modeling control optimization fixed bed reactor

MODELING, CONTROL, AND OPTIMIZATION

OF FIXED BED REACTORS

by

KISHOR G. GUDEKAR, B.S.

A DISSERTATION

IN

CHEMICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

December, 2002

ACKNOWLEDGEMENTS

I would like to express my sincere thanks to my advisor Dr. James B. Riggs for

his financial support, guidance, and patience throughout the project. I would like to

express my thanks to Dr. Karlene A. Hoo for her guidance in the project. I would also

like to thank Dr. Theodore F Wiesner, and Dr. Surya D. Liman for being a part of my

dissertation committee.

There are many people who have influenced my life. I am grateful to the

Kawathekar family (Rohit, Gouri, and Anuya) for their constant support, love and care.

Special thanks to Govindhakanan for his constant encouragement and motivation during

the times of frustration and disappointment and enlightening the views about life.

I am also grateful to the centaur2 group (Shriram, Shree, Parag, Alpesh, Namit,

Satish, Mukimd, Makrand, Kulin, and Dungar) for making my stay in Lubbock pleasant. I

am thankful to the rapchick group (Rahul, Sameer, Milind, Sachin, Simil, Puru, Doctor

Sunil, Vijay, Robin, Kirti, and Vinay) for making my stay memorable in Lubbock. I

cannot forget those late night parties and oxir regular visits to the recreation center.

I would like to thank my fellow graduate students Dale Slaback, Eric Vasbinder,

Danguang Zheng, and Tian for making my stay pleasant in the department. I wish to

express my thanks to Matthew Hetzel for his help with the computer problems.

Most importantly, this could not have been possible without constant support,

love, and encouragement from my parents, my brother and sister, and other family

members and friends back home.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS n

ABSTRACT vi

LIST OF TABLES viii

LIST OF FIGURES ix

CHAPTER

1 INTRODUCTION 1

2 LITERATURE SURVEY 5

2.1 Modeling of Fixed Bed Reactor 5

2.2 Solution Procedure 9

2.3 Fixed Bed Reactor Control 10

2.4 Optimization 12

2.5 Multiplicity, Bifurcation Theory and Stability 12

3 MODEL DEVELOPMENT FOR A VINYL ACETATE REACTOR 18

3.1 Generalized Dynamic Model for a Fixed Bed Reactor 20

3.2 Steady State Vinyl Acetate Reactor Model 24

3.3 Orthogonal Collocation 30

3.4 Catalyst Deactivation Model 36

3.5 Nomenclature 38

111

OPTIMIZATION OF A VINYL ACETATE REACTOR 44

4.1 Model VaUdation 45

4.2 Offline Optimization Approach 54

4.3 Sensitivity Analysis 56

4.4 Onhne Optimization 58

4.5 Nomenclature 62

MODEL DEVELOPMENT FOR ETHYLENE OXIDE PROCESS 64

5.1 Process Description 64

5.2 Reaction Chemistry and Mechanism 67

5.3 Kinetics 68

5.4 Mathematical Modeling Assumptions 69

5.5 Mathematical Model of Ethylene Oxide Reactor 74

5.6 Orthogonal Collocation 77

5.7 Modeling Equations for Steam Generator 85

5.8 Modeling Equations for Gas-Gas Heat Exchanger 86

5.9 Modeling Equations for Separation System 88

5.10 Catalyst Deactivation Model 88

5.11 Nomenclature 89

OPTIMIZATION AND CONTROL OF ETHYLENE OXIDE PROCESS 94

6.1 Model Validation 94

6.2 Offline Optimization Approach 103

6.3 Contirol of Ethylene Oxide Reactor 106

IV

6.4 Nomenclature 110

7 BIFURCATION ANALYSIS OF ETHYLENE OXIDE PROCESS 112

7.1 Bifurcation Study of an Industrial Ethylene Oxide Process 112

7.2 Continuation Algorithm to Develop Bifurcation Diagram 113

7.3 Stability of Steady State Solutions 114

7.4 Results and Discussions 115

7.5 Runaway Boundary 120

7.6 Closed-Loop Nonlinear Bifurcation Analysis 124

8 CONTRIBUTION 132

9 DISCUSSION, CONCLUSIONS, AND RECOMMENDATIONS 135

BIBLIOGRAPHY 139

ABSTRACT

In this work, modeling and optimization of an industrial vinyl acetate reactor, and

modeling, optimization, control and bifurcation analysis of industrial ethylene oxide

process is performed.

For a vinyl acetate reactor, a steady state two-dimensional homogeneous model is

developed. The catalyst activity is expressed as a nonlinear function of catalyst age, shell

side coolant temperature and the moderator used in the reaction. Offline optimization is

carried out for the vinyl acetate reactor using a steady state reactor model to find an

optimal operating temperature profile, which maximizes the profit of the process.

Updating the model parameters online does online optimization.

The ethylene oxide process studied consists of a feed effluent heat exchanger, a

multitubular fixed bed reactor, a steam generator, and a separation system. The

exothermic heat of reaction from the reactor is removed by passing coolant on the shell

side of the reactor. A portion of the heated coolant is passed through a steam generator to

produce steam, and the total coolant stream is recycled back to the shell side of the

reactor. A single-loop PID control system uses the flow rate of the coolant that is passed

through the steam generator to maintain the inlet temperature of the coolant to the

reactor.

A rwo-dimensional heterogeneous dynamic model is developed for a catalytic

multitubular ethylene oxide reactor. The catalyst deactivation is modeled as a nonlinear

function of operating time and temperature of the reactor. Sequential quadratic

VI

programming (SQP) is used to solve this nonlinear programming problem. An optimal

temperature profile is found which maximizes the profit over the existing operating

conditions for the fixed run length of the reactor.

The open-loop and closed-loop stability studies are conducted using the

benchmarked model of an ethylene oxide reactor system. Steady-state nonlinear

bifurcation analysis is performed to identify the multiplicity in the heat integrated

ethylene oxide reactor system. The effect of manipulated (flow through steam generator)

and disturbance (reactor inlet carbon dioxide composition) variables are addressed. An

analysis of the stable control region of the system is developed as a function of operating

temperature, catalyst activity, and disturbance direction and magnitude.

Vll

LIST OF TABLES

2.1 Main fixed bed catalytic processes 5

2.2 Classification of fixed bed reactor models 6

3.1 Comparison between reactor model with and without pressure drop 26

3.2 Deactivation rate forms: Power law forms 37

4.1 Comparison between industrial data and model prediction for reactor outlet composition 47

4.2 Comparison between industrial data and model prediction for temperature along the reactor 48

4.3 Comparison between industrial data and model prediction over the catalyst hfe.50

4.4 Model Parametric Sensitivity Analysis 57

5.1 Comparison between reactor model without pressure drop and model with pressure drop 70

5.2 Effectiveness factors, rj^, r]^ for catalyst activity a = \ 72

5.3 Effectiveness factors, r]^, t)^ for catalyst activity a = 0.93 73

5.4 Effectiveness factors, r]^, 772 for catalyst activity a = 0.11 73

6.1 Comparison between industrial data and model prediction for reactor outiet composition (mole %) 97

6.2 Comparison between industrial data and model prediction for temperatures 98

6.3 Percentage profit improvement over the base case for different production

rates. .106

VIU

LIST OF FIGURES

4.1 Comparison between model prediction and industrial data for the average radial temperature along the reactor 48

4.2 Comparison between industrial data and model prediction (ethylene reactor outlet composition wt%) 51

4.3 Comparison between industiial data and model prediction (oxygen reactor outlet composition wt%) 51

4.4 Comparison between industrial data and model prediction (acetic acid reactor outlet composition wt%) 52

4.5 Comparison between industrial data and model prediction (vinyl acetate outlet composition wt%) 52

4.6 Comparison between industiial data and model prediction (carbon dioxide reactor outlet composition wt%) 53

4.7 Comparison between industrial data and model prediction (reactor outlet temperature) 53

4.8 Optimization Procedure 55

4.9 Comparison between base case temperature profile and optimum temperature profile 56

4.10 Schematic of online optimization implementation 59

4.11 Comparison between offline and online temperature profile 60

4.12 Comparison between offline and online temperature profile using a filter 61

4.13 Comparison between offline and online temperature profile 62

5.1 Schematic of ethylene oxide process 66

5.2 Collocation element 82

5.3 Collocation on finite element 84

IX

6.1 Comparison between industrial data and model prediction for ethylene reactor outlet composition 100

6.2 Comparison between industrial data and model prediction for oxygen reactor outlet composition 100

6.3 Comparison between industrial data and model prediction for ethylene oxide reactor outlet composition 101

6.4 Comparison between industrial data and model prediction for carbon dioxide reactor outlet composition 101

6.5 Comparison between industrial data and model prediction for reactor outlet gas tempeature 102

6.6 Comparison between industrial data and model prediction for reactor outlet coolant temperature 102

6.7 Comparison between industrial data and model prediction for reactor inlet coolant temperature 103

6.8 Comparison between base case temperature profile and optimum temperature profile 105

6.9 Schematic of the reactor inlet coolant temperature control system 106

6.10 Response of controUed variable to 1 deg C increase in set point 107

6.11 Response of manipulated variable to 1 deg C increase in set point 108

6.12 Response of controlled variable to 0.5% change in the disturbance 109

6.13 Response of manipulated variable to 0.5% change in the disturbance 109

7.1 Bifurcation diagram using the flow through steam generator as a continuation parameter 116



7.4 Bifurcation diagram using the reactor inlet CO2 mole fraction as a

contmuation parameter 11 g

7.5 Bifurcation diagram using the reactor inlet CO2 mole fraction as a continuation parameter 119

7.6 Bifurcation diagram using the reactor inlet CO2 mole fraction as a continuation parameter 119

7.7 Temperatiare of the catalyst particle and its normalized sensitivity with respect to the flow through steam generator 122

7.8 Boundary of the runaway region 123

7.9 Locus of bifurcation points for different catalyst activity 123

7.10 Closed-loop stability region for Carbon Dioxide disturbance change in the positive direction 125

7.11 Closed-loop stability region for Carbon Dioxide disturbance change in the negative direction 126

7.12 Effect of detuning factor on the runaway boundary 127

7.13 Effect of operating temperature on the stability region 128

7.14 Comparison between runaway boundary for different catalyst activity 129

7.15 Temperature slope for different catalyst activity 130

7.16 Response of the outlet temperature to eliminating oxygen in the feed when runaway observed 131

CHAPTER 1

INTRODUCTION

The discovery of solid catalysts and their apphcation to chemical processes led to

a breakthrough of the chemical industiy. The major part of the catalytic processes of

today's chemical and petroleum refining industries is carried out in fixed bed reactors.

For economical production of large amounts of product, they are usually the first choice,

particularly for gas-phase reactions. Many catalyzed gaseous reactions are amenable to

long catalyst life (1-10 years); and as the time between catalyst change out increases,

annualized replacement costs decline dramatically, largely due to savings in shutdown

costs (Rase, 1990). It is not surprising, therefore, that fixed bed reactors now dominate

the scene in large-scale chemical product manufacture.

Inherent in the design of fixed bed reactors rests some of the most difficult control

problems found in the industry. These problems are due to the process being distributed

and nonlinear, and having nonminimum phase characteristics and dead time. In addition,

other complicating phenomena such as exfreme parametric sensitivity of the steady state

profiles may occur in some reactor. The incentive for online optimization is obvious on a

fixed bed reactor with slowly varying catalyst activity. Here constant conversion can be

achieved by increasing the inlet temperature.

The objectives of this research work are to study the modeling, control,

optimization, and stability of industiial multitubular fixed bed reactors for gas phase solid

catalytic reactions with or without heat integration. In this regard two very important

reaction systems, production of vinyl acetate and ethylene oxide, are shidied.

Vinyl acetate is mainly used for making poly-vinyl acetate (PVAC) and vinyl

acetate copolymers, which are widely used in water-based paints, adhesives, paper

coatings and applications not requiring service at high temperatures. The features of the

vinyl acetate process are:

1. Nonlinear partial differential equation (PDE) mathematical model;

2. Tradeoff between vinyl acetate selectivity and reaction rate.

Ethylene oxide (EO) is the world's second most important ethylene-derived

chemical (after polyethylene) based on ethylene consumption. Currently, the major use

for EO is in the manufacture of ethylene glycol, which is mainly used as a raw material

for producing polyethylene terephthalate. The features of the ethylene oxide process are:

1. Nonlinear partial differential equation (PDE) mathematical model;

2. Tradeoff between ethylene oxide selectivity and reaction rate;

3. Runaway reactions that produce carbon dioxide.

In both of the processes, the solid catalyst loses activity over a period of operation

time due to impurities in the feed and sintering of the catalyst. The operating temperature

in the reactor is increased to compensate for the loss of activity, but increased operating

temperatures favor the complete oxidation over the partial oxidation of ethylene and thus

decreases the selectivity of the desired product by producing more byproducts (i.e.,

carbon dioxide and water). Therefore, there is a need for the optimization of the reactor

operating temperature profile over the length of the operation that will improve the

selectivity of the desired product thus maximizing the profitability of the process. This is

true for both the ethylene-oxide and vinyl-acetate processes.

In case of ethylene-oxide production, the complete oxidation of ethylene (which

produces carbon dioxide and water) liberates eleven times more heat than the partial

oxidation of ethylene (which gives ethylene oxide) at high operating temperatures. At

high operating temperatures, the ethylene-oxide reactor is sensitive to operating

conditions. Therefore, a stability analysis of the heat-integrated ethylene-oxide reactor

system will be useful in understanding the different stable/unstable operating regimes.

This kind of study will be helpful to carry out the operation in a safe manner. Bifurcation

theory has been recognized as a very useful tool to address the stability analysis of this

nonlinear system.

The organization of the dissertation is outlined below.

In Chapter 2, literature is reviewed for modeling, control, optimization, and

stability analysis of multitubular fixed bed reactors. The generalized model of a

multitubular reactor for a gas-phase, solid-catalyzed reaction is discussed in Chapter 3. In

the same chapter, the vinyl-acetate reactor model and catalyst-deactivation model are

described. Chapter 4 discusses vinyl-acetate reactor model and catalyst-deactivation

model benchmarking, model verification, offline and online optimization procedure, and

results. A detailed model of the ethylene-oxide process, which includes reactor, catalyst

deactivation, steam generator, heat exchanger, and separation modeling, are discussed in

Chapter 5. Ethylene-oxide process model benchmarking, model verification, optimization

results and confrol study are discussed in Chapter 6. Chapter 7 discusses the open-loop

and closed-loop bifurcation analysis of the ethylene oxide reactor. Chapter 8 discusses

the contribution of this work to science and technology Finally, conclusions and

recommendations are discussed in chapter 9.

CHAPTER 2

LITERATURE SURVEY

2.1 Modeling of Fixed Bed Reactor

Catalytic gas-phase chemical reactions play an important role in chemical

industry. Such reactions are carried out in a multitubular reactor, in which each tube is

packed with a catalyst. Some of the main fixed bed catalytic processes are listed in Table

2.1 (Froment, 1974).

Table 2.1 Main fixed bed catalytic processes

Basic chemical industry Petrochemical industry Petroleum refining

Steam reforming

Carbon monoxide conversion

Carbon monoxide methanation

Ammonia synthesis

Sulfuric acid synthesis

Methanol synthesis

0x0 synthesis

Ethylene oxide

Ethylene dichloride

Vinylacetate

Maleic anhydride

Phthalic anhydride

Cyclohexane

Styrene

Hydrodealkylation

Catalytic reforming

Isomerization

Polymerization

Hydrodesulfurization

Hydrocraking

Steady-State modeling and simulation stiidies of fixed bed reactors have been

covered in a number of surveys by Froment (1974), Schmitz (1976), Hofinann (1979),

and Hlavacek (1977, 1981). Froment (1974) proposed a general classification of fixed

bed reactor, which is shown in Table 2.2.

Table 2.2 Classification of fixed bed reactor models

Pseudo homogeneous models Heterogeneous Models

r = 7;,c = c. T ^T^,C^C^

One-dimensional basic, ideal

axial mixing

Two-dimensional radial mixing

interfacial gradients

intraparticle gradients

radial mixing

where

C - Catalyst surface concentration,

C Gas bulk concentration,

T^ Surface temperature of the catalyst,

T Bulk temperature of gas.

Pseudo-homogeneous models lump the gas and solid (catalyst) phases together in

the reactor modeling mass and energy balance equations. When it is assumed that reactor

temperature and composition only change in the axial direction of the reactor and do not

change in the radial direction, it is caUed a one-dimensional model. When it is assumed

that the flow velocity is constant across the reactor and does not depend on the radial

direction, the resulting model is called a one-dimensional plug-flow reactor model. In

addition, A'ery few data are available to date and no general correlation could be set up for

the velocity profile (Froment, 1990). Sometimes in addition to plug flow, axial mixing is

considered to account for non-ideal conditions in the reactor, then the resulting reactor

model is called the dispersed plug-flow model or simply the axial-dispersion model. The

length of the industiial fixed bed reactors removes the need for reactor models with axial

diffusion. Hlavacek and Hofmann (1970) have shown that for ammonia, methanol, and

oxo-synthesis and in ethylene, naphthalene, and o-xylene oxidation, there is no need to

account for the effect of axial mixing. The overall model is still one-dimensional in

nature. For nonadiabatic reactors, there is a heat transfer across the wall of a tube, which

generates radial temperature and concentration gradients. When these radial gradients are

considered in the model, the model becomes two-dimensional.

For very rapid reactions with important heat effects, it may be necessary to

distinguish between conditions in the fluid and on the catalyst surface or even inside the

catalyst. In case of heterogeneous models, gas and solid phases are modeled as separate

mass-balance and energy-balance equations by considering interfacial gradients of

temperature and concentiation. The heterogeneous model becomes more complicated

when the temperature and concentiation gradients (i.e., intraparticle gradients) inside the

catalyst are accounted for. Fortunately, even with strongly exothermic reactions, the

catalyst is practically isothermal. The main resistance inside the catalyst is to mass

transfer, and the main resistance in the film surrounding the catalyst is to heat transfer

(Weisz, 1962, Carberry, 1961).

2.1.1 Catalyst deactivation

Catalysts frequently lose an important fraction of their activity while in operation.

There are primarily three causes for deactivation.

a. Structural changes in the catalyst: There are different kinds of solid-state

transformations that can occur in the variety of catalysts used in chemical industry. For

example, in the case of a catalyst in which alumina is used as a carrier in y -modification

can transform into a -modification due to prolonged effect of temperature. Sometimes

amorphous silica is used as a carrier in the catalyst, which can deactivate by changing

into crystalline form by the effect of temperature or the presence of impurities in feed. In

some cases, the texture of the catalyst is changed, which can be revealed by a change in

the pore-size distribution in a catalyst. In chromia/alumina catalysts, segregation of the

components has been shown to occur, but deactivation has also been shown to occur

through the formation of solid solution. Sintering of metals loaded on a support also leads

to deacti\ation, for example, with Pt/alumina catalysts used in the reforming of

hydrocarbons. When a V2O5/M0O3 catalyst is used for the oxidation of benzene into

maleic anhydride in a fixed bed reactor, the M0O3 has been shown to migrate down

stream from the hotspot.

b. Poisoning: Essentially irreversible chemisorption of some impurity in the feed

stream is termed as poisoning. Metal catalysts are poisoned by a wide variety of

compounds. For example, in case of hydrogenation reaction, Pt is poisoned by sulfur.

Basic compounds can readily poison acid catalysts.

c. Coking: Deposition of carbonaceous residues from the reactant, product, or

some intermediate is termed coking (Froment, 1990). Many petroleum refining and

petrochemical processes, such as the catalytic cracking of gasoil, catalytic reforming of

naphtha, and dehydrogenation of ethyl benzene and butene hydrofining are accompanied

by the formation of carbonaceous deposits, which are strongly adsorbed on the surface,

somehow blocking the active sites. Levenspiel (1972) mentioned simple equations to

describe deactivating porous catalyst particles.

2.2 Solution Procedure

Both the homogeneous and heterogeneous dynamic fixed bed reactor models are

described by a set of partial differential equations which may be solved using one of the

several types of techniques: Crank-Nicolsen (Eigenberger and Butt, 1974), orthogonal

collocation (Michesen et al., 1973), and orthogonal collocation on finite elements (Garey

and Finlayson, 1975). Froment (1961, 1967) integrated a two-dimensional homogeneous

model using a Crank-Nicholson procedure to simulate a multitubular fixed bed reactor.

However, orthogonal collocation has been the dominant method used for solving the

fixed bed reactor model equations. The orthogonal collocation method proves to be faster

and more accurate than the finite-difference method used (Finlayson, 1971). To this end,

various authors have applied collocation using two or three radial collocation points to

the solution of the reactor model equations. Bonvin et al. (1983) has applied collocation

to nonadiabatic tiibular reactors using quadratic radial profiles (i.e., two radial collocation

points). Jutan et al. (1977) solved the steady-state reactor equations using two collocation

points for temperatiire in radial direction and three collocation points for concentration in

the radial direction. Ampaya and Rinker (1977) and Finlayson (1971) examined the

convergence of collocation solutions to steady-state reactor equations as the number of

radial collocation points is increased. Both of these papers show that collocation points

placed at the roots of the Jacobi polynomials lead to faster convergence of the solutions

as the number of collocation points is increased.

2.3 Fixed Bed Reactor Control

Fixed bed reactor control has been reviewed in Denn (1969), Padmanaban and

Lapidus (1977), and McGreavy (1983). Fixed bed reactors are more difficult to control

because of the process being distributed and nonlinear and having nonminimum phase

characteristics and deadtime. Control of fixed bed reactors is also important because of

sensitivity towards disturbances, or more seldom because of lack of stability of the

reactor. Jorgensen (1986) reviewed the contiol-design method, which has been apphed on

fixed bed reactors to fulfill one or more of the confrol objectives. For tubular fixed bed

reactors with exothermic reactions, it is necessary to control simultaneous peak

temperature and exit concentration for preventing bed temperature from being excessive

or runaway. Jutan et al. (1977) and Jorgensen et al. (1990) have stiidied control of fixed

bed reactors based on a local linearized confrol. However, a linear confroUer works better

if the reactor is operated in a small range around a nominal steady state. The fixed bed

10

reactor may experience disturbance or significant set-point changes from an online

optimizer. Therefore, it is important to develop and implement nonlinear control

strategies for fixed bed reactors to allow tight operation of both controlled variables

available over a wide range of conditions. Xiangming Hua et al. (2000) has discussed the

procedure for developing a nonlinear inferential cascade control of exothermic fixed bed

reactors. The developed cascade structure provides some important benefits for control of

fixed bed reactor such as allowing multiple control objectives, hot-spot position

movement, hard constraint handling on both state and control variables, reduction of

dynamic coupling between loops, and effects of disturbances. The cascade structure

combined with inference of output variables can greatly improve reliability and

robustness. The proposed control strategy was applied to a commercial-scale fixed bed

reactor for phthalic anhydride synthesis. It was shown that the proposed control strategy

could achieve tight control of exit conversion and stabihzation of hot-spot temperature

over a wide range of operations. K. S. Lee et al. (1985) have devised a scheme of online

optimizing control for a nonadiabatic fixed bed reactor in which the partial oxidation of

n-butane to maleic anhydride was carried out. The optimizing control scheme was

designed to perform two functions concurrently: adaptive control of a bed temperature

and online identification and optimization.

11

2.4 Optimization

The optimization of the fixed bed reactor over the length of catalyst life is

important because the catalyst activity changes over the course of the operation. The

operating temperatiire is increased over the life of the catalyst to compensate for the loss

of catalyst activity. Taskar (1995) discussed the optimization of catalytic naphtha

reformer, which is a fixed bed reactor, and described the formulation of the dynamic

optimization problem. Orthogonal collocation (Biegler and Cuthrell, 1985) and control-

vector parameterization (Biegler, 1990) can be used to solve a nonlinear dynamic

optimization problem.

2.5 Multiplicity. Bifurcation Theorv, and Stabihty

2.5.1 Multiplicity

The occurrence of more than one steady-state solution for the same operating

conditions can be demonstrated theoretically for all models, except for a one-dimensional

plug-flow reactor model. The possibility of more than one solution is due to interaction

between dispersion and reaction. This subject has been treated by Hlavacheck and Van

Rompay (1981) and Jensen and Ray (1982). They have both reviewed experimental

findings of multiple steady states. In the cases where multiple steady state occurs, it is

possible to have high sensitivity of the reactor conditions towards changes in inlet

conditions or in parameter values. Cases of high parametric sensitivity occur near

bifurcation of steady states, i.e., where multiplicity of steady state shows up. Puszynski et

al. (1981) show that multiple steady states may occur in nonadiabatic packed beds for

12

strongly exothermic reactions even when the Peclet (Pe) number is large, as in most

mdusfrial reactors. In adiabatic reactors, a multiplicity of three ,s possible, whereas more

steady states can be found m the nonadiabatic fixed bed reactors (Jorgensen, 1986).

Schmitz et al. (1987) have reviewed multiplicity and mstabilities m chemically reacting

systems.

2.5.2 Bifurcation Analysis

Several articles dedicated to the bifiircation analysis of fixed bed reactors have

been published in the past years. Jensen et al. (1982) apphed static and Hopf bifurcation

theory for PDEs for the special case of a first-order, irreversible reaction in tubular

reactor with axial dispersion. The bifurcation behavior was classified and summarized in

parameter space plots. Although the analysis was based on the pseudo-homogeneous

axial-dispersion model, it can readily be applied to other reaction-diffusion equations

such as the general two-phase models for fixed bed reactors.

The mathematical models of many lumped-parameter, chemically reacting

systems consisted of a set of algebraic equations that could not be reduced explicitly to a

single equation. The Liapunov-Schmidt procedure (Balakotaiah et al., 1985), reduced the

prediction of the local multiplicity features of a system of algebraic equations to the

analysis of the features of a single equation even though the original set of equations

could not be reduced to a single equation. The technique may be used also to analyze the

behavior of distributed-parameter and dynamical systems.

13

Balakotaiah et al. (1989) indicated that the presence of small axial thermal

conductivity in a packed bed reactor and small impact of thermal expansion on the

pressure change in the reactor enable thermoflow multiplicity to occur under practical

conditions. Thus, it is important to check for its occurrence in the design of muhi-tube

reactors, as its presence can lead to undesired radial gradients and corresponding

mechanical stresses.

The multiplicity behavior of a non-isothermal, heterogeneous axial-dispersion

reactor model was examined by Juncu et al. (1994). This model took into account both

external and internal heat and mass transfer, with different spatial directions for the fluid

and solid phases. An irreversible first-order reaction in the adiabatic and the

nonisothermal, nonadiabatic regimes was considered. The method of orthogonal

collocation was used to discretize the boimdary-value problem. The analysis was based

on a continuation technique with local parameterization performed on the orthogonal

collocation discrete approximation of the model. The effect of Peclet {Pe) number on the

multiplicity behavior was investigated for a fixed value of adiabatic rise parameter. It

appeared that the introduction of thermal-axial dispersion and the intraparticle mass and

heat transfer increased the total number of steady states to eleven. Similarly, the

homogeneous axial-dispersion model admitted, at most, five solutions. Thus the

combination of the two independent multiplicity sources (i.e., axial dispersion and

heterogeneity) produced more steady-state solutions than either one of them.

Bifurcation theory and numerical continuation techniques were used by Wagialla

et al. (1995) to investigate the complex static and dynamic characteristics of fixed bed

14

reactors modeled by a heterogeneous cell model. The cell model discretized the length of

the reactor to cells each having a length equal to the catalyst pellet diameter (or

characteristic length). The nonlinear kinetics, high activation energy, thermal coupling of

reactor cells and the heterogeneous nature of the reactor were all interacting and

confributing factors to this complex behavior. For a particular set of system parameters,

three general regions of static bifiircation behavior exist in the heterogeneous fixed bed

reactor. In the first region, below a critical feed temperatiire value, single extinguished

steady states exist. In the second region, a multitude of steady states exists, ranging in

number from 3 to a maximum of 29 states. In the third region, above a critical feed

temperattire, three steady states exist: the upper one as an ignited state and the other two

as low extinguished states.

Balakotaiah et al. (1996) have shown that for the case of distributed models in one

spatial dimension, the singularity theory combined with the Liapunov-Schmidt reduction

and shooting method can serve as a useful tool, but for systems described by more than

one spatial dimension, this method cannot be used.

A complete bifurcation analysis of a general steady-state, two-dimensional

catalytic monolith reactor model that accounted for temperature and concentration

gradients m both axial and radial directions is studied by Balakotaiah et al. (2001) A

single exothermic first-order reaction was considered. The analytical results given here

could lead to a quick order of magnitude estimation of the influence of various design

and operating parameters on the monolith behavior.

15

Chang (1984) presented a bifurcation approach to nonlinear systems stabilized by

a conventional proportional-integral-derivative (PID) controller.

2.5.3 Stabihty

Several researchers investigated parametric-sensitivity behavior of fixed bed

catalytic reactors. They used different criteria to find a critical boundary between stable

and runaway operating regions. The pioneering theory of thermal explosions by Semenov

(1928, 1959) was developed originaUy on the assumption of negligible reactant

consumption. This assumption was obviously violated in most real systems; however, its

simplicity and explicitness allow one to have a fiindamentally correct and synthetic view

of the mechanism of thermal explosion. Based on physical intuition, Thomas and Bowes

(1961) proposed to identify thermal runaway as the situation in which a positive second-

order derivative occurs before the temperature maximum in the temperature-time plane.

The criterion proposed by Thomas and Bowes was examined further by Adler and Enig

(1964), who found that it was more convenient to work in the temperature-conversion

plane than in the temperature-time plane. The runaway criterion derived by Welsenaere

and Froment (1970), originally for runaway in a homogeneous tubular reactor, defines

criticality using the locus of temperature maxima in the temperature-conversion plane.

All the above mentioned criteria are based on the idea of defining runaway operations

using some geometric feature of the temperature profile in time or in conversion.

Morbidelli and Varma (1988) developed a generalized criterion for runaway, which is

based on the concept of parametric sensitivity. Criticality is defined as the sitiiation where

16

the normalized sensitivity of the temperature maximum to any of the physicochemical

parameters of the model, is a maximum. Thus, this criterion predicts a parametrically

sensitive or runaway region, which may be called "generalized" since the maximum

temperature becomes simultaneously sensitive to small changes of any of the model

inputs. Along the lines of using parametric sensitivity to identify the boundary for

runaway or explosive behavior, Vajda and Rabitz (1992) have considered the sensitivity

of the temperature trajectory to arbitrary, unstructured perturbations applied at the

temperature maximum. Another sensitivity-based cnterion has been presented by Strozzi

and Zaldivar (1994), which uses the Lyapunov exponents to define sensitivity.

17

CHAPTER 3

MODEL DEVELOPMENT FOR A VINYL ACETATE REACTOR

In this chapter, a generalized mathematical model for a multi-tubular fixed bed

reactor is discussed. A multi-tubular fixed bed reactor is used mainly for gas-phase

catalytic reactions. A detailed mathematical model of the fixed bed reactor consists of the

following aspects in its mass and energy balance equations.

1. Axial dispersion: Mixing of the components in the reactor in the axial direction is due

to the turbulence and the presence of catalysts in the fixed bed reactor. It is accounted

for by superimposing an effective transport mechanism on the overall transport by

plug flow. The flux due to axial dispersion is described by a formula analogous to

Pick's law for mass transfer or Fourier's law for heat transfer by conduction. The

proportionality constants are effective diffusivity and conductivity in the axial

direction (Rase, 1990).

Fick's law for mass fransfer:

ac,

Fourier's law for heat transfer:

N- =-D ' dz

, dT

2. Radial dispersion: Adiabatic commercial reactors exhibit no significant radial

gradients since no heat transfer is involved across the wall of the tube. Nonadiabatic

fixed bed reactors can develop significant radial-temperatiire gradients because of the

heat transfer at the wall. Temperature gradients will produce radial-concentration

gradients as well. These gradients of temperature and concentration occur when large

maxima (hot spots in exothermic reactions) or large minima (low temperatures in

endothermic reactions) occur. To formulate the flux of heat or mass in the radial

direction, it is superimposed on the transport by overall convection, which is the plug-

flow type (Rase, 1990).

Fick's law for mass transfer:

^jr = - A I dCj \

Fourier's law for heat transfer:

3. Radial velocity gradients: The flow in a fixed bed reactor deviates from the ideal

pattern (plug flow) because of radial variations in flow velocity and mixing effects

due to the presence of catalyst.

4. Interphase mass and heat transfer resistance: Since the components (reactants,

products) and energy must move from the catalyst interior, the exterior catalyst

surface, and the bulk fluid phase, there exists a resistance to mass and heat fransfer

between the gas and sohd phases. Since both heat and mass transfer coefficients are

increased by increasing the mass velocities, it is possible and practical in most

industrial reactors to avoid significant interphase gradients.

5. Infraparticle mass and heat fransfer resistances: As the catalyst-pellet size increases, a

significant concentt-ation gradient between the surface and the interior can develop for

19

all the components. Conversely, if the catalyst size is held constant and the

temperatiire is increased, a similar gradient may occur due to more rapid consumption

of the reactants because of the exponential rate increase caused by increased

temperature. In both cases, the rate varies with position inside the catalyst. The

effectiveness factor can be used to calculate the actual reaction rate that would be

observed. The effectiveness factor is defined as follows:

r] = rate of reaction with catalyst pore diffusion resistance

rate of reaction with surface conditions

6. Pressure drop in fixed bed reactor: Pressure drop through a reactor is seldom more

than 10% of the total pressure and thus is not a major factor in changing the chemical

reaction rates in a gaseous reaction. The Ergun equation has been preferred for

several years to calculate the pressure drop in the reactor (Rase, 1990):

dP_

dz a {\-ef fx

D. u + iS

(}-ey- Pf D.

3.1 Generalized Dynamic Model for a Fixed Bed Reactor

3.1.1 Material balance

The material-component balance for a fixed bed reactor in terms of the

concentration can be written as follows (Rase, 1990):

Bulk gas-concenfration balance:

dt dz dz

(d'C J \dCj\

dr' r dr ^Ac^-cJ (3.1)

20

Surface concenfration balance:

When the reaction throughout the catalyst is uniform we can write the following

equation for the surface concenfration:

ldC^\

dt (3.2)

3.1.2 Energy balance

The energy balance for a fixed bed reactor can be written as follows:

Bulk temperature balance:

PfC pf dT_

dt dz ^PfCpfT)+\--Y + K

dz'

(d'T \ dT\ — r + dr r dr

-KA-T,y^{T-T„)

(3.3)

Surface temperature balance:

When the reaction rate throughout the catalyst is uniform we can write the

following equation for the surface temperature of the catalyst:

P-^CA-^\ = Kp''^-Tpypst{-^\r)r„ V 5^ /

(3.4)

Metal temperature balance:

In some cases, the heat transfer resistance through the metal wall can be very

significant; therefore, it is important to write a metal temperature balance in the fixed bed

reactor modeling:

^ ^ (^/ - dr y^-dx (r, - r„)- d,K (T^-T) (3.5)

21

Shell-side temperature balance:

Most of the industrial reactors use a coolant to remove the exothermic heat of

reaction. The coolant temperature profile on the shell side of the reactor is modeled as

foUows:

3.1.3 Initial and boundary conditions

The following initial and boundary conditions can be used to solve equations 3.1

through 3.6:

dC^ dT z = L, —^ = 0, — = 0.

dz dz

The above flux conditions at the reactor entrance and exit is termed as the

Danckwerts boundary conditions (Danckwerts, 1953).

z = 0, T=T ^3 C C.O

, = 0, ^ = 0, ^ = 0 (3.7) dr dr

dr dz

= 0,z = 0, C . =C^o= T = T^, T^=T^,,, T^=T„,o^ T. T =T

22

3.1.4 Intraparticle resistance

When the resistance to mass and heat transfer inside the catalyst (i.e., intraparticle

resistance) is important, the above equations do not adequately describe the system.

Instead of equations 3.2 and 3.4 the following equations are used for surface

concentration and surface temperature. The rest of the equations (3.1, 3.3, 3.5 and 3.6)

remain the same.

(dC,:\ D^ d ^ ,dC^j s,J

\ ^ / r^ dr^ dr^ ^ V. r (3.8)

p.c4^14:^('--=l^)-''.2(-^)"'-- <^'> dt j r^ dr^ dr^ jz^

3.1.5 Initial and boundary conditions

The equations are subject to the following initial and boundary conditions.

z = 0, u ( c , . „ - c J = - £ - D . ^ , p,-u-C^,{r„-T)=-K^

2 = 0, 2:=r,,„

dC^ ^ dT ^ z = L, — ^ = 0, — = 0

dz dz

, = 0 ^ = 0, ^ = 0 (3-10) dr dr

dr dz

, . 0 iE^.^.O ' dr. dr.

c

23

d „ riC

d „ HT

2 ^ ' 'dr^

t = Q, z = 0, C.=C.„C^^=C.^„ T = T„ T^=T„^^, T^=T^^^

22 Steady-State Vinyl Acetate Reactor Model

3.2.1 Vinyl acetate process

Vinyl acetate is mainly used for making poly-vinyl acetate (PVAC) and vinyl-

acetate copolymers, which are widely used in water-based paints, adhesives, paper

coatings, and applications not requiring service at high temperatures. Vinyl acetate is

produced by vapor phase oxidation of ethylene and oxygen in the presence of acetic acid

on a silica-supported catalyst in a multi-tubular fixed bed reactor. Water and carbon

dioxide are the byproducts. The reaction is exothermic, and the heat liberated from the

reactor is removed by generating steam on the shell side of the reactor. The reactions are

irreversible and the reaction rates have an Arrhenius-type dependence on temperature

(Luyben, 1998).

3.2.2 Reaction chemistiy and mechanism

Ethylene and acetic acid are activated as the result of the abstraction of hydrogen

by palladium in the catalyst. Vinyl acetate is produced as the result of the combination of

dissociately adsorbed acetic acid with dissociately adsorbed ethylene, and this surface

24

reaction is the rate determining step in industrially important reactions. Co-catalysts such

as potassium or other alkali metals promote the abstraction of hydrogen from acetic acid

and weaken palladium-oxygen bonds in dissociatively adsorbed acetic acid (Nakamura et

al., 1970). There are four possible reactions in this system:

C^H, + CH.COOH + O.5O2 ^ CH^ = CHOCOCH, + H,0

0.5C,H, +1.50, -* CO2 + H^O

CH^ = CHOCOCH, + +2.5O2 -^ 2CO2 + H^O + CH.COOH

CH^COOH + 2O2 -^ 2C0^ + 2H,0

Based on the kinetics developed in the industrial facility, the last two reactions are

eliminated and not considered in developing the rate expressions.

3.2.3 Kinetics

The rate expressions for both reactions were obtained from the industrial facility.

Due to the confidentiality of data the kinetic rate expressions are not given here.

3.2.4 Mathematical modeling assumptions

The vinyl-acetate reactor model is derived from the generalized fixed bed reactor

model, with the following assumptions:

1. Neghgible axial dispersion:

When the catalyst bed depth exceeds about 50 catalyst particle diameters, the

effect of axial dispersion of heat and mass on conversion is neghgible (Rase, 1990). For

25

the industrial reactor system stiidied, this criterion is satisfied, and therefore the axial

dispersion term is not considered in the modeling equations.

2. Negligible pressure drop:

Pressure drop through a reactor, though seldom more than 10% of the total

pressure, is not a major factor in changing the chemical reaction rate in a gaseous

reaction. In the indusfrial case stiidied, the pressure drop is approximately 5%; therefore,

the effect of pressure drop is neglected in the modeling equations. This assumption can

be verified by comparing the reactor outlet composition and temperature for a model with

and without pressure drop. The Ergun equation, as described above, is used to calculate

the pressure drop in the reactor. The following table shows the error between the model

with and without pressure drop for the reactor outlet composition and temperature (Table

3.1).

Tables 3.1 Comparison between reactor model with and without pressure drop

Process variables at reactor outlet Absolute error

Ethylene composition (mole%) 0.0654

Oxygen_composition (mole%) 0.0498

Acetic acid composition (mole%) 0.0138

Vinyl acetate composition (mole%) 0.0202

Carbon dioxide composition (mole%) 0.0133

Gas temperature composition (mole%) 0.0990

26

3. No concenfration and temperatiire gradient betiveen the catalyst particle and the bulk

gas:

Concentration and temperatiire gradients betiveen the catalyst exterior surface and

the fluid are usually negligible in commercial reactors. For very moderate reactions with

moderate heat effects, it may not be necessary to distinguish between the conditions in

the gas and on the catalyst surface, or even inside the catalyst. In pseudo-homogeneous

models, it is assumed that inter-phase composition and temperature gradients are

sufficiently small. Nakamura et al. (1970) have shown that the catalyst surface reaction

(i.e., production of vinyl acetate) is the rate-determining step. Therefore, m the present

work, it is assumed that diffusion effects inside the catalyst are lumped into the kinetic

parameters of the reaction rate expressions. The above assumption can be justified by

using Mears' criterion (1971a, b) for detecting the onset of interphase gradients. If the

following Mears criterion is satisfied, then the observed reaction rate will deviate less

than 5 % from the true chemical rate,

(-AH)-r-ps-d^ RT ——^<0.15-^^.

2-h^-T E

In the vinyl acetate reaction system,

^ - ^ ^ • ^ • ^ - • ^ - = 0 . 0 0 3 1 6 6 , a n d 0 . 1 5 - ^ = 0.011. 2-h^-T E

Thus, in this case, the Mears criterion is satisfied. Therefore, the interphase gradients

can be easily neglected in the model equations.

27

4. Flat velocity profile:

A reasonably flat profile can be assumed when the ratio of ^ ^ 30 ( Rase

p

1990). This criterion is readily met in the industiial reactor stiidied.

Based on the above assumptions a homogeneous model is developed for the vinyl

acetate reactor, which is described by the following 10 steady state partial differential

equations.

Defining the following dimensionless variables.

/ - t / * _ ^ 2 " 4

CO*. ''°'

Y * _ ^2

lor.

r

0^*

H^O"-

Ar* =

. )

r

'1

O2

Ar

T * _ stm

Z =

(T.

z

1

CH,*- ^"\

-T ) Urn ^ R /

lor.

Dimensionless Equations:

EthyleneBalance:^§?^ = i ^ - - ^ ( / - ^ § ? ^ ) - - M _ . ( , , ^ 0 . 5 - r , ) (3.11) dz R Vr dr dr VCjH^

O x y g e n B a l a n c e : ^ = i ^ - ^ ( / ^ ) - - ^ - ( 0 . 5 . , + 1 . 5 . , ) (3.12) dz R Vr dr dr VCjH^

28

AceticAcid Balance : ^ ^ ^ = i ^ - i _ ( / ^ ^ ^ P,L dz R'V/d/^ dr' ^~~VCW' ^^-^^^ -2^^^

Vinyl Acetate Balance: ^ = i ^ - i - ( / ^J^. ^ P,L dz R'Vr' dr' dr' ^ VC H °

-I'-^H

CarbonDioxide Balance: ^ ^ = l£rLJ_(/ ^£91^^ . P^L

2." ."4

(3.14)

IT^^V/^^' -l/^^WW^'^ ^^-''^

Water Balance : ^ ^ = - ^ - L ( / M 2 ^ ) ^ _ P ^ . , ^ , . ., ,,, dz' R'Vr'dr'^ dr' ^ VC,H;^' '^ ^^ ^

x:^u D I dC,H^* sDL d , . dC,H' Ethane Balance:—=—— = (r —^ ^) t7 M\

dz' R'-Vr'dr'^ dr' ^ ^^'^^^

Nifrogen Balance :—^ = — ^ — . ( r — ^ ) (3.18) dz R Vr dr dr

- , , ^ , dCH,' sD^L d , . dCH,', Methane Balance: — = —-^ (r =-) G 19'i

dz R'Vr dr' dr' ^ ^ ^

. ^ , dAr' sD^L d , . dAr', Argon Balance :—^ = - — ^ — - ( r -—-) (3.20)

dz R'Vr dr dr

Temperattire Balance:

^=(_4^x,j_r).—^^Ac^-^)-^^'^"^' ''^'^^'^-dz pX^jr R P,C,,Vr dr dr lOT.Vp^C^ lOT.VpX,

(3.21)

The above partial differential equations are subject to the following boundary conditions:

z=o, T'=T:, C;=C.;

29

dC. dT' r =0, - 4 - = 0, ^ = 0

dr dr

r * = l .

r' =\.

dC. ^ 0 dr' "°'

dC' — ^ = 0. -

dr ^ """ '

dr hR dr dr A, k'-'^.:\Bk-TSr

3.3 Orthogonal Collocation

Equations (3.11)-(3.21) are converted into ordinary differential equations by

orthogonal collocation (Finlayson, 1980). Two radial collocation points are considered

for temperature and three collocation points are considered for concentration. The

reduced ordinary differential equations are solved using LSODE (Hindmarsh, 1986)

integrator. The developed model predicts composition and temperature profile in the axial

and radial directions of the reactor.

3.3.1 Orthogonal collocation for radial diffusion

The orthogonal collocation method has proved to be a useful method for problems

of diffusion (Finlayson, 1980). In many of these problems, it is possible to prove that the

solution is a symmetric function of r (where r is the radial coordinate), i.e., a fimction

of only even powers of r . To do this we construct orthogonal polynomials that are

functions of r ' . One choice is

y{r') = yi^)^{\-r')\a,P,_M''). (3-22) /=1

30

An equivalent choice is

y{r') = -2b,P^_,{r'). (3.23)

Equation (3.23) can be simplified as follows:

y{r') = J^d,r''-\ (3.24) / = 1

We define the polynomials to be orthogonal with the condition

1

fWir')P,{r')P^irydr = 0 k^m. (3.25)

We take the first coefficient of the polynomial as one, so that the choice of the weighting

function W{r') completely determines the polynomial, and hence the trial function and

the collocation point (Finlayson, 1980). Differentiating equation (3.24) we get:

^ = y j , ( 2 ^ - 2 ) r ^ ' - \ (3.26) dr ,,1

Now differentiating equation (3.26), we get

V ^ = J-4f . ^ ] = yj,.(2/-2l(2/-3>lp- (3.27)

r dr\ dr j jr(

Now the collocation points are N interior points in the range 0<r. < 1 and one boundary

point r . , = 1. The point r = 0 is not uicluded because the symmetry condition requires

that the first derivative be zero at r = 0 and that condition is already buih into the frial

function. Expressing equations (3.24), (3.26) and (3.27) at a collocation point we get.

31

ff,)=|V-(2,-2X (3.29)

N + l

^'yirj) = ^W\r--\^_^d^ (3.30) i-\

Writing equations (3.28), (3.29) and (3.30) in mafrix notation we have,

y = Qd, ^ = Cd, V-y = Dd, (3.31)

where

Q,-r;^-\ C,=(2z-2>/-, Dj,.=V'-(^--%. (3.32)

Solving for d gives,

^ = CQ-'y = Ay ^ = DQ-'y = By . (3.33)

dr dr

Thus, orthogonal collocation can be used to convert differential equations into algebraic

equations.

3.3.2 Collocation solution of the steady state Vinyl Acetate Reactor model

3.3.2.1 Temperatiire collocation. A quadratic polynomial is used to collocate

radial temperature derivatives. To find quadratic orthogonal polynomials, let

W{r) = (1 - r " ) , which gives the Jacobi polynomial. The polynomials are

PQ =1, P^ =l + br' .To find b we use the following orthogonality condition,

1 1

CW(r)P,P^rdr = 0 or f(l-r'){l + br')rdr = 0 0 0

which makes b = -3.

32

Therefore P, = 1 - 3r^ The positive root of the quadratic polynomial (a collocation

point) P, IS r, = 0.57735 which is an interior collocation point. The boundary collocation

point is selected asr^ = 1.0

From equation (3.32),

e = 2 "1

Q-' =

1 0.3333

1 1

1.5 -0.5

-1.5 1.5

C = 0 2r,

0 2r.

D =

0 1.1547

0 2

ro 4' 0 4

This gives

A = •1.732 1.732

- 3 3 B =

- 6 6

- 6 6

The resulting polynomial can be written as follows:

T{r,z) = Q-T{r„z)

nr„z)

r(r ,z) = (-1.5r'+1.5)r(r,,z) + (1.5r--0.5)7(^2,z).

Thus the temperature at any radial location can be found as a function of the temperatures

at the two collocation points.

33

The radial partial derivative of temperature at the collocation points can now be

replaced using the collocation derivative formula, obtained by differentiating the

interpolating polynomial

I d ( dT\ 7 a ^ i ' " a 7 j _ =^^'^(^>'^)-^^^2n^2,^) K = I,..,N.

The boundary condition of heat removal at the wall is approximated using the

derivative formula for the collocation:

hJT^-T{r„z)) = -XJA,Jir„z) + A,,T(r„z)).

The equation can be solved for r(r,, z) in terms of Tir^, z), or vice versa, so that one of

the temperatures can be ehminated.

3.3.2.2 Concenfration Collocation. A quartic trial function is the lowest order

polynomial that satisfies the boundary conditions and still allows a non-constant profile

for concentration along the radial direction. If a quadratic polynomial were used for the

concenfration profiles, it would reduce to a uniform radial concenfration profile, because

this is the only quadratic that satisfies the boundary conditions of zero slope at r = 0 and

r = 1. Hence, a quartic polynomial was chosen for the radial concentration profiles.

To find quartic orthogonal polynomials, let W{r) = (1 - r^) The polynomials are

P^ = l, P, = 1 - 3/-^ P, = 1 + cr' + dr''. To find c,d we use the following orthogonality

condition:

34

JW(r')P^P^dr = 0 and CW(r')P^P^dr = 0 0 0

which makes c = - 8 , J = 10. The results are Po =1, P 2 = l - 3 r ' , P = l - 8 r ' + 1 0 r '

The positive roots of the quartic polynomial (collocation points) P are

/-] = 0.393765, r, = 0.803087, these are interior collocation points. The boundary

collocation point is selected as rj = 1.0 . The resulting interpolating polynomial for

concentration is:

C(r,z) = Q-

C(r„z)

C{r^,z)

C{r„z)

C(r,z) = (2.415816r'-3.973894r'+1.558078)C(r,,z)

+ (-5.749150r^ +6.640561r' -0.891412)C(r2,z)

+ (3.333333/ -2.666667r- +0.333333)0(^3,z).

The radial partial derivative in concenfration at the collocation points can now be

replaced using the derivative formula by differentiating the above equation:

- T - f ^ ^ l =5^,C(r„z) + 5^,C(r„z) + 5^3C(r3,z) r dr\ dr)^^^^

K = l..M.

The boundary condition of zero mass flux at the wall is incorporated using the

collocation formula for the derivative to give:

dC r = R,

dr = 0,

r = R, A,,C{r„z) + A,X{r^,z) + A,,C{r„z) = 0

35

This equation can be used to eliminate one of the concentrations at a collocation

point in terms of the other two, so that each concentration derivative can be expressed as

a function of only two radial concentration values.

3.4 Catalyst Deactivation Model

For gas-phase sohd catalyzed reactions, the catalyst deactivation process is very

slow and the catalyst is used for several years. The change in the catalyst deactivation is

offset by increase in the operating temperature, which causes the increase in the catalyst

deactivation rate. This is due to sintering and agglomeration of catalyst at high

temperatures. Simple empirical forms (Table 3.1) have been proposed (Levenspiel, 1972)

and shown through numerous experimental observations to correlate the observed data

satisfactorily. It is always advisable to try several forms that make sense for the situation

at hand and then use the one providing the best fit.

36

Table 3.2 Deactivation rate forms: Power law forms

Deactivating Process Rate form

Sintering — = -k.a" dt '

Poisoning and fouling — = -k^a" dt

da

A-^R^P i

dt Parallel: — = -k^P^ a

A^R

A^ P i

Series: ^ = -k,Pj,'"a" da

dt

da , ^m n Impurity Deactivation: — = -^d^p ^

A^R

P^P i

In the vinyl acetate reactor, it is beUeved that the deactivation is due to sintering

of the catalyst. Also, since the moderator is used to promote the vinyl acetate reaction, the

amount of moderator also affects the activity of the catalyst. Therefore, catalyst

deactivation is expressed as a fimction of the operating time, shell side temperattire T,

and moderator feed rate F^ .

37

a = Qxp{-a-tyJ whent^^^t^

a = exp(-^ -t^^)- exp(-y • T) • Fj" when t^^>t^J> T^

a = exp(-^ • f^J • exp(-r7 • T) • F^" when t^^>t^,T^ T^

3.5 Nomenclature

A - Limiting reactant

Ar - Dimensionless argon concenfration

a - Catalyst surface area per unit volume of the reactor ( )

m^

Bi Biot number

*

C2//4 -Dimensionless ethylene concenttation

C,Pfg -Dimensionless ethane concentration

C//4 - Dimensionless methane concenttation C-,H4 - Ethylene concenttation at the reactor inlet (—r- ) m^

CO2 - Dimensionless carbon dioxide concenfration

C^„ - Heat capacity of bulk gas (—-—) ' ° kg- K

Cp^ Average heat capacity of gas ( —)

C„, Heat capacity of the coolant ( —) PC i^ y kg°K

38

C Heat capacity of the metal ( ) k g ° K '

Cp Heat capacity of catalyst ( —) kg °K

Cj Bulk Concentration of component j in the gas phase ( ^ )

C^j - Surface Concenfration of component j ( ^ ^ ^ ) m^

2

D^ Axial dispersion coefficient ( ) sec

2

D^ Radial dispersion coefficient ( ) sec

2

D^ Effective diffusivity ( ) sec

2

D^ -Radial diffusivity ( ) s

m' D^ Radial diffiisivity of component j ( )

sec

Dp - Equivalent diameter of catalyst particle (m)

dp - Catalyst diameter (m )

d^ Diameter of reactor (m)

d. - Outside diameter of tube (m)

d„ - Outside diameter of tube (m)

E Activation energy ( ) mole

39

Ih F^ Moderator feed rate (—)

hr

h^ - Heat transfer coefficient between catalyst and gas ( J m' -sec-K

J /z„ Heat transfer coefficient between solid and gas (—:p— ) m~ sec °K

h^ Shell side heat transfer coefficient ( ) m^sec^K^

h Tube side heat fransfer coefficient ( ) m^sec°K^

h^ - Wall heat fransfer coefficient ( ) m' sec °K

AH„ Heat of reaction of n* reaction ( ) mole

AH Heat of reaction ( ) mole

HjO' - Dimensionless water concenfration

HAc' - Dimensionless acetic acid concentration

k^ Deactivation rate constant

k - Mass transfer coefficient between catalyst exterior and bulk fluid ( — ) ^ sec

L - Lengtii of the tube (m)

.r ^, n • • , • , 1- • ^ moles , A'. - Flux of component J m the axial direction (— ) m sec

A'' Heat flux in the axial direction (—; ) *" m'sec

40

Nj^ - Flux of component j in the radial direction (— ) m sec

N^^ Heat flux in the radial direction (—-—) m sec

A2 - Dimensionless nifrogen concentration

n Number of tubes in a reactor

n^ Number of reactions

Oj - Dimensionless oxygen concenfration

P Partial pressure of limiting reactant

Pp - Partial pressure of poison

P^ - Partial pressure of product

P Poison and fouling precursor

P i Adsorbent poison or foulant (coke or inorganic deposit)

R - Gas constant ( )

mole • K

R Product (in Table 3.2)

R - Radius of the tube (m) ^ th • / moles ,

r Rate of n reaction ( ; ) kg of catalyst sec

moles , r Reaction rate ( ; )

kg of catalyst sec

r^ Effective radius of the catalyst particle (m)

7; Shell side temperature (°K)

41

T Bulk gas temperature (K)

T - Dimensionless gas temperature

r^ - Reference temperature,

t^.^ Catalyst life in years

U - Overall heat fransfer coefficient (———) m • K

u^ Velocity of the coolant ( — ) sec

u Superficial velocity ( — ) sec

Vj^ Stoichiometric coefficient of component / in reactions

V - Superficial tube velocity ( — ) sec

VAc' - Dimensionless vinyl acetate concenttation

z - Axial coordinate (m)

z* - Dimensionless reactor length

Greek Letters

a, P,y,r],n Empirical constants

a Activity factor ( s i )

A Axial thermal conductivity ( —) msec K

A. Radi a 1 thermal conductivity ( —) ' m sec K

42

A - Effective thermal conductivity ( —) msec K

A - Wall thermal conductivity ( —) msec K

kg p Dens ity of coolant (—-)

ks Pg Bulk density ( ^ )

m

kg p Density of metal (—-)

'"" m

kg p f - Average density of gas (—r-)

' m

kg p -.Dens ity of the sohd (—r-)

m

r\ Effectiveness factor

£ Bed void fraction

a - Gas viscosity ( ) msec

43

CHAPTER 4

OPTIMIZATION OF A VINYL ACETATE REACTOR

One of the main objectives is the optimization of an industrial vinyl acetate

reactor. The data required for this study were obtained from Celanese Chemicals (Clear

Lake, TX). The initial estimates of the model parameters in the vinyl acetate reactor

model are calculated from the correlations available in the open literature. But it is

necessary to benchmark the model against the industrial data to closely represent the

industrial process. This is essential to ensure that the problem being studied deals with

issues faced in the industrial practice. The data supplied from Celanese contained the

information about the following operating variables from the start to end of the operation:

1. Inlet and outlet composition (wt.% of ethylene, oxygen, acetic acid, vinyl acetate,

carbon dioxide, water, inert);

2. Inlet and outlet gas temperature;

3. Coolant temperature;

4. Feed flow rate;

5. Reactor bed inlet and outlet pressure.

44

4.1 Model Validation

A base case operating condition from the industrial data is selected at which the

catalyst activity is assumed to be unity. There are eight model parameters to be found to

obtain a reasonable fit at the base case operating point. The model parameters are as

follows:

1. Rate constant for the 1'* reaction k^;

2. Rate constant for the 2" ^ reaction k^;

3. Adsorption coefficient for reaction 1 K^;

4. Adsorption coefficient for reaction 2 K^;

5. Overall heat fransfer coefficient U;

6. Radial dispersion coefficient D^;

7. Radial thermal conductivity A ;

8. Biot number Bi.

The above model parameters are estimated in such a way that for a given set of

input variables the weighted errors between the model predictions and the industrial data

are minimized. The input variables to the parameter estimation problem are:

1. Total feed rate to the reactor;

2. Reactor inlet composition (ethylene, oxygen, acetic acid, vinyl acetate, carbon

dioxide, water);

3. Reactor inlet gas temperature and pressure;

4. Coolant temperature.

45

There are a total of 12 operating points available to fit the model (6 outlet

compositions, 5 temperature measurements along the reactor, and one outlet

temperature). It is important to mention that the catalyst is supposed to be fresh (catalyst

activity equal to one) and deactivation of the catalyst does not feature in this step of

benchmarking.

NPSOL (Gill, 1986) solver is used to find the model parameters by minimizing

the following weighted objective function:

u wt. n T,

By minimizing the above weighted objective function we get the following values

of the model parameters.

mole atm°' Rate constant for the 1' reaction k^ = 3700.0

m sec

. g mole atm°' Rate constant for the 2" reaction A;2 = 1.3 • 10 ^

m sec

Adsorption coefficient for reaction 1 K^ = 0.1 attn"

Adsorption coefficient for reaction 2 K-^^'d.l atin

W Overall heat ttansfer coefficient U = 466—y-^—

m^ "K

Radial dispersion coefficient Z) = 10' - ^

W Radial thermal conductivity A = 0.09 j —

m iv

Biot number Bi = 0.8

46

Tables 4.1 and 4.2 show the relative error between the mdusfrial data and model

prediction. It can be seen that the model predicts the industrial data quite well for base

case condition except for the oxygen outiet composition, which is due to the kinetics of

the reactions.

Table 4.1 Comparison between industrial data and model prediction for reactor outlet composition

Component Relative Error in Prediction (%)

Ethylene 0.44

Oxygen 4.17

Acetic Acid 1.84

Vinyl Acetate 0.00

Carbon Dioxide 0.00

Relative eiTor is defined as follows:

, . Model prediction - Industrial data ,^^ Relative error = • 1UU.

Industrial data

Because of confidentiahty reasons, we are not able to provide actual values of the

operating conditions (e.g., temperature and composition) in the figures and tables

47

Table 4.2 Comparison between industrial data and model prediction for temperature along the reactor

Length (m) Absolute Error in Prediction ( C)

1.0922

2.0066

2.9210

3.8354

4.7498

-0.10

-0.50

2.40

0.17

-1.05

TO+ 25

TO+ 20

TO+ 15 E <u

a> > < TO+ 5

TO 0.2 0.4 0.6

Dimensionless Reactor length

0.8

• Model Prediction • Industrial data

Figure 4.1 Comparison between model prediction and industrial data for the average radial temperature along the reactor

48

4.1.1 Catalyst deactivation model benchmarking

The catalyst used in the fixed bed reactor deactivates over time due to impurities

in the feed and sintering of the catalyst. Therefore, it is required to model catalyst

deactivation which will predict the reactor outiet temperatiire and composition profiles

over the life of the catalyst. Catalyst deactivation is a nonhnear function of operating time

and temperature, since higher temperatures within the ttibe promote deactivation

(Froment, 1974). Here, the catalyst activity at the start of operation is assumed to be unity

(a = 1), and then it decreases exponentially with time t^^. It is expressed as a function of

shell side temperature T and moderator feed rate F^ .

a = exp(-a • t^^) when t^^ s t^

a = exp(-/3 -t^^)-exp(-7 •T)-F^" when t^^>t^J> T^

a = exp(-^ -ty^)- exp(-r7 • T) • F^" when t^^>t^,T^ T^

The catalyst deactivation model contains 5 empirical constants. There are total of

12 X Age (where Age - catalyst age in days) operating points available to fit the model (6

outlet compositions, 5 temperature measurements along the reactor and one outlet

temperature for each day). NPSOL (Gill, 1986) solver is used to find the empirical

constants by minimizing the following weighted objective function,

^ Wtr.. -Wt..^, . . __.. 4^.T,„, -Pp, Age

min obj = V Jay=0

49

By minimizing the above weighted objective function we get the following values

of the model parameters.

a =1.8207

13 =0.1229849

Y =0.0125

17 = 0.0155

n = -0.5

4.1.2 Catalyst deactivation model validation

Table 4.3 and Figures 4.2 to 4.7 show the comparison between the model

prediction and indusfrial data, over the entire hfe of the catalyst. It can be seen that the

catalyst deactivation model predicts the industrial data quite well.

Table 4.3 Comparison between industrial data and model prediction over the catalyst life


Ethylene Consumed 3.42

Oxygen Consumed 2.86

Acetic Acid Consumed 2.71

Vinyl Acetate Produced 0.53

Carbon Dioxide Produced 9.24

50

M + 10

o M + 8 o. E o o ^ M + 6

^ M + 4 a IT <u

t M + 2

M

Time

• Industrial data A Model Prediction

Figure 4.2 Comparison between industrial data and model prediction (Ethylene reactor outlet composition wt%)

M + 3

o a. E o u

3

o

0)

M + 2

M + 1

M

Time


Figure 4.3 Comparison between industrial data and model prediction (Oxygen reactor outlet composition wt%)

51

M + 5

a M + 4 E o O

I M + 3

M + 2

M + 1

M

Time


Figure 4.4 Comparison between industrial data and model prediction (Acetic acid reactor outlet composition wt%)

M + 9

Time


Figure 4.5 Comparison between industrial data and model prediction (Vinyl acetate reactor outlet composition wt%)

52

c M + 8 o in o Q. E O M + 6

tS M + 4

o Q c o .o TO O

M + 2

M

Time


Figure 4.6 Comparison between industrial data and model prediction (Carbon dioxide reactor outlet composition wt%)

TO+ 40

« TO+ 20

<D T O + 1 0

TO

Time


Figure 4.7 Comparison between industrial data and model prediction (Reactor outiet Temperature)

53

4.2 Offline Optimization Approach

Since the catalyst deactivates over time, the shell side temperature is increased to

compensate for the loss of catalyst activity. The manner in which the shell side

temperature is changed affects the net profit of the process. This motivates us to carry out

an optimization study for this process. The optimization problem uivolved determining

the optimal shell side temperature profile over the run length of the operation by

maximizing the net profit of the process. This kind of optimization study can be used to

identify the benefits to implement real time optimization on vinyl acetate process.

Figure 4.8 illustrates the optimization procedure. The continuous temperature

profile over the length of the operation is expressed in terms of node values at specific

points in the time domain connected together by smoothly varying interpolating

polynomial. The optimizer uses these node values as decision variables. The intermediate

values for any particular temperature required by the simulator are provided by applying

cubic spline interpolation (Riggs, 1994).

The optimization was carried out for fixed catalyst Hfe and fixed reactor inlet

conditions. The optimizer sought the ten decision variables such that the profit function

(<I)) value over the entire run period was maximized.

<^ = Py,-VA-P,-E-Po-0-P^,,-HAc-P,„,,^-VA

The carbon dioxide separation cost is insignificant and not considered in the profit

function. The optimization is subjected to two consttaints:

1. Maximum amount of Vinyl acetate that can be removed in the separation system

2. Maximum amount of Carbon Dioxide that can be removed in the separation system.

54

Initial values of the decision variables

Objective function & Constraint values

Optimizer Simulator

Decision variables

Figure 4.8 Optimization Procedure

4.2.1 Offline optimization resuhs

The optimizer (NPSOL, Gill 1986) calculates the optimum temperature profile in

such a way that the vinyl acetate (VA) produced in the optimum case approaches the

upper consfraint on VA that can be removed as compared to the VA produced at base

case conditions. Thus, the profit improvement of 7.86% over base case is achieved by

pushing the process to the most profitable constraint of vinyl acetate that can be removed.

From Figure 4.9, it can be seen that the optimum temperature for the initial period

is higher than the base case temperature. For the remaining period of time, the optimum

temperature is greater than or almost equal to the base case temperature profile. Thus, an

increase in temperature contributes to an increase in the production of vinyl acetate.

55

TO+ 35

TO+ 30

£ TO+ 25 15

I TO+ 20 (U

K S TO+ 15

w TO+ 10

TO+ 5

TO

Time

— ^ — Optimum Base Case a Decision Variables

Figure 4.9 Comparison between base case temperature profile and optimum temperature profile

4.3 Sensitivity Analysis

The optimization studied here illustrates the potential for improving the

profitability of the unit by conducting analysis of this sort. However, the results are

subject to the accuracy of the model parameters. A parametric sensitivity study was

conducted to assess the effect of errors in some of the main parameters (rate constants

it,,A;,) on the objective-function value. A 10% relative error was introduced in each

parameter separately. The decision variable values found after the optimization analysis

on this model were substituted back into the original model (i.e., model without any

error) and the objective function value was re-evaluated. The results of the sensitivity

analysis are presented in Table 4.4. As expected, this profit value shows some change

56

from the original profit value, which was found by performing the optimization analysis

on the error free model.

Table 4.4 Model Parametric Sensitivity Analysis

Parameter Relative Error Infroduced Absolute error in profit prediction

k, +10 % 3.7 %

k, +10% 1.6%

4.4 Online Optimization

No practical model of a plant, no matter how rigorous, can provide an accurate

long-term projection of the plant's responses through its normal evolution. The plant

response changes with time. As an example catalyst ages ttace feed impurities and feed

source changes. Thus, the predictions from the model may not be accurate. The online

optimization adjusts the model parameters according to discrepancies between model-

predicted values and measurements. The online optimizer uses the updated model to

recalculate the optimal temperatiire profile. A schematic of the online optimization

implementation is shown in Figure 4.10.

A process simulator represents the real vinyl acetate process in this study. The

simulator is constructed using the original kinetic model. The measurable variables

include composition of ethylene, oxygen, acetic acid, vinyl acetate, carbon dioxide and

the outlet reactor temperattire. The frequency of the model updating is once per day.

57

One variable A:, (rate constant of first reaction) is adjusted online based on the

errors between predicted and measured composition and temperature. Then the predicted

compositions and temperature are compared to current measured values, k^ is regressed

based on least squared errors using Nelder Mead (Riggs, 1994) optimization method.

The temperature profile is recalculated to maximize the profit using the updated

model. NPSOL (Gill, 1986) optimizer is used to find the optimal temperature profile.

58

•D 0)

re a

L . Oi

* J

Oi

E (0

RS CL ^ K

0) •D

o 2

c o CD c CD E (D Q. E c o '^

iza

t -*—' a O <D C

c O M— O o

.4_-i

cc E CD

-C o

C/3 o • T ~

• ^

(D i—

3 D;

4.4.1 friaccurate model

1. In this case, the rate constant for reaction 1 ki in simulator is 2.5% greater than

ki in the model. The comparison of onhne optimization with offline optimization is

shown in Figure 4.11.

TO+ 30

TO+ 25

2 TO+ 20 CD

oi Q .

E I- TO+ 15

CO

"53 TO+ 10

TO+ 5

Offline Optimization

i ^ f

Online Optimization

TO

Time

Figure 4.11 Comparison between offline and online temperature profile

From Figure 4.11, it can be seen that the temperature profile calculated by the

online optimizer is different from that of the offline optimizer. Following the temperature

profile calculated by the online optimizer gives 3.05% profit improvement over the

offline optimization. It can be seen that the optimal temperatue profile is very noisy and

will be difficuh to implement in the plant. Therefore, the online optimization is re-run

with a first-order filter on shell-side temperature. Figure 4.12 shows the optimal

60

temperature with a filter. Following the temperature profile calculated by online

optimizer gives 1.18% profit improvement over the offline optimization.

TO+ 25

Time

— f=0.05

Figure. 4.12 Comparison between offline and onhne temperature profile using a filter

2. In this case, the rate constant for reaction 2 k, in simulator is 2.5% greater than

k, in frie model. The comparison of online optimization with offline optimization is

shovm in Figure 4.13.

From Figure 4.13, it can be seen that the temperatiire profile calculated by the

online optimizer is different from the offline optimizer. Following the temperature profile

calculated by the online optimizer gives 1.06% profit improvement over offline

optimization.

61

TO+ 25

TO+ 20

Time

•Offline Online

Figure 4.13 Comparison between offline and online temperature profile

4.5 Nomenclature

Age Age of the catalyst in days

F^, Moderator feed rate (—) hr

P^ -Cost of ethylene ($/lb)

PEnergy ' ^otal cuergy cost ($/lb of vinyl acetate)

PHAC - Cost of acetic acid ($/lb)

PQ - Cost of oxygen ($/lb)

Py^ - Price of vinyl acetate (S/lb)

T - Shell side temperature (°C)

62

TI"^ Industrial reactor temperature at the r"" axial location in the reactor

T^""' Model Predicted reactor temperature at the i'^ axial location in the reactor

t^.^ - Catalyst life in years

W - weight, 0 < < 1

Wtl"'^ - Industrial reactor outlet wt% of i"' component

Wt^"'' Model Predicted reactor outlet wt% of r"" component

Greek Letters

a, P,y,r], n Empirical constants

63

CHAPTER 5

MODEL DEVELOPMENT FOR ETHYLENE OXIDE PROCESS

Ethylene oxide is one of the most important petrochemical intermediates and one

of the raw materials used for the production of glycol, polyethylene glycol, and glycol

ethers. The production of ethylene oxide is a critical process because the reactor can

generate eleven times as much heat in a runaway condition as under normal operating

conditions (Piccinini, 1984). Therefore, the safety issues for an ethylene oxide reactor

system are very important, as industry must attempt to operate them in an economically

advantageous manner. This motivates us to study the control, optimization, and

stabilization of the ethylene oxide process.

5.1 Process Description

Figure 5.1 shows the schematic of the ethylene oxide process. The process studied

consists of a gas-gas heat exchanger, a multi-tubular fixed bed reactor, a steam generator,

and a separation system. The exothermic heat of reaction from the reactor is removed by

passing coolant on the shell side of the reactor. A portion of the heated coolant is passed

through a steam generator to produce steam, and the total coolant stream is recycled back

to the shell side of the reactor. A single-loop PID control system uses the flow rate of the

coolant that is passed through the steam generator to maintain the inlet temperature of the

coolant to the reactor. The interesting features of this system are: (1) possibility of

64

runaway reactions that produce carbon dioxide, and (2) tradeoffs between selectivity and

reaction rates.

5.2 Reaction Chemistry and Mechanism

Ethylene oxide is produced by the direct vapor- phase oxidation of ethylene over

a silver catalyst at 10 to 20 atm and 200 to 300°C. The main reaction is as follows:

CH, =CH,+- O, ° ' ' ' > CH.OCH,.

Carbon dioxide and water are produced by the side reactions and represent the

only significant byproducts formed. The reaction also produces very small amounts of

acetaldehyde (usually less than 0.1 wt percent of the ethylene oxide product) and trace

amounts of formaldehyde. The stoichiometric equations for these reactions are:

CH^ = CH^ + 3O2 -^ 2CO2 + 2H,0

CH.OCH, +~0, -> 2CO,+2H,0

CH^OCH^ -^ CH.CHO

CH.CHO + -O2 -> 2CO2 + 2H,0 3 ^ 2 2 .

CH^ =CH,+0, ^ 2 C / / , 0 .

The reaction mechanism has been thoroughly studied and is discussed in detail in

the literature (Meyers, 1986). It is generally agreed that the fimction of the silver catalyst

in the oxidation reaction is the activated adsorption of oxygen on its surface, hi the most

accepted theory, the adsorbed oxygen is in a molecular, ionized form (O, ) known as

65

o

O

to ^ - » 1/3

c _o

Ui c« CL, CD

(/3

o Q .

m

c/3 j : ;

J

a C3 .*—» O CO (D

a.

cd

B:: uj <: Li fc- O fS

^,

CO

C/3

O cS ;-! (U C

a cc3

00 ^ 5

CD

CD T3

5

CO en CD o O i _

CL 0

• g x o 0 c _0

LIJ

o o

"TO E

O CO

66

superoxide. Ethylene reacts selectively with superoxide to form ethylene oxide and

atomic oxygen, as shown in the following reaction:

O, (ads.) + QPf, -^ CH.OCH, + (9(ads.).

The adsorbed oxygen atoms are then believed to react mainly with ethylene to

produce carbon dioxide and water. The addition of chlorine in controlled amounts

inhibits the initial adsorption of atomic oxygen while permitting superoxide to be

adsorbed, and thus improves the selectivity of ethylene oxide.

5.3 Kinetics

Westerterp et al. (1992) studied the kinetics of the selective oxidation of ethylene

over a silver a -alumina catalyst. The relevant reactions are:

C,H,+-02^C,H,0 (5.1)

C^H, + 3O2 -^ 2CO2 + 2H,0. (5.2)

The following equations were obtained for the production rates of ethylene oxide and

carbon dioxide:

k' -P • P"' i?, = ^ ' ' \ • ,0- = l,2) (5.3)

l + K,' -P.+K^' -PC+K; •P^+K.o' P,o

67

^ ; =0.67exp(-8068/r),;t/ =75exp(-11381/r)

K^' - . 30 .10 - \ i ^ / =.49.10-°'

K^' =.87.10-°\^c' =1.14.10-°^

i:ô'=-90.10-"\^ô'=-49.10-°'

K^' =3.68.10-êxp(2370/r),/r„' =4.04.10-êxp(3430/r)

«i =0.13,n2 =0.14

5.4 Mathematical Modeling Assumptions

The ethylene oxide reactor model is derived from the generalized fixed bed

reactor model (Chapter 3), with following assumptions.

5.4.1 Negligible axial dispersion

When the catalyst bed depth exceeds about 50 particle diameters, the effect of

axial dispersion of heat and mass on conversion is negligible (Rase, 1990). For the

industrial reactor system studied, this criterion is satisfied; therefore, the axial dispersion

term is not considered in the modeling equations.

5.4.2 Negligible pressure drop

Pressure drop through a reactor, though seldom more than 10% of the total

pressure, is not a major factor in changing the chemical reaction rate in a gaseous

reaction. Therefore, effect of pressure drop is neglected in the modehng equations. This

68

assumption can be verified by comparing the reactor outlet composition and temperature

for model with pressure drop and without pressure drop. Ergun equation as described in

Chapter 3 is used to calculate the pressure drop in the reactor. The following table (Table

5.1) shows the error between the model without pressure drop and model with pressure

drop for reactor outlet composition and temperature.

Table 5.1 Comparison between reactor model with and without pressure drop

Process variable at reactor outlet Absolute error

Ethylene composition (mole%) 0.0086

Oxygen composition (mole%) 0.0098

Ethylene oxide composition (mole%) 0.0079

Carbon Dioxide composition (mole%)) 0.0045

Gas temperature (° C) 0.0919

Coolant temperature (° C) 0.0413

5.4.3 No concentration gradient between catalyst particle and gas

Rase (1990) illusfrated the various gradients established in a fixed bed reactor for

the production of ethylene oxide by oxidation of ethylene. For ethylene oxide system at

moderate chemical reaction rate, the concenfration gradients between the fluid and the

exterior of the catalyst are small. This situation is caused by the easier transport of

69

molecules in the fluid phase, even near the catalyst surface, than the more difficult and

constricted region within the catalyst. By contrast, the temperature gradient is greater

exterior to the particle. In fact, in most cases, the temperature profile inside the catalyst is

essentially flat (Rase, 1990). Heat is transferred by several means in the catalyst pellet:

conduction along with some bulk movement in the fluid, conduction in the solid, and

radiation at high temperatures. Although upon initial operation, the catalyst may assume

rather high temperature differences due to changes in concentration of reacting

components with depth of penetration, at steady state, conduction in the pellet will cause

all areas to reach essentially the same temperature. The temperature gradient between the

exterior of the particle and the fluid is more significant than the concentration gradient. In

the present study, only the temperature gradient between the bulk gas and catalyst

exterior surface is considered, and the concentration gradient between the bulk gas and

catalyst exterior surface is neglected.

5.4.4 Flat velocity profile:

A reasonably flat profile is assumed for — values of 30 or more (Froment et al., p

1990). This criterion is readily met in the industrial reactor studied.

5.4.5 Intraparticle resistance

70

When the resistance to mass and heat transfer inside the catalyst (i.e., intraparticle

resistance) is important following equations are used for catalyst surface concentration

and catalyst surface temperature.

'dC^,^

dt ^ ^ ( ^ / ^ ) - A £ ^ . . r„ dr dr (5.4)

P c fQj\

\St J h^ir:-^).pX{-AHlr„ fc dr, dr t f

The equations are subjected to the following initial and boundary conditions.

dC dT re=0, -D,-^ = 0 - ^ - ^ = 0

dr^ dr^

re=-f^ Cs=C, T.=T,

(5.5)

(5.6)

t = 0, r, = ^ , C, =C,,o T,=T^^,

The effectiveness factor for a reaction is defined as follows:

rate of reaction with catalyst pore diffusion resistance 71 = • •

rate of reaction with surface conditions

Equations (5.4) and (5.5) are solved to find the effectiveness factors 7, and 77, for

reactions (5.1) and (5.2), respectively. The effectiveness factors are calculated for

different catalyst activities (acrivz(y a = 1.0,0.93and0.77) and at three different

locations in the reactor as shown in Tables 5.2-5.4, respectively.

71

Table 5.2 Effectiveness factors, 7,, T/ for catalyst activity a =

Position in the rector 77 V2

hilet

Center

Exit

1.0179

1.0366

1.0425

1.0258

1.0529

1.0613

Table 5.3 Effectiveness factors, T]^ , r]^ for catalyst activity a = 0.93

Position in the rector 77, 77 ,

hilet

Center

Exit

1.0306

1.0417

1.0476

1.0441

1.0602

1.0686

Table 5.4 Effectiveness factors, 77,, 77, for catalyst activity a = 0.77

Position in the rector 77, 77 ,

iiiiet 1.0381 1.0549

Center 1.0543 1.0785

Exit 1.0628 1.0908

From Tables 5.2 to 5.4, it can be seen that both 77, and 77, are close to unity, which

means no appreciable resistance inside the catalyst particle. Therefore, we can neglect the

intraparticle resistances.

72

Based on the above assumptions, a heterogeneous model is developed for the

ethylene oxide reactor, which is described by the following 10 dynamic partial

differential equations.

5.5 Mathematical Model of Ethylene Oxide Reactor

The material and energy balance equations for the ethylene oxide reactor is given below.

Material balance:

dC^H, dC^H^^aD^d.oC^. . ^ . ,, _. s—^ = -u - +—^T-(^—7 )-Pbi^x+f2) (5-7)

dt dz r or or

s—^ = -u-—^ + — ^ — ( r — - ) - / ? , (0.57-1+2.5rJ (5.8) dt dz r dr or

dEO dEO sD^ d , dEO. ,. Q.

£ =-u-—- + —^—{r^) + p,r, (5.9) dt dz r dr dr

dCO, dCO, sD^ d . dCO,. .. . . . ,^.

dt dz r dr dr

dt dz r dr dr

dCH, . _ ^ ^ ^ . + ^ A ( , ^ ^ ) (5.12) dt dz r dr dr

Energy balance:

Gas phase

9r 4A,,^ _ r ^ ^ i ^ A r . ^ l c . . . .? = -c....«f.^(r„-m^^(^^)-M(r-r,) (5.n, dt ' ^ dz d

73

Catalyst Surface Temperature.Balance

(QT.^

V dt J PsC,. -f- =h^a{T-T^)+p^Y.i-^)j. (5.14)

n=l

Metal Temperature Balance

Cp„P pm r pm 4 (^°'-<')-|' = A(^.-^J-^A(r„-r) (5.15)

Shell Side (coolant) Temperature Balance

c...f = -.c,.f.p^fc-,) (5.16)

The above equations can be made dimensionless by using the following

dimensionless variables.

Dimensionless variables:

r H *= ^-^^ C,H°

O'- °^ C^H^

E0* = EO

C^H^

co* = CO,

C-,H^

^,^(T-T,) T* =

lOP.

H,0* = H,0

C^H^

Z lor.

C,H;

(Te-T,) T * =

lOP.

T * — •M ior„

r = R z = L u

T = —t L

74

Dimensionless Equations:

Material balance:

dz dz R^ur dr dr uC,H°

f^ = -f=^.f4f(/f^)-^.(0.5,.2.5,) (5.18) dz dz R ur dr dr uC^H^

dEO' dEO' sD^L d , . dEO', p,L .^ ., „,

dr dz R'ur dr dr uC-,H,

dCO' dCO,' sD^L d , . dCO^, p,L ,. ^^. — = r - + -^^ r(^ 7^)+ 0 2 (5-20)

dz dz R'ur dr dr uC,H,

dJi20i_J_H20i^^D^^^>dH^ _p,L_

dr ~ dz R\rdr'^ dr' ^ UC.H''' '' 2^^ 4

dCH: dCH: sDL d , , dCH, 4 _ ^^^•••4 Ar'^-^^) (5-22)

dz dz' R^ur' dr' dr

^ = _ : ^ (^hIl—)(T' T')\ ^'^ ^(/-^) dz dz'^^p^C^j/^' R'-PgC,ydr' dr' ^^^^^

haL [T'-T:] PbCPs^

Surface Temperature Balance

dr; v^ ir-rV ^^^ - ,+—^^2^—n (5.24)

Metal Temperature Balance

75

dT^ 4dh£L

Shell Side Temperature Balance

^ = _ M ^ , ^-K-n-d^eL ( . . 5 - " " & ^ ; : ^ : ; ^ p ^ ^ ^ -"^^ ^ (5.26)

The above partial differential equations (5.17)- (5.26) are subjected to the following

initial and boundary conditions:

z'-o, r=T:, T:=TJ, C; = C;

dC- dT' r =0, - ^ = 0, ^ = 0 (5.27)

dr dr

^ = 0, z = 0, T = T, T =T T =T T =T C =C

5.6 Orthogonal Collocation

The above partial differential equations are converted into ordinary differential

equations by orthogonal collocation (Finlayson, 1980). A cubic polynomial (i.e., three

internal collocation points) is used for axial derivative approximation. Two radial

collocation points are considered for temperature and three collocation points are

considered for concentration (as illustrated in Chapter 4 for the vinyl acetate reactor). The

reduced dynamic ordinary differential equations are solved ushig LSODE integrator. The

76

developed model predicts composition and temperature profile in the axial and radial

directions of the reactor.

5.6.1 Orthogonal collocation for axial derivatives

In the orthogonal collocation, the trial function is taken as a series of orthogonal

polynomials and the collocation points are taken as the roots to one of those polynomials.

In many of these problems the solution is not symmetric function of z (where z is the

axial coordinate), i.e., it is a function of odd and even powers of z To do this we

construct orthogonal polynomials that are functions of z", where 72 = 1,2,3...,//,

N - order of polynomial. One choice is

y^z + z{\-z)Y^a,P,_,(z). (5.28) 1=1

Equivalent choice for Eq. (5.28) is

y-fj^A-A-)- (5.29) 1=1

Eq. (5.29) can be simphfied as follows:

y'td,^". (5-30) /=1

We define the polynomials to be orthogonal with the condition

jPF(z)P,(z)P„(z)Jz = 0 k<m-\. (5.31)

77

Again, we take the first coefficient of the polynomial as one, so that the choice of the

weighting function ff(z) completely determines the polynomial, and hence the trial

function and the collocation points.

We take the collocation points as the A'' roots of the polynomial P^{z) = 0.

These roots are between zero and one. The collocation points are then z, = 0,

Z2,Z3,...,z^^, and z^^, = 1 , where, Zj = 0 and z^^, =1 are the boundary collocation

points and z,,Z3,...,z^ are interior collocation points. Eq. (5.30) can be written at a

collocation point / ,

N+\

y{z.)^Y.d^zr . (5-32) ;=i

Differentiating Eq (5.32) with respect to z , we get

^ ( z , ) = X ( z - l ) J , . z ; - . (5.33) dz ,=1

Now differentiating Eq. (5.33) with respect to z, we get

f ^ = f ; ( . - i X , - 2 y , z r . (5.34) dz ,=1

We can write Eqs (5.32), (5.33) and (5.34) in matrix form as follows:

y = Od ^ = Cd ^ = Dd, (5.35) dz dz'

where

Q,-,:- C,=0-l>r D,={i-l)ii-2)zr (5.36)

where

78

/ = 1,2,3..., TV+ 2

7=l,2,3...,7V + 2

Solving for d gives,

^ = CQ-'y = Ay i ^ = DQ-'y = By. (5.37)

dz dz'

Thus, orthogonal collocation method can be used to convert the differential equations

into the algebraic equations. Stiff problems are solved by using multiple sub intervals

along the axial direction. Dependent variables values are equated at the first and last

collocation points of consecutive intervals. Low order polynomial (e.g., quadratic or

cubic) is used for approximation to the axial derivative. This is because higher order

polynomial tends to oscillate in the intervals between the collocation points. Therefore in

ethylene oxide reactor modeling a cubic polynomial is used for axial derivative

approximation. Six subintervals are considered along the axial direction.

5.6.2 How to find the cubic polynomial

Let W{z) = 1, and the polynomials be

Po=l , P, = l + fe, P2 = l + cz + Jz ' , P3 = l + ez + / z ' + g z ' .

P, is found by requiring the orthogonality condition,

1 \

fPF(z)PoPi Jz = 0 or J(l + fe)Jz = 0, 0 0

which makes b = -2. Then Pj is found from

79

\Wiz)P^P^_dz = 0 ]wiz)P^P,dz = 0.

I.e.

/(I + cz + dz'-)dz = 0 f(l - 2z)(l + cz + z' )rfz = 0 0 0

which makes c = -6 , cf = 6.

Then P3 is found by requiring the orthogonality condition,

\W{z)P,P^dz = 0 \W{z)P,P^dz = 0 ]wiz)P,P^dz = 0,

I.e.,

i

\(l + ez + fz^+gz')dz = 0 0

1

j(l - 2z)(l + ez + fz' + gz')dz = 0 0

1

J(l - 6z + 6z')(l + ez + fz-+ gz')dz = 0

which makes e = -12, / = 30, g- = -20.

The polynomials are PQ = 1, P ; = l - 2 z , P 2 = l - 5 z + 5z', P3 = l -12z + 30z ' -20z '

The roots of the cubic polynomial P3(z) = 0 are 0.1127,0.5,0.8873, so these are the

internal collocation points along with z = 0 and z = 1 as the boundary collocation points.

In this study, all the axial derivative computations are performed using three internal

collocation points and one on the boundaries of each finite element. Three internal

collocation points are illustrated in Figure 5.2. If the element is scaled so that z, = 0 and

80

Z5 = 1 , then the node points are given by{z,,Z2,Z3,z4,Z5} = {0,0.1127,0.5,0.8873,1.0},

which are the roots of the third-order Lengendre polynomial, augmented by element

endpoints.

Figure 5.2 Collocation element

From equation (5.36),

e=

1 Zi z,-

1 Z, Zj"

1 z , z •

1 z,

1 z.

Z4 Z4 Z4

Therefore,

C

0

0

0

0

0

1

1

1

1

1

2z,

2z2

2Z3

2Z4

2Z5

3zr 3z;-

3Z3^

3z /

3zs^

4z/

4z.^

4Z3^

4 z ;

4 z ;

81

D =

0 0 2 6z, 12z,'

0 0 2 6z2 I2Z2'

0 0 2 6Z3 12Z3-

0 0 2 6z, 12z,'

0 0 2 6z, 12z.'

is gives

A =

-13

-5.3238

1.5

-0.6762

1

14.7883

3.8730

-3.2275

1.2910

-1.8784

-2.6667

2.0656

0.0

-2.0656

2.6667

1.8784

-1.2910

3.2275

-3.8730

-14.7883

-1

0.6762

-1.5

5.3238

13

Thus, using the matrix A, we can approximate the axial derivative formula for

temperature as follows:

dT:

dz : =I4,J;

where

Tj Temperature at /•"• collocation point (7 =1 to 5),

,nd e.g., at 2 collocation point ( 7 = 2 ) , the above equation becomes

dT,

dz = A,,T, + ^22^2 + ^23?; + ^ 2 4 ^ ; + ^25^5 •

Since we cannot use only three internal collocation points to calculate the

temperature and composition profiles for the entire reactor, we extend the orthogonal

collocation method to a set of finite elements in axial direction, with axial derivative

approximated by cubic polynomials defined on each element. This situation is illustrated

82

in Figure 5.3. Thus at the end of each interval, all the dependent variables values are

equated at the first and last collocation points of consecutive intervals. In the present

reactor model, we have chosen 6 subintervals to represent the entire reactor. This is

because by increasing the subintervals do not change the model-predicted values.

83

CO

"TO

"c

1

-10

-4{s-1)+1

c o c

_o

a _o "o U

<D

OX)

-4s+ 1

84

5.7 Modeling Equations for Steam Generator

From Figure 5.1, it can be seen that the exothermic heat of reaction from the

reactor is removed by passing the coolant on the shell side of the reactor. A portion of the

heated coolant is passed through a steam generator to produce steam, and the total

coolant stream is recycled back to the shell side of the reactor. The steam generator,

which is a heat exchanger, is modeled as a distributed parameter system. The following

are the two energy balance equations for the steam generator.

Metal temperature balance

^^(l'-dr)^ = dA(t't)-dMt-fJ (5-38)

Tube side coolant temperature balance

Eqs. (5.38) and (5.39) are converted into dimensionless equations using following

dimensionless variables.

stm • I 1 1

•R WT ' 10 . 10 ^

. z u z =- r = -t

L L

85

Dimensionless Metal Temperature Balance

dT^ _ ^d^L / . . ..\ 4dir dr Co

pm r pm

Dimensionless Shell Side Coolant Temperature Balance

df^ u^Ldt; 4-h-n-dL (., . .\

Eqs. (5.40) and (5.41) are converted into ordinary differential equations (ODEs)

using finite difference formula. The axial derivatives are discretized using 10 axial points.

The resulting dynamic ODEs are solved simultaneously along with fixed bed reactor

equations using LSODE integrator.

5.8 Modeling Equations for Gas-Gas Heat Exchanger

Heat exchangers have fast dynamics compared to reactors and other unit

operations in a process. Normally, the time constant is measured in seconds but could be

up to a few minutes for large heat exchangers. Process to process heat exchangers should

be modeled rigorously by partial differential equations since they are distributed

parameter systems, in order to estimate correct amount of dead time and time constant in

the exit stream temperatures. However, the resulting models are inconvenient to solve,

especially in large-scale plant wide process simulations. Luyben et al. (1998) found that

for the purpose of plant wide studies it was not necessary to build such detailed models of

heat exchangers, since these units rarely dominate the process response. They

recommended a simplified approach, in that one can use the effectiveness method to

86

calculate the steady state exchanger exit temperature and then delay these temperatures

by first-order time constant to capture the dynamics. This approach is adopted to model

the gas-gas heat exchanger in this study.

The effectiveness of a heat exchanger is defined as

_{mCp)„{f^-f^ {mCp),(T^-T,) Q

imCp)^ST,-T,) (mCp)^aT,-T,) {mCp)^^{T,-f,)' ^ ' ^^

For a given values of inlet flows and temperatures, the exit temperattires are explicitly

calculated for a known exchanger effectiveness:

{mCp)^

T,-.T,.^^'"^ffy'\ (5.44) (mCp)„

The effectiveness is determined by the heat exchanger's design parameters

through the following equation:

l-e -(\-r)NTU

Both r and NTU are weak functions of temperatures in most cases. Therefore,

for dynamic simulation around some nominal operating condition, the effectiveness can

be assumed constant, and calculated from the initial condition.

The dynamics of the gas-gas heat exchanger is modeled in the following way

1. At each step calculate the steady state exit temperatures using Eq. (5.43) and (5.44).

2. To get the current exit temperature, delay the exit temperatures calculated in step 1 by

a first-order filter with time constant of 6 seconds.

87

5.9 Modehng Equations for Separation System

In the separation system, ethylene oxide and carbon dioxide are removed, and the

unreacted ethylene and oxygen are recycled back which is then mixed with fresh

ethylene, oxygen and inert methane. The mixture is then passed through the gas-gas heat

exchanger where it is heated to the reactor temperature. The separation system is modeled

as a first-order system as shown below.

dC,H^ {CjH ^ —CjH^)

dt

dO,

dt

dEO

dt

dCO,

dt

dT,

dt

z,

(O2" -

^sp

(EO" -

^sp

(CO," •

V

02)

-EO)

-CO,)

''sp

(T" -

^sp

T,)

5.10 Catalyst Deactivation Model

fri the ethylene oxide reactor, it is believed that the deactivation is due to sintering

of the catalyst. And the catalyst deactivation is expressed as a function of the operating

time t, inlet shell side temperature T.

a = exp(-C!; • t^^) when T<Z

a = exp(-a . r J • exp(-y5 • T) when T>T^

a = exp(-Qr • r J • exp(-7 • T) when T >T^

5.11 Nomenclature

a Activity of the catalyst (0 < <3 < 1)

CjH^ Amount of ethylene coming out of separation system at steady state

CO" - Amount of ethylene coming out of separation system at steady state

C,H^ -Dimensionless ethylene concentration

CO, - Dimensionless carbon dioxide concentration

CH, - Dimensionless methane concentration

C2p 4 - Ethylene concentration at the reactor inlet (——) m

C„„ - Gas heat capacity (—-—)

Cp, Heat capacity of the coolant ( ^—) kg°K

C , Heat capacity of catalyst ( —) PS F J ^ ' k g ° K

C„ - Heat capacity of the metal ( —) pm V ^ kg°K

Cp, - Heat capacity of the coolant ( 5—) (in steam generator model)

89

C^„ Heat capacity of the metal (——-) (in steam generator model) kg K '

2

D^ Effective diffusivity ( ) sec

d^ Diameter of reactor (m)

d- - Outside diameter of tube (m)

d^ - Outside diameter of tube (m)

d^ Diameter of shell (m) (in steam generator model)

d. - Outside diameter of tube (m) (in steam generator model)

d^ - Outside diameter of tube (m) (in steam generator model)

EO' - Dimensionless ethylene oxide concentration

EO" Amount of ethylene coming out of separation system at steady state

A//„ - Heat of reaction of n* reaction ( —) mole °K

H,0' -Dimensionless water concentration

W h^ - Shell side heat transfer coefficient (———)

m"- K

W h, Tube side heat transfer coefficient (———)

m • K

h Shell side heat transfer coefficient (—; r—) ' m" sec K

h - Shell side heat transfer coefficient (——. ) (in steam generator model) m" K sec

90

J h, Tube side heat transfer coefficient (—— ) (in steam generator model)

m" Ksec

k^' Reaction rate constant, reaction i, kg sec bar

Kj' Absorption rate constant, component j , reaction/, Pa"'

L - Length of the tube (m)

L - Length of the tube (m),

('^Q')m.x The lager of (mCp)^, and (mCp)^

(mCp)c Product of flow rate and specific heat capacity of the cold stream

(mCp)j^ Product of flow rate and specific heat capacity of the hot stream

(mCp)„^i„ - Smaller of (mCp)„ and (mCp)^

NTU^ "^ ('«Q')min

n Number oftubes in the reactor

h - Number oftubes in steam generator

O,* -Dimensionless oxygen concentration

O," Amount of oxygen coming out of separation system at steady state

P Partial pressure of component j , Pa

Q Heat transferred

r = ('"Q')max

91

r. Effective radius of the catalyst particle (m)

T - Inlet shell side temperature (° C)

r, - Catalyst temperature, °K

T" Temperature of stream coming out of separation system at steady state

T' - Dimensionless bulk gas temperature

P] Hot sfream inlet temperature

r2 Hot stream exit temperature

r3 - Cold stream inlet temperature

T^ - Cold stream exit temperature

ty^ Catalyst life in years

UA Product of overall heat transfer coefficient and heat transfer area

u - Superficial tube velocity ( — ) sec

u Velocity of the coolant ( — ) sec

u Velocity of the coolant (—) (in steam generator model) ' s

z* - Dimensionless tube length,

z* - Dimensionless reactor length

Greek letters

a, P,Y - Empirical constants

92

z^p Time constant for separation system

z Dimensionless time

kg p. Density of coolant (—^ )

m

k< Pp^ Density of metal (—^)

m

p. Density of coolant (—y) m

kg p Density of metal (—2-)

m

kc /9j, - Bulk density (—^)

m

s - Void fraction

W X^ -Radial thermal conductivity (—-—)

m- K

X^ Effective thermal conductivity ( —) msec K

93

CHAPTER 6

OPTIMIZATION AND CONTROL OF ETHYLENE OXIDE PROCESS

The data required for this study was obtained from Huntsman Chemicals. The

initial estimates of the model parameters in the ethylene oxide reactor model were

calculated from the correlation available in the open literature. But it was necessary to

benchmark the model against the industrial data to closely represent the industrial

process. The data supplied from the industry for the model development contamed the

information about the following operating variables from the start of operation (fresh

catalyst) to end of operation:

1. Inlet and outlet composition (wt.% of ethylene, oxygen, ethylene oxide, carbon

dioxide);

2. Reactor inlet and outlet gas temperature;

3. Reactor inlet and outlet coolant temperature;

4. Gas feed rate;

5. Coolant feed rate;

6. Reactor bed inlet and outlet pressure;

7. Gas -gas heat exchanger inlet and outlet temperatures.

6.1 Model Vahdation

The model developed in Chapter 5 for fixed bed multittibular reactor, steam

venerator and gas-gas heat exchanger have to be benchmarked against the industtial data

94

to represent the real process. A base case operating condition is selected at which the

catalyst activity is assumed to be unity. There are seven model parameters to be found for

the rector model, two model parameters for the steam generator, and one parameter for

the gas-gas heat exchanger.

Model parameters for reactor:

1. Rate constant for the 1 ' reaction k^

2. Rate constant for the 2"'' reaction k,

3. Shell side heat transfer coefficient h s

4. Tube side heat transfer coefficient /z,

5. Radial dispersion coefficient D^

6. Radial thermal conductivity X

1. Biot number Bi

Model parameters for the steam generator:

1. Shell side heat transfer coefficient h S

2. Tube side heat transfer coefficient h.

Model parameters for gas-gas heat exchanger:

1. Effectiveness factor

There are a total of 8 operating points available to fit the model (4 outlet

compositions, 2 reactor inlet temperatures, and 2 reactor outlet temperatures). It is

important to mention that the catalyst is supposed to be fresh and deactivation of the

catalyst does not feature in this step of benchmarking.

95

NPSOL (1986) solver is used to find the model parameters by minimizing the

following weighted objective function.

Above parameters can be estimated by minimizing the following weighted

objective function:

^ Ind ^

min obj = WY^( '• " ; • f ^(\-W) /=1 C;

yi Ind rp ?Ted / cin cm \ 2 , /

rp Ind rp Pred cout cout

Z:. T Ind y- +

rp Ind rp Pred rp Ind rp Pred r in m \ 2 , / out ou(_ \2

p., P„

By minimizing the above weighted objective function, we get the following

values of the model parameters.

Model parameters for reactor:

Rate constant for the 1^'reaction^, =1.831 (-; ;—) kg sec bar

„. . , „ , mole Rate constant for the 2"^ reaction k, = 290 (- — )

kg sec bar

Shell side heat transfer coefficient h^ =1300 ( ^—) msec K

Tube side heat transfer coefficient /z, =1300 ( ^—) msec K

Radial dispersion coefficient D = 10 ° ( ) sec

W Radial thermal conductivity X = 0.0825 (——)

m K

Biot number Bi = 0.1

Model parameters for the steam generator:

96

Shell side heat transfer coefficient h = 1600 0 { - ^ msec K

Tube side heat transfer coefficient A, = 1585.8 ( ) msec °K

Model parameters for gas-gas heat exchanger:

Effectiveness factor s = 0.79588

Table 6.1 and Table 6.2 show the relative error between the industrial data and

model prediction. It can be seen that the model predicts the industrial data quite well for

base case operating condition.

Table 6.1 Comparison between industrial data and model prediction for reactor outlet composition (mole %)


Ethylene 0.82

Oxygen 2.19

Ethylene oxide 0.01

Carbon Dioxide 0.47

Table 6.2 Comparison between industrial data and model prediction for temperatures

97

Temperature Absolute Error in Prediction (°C)

Reactor gas inlet -0.0

Reactor gas outlet -1.3

Reactor coolant inlet -0.0

Reactor coolant outlet -0.5

6.1.1 Catalyst deactivation model benchmarking

The catalyst used in the fixed bed reactor deactivates over the period of time due

to impurities in the feed and sintering of the catalyst. Therefore, it is required to model

catalyst deactivation which will predict the temperature and composition profiles over the

life of the catalyst. This catalyst deactivation is a nonlinear function of operating time and

temperature, since higher temperatures within the tube promote the deactivation. Here the

catalyst activity at the start of operation is assumed to be unity (a = l). Then it decreases

exponentially with time t^^. It is expressed as a function of outlet tube side temperature

T.

a = exp(-a • t^^) when T<T^ for reaction 1 & reaction 2

a = exp(-a-f^,)-exp(-;5-P) T>T, for reaction 1

a = exp(-a-V)-exp(-r -P) T>T^ for reaction 2

98

The catalyst deactivation model contains 4 empincal constants. There are total

8 X Age (where Age - catalyst age in days) operating points are available to fit the model

(4 outlet compositions, 2 reactor inlet temperatures and 2 reactor outiet temperatures).

NPSOL (1986) solver is used to find the empirical constants by minimizing the following

weighted objective function.

The model parameters are found by minimizing the following objective function.

mm Age

Obj = j ; day^O

4 Wf -Wt

/=! Wt

(\-W)

Ind,

rp Ind rp Pred f cin - cin \ 2 /

rp Ind rp Pred cout -' cour

L Ind

P„ Ind )'-

+ (•

rp Ind rp Pred

T:. Ind

rp Ind rp Pred \2 . (out •'•out \2

P„ Ind

By minimizing the above weighted objective function, we get the following

values of the model parameters:

a = 0.0125

y5= 0.0133

Y = 0.020.

6.1.2 Catalyst deactivation model validation

Figures 6.1 to 6.7 show the comparison between the model prediction and

industrial data over the entire life of the catalyst. It can be seen that the catalyst

deactivation model predicts the industrial data quite well.

Because of confidentiality reasons, we are not able to provide actual values of the

operating conditions (e.g., temperature and composition) in the figures and tables.

99

M + 4

o Q. E o o

M + 2

a:

4 M

• • A A

• J

A

•

Time

• Industrial data A Model Predicrtion

Figure 6.1 Comparison between industrial data and model prediction for ethylene reactor outlet composition

o M + 3

o Q. E o o g M + 2

M + 1

Time


Figure 6.2 Comparison between industrial data and model prediction for oxygen reactor outlet composition

100

M + 2.0r-

o I M + 1.6 o (J

g M+1.2

f M + 0.8

cr

M + 0.4

M

Time

• Industrial data A Model Predic:tion

Figure 6.3 Comparison between industrial data and model prediction for ethylene oxide reactor outlet composition

o I M + 12 o o 0)

"D x o ''^ M + 8

o

3 M + 4 o

^ ^

<D

M

Time

• Industrial data A Model Predic:tion

Figure 6.4 Comparison between industrial data and model prediction for carbon dioxide reactor outlet composition

101

TO+ 50

3

n (1)

u. b £ m ro O)

(I)

o

tor

<J

0)

ce

TO+ 40

TO + 30

TO + 20 1

i *

T0 + 10

£i

P *m -y • •

A

f^

•

TO

Time

• Industrial data A Model Predit:tion

Figure 6.5 Comparison between industrial data and model prediction for reactor outlet gas tempeature

TO+ 40

CO

I TO+ 30 E <u

§ TO+ 20 o

2 TO + 10 u ro 0)

a:

i ^ t^cM^

TO

S 6

A •

Time


Figure 6.6 Comparison between industrial data and model prediction for reactor outlet coolant temperature

102

TO+ 30

ro 0)

I TO+ 20

J5 o o u

= TO+ 10

^.

A „

^

TO

A A A A &^

Time


Figure 6.7 Comparison between industrial data and model prediction for reactor inlet coolant temperature

6.2 Offline Optimization Approach

In the ethylene oxide reactor, higher temperatures lead to excessive formation of

carbon dioxide and water which resuhs in loss of selectivity. Also, the lower

temperatures result in lower conversion and loss of productivity. There is a need for

conversion and selectivity balance which will be met by careful control of optimal shell

side inlet temperature profile. Since the catalyst deactivates over the period of time, shell

side inlet temperature is increased to compensate for the loss of activity. The manner in

which the shell side inlet temperature is changed affects the net profit of the process. This

motivates us to carry out an optimization study for this process. The optimization

problem involved determining the optimal shell side inlet temperature profile over the run

length of the operation by maximizing the net profit of the process.

103

Figure 4.8 illustrates the optimization procedure. The continuous temperature

profile over the length of the operation is expressed m terms of node values at specific

points in the time domain comiected together by smoothly varying mterpolatmg

polynomial. The optimizer uses these node values as decision variables. The intermediate

values for any particular temperature required by the simulator are provided by applying

cubic spline interpolation (Riggs, 1994).

The optimization was carried out for fixed catalyst life and fixed reactor inlet

conditions. The optimizer sought the four decision variables such that the profit fimction

(O) value over the entire run period was maximized.

^ = P,o-EO-P,-E-P^-0

The carbon dioxide separation cost is insignificant and not considered in the profit

function.

6.2.1 Offline optimization results

The optimizer (NPSOL, Gill 1986) calculates the optimum temperature profile,

which gives the profit improvement of 8.56% over base case.

From Figure 6.8, it can be seen that the optimum temperature for the initial period

is higher than the base case temperature. For the remaining period of time, the optimum

temperature is greater than or almost equal to the base case temperature profile.

104

TO+ 50

TO+ 45 (U

a TO+ 40 ro

I TO+ 35

*Z TO+ 30

Z TO+ 25 c g TO+ 20 O o TO+ 15 •o ro <u a:

TO+ 5

TO

Time

• Base case • Optimal A node values

Figure 6.8 Comparison between base case temperature profile and optimum temperature profile

6.2.2 Optimization for different production rates

Above offline optimization procedure is used to calculate profit for different

production rates. Three different production rates are considered: (a) 10% decrease over

the base case production rate, (b) 10% increase over the base case production rate, and (c)

20% increase over the base case production rate. Table 6.3 shows the percentage profit

improvement over the base case for different production rates

105

Table 6.3 Percentage profit improvement over the base case for different production rates

% Over the base case production rate % profit increase over the base case

10% increase

10% decrease

20% decrease

9.02

8.96

9.43

6.3 Control of Ethylene Oxide Reactor

As described earlier, the process studied consists of a feed effluent heat

exchanger, a muhitubular fixed bed reactor, a steam generator, and a separation system.

The exothermic heat of reaction from the reactor is removed by passing coolant on the

shell side of the reactor. A portion of the heated coolant is passed through a steam

generator to produce steam and the total coolant stream is recycled back to the shell side

of the reactor. Figure 6.9 shows a single-loop PI control system that uses the flow rate of

the coolant that is passed through the steam generator to control the inlet temperature of

the coolant to the reactor.

Steam Generator

r-><}^X

R E A C T O R

Heat Excli. Separation System

TC K-

Fiaure 6.9 Schematic of the reactor inlet coolant temperattire control system

106

The PI controller is tuned for a change of 1°C in the set point of the reactor inlet

coolant temperature. Figure 6.10 shows the response of the reactor inlet coolant

temperature. Coolant flow through the steam generator is manipulated to control the

coolant inlet temperature to the reactor. Figure 6.11 shows the response of the flow

through the steam generator. Coolant flow through the steam generator and bypass

coolant flow is mixed before sending it to the reactor. Since we want an increase in the

coolant inlet temperature to the reactor, flow through the steam generator decreases first

(this means an increase in bypass flow) to increase in the coolant inlet temperature to the

reactor.

TO+ 2.

TO+ 1.6

<u 3 ro » TO + 1.2 Q. E o o

Z TO+ 0.8 c j5 o o O

TO + 0.4

TO

Time

Figure 6.10 Response of controlled variable to 1°C increase in set point

107

TO+IOOOT

o TO+ 800 4-1

ro k. o c o u> E TO+ 600 ra 0)

In

o TO+ 400

0)

ro

5 TO+ 200

u.

TO

Time

Figure 6.11 Response of manipulated variable to 1*C increase in set point

The performance of the controller is checked against step in a disturbance (i.e.,

change in carbon dioxide inlet composition). A 0.5% mole increase in the carbon dioxide

reactor inlet composition is made. Figure 6.12 and Figure 6.13 show the response of the

reactor inlet coolant temperature and flow through the steam generator, respectively.

108

TO+ 0.12

£ 3

4-»

ro g. TO+ 0.08 E o

4->

c _ra o o o

.E TO + 0.04

0)

on

TO

Time

Figure 6.12 Response of controlled variable to 0.5% change in the disturbance

F + 160 T

o F + 140

Time

Figure 6.13 Response of manipulated variable to 0.5% change in the disturbance

109

6.4 Nomenclature

Age Age of the catalyst in days

a Activity of the catalyst (0 < a < 1)

C;" Industrial reactor outlet mole % of i"' component

C^""' Model Predicted reactor outlet mole % of /"• component

$ P^ Cost of Ethylene (—)

$ P^o Price of Ethylene Oxide ( —)

$ PQ Cost of Oxygen (—)

T - Outlet tube side temperature

T^J"'' Industrial reactor coolant inlet temperature

T^J'"' Model predicted reactor coolant inlet temperature

T '""^ Industrial reactor coolant outlet temperattire cout

Tcou^'^ - Model predicted reactor coolant outiet temperature

TJ"^ Industrial reactor gas inlet temperature

TJ'"' Model predicted reactor gas inlet temperature

T '"^ - Industrial reactor gas outlet temperature out

TJ''^ Model predicted reactor gas outlet temperature

t ^ - Catalyst life in years

110

W - weight, 0<W <l

Greek Letters

« , / ? , / Empirical constants

111

CHAPTER 7

BIFURCATION ANALYSIS OF ETHYLENE OXIDE PROCESS

7.1 Bifiircation Study of an Industrial Ethylene Oxide Process

For a long time, it has been recognized that the nonlinear behavior (i.e.,

input/output multiplicities) of chemical reactors might have an important effect on the

operation difficulty of such process (Seider, 1990). Bifiircation theory has been

recognized as a very useful tool to address the nonlinear pattern behavior of processing

systems subject to the variation of some parameters (Kuznestov, 1998).

In this work, the open-loop and closed-loop nonlinear bifurcation analysis of an

industrial ethylene oxide reactor is performed. Ethylene oxide is one of the most

important pefrochemical intermediates and one of the raw materials used for the

production of glycol, polyethylene glycol and glycol ethers. The production of the

ethylene oxide is a critical process because the reactor can generate eleven times as much

heat in a runaway condition as under normal operating conditions. Therefore, the safety

issues for an ethylene oxide reactor system are dominant as industry tries to operate them

in an economically advantageous marmer.

The aim of this work is to provide a first look into the operability problems faced

by ethylene oxide reactor and to perform open-loop and closed-loop bifurcation studies

using the benchmarked model of the ethylene oxide reactor system. The steady state

operability problem is addressed by using nonlinear bifurcation techniques. The ethylene

oxide commercial facility that we studied did not operate at higher operating

112

CHAPTER 7

BIFURCATION ANALYSIS OF ETHYLENE OXEDE PROCESS

7.1 Bifurcation Study of an Industrial Ethylene Oxide Process

For a long time, it has been recognized that the nonlinear behavior (i.e.,

input/output muhiplicities) of chemical reactors might have an important effect on the

operation difficulty of such process (Seider, 1990). Bifurcation theory has been

recognized as a very useful tool to address the nonlinear pattern behavior of processing

systems subject to the variation of some parameters (Kuznestov, 1998).

In this work, the open-loop and closed-loop nonlinear bifurcation analysis of an

industrial ethylene oxide reactor is performed. Ethylene oxide is one of the most

important petrochemical intermediates and one of the raw materials used for the

production of glycol, polyethylene glycol and glycol ethers. The production of the

ethylene oxide is a critical process because the reactor can generate eleven times as much

heat in a runaway condition as under normal operating conditions. Therefore, the safety

issues for an ethylene oxide reactor system are dominant as industry tries to operate them

in an economically advantageous maimer.

The aim of this work is to provide a first look into the operability problems faced

by ethylene oxide reactor and to perform open-loop and closed-loop bifurcation studies

using the benchmarked model of the ethylene oxide reactor system. The steady state

operability problem is addressed by using nonlinear bifurcation techniques. The ethylene

oxide commercial facility that we studied did not operate at higher operating

112

temperattires because of the associated risks of a reactor runaway. Therefore, stable

temperature control of the ethylene oxide reactor is important. An analysis of the stable

control region of the system is developed as a function of the operating temperature,

catalyst activity, and disturbance direction and magnitude. In the open literature there are

no published papers on the nonlinear bifurcation analysis of the ethylene oxide

manufacturing process.

7.2 Continuation Algorithm to Develop Bifiircation Diagram

Elementary catastrophe theory might be used in order to detect analytical

conditions under which input/output multiplicities could emerge. However, one of the

major problems related to the use of catastrophe theory is that it requires collapsing the

entire mathematical model into a single algebraic equation. The procedure is totally

impractical for large-scale models. Because of the complexity (higher dimensionahty) of

the ethylene oxide process model equations, a purely numerical procedure is used to

characterize the multiplicity behavior (bifurcation study) of the ethylene oxide process. A

numerical technique enabling us to obtain one branch of solutions (or more branches of

solutions mutually connected at branch points) is called the continuation technique

(Kubicek, 1983). The continuation algorithm can produce a continuous curve (consisting

of branches of solutions). For any other curve (branch of solutions), we need to obtain an

initial estimate of the solution in order to begin the continuation procedure. How can we

obtain all steady state solutions of tiie given set of equations? The task may be

particulariy demanding in situations where we have poor preliminary estimates of the

113

solution and chosen iteration method diverges. In such situations, we can use randomly

generated initial estimates of solution for every initial estimate of the solution: either the

iteration algorithm converges or diverges. The new converged solutions thus, obtained

are subsequently stored into a memory of solutions. If a sufficiently high number of

initial estimates are chosen, the probability of solving the problem is high. The number of

random initial estimates necessary can sometimes be rather high. In principle, this

technique enables us to start from a known solution and continuously compute solutions

along a chosen branch.

7.3 Stability of Steady State Solutions

Stationary state x of the differential equation is called locally stable if

l im | |x (0-x 11=0,

for x(0) chosen in a sufficiently small neighborhood of x , i.e., for x(0) such that

\\x(t)-x\\<5,

where (5 is a conveniently small number.

Consider a system of linear differential equations with constant coefficients

dx — = Ax dt

where x € i?". Let the only stationary solution be 3c = 0. If the eigenvalues of matrix A

are known, we can determine the stability of this unique stationary solution (Kubicek,

1983). If for every eigenvalue X-,

Re(l,.)<0 z = l,2,...,7z

114

the zero solution is stable and all frajectories approach it for r ^ oo . If on the contrary, at

least one eigenvalue has a positive real part, the solution is unstable since there are

trajectories that approach oo as r -^ oo .

7.4 Results and Discussions

7.4.1 Effect of manipulated variable

In Figures 7.1-7.3, a bifurcation diagram of the reactor coolant inlet temperature

using the coolant flow through the steam generator as the continuation parameter is

shown. The bifurcation diagram is obtained by solving the 428 nonlinear algebraic

equations for a given coolant flow through steam generator. First, a continuation

algorithm is used to generate the continuation (bifurcation) diagram. For this system of

equations to converge, it required more number of initial guesses. Therefore, the problem

is formulated in a different way to obtain the bifiircation diagram starting only with a

single initial guess. One more equation is added to the existing 428 nonlinear equations

for the reactor inlet coolant temperature. Instead of specifying the coolant flow through

the steam generator, reactor coolant inlet temperature (set point) is specified and 429

nonlinear algebraic equations are solved using MINPACK. This formulation of the

problem was found to be more efficient than the earlier one. Under nominal operating

conditions, the reactor displays output muhiplicities, i.e., for a given coolant flow through

the steam generator. There are two different reactor coolant inlet temperattires. The

nominal upper steady state temperature is unstable while the lower steady state is stable.

This type of bifiircation behavior is called saddle node bifurcation. In saddle node

115

bifurcation, there is only a single turning point and at this turning point the solution

from stable to unstable.

The numbers (i.e., temperature and composition) in the figures and tables are

scaled.

TO+ 30

5 TO+ 25

E TO+ 20

"5 TO+ 15 o

Z TO+ 10

•o ro

tc TO + 5

TO

,A

0.4 0.5 0.6 0.7 0.8

Fraction of flow through steam generator

0.9

- • stable • - - A- - - unstable

Figure 7.1 Bifiorcation diagram using the flow through steam generator as a continuation parameter

116

XO + 12 T

XO + 10

XO + 8

-A -

- A ' . . A-

c

XO + 6

XO + 4

XO + 2

XO

0.4 0.5 0.6 0.7 0.8


0.9

- stable unstable

Figure 7.2 Bifurcation diagram using the flow through steam generator as a continuation parameter

iper

atur

e

)rte

n

Max

imum

rea

ctc

TO+ 50

TO+ 45

TO+ 40

TO+ 35

TO+ 30

TO+ 25

TO+ 20

TO+ 15

TO+ 10

TO+ 5

TO 0.4 0.5 0.6 0.7 0.8


0.9

• stable unstable

Figure 7.3 Bifiircation diagram using the flow through steam generator as a parameter

117

7.4.2 Effect of disturbance

Here, we analyze the open-loop bifurcation behavior of the ethylene oxide reactor

process with respect to a process disturbance. Reactor inlet carbon dioxide mole fraction

is a major disturbance to the ethylene oxide reactor inlet coolant temperature control

system. Figures 7.4-7.6 show bifuration diagrams using reactor inlet carbon dioxide mole

fraction as the continuation parameter. The same procedure as described in section 7.4.1

is used to obtain the bifiircation diagram. Similar to the effect of manipulated variable,

the reactor displays output multiplicities (saddle node bifurcation). Thus, the above

bifurcation diagrams can be used to understand changes in stability on a given branch of

solutions.

TO+ 30

Si TO+ 25

ZJ

"TO

0}

f TO+ 20 B "c ro o TO+ 15 o o - TO+ 10

t3 ro 0) CC TO + 5

TO 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14

Carbon Dioxide Reactor inlet mole fraction

0.15 0.16

- • — stable - • - A- - - unstable

Figure 7.4 Bifiircation diagram using the reactor inlet CO2 mole fraction as a continuation parameter

118

TO+ 50

TO+ 45

S. TO+ 40

% TO+ 35 Q.

i TO+ 30

% TO+ 25 ro (U ^ TO+ 20

I TO+ 15 X

i TO+ 10

TO+ 5

TO

.A'

0.07 0.08 0.09 0.1 0,11 0.12 0.13 0.14

Carbon Dioxide reactor inlet mole fraction

0.15 0,16

- stable - - • A- • - unstable

Figure 7.5 Bifurcation diagi"am using the reactor inlet CO2 mole fraction as a continuation parameter

XO + 7

XO + 6

g XO + 5 c: o '2 XO + 4 >

^ XO + 3 JJ

£ XO + 2 <u

XO + 1

XO

., ^•

0.07 0.08 0.09 0.1 0.11 0.12 0.13

carbon Dioxide reactor inlet mole fraction

0.14 0.15

• stable unstable

Figure 7.6 Bifurcation diagram using the reactor inlet CO2 mole fraction as a parameter

119

7.5 Runaway Boundary

In the case of a fixed bed reactor for an exothermic reaction, a temperature

maximum may be exhibited at some location along the reactor, which is generally

referred to as a "hot spot." The magnitude of this hot spot must be bounded within

specific limits, because it may seriously affect reactor safety and performance. The

magnitude of the hot spot depends on the system parameters, such as operating

conditions, physicochemical properties, and reaction kinetics. For specific values of the

system parameters, the hot spot may undergo large variations relative to small changes in

one or more of the operating conditions or system parameters. In this case, the reactor is

said to operate in a parametrically sensitive region. In practical applications, it is

desirable to avoid this operating region for safety of the process. This provides the

motivation to develop a runaway boundary for ethylene oxide reactor. In particular, we

will identify the runaway region for this reactor by applying the generalized runaway

criterion (Morbidelli and Varma, 1986b) using the maximum in the catalyst temperatiire,

e *, profile along the reactor as the objective. For this we need to define the objective

sensitivity, s(Oj;^), which is defined as

dd„' s(0, •,<!>) =

d(p

where represents the model-input parameter or operating condition such as coolant

flow through steam generator. But the more appropriate quantity m sensitivity analysis is

120

the normahzed objective sensitivity, S(0*;^). The normalized objective sensitivity of

the catalyst temperature maximum, 6 J, along the reactor length is given by

e, ' e d$ p " p r

In the present ethylene oxide process, the fixed bed reactor is too short for

developing a local temperature maximum (hot spot). Here, the axial temperature profile is

monotonically increasing, and so the maximum catalyst temperature value considered in

the sensitivity analysis is that at the reactor outlet. The critical conditions for reactor

runaway, according to the generalized sensitivity criterion, are then identified as the

situation in which the normahzed objective sensitivity, 8(0^ ; ^ ) , is maximized

(Morbidelli and Varma, 1986b).

As shown in Figure 7.7, the normalized sensitivity S(0* •,(!>) for the temperature

of the catalyst particle operating on the lower temperature branch (open-loop stable

operating point) decreases with a decrease in the flow through the steam generator and

becomes large at the bifurcation point. Thus, at the bifiircation point, the catalyst

temperature becomes very sensitive to the change in the coolant flow through the steam

generator. It is worth noting that when there exist multiple steady states for the reactor, its

runaway boundary is always coincident with the bifurcation point of the multiple steady

state regions.

Note that the sign of the normahzed sensitivity value has a particular meaning. A

positive (negative) value of the normalized sensitivity of the temperature maximum with

121

respect to coolant flow through the steam generator indicates that the temperature

maximum increases (decreases) as the magnitude of coolant flow increases. Thus, m this

case, the normalized sensitivity is negative, which indicates that the fransition from non-

runaway to runaway behavior occurs as the coolant flow through the steam generator is

decreased.

CO

73 CU N

0,2

0,1

0

-0.1

-0.2

-0.3

-0.4

-0.5

0.75

. • - •

H i _ • -It^

0.8 0.85 0.9 0.95

Fraction of Flow through steam generator

TO + 10

TO+ 8

TO+ 7

TO+ 6

TO+ 5

TO+ 4

TO+ 3

TO+ 2

TO + 1

TO

• Normalized sensitivity S - - - - - - - coolant inlet temperature

o o o

3 T3 CD

Figure 7.7 Temperature of the catalyst particle and its normalized sensitivity with respect to the flow through steam generator

Figure 7.8 shows the open-loop runaway region for the different catalyst activity.

Any arbitary size disturbance near the runaway boundary will make the reaction runaway

in the open-loop. The runaway region was obtained by calculating the bifurcation points

for different catalyst activity as shown in the Figure 7.9.

122

TO + 2.4

TO+ 2.

ro (1)

emp

c ro

coo

(U c

o ro C) C

10

10

TO

+ 1.6

+ 1.2

+ 0.8

TO + 0.4

TO

0.85 0.9

Activity of the catalyst

0,95

Figure 7.8 Boundary of the runaway region

TO+ 30

<u 3

ro n> F <i)

o

tin

c ro o o o o (J ni u> 01

TO+ 25

TO+ 20

TO+ 15

TO+ 10

TO+ 5

TO

I I X

X X X X ^ ?-X X X X

, X X .' X X • — • — ( X ,', X -•. X > X

X • X X X

X X

x" ^

X

B

• • E

• • • • • #

f -f—:

I \ 1

• • • • • • • X • X X X

a

• •

•

•

•

• •

•

•

•

0.1 0.2 0.3 0.4 0,5 0.6 0.7 0.8 0.9


• a=1,0 a=0.95 a=0,9 X a=0.85 X a=0.8 • locus of bifurcation points

Figure 7.9 Locus of bifiircation points for different catalyst activity

123

7.6 Closed-Loop Nonlinear Bifurcation Analysis

The ethylene oxide reactor is always operated at higher coolant inlet temperature

(open-loop unstable operating point) because it gives higher ethylene conversion than the

lower coolant inlet temperature (open-loop stable operating point). Although the

operating point is open-loop unstable, a controller may be able to stabihze it.

7.6.1 Effect of disturbances

Inlet carbon dioxide composition (mole %) is the primary disturbance for the

reactor inlet coolant temperature control loop. The disturbance magnitude is changed in

the positive and negative direction for different values of the control loop dead time to

evaluate the performance of the PI controller. Figure 7.10 shows the runaway region for a

positive change in the carbon dioxide reactor inlet composition. It can be seen that for

dead times below 20 seconds the system goes unstable for almost a +10mole% change in

the carbon dioxide inlet composition. This is because the increased carbon dioxide partial

pressure (which is due to increased inlet carbon dioxide mole%) favors the partial

oxidation of ethylene and does not favor the complete oxidation of ethylene, which has

more heat of reaction.

124

o £

in o Q . E o o

CM

o o .c CD cn c

Q) > +

10,5

10

9.5

8,5

7,5

6,5

Stable

Runaway

12 14 16 18 20

Dead time (seconds)

22 24 26

Figure 7.10 Closed loop stability region for Carbon Dioxide disturbance change in the positive direction

Figure 7.11 shows the closed-loop runway region for a negative change in the

carbon dioxide reactor inlet composition. It can be seen that the system is very sensitive

to negative change in the reactor inlet carbon dioxide composition. This is because the

decrease in the partial pressure of the carbon dioxide in the reactor favors the complete

oxidation reaction, which has a high heat of reaction.

From Figures 7.10 and 7.11, we conclude that a decrease in the carbon dioxide

reactor inlet composition from nominal is a major disturbance as compared an increase in

the carbon dioxide reactor inlet composition. Only a decrease in the carbon dioxide

reactor inlet composition is studied.

125

10 15

Dead time (sec)

20 25

Figure 7.11 Closed-loop stability region for Carbon Dioxide disturbance change in the negative direction

7.6.2 Effect of detuning factor

Here we study how the controller aggressiveness and sluggishness affect the

runaway region. First, the controller is tuned for set point changes using 1/6' decay ratio

as the tuning criterion. Then the detuning factor is varied from 0.1 to 5.0, and each time

the disturbance magnitude is varied till the conttoUer goes unstable. Figure 7.12 shows

the effect of the detuning factor on the stability region for zero dead time in the control

loop. It can be seen that runaway boundary is sensitive to the detuning factor.

126

3.5

.2J o E ^ 3 o

o a. E 8 2,5

CN o o '3) ,E 2

1,5

o Oi

>

Unstable

Stable

2 3

Detuning factor ft

Figure 7.12 Effect of detuning factor on the runaway boundary

7.6.3 Effect of operating temperature

Here we study, how the disturbance affects the stability of the controller for

different operating temperatures (i.e., reactor inlet coolant temperature). For a fresh

catalyst (catalyst activity equal to one), the range of the operating temperature is

estimated by solving the steady state nonlinear algebraic equations. But as discussed

earlier, in industry, the reactor is operated above the bifiircation point; therefore, we also

considered the operating temperature range from the bifurcation point to the maximum

achievable reactor inlet coolant temperature. The following procedure is used to get the

runaway region. First, the PI controller is tuned for a given operating temperature set

point for 1/6" decay ratio as a tuning criterion. Then the disturbance magnitude is varied

127

until the controller goes unstable. From Figure 7.13, it can be seen that the system is very

sensitive to disturbances at higher operating temperatures.

3.5

2.5

2

o Q . E o o

CM o o ^ 1.5

.£ 1

0.5

Runaway

Stable

TO TO + 1 TO + 2 TO + 3 TO + 4 TO + 5 TO + 6 TO + 7 TO + 8

Operating temperature

Figure 7.13 Effect of operating temperature on the stability region

7.6.4 Effect of catalyst activity

Figure 7.14 shows the effect of operating temperature on the runaway boundary

for different catalyst activities. The same procedure as described for Figure 7.13 is

applied to get the runaway boundaries for different catalyst activities. It can be seen that

the sensitivity of the operating temperature to the change in the reactor inlet carbon

dioxide (mole%) decreases as the activity of the catalyst decreases.

128

TO+ 5 TO+ 10 TO+15 TO + 20 TO + 25 TO + 30 TO + 35

Operating Temperature

-a=0.93 a=0.85 --X ••a=0.75 ^ ^ a = 0 . 6 5

Figure 7.14 Comparison between runaway boundary for different catalyst activity

7.6.5 How to detect runaway

The runaway situation can be detected from the gas temperature measurements

along the length of the reactor. Here, we have considered three different temperature

measurements at 60%, 80%, and 100% (i.e., reactor outlet) of the reactor length. The

slope is calculated between these temperature measurements. Figure 7.15 shows the

runaway boundary for different catalyst activity. J t is remarked that whenever the

temperature slope exceeds the runaway boundary, the reactor will become unstable.

129

Runaway

0.85 0.9

Catalyst activity

0.95

- • — Temp slope between measurement at reactor outlet and 80% of the reactor length

-A— Temp slope betw een measurement at 80% and 60% of the reactor length

Figure 7.15 Temperature slope for different catalyst activity

7.6.6 How to prevent runaway

The runaway situation can be avoided by setting the make up oxygen to zero

flow. One such simulated runaway situation is shown in Figure 7.16. In this figure, the

disturbance is infroduced after 5 seconds, which causes a runaway reaction (i.e., the

reactor temperature shoots up exponentially). The 25 C increase in temperature is caused

by the change in the heat transfer coefficients, which are modeled as a function of

temperature through thermal conductivity and viscosity calculations. The make-up

oxygen is reduced to zero after 1.4 min once runaway is detected, since the oxygen

partial pressure becomes low the reaction ceases and the temperature drops immediately.

After 1.5 mins, there is no oxygen in the reactor, thus no reaction. There is only heat

130

transfer between the coolant and the reaction medium. Thus, reducing the oxygen can

always prevent the runaway reactor.

TO+ 45

TO+ 40

& TO+ 35 "ro

g_ TO+ 30 E 0)

•;;; T O + 25 ro

I TO+ 20 "3 o o TO+15 "o ro

l2 TO+10

TO+ 5

TO

Oxygen set to zero

0.5 1 1.5

Time (min)

Figure 7.16 Response of the outlet temperature to eliminating oxygen in the feed when runaway observed

131

CHAPTER 8

CONTRIBUTION

Quina et al. (1999) presented the steady state analysis of the region of parametric

sensitivity and the range of operating conditions leading to the phenomenon of

temperature runaway for a fixed bed reactor (the selective oxidation of methanol to

formaldehyde), where the catalytic bed is partially diluted with inert packing. A complete

bifurcation analysis of a general steady state two-dimensional catalytic monolith reactor

model that accounted for temperature and concenfration gradients in both axial and radial

directions is studied by Balakotaiah et al. (2001). A single exothermic first-order reaction

was considered. Garcia et al. (2000) studied the steady state nonlinear bifurcation

behavior of a high impact commercial polystyrene continuous stirred tank reactor. Chang

(1984) presented an analysis of the various types of bifurcation that are caused by a

conventional, SISO PID controller on a general nonlinear system. Thus, most of the

studies are centered on the steady state bifurcation analysis of fixed bed catalytic reactors.

The steady state bifurcation analysis helps to understand the input/output multiplicity in

the reactor and the stable/unstable operating points with respect to certain

physicochemical paramefrs. For a reactor to operate in a rehable and safe marmer, not

only the steady state nonlinear bifiircation analysis but also the closed loop stability

analysis is important. Also the effect of important parameters e.g deadtime, disturbances

which affects the closed loop performance, can not be stiidied in steady state bifurcation

132

analysis. The effect of controller aggressiveness can be stiidied in a closed-loop stabihty

analysis by using a detuning factor.

The aim of the present work was to contribute to the open-loop and closed-loop

stability analysis in a heat integrated multitubular fixed be reactor used for ethylene oxide

production. In this regard, the bifurcation theory is used to study the stability issues.

The open-loop bifurcation study showed that under nominal operating conditions,

the ethylene oxide reactor system displays output multiplicities (saddle node bifiircation).

The nominal upper steady state is unstable while the lower steady state is stable. The

bifiircation plots were obtained by varying flow through steam generator (a manipulated

variable) and inlet carbon dioxide inlet composition (a disturbance to the reactor inlet

coolant temperature control loop). These results are particularly interesting because they

enable one to conclude that bifurcation analysis can be based on simple parameters such

as COj'"'"' (inlet carbon dioxide composition) and F^ (flow through steam generator),

therefore allowing their manipulation in order to avoid the risky operating conditions.

The analysis is based on a heterogeneous two-dimensional model of the reactor, which

predicts the temperature and composition in the radial and axial direction of the reactor.

Also the coolant temperature is not constant, but it varies in the axial direction on the

shell side of the reactor.

The inlet coolant temperature control of the ethylene oxide reactor is

economically important. This is because at higher temperature the selectivity of ethylene

oxide decreases and at lower temperatures the conversion of ethylene to ethylene oxide

decreases. So there is an optimal temperature profile at which both conversion and

133

selectivity is maintained by careful control of the reactor inlet coolant temperature. But at

the higher reactor coolant inlet temperature, the reactor is susceptible to runaway due to

high heat of reaction of complete oxidation of ethylene. Therefore, the closed-loop

stability analysis of this system is very important and will be useful in understanding the

safe operating regions. An analysis of the stable confrol region of the system is developed

as a function of operating temperature, catalyst activity, detuning factor, and disturbance

(reactor inlet carbon dioxide composition) direction and magnitude. The closed-loop

stability region was found to be sensitive to the negative change in inlet carbon dioxide

composition. The reactor system was also found to be more prone to instability at higher

operating temperatures and higher catalyst activity.

Based on this study, the ethylene oxide reactor can be operated at higher

temprature which will improve the profitability of the system without substantially

increasing the risk of a reactor runaway. It is also shown that shutting off oxygen feed to

the reactor can always prevent the runaway reactor. This sttidy represents the first open-

loop and closed-loop bifiircation sttidy of an industtial reactor system.

134

CHAPTER 9

DISCUSSION, CONCLUSIONS, AND RECOMMENDATIONS

The primary objectives of the research work were as follows:

1. For the ethylene oxide process:

a. develop a detailed mathematical model for the process,

b. benchmark the developed model against industrial data,

c. study offline optimization and control of the process based on economic objective

function,

d. study open-loop and closed-loop bifurcation of the process;

2. For the vinyl acetate process:

a. develop a detailed mathematical model for the reactor,

b. benchmark the developed model against industrial data,

c. study offline and online optimization of the reactor based on economic objective

function.

In both cases, the developed mathematical model was derived from a generalized

model for a multitubular gas phase solid catalyzed reaction based on certain assumptions.

For a vinyl acetate reactor, a steady state two-dimensional homogeneous model was

developed. For an ethylene oxide reactor, a two-dimensional dynamic heterogeneous

model was developed. Radial derivatives are approximated by orthogonal collocation and

axial derivatives were approximated by orthogonal collocation on finite elements. Most

of the assumptions were justified by satisfying existing criterion and some assumptions

135

were by actual calculations. For example, in the case of ethylene oxide reactor, the

effectiveness factors for both the reactions are calculated by simulating temperature and

concenfration profiles inside a catalyst. From the results, it was shown that the

intraparticle resistance could be neglected since the effectiveness factor is closed to unity

for both the reaction systems.

A base case operating condition was chosen from the industrial data at which the

catalyst was assumed to be fresh (activity =1). The developed model was benchmarked

against the industrial data. The model parameters were estimated through regression

analysis by minimizing the weighted error between the model predicted output values

(e.g., reactor outlet temperature and composition) and the industrial data for the base case

operating point.

Since the catalyst loses activity due to sintering and impurities in the feed, it was

required to take care of this effect in the model through catalyst deactivation. Based upon

the catalyst deactivation models available in the literature, a suitable model was selected.

The catalyst deactivation model was benchmarked against the industrial data over the

period of operation to represent the real process. In the case of deactivation model, the

model parameters were estimated through regression analysis by minimizing the total

error between the model predicted output values (e.g., reactor outlet temperatures and

compositions) and industrial data over the length of the operation. A comparison between

the industrial data and model prediction showed that the model predicted the industrial

data quite well for both the processes.

136

Since the catalyst deactivates over the period of operation, operating temperature

is increased to compensate for the loss of activity. But the increased operating

temperature can affect the selectivity of the desired product. Therefore, offline

optimization is done for both the vinyl acetate and ethylene oxide process, using a steady

state process model to find an optimal operating temperature profile which maximizes the

profit of the process.

To see the effect of the model parameter's uncertainty, sensitivity analysis is

carried out by perturbing the model parameter values by 10% and re-running the

optimization algorithm for offline optimization. The sensitivity analysis for vinyl acetate

process optimization showed that the results were sensitive to the uncertainty in the

model parameters.

For the vinyl acetate reactor, online optimization is done by updating the model

parameters online. Online optimization showed more profit improvement as compared to

offline optimization.

Nonlinear bifiircation analysis of ethylene oxide reactor has been carried out to

study the safe operating regimes. Under nominal operating conditions, the ethylene oxide

reactor system displays output multiplicities (saddle node bifiircation). The nominal

upper steady state is unstable, while the lower steady state is stable. The closed-loop

stability of the reactor was found to be sensitive to the negative change in inlet carbon

dioxide composition. The reactor system was also found to be more prone to instability at

higher operating temperatures and higher activity catalyst. Based on this sttidy, the

reactor can be operated at higher temperature which will improve the profitability of the

137

system without substantially increasing the risk of a reactor runaway. It is also shown that

shutting off the oxygen feed to the reactor can always prevent the runaway reactor.

The following recommendations are made for possible future study.

In case of the ethylene oxide process, the carbon dioxide inlet compositon

disturbance comes from the separation system; therefore, it can be measured before it can

affect the reactor inlet coolant temperature control system. Thus, a feedforward control

system will be more useful to take care of reactor inlet carbon dioxide composition

disturbance.

A confroUer with a nonliear control law can be designed for a reactor inlet coolant

temperature control system (in case of ethylene oxide process) in the vicinity of the

closed-loop runaway boundary. This will allow to operate the reactor at higher operating

temperatures especially when the catalyst activity decreases.

In case of the ethylene oxide production, a moderator (e.g., chlorine compound) is

added in controlled amounts which improves the selectivity to ethylene oxide while

inhibiting the total oxidation reaction (ethylene to carbon dioxide and water). The catalyst

activity depends on the amount of moderator added to the reactant mixture. Therefore,

the effect of the moderator on the actalyst activity can be considered in the catalyst

deactivation model.

138

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