**Mathematical Modeling and Control of a Non-isothermal Continuous Stirred Tank Reactor, CSTR**

**ABSTRACT**

Mathematical Models describing the variations in the volume of the system, concentration of reactant (s) yet to react, temperature of the system, and the temperature of the cooling jacket over time in a non-isothermal CSTR that handles a simple, irreversible, first order or second order exothermic reaction in liquid phase were formulated. This work is with a particular reference to the synthesis of propylene from cyclopropane and that of cumene (isopropyl benzene) from benzene and propylene. The models were solved simultaneously by analytical approach rather than the normal numerical approach employed for solving non-linear differential equations. We noticed that the major determinants of the reactants conversion level and the extent of reaction are the feed concentrations, feed temperature and the cooling jacket inlet temperature. The system is found to have a single, locally stable, steady state with periodic (underdamped) behaviors due to the existence of both inherent negative and positive feedback in it. Nonlinear feedforward control equations show that feed flow rate does not have to be changed when feed temperature Ti changes, rather its changes inversely with feed concentration CAi. Again, the cooling -jacket temperature Tc changes linearly with feed temperature Ti and nonlinearly (inversely) with feed concentration CAi.

The models were utilized to explore the dynamic response and the controllers design equations of the system. We noticed from the dynamic response that the system is self regulatory. Also feed forward controller is physically realizable and has two (Gc1 and Gc3) lead elements and one gain-only element (Gc2) controller for the control of concentration CAO and CBO, and a lag element (GC1), gain-only element (GC2) and a lead-lag element (GC3) controller for the control of temperature, To.

**TABLE OF CONTENTS**

TITLE PAGE i

CERTIFICATE OF APPROVAL ii

DEDICATION ACKNOWLEDGEMENT iv

ABSTRACT vi

TABLE OF CONTENTS vii

LIST OF FIGURES xi

LIST OF TABLES xii

LIST OF APPENDICES xiii

CHAPTER ONE

INTRODUCTION 1

1.0 Mathematical Modeling And Control 1

1.1 Aims / Objectives 2

1.2 Significance Of The Study 2

1.3 Scope Of The Study 3

1.4 Limitations Of The Study 4

1.5 Definitions Of Terms 5

CHAPTER TWO

LITERATURE REVIEW 7

CHAPTER THREE

THE THEORY 12

3.1 Chemical Reactions 12

3.1.1 Types Of Chemical Reactions 12

3.1.2 Phase Criterion 12

3.1.3 Reaction Mechanism Criterion 12

3.1.4 Molecularity of Reactions 14

3.1.5 Order of Reaction Criterion 14

3.1.6 Temperature Conditions 15

3.1.7 Heat Energy Requirement 15

3.1.8 Catalysis Criterion 15

3.2 Reaction Progress Variables 15

3.2.1 The Molar Extent Of Reaction 16

3.2.2 Fractional Conversion 16

3.2.3 Rate Of Reaction 16

3.3 Factors That Affect Rate Of Chemical Reactions 17

3.3.1 Effect Of Concentration 17

3.3.2 Effect Of Temperature 18

3.3.3 Effect Of Surface Area Of Reactants 18

3.3.4 Effect Of Pressure 18

3.3.5 Effect Of Catalyst 18

3.4 Chemical Reactors 19

3.4.1 Types Of Chemical Reactors 19

3.4.2 Batch Reactors 19

3.4.3 Steady State Flow Reactors 19

3.4.4 Semi-Batch Reactors 19

3.4.5 Isothermal Reactors 19

3.4.6 Non-isothermal Reactors 20

3.4.7 Continuous Stirred Tank Reactors (CSTR) 20

3.4.8 Plug Flow Reactor (PFR) 20

3.4.9 Fixed Bed Reactors (FBR) 20

3.4.10 Packed Bed With Counter-Current Flow Reactors (PBCCFR) 20

3.4.11 Fluidized Bed Reactors (FLBR) 20

3.4.12 The Case Study 20

3.5 The Principles Of Conservation Of Fundamental Quantities 20

3.5.1 Total Continuity Equation 21

3.5.2 Component Continuity Equation 21

3.5.3 The Equations Of Motion 22

3.5.4 The Energy Equation 23

3.6 Constitutive Balance Equations For Fundamental Quantities 23

3.6.1 Transport Equations 24

3.6.2 Equations Of State 25

3.6.3 Chemical And Phase Equilibrium 25

3.6.4 Chemical Kinetics Rate 26

3.6.5 Dead Time 27

3.6.6 The Case Study 27

CHAPTER FOUR

THE MODELS AND SOLUTIONS 29

4.1 The Models 29

4.1.1 Assumptions 29

4.2 First Order, Simple, Irreversible, Exothermic Reactions 30

4.2.1 Total Mass Balance 30

4.2.2 Mass Balance On Components 31

4.2.3 Total Energy Balance 31

4.3 Characterization of Its State Variables 32

4.4 Second Order, Simple, Irreversible, Exothermic Reactions 36

4.5 Characterization of Its State Variables 37

4.6 Empirical Nth Order Reactions 39

4.7 Solution of The Models 40

4.7.1 Solution of First Order Reaction Models 40

4.7.2 Solution of Second Order Reaction Models 47

CHAPTER FIVE

APPLICATIONS, ANALYSIS, DISCUSSIONS AND CONCLUSIONS 55

5.1 Applications of the Models 55

5.2 Transfer Function of the Linearized Models of The CSTR 55

5.2.1 Transfer Function of First Order Reaction Models 55

5.2.2 Transfer Function of Second Order Reaction Models 57

5.3 The Response of the CSTR System 61

5.4 Steady State Techniques 61

5.4.1 Steady State Techniques for First Order, Nonlinear Models 61

5.4.2 Steady State Techniques for Second Order Nonlinear Models 63

5.5 Dynamic Behaviour of the Linearized Non-isothermal CSTR 64

5.5.1 Dynamic Response for the First Order Reaction Systems 66

5.5.2 Dynamic Response for the Second Order Reaction Systems 69

5.5.3 Characteristics of an Underdamped Response for First and SecondOrderReactions 73

5.6 Design of Feed Forward for the Non-isothermal CSTR 74

5.6.1 Design of Steady State Nonlinear Feed forward Controllers 74

5.6.2 Design of Dynamic Feed forward controllers for the CSTR 76

5.7 Analysis and the Method of Analysis 80

5.8 Discussion of Results 83

5.9 Conclusions 86

5.10 Appendices 87

Appendix A 87

Appendix B 106

References 121

**CHAPTER ONE**

**INTRODUCTION**

**1.0 MATHEMATICAL MODELING AND CONTROL**

This research work is based on the characterization of a processing system, non-isothermal continuous stirred tank reactor (CSTR) and its behaviour using a set of fundamental dependent quantities (mass, energy, and momentum) whose values describe the natural state of the system, and the modeling of a set of equations in the dependent variables which describe how the natural state of the system changes with time.

The study is carried out by the use of mathematical models which are built based on the knowledge of the constitutive equations namely; transport rate equations, equations of state, chemical and phase equilibrium, kinetic rate equations, and dead-time. These were used in characterizing the conservation balances on mass, energy, and momentum. This is done with particular interest on a non-isothermal CSTR reactor for simple irreversible, exothermic reactions in the same phase. Every physical and chemical phenomenon applied, and the balance equations developed were all from the macroscopic viewpoint so as to moderate the size and complexity of the emerging models.

Since the fundamental variables cannot be measured conveniently and directly, other variables which can be measured conveniently, and when grouped appropriately determine the values of the fundamental variables, were selected. Thus mass, energy, and momentum can be characterized by variables (state variables) such as density, concentration, temperature, pressure and flow rate which define the state of the system. The equations that relate the state variables (dependent variables) to the various independent variables are derived from application of the conservation principle on the fundamental quantities and are called state equations.

The study was considered for a single, irreversible, unimolecular first order reaction, bimolecular second order reaction, and the empirical nth order reaction occurring exothermically in the same phase. The developed models were solved simultaneously by numerical and analytical approach employed for solving non-linear differential equations.

The dynamic responses of the system were analyzed and the steady state and the dynamic feed forward controllers design equations were derived. Such other information that may be very necessary for the proper understanding of our mathematical models was also treated.

**1.1 AIMS/OBJECTIVES OF THIS STUDY**

This work is aimed at the formulation of a mathematical representation of the physical and chemical phenomena (temperature, density, pressure, concentration, flow rate, etc). taking place in a non isothermal continuous stirred tank reactor, CSTR which handles a simple, irreversible first or second order exothermic reaction in same phase. The mathematical model is to describe the variations in the volume of the system, concentrations of the reactants, temperature of the system and the temperature of the coolant over time.

Other objectives that this model is called on to satisfy or perform are to ensure the stability in the operation of the chemical reactor, and to suppress the influence of external disturbances on the reactor. The purpose at this stage is to translate all the important phenomena occurring in the physical and chemical processes into quantitative mathematical equations. The models give the understanding of what really make the process “tick”, enable one get to the core of the system to see clearly the cause-and-effect relationships between the variables. The work also gives a physical application of the solution of the models obtained.

**1.2 SIGNIFICANCE OF THE STUDY**

Mathematical models can be useful in all area of life, and as in chemical engineering, it is useful in all phases of chemical engineering, from research and development to plant operations, and even in business and economic studies. Most often the physical equipment of chemical process we want to design and control has not been constructed. Consequently, we cannot experiment to determine how the process reacts to various inputs and therefore we cannot deign them and their appropriate control system. But even if the process equipment is available for experimentation, the procedure is usually very costly. Therefore, we need a simple description of how the process reacts of various inputs, and this is what the mathematical models can provide to the process and control engineer.

Uses Of This Mathematical Model Are As Follows.

(1) Research and development.

(2) Design of chemical processing equipment and their control.

(3) Plant operation and optimization is cheaper, safer, and faster done on a mathematical model then experimentally on an operating unit.

**1.3 SCOPE OF THE STUDY**

The investigation reported in this project bothers on the formulation of a mathematical representation of the physical and chemical phenomena occurring in the state system; non-isothermal CSTR from a microscopic view point. In the work, the principles of conservation of fundamental quantities (mass, energy, and momentum) were applied using already well-developed constitutive models. We did not go into driving a reaction mechanism, kinetic rate equations, transport rate equations, equations of stage, chemical and phase equilibrium equations but the well developed models from a number of postulated mechanisms were used to correlate the constitutive fundamental quantities.

The verified mathematical models for the temperature and concentration-dependent terms of the rate equation for first order, second order, and nth order reactions were used and not developed in this investigation Levenspiel (1972). The models were derived for a unimolecular first order, bimolecular second order and nth order reactions respectively.

The emerged models were simultaneously solved by analytical and numerical approach employed for solving nonlinear differential equations, and the comparison of the solutions and the subsequent analysis were properly carried out. We considered the dynamic responses of the system, developed its transfer functions (inputs-output interaction), and subsequently the steady state and dynamic feed forward controllers design equations.

The research is particular to a non isothermal continuous stirred tank reactor, CSTR that handles a simple, irreversible exothermic reaction in liquid phase to enable us predict the rate mechanisms, rate equations and reaction occurring in it. The result obtained may be extrapolated to cover a simple, irreversible endothermic reaction in same phase to a good accuracy.

**1.4 LIMITATIONS OF THE STUDY**

There are series of difficulties that were encountered in an effort to develop a meaningful and realistic mathematical description of this chemical process – a non-isothermal continuous stirred tank reactor, CSTR. Serious difficulties occurred due to incomplete knowledge of the physical and chemical phenomena taking place in the reactor. Even an acceptable degree of knowledge is at times very difficult. How to account for the effect of alteration of the value of the overall heat transfer co-efficient caused by scaling, fouling etc during the operation of the reactor became a limitation.

Also we have considered only the first, second, and empirical nth order kinetics to describe the reaction rate.

**Cite this article:**Project Topics. (2021).

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