3.6)

38




This is regarded as the statement of Gibbs's rule of stoichiometry. In the meanwhile, it

provides important information for degree of system freedom. In theory Njndreactions

are enough to describe the reaction system. But in most cases, t doesnot mean only Nand

reactions occur in the reaction system, and also cannot determine which Nindreactions

occur.

In this work, mild is used as an important constraint for constructing a reaction scheme,

which will be described in the section 3.3, to represent the main skeleton of the reaction

system.

For

most complex reaction systems,

it is difficult to describe therelationship

among reacting species and reactions.

As mentioned before, functional groups can be separated in the construction of atom-

molecule matrix. Not only molecular formula but also the isomeric form, the phase in

which the species is present, functional groups and even the charge of ionic species can

be used to distinguish a species if applicable. In this situation, the rank of the atom-

molecule matrix will be increased, accordingly the number of independent reactions Nind

is decreased. It aims to reduce the number of feasible reactions to be used for the

construction of reaction schemes.

For example in the dimethylbenzene oxidation, 1,2-dimethylbenzene can only be

converted to 2-methyl benzoic acid not 3-methyl benzoic acid, so does 1,3-

dimethylbenzene (in Fig. 3.3). If the isomeric form is added into the atom-molecule

matrix, the infeasible reaction reactions will be avoided.

CH3

ul - CH3

CH3

CH3

CH3

COOH

CH3

COOH

0 Feasible reaction

....................nfeasible reaction

Fig. 3.3 Dimethylbenzene oxidation

39




In a reaction system, according to linear algebra, there are infinite solutions for this

linear problem, AX =0 (see Appendix B). In order to obtain all possible solutions that

satisfy reaction constraints and have the smallest values of coefficients, a two-stage

strategy is proposed to guarantee all solutions are explored. This algorithm is basedon

the pretreatment of a stoichiometric coefficient vector in order to make the

mathematical equation have a practical reaction meaning. The sign of the elements in

the vector is determined during the reaction set-up, positive for products, negative for

reactants and zero if not involved. The smallest integer values can be obtained through

linear programming if a feasible solution is available, otherwise the reaction does not

exist.


Given all stable species involved in the system, a strategy is needed to identify single

reactions step-by-step to represent the situations usually encountered inwhich more

than one reaction step is involved. A constraint of feasible reactions is that it should

have an atomic balance.

For the benefit of the reaction scheme construction, a two-stage method is used in the

reaction generation, including reacting species classification, reaction set-up, atomic

balance and generation of reaction list for each stage. The whole framework for the

procedure is shown in Fig. 3.4. From this, the two reaction lists are obtained for the

construction of the reaction scheme.

In any reaction system, regardless of reaction kinetics, reactions whose reactants are

either products or combinations of raw materials and products can only happen only

after reactions whose reactants are purely from the combinations of raw materials. For

example, the reaction scheme in Fig. 3.2, requires the first hydrogenation reaction to

occur for the reaction schemeto exist.

In any reaction scheme, one set of reactions can be obtained whose reactants are

obtained purely from the combinations of raw materials. The other set of reactions

involve reactants that are either products or combinations of raw materials and products.

Accordingly, reactions are classifiedinto

thesetwo stagesbased

on reactants.

40




At least one reaction from Stage I can guaranteethat the reaction schemehas a feasible

start. The two-stage method can also simplify the feasibility checks for the reaction

scheme.

In the construction of a reaction scheme, reactions are selectedfrom the reaction list of

each set. Stage I of the procedure identifies reactions with reactantsinvolving only raw

materials. Stage II of the procedure identifies reactions with reactants involving

products, or combinations of raw materials and products. A subsequent step in the

procedure combines reactions from Stage I and Stage II into feasible reaction schemes.

41

Fig. 3.4 Reaction generation framework




3.2.2.1 Reacting species classification

The proposed two-stage method is based on the classification of the reacting species

into reactants and products. A reaction system consists of N species, represented as

Species Set S, which has two subsets: the Reactant Subset R and the Product Subset

P. R and P are dynamic sets, changing with the stage in the procedure. Eq. (3.7)

gives the relationship between the set and subsets.

S=RUP Where RcS, PcS (3.7)

In any reaction, reactants must be chosen from Reactant Set R, whereas products must

be chosen from Product Set P.

For Stage I, the Reactant Set is Ro= {Sl, S2,""", S,.11which includes all the raw materials

fed to the reactor. Consequently, the Product Set is 1={ ST+,,S+2,..,

SN15 which

includesall the new species generated

nthe reactor.

For Stage I:

S=Ro+Po (3.8)

For Stage II, the Reactant Set is Rl =S and includes all the species.The Product Set is

the same as in Stage I, Pl = Po.No reacting speciesthat are raw materials are allowed to

be produced in any reaction of this two-stage strategy. When determining the reaction

scheme, all reactions are assumed o be irreversible.

3.2.2.2 Reaction set-up

An algorithm for selecting reactants and products from the different sets is used to set

up reactions. There are two issues regarding reaction set-up: whether species are

reactants or products, andthe

number of reactants/products.

In each Stage of the

procedure, reactants selected from Reactant Set R are combined with products selected

from Product Set P to build reactions. Y and 0 are the sets of combinations of

reactants and products.

Vf={(Sl),

(S2),..., (Sl,

`'2),

(S,

)S3/,... 9(S1IS2)S3)...

1 (3.9)

42




O= {(ST+1 )' (ST+2

/, .. ''(

T+1 IST+2 ), (ST+1, ST+3 ),

" .. '

(ST+1'ST+2, ST+3 )... } (3.10)

According to the combinatorial theory, for any set W (n dimensions),

an unorderedchoice of r different elements from a set of W is called an r-combination of W. Note that

an r-combination is just a subsetwith r dimensions.

nWe denote the number of r-combinations of an n-dimension set W byr

n!

r r! (n-r) !(3.11)

So the total number of reactions through the match of combinations of reactants and

products is:

T N-T(3.12)

rr

where N is the total number of reaction species,and T is the number of reactants.

Fig. 3.5 shows the match of reactant and product combinations.

Fig. 3.5 Reaction setup

The reaction set-up is carried out using an index vector I. This allows three different

statusesfor reacting species: reactant,product, or none.

43




ITlul

,u2 IN (3.13)

+1 for productsin which ,uj _ -1 for reactants

0 not involved

In one reaction, any speciescannot be a reactant and a product at the sametime. That is,

any element in the index vector I has a unique value.

For example, an index vector

A B """ F GIT = [-1

-1 """ 1 0]

representsthe following reaction:

A+B--

F

3.2.2.3 Reaction feasibility: linear programming

After obtaining all combinations of reactions, the reactions must be tested to determine

whether they are in stoichiometric balance or not. At the same time, the smallest value

of the sum of absolute reaction coefficients for every combination is chosen.

If any reaction exists in the reaction system, species vectors are linearly dependent.That

is, the atom-species matrix will be singular. Stoichiometric coefficients that are any

bigger than the smallest value of stoichiometric coefficients have the same physical

meaning. The smallest values for the steichiometric coefficients are required for each

reaction. Any method that is chosen to solve this problem must have the ability to match

these two requirements. A linear equation solver and linear programming both can be

used for atom balancing, but only linear programming can easily solve the problem

regardless of singularity, and gives a minimum solution in order to avoid duplicate

reactions. A reaction is feasible only if linear programming can find a solution, then it is

added to the reaction list with the reaction stoichiometric coefficients given by the

solution.

44




A reaction can be described by the following reaction equation:

Ax=O

v is the reaction stoichiometric coefficient vector for one reaction

vT =[ vl v2 ... VNJ

which vj is the non-negative integer variable. So the stoichiometric vector is:

x=I"v

(3.14)

(3.15)

(3.16)

If a combination of reactions is feasible, then Eq. (3.14) must be satisfied and the

relationship between x and I is:

xi>0, ifp=1

xi=0, ifuj =0

xi < 0, if uj = -1

The objective function for linear programming is:

N

Min Z=>, v,

;_l

St. A(I "v) =0

vi - clpjI<_

vý >_

(3.17)

where e is the upper bound of the coefficient value. If a feasible solution exists, then

the values of x=I"v become the column vector of the stoichiometric coefficient matrix

0. The feasible reaction is addedto the reaction list.

45



Chapter 3Construction of Reaction Schemes

3.2.2.4 Stage II reaction generation

After the reaction list of all the feasible reactions in Stage I is complete, the Reactant

Set R for Stage II is updated accordingly (shown in Fig. 3.6). The procedure of reaction

Reaction list 1

Reaction list 2

Fig. 3.6 Changesof reactantand product set

setup and feasibility check in Stage I is repeated for Stage II to complete the reaction

list. Anycombination of reactants in Stage II cannot be the same as reactants of any

reaction in Stage I. So

NT

j>1 µj1 j=l

where =1,2.... T,...N (3.18)

must be satisfied.

When all the feasible reactions have been obtained, reaction lists for the two stagesand

the stoichiometric coefficient matrix 0 are complete.

3.2.2.5 Simplification

If possible, the size of the reaction lists should be reduced by deleting some reactions.

There are several different situations that need to be taken into account, even though

some constraints have already been addedin the previous step. In practice, it is often the

case that reactions with greater than 3 reactants and 3 products are very rare. Also, the

value of the absolute reaction coefficients for many reacting speciesis usually less than

10. Furthermore, specific knowledge that particular reactions cannot take place allows

them to be eliminated.

46




Once any possible simplification of the reaction lists has been carried out, the two

reaction lists are complete. Assuming there are U reactions generated from Stage I and

Vreactions

from StageII, the stoichiometric coefficient matrix 0 is aNx (U + V)

matrix.

I II

al

a2

rl r2 ru : ru+1

O/R

ru+2 "' ru+v

= aT

aT+l

aT+2

aN

OUR

oIIP OIIP

3.3 Reaction scheme construction

For a complex reaction system, the number of possible routes from raw materials to

products is potentially very large; and it is impossible to tell which one is more

appropriate than others. So choosing a reaction schemefor a complex reaction is always

a difficult task, especially for a new reaction process. The most appropriate reaction

scheme can only be identified when combined with kinetic equation analysis, as will be

carried out in the following chapter. With limited preliminary experimental information,

the second step of the reaction scheme construction methodology is to provide all

plausible reaction schemes n order to avoid potential important schemesbeing missed.

The reactions generated in Stage I and Stage II provide useful information about

possible reactions that can be used to describe reaction systems,but it is still difficult to

determine which reaction and how many reactions are involved in the reaction scheme.

In general, the relationship among reaction steps in one reaction scheme can be

described as different types: series reaction, parallel reaction and competitive and

consecutive reactions.

Series reactions:{A

-> R -> S

47




1--Parallel reactions:

ARor

AR

AS B-*S

A+B -> RConsecutive and competitive reactions:

B+R -*S

For more complex reaction systems, the definition is not clear. As there might be no

appropriate type to describe it by, for example the reaction schemesshown in Fig. 3.7.

With respect to different reacting species, a parallel reaction can become a series

reaction.

A""ý

R

yT

S

A<R)T

S/

B

U

A-ºSý V

TW

Fig. 3.7 Complexmonomolecular reaction schemes

If the reaction is a bimolecular reaction, or a trimolecular reaction, the reaction schemes

will be more complicated.

The most important feature of a strategy to solve this problem involves the ability to

generate feasible reaction schemes to include all plausible reaction schemes, and to

provide basic information about the structure of any reaction scheme. A reaction

scheme constructed through randomly choosing reaction steps provides no guarantee

that the reaction scheme is feasible. Also, the total number of all possible reaction

schemes that can describe a reaction system cannot be calculated from existing theory.

Relationships among reaction steps in a complex reaction system cannot be simply

described as consecutive reactions or competitive reactions.

So in this work, the concept of level is introduced to represent the relationships among

reaction stepsin one reaction schemefor reaction schemeconstruction. The first level of

a reaction scheme (Level 1) involves only reactions between the raw materials (and

corresponds with Stage I reactions). The next level of reactions (Level 2) involves the

species generated from the first level. The third level of reactions (Level 3) involves the

species from the previous two levels, and so on.

48



Chapter 3 Constructionof ReactionSchemes

There are two possible extreme Situations that might exist for a possible reaction

scheme: all reactions occur in parallel, or all reactions occur in series. The number of

parallel reactions or series reactions is restricted to the number of independent reactions

Ni,,,d. Correspondingly, in a feasible reaction scheme, the number of reactions selected

from Stage I must be equal to or less than Ni,,d, and also the total levels of any reaction

scheme cannot exceed Nind-

Any feasible reaction scheme is a feasible route to connect raw materials with products

by selecting reaction steps generated in the two-stage strategy. So all the species

involved in the reactions must be either consumed or produced in the reaction scheme.

Only after reactions occur between raw materials, can the whole reaction system be

initiated. The number of reactions selected from Stage I must be at least 1 to guarantee

that one reaction occurs from raw materials. Whether a product of the reaction system

can be a reactant in some reaction steps inside a reaction scheme or not, cannot be

determined bysimple observation

that this product has already been produced in

another reaction step. The possibility exists that two reaction stepsmaybe dependentof

each other without connection to the others. So the level concept can also help to test

the feasibility of reaction schemes.

Reactions in the first level only start from raw materials, so they can be determined

from the reaction list of Stage I. The reactions in the following level depend on the

products of reactions in the previous level. Their reactant set is the sum of the raw

materials set and products from reactions of the previous levels. If there is any reaction

that has not been included in any level, this means this reaction scheme s not feasible,

and it should be deleted.

3.3.1 Incidence matrix

The stoichiometric coefficient matrix 0 not only provides stoichiometric information

on the relationships between reactants and products, but can also be used to

systematically check the feasibility of a reaction scheme.

49




An index vector qp s used to represent a specific reaction scheme,in which elementsof

the vector are integer variables. If reaction k is selected in this reaction scheme, t is set

to be one, otherwise zero.

ýT =[(o

i ý9a (iou CPU+i (Pu+2 ... 4Pu+v (3.19)

So as aNx (U + V) matrix, the reaction step space 0 is divided into four submatrices:

O,R1

011

OR and 0,1p,according to the classification of reacting speciesand different

reaction lists.

rl r2 ru rU+l rU+2 ru+v

al

a2

OIR OIIR

(D

aT

... ... ... ... ... ... ... ... ... ...

aT+l

aT+2

oIIP oIIP

aN

(3.20)

Reaction scheme construction finds the possible routes between reacting species

through the information obtained from the two-stage strategy. The stoichiometric

coefficient matrix 0 can be used as the solution space of feasible reaction schemes.

An incidence matrix in graph theory is used to represent the incidence of labelled

vertices and labelled arcs (Dolan & Aldous, 1993). The value of the matrix element is 1,

-1, or 0, which depends on the orientation of arcs toward vertices. Similarly, the incident

matrix can also be used to describe a reaction system with U+V labelled reactions and N

labelled species. The incidence matrix D of a reaction system is defined as a

Nx (U + V) matrix,

1 if species i is a product in reaction j;

d17_ -1if species i is a reactant in reaction j;

0 otherwise.

50




The incidence matrix D can easily be derived from the stoichiometric coefficient matrix

0 by restricting the elements of the matrix to be no lessthan -1 and no greater than 1.

a,

a2

aT

a T+l

aT+2

aN

+1 ojk >0

where djk = -1 oak<0

0 ojk=0

d1 dj1

(3.21)

The incidence matrix no longer represents the molar balance within the stoichiometric

equations, but the connectivity between certain reactants and certain products.

The incidence matrix D is divided into 4 submatrices, d.,

d,,m,

d,,p and d1p

according to the classification of reacting species and reaction lists. Through matrix

transformations, it aims to:

1. Calculate the sum of row elements. This should be greater than 0 for products

and less than 0 for raw materials for a feasible reaction scheme.

2. Make the values of all elements in submatrix d1p greater than 0 through

addition of a reaction column in the previous level to those in the remaining

reactions.

3. Classify the reactions into different levels.

If any of the three cannot be satisfied, this is not a feasible reaction scheme.

rl r2 ru ru+l ru+2 ru

dIR duR

+v

51



Chapter 3 Constructionof Reaction Schemes

3.3.2 Feasibility check procedure

Before the feasibility check of reaction schemes s carried out, it must be ensured that

any reaction step cannot be the reverse reaction of any other reaction steps in one

reaction scheme. It aims to avoid the duplication of reaction reversibility when kinetic

expressions are combined. In order to test the feasibility of a reaction scheme,the basic

procedures and constraints are summarised asthe following 7 steps:

1. Generate the incidence matrix D from the stoichiometric coefficient matrix (D.

2. Modify the matrix D based on any specific reaction scheme. From this, D(l) is

obtained:

djk(l) = dJk (Pk (3.22)

3. Start from any reaction in Stage I, whose column is not equal to 0, add to every

reaction in Stage II one by one.

4. If all the elements of any addition column in sub-matrix d11 are equal to or greater

than 0, then this reaction is listed following all the reactions in StageI. It belongs to

Level 2. If there is no reaction that can be classified into Level 2, this reaction

scheme is not feasible and therefore stops. Otherwise the incidence matrix is

updated and marked as D(2), where the superscript represents the level, then

continue.

5. If there is any reaction whose elements in sub-matrix dj1Pare still less than 0, start

from any reaction in Level 2 and repeat Steps 3 and 4. The mark of the incidence

matrix is changed accordingly, and continue the procedure.

6. If the total level reaches Nid, but there are still some reactions outside the level,

delete this reaction scheme.

7. If all the reactions are classified in Level N< Nind, and the following 3 basic rules

are satisfied, this reaction scheme s feasible.

52



Chapter3 Construction of ReactionSchemes

i)

ii)

iii

The total number of reactions selectedfrom Stage I is less than the number

of independent reactions, and more than 1.

U

1_<L cpj

<_ Nina

;_l

(3.23)

Every species in the raw materials must be consumed at least once in the

reaction scheme,except catalyst, solvent or diluent.

U+V

D(L) <_-1, Vj =1,2,..., T (3.24)

Every speciesexcept raw materials must be produced by the reaction step in

the reaction scheme.

U+V

D(L) >_1, Vj=T+1, T+2,..., N

j=ý

(3.25)

Because the elements of the index vector combined with the incidence matrix can only

represent one orientation of any reaction step, it is straightforward to represent

irreversible reaction schemesbut not for reversible reactions. In order to avoid affecting

the feasibility check of reaction schemes, missing the possibility of the reaction

reversibilityand causing the duplication of representations, the reaction reversibility

will be represented through kinetic expressions later in the procedure. This will be

discussed n the following chapter.

In addition, some empirical constraints can also be used, according to the specific

system. For instance, some products cannot appear as reactants in reactions, or some

reactions cannot appear in the same reaction scheme. The procedure will be

demonstrated using an example.

3.4 Illustrative example

Oleic acid epoxide is the desired product of the net reaction between oxygen,

benzaldehyde and oleic acid, dissolved in acetone as solvent. Through the chemical

53



Chapter3 Constructionof ReactionSchemes

analysis, all the species, including by-products and intermediates involved in the

reaction system are:

A: benzaldehyde, C6H5CHO

B: oxygen, 02

C: oleic acid, CH3(CH2)7-HC=CH-(CH2)7-COOH

D: perbenzoic acid, C6H5CO3H

E: benzoic acid, C6H5000H

F: oleic acid epoxide, CH3(CH2)7-HC-O-CH-(CH2)7-COOH

A detailed model for oleic acid epoxidation in terms of a free radical mechanism

consisting of 11 elementary reactions was developed by Kuo and Chou (1987) from

initial rate data. Here it will be assumed that any intrinsic mechanism and kinetic

information is unavailable. Without a model, reactor design and optimisation cannot be

carried out. So a reaction model that can interpret experimental data well and can be

usedfor reactor design and optimisation must be obtained.

The atom-molecule matrix for oleic acid epoxidation is:

A B C D E F

M1 0 0 1 0 0 1

M2 0 0 1 0 0 1

AC 7 0 3 7 7 3

H 6 0 3 6 6 3

0 1 2 2 3 2 3

in which M1 representsCH3(CH2)7,M2 represents(CH2)7.

After the SVD calculation from the atom-molecule matrix A, the rank of the matrix is 3,

so the number of independent reactions is NNnd=6-3=3.

For oleic acid epoxidation, the Reactant Set and the Product Set for Stage I are

Ro ={A, B, C} and Po=

{D, E, F} respectively. The reaction is set up from the

combinations of Ro and Po.

Reaction list:

54




AHD

A-E

A<->F

StageI: AHD +E

At->E+F

At->D+F

After checking the stoichiometric feasibility of all reactions using integer linear

programming, only three are feasible. The third reaction is stoichiometri cally balanced,

but is impossible in practice from knowledge of chemical thermodynamics, and is

deleted.

1 (A+B=D1 A+B=D

2 2A+B=E2 2A+B=E

3B+ 2C = 2F

In Stage II the same procedure for the reaction step-up and feasibility check is carried

out after updating the Reactant Set and the Product Set. The reaction list from StageII

includes three reaction steps:

3 C+. D=E+F

4 A+D=2E5 B+2E=2D

The stoichiometric coefficient matrix for this reaction systemis:

rl r2 r3 r4 r5

A -1 -2 0 -1 0

B-1 -1

0 0 -1

C 0 0 -1 0 0

D 1 0 -1 -1 2

E 0 1 1 2 -2

F 0 0 1 0 0

55




Accordingly, the incidence matrix is generated from the stoichiometric coefficient

matrix by restricting the elements to be -1: - x<<-1:

rl r2 r3 r4 r5

A -1 -1 0 -1 0

B -1 -1 0 0 -1

C 0 0 -1 0 0D=

... ... ... ... ... ... ...

D 1 0 -1 -1 1

E 0 1 1 1 -1

F 0 0 1 0 0

In this system, the number of possible combinations of reaction steps is 21. After

feasibility check of reaction schemes, only 10 reaction schemes remain feasible.

Now three reaction schemes are selectedto demonstrate how they are determined to be

either feasible or infeasible. The level of the reaction system is shown using matrix

transformation. Finally the reaction scheme s drawn as a flowsheet.

Reaction scheme 1:

rpT=[1 011 0]

rl r2 r3 r4 r5

A

-1

00

-1

0

B -1 0000

C00 -1 00

... ... ... ... ... ... ...

D1 011 0

E00110

F00100

A

B

I>D(2)=C

D

E

F

rl r2 rl + r3 rl + r4 r5

-10 -1 -2 0

-10 -1 -1 0

00 -1 00-1

...

1

... ...

0

...

0

...

0

...

0 1

0 0 1 1 0 2

0 0 1 0 0 1

In the original form of the incidence matrix D(1), two elements of sub-matrix d11Pare

less than 0 (highlighted by circles). So add column rl to column r3 and add column rl

to column r4. Check the value of all elements in sub-matrix d11.There is no element

whose value is less than 0, so stop. Check the sum of any reacting speciesbelonging to

R0, its value is less than 0 and the sum of any speciesbelonging to Po is greater than 0

(right side of the matrix). For a reaction scheme to be feasible, all reactants mustbe

56




consumed, i.e. the sum must be negative, and all new products generated, i.e. the sum

must be positive (from the summation on the right of matrix). So this reaction scheme s

feasibleand

thenumber of

levelsof the reaction scheme is 2. This reaction schemeis

also obtained by Rastogi (1990).

Reaction scheme 2:

(OT=[l 011 1]

rl r2: r3 r4 r5 rl r2 rl + r3 rl + r4 rs

A -1 0 0 -1 0 A -1 0 -1 -1 0

-1 0 0 0 -1 B -1 0 -1 -1 -10 0 -1 00

C*D(2)

-C

...

0

---

0 -1

-. .:....

0

... ...

0

...D 1 0 (9 01

D 1 00 0 1

E 0 0 1 1@ E 0 01 1:G

F 0 0 1 00 F 0 01 0 0

rl r2 rl + r3 rl + r4 rl +r4+r5

A -1 0 -1 -1 -1 _4B -1 0 -1 -1 -2 -5

Dý3ý =

C 00 TILL 0 0_1

D 10 0 0 : 1 2

E 00 1 1 0 2

F 00 1 0 0

1

In the original form of the incidence matrix D"), three elements of sub-matrix d11Pare

less than 0 (highlighted by circles). So add column rl to column r3, and column r4 and

column r5 .Check the value of al elements in sub-matrix d11P

.There is still one

element whose value is less than 0, so put reaction r3 and reaction r4 nto Level 2.

Continue to add column r3 to column r5. There is no element whose value is less than

0, so stop. Check the sum of any reacting species belonging to Ro that its value is less

than 0 and the sum of any reacting speciesbelonging to Po is greater than 0. The criteria

are satisfied, so this reaction scheme is feasible and the number of levels of the reaction

scheme is 3.

57




Reaction scheme 3:

(pT =[0 111 0]

rl r2 : r3 r4 r5

A0-1 0

-1 0

B0 -1 000

Df\-C00 -1 00

... ... ... ... ... ... ...

D00ýJ] G0

E01110

F00100

rl r2 r3 r4 r5

A 0 -1 0 -1 0

B 0 -1 000

D(2)-C

00 -1 00_1

D 00QG

0

E 01110 3

F 00100 1

In the original form of the incidence matrix D(1),

two elements of sub-matrix d11pare

less than 0 (highlighted by circles). So add column r2 to column r3 and column r4.

Checking the values of the elements in sub-matrix d11,there are still two elements less

feasible species/reactic

Reaction scheme 1

""""""""""""""nfeasible species/reactions

Reaction scheme 3

Fig. 3.8 Reaction schemeexamples

than 0, so this is an infeasible reaction scheme. Because the reacting speciesD is not

produced in the first level, then it cannot be a reactant in the following level.

58

Reaction scheme 2



Chapter 3 Constructionof Reaction Schemes

Finally, the reaction schemes are drawn in the form of flowsheet in Fig. 3.8 to clearly

show the feasibility of reaction schemes.Dotted lines in reaction scheme3 representthe

impossible appearanceof reacting species, hen reactions.

3.5 Conclusions

Conventional methodologies of reactor design and optimisation for speciality chemicals

applied to complex reaction systems lead to the process being operated under non-

optimal conditions. A new systematic method to combine the work of chemists and

chemical engineers is necessary,to make full use of experimental information, to extract

an optimal model suitable for process design and with the minimum number of

experiments to reduce the expense of laboratory and pilot experiments.

In this chapter, a new methodology of reaction scheme construction is proposed. After

all detectable and stable species (reactants, products and reaction intermediates) are

known, a matrix formulation allows all participating atomic and molecular species o be

systematically represented. The rank of the matrix as a very important factor and is

calculated through the SVD (singular value decomposition) method to obtain the

number of independent reactions for a specific reaction system.

Integer linear programming is used to test the stoichiometric feasibility of the reactions

generated through random combinations of reactantsand products in the reaction set-up

step. Each feasible reaction is entered into the reaction list. A reaction stoichiometric

coefficient matrix is built from the two-stage reaction list, which is used to provide

guidance for generating all possible reaction schemes.

An incidence matrix derived from the reaction stoichiometric coefficient matrix is used

to test the feasibility of combination of reaction stepsby following a 7-step procedure.

At the same time, the levels of reaction scheme are obtained. The whole strategy can

guarantee all feasible reaction schemesare generated.

59



Chapter 4 Hybrid Optimisation of Rival Models

Chapter 4: Hybrid Optimisation of Rival Models

4.1 Introduction

4.2 Problem description

4.2.1 Preliminary experimental design

4.2.2 Data fitting

4.2.3 Reaction system modelling

4.3 Rival models

4.3.1 Optimisation framework

4.3.2 Objective function

4.3.3 Hybrid optimisation

4.3.3.1 Simulated Annealing (SA)

4.3.3.2 Nonlinear Programming (NLP)

4.3.4 Implementation of the optimisation

4.3.4.1 Simulated Annealing framework

4.3.4.2 Optimisation moves

4.3.4.3 Parameterprecision

improvement

4.4 Illustrative cases

4.4.1 Base case

4.4.1.1 Reaction list generation

4.4.1.2 Parameterestimation

4.4.1.3 Results and analysis

4.4.2 Oleic acid epoxidation reaction system

4.4.2.1 Model building

4.4.3 Discussion

4.5 Conclusions

60



Chapter4 Hybrid Optimisation of Rival Models

In this chapter, a new approach is discussed using a hybrid optimisation method to

obtain a set of reaction schemes and kinetic expressions as rival models for model

discrimination. This will efficiently make use of experimental information.

4.1 Introduction

As a complete model for describing the reaction system, it includes two aspects: the

reaction scheme and associated kinetic expressions. In chapter 3, the strategy for the

reaction scheme construction was discussed in detail. In this chapter, rival models for

the same reaction system will be derived as a part of the model building procedure.

Stoichiometric

analysis

Reaction scheme andkinetic rival models

Model discrimination

and improvement of

model accuracy

Model Derivation

Model Discrimination

Fig. 4.1 Generalmodel building framework

Further

experimentaldesign

For a complex reaction system, the number of possible reaction schemes is very large.

Also it is impossible to tell which one is a better representation than another without any

information from experimental data and kinetics. The experimental measurementsmust

go hand in hand with the model building (Horak & Pasek, 1978) to make use of

experimental information efficiently and to reach the target for minimum experiments.

As shown in Fig. 4.1, preliminary experiments are to provide information that is

essential for rival model derivation.

Chemical kinetics consists of a set of equations to describe the dynamics of reaction

systems, whose parameters are estimated from data fitting. For most industrially

Preliminary

experiments

61



4Hybrid Optimisation of Rival Models

relevant reactions, the kinetic parameters cannot be estimated reliably from theory and

must be determined by experiments.

In practice, the kinetic description is still amongst the most difficult tasks in the modelbuilding procedure. In particular, detailed models for complex reaction systemsconfront

the following difficulties: the reaction mechanism remains unknown or too many

assumptions; a large number of reacting species and possible reaction steps are

involved; there is a large number of possible reaction schemes;a large number of model

parametersneed to be estimated. If the reaction system involves heterogeneouscatalytic

reactions, there are also a large number of variables available to represent the external

and internal diffusion effects.

Becauseof the complexity of chemical reaction processesand the short market window

of chemical products, there is always a compromise between the desire to spend a lot of

time to study the detailed mechanism of complex reactions and the requirement that an

approximate model should have ability to represent the key features of the reaction

system to avoid major opportunities being missed. Hence, it is necessary to find

approximate models for these complex reaction systems by the fast and effective

evaluation of experimental data, to shorten development time to market.

With limited preliminary experimental data, there are many applications in which an

experiment can conform to more than two models within a certain confidence level. So

no reaction model can be determined without further experimental validation. At the

early stage of reaction model building, a premature decision on which reaction is the

best model might lead to the wrong research direction and process scale-up failure. In

order to obtain the most reliable reaction model, it is appropriate to explore all potential

models that have good agreement with preliminary experimental data. It is essential that

themethod selected

for data fitting has the ability to provide multiple solutions as rival

models, including reaction scheme and kinetic expressions.

4.2 Problem Description

Kinetic information can be obtained from experimental data fitting directly. It will also

provide useful information to help to screen rival reaction schemes.For some reaction

62




schemes, their kinetic models might not be in good agreement with experimental data

within the range of feasible kinetic parameters and thermodynamic constraints.

4.2.1 Preliminary experimental design

The primary objective of the preliminary experiments is to collect enough data to begin

the model identification and parameter estimation procedure. It is also desired to

determine which variables have a significant effect on the responseof the system, and

what are the ranges of these variables that will give the most desirable value of

response. The results from this stage of experimentation are expected to provide a

qualitative insight into the system behaviour. For example results from the initial

experimentation may help in deciding upon an appropriate optimisation variable, or may

provide valuable information on the kinetics of the system.

Many techniques are available for the design of preliminary experiments, e.g. the

classicalone-variable-at-a-time approach, factorial design and nonlinear/optimal

sequential design. A detailed discussion of the methodology and relative advantagesor

disadvantages of the various approachescan be found in the many books (Box et al.,

1978; Cochran & Cox, 1957; Montgomery, 2000) on the design of experiments. The

nonlinear and optimal design techniques (Atkinson & Donev, 1992) require knowledge

of the model relating the response to the operating variables, and approximate

knowledgeof the model parameters.

Sinceat

thestage of

thepreliminary

experiments,

no clear knowledge of kinetics and thermodynamics is available, the commonly used

method of factorial design is usually the best choice. It is more efficient for studying the

effects of two or more factors than one-variable-at-a-time experiments.

The method of factorial experimental design forces the data to be orthogonal and allows

us to determine the relative importance of each input variable to be determined and thus

to develop a parsimonious model. Factorial experiments also represent efficient

experimentation. All the variables are changed simultaneously, rather than one at a time,

so the number of experiments needed is reduced. Becauseof the orthogonality property

of the factorial design, coefficient estimates have a lower variance than can be obtained

with a nonorthogonal experimental design (Box et al., 1978).

63




From a practical standpoint, the decision on which input variables should be studied will

determine the number of tests that must be carried out. In a standard factorial design, 2°

tests are required, where n is the number of input variables to be studied. The

determination of the region of experimentation requires process knowledge. The

experimental range should be chosen so that the resulting measurementsof the response

do not involve errors in the sensors hat are greater than the typical noise level.

After the experimental design method is chosen, variables expected to influence the

response (yield, selectivity, conversion) must be analysed carefully. For a reaction

system, temperature, initial concentrations, pressure, catalyst concentration etc. all can

be factors.

4.2.2 Data fitting

Once the experimental data are available, extracting as much information as possible

from the limited data for the reaction scheme and reaction kinetics is the main aim for

data regression. Finding the model that closely matches the experimental data becomes

an optimisation problem.

In order to measure or evaluate the agreement between the data and the model with a

particular set of parameters, an estimator needsto be chosen or designedin which small

values represent close agreement. The parameters of the model are then adjusted to

achieve a minimum in the estimator, yielding best-fit parameters. The adjustment

process is thus a problem in minimisation in many dimensions.

The most commonly used estimator is the least square, which is based on the principle

of the maximum likelihood, assuming the measurement errors are independent and

normally distributed with constant standarddeviation.

In general, kinetic experimental data can be interpreted in several ways. There are some

conventional methods of estimating kinetic coefficients for the kinetic model from

experimental data: analytical (or numerical) integration of the set of differential

equations, or differentiation of the empirical data directly. Both are straightforward to

apply to a single reaction system.

64




For a complex reaction system, the reaction model most likely dependsnonlinearly on

model parameters, while the number of parameters to be estimated increases very

rapidly with the complexity of the physico-chemical model. The application of iterative

non-linear least-squares regression techniques is necessary.The Levenberg-Marquardt

method (also called the Marquardt method) works very well in practice and has become

the standard for nonlinear least-squaresroutines.

The nature of the problem of obtaining a set of reaction schemes rom a large number of

feasible reaction schemes and rival kinetic models is aM NLP problem. The approach

used to solve this problem should have the ability to cope with the selection of the

reaction scheme and kinetic expressions simultaneously. For a complex reaction system,

the problem size will be increased, thus the approach should be able to solve a large

scale MINLP also. Furthermore, initial guesses or a large number of parameters will

present difficulties to the optimisation. In addition, multiple solutions are required to

represent the phenomenon that more than one model can fit the sameexperimental data

within a certain confidence level, due to experimental errors.

The drawback with deterministic methods is that they are only able to find the solution

in the neighbourhood of starting point. The solution depends on good initial value for

starting point, so it is very easy to be trapped in the local optimum. If a better starting

point can be obtained using other methods, the result will be more accurate.

Deterministic methods have difficulties in handling such a complex MINLP problem. It

is necessary to use a robust estimation method, allowing the whole parameter spaceto

be explored in order to determine multiple solutions to the problem, and not merely a

single solution. In this work, a hybrid optimisation is chosen, which allows the search

through large parameter spaces.This is able to converge including the correlation of rate

parameters and provides multiple solutions.

4.2.3 Reaction system modelling

In order to obtain the model prediction on concentration, mass balancesof all reaction

species in the system should be carried out. The rate of disappearanceor formation of

component A, - rA or rA depends on temperature and composition. In general, when

65




one species in a specific reaction is selected, its concentration is monitored as a function

of time or flowrate, depending on whether the reactor is batch or continuous.

Commonly, two kinds of kinetic equations are used: power and hyperbolic equations.

For instance, the reaction rate of the following reaction:

A -> products

may be a linear function of the speciesconcentration,

- rA = kCA (4.1)

or a general form of the algebraic equation, such as:

- rA = kACaCB....-k'

CSCT.... (4.2)

The second general form is hyperbolic:

kACACB....-r= (4.3)A 1+kACÄCB....

which are determined from experimental observation. k is termed as the reaction rate

constant. These kinetic expressions can be used to represent both irreversible and

reversible reactions, determined by the appearanceof the reversible term.

The temperature effect is accounted for within k,

k=A exp(-

Ea

)RT(4.4)

where A is the frequency factor, Ea is the activation energy, R is the gas constant, and T

is the temperature.

It is often necessary to convert the rate of reaction of a species o that of another.Thus a

stoichiometric reaction can provide the relationship between reaction rates of all

species. Considering the general reaction:

VAA+VBB--4VCC+VDD

it can be easily written:

-RA - RB=Rc =RD=

VA VB Vc VD(4.5)

66




where r is the extent of this reaction.

In complex chemical reaction systems, typically multiple reactions will occur, some

desired and some undesired. The net rate of formation of each species s required for the

massbalance. If R reactions are taking place

Reaction 1: A+B k' > 3C+D

Reaction 2: A+ 2C k2 > 3E

Reaction 3: 2B + 3E k3 4F

Reaction R: A+ 1/2B -kR

)G

Then the net rates of reaction of all speciesare

RA =( -rl )+( -r2 )+ (-r3 )

RB =(-rl )+(-2r3 )+(-1/2rR )

Rc =(3r1)+(-2r2 )

General chemical equations for a complex reaction systemare given by:

xkAi =0k=1,2,..., R (4.6)

The net rate of reaction for speciesAi is the sum of all rates of reaction in which species

Al appears,that is:

R

RA; = xik rk

k=1

(4.7)

All reaction systems, batch or continuous, should satisfy the mass conservation law. A

mole balance for speciesAt at any instant in time, t, yields the following equation:

In-

Out + Generation = Accumulation

That is, for a homogeneous reaction system:

Fa,'in-F..

+ RA; V_dN4dt

(4.8)

67



Chapter 4Hybrid Optimisation of Rival Models

For batch reactor FAQ = FA = 0, so the general mole balance equation becomes:

RA,dV =

dNA,(4.9)

dt

For a CSTR operated at steady statedN`

A.=0, RA. V = VRA,

it becomes:dt

f

`ý

FA° - FA'(4.10)

RAi

Similarly, for PFR the mole balanceequation is:

dFA,

dVRAj (4.11)

When the mole balance equations are combined with reaction rate equations, a set of

ordinary differential equations (ODEs) is obtained. Also, masstransfer and heat transfer

equations are coupled, leading to an ODE/algebraic, initial value problem (IVP).

There are several methods for solving ODE problems. The fourth-order Runge-Kutta

method is most often used. Before any method is used, system analysis must be done.

For a fixed reaction scheme and associatedkinetic equations, the ODEs present either a

stiff problem, or a non-stiff problem. When a stiff problem is integrated in some

interval, such as [0, b], the stepsize needed to maintain stability is much smaller than

that needed to meet the accuracy requirements. This is a criterion to test if the problemis stiff or not. So the integration method should be chosenaccordingly.

In this research work, the reaction scheme and kinetic equations are moved randomly

during the optimisation, so sometimes the ODEs are non-stiff, at other times they are

stiff, and the method must switch with the construction of the ODEs. Also, it is possible

that in some situations, the ODEs are ill posed, which should be deleted to avoid

inaccurate physical meaning. The method initially assumes he system of ODEs to be

non-stiff and uses the fourth-order Runge-Kutta method for integration. If this does not

allow the required numerical accuracy to be achieved then the integration switches to a

method suitable for stiff systems, in this case a backward difference method. The

calculation procedure is describedin Fig. 4.2.

68




4.3 Rival Models

There are many applications in which an experiment can conform to more than two

known probability models, all within the same sample space.

For instance, maxima on concentration versus reaction time plots are a typical feature of

consecutive reactions. In reverse, the appearance of a maximum might not only indicate

that (Horak & Pasek, 1978) the system of reactions taking place involves consecutive

reactions, or consecutive reactions as a part of a more complex reaction scheme, but

also some combinations of reversible and irreversible, slow and fast reactions.

The two reaction schemes shown in Fig. 4.3 have the same features, and fit

experimental data well within a certain confidence level. Without further information, it

is impossible to select one reaction model to describethe reaction system.

Through an atomic balance, the number of all possible reactions is larger than the

number of independent reactions. Selection of reactions to construct a reaction scheme

69

Fig. 4.2 Simulation method framework



ter 4Hybrid 0 n of Rival Models

Model 1 Model 2

A+ B---> C A+ B<-4 CA+ C----> D

{A+

B-ý D

(i = k, Cý a2C 73CC3i = k1Cq'Cgz

-k- ýýZ = k2`"f+1C k/, )

r2 = k2CA CC 2

80" Exp. Data

j---Model 1

60 Modelt

'

:

/0246

Time

Fig. 4.3 Data fitting curve

can be done in two ways, sequentially or simultaneously. For example, in a 6-species

reaction system, the number of possible reactions is 7, but the number of reaction

schemes exceeds 20, and every scheme includes more than 3 single reaction steps. Then

data fitting can be used for the parameter estimation to obtain kinetics for all reaction

schemes, so the model space is too big for the next step of model discrimination.

The new approach uses an optimisation method for reaction scheme construction and

selecting a kinetic expression, while obtaining a set of suitable schemes and kinetics as

rival models. The stochastic method enables consideration of complex non-linear

formulations, closed or sequential simulation models, with many decisions addressed as

stochastic variables. The first advantage is that a group of solutions is obtained near the

optimal region.

4.3.1 Optimisation framework

After the reaction system is formulated as an MJNLP problem, hybrid optimisation is

carried out in a two-stage optimisation scheme, shown in Fig. 4.4, in order to obtain a

set of rival reaction models.

70




Problem Statement

MINLP Problem

StochasticOptimisation

NLP Optimisation

Fig. 4.4 Optimisation method

The initial stage uses stochastic optimisation techniques in the form of simulated

annealing, which guarantees the optimum performance of the final solution within a

certain confidence level.

The second stage makes use of mathematical programming techniques to fine tune the

solution obtained from stochastic optimisation by varying the continuous variables.

4.3.2 Objective function

In order to fit the model to experimental data, a non-linear multi-parameter estimation

approach is followed. Thus all parameters (i.e. frequency factors, activation energies,

reaction orders with respect to reacting species)are evaluated simultaneously.

For a reaction system, the objective function of data fitting needs to evaluate the

difference between measured and predicted concentrations for the different species.

These different species might be at high or low concentrations. Yet they must somehow

be combined to give a single objective function.

Suppose we have N data points yi, i=1,2, ... N, to a model that has M adjustable

parameters ai ,i=1,2, ...M. The model predicts a functional relationship between the

independent variables and dependentvariables:

yi =yj xi;at,a2,...N) (4.12)

71




The data have a different precision due to the different analysis methods. This would

normally be quantified by an objective function of the type of general least squares

(GLS) :

Min a2 =jjjk,

j,k(eXPtl)-Yi, j, k(calc)p(4.13)

i=1 j=1 k=1

or

Min 62 =y,,fik

(exp tl) - yi jk(calc)

2

(4.14)

i=i j=i k=1 yi, j,k(exp tl )

where the indices i, j, k relate to experimental runs, samples and reacting species, and

yt.j_kis the concentration of the kth sample of jth experiment in ith experimental

condition.

However, the number of reaction steps in one reaction scheme used to describe a

specific reaction systemis

anotherimportant factor for

modelbuilding. More

reaction

steps mean more equations need to be solved and pose difficulties in the simulation and

control of the system.

A new objective function is proposed to account in thesefactors:

EE Y-[Yi,

i, k(exptl )-yi,l, k(calc )ý

Min F= N`=''='k=l _1(4.15)

where N is the number of stoichiometric equations used in the model.

Thus, the new objective function minimises the difference between the measured and

predicted concentrations and, at the sametime, reducesthe complexity of the model.

4.3.3 Hybrid optimisation

4.3.3.1 Simulate Annealing (SA)

Simulated annealing was introduced by Kirkpatrick et al. (1983) and is a generalisation

of a Monte Carlo method that exploits an analogy between the way in which a metal

cools and freezes into minimum energy crystalline structure.

72




In an annealing process, a melt, initially at high temperature and disordered, is slowly

cooled so that the system at any time is approximate in thermodynamic equilibrium. As

cooling proceeds, the system becomes more ordered and approachesa "frozen" ground

state at T=0. Hence the process can be thought of as an adiabatic approach to the

lowest energy state. If the initial temperature of the systemis too low or cooling is done

insufficiently slowly the system may become quenched, forming defects or freezing out

in metastable states(i.e. trapped in a local minimum energy state).

By analogy the generalisation of this Monte Carlo approach to combinatorial problems

is straightforward (Kirkpatrick et al., 1983; Cerny, 1985). The current state of the

thermodynamic system is analogous to the current solution to the combinatorial

problem. The energy equation for the thermodynamic system is analogous to the

objective function, and the ground state is analogous to the global minimum.

The direct precursor to the SA algorithm was the Metropolis algorithm (Metropolis et

al., 1953). The Metropolis algorithm moves from an arbitrary point (with energy E! ) to

the next state (with energy E1) and finds the equilibrium energy change (Ej - E; ) at

that state. Whether moves in state spaceare to be accepted or rejected dependson the

probability criterion provided by the Metropolis algorithm.

If the change in energy is negative the new state is accepted mmediately. If the change

in energy is positive, it is acceptedwith a probability:

E. -E.pi,; (T) = min 1, exp(-

kT`) (4.16)

where T is the temperature of the system, k is the Boltzmann constant, E is the energy

value, and i, j are the index of old and new states,respectively.

The Markov chain theory provides the essential background for the development,

behaviour and convergence of Monte Carlo based algorithms. A Markov Process is a

stochastic sequence of states, where the probability of the outcome of a given state

depends only on the outcome of the previous state.

73




Simulated Annealing is a Monte Carlo method that can be modelled mathematically

using the theory of finite Markov chains. In the case of SA, a state corresponds to a

move, and the set of outcomes is given by a finite set of neighbouring states.Each move

depends only on the outcome of the previous attempt, so the concept of Markov chains

applies. Since the number of the statesdoesnot affect the probabilities, this feature will

help to remove the effects of initial guesses.

This process is then repeated sufficient times to give good sampling statistics for the

current temperature, and then the temperature is decreased ollowing a cooling schedule

until the final temperature is reached.

To govern the convergence of the algorithm, a set of parameters,known as a cooling

schedule, are specified by:

" an initial value of the control parameter (i.e. temperature)

"a finite length of eachhomogeneousMarkov chain

"a final value of the control parameter or a termination criterion

"a decrement function for lowering the value of the control parameter.

Determining these parameters is a challenge for implementing Simulated Annealing.

There has been much research on the topic, dealing mostly with heuristic schedules.

There are two main categories of heuristic schedules: static and dynamic. In a static

cooling schedule, the parameters are fixed and cannot be changedduring the execution

of the algorithm. With a dynamic cooling schedule, the parameters are adaptively

changed during the execution.

A suitable initial temperature is one that can make the average acceptanceprobability

about 0. Thus, the value of the initial temperature will dependon the specific problem.

The Markov chain length Lk is a very important parameter for stochastic optimisation

implementation. if it is too large, the optimisation is very slow, as many simulations

need to be performed at each annealing temperature level. If it is too small, the system

will be "quenched" and no reliable results will be obtained.

74




In some implementations of the SA algorithm, the final temperature is determined by no

improvement (i.e. no new best solution) being found at this temperature or the

acceptanceratio falling below a given smallvalue.

There are two widely used cooling schedules:

1. The simplest and most common temperature decrement rule is:

tk+, = a' tk k =0,1,2.... (4.17)

where a is a constant close to, but smaller than 1. This exponential cooling scheme

(ECS) was first proposed by Kirkpatrick et al. (1983), a=0.95.

2. Another cooling schedule is the Linear Cooling Scheme (LCS) in which tk is

reduced in every trial (while avoiding negative temperatures):

tk = tk - AT k=0,1,2,... (4.18)

Randelman & Grest (1986) found the reductions achieved using the two schedules o be

comparable, and also noted that the objective function was, in general, improved with

slower cooling rates at the expense of greater computational effort. Finally, they

observed that the algorithm performance dependedmore on the cooling rate than on the

individual values of control parameter and Markov chain length.

Some general guidelines exist when choosing an annealing schedule.For instance, there

is a trade-off between choosing small decrements of the control parameter (tk) and

choosing long Markov chain lengths (Lk ). Usually small decrements of the control

parameter (tk) are chosen to avoid long chains. Apparently, when a sufficiently long

schedule is used, simulated annealing replacesiterative improvement

asthe

optimal

schedule.

Aarts and van Laarhoven (1985) suggestedthat the annealing temperature is updated

according to:

ln(1+ y)tktk+l

-tk '

(1 +)

3ý(tk

(4.19)

75




variables are unlikely to reach the exact optimal points, due to the discrete nature of the

moves.

Optimisation tools start from an initial guess, search through the solution space, and

converge to an "OPTIMAL" state. Unlike deterministic methods, where gradient

information is used to evolve the search, in stochastic optimisation the moves are

random. In process synthesis, each stochastic move outputs a new design instance,

which is assessedto guide the evolution of the search. The assessment calls for

simulation of the design and calculation of the overall performance. The search

generates a chain of design instances, thus the programming effort is minimised to a

seriesof simulation tasks.The stochastic evolution enablesthe consideration of:

" Complex non-linear formulations

" Closed or sequential simulation models

" Many decisions addressedas stochastic variables

"A set of solutions close to targets

The convergence properties of the stochastic searchdepend on the strategies adopted to

guide the evolution of the process. One extreme is to accept every design generated

through the search, which leads to a completely random process of no use for

optimisation. The other extreme is iterative improvement, where only moves that

present immediate benefits to the objective are accepted. Stochastic algorithms, based

on Monte Carlo simulation, assume a probabilistic acceptance scheme that allows

deterioration of the system as well as improvement, starting from the completely

random mode and smoothly switching to iterative improvement.

The choice of SA allows flexibility in the way that the solution space s represented and

searched hat is suitable for complex reaction systemswith large parameter spaces.

4.3.3.2 Nonlinear Programming (NLP)

NLP deterministic method is used in the parameter precision improvement, after

multiple near-optimal solutions are obtained. Discrete variables that represent reaction

scheme composition can be fixed from stochastic optimisation. Continuous variables

77




that represent kinetic parameters need to be further optimised. In this work, a successive

quadratic programming (SQP) algorithm (NAG subroutine) is adopt from the available

NLP methods. SQP uses an implementation of Powell's successive quadratic

programming algorithm and is aimed specifically at large, sparse nonlinear programs. It

solves the quadratic programming subproblems by using a sparsity-exploiting reduced

gradient method. Sparse data structures are used for the constraint Jacobian, and there is

an option to represent the approximate Hessian as a small set of vectors using a limited

memory-updating scheme.

Even though SQP is likely to be trapped in the local optimum in the neighbourhood of

78

Fig. 4.5 Simulate Annealing optimisation framework




the starting point like other NLP optimisation methods, with good initialisation basedon

results of SA optimisation method, it can guaranteethat optimum points can be reached

and multiple solutions obtained as rival models for model discrimination.

4.3.4 Implementation of the optimisation

This specific reaction scheme and kinetic model optimisation problem involves

combinations with a large number of continuous variables and discrete variables. Which

reaction is selected, which type of rate equation is used, and which species affect the

reaction rate, etc. are all addressed as discrete variables. Reaction order, activation

energy and frequency factor are described as continuous variables, moving between

lower bounds and upper bounds assignedby the user.

4.3.4.1 Simulated Annealing framework

During SA optimisation, starting from an initial state and annealing temperature, a new

state is generated and its objective is evaluated, as shown in Fig. 4.5. When the

acceptance criterion stipulates rejection of the new state, a different state is generated.

The acceptance criterion is controlled by move probabilities. When the new state is

accepted, it will be the basis for the next modification. As long as the end of the Markov

chain is not yet reached or the standard deviation of the elements of the Markov chain is

too low, the Markov loop will be continued. If no acceptable standarddeviation can be

found, the optimisation ends abnormally. Otherwise the cooling schedule is applied at

the end of the Markov loop to generate a new annealing temperature. The stochastic

optimisation ends when any of the following termination criteria are satisfied.

1. The annealing temperature falls beneath a lower bound, indicating that the

Simulated Annealing terminates properly

2. No transitions are acceptedfor a certain number of Markov steps, ndicating that

the system is frozen

3. The standard deviation does not satisfy Eq. 4.19 for a certain number of Markov

steps, indicating that the system is frozen

79




4. A maximum number of Markov chains are evaluated, indicating that the

Simulated Annealing did not reach a stable solution with minimum energy yet.

In order to obtain reliable results, annealing parameters should be adjusted to obtain fast

convergence based on specific problems. It is also suggested that multiple runs can be

used to analyse the average value of the objective function, avoiding the result of one

run beyond a certain confidence level according to the normal distribution of the

objective value.

4.3.4.2 Optimisation moves

Modifications in SA optimisation include two kinds of moves: reaction scheme moves

and kinetic model moves as shown in Fig. 4.6.

Reaction scheme moves are discrete moves, including:

" Add reaction

9 Delete reaction

" Change reaction

Reaction scheme moves are subject to the construction strategy of reaction schemes.

On the other hand, kinetic model moves can be both discrete moves to modify the form

Optimisation Moves

Kinetic model

changes

0

Reaction scheme

changes

Change Frequency Reaction Activation Number of Reaction

equation factor order energy reactions involved

Fig. 4.6 Optimisation moves in SA

80




of the equation and continuous moves to modify the reaction parameters, ncluding:

" Change of the form of kinetic equation

9 Activation energy modification

9 Frequency factor modification

" Reaction order modification

If the kinetic expression for any reaction step in a certain reaction scheme is in the

reversible form, the feasibility of the reaction schemewill not be affected.

Moreover, kinetic parameters as continuous variables have constraints to satisfy the

physical meaning. For instance, the frequency factor should be lower than the collision

frequency of the molecules in the gas phase, and normally ranges from 10-6 o 1016.The

activation energy must be a positive number greater than 5 kcal/mol, and lower than 50

kcal/mol, the order of magnitude of the energy for the chemical bond breaking

(Santacesaria, 1999). The reaction order for individual species is between -2 and 2.

In Fig. 4.7, three reaction models are used to show how the variables move from state to

state using discrete and continuous moves.

A+ B-> D

A+D-2E

C+D-> E+F

r, =k, CACB

r2=k2CACo

r3=k3CCCD

ME*

A+ B--> D

A+D->2E

C+ D-> E+F

r, = k, CACB

r2 =k/$

'0

r3= k3CCCD

=I*

A+ B---> D

C+D ---*E+F

+2E---> 2

r, =k, CACB

r2=k2CCCor3 =

k3 CBCE

Fig. 4.7 Example of state-to-state moves

During the stochastic optimisation, the search s driven by random modifications. These

modifications from an old state into a new state are called perturbation moves, which

are controlled by probabilities of structural and kinetic parameters.These probabilities

can either be fixed or modified during optimisation to reflect preferences towards

certain reaction schemesand kinetic parameters.

81




In practice, the sum of move probabilities is normalised to 1. The value is distributed

among all the moves, the bigger the probability is, and the more important the decision

is.

4.3.4.3 Parameter precision improvement

In order to searcla or local optima near SA optimisation results, the searchregions are

narrowed. In this work, ± 10% is allowed while still in the range of kinetic parameter

constraints.

In order to obtain the multiple solutions, multiple runs of optimisation should be carried

out. In each run, simulated annealing and NLP are running sequentially.

4.4 Illustrative cases

In this chapter, a base case and catalytic epoxidation of oleic reaction are used as

examples to elucidate the method of model building. First, the hybrid optimisation was

tested on a case for parameter estimation, to see if this method is suitable for continuous

variable problems when ':he reaction scheme is fixed. In this method, the reaction order,

activation energy and frequency factor are known through optimisation moves.

4.4.1 Base case

Time

A, D

A, B, C, D, E

Fig. 4.8 Base case

A: R1R2

B: RIRZR3

C: RIR2(R3)2

D: R3

E: (R3)2

R1,R2,R3:Functional b oups

In this homogeneous constant-volume reaction system (Coker, 2001), raw materials A,

D are fed into a batch reactor, then B, C, E are observed, in which components A, B, C,

D are measured using a suitable analytical method for the kinetic study. The

compositions of these reacting speciesare listed in Fig. 4.8, which are assumed n order

to generate feasible reaction stepsusingfunctional

groupsR1,R2,R3.

82




Reaction conditions are:

Experimental reactor: batch reactor

Reaction temperature: 25 °C

Reaction time: 10 hours

Experimental data are listed in Table 4.1 and Fig. 4.9.

Table 4.1 Experimental data for basecase

Time(hr. ) CA(moUm3) CB(moUm3) Cc(mol/m3) CD(mol/m3)

0 0.2 0 0 0.4

0.5 0.185 0.0147 0.0003 0.3809

1 0.1717 0.0272 0.0011 0.3634

1.5 0.1599 0.0378 0.0022 0.3473

2 0.1494 0.0469 0.0037 0.3324

2.5 0.14 0.0547 0.0053 0.3187

3 0.1315 0.0613 0.0071 0.306

3.5 0.1239 0.067 0.0091 0.2942

4 0.1169 0.072 0.0111 0.2831

4.5 0.1106 0.0762 0.0131 0.2728

5 0.1048 0.0799 0.0152 0.2631

5.5 0.0996 0.0831 0.0173 0.2541

6 0.0947 0.0858 0.0195 0.2456

6.5 0.0902 0.0882 0.0216 0.2375

7 0.0861 0.0902 0.0236 0.2300

7.5 0.0823 0.092 0.0257 0.2228

8 0.0788 0.0935 0.0277 0.2160

8.5 0.0755 0.0948 0.0297 0.2096

9 0.0724 0.0959 0.0317 0.2035

9.5 0.0696 0.0968 0.0336 0.1977

[-l0 0.0669 0.0975 0.0355 0.1922

83




04

0.35

0,J

ES

0.25

0,2

ö 015

Ev 0,1

0.05

0.0D. 0

D_4

o ob

D. 3

X B(ExP)2(EXP)C

ý-ýC(EXP

D(ExP)

D. D -1.0 20704D5D6.0 7.0 8.0 90 10D D.0 1,D 2.0 3.0 4.0 5, D 6.0 7,0 6.0 9.0 1D.

Time [hr] Tlme [hr]

Fig. 4.10 SA and NLP optimisation results

Fig. 4.11 is the result of residual analysis after NLP optimisation.

2. E-04

1. E-04

0.E+00

-1.E-04

-2.E-04

SA

A B fC XD

x 15 "

x xxxx

Fig. 4.11 Residual analysis

From the results, it was observed that the model gives good agreement with the data

using SA to provide a good initialisation for the NLP. So the SA/NLP hybrid method is

suitable for problems with continuous variables and can access the global minimum,

even though the ranges of parameters are extremely wide. Also, the reaction simulation

method can cope with both stiff and non-stiff reaction rate equations. So this hybrid

method is now extended to more complex problems in which the reaction schemeand

kinetics are to be obtained simultaneously.

Finally, three rival models are generated from the preliminary experimental data fitting

if reaction scheme moves are allowed. They are screened from the most frequent model

appeared in the multiple optimisation results. Results show that all the rival models fit

0

0 25

c

0.1

0.15

aE

o. 1U

0 05

NLP

86




the experimental data well in Fig. 4.12. Discrimination between them will be left to the

next chapter.

a. 4

o 35

0.3a

D 25

c

0,2

ä 0,15

CLE

a 0.1U

Model 1

x_ä(EXP)C(ExP)

C(EXP)

---ý DD(EXP)

-*

Time. hfl .... ý ý. ý .ý .ý

D0D5 L

.0D. 0

D. 4

0.35

D. l

E

0 25

c° D. 2

Y

0 15

a

Model 2

D.

0,05

D. DD, 0 1.0 Z. 0 3, D 4. D 5, D 6, D 7.0 0.0 9, D 10.0

Ttme [hr]

Model 3D. 4

0.33

D. 30E

0.25

c

0.2

0 15nO D. 1

V

(EXP)

(EXP)

(EX P}.

(EXP)

0 05

0.0 =-- --s--v 71

D.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 0.0 9.0 10,0rime [ter]

Fig. 4.12 Rival models

4.4.2 Oleic acid epoxidation reaction system

The net reaction is between oxygen and oleic acid, both dissolved in acetone. The

desired product is oleic acid epoxide. In Chapter 3, from information of species

involved in the reaction system, two reaction lists are obtained as follows:

3 1C+D t- E+F1 A+B F--> 4

A+D H 2E2 2A+BHE

5 B+2EH2D

During the optimisation of rival models, reaction schemes will be screened by the

strategy described in the last chapter.

Experimental design and experimental data (Rastogi et al., 1990,1992) for the oleic

acid epoxide reaction are

listed in Table 4.3,4.4 and 4.5.

87




Table 4.3 A 24-1ractional factorial design

FactorOrder of Run

1 2 3 4 runsID

- - - - 4 RF4

+ - - + 1 RF1

- + - + 8 RF8

+ + - - 3 RF3

- - + + 2 RF2

+ - + - 7 RF7

- + + - 5 RF5

+ + + + 6 RF6

Table 4.4 Operating levels for variables of interest

Factor Variables Lo-level(-) Hi-level(+)

1 Temperature 25°C 35°C

2 Catalyst concentration 1x10-5 M 4x10"5 M

3 Initial oleic acid concentration 0.24M 0.48M

4 Initial benzaldehyde concentration 2.5M 3.5M

Table 4.5 Experimental datafrom the factorial experiments

Run Time [A] [D] [C] [E] [F]

ID (min. ) (M) (M) (M) (M) (M)

RF1 0 3.5663 0.0042 0.2346 0.0198 0.0000

40 3.4603 0.5961 0.1799 0.0708 0.0487

80 3.2676 0.0876 0.0845 0.2349 0.1308

RF2 0 3.4290 0.0047 0.4800 0.0236 0.0000

40 3.4082 0.0168 0.4706 0.0322 0.0087

80 3.4042 0.0188 0.4669 0.0343 0.0117

RF3 0 2.4753 0.0057 0.2427 0.0331 0.0000

40 2.3697 0.0479 0.1741 0.0966 0.0623

80 2.1452 0.0911 0.0616 0.2779 0.1595

RF4 0 2.4635 0.0057 0.2366 0.0210 0.0000

40 2.4470 0.0136 0.2309 0.0296 0.0074

80 2.4416 0.0155 0.2247 0.0331 0.0111

RF5 0 2.6492 0.0059 0.4781 0.0010 0.0000

40 2.6233 0.0156 0.4719 0.0184 0.0077

80 2.5969 0.0243 0.4632 0.0340 0.0139

RF6 0 3.5106 0.0049 0.5019 0.0373 0.0000

40 3.4089 0.0183 0.4080 0.1255 0.0765

80 3.2683 0.0664 0.3066 0.2181 0.1721

88




RF7 0 2.5543 0.0046 0.4810 0.0263 0.0000

40 2.5353 0.0096 0.4649 0.0403 0.0131

80 2.5144 0.0145 0.4541 0.0563 0.0278

RF8 0 3.5043 0.0037 0.2412 0.0432 0.0000

40 3.4442 0.0160 0.2139 0.0837 0.026180 3.3305 0.0414 0.1832 0.1329 0.0569

Without a reaction model, reactor design and optimisation will be based on factorial

experiments from the laboratory. From Table 4.5, if the goal is maximum yield from the

reactor or maximum reactor conversion, then the chosen conditions would be: high level

of temperature (35°C) and catalyst concentration (4x10-5M) with low level of initial

oleic acid (0.24M) and benzaldehyde concentration (2.5M). This would indicate a

maximum conversion of oleic acid for the reaction of 65.69% and a maximum yield of

epoxide of 6.44% with respect to the initial benzaldehyde concentration. Scale-up from

the laboratory data could then lead to the first reactor design. Clearly, further

experiments could be carried out using factorial experimental design in a narrowed

searchspaceto gradually refine the optimum conditions.

However, this creates greater risk that the potential optimum conditions will be never

reached and delays the development program. Instead, In this thesis, a model that can

interpret experimental data well and can be usedfor reactor design and optimisation will

be derived by the proposed methodology.

4.4.2.1 Model building

Before the data were processed,assumptionswere made to simplify the reaction system.

It was assumedthat there was no evaporation of the liquid phase,reaction occurs only in

the liquid phase, and reaction rate was not affected by gas flow rate. Thus no mass

transfer resistance exists in the gasphase.Also, the saturatedconcentration of gas in the

liquid phasewas assumedat the gasflow rates used to obtain the experimental data.

After optimisation, three rival models, Model 1, Model 2 and Model 3, whose

objectives were 0.364191,0.36161 and 0.374843, were obtained from the multiple

optimisation results with the relative minimum objective functions.

89




Model 1

Reaction steps Kinetic equations Kinetic parameters

Reaction 1(I) Ea1rl = Al exp(- )CACcat

Al = 78.6867

A+B ->DRT

Eat 909.392

Reaction 3(II)=2 A2exp(-

Eat)CDCl

A2 = 0.0148583

C+D->E+F RTEat 60.8269

Reaction 5(II) Ea3

r3= A3exp(- ýCECýat

A3 = 0.100000E-05

B+2E-> 2D RTEa3 2000.00

Model 2


Reaction 1(I) Ea1rl = Al exp(- )CACýar Al = 19.2384

A+B -ýDRT

Eat= 56.6036

Reaction 3(11)=2 A2 exp(-

Ea2)CDC1 A2 = 0.285781

C+D ->E+FRT

Ea2 2000.0

Reaction 4(II)r=A exp(-

Ea3)C CD3

A3= 0.00708276

A+D -> 2E

RT

Ea3 673.469

Model 3


Reaction 1(I) Eal 1rl = Al exp(- t

Ca CAl = 166.795

A+BD

oaTEat = 50.4892

Reaction 3(II)r2 = A2 exp(-

Ea2)CDC1 A2 = 0.134936

C+D --> E +FRT

Eat= 1665.0

Reaction 4(11) Ea3r3 = A3exp(- )CACD

A3 = 0.0585312

A+D ->2E RT

Ea3= 2000.00

90




These three models all fit experimental data reasonably well, but have different reaction

schemesand kinetic expressions.

4.4.3 Discussion

From the results of two cases,the rival models which fit the same experimental data

have similar values of objective functions, but give different reaction models that satisfy

the statistical screen. Hybrid optimisation can easily deal with the large scale MINLP

problem and provide robust solutions. So far rival models cannot be further validated

with currently available experimental data.

4.5 Conclusions

The experimental data can be used to identify which reaction network is most

appropriate and to derive kinetic parameters for the reactions within the networks

simultaneously by an optimisation method. A stochastic method is designed to obtain a

set of solutions in the region of the optimum by setting up reaction network moves and

kinetic moves to generate alternatives and monitor the random search. The SA

algorithm can reduce the risk of model mismatch through avoiding premature deletion

of a model in the preliminary stage. Also, SA provides a good initialisation for NLP.

Conclusively, SA optimisation combined with NLP provides good solutions for the

MINLP problem and a set of rival models is determined for model discrimination and

optimal experimental design.

91



Chapter 5 Model Discrimination and Optimal Experimental Design

Chapter 5: Model Discrimination and Optimal

Experimental Design

5.1 Introduction

5.2 Model discrimination criteria

5.3 Optimal experimental design

5.3.1. Laboratory reactors

5.3.2. Experimental conditions

5.3.3. Reactor superstructure

5.3.4. Simulated Annealing (SA) optimisation

5.4 Case studies

5.4.1. Basecase

5.4.2. Oleic acid epoxidation

5.5 Conclusions

92



Chapter 5ý, ýiýDýý,ý. iýrs ,atý ýý--1 ý.rý_ ýý._1_ý-rimental

Design

A set of rival models can be obtained within a certain confidence level to fit the same

set of experimental data. However, they cannot be discriminated without further

experimental data to find the best model. In this chapter, a methodology that can obtain

optimal experimental design for rival model discrimination to reach the minimum

experimental measurement target is proposed and validated by two case studies, whose

rival models are derived in the previous chapter through hybrid optimisation.

5.1 Introduction

ti............................

Stoichiometric

analysis

PreliminaryModel Derivation

..ý;ý

experiments....................................... I.......... ti

Reaction scheme and,...

""""""""""""

kinetic rival models

....................................................


and improvement of

model accuracy

Model Discrimination

Fig. 5.1 General model building framework

Further

experimentaldesign

There are many applications in which one set of experiment data can conform to more

than two models, all within the same sample space, as discussed in the previous chapter.

In the first step of the model building methodology, model derivation aims at providing

multiple solutions to avoid any premature decision. On the other hand, a set of

competing models without further screening can only cause chaos for process design

and optimisation. As shown in Fig. 5.1, the remaining work of the model building

procedure is model discrimination.

There is a probability that at other operating conditions theserival models have different

performance, which can be used to discriminate among rival models. How to decide the

next experimental point is a critical task for model discrimination. For instance, the

following two reaction models fit experimental data very well from experimental time

93




zero to t, as shown in Fig. 5.2. If further experiments are carried out within the same

time range, no extra information can be provided except parameter precision

improvement. So accurate model fitting is not enough for model discrimination.

Consider again the example in Fig. 5.2, if the experimental time is extended, the two

models start to exhibit significance different performance. Alternatively, if experiments

are carried out in different types of reactors, the two models might produce different

product distributions. This concept can also be used to distinguish among rival models.

70

60

50

CIO 400

30

20

10

0

13579 11 13

Exp. Time

Fig. 5.2 Data fitting and model performance

The objective of model discrimination is that, given that several different kinds of

models (i. e. polynomial vs. exponential) are available, how should experimental points

be chosen to distinguish between them? In other words, some experimental data have

been used to derive rival models, where should the next experimental point be located to

learn the most about which is the right model? It is desirable to build the relationship

between model discrimination and experimental design.

The concept of optimum experimental design was first presentedby Kiefer in 1959 at

the Royal Statistical Society. Optimal experimental design can provide the

mathematical tools for producing the most relevant information in order to reach well-

predefined objectives with high efficiency and at minimum cost. The aim of optimal

experimental design is either for parameter estimation, or for model discrimination.

94



Chapter 5 Model Discrimination andOptimal Experimental Design

There are two general groups of optimal experimental design: simultaneous (statistical)

or sequential.

Optimal experimental design has received a considerable amount of attention in the

literature, for example Anderson (1962), Hunter and Reiner (1965), Box and Hill

(1967), Hunter and Mezaki (1967), Pazman and Fedorov (1968), Froment and Mezaki

(1970), Meeter et al., (1970), Atkinson (1972), Atkinson and Cox (1974), Atkinson and

Fedorov (1975a, b), Hill (1978), Atkinson (1981), Spruill (1990), Dette (1994,1995),

Dette and Ro der (1997), Dette and Haller (1998), and Pukelsheim (1993). Commonly,

optimal experimental design methods are also based on statistics, such as factorial

design, fractional factorial design, orthogonal design and uniform design.

Experimental design theory for precise estimation of the model parameters has been

well developed in recent years. For example, D-optimum, T-optimum and other

alphabetic optimum designs have been successfully applied in many subjects, which

belong to the first group (Atkinson & Donev, 1992; Pukelsheim, 1993; Fedorov &

Hackl, 1997).

However, the design problem for discrimination between models hasreceived much less

attention and has been developed for simple models only. Various criteria were

considered by Box and Hill (1967), Atkinson and Fedorov (1975a,b), Burke et al.

(1994,1995), Stewartet al.

(1998), Ponce de Leonand

Atkinson (1991), Müllerand

Ponce de Leon (1996), Felsenstein (1992), Fedorov and Khabarov (1986).

The best design for discriminating among several rival models may be quite different

from the design that is optimal for estimating all the parameters in the unknown true

model. If an optimum design method, such as a D-optimum design, an A-optimum

design or an E-optimum design (Pukelsheim, 1993), is chosen, efficient estimation of

model parameters is guaranteed. But for model discrimination, the resulting design may

even be singular for some of these models. Furthermore, many optimal design criteria

for model discrimination depend on the specific ordering of competing models and

which one of the models is the true one (Atkinson & Fedorov, 1975a,b).

95




In this work optimal experimental design will be consideredthat simultaneously achieve

two objectives: generation of optimal design points so as to discriminate among rival

models and to improve estimation of all the parameters n that model.

5.2 Model discrimination criteria

To identify which is the best model to describethe reaction system, further experimental

data are needed. The basis of experimental design is that, at other operating conditions

than those at which rival models are derived, rival models might have different

performance. The point where rival models have the biggest difference, provides the

optimal operating conditions for the experimental design. In other words, it provides the

Preliminary

experimental data

Rival models

Further Experimental

design

Model precisionimprovement


Onemodel

Yes

Final model

Fig. 5.3 Model discrimination framework

96



Chapter 5 Model Discrimination andOptimal Experimental Design

best opportunity to determine which model is the best model, if further experiments are

carried out at the optimal operating conditions.

In principle, it is necessary to avoid a routine approach by making extensive

measurements without simultaneousevaluation. The basic problem is to select the set of

experimental points that will provide the best discrimination amongst the models.

The whole procedure of model discrimination used in this chapter is shown in Fig. 5.3.

Further experimental design is determined according to the predictions of existing rival

models to provide maximum discrimination, at the same time aiming to improve the

adequacy of fitting with additional experimental data.

Several alternative models are often proposed to explain the same data, and objective

criteria are neededto choose amongst models. The choice of objective criteria can affect

the method of data evaluation.

The rival models may be nestedor non-nestedbasedon their origins. Nested models are

a series of models that allow a simpler model to be obtained from a more complex

model by eliminating one or more parameters from the more complex model. Thus

choosing among models is reduced to determining the appropriateness of the additional

parameters.Non-nested models are not related in this way, and any model is valid under

a certain hypothesis. In this thesis, the rival models have the feature of non-nested

models. Accordingly, the discrimination criteria are different from those for nested

models.

There is a large number of possible optimality criteria, depending on the particular task

at hand, named by letters of mnemonic value: maximizing the determinant is D-

optimality, minimising the variance of predictions is G-optimum. There are A-, c-, C-,

DA-, D, beta-, D's-, E-, L-, T- and V-optimum also, and there are equivalence

theorems connecting many of these.

Sequential design of experiments (Hunter & Reiner, 1965; Box & Hill, 1967) is the

most widely used method in kinetic studies and for discrimination. Experiments are

successively performed in a direction of improvement until the optimum is reached. The

sequentialdesign

methodis

carried out while an optimisation methodis

applied.

97




Supposethere are m rival model after fitting to N experiments.

Model H1: 17,,= f, (e1,ý, )

Model H2: i7 = f2(82, ý2 )

Mode H: 17 =.fm(0m, ým) u =1,2,...N (5.1)

where i is the hypothesised model response; 0= (0,02,"""9p) is the vector of p model

parameters; ;= (t1, J2,"""

Jk) is the vector of k dependentvari ables.

The observed response y should be:

y=11±e (5.2)

wherethe

errore is

normally andindependently distributed

with constant variance 62 .

It is desired to find the best setting of tN+1 for N+1 experiments that will best

discriminate between rival models.

According to Hunter and Reiner (1965), the best experimental condition for model

discrimination is the one that maximises the divergence among the responses of the

models.

It has been applied to the system with two rival models (m = 2). Assume that either

model H1 or H2 is correct (not both will hold), the design criteria is either:

N+1

Si =ý (YU- f2U 2 (5.3)u=1

where YN+1 _N+l

_1le1N+1' ý1N+1 l' N+1

=f2 (eN+l'1 )f

or

N+1

s2= (Yu-fiu)2 (5.4)

U=1

98




+1 )where yN+l =fN+1

=fa (e2 +1 t +1) fN+i

=fi (8N1 J,

Box and Hill (1967) showed that the design procedure should takeexperimental errors

and model prediction variances into account in order to improve the performance of the

discrimination procedure. They extended the method to general problems (m >_ ).

A multiple hypothesis test is a generalisation of a binary hypothesis test. Suppose m

rival models: Ho, HI,....

Hm_j are obtained from preliminary experimental data and

assumethat a certain model Hi is the best model based on the performance of further

experiments.

Then the problem arises: how to identify which model is the best model that closely

matches the experimental data? This is an optimisation problem again. Choice of the

objective function, i.e. the choice of the quantities to be compared, is a problem for the

engineer who is responsible for the kinetic measurements and reactor design. It must be

selected so as to achieve an accurate description of data in those regions that are

important.

Experiments are selected on the basis of an entropy criterion suggestedby Box and Hill

(1967) that measures the information increment provided by each of the experiments.

The concept of entropy commonly used in the thermodynamics is adopted in the

informationtheory

(Shannon, 1948), tomeasure

theamount of

informationcontained

in

a message.Therefore, in order to obtain the maximum information from the system it is

desirable to have a maximum changein entropy between input and output.

As a discrimination criterion, entropy has also been used in the work of Reilly (1970)

and Fedorov (1972) in statistics and in the work of de Kleer and Williams (1987) and

Struss (1994) in model-based diagnosis.

D=II ][IýN-ill jN-1

JPi Inp`

dyN +JP; In

p'dYN (5.5)

i=1 =i+l P; A

where i, j index of models, i, j=1,2,.... m

N index of experiments

A probability density function of the n-th experiment under model i

99



Chapter5 Model Discrimination and Optimal Experimental Design

YN expenment

11N-1 prior probability

II N posterior probability

This criterion D is called Kullback's total distance measurement of information

(Kullback, 1959) when discriminating between hypotheses Hi and Hj. It was derived

from statistical theory when experimental errors are considered. The minimum mutual

information is corresponding to the maximum function D, representing the maximum

difference of two hypotheses.

This concept is now applied to the discrimination among m models where it is desired to

go from a non-informative situation to a more informative situation.

The predicted value at the next operating conditions is determined from the probability

within a certain confidence level.

Pt -1 exp -1(YN

- YN (5.6)6c2

-t- 6l2ý2(62 62

i is the variancehere 62 is the known variance form the first N -1 experiments; 62

of predicted value of N-th experiment yN under model i.

The objective function for model discrimination among the m rival models is to

maximise

mm22 I2cTi

-6'l`'11 ()= 1I

Y, HEN-1fljN-1

(2+22+2) + yN -ya 2+ 225.7j=i+1

6i 6 -I- 6i 6 -I- 6i

It can be simplified according to the specific problem. Normally it is an unconstrained

optimisation problem.

100




5.3 Optimal experimental design

A model generally includes a set of independent variables ý, and a single scalar

dependent variable, called the single response model. Or a model with a vector of

dependentvariables is called multiresponse model.

17u f(0 ,)(5.8)

where 0 is the vector of estimated model parameters.

The purposes of experimental design are, either estimating parameters of the model, or

discriminating among models. An experimental design is, for these purposes, a

distribution (discrete or continuous) over the independent variables, at which the

response is measured.

Based on the concept that either within different reactor types or at different operating

conditions, the predictions of rival models that describe the same reaction system can

have different performances, which is preferred for model discrimination. The

independent variables are composed of discrete variables to represent reactor

configurations and continuous variables to represent operation conditions, such as

temperature, pressure, and so on.

Rival models

Model 1

Model 2,,,,,,

Model m-

4Reactor superstructure Optimise

Operating Experimental J

conditions+

reactors

.--.Model 1Model 2

160 =Modeli

120

C

'ý-

aj80

"

40

0246

Fig. 5.4 Optimal experimental design framework

101



Chapter 5 Model L)i; Lrimination and Optimal Experimental Design

In order to achieve the maximum objective, an optimisation approach is required to

provide a robust solution, while handling both discrete and continuous variables. Here

Simulated Annealing (SA) optimisation is applied. A reactor superstructure is necessary

for effective optimisation, in which all possible combinations of reactor configurations

and operating conditions are embedded(as shown in Fig. 5.4).

5.3.1 Laboratory reactors

The type of reactor to be used in laboratory experiments to obtain data for process

evaluation or kinetic modelling depends on the nature of the reaction. Laboratory

reactors can generally be used in two ways: reactors for gathering data in a chemist's

laboratory with the objective of developing a feasible synthetic routine for a chemical,

and reactors used to obtain precise kinetic data under isothermal conditions, which also

take into account the mass and heat transfer features of the reactions.

Alternatively, Laboratory reactors can be divided into two categories: reactors for

homogeneous reactions and reactors for heterogeneous reactions. Choosing the type of

laboratory reactor for evaluating process kinetics may be the most crucial step in an

industrial process development program. Not only would a wrong choice result in

expensive delays, but also data may be obtained which would scale-up erroneously,

leading to a disastrous commercial design.

The most common reactor types used in the laboratory are shown in Fig. 5.5 and Fig.

5.6.

Time

Feed

ProductBatch

Time

Feed

ProductSemi-batch

Feed

Feed Product

ProductPFR Stirred tank

Fig. 5.5 Laboratory reactorsfor homogeneousreactions

102




Feed

Product

Stirred tank

Fluid-Solid Reaction

Feedle-O

Product

Stirred basket tank

Fluid-Fluid Reaction

Feed

Fee Pro uct

Feed 2

Feed Produ t

R

loop reactor

Fluid-Fluid-Solid Reaction

"Q"G( O: O

Feed "P :ö

Prnrinrt Product

Stirred tank Tower reactor Stirred slurry reactor

Fig. 5.6 Laboratory reactors for heterogeneous reactions

5.3.2 Operating conditions

Feed

Product

Stirred basket reactor

It is obvious that the inclusion of operating conditions is dependent on the specific

reaction system and reactor chosen. The following operating conditions cannot usually

be ignored:

9 Temperature

" Pressure

" Reaction/residence time

" Phasecontacting type

" Mixing type

" Feeding policy

" Recycle or bypass

" Etc.

Operating variables can be addressed as continuous variables, optimised within

reasonablebounds to achieve maximum divergence.

103



Chapter 5 i( del 1ýiý: irninetil; ti and L.-yperimental Design

5.3.3 Reactor superstructure

The optimal experimental design requires the evaluation of the most promising

...............................................................................................................................................

FEED`r

Fig. 5.7 Homogeneous experimental reactor superstructure

experimental conditions. If a new experimental point seems to be promising, it should

be investigated and compared to previous conditions. Completely different reactor

arrangements should also be included, so that novel but promising solutions are not

missed. Therefore the superstructure for the optimal experimental design should

accommodatedifferent

reactor configurations and operating conditionsduring

optimisation.

Fig. 5.7 is the reactor superstructure for homogeneous reactions with one feed and one

product used in this work. Also, the reactor superstructure can be extended to multi-

phase reactions. Two-phase systems can be described by one pair of reactor

compartments (Fig. 5.8).

> .................................... ................................... ....................................................... "

FEED L.

"`"R

....... .........................:.......:......................

............ ...................................................................................................

r....... j .......................... .... .......................... ...... j...............................................

FEED

ýr

PRO]

.....................................y................................

......................................... ............................................................................

Fig. 5.8 Multiphase experimental reactor superstructure

104




The superstructurerepresentation is basedon reactor units and splitter/mixer units. The

representation follows the concepts developed by Mehta (1998) for multiphase reactor

networks. By using different simulation models, the same generic representation can be

used for different scopes and applications. The models applied by Mehta (1998) for

multiphase reaction systems can be modified to address the issues required in

experimental design.

The presentation of reactor units is based on ideal reactors: CSTR or PFR. Isothermal

and steady state s assumedwithin the unit of superstructure

The superstructure accommodates each phases independently. The two phases are

coupled by mass transport within a reactor unit. For each reactor unit, the options for

mass transfer, reaction, number of compartments, flow direction can be controlled, so

that each unit either represents a well-mixed reactor (CSTR), a cascade of CSTRs, a

plug-flow reactor (PFR), in co- or counter-current mode.

In the reactor superstructure, the splitter unit can facilitate connections between the

reactor compartments of a given phase as feed condition, bypass or recycle. The mixer

unit can function as feed condition controller. The models associated with the various

options included in the superstructure of this work are described in more detail as

follows:

0 Reactor configuration

All commonly used reactor configurations in the laboratory can be represented in the

reactor superstructure, but the reactor configurations for a specific reaction system are

restricted to certain groups. For example, the reactor configurations for the fluid-solid

reaction system are not suitable for fluid-fluid reaction systems. In order to reduce the

size of the superstructure and to avoid incremental modifications to the superstructure

with little effect on the objective function, the range of valid unit numbers between one

and a maximum value is discretised into a series of valid numbers (e.g. 1,2,3. ). In

practice, generally no more than three different reactors are used to study one reaction

system

0Mass

transfer

105




Mass transfer is reflected by mass transfer coefficients. Reactions only occur in one

phaseand no phase-transferhappensbetween any two phases.

0 Feeding policy/recycle/bypass

By specifying flow fractions for each splitter or mixer and other operational variables

like mass transfer coefficient, flow direction of a compartment, etc., a unique reactor

will be generated.

" Flow direction

The flow direction can be varied for a unit, so that both the counter-current and co-

current flows are included in the superstructure.

0 Phasecontacting type

For multiphase reactors, which type of the phase contacting pattern, plug flow/plug flow,

or plug flow/mixed flow or mixed flow/mixed flow is applied depends on whether

CSTR or PFR is chosen as the ideal unit.

0 Continuous parameters

All continuous parameters such as temperature, pressure, reaction/residence time, feed

flow rate and initial concentrations, can be varied also.

5.3.4 Simulated Annealing (SA) optimisation

During the optimisation step, the superstructure is reduced to a final structure with a

near-optimal performance. Because of the severe non-linearity of the equation system

and the large number of discrete decision variables, deterministic optimisation

approachesare unlikely to be successful.Instead a stochastic optimisation scheme n the

form of Simulated Annealing is applied.

When variables of experimental reactor superstructure are optimised, all possible

experimental reactors and operating conditions are generated to achieve maximum

divergence for model discrimination.

106




5.4 Case Studies

In Chapter 4, two examples were used to illustrate the first step of the model building

methodology. Three rival models were generated for each example. In this Chapter, the

procedurefor model discrimination will be applied to thesetwo cases.

5.4.1 Base case

Three rival models were generated from the preliminary experimental data fitting in

Chapter 4.

Model 1

1) 2D --> E r1= 0.0385CD2

2) A+C-->2B r2 =3.0168CACC

3) A+E-->

C r3 = 13.356CACE

Model 2

1)A+2D-->C rl = 0.0311CACD

2) 2C --> 2B +E r2= 1.2435x105CC

3) 2B +E -->2C r3 = 5.5563x105CBCE

Model 3

1) 2D --> E rl = 0.1177CD22) A+C

-->2B r2 = 1.854CACc

3) 2A +E --> 2B r3 = 7.5032x106CACE

4) 2B +E -->2C r4 = 1.3552x106CBCE

In this case,there are two feeds, each comprising one of the pure raw materials (A and

D). Only the feed composition can be optimised in order to design

a new experimentto

discriminate between the three candidate models. So during the optimisation, the feed

charge size of one of the feeds is modified asthe optimisation parameter.

The initial guessfor the feed of component A is 0.2 kmol; 0.4 kmol for component D.

After optimisation, the optimal charge size for component A is 0.107 kmol, while the

feed of component D is fixed. The performances of the different model are shown in Fig.

107




5.9 under optimal operating conditions. It is concluded that the predicted responsesof

the models to the new input conditions are easily distinguished from each other based

on the prediction of D in different models.

ý- Model 1 -1 -- Model 2

0.05E

0.04

0.03

'ä 0.02

ö 0.01

00

--t- Model 1 Model 2 Model 3

0.15

B0.12

ö 0.09

ö 0.06

0.03

0

02468 10Reaction time (hr. )

Model 3

-f- Model 1 Model 2

0.05 --ý-C

0.04

0.03

0.020

ö 0.01

U0º

Model 3


E

czE0U

-ý- Model 1 Model 2

0.5

D0.4

0.3

0.2

0.1

0

0

Model 3

Fig. 5.9 Model performance

By creating new experimental data at the optimal conditions sequentially, this will allow

to be selected the best model. In the design of new experiments, only the effect of the

initial composition was considered. In principle, other network parameters (reactor

volume, number of reactors, etc. could also be considered.

108






5.4.2 Oleic acid epoxidation

Again, oleic acid epoxidation is used as an example to demonstrate how the optimal

experimental design is obtained using optimisation. Saturated concentration of gas in

the liquid phase will be assumedfor the three rival models. In this case,the reactor type

and size are fixed, but the operating mode can be changed from batch to semi-batch.

Optimisation parameters are temperature, reaction time, and feeding rate. Maximum

allowable operating temperature is 48 °C (5°C less than the flash point of the acetone

solvent).

After optimisation, the optimal experimental design for model discrimination is:

Reactor type:

Batch time:

Temperature:

Feeding rate/time:

batch

83.177 min.

48°C

0.25 litre/min

Unfortunately experimental data are not available at these conditions, and so the

methodology cannot be pursued further. However, the performance of the different

models can still be compared. Assume that all three models are used to design the

reactor for maximum oleic acid conversion (which is corresponding to maximum

epoxide yield).

The change of oxygen partial pressurehas no effect on the forming of epoxide product,

which makes the reaction scheme of Model 1 irrational. Model 2 favours low initial

oleic acid concentration and high initial benzaldehyde concentration for the best design,

which is consistent with that initial oleic acid concentration is 5 to 15 times lower

compared to initial benzaldehyde concentration in the experimental data. Model 3 with

the samereaction scheme cannot give the same design. So Model 2 is assumed o be the

best model, which is consistent with the model developed by Rastogi (1992).

Reactor design and optimisation are carried out using Model 2. The highest conversion

of OA is 92.38 %, at the following operating conditions:

Batch cycle time: 32.17 mins

Temperature: 48°C

109




Feeding rate (OA): 0.497 litre/h.

Now check the performances of the other models at theseconditions. The conversion of

oleic acid is 57.9% using Model 1, around 37% lower than the Model 2 prediction. The

conversion of oleic acid is 0.36% using Model 3, around 99% lower than the Model 2

prediction.

So if the model is not chosen and tested carefully, a reactor designed on the basis of an

inappropriate model cannot operate under optimal conditions.

5.5 Conclusions

The methodology for reaction model building is completed in this Chapter through the

model discrimination procedure described. Further experimental design is carried out

from the rival model optimisation method. All kinds of laboratory reactor and operating

conditions are embedded in the reactor network superstructure. SA optimisation for the

MINLP problem and the most appropriate reactor configuration, along with operating

conditions for experimental design, are determined for model discrimination and

optimal experimental design.

110



Chapter 6 Model Building for Refinery Heterogeneous Catalytic Reactions

Chapter 6: Model Building for Refinery Heterogeneous

Catalytic Reactions6.1 Introduction

6.2 Heterogeneouscatalytic reactions

6.2.1 General features

6.2.2 Hydrodesulphurisation (RIDS)

6.3 Model building methodology for HDS processes

6.3.1 Catalyst characterisation

6.3.2 Catalyst kinetics

6.3.3 Model discrimination

6.4 Casestudies

6.4.1 Hydrodesulphurisation of thiophene

6.4.2 Hydrodesulphurisation (HDS) of diesel

6.4.2.1 Temperature effects

6.4.2.2 Sulphur compound addition effects

6.4.2.3 Catalyst effects

6.4.3 Discussion

6.5 Conclusions

111



Chapter6 Model Building for Refinery HeterogeneousCatalytic Reactions

The model building methodology discussed so far has been restricted to homogeneous

reactions. Heterogeneousreactions other than those involving solid catalysts will not be

in thescope of

thisstudy.

Heterogeneouscatalytic reactions are much more complex

than homogeneous reactions, so the general procedure of model building is more

complex for heterogeneous catalytic reactions. In this chapter, the model building

methodology is extended to apply to heterogeneous catalytic reactions in accordance

with its special features.This work aims at exploring general model building procedures

from aspects of catalyst characterisation,kinetic studies and model discrimination with

minimum experimental effort.

6.1 Introduction

Heterogeneous catalytic reactions generally involve a solid catalyst and a fluid phase

(gasand/or liquid) that supplies reactantsto the site of catalysis, the fluid-solid interface

(Carberry, 1976). As mentioned in Chapter 2, catalytic reaction depends on the

interrelationship between chemicals and catalysts, from the nature of catalytic reactions,

aswell as on other operating factors (temperature,pressure,concentration, etc. .

Hence, properties of the contact area between the phases are important reaction

variables. Diffusional steps are implicit components in heterogeneous systems, since

one or more reactants and/or products must be transported from their phase to another

phase, where, for reactants the reaction actually occurs. Factors that govern interphase

heat and mass transport therefore become important reaction parameters, which need to

be accounted for simultaneously in order to model heterogeneouscatalytic reactions.

Typical heterogeneous catalytic reaction systems in the chemical industry include

hydrodesulphurisation, methanol synthesis,methanation and Fischer-Tropsch synthesis,

ammonia synthesis, toluene disproportionation, and ethyl benzene isomerisation, etc. To

illustrate the model building methodology for heterogeneous catalytic reactions, the

hydrodesulphurisation (HDS) process from petroleum refining is selected becauseof its

importance in the refinery industry.

Sulphur reduction in motor fuel has been prompted by several factors. First, air

pollution control standards require removal of up to 80% or more of the sulphur that

112




would be present in various fuels. Second, many catalysts in down stream units are

sensitive to the amount of sulphur in the feed. Third, the reduction of sulphur reduces

the amount of corrosion in the refining process, improves the odor of the product, and

reduces he amount of sulphur that can poison the catalytic converter in a motorcar.

Table 6.1 Quantity of sulphur in the various distillation fractions

Fraction Boiling Point Percentage sulphur (wt %)

Light gasoil 0-70 0.001-0.02

Naphtha 70- 140 0.002-0.02

Kerosene 140- 250 0.01-0.2

Diesel 250 - 350 0.1-1.4

Residue > 350 0.3-4.1

Sulphur compounds are present in all distillation fractions of crude oil. Crude oil itself,

depending on type and origin, contains 0.1- 2.5 wt% sulphur. After distillation, the

quality of sulphur increaseswith the boiling point of the different fractions, the average

amount of sulphur content in the various distillation fractions of crude oil in shown in

Table 6.1.

In recent years, the supply of heavy types of crude oil on the world market has increased

significantly. At the same time, the specifications for liquid fuel products have become

more and more strict.

Table 6.2 Europeandiesel specifications*

Unit Year: 2000 Year: 2005 Year: 2010

Sulphur ppm (max) 350 50 10

Cetane number min 51 51 51

T95 °C 360 360 360

PNA %wt (max) 11 11 11

Density Kg/m3 (max) 845 845 845

* Directive of the European Parliament and of the Council 98/70/EC

113



Chapter 6 Model BuilJinLý fier R.ýA'ineryHeterogeneous Catalytic Reactions

Sulphur content is the main concern of the diesel quality. Take the EU market as an

example (Table 6.2). The sulphur content of diesel fuels should be reduced to 50 ppm

by 2005, further to 10 ppm by 1st January 2011, to finally achieve a zero-sulphur target.More strict markets with low-emission fuels, such as UK, Germany, and Norway have

reached he 50 ppm S diesel target already.

If reducing the sulphur level from 500 to 50 ppm by conventional HDS processes, the

volume of catalyst beds will have to be increased 3.2 times, compared with the current

SIDS catalyst bed, or the temperature of reaction will have to be increased by 38°C

(Knudsen et al., 1999), as shown in Fig. 6.1. Furthermore, a 200% increase in catalyst

volume will be required if the sulphur content is reduced to 10 ppm.

f Relative activity Delta T

40

30

E-

M 20a)

10

0

500

400

300

200

100

0

500 40Q 300 200 100 0Sulphur content (ppm)

Fig. 6.1 Operating conditions change with sulphur content target

These options result in accelerating catalyst deactivation with no additional profit to the

process. Choosing a more active catalyst, or process revamping is required to meet the

profit target. Accordingly, improving the accuracy of the model prediction and speeding

up the time for model building for new processes and catalysts becomes increasingly

important for process profitability.

6.2 Heterogeneous catalytic reactions

6.2.1 General features

Generally, heterogeneous catalytic reactions comprise of adsorption, surface reaction

and desorption. There are 7 steps in a heterogeneous catalytic reaction (Fogler, 1992):

mass transfer of the reactant in the bulk phase, diffusion of the reactant inside the

114



Chapter 6 IVIý, d Buildin ur Refinery Heterogeneous Catalytic Reactions

catalyst, adsorption of reactant onto the catalytic surface, reaction on the surface of the

catalyst, desorption of the products from the surface, diffusion of the products from the

interior of the pellet to the external surface, and mass transfer of the products from the

external pellet surface to the bulk fluid, as shown in Fig. 6.2.

nFl

AB

6

----_E lJ -ýýýýý--

11 Eitetnal

diffusion

---------- --

EiInternal

diffusiun

------------------- -----w---- --------

Gc iR1yfiC 5urfDI6

Fig. 6.2 Steps in a heterogeneous catalytic reaction

Understanding of the reaction mechanism has always been the most difficult task for

model building. This has attracted significant attention, but still there is a lack of

prominent theories. The ability to reliably predict the structure of catalyst or catalyst

intermediate is a very important aspect to the model, design and catalyst improvement

(Cundari et al., 1998).

Studies show that through a reaction mechanism in a stabilized catalyst, better activity

depends on the ability of the transition metal components of formation of the active site

on the surface, and the effect of geometry is important, as it affects the accessibility of

active sites.

This information can be obtained directly from a variety of diffraction experiments (X-

ray, neutron, and electron being the most common) or can be inferred from

spectroscopy (NMR, EXAFS, infrared, etc.). Spectroscopic techniques are valuable in

two aspects, firstly, to confirm/exclude potential catalytic intermediates, and secondly

as a tool to reveal chemical information about catalytic intermediates.

115



Chapter 6 Model Budding fur Refinery Heterogeneous Catalytic Reactions

Molecules could be adsorbed on the crystal surfaces, and their structures and bonding

could be studied with the available techniques. For example, low-energy electron

diffraction (LEED), Auger electron spectroscopy (AES), X-ray photoelectron

spectroscopy (XPS), high-resolution electron energy loss spectroscopy (HREELS),

could be used.

Recent progress in developing and improving these techniques for examining surface

and bulk properties of solid catalysts provides encouragement. However, it is unlikely

that a universal theory can be applied to a wide variety of reactions.

A wide range of catalysts has been used in industry applications. Commonly they are

classified into two categories: metal and metal oxide. Hytrotreating catalysts belong to

metal oxide catalysts.

Catalysts are generally composed of three main constituents: supporter, active

ingredient, and promoter.There

are three methodsfor

preparation of catalysts:

precipitation, deposition or impregnation, depending on how the active ingredients are

deposited to the support surface.

Fig. 6.3 Method for special distribution of active ingredients

For hydrotreating catalysts, impregnation is the most widely used method. This method

is faster,and allows

the finalproperty and configuration

to becontrollable.

It is,

116




however, more difficult to prepare high concentration catalyst and to obtain an even

dispersion of catalyst components on the surface. Special techniques (in Fig. 6.3) are

developed todeposit

the catalyst on the surface skin of the pore structure (egg shell) or

on to the inner pore structure (egg yolk) by a competitive chemisorption of special

adsorbate, e.g., citric acid, formic acid or HCI. The eggshell arrangement of catalyst

components is desirable in a diffusion-controlled reaction. The egg yolk arrangement

allows the smaller reactant molecules to contact with the active component of the

catalyst. This is sometimes used in the case where the reactant stream contains impurity

of high molecular weight substances.Both arrangementsenable the saving of precious

metal.

Catalytic reaction-rate expressions can be derived for ideal surfaces by the Langmuir-

Hinshelwood-Hougen-Watson (LHHW) approach. This approach derives rate equations

in terms of surface concentration of adsorbed speciesand free sites and then expresses

theseconcentrations in terms of Langmuir isotherms.

The formulation of LHHW equations involves the postulation of steps3,4 and 5 of the

7-steps and their corresponding rate equations. First, the rate-determining step (RDS)

assumption should be made to simplify the reaction expression for subsequent steps.If

one of the steps is significantly slower than the others, then this determines the overall

rate of reaction and it is known as the rate-controlling step or rate-determining step

(RDS). In the derivation of LHHW models, the RDS is often specified prior to

regression of kinetic data.

It is also usually assumed that the conditions on the surface are stationary. The

combination of the rate equations, the balance on the catalytic sites and equality of the

surface rates indicated by the steady-state assumption allows the elimination of all

dependentterms.

Every reaction system has its own distinguishing features. To be more specific, hydro-

desulphurisation (HDS) processes will be discussed.

117



Chapter 6 Mt, dei f', r 11_cfinery eterogeneou" ýI'atalvtic Reactions

6.2.2 Hydrodesulphurisation (HDS) processes

A refinery is a complex network of interdependent processes,reflecting the result of

advanced chemistry, engineering, and metallurgy. Hydrodesulphurisation processesare

very important units in the modem refinery shown in Fig. 6.4.

Primary Conversion Impurities Light endsSeparation removal upgrading

Naphtha Hydro-

VacuumHigh

DistillatesVacuumDistillation

Vacuum

Product

LPG

NAPHTHA

GASOLINE

JET FUEL

DIESEL

LUBE OIL

FUEL OIL

ASPHALT

Fig. 6.4 Modern refinery

The hydrodesulphurisation process involves net reactions on the catalyst surface to

Feed: +/- 5000 ppm Sulfur

Counts

250000

200000

150000

100000

50000

0

I Propane/Butane

Isomeri-Light Gasoline sation

118

5 10 15 20 25 30

Fig. 6.5 Feed and product chromatogram min




convert the various sulphur compounds to hydrogen sulphide by reactions with

hydrogen. The sulphur compounds in crude oil are present largely in the form of thiols,

sulphides, and various thiophenes and thiophene derivatives, which are difficult to

detect and to measure.For example, the number of different sulphur compounds that are

detected n the chromatogram shown in Fig. 6.5 is enormous.

Each single species goes through a complex reaction schemein hydrodesulphurisation

(HDS), mainly composed of hydrogenolysis and hydrogenation. Fig. 6.6 gives the

reactionschemefor dimethyldibenzothiophene.

SH3C CH3

Hydrogenation

Hydrogenolysisss

H3C CH3 H3C CH3

SH3C CH3 H3C CH3 H3C CH3

Fig. 6.6 Dimethyldibenzothiophene HDS reaction scheme

The general approaches to characterise the fuel mixture are to classify sulphur

compounds into lumping groups, which have similar structure and reactive behaviour.

Every group is represented by a typical component, in the form of pseudo-component.

Fig. 6.7 shows the relative reaction rate of six typical pseudo-components.

119



Chapter 6 Model Building for Refinery Heterugeneu s

100

80

aýcz

c 600U

40aý

czT 20

0

Fig. 6.7 Relative reaction rate of sulphur compounds

The properties of two commercial HDS catalysts are listed in Table 6.3. Ni-Mo/y-A1203

has higher activity for hydrogenation, which is typically used for hydrodenitrogenation

(HDN). The ratio of molybdenum to cobalt is always considerably greater than 1.

Table 6.3 Typical commercial hydrotreating catalysts properties

Chemical content and properties A B

Chemicals (wt % dry basis)

MoO3 15.0 18.5

CoO 3.2

NiO 3.3

Physical properties

Surface area (m /g) 310 180

Pore volume(cm3/g) 0.80 0.53

Diameter (in. ) 0.125 0.062

Average length (in. ) 0.23 0.18

Compactedbulk density 36 52

Average crush strength 4.2 3.1

Fig. 6.8 shows the procedure of catalyst synthesis for hydrodesulphurisation (HDS)

catalysts. This method starts with a support in the desired specification, size and shape;

the precursor of the catalyst is then coated or impregnated onto the surface of the

support by an incipient wetting with a minimum amount of saturated solution of the

precursor mixture. This process may be repeated several times in order to reach a

desirable level of catalyst concentration on the support if all the component salts cannot

120




dissolve at one time. The procedure for the promoter is similar. Once the catalyst is

impregnated onto the support, it is then dried, calcinated or reduced as required. During

the whole procedure, the volume of solution should be controlled to be about equal to

the pore volume of support to ensure total absorption of the solution into the pore

Fig. 6.8 Catalyst synthesis procedure

structure of support and to avoid waste of the active ingredients and avoid causing an

error in composition. Optimum pH of the dissolving solution may differ from one salt to

the other. For example, Co-salt prefers low pH whereasMo-salt prefers high pH.

The catalytic activity and selectivity of hydrotreating catalysts may be very sensitive to

even small changes in the preparation procedures employed. The major factors are:

metal loading (concentration, metal ratio), impregnation procedure (pH, time, order of

addition of metals) and calcination temperature. The effect of preparation method on the

structure, stability and metal/support interaction in Co-Mo/Al203 can be studied by

different methods (Sie, 1993; Khorashen et al., 1998; Adachi, 1996; Leliveld et al.,

1998; Farag, 2002).

Industrial HDS is generally carried out between 573 K and 698 K and pressure from 10

to 200 atm, depending on the feedstock and process requirements. HDS processes are

121




highly exothermic. Hydrogenation is favoured only at low temperature and at high

pressure.

6.3 Model building methodology for HDS processes

Any factor in each of the three aspectsof chemistry, catalyst and the process cannot be

ignored in model building for heterogeneous catalytic reactions. Accordingly,

difficulties and challenges arise in the HDS model building process. For example,

complex catalytic chemistry needs further exploration on the choice of the reaction

scheme, mechanism assumptions or different mixture representations. Effects of

different catalysts are unclear, and no quantitative correlation is available for structure

sensitivity. Accurate data fitting and changing different type of reactor or reactor size

(batch, continuous fixed bed reactor, CSTR) cannot provide future information for

mechanismdiscrimination.

Hytrotreating catalysts are presently going through a major development program. The

primary reason for this development is that the catalysts presently used for

hydroprocessing are quite satisfactory when employed for the lighter crude petroleum,

but the United States and much of Europe may find the lighter petroleum crudes to be

less available, either because the supplies are exhausted, or because of geographic or

political problems. As a consequence, he need to develop processes hat are satisfactory

for heavier crudes, coal-derived liquids, and shale and tar sands oil is evident. In

general, the model building for heterogeneouscatalytic reactions will have duration of

up to 5 years, with a large number of experiments being carried out.

Hydrodesulphurisation has been studied by numerous researchers. However, most

studies have used one or more of the following methods or assumptions to simplify the

model building:

1. Sulphur model compounds are dissolved in pure solvents to simulate

petroleum fractions

2. The reaction schemes are simplified as parallel reactions: hydrogenation and

hydrogenolysis (Studies have indicated that the hydrogenolysis and

hydrogenation reactions occur on separatesites).

122




3. Surface reaction is assumedas the rate-determining step

4. Sulphur compounds are lumped as groups

However, there are some unanswered questions: which kind of adsorption of hydrogen,

molecular or dissociative and which kind of surface reaction, single site or dual site or

Eley-Rideal? Should catalyst active site concentration be constant or variable? Also,

exactly how the catalyst geometric properties affect the reaction rate still cannot be

determined easily. A large number of rival models from the combinations of all possible

assumptionsare inevitable for heterogeneouscatalytic reactions.

For instance. there are 174 rival models for Dibenzothiophene HDS (Vabrysselberghe,

1996) or 15 rival models for Diesel HDS (Hidalgo, 1999) in the literature. It is difficult

to carry out mechanism discrimination without information from catalyst

characterisation.

HDS kinetics of a mixture of sulphur compounds in an industrial feedstock is far more

complicated than that of a pure substance. However, although for process engineers

these power rate equations are simple and can easy predict the effect of the reactor

operating changes on performance, they fail to predict interactions between reactants,

products and the catalyst. The LHHW approach for expressing the reaction rate is used.

Inthis work, reaction model

buildingwill

be divided intothree

levels, inorder to

separate kinetic from diffusion effects. Catalyst characterisation and optimal

experimental design are used in each level to help discriminate among rival models.

Model discrimination in the early stages can help to save experimental effort in the later

stages. Practical operating conditions are preferred. Thus, the best model will be

obtained to provide reliable parameters for process development. The whole procedure

shown in Fig. 6.9 will be discussedin detail in the following sections.

123




Low

External ExtrinsicDiscrimination

,>1 diffusion level400

iscrio,

Internal Apparent

diffusion level

U

No diffusio

Lffect

High

Optimal

experimentaldesign

Catalyst

characterisation

Discrimination

Intrinsic

level

Fig. 6.9 Model building framework

6.3.1 Catalyst characterisation

Heterogeneous catalytic reaction occurs on the surface of catalysts. Therefore, the

surface status is crucial for the performance of catalysts.

There are three main properties related to the catalyst that need to be characterised: bulk

properties, texture properties, and surface properties, as shown in Fig. 6.10.

Bulk properties include:

1) Composition and constituents of a catalyst

2) Pellet size and shape

3) Mechanical strength

4) Bulk density

Texture properties include:

1) Surface area

2) Pore volume

3) Pore size and distribution

Surfaceproperties

include:

124



Chapter 6 Model Building for Refinery Heterogeneous 1_. «tions

1) Adsorptive properties

2) Dispersion

3) Chemical activity

4) Bonding and oxidation states of active components

5) Atomic and crystal structure of active components

Texte re Surface area

Pore size distribution

Bed density Active material distribution

Pore volume 1, ý , Void volume xý .

Pellet shapeBulk Morphology

Pellet sizeStructure

Dispersion

Acidity/

Fig. 6.10 Catalyst properties

In order to have a comprehensive understanding of catalyst properties to study a

reaction mechanism including the function of active components on the surface, to

guidethe design

and synthesis of catalysts andto

choosethe best

performance catalystfor a certain feedstock, it is desirable to obtain a complete description of both the

structures and active sites where the catalysis takes place. Such information can also

serve the purposes of reproducing a known catalyst and providing a specification for

future reference or for a business transaction.

For the purpose of obtaining a clear knowledge of the catalyst properties related to the

kinetic study, characterisation of the catalyst is necessary at every stage of the

development. Characterisation can provide the link between catalyst properties and

kinetic models through catalytic activity to validate the mechanism assumptions by:

1. Reducing assumptions relating to the mechanism

2. Helping to determine kinetic parameters for single reaction steps.

125




With the aid of chemical and physical methods for catalyst characterisation, bulk,

texture and surface properties can be obtained, but not all are necessary for a specific

application. The surface properties determine the nature of the catalysts, while the bulk

and texture properties affect the catalyst performance through masstransfer.

For kinetic studies, information that can clarify the reaction mechanism and accurately

determine the heat of adsorption and reaction intermediates is useful. Recently, more

and more new techniques have been introduced to identify or accurately measure

catalyst properties. These should be incorporated in the kinetic framework of the

heterogeneouscatalysis.

There are a large number of techniques available to characterisecatalyst properties and

measure activities. These include spectroscopic methods, diffraction methods,

porosimetry methods, isotherm adsorption methods, temperature programmed methods,

etc. The experimentalist must be discriminating and select the technique that yields the

most information with accuracy,speed, and economy.

To achieve this, it is necessary o know precisely the objectives of the investigation, the

capabilities and limitations of eachtechnique, and the expected value of the results.

General rules for selecting characterisation techniques are:

" Applicability: real catalysts

" Easy access: common equipment

" Low cost: normal working conditions

" Informative: multipurpose techniques

" Versatile: many different systems

For the purpose of kinetic studies and model discrimination of the HDS process, there

are three useful properties. These are the sulphided state of the metal, which is related to

the active sites of the surface reactions, metal-sulphur binding energy that also plays a

role in the adsorption and desorption, and identification of intermediates that can help to

describe the reaction mechanism.

126



Chapter 6 Model Building for Refinery Heterogeneous Catilytic Reactions

The widely used characterisation techniques that can provide the above information are

compiled in Table 6.4 for -IDS process catalysts. For the time being, building

quantitative correlations between activity and catalytic characteristics still needs further

research. These could be used for design and optimisation of catalysts for a certain

feedstock in the future.

Intrinsic

level No diffusion effect

Apparent

level Internal diffusion control

Extrinsic

level External diffusion control

-Vi l7i ' 77erwd(W Fo

Kinetic model

Fig. 6.11 Catalytic kinetic model composition

Table 6.4 Compilation of techniques for HDS catalysts

T h i

Information

ec n quesSurface composition Intermediates Adsorption heat

XPS

TPD

Calorimetry

EXAFS*

IR

Mossbauer*

127




6.3.2 Catalyst kinetics

Anything related to reaction kinetics cannot be estimated from theory, and must be

determined by experiments. Experimental data is essential for model derivation of

heterogeneouscatalytic reactions, asis the casefor homogeneous reaction systems.

In comparison with homogeneousreaction systems,catalysts play a very important role

in the model building for heterogeneous catalytic reactions. So in catalytic kinetic

studies,catalyst effects are reflected by internal and external diffusion for a preselected

catalyst.

The kinetic studies can be carried out under different sets of experimental conditions,

dependingon whether the kinetic data is obtained in the presence or absenceof internal

or external mass transfer limitations. Therefore, experiments can be conducted in three

levels, intrinsic level, apparent level and extrinsic level, to represent the different mass

transfer effects. Accordingly, reaction models are developed level-by-level, while the

kinetic expression is written in the form of separate erms that represent different levels

(shown in Fig. 6.11).

The Weisz-Prater (1954) and Carberry criteria are used to test if the experimental data

are in the kinetically controlled regime, internal diffusion controlled or external

diffusion controlled regimes. Thus, different experimental data are responsible for the

parameter estimation in different terms of the kinetic expression.

(i) In the intrinsic level of kinetic studies, both internal and external mass transfer

limitations should be eliminated. Generally, experiments are carried out using crushed

fine catalyst particles and under appropriate conditions to achieve the target when the

Weisz-Prater criterion is satisfied.

The Weisz-Prater criterion for internal diffusion control is:

2

(D_ 17102=rv "L n+1

)<O.15

Def" Cb 2

where i7 is internal effectiveness factor

is Thiele modulus

(6.1)

128




Deff is diffusion coefficient

n is reaction order

Cb is bulk concentration

rv is reaction rate per unit particle volume

L is pore length

Concentration profiles inside and outside the catalyst particle are shown in Fig. 6.12.

(ii) Experiments that are conducted using commercial size catalyst, but eliminating only

the external mass transfer resistancesprovide the experimental data for apparent kinetic

study.

In this level, the Carberry criterion for external diffusion control is used. Conditions are

chosen or the experiments suchthat:

C =Cb-CS < 0.05a Gb Inj

where Ca is Carberry number

CS is catalyst surfaceconcentration

Cb is bulk concentration

n is reaction order

(6.2)

When this criterion is satisfied, experimental data are obtained for the parameter

estimation in apparentkinetic term.

(iii) In the third level, experiments are carried out in such a way that neither internal

diffusion resistancesnor the external masstransfer limitations are eliminated. This gives

the extrinsic kinetic model of a reaction where neither the Weisz-Prater criterion nor theCarberry criterion are satisfied.

Diffusion effects are accounted for using effectiveness factors.

rate at catalyst surfaceconditions observedrate= r?e "77i (6.3)

rate at bulk conditions rate at catalyst surface conditions

129




CS CS

Fig. 6.12 Concentration profile

During the model derivation for heterogeneous catalytic reactions, some physico-

chemical constraints must be satisfied.

Accepting that the rate coefficient should obey Arrhenius' law, Boudart et al. (1967)

also derived constraints on the adsorption enthalpies and entropies, which are too often

overlooked. Since adsorption is exothermic, the standard adsorption enthalpy AH° must

satisfy the inequality:

-AH° >0 (6.7)

while -4Sää ,the difference in standard entropy between an adsorbed component at

surface coverage 0.5 and the same component in the gas phase, must satisfy:

0<-ASä <AS°

As a rule, the following limits for the adsorption entropy should be observed

41.8<-AS0 <51.04+1.4x10-3(-AHäý

where the unit of - ASä s J mol"1K-1; the unit of - AH° is J mol-1.

(6.8)

(6.9)

131



Chapter6 Müclýýýui1 iý,ýýk- It_,ineryHeteýýPýýýý,ýu

Fig. 6.13 Laboratory reactors for heterogeneous reactions

6.3.3 Model discrimination

As discussed earlier, catalyst characterisation can provide information to discriminate

mechanisms at the early stage of model building to reduce the number of rival models.

After a set of rival models is obtained, further experimental data are needed for model

discrimination. It should follow the same procedure as for homogeneous reaction

systems to achieve the maximum difference among model performance.

Optimal experimental design for model discrimination is obtained through combinations

of reactor types and operating conditions. The operation modes of laboratory reactors

used for heterogeneous reaction systems are classified and listed in Fig. 6.13. The same

reactor superstructure and hybrid optimisation approach are used as for homogeneous

reaction systems.

In conclusion, to deal with complex heterogeneouscatalytic reactions, kinetic studies

and model discrimination are carried out in three levels, to separatekinetics from the

diffusion effects. Combining the information from catalyst characterisation, the number

of rival models can be reduced in the early stage.The priority for model discrimination

in the three levels is in the order from the intrinsic level, apparent level, to the extrinsic

132




level, in order to save experimental effort. To be more specific, the model building

procedure is interpreted by hydrodesulphurisation (HDS) processes.

6.4 Casestudies

In this section, two HDS processes will be used to illustrate the model building

methodology for heterogeneouscatalytic reactions. These are -IDS processes of typical

single sulphur compounds and a mixture of sulphur compounds.

6.4.1 Hydrodesulphurisation of thiophene

Firstly, a single compound thiophene HDS is used to illustrate the methodology. The

hydrodesulphurisation of thiophene is a well-studied reaction, and a large amount of

experimental data is available. Different researchershave tried to reveal the reaction

mechanism of thiophene and have built different models to represent the reaction

system.

The experimental data (Satterfield, 1968; Lee, 1977) used in this case study to develop

model are listed in Fig. D-1 and D-2 of Appendix D. A small proportion of

experimental data will be held in reservefrom the model building procedure and will be

used ater in the model discrimination stage.

Model 1,2,3,4 (Eq. 6-10,11,12,13) have been developed by Satterfield & Roberts

(1968), Lee & Butt (1977), Morooka & Hamrin (1977), and Van Parijs & Froment

(1986b) respectively. Kinetic parametersfor Model 1 to Model 4 are listed in Table D-1

to D-4 of Appendix D.

Y1=k'PT ' PH2

(6.10)(1+KTpT

+K 2H2S PH2SS

r2 =VPT pH'-

(6.11)(l+KTpT

+K HZS pH2S

3k"pT

"pH2

r=2 (6.12)(i+

KTPT +KH2SPH2s

133




4

r= (1+(KH2PH2

P.s+ KT PT + KH

SPH S/ PH

-where k= k° "exp(

Ea)

RT

0-ýT

KT = KT "exp(RT

)

HZS

KHZS=Kp

HZS

exp(

RT

)

KHZ=K0H,

"exp(H2

RT

The thiophene (T) BIDS reaction scheme s describedas:

J 4H4S+ 3H2 - C4H8+ H2S

C4H8 + H2 ->C4Hlo

The following assumptionsare possible:

1. Surface reaction is the rate determining step

2. Hydrogen adsorption type: either molecular or dissociative

(6.13)

3. Hydrogen and sulphur compounds adsorption: either in the same site or not

4. Adsorption of thiophene: either one point or two points

Eight models are obtained through combinations of the above mechanism assumptions,

Model A to H (Eq. 6.14). Values of the model parameters,m, n, a, b, are listed in Table

6.5.

ra-H = (1+(KH2"PH2

)"+ KT pT + Kx2sPH2s "ý1+(KH2 "PH2)b

Table 6.5 Factorsof Eq. 6.14

k'KT'KH2 *PT'PH2

ode M n a b

A 2 1 0 1

B 1 1 0 1

k'KT''H2'PT'PH2

(6.14)

134



Chapter 6 Model BuiIdiriý, for Refinery Heterogeneous Catalytic Reactions

C 3 0 1 0

D 2 0 1 0

E 2 1 0 0.5

F 1 1 0 0.5

G 3 0 0.5 0

H 2 0 0.5 0

The kinetic parameters of each model obtained from data fitting are listed in Table D-5

of Appendix D. The parity plot of Model A is shown in Fig. 6.14. Model A has a good

agreement with the experimental data, as do the other models.

Model A

500

400

300

200

100

0

"f

ff /

ýy100 200 300 400 500

Fig. 6.14 Parity plot for Model A

Discrimination of Model A to H needs further experimental data. However, extra

experimental data are only available at the following conditions:

" Temperature:623K

0 Total pressure: 780 mmHg

0 Thiophene partial pressure range: 6 mmHg to 18 mmHg

It is further assumedthat experiments will be carried out under the above conditions in a

differential reactor. Therefore, the reaction rate (units: mol "min-1 "gear-' ) can be

135



`hapter6 MOýý.: adding for Refinery Heterogeneous Catalytic ReactionsLIJ

obtained and compared directly instead of conversion. From Fig. 6.15, it can be

observed that model predictions of the 12 models, Model 1 to 4 and Model A to H, have

quite different performances. The maximum point of difference is at the highest

thiophene partial pressure of 18 mmHg. When compared with actual experimental data

at the optimal experimental point, measurement error cannot be ignored. In Fig. 6.16

five rival models (circled) can be eliminated because their predictions are far from the

actual performance.

A B C D

ýE F G H

Satterfiled Morook ^ ý^ Van Parijs Lee

1000

900

800

700

+ 600

° 500

400

300

200

100

0

Fig. 6.15 Model predictions

--ý-ý--- ABCDEFG -HSatterfiled Morook -- -Van Parijs Lee

f data

1000

900

800

700

+ 600

500

400

300

200

..

100

0

5 10 15P(T) mmHg

20

Fig. 6.16 Optimal operation conditions for model discrimination

136

579 11 13 15 17 19

P(T) mmHg



Chapter 6 M( dc 1B, L,

The other models, whose model parameters are updated with the new data, can be

further discriminated by using the data from the next optimal operating conditions.

6.4.2 Hydrodesulphurisation (HDS) of diesel

In the second case study, diesel HDS is chosen because of its importance in the refinery

industry. The sulphur compounds in diesel are represented here by four pseudo-

components in the model building of the diesel hydrodesulphurisation processes:

Benzothiophene (BT), Dibenzothiophene (DBT), Methyldibenzo-thiophene (MDBT),

Dimethyldibenzo-thiophene (DMDBT). Their molecular structures are shown in Fig.

6.17. The reason for choosing these four pseudo-components is to make use of current

experimental data in the literature, because either the experimental data for the single

components are available, or the data for diesel is based on this four pseudo-component

assumption.

\ IS\

ss sCH3

sH3Ci CH3

Fig. 6.17 Pseudo-component of sulphur compounds

In order to maintain consistency, the same feedstock is used during model

discrimination, and other compounds, except sulphur compounds, remain unchanged.

The boiling curve and properties of diesel used in this case are listed in Fig. 6.18 and

Table 6.6.

137




Boiling Curve400

380

360

340ö

320a

300

ö 280as

260

240

220

200

Fig. 6.18 Boiling curve of diesel fuel

Table 6.6 Diesel properties

Total sulphur, wt% 0.706

Density at 15°C, kg/m3 840

Pour point, °C -10

ASTM boiling range, °C 240-380

Aromatics 17

Composition, vol. % Olefins 5

Saturates 78

In HDS processes, the sulphur compound reaction scheme is simplified to be parallel

reactions, hydrogenation and hydrogenolysis. Each reaction takes place on a different

site of the catalyst surface, denoted 6 for hydrogenolysis and i for hydrogenation, which

is agreedby most researchers (Delmon, 1979; Broderick & Gates, 1981; Vrinat, 1983;

van Parijs & Froment, 1986b; Van Parijs et al., 1986)

The options for the reaction mechanism are:

1. a -site:hydrogenolysis

Hydrogen adsorption: molecular or dissociative

Sulphur compound/ H2 adsorption: non-competitive or competitive

138

0 10 20 30 40 % 50 60 70 80 90 100




Sulphur compound adsorption: two adjacent or-sites or one a -site

Rate-determining step: surface reaction and is irreversible

2. r -site:hydrogenation

Hydrogen adsorption: molecular or dissociative

Sulphur compound/ H2 adsorption: non-competitive or competitive

Rate-determining step: surfacereaction and is irreversible

If all plausible reaction mechanisms had been considered, the number of models

formulated would have been of the order of 102.Furthermore, the number of kinetic

parameters s large and their estimation is a difficult task.

So, only two models for the HDS process are developed to illustrate the nature of the

problem. The kinetic models are used to fit the experimental data of conversion against

time reported by Kabe et al. (1992), Bartsch & Tanielian (1974), Singhal et al. (1981),

Ledoux et al. (1990), Kabe et al. (1994,1997), Ishihara & Kabe (1993).

In Model 1, the combination of assumption options is that on o sites, hydrogen

adsorption is molecular; sulphur compound adsorption is non-competitive with H2

adsorption, two sulphur compounds are adsorbed on one o' site. On z sites, hydrogen

adsorptionis

assumed tobe

molecular; sulphur compound adsorptionis

assumednon-competitive with H2 adsorption.

Eq. 6.15 and Eq. 6.16 are the kinetic models for the two sites. The kinetic parameters

are listed in Table 6.7.

Kl6PKH2 O.

H2r= k1

42

1+ KiaP

+ KH2SPH2S "(1+ H2,07H2i=1

rz=ký . (1+

KlrFKH2, rPH2

4

Ki,zP

+ KAPA

(6.15)

(6.16)

139




Table 6.7 Kinetic parameters or Model 1

U -site Z -site

E Ea AH E AHk a K k a K

(kJ/mol) (kJ/mol) (kJ/mol) (kJ/mol)

BT -0.14; 47000.1 -10.7 -104.50 -4.7 89000.2 -7.94 -102.50

DBT -0.2;165000.

-10.81 -166.30 -5.5140000.

-7.36 -97.60

MDBT -2.0;188000.

-10.50 -190.60 -5.8136000.

-7.28 -158.00

DMDBT-3.3;

201000.

-11.64 -217.70 -5.8

222000.

-7.93 -204.80

H2 - - -7.5 -145.07 - - -7.79 -223.36

H2S - - -7.59 -92.46 - - - -

Aromatic - - - - - - -10.06 -120.56

In Model 2, the combination of assumption options is that on or sites, hydrogen

adsorption is molecular; sulphur compound adsorption is non-competitive with H2

adsorption, and two sulphur compounds are adsorbed on two adjacent 6 sites. On z

sites, hydrogen adsorption is assumed to be molecular; and sulphur compound

adsorption is assumed non-competitive with H2 adsorption.

Eq. 6.17 and Eq. 6.18 are the kinetic models for the two sites. The kinetic parameters

are listed in Table 6.8.

K PK Pr=k"ý,

ý x2,ß H2 (6.17)

1+ Ki,, P + K;12SPH2S

(1+ KH2,

ýPH2

KirFKH2 rPH2i. =ki,.

41+KZ,

zP+KAPA

=i

Table 6.8 Kinetic parameters or Model 2

(6.18)

U -siteT -site

kEa

KAH

KEa AH

(kJ/mol) (kJ/mol) 0l)kJ/mol) (kJ/mol)

140



Chapter 6 N1o;i: Building for Refinery Heterogeneous Catalytic Reactions

BT -0.15 77000-10.5 -104.50 -2.0

89000 -7.44 -100.50

DBT-0.19 165000

-10.8 -146.30 -5.0140000

-7.36 -120.60

MDBT -1.9 198000-10.50 -190.60 -5.8

186000-7.28 -158.00

DMDBT-3.5

201000-11.64 -217.70 -6.9

222000-8.03 -204.80

H2 - - -7.5 -140.07 - - -7.79 -223.36

H2S - - -7.59 -102.46 - - - -

Aromatic

- - - - - - -10.06 -220.56

In general, optimal experimental design is to optimise all parameters together, such as

sulphur content distribution, temperature, H2, H2S partial pressure, reactor type and size,

etc. Unfortunately, for such complex reaction systems, the data points cannot be easily

obtained, so it is impossible to compare two model predictions at the randomly

calculated optimal point. Single parameter effects are discussed here to clearly show the

methodology. Firstly, temperature and sulphur compound content effects are considered

without diffusional effects.

6.4.2.1 Temperature effects

The only available experimental data for diesel IDS is from Ma (1994). This tested the

change in conversion with temperature. Two rival models are used to calculate the

behaviour of four pseudo-components (BT, DBT, MIDBT, DMDBT) with the change of

temperature. The objective function is to calculate four pseudo-component prediction

differences of the two models separately.

1

o.8w

0.6

00.4

ö 0.2U

n

f data --- Modell ---A---Mode12

500 550 600 650 700 750

Temp erature(K)

1F-

0.8Q

° 0.6

0Cl) 0.4a)

0.2

0

f data a Model l--- A- - -Model 2

41

500 550 600 650 700 750

Temp erature(K)

141




" data -- -"F----- Model 1--"A---Model 2" data ---*-Model 1---A--- Model 2

0.8

o O.6

,, 0.4

0.20U0

500

E, 1

0.8

0.6

0ö 0.4

0.2

rý 0

"

""i

VAL-

500 550 600 650 700 750

Temperature(K)

Fig. 6.19 Pseudo-componentperformance of different models

It is observed that temperature effects for the two models on four pseudo-components

are similar. Model prediction has the biggest difference on the BT conversion (Fig.

6.19)

6.4.2.2 Sulphur compound addition effects

Ideally, it is preferable to know the effect on model prediction if the sulphur compound

distribution profile is changed. Because changing the content of four sulphur

compounds simultaneously is impractical in the laboratory, optimization of the sulphur

compound content is based on single sulphur compound addition, which can easily be

carried out in the laboratory. The difference of the two models is calculated when the

content of four sulphur compounds is changing individually. It is shown in Fig. 6.20

that the difference of BT between model predictions is more sensitive than the other

compounds. Thus, if change of the sulphur compound content is used for model

discrimination, BT addition is the best choice.

0 BT DBT

0.45

0.4

0.35

0.3

0.25

Q 0.2

ö 0.15

0.1

0.05

0

MDBT DMDBT

140% 180% 260% 420%

Sulphur Compound Addition

Fig. 6.20 Model difference of each pseudo-componentaddition

142

550 600 650 700 750

Temperature(K)




6.4.2.3 Catalyst effects

Catalyst effects on kinetic model building are reflected in the diffusional terms of the

rate expression. If rival models cannot be discriminated from the operating conditions

(T, P, sulphur content distribution, etc.), catalyst effects should be considered for further

model discrimination. Catalyst pellet size, shape, active material density, location, pore

size distribution can all be used for model discrimination. In this thesis, only active

material location inside the pellet and pellet size will be discussed.

1. Active material location

There are several different methods to model active material location, step function, 6-

dirac function, and active material distribution profiles. Here distribution of active

material inside the pellet is represented by a step function, as in Fig. 6.21. The idea is

based on the different diffusion characteristics of the four pseudo-components inside the

catalyst pellet.

0

b

U

d

0Pellet dimension x1

Fig. 6.21 Catalyst active material distribution profile

Diffusivity of BT, DBT, MDBT, DMDBT will be calculated by assuming that the

diffusion of sulphur compounds in the catalyst is purely Knudson diffusion.

Deff = 9700rprýw

where rp is pore radius

T is the temperature K

M is the molecular weight of diffusing species

(6.19)

143




The relative diffusivity can be obtained easily through:

D1_ M2

D2 M1

1

0.96

9.1°0.92

r'0.88

U

0.84

0.8

Model 1

f- BT

--- DBT

MDBT

DMDBT

Model 2

" BT

0.995 ---K- DBT

MDBT

0.99 DMDBT

0.985

j

0.98

0.975

0.970 0.2 0.4 0.6 0.8 100.2 0.4 0.6 0.8

Active material location Active material location

Fig. 6.22 Model performance changing with active material location

(6.20)

From Fig. 6.22, it is observed that the change of active material location from the

surface to the core of the catalyst pellet, has less effect on the compounds with higher

diffusivity because the compounds with the higher diffusivity can easily reach the centre

of pellet. So model discrimination can be done by comparing refractory compounds,

which have a quite different performance.

2. Catalystpellet size

While catalyst size is decreased to reduce the internal diffusion effects, Model 1 shows

the changing point of conversion in the experimental range, but Model 2 shows no such

point (Fig. 6.23). It has been used in the experiments to determine the change of

diffusion-controlled regime. This point can also be used for model discrimination

through changing the catalyst size.

144



Chapter 6.I_

11-Iery etero«en,--()w

i(ý 1ýeactiýýrýý

Model 1

0.998

0.997

0.996

.O. 95

0.994

00.993

0.992

0.991

0.99

-f- BT

--+-- DBT

MDBTDMDBT

0.982

0.981

0.98

.20.9790.978

Ü 0.977

0.976

0.975

0.974

Model 2

3010 20 30 0 10 20(mm)artical size (mm) Partical size

Fig. 6.23 Model performance changing with particle size

In brief, through the above calculation results, experimental conditions for diesel model

discrimination can be recommended as:

" Temperature: 633 - 673 K

0 BT addition: more than 40% of original content

0 Catalyst active material location: centre to surface

0 Catalyst size: 0.1 mm to 5 min

At the above operating conditions, rival models stand the biggest chance to be

discriminated.

6.4.3 Discussion

Two hydrodesulphurisation reaction systems have been used to illustrate the model

building methodology for heterogeneous catalytic reactions: thiophene and diesel HDS.

From no diffusional effects to internal diffusion effects, different ways to discriminate

between rival models have been explored to provide the basic view for model

discrimination of HDS models.

In the future, the relationship between catalyst characterisation and catalyst properties

should be included to effectively develop models and discriminate amongst them.

145




6.5 Conclusions

In this Chapter, a systematicmethodology

formodel

buildingand

discrimination for

heterogeneous catalytic reactions has been explored. Due to the complex nature of

catalysis, a large number of rival models pose difficulties for model building and

discrimination. A three-level kinetic study method has been introduced to reduce the

model complexity by separating diffusion effects from kinetic equations. In addition,

catalyst characterisation has been suggested to assist model discrimination. A

classification of those techniques discussed here provides guidance for selectingtechniques to characterise HDS catalysts with accuracy, speed, and economy. Because

of the importance of hydrodesulphurisation (HDS) processesin the refinery industry,

plausible ways for model discrimination and model improvement for thiophene and

diesel HDS have been explored, including operating conditions, feedstock and catalyst

effects.

146



Chapter7 ConclusionsandFuture Work

Chapter 7: Conclusions and Future Work

7.1 Conclusions

7.2 Future work

7.3 Remarks

147



Chapter 7 ConclusionsandFutureWork

Reaction models for complex reaction systems involving detailed chemistry and

mechanisms are crucial for the purpose of reactor design and optimisation. However,

complex reaction systems, such as processesfor fine

and speciality chemicals and

pharmaceutical chemicals, feature high value and short market window, which leads to

the development of detailed models being unattractive. Processes involving

heterogeneous catalytic reactions also need a systematic approach to quickly and

accurately predict the behaviour, to design and optimise processes for a certain

feedstock to meet the product specifications.

To date, generic model building methodologies applied to complex reaction systems

have mostly been based on empirical rules, where the key features of reaction systems

might be missed. This can lead to the process being operated under non-optimal

conditions, due to the limitation of costs and development time in the procedure of

model building.

In this work, the research objective focuses on providing a new approach to model

building from the chemical engineer's point of view, to avoid important information

being missed and reduce the risk of processscale-upfailure.

A new systematic method to combine the work of chemists and chemical engineers is

proposed in this thesis, aiming to make full useof experimental information, extract the

optimal model suitable for process design, and arrange the minimum experimentsto

save expenseof laboratory and pilot experiments.

This chapter will summarise the proposed methodology, discuss the limitations of the

methodology and recommend future work and potential extension to areas closely

linked to this project.

7.1 Conclusions

With little knowledge of reaction chemistry, feasible reaction schemes are derived

directly from information of the inlet and outlet species of reaction systems and

screened with preliminary experimental data. The participating species are used to

generate the set of all reactions in a two-stage method. An atom-molecule matrix

formulation allows all species to be systematically represented for reaction system

148



Chapter 7 Conclusionsand FutureWork

analysis and can be used to describe reactions in the form of reaction equations. In

addition, the rank of the matrix is an important constraint, which is calculated through a

SVD (singularvalue

decomposition)method.

After generating all possible reactions by a two-stage method, integer linear

programming is used to test the stoichiometric feasibility of the reactions identified

satisfy the mass conservation law. It is guaranteed hat all possible reaction steps and all

feasible reaction schemes are generated. Feasible reaction lists in each stage through

two-stage method are used to construct reaction schemes.

Construction of reaction schemesprovides reaction schemes rom the combinations of

feasible reactions generatedfrom the two-stage procedure.The strategy that searchesall

possible combinations of reaction steps can guarantee that all plausible reaction

schemes are obtained, and has beendiscussed n detail with an example.

Theexperimental

datacan

beused

to identifywhich reaction schemes are of more

interest, and derive kinetic parameters for the reactions within the schemes

simultaneously by optimisation.

Becausemore than one model can fit the sameexperimental data set, it is essential that

all reaction models are obtained before further experiment information is available.

Extracting reaction models from experimental data, including reactions schemes and

kinetic expressions, is a mixed integer nonlinear programming problem (MINLP).

Appropriate optimisation methods should be chosen to guarantee the global optimal

solution.

A hybrid optimisation approach is proposed to identify suitable reaction schemes and

kinetic equations with the aid of experimental data. This approach combines a stochastic

method - Simulated Annealing (SA) algorithm with a deterministic optimisation method

(SQP). The former provides potential candidates as rival models, and the latter is used

to fine-tune model parameters.

Stochastic optimisation is designed to get a set of solutions close to the optimal region

by setting up reaction scheme moves and kinetic moves to generate alternatives and

monitor the random search. The SA algorithm can reduce the risk of model mismatch

149




through avoiding model elimination in the preliminary stage. Through the test of two

cases,SA optimisation combined with NLP has been shown to provide good solutions

for the MINLPproblem.

Further experimental data is required for model discrimination among rival models.

Optimal experimental design should be applied to achieve the target of minimum

experimental effort. Based on the phenomena that reaction models might have different

performances in different types of reactors and under different operating conditions, an

experimental reactor superstructure is used, in which all kinds of laboratory reactor and

operating conditions are embedded. Therefore, all kinds of operating conditions,

including feeding policies, phase contacting patterns, mixing types, recycles and

discharges can be generated within the experimental reactor superstructure during the

maximisation of model differences. In this work, the stochastic optimisation method is

used again to guaranteerobust and fast global solutions.

This approach has been applied to two reaction systems. A small set of models are

readily obtained for model discrimination and optimal experimental design. Becauseof

the lack of experimental data, further validation of models is restricted in the available

experimental data.

A model building methodology for heterogeneous catalytic reactions is much more

complex, due to the complex nature of catalysis, in which mass transfer effects cannotbe ignored. A large number of models exist from the combination of mechanism

assumptions, so the decomposition of complex reaction steps in catalytic reactions for

model building is used. Model discrimination is carried out in the early stages, along

with model building to reduce further experimental effort.

Firstly, characterisation of catalyst property effects on reaction models provides good

insights for discrimination of mechanism assumptions on diffusion, adsorption and

desorption. However, there are a large number of techniques with different capabilities

and limitations available to connect catalyst properties, catalyst activities with model

performance, but not all of thesewill be useful in a given application. A classification of

those techniques discussed here provides guidance for selecting suitable techniques to

yield the mostinformation

withaccuracy, speed, and economy.

150



Chapter 7 Conclusions and Future Work

Secondly, a three-level kinetic study method has beencombined into the model building

methodology to reduce the model complexity by separating diffusion effects from

kineticequations.

Three levels in kineticstudies serve the samepurpose of simplifying

the procedure of reaction model development.

Hydrodesulphurisation (HDS) is an important process in the refinery industry.

Thiophene HDS and diesel HDS have been used as illustrative examples. Plausible

ways for model discrimination and model improvement for thiophene and diesel HDS

have been explored, including operating conditions, feedstock, and catalyst effects.

Nevertheless, there are some limitations when the methodology is applied. The main

limitation of the methodology is that all the reacting species nvolved are assumed o be

detectable. However, it is not always true in some cases,in which either species are

missing or species to be known existing but undetectable. It will add uncertainties for

reaction model building, which might be crucial for systemscale-up and control.

Another limitation involved in the new approach is that thermodynamic feasibility of

reaction systemsis not applied as constrains while reaction stepsare obtained. Only the

wide ranges of general limitations are applied, when the kinetic parameters are

optimized. Thermodynamic feasibility of a reaction is very important as it would let us

know in the preliminary stage f further evaluation of the reaction should be carried out

or not.

The feasibility and condition for the chemical equilibrium is derived on the basis of

energy, enthalpy and entropy. The Gibbs Free Energy (OG ) predicts the feasibility and

equilibrium condition at constant temperature. Hence the Gibbs Free Energy of

formation is very important in the analysis of chemical reactions.

If the change in the Gibbs Free Energy of the reaction is Negative, the thermodynamics

for the reaction is favourable. If the change in the Gibbs Free Energy of the reaction is

Highly Positive, the thermodynamics for the reaction is not favourable.

Values of individual species (reactants and products) are required to determine the

change n the Gibbs Free Energy of the reaction. However, for most of complex reaction

systems thermodynamics properties are lacking especially when biologic systems are

151



Chapter 7 Conclusions and Future Work

considered. To overcome this limitation, methods such as a group contribution method

for the estimation of the thermodynamic properties of reacting species should be

implemented.

Also, it is need to be aware that well-studied reaction stepsand kinetics in the reaction

systemscan help to reduce the searching spaceand improve model accuracy.

7.2 Future work

There isstill a considerable amount of work that needs to

be done inorder to

findthe

best model for reactor design and optimisation.

Some aspectsneed to be considered in the future are:

a) The methodology has been applied to homogeneous reaction systems and

heterogeneous catalytic reaction systems. It would be possible to refine the

method for multiphase reaction systemsthat includes masstransfer effects.

b) Factor sensitivity analysis should be carried out during the optimisation of rival

reaction models. This information can provide the wiser optimisation in order to

reducethe large searchspace.

c) The hybrid optimisation approach can be applied to model reduction of complex

reaction systems, where the strategy of the reaction schemeconstruction is able to

provide the feasibility check of the reduced reaction schemes.

d) For HDS processes, the models proposed here are based on simple pseudo-

component lumping. In most work, analysis, evaluation and quantification of the

real feedstocks have been carried out separately from catalyst characterisation.

Even through much effort has been devoted to establishing fundamental

relationships between the structure of the catalyst and the reactivity, the

correlation is still far from mature. In the future, the characterisation of real

feedstocks (Ho, 2003; van Looij et al., 1998; Shih et al., 1992), such as the feed

density, nitrogen compounds and aromatics contents and boiling point distribution,

152




together with catalyst characterisation, should also be included in the model

building.

e) Catalyst preparation and complex profiles (egg shell, egg yolk, non-uniform, etc)

for the active material location inside a catalyst should also be used for model

discrimination.

7.3 Remarks

Themethodology proposed

in this thesis has beensuccessfully applied to

homogeneous

reaction systems and refinery heterogeneous catalytic reaction systems. The most

significant concept is that of using chemical engineering approaches to provide

guidelines for chemists to develop reaction models for the purpose of reactor design and

optimisation. Furthermore, hybrid optimisation methods can meet the requirement that

more than one model can fit with the same set of experimental data and can avoid the

fact that decisions made early might lead to missing the best options for model building.Finally, organic synthesis chemists can also find the importance of the strategy of

reaction scheme construction in automated reaction scheme generation from existing

elementary reactions.

153



i, j, k integer indices

r reaction rate for ith reaction

Rj net reaction rate forjth species

Ci concentration forjth species

(p index vector

If

setsof combinations of reactants

O setsof combinations of products

0 stoichiometric coefficient matrix

,u1 element of the stoichiometric coefficient matrix

77 hypothesisedmodel response

0=(01,02,

'' *OP)vector of model parameters

(i'2''vector of k dependentvariables.

E upper bound of the coefficient value; error

62 covariance

Ea activation energy

Fj molar flow rate

k Boltzman constant

tk annealing temperature

Lk Marko v chain length

155



y control parameter

ri internal effectiveness factor

17e external effectiveness factor

0 Thiele modulus

Deff diffusion coefficient

n reaction order

Cb bulk concentration

r,, reaction rate per unit particle volume

L pore length

CS catalyst surface concentration

Ca Carberry number

n reaction order

X conversion

W catalyst weight

FO flow rate

Vi stoichiometr is coefficient

rw intrinsic reaction rate.

- AH° standard adsorption enthalpy

156



-ASä

6

difference in standardentropy

hydrogenolysissite

hydrogenation site

157



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171



Appendix A

Calculationof rank of matrix using

Singular Value Decomposition (SVD)

For any matrix A, the column rank is the number of linearly independent columns. The

row rank is the number of linearly independent rows. The row rank and column rank of

amatrix are equal. Hence, the rank of a matrix A is its row rank or column rank.

There are a number of ways to compute the rank of a matrix, Gaussianelimination and

LU decomposition, but when the matrix that is either singular or else numerically very

close to singular, the technique, known as singular value decomposition, or SVD, is the

most reliable. Even though sometimesSVD is time-consuming.

The SVD method is based on the following theorem of linear algebra. Any MxN

matrix A whose number of rows M is greater than or equal to its number of columns

N, can be written as the product of an MxN column-orthogonal matrix U, and

diagonal NxN matrix W with positive or zero elements, and the transpose of an

NxN orthogonal matrix V.

A U

11

=

ý

JL Jý

VJ

The matrices U and V are each orthogonal in the sense that their columns are

orthonormal,

M 1<k<NUikUin Skn

1<n_<N1=

1<k<NUikV

jn=Skn1n<N

i=1

The element wj in matrix W is the singular values of W.

Wl>W2>. WN>O

172



The rank of matrix A, rank(A), is the number of non-zero elements of matrix W. The

SVD decomposition can also be carried out when M<N.

Under any condition, rank(A) <_Min(M, N) is always true. If rank(A) = Min(M, N),

the matrix is a full-rank matrix, otherwise it is a rank-deficient, singular matrix.

173



Appendix B

Linear system solutions

A system of linear equations

al, x, +a12x2+... +alNxN= b,

a21x,+a22x2 +"""+a2NxN= b2

aM1xl + aM2x2 +... +aMNxN= bN

canbe written in matrix form as

Ax=b

where A is aMXN matrix of coefficients ,

al a1., ... aIN

_

a21 a22 "". a2NAM

xNaj

aM1

am2 ... aMN

and x is a column vector of N unknowns

x1

x2X

x N

b is the right-hand side written asa column vector,

bl

bb2

=.

bM

The augmented matrix of A is (Alb):

174



Appendix C

The incidencematrix of a digraph*

A digraph G= (X, U) consists of a set of elements, called vertices, and a list of

ordered pairs of these elements, called arcs. The set of vertices X= (xl,x2,...xn) is

called the vertex-set of G, and the list of arcs U= (ul,u2,...u,n)

is called the arc-list of

G.

The incidence matrix of a digraph D(G) involves the incidenceof vertices and arcs.

It

is defined as anxm matrix in which the entry in row i and column j is

I arc j is incident from vertex i

d13_ -1 arc j is incident to vertex i

0 otherwise

Analogously, the reaction scheme for reaction systems is also a digraph with reacting

speciesset and reaction lists. The incidence matrix is obtained from the stoichiometric

1 výý>0

coefficient matrix where dtv _ -1 vlý <0.

0 vii =0

The incidence matrix and stoichiometric matrix all can be used to describe the

relationship between reactants and products, while the latter one provides quantitative

information.

*Dolan, A., & AIdous, J. (1993). Networks and algorithms: an introductory approach.

John Wiley &sons, London

176



Appendix D

Experimental data for hydrodesulphurisation

ofthiophene

1.Thiophene BIDS model and experimental datafrom Satterfield & Roberts (1968)*

Operating Conditions:

Reactor: differential recycle flow reactor

Total pressure: 780

-

840 mmHg

Temperature range: 235- 265 °C

Catalyst: CoO-Mo03/A1203

Experimental data:

f 235

70

60

° 50

'0 40

30

20

l0

n

®251 :265

;u.

i

ff

ýM

v

0 10 20 30 40 50 60

Thiophene Partial Pressure,mmHg

Fig. D-1 Experimental datafor Model of Satterfield, C. N. (1968)

Kinetic model:

k'pT'PH2

r 12(1+KTpT+KH2SPH2S)

where k=k°" exp(E)

177



KT = K7° "exp(RTT

)

0 -AHHS

KH`S = KHs "exp( RT )

Kinetic parameters

Table D-1 Kinetic parameters or Model of Satterfield, C. N. (1968)

Temperature k KT KH2S Ea- AHT - AHHZS

Omol (mmHg

)-1 (mmH )-1( C) 2goat. min (mmHg) g kcaUmol kcal/mol kcaUmol

235 0.156E-08 0.056 0.042

251 0.164E-08 0.031 0.018 3.7 24 19

265 0.180E-08 0.033 0.0074

* Satterfield, C. N., Roberts, G. W. (1968). Kinetics of thiophene hydrogenolysis on a

cobalt molybdate catalyst, AIChE, J., 14(1), 159-168.

2. Thiophene HDS model and experimental data from Lee & Butt (1977)**

Operating Conditions:

Reactor: internal recycle reactor

Catalyst: COO-Mo03/A1203

Temperature range: 250-313 °C

Partial pressure of thiophene: 20-160 Torr

Partial pressure of H2: 550-750 Torr

Experimental data:

178



f 313 Ilt 300

7

6

500

ö4

0

.

.

.

ý(

275,>( 249

0 20 40 60 80 100 120 140 160

P(T) torr

Fig. D-2 Experimental datafor Model of Lee, H. C (1977)

Kinetic model:

k'pT'PH2

r=1+ KTpT + KH2s

pH2s

2

where k=k°" exp(-E°

)RT

KT = K7° "exp(-RTOHT)

KH= Ko exp(-OHH? s

ZS HZs RT)

Kinetic parameters

Table D-2 Kinetic parameters or Model of Lee, H. C (1977)

%C0VOT

0H2S Ea-

AHT -

ýH2S

mol

öcat. " min- (Torr)2 (Torr)-1 (Torr)-1kcal/mol kcal/mol kcal/mol

3.40E-05 4.91E-08 3.93E-11 11.9 12.2 20.6

** Lee, H. C., Butt, J. B. (1977). Kinetics of the desulfurization of thiophene: reactions

of thiophene and butene, Journal of Catalysis, 49,320-331.

179



3. Thiophene HDS model from Morooka & Hamrin (1977)***

Kinetic model:

k'PT'PH2

r= (l+KTpT+ KHZS'PH2S

Kinetic parameters

Table D-3 Kinetic parameters or Model of Morooka, A. (1977)

Temperature k KT KH2s

kmol(ýC)

kgcat.'s" (kN/ m2)2(kN/m2 )-1 (kN/m2 )-1

250-350 1.43E-08 0.0713 0.0272

***Morooka, A., Hamrin, C. E. Jr. (1977). Desulfurization of model coal sulfur

compounds by coal mineral matter and a Cobalt Molybdate catalyst - I. Thiophene.

Chem. Eng. Sci., 32,125-133

4. Thiophene HDS model from Van Parijs & Froment (1986b)****

Kinetic model:

k"KT 'KH2 PT 'PH2r= (1+ (KH2

'PH2 P.

5+KTPT +KH2SICH2S /PH2

\\

-where k= k° "exp(

E)

-OHKT = KT "exp(

RTT)

Kinetic parameters

Table D-4 Kinetic parameters or Model of Van Parijs I. A. (1986)

ko KT KHs

Ea I KH2

180



kmol

kgcat.'h" (bar)2

(bar)-1 (bar)-1 kJ/mol kJ/mol kJ/mol

5.22E+07 0.00056 91.2 29.9 10.7 0.536

****Van Parijs, I. A., Froment, G. F. (1986). Kinetics of Hydrosesulfurization on aCoMo/y-A1203 catalyst. 1. Kinetics of Hydrogenolysis of thiophene. Ind. Eng. Chem.

Prod. Res. Dev., 25(3), 431-436.

5. Kinetic parameters for Model A to Model H

Table D-5 kinetic parameters

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