Low-temperature Combustion and Autoignition, Volume 35 (Comprehensive Chemical Kinetics)

COMPREHENSIVE CHEMICAL KINETICS COMPREHENSIVE Section 1. THE PRACTICE AND THEORY OF KINETICS (3 volumes) Section 2...

Author: M.J. Pilling

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COMPREHENSIVE CHEMICAL KINETICS

COMPREHENSIVE Section 1.

THE PRACTICE AND THEORY OF KINETICS (3 volumes)

Section 2.

HOMOGENEOUS DECOMPOSITION AND ISOMERISATION REACTIONS (2 volumes)

Section 3.

INORGANIC REACTIONS (2 volumes)

Section 4.

ORGANIC REACTIONS (5 volumes)

Section 5.

POLYMERISATION REACTIONS (3 volumes)

Section 6.

OXIDATION AND COMBUSTION REACTIONS (2 volumes)

Section 7.

SELECTED ELEMENTARY REACTIONS (1 volume)

Section 8.

HETEROGENEOUS REACTIONS (4 volumes)

Section 9.

KINETICS AND CHEMICAL TECHNOLOGY (1 volume)

Section 10.

MODERN METHODS, THEORY AND DATA

CHEMICAL KINETICS EDITED BY

R.G. COMPTON M . A . , D . Phil. (Oxon.) Oxford University The Physical and Theoretical Chemistry Laboratory Oxford, England

G.HANCOCK Oxford University The Physical and Theoretical Chemistry Laboratory Oxford, England

VOLUME 35

LOW-TEMPERATURE COMBUSTION AND AUTOIGNITION M.J. PILLING (Editor) University of Leeds School of Chemistry Leeds, England

AMSTERDAM-LAUSANNE-NEW

1997 ELSEVIER YORK-OXFORD-SHANNON-SINGAPORE-TOKYO

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

ISBN: 0-444-82485-5 (Vol. 35) © 1997 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. pp. 1-124, 661-760: copyright Shell Research Ltd. This book is printed on acid-free paper. Printed in The Netherlands.

COMPREHENSIVE CHEMICAL KINETICS

ADVISORY BOARD Professor Professor Professor Professor Professor Professor Professor Professor Professor Professor

C.H. BAMFORD S.W. BENSON LORD DAINTON G. GEE G.S. HAMMOND K.J. LAIDLER SIR HARRY MELVILLE S. OKAMURA Z.G. SZABO O. WICHTERLE

Volumes in the Series Section 1.

Volume 1 Volume 2 Volume 3

The Practice of Kinetics The Theory of Kinetics The Formation and Decay of Excited Species Section 2.

Volume 4 Volume 5

8 9 10 12 13

OXIDATION AND COMBUSTION REACTIONS (2 volumes)

Liquid-phase Oxidation Gas-phase Combustion Section 7.

Volume 18

POLYMERISATION REACTIONS (3 Volumes)

Degradation of Polymers Free-radical Polymerisation Non-radical Polymerisation Section 6.

Volume 16 Volume 17

ORGANIC REACTIONS (5 Volumes)

Proton Transfer Addition and Elimination Reactions of Aliphatic Compounds Ester Formation and Hydrolysis and Related Reactions Electrophilic Substitution at a Saturated Carbon Atom Reactions of Aromatic Compounds Section 5.

Volume 14 Volume 14A Volume 15

INORGANIC REACTIONS (2 volumes)

Reactions of Non-metallic Inorganic Compounds Reactions of Metallic Salts and Complexes, and Organometallic Compounds Section 4.

Volume Volume Volume Volume Volume

HOMOGENEOUS DECOMPOSITION AND ISOMERISATION REACTIONS (2 volumes)

Decomposition of Inorganic and Organometallic Compounds Decomposition and Isomerisation of Organic Compounds Section 3.

Volume 6 Volume 7

THE PRACTICE AND THEORY OF KINETICS (3 volumes)

SELECTED ELEMENTARY REACTIONS (1 volume)

Selected Elementary Reactions

VII Section 8. Volume Volume Volume Volume

19 20 21 22

Simple Processes at the Gas-Solid Interface Complex Catalytic Processes Reactions of Solids with Gases Reactions in the Solid State Section 9.

Volume 23

24 25 26 27 28 29 30 31 32 33 34 35

KINETICS AND CHEMICAL TECHNOLOGY (1 volume)

Kinetics and Chemical Technology Section 10.

Volume Volume Volume Volume Volume Volume Volume Volume Volume Volume Volume Volume

HETEROGENEOUS REACTIONS (4 volumes)

MODERN METHODS, THEORY, AND DATA

Modern Methods in Kinetics Diffusion-limited Reactions Electrode Kinetics: Principles and Methodology Electrode Kinetics: Reactions Reactions at the Liquid-Solid Interface New Techniques for the Study of Electrodes and their Reactions Electron Tunneling in Chemistry. Chemical Reactions over Large Distances Mechanism and Kinetics of Addition Polymerizations Kinetic Models of Catalytic Reactions Catastrophe Theory Modern Aspects of Diffusion-Controlled Reactions Low-temperature Combustion and Autoignition

Contributors to Volume 35

D.L. BAULCH

School of Chemistry, University of Leeds, Leeds LS2 9JT, United Kingdom

D. BRADLEY

Department of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, United Kingdom

J.F. GRIFFITHS


C. MOHAMED


C. MORLEY

Shell Research Ltd., Thornton Research Centre, P.O. Box 1, Chester CHI 3SH, United Kingdom

M.J. PILLING


S.H. ROBERTSON Molecular Simulations Inc., 240/250 The Quorum, Barnwell Road, Cambridge CB5 8RE, United Kingdom S.K. SCOTT


P.W. SEAKINS


A.S. TOMLIN

Department of Fuel and Energy, University of Leeds, Leeds LS2 9JT, United Kingdom

T. TURANYI

Department of Physical Chemistry, Eotvos University (ELTE), H 1518 Budapest 112, P.O. Box 32, Hungary

R.W. WALKER

School of Chemistry, University of Hull, Hull HU6 7RX, United Kingdom

Introduction

Combustion has played a central role in the development of our civilization which it maintains today as its predominant source of energy. It is not surprising that it has long been the object of scientific, and indeed prescientific study. Alchemists referred to the flame as the "transforming element" because it provided them with all the power required at the time. Combustion was the virtually unique tool for establishing the structure of matter through the seminal experiments of Lavoisier and Priestley. The classic treatise by Faraday, "The Chemical History of a Candle", demonstrated a fascination with unravelling the interactions between the many complex processes that comprise combustion. There is an intrinsic and continuing interest in being able to explain the rich variety of phenomena, such as flames, explosions, detonations and cool-flames, which can occur during combustion and oxidation, depending on the temperature, density and the chemical nature of the hydrocarbon. Since the 1930s it has been recognized that the chemical changes in combustion and oxidation are caused very largely by reactions of free radicals. These are not present in the reactants (fuel and oxidant), and their supply largely governs the rate of combustion processes. In autoignition, the main emphasis of this book, the radicals are generated from the reactants alone, in a process which develops with time and, at its simplest, independently at each point in space. It is appropriate to compare autoignition with other types of combustion.

COMBUSTION PHENOMENA

In flames, the fuel and oxidant are consumed in a region with high temperature and radical concentration gradients and the radicals are supplied to the reacting mixture by diffusion from the burnt products. The rate at which the flame burns is, to a large extent, controlled by the processes which occur in the higher temperature regions of the flame,

X

Introduction

which tend to dominate because the rate of chemical change is greater there. Broadly, the conditions in these regions are decided by thermodynamic inputs, principally the heat of combustion, rather than by chemical kinetics. Consequently, in hydrocarbon flames, chemical structure, except in as much as it affects the heat of combustion, tends to have only a small influence on the reaction rate, as measured by the burning velocity (see Section 7.4). While there are differences in the burning velocities of alkane hydrocarbons, they are nowhere as large as the differences in their autoignition behaviour. By contrast, in most other forms of combustion the reacting system has to generate its own radicals in order for the oxidation to occur. The rate of generation of radicals is controlled by chemical kinetics and, consequently, varies much more between fuels of differing structure. The oxidation can proceed in a controlled manner in a slow reaction or through a rapid (exponential) increase in rate which leads to an explosion or ignition. The increase in rate can occur through multiplication of the number of radicals through chain branching in elementary chemical reactions, which can sometimes occur under essentially isothermal conditions. Because the oxidation chemistry is invariably exothermic and the rate of reaction increases with temperature there is also the possibility of a runaway increase in reaction rate - a thermal explosion. Usually, an explosion involves a mixture of the two mechanisms and is referred to as thermo-kinetic. These concepts of chemical and thermal feedback are discussed in Chapter 5. Another form of feedback in combustion systems is through energy released by compression, which can lead to the phenomenon of detonation. Small portions of the reacting gas autoignite and generate shock waves which interact causing local heating and further autoignition. A self-sustaining combustion wave with internal cellular structure is generated which propagates at supersonic speeds and has very high (and potentially damaging) peak pressures. Although essentially an autoignition phenomenon, the chemistry must occur on a time-scale smaller than that associated with the speed of sound. Consequently, chemistry in true detonations occurs at very high temperature (>1500K) and is not discussed further here. However, it has recently become apparent that the high pressure autoignition that occurs in engines, and possibly other systems, can generate energy at a rate sufficient for pressure wave generation to be significant. This process is probably responsible for the damage caused by extreme knock in sparkignition engines and is discussed in Chapter 7.

Introduction

XI

Although the rates of most reactions increase with temperature, there are conditions in the oxidation of alkanes (and some other classes of organic compound) where the rate decreases with temperature. This negative temperature region, which is illustrated in Fig. 6.3, is responsible for the phenomenon of cool-flames. These have the appearance of curtailed ignitions: the rate increases sharply, with an accompanying temperature rise (100-200 K) and usually a pale blue glow, but then falls away rapidly without leading to complete oxidation. In static systems the cool-flame can be repeated several times in a periodic fashion with perfect reproducibility under fixed conditions. A cool-flame occurs essentially because the temperature rise associated with it takes the system from a region where the rate is increasing exponentially to one where, because of the negative temperature coefficient, the chemistry is slower and only a steady reaction is possible. Multiple cool-flames occur because heat losses cause the system to cool back into the accelerating region. The dynamics of such systems are discussed in Chapter 5 and the experimental aspects in Chapter 6. Two-stage ignition is when a cool-flame is followed by a true ignition where the temperature rises to above 1400 K and flame chemistry can take over.

MECHANISTIC INTERPRETATION

Chain reactions Prior to 1930, attention of fundamental combustion scientists was focused mainly on the morphology of the cool-flame and ignition regions. The acceptance of free radicals, followed by the masterly and elegant Semenov theory (outlined in Chapter 5), which established the principle of branched chain reactions, provided the foundation for modern interpretations of hydrocarbon oxidation. The significant processes are: (i) (ii) (hi) (iv) (v)

primary initiation (radicals from parent molecules), propagation of the chain (no change in the number of radicals), termination of the radical chain (removal of radicals), branching (multiplication of the number of radicals), secondary initiation (or degenerate branching), where new radicals are formed from a "stable" intermediate product.

XII

Introduction

These were first demonstrated in the H 2 + 0 2 system but are applicable to hydrocarbon oxidation. The processes which change the number of radicals are of special importance in influencing the behaviour. Chemical mechanisms The study of the products of an oxidation and how these depend on the conditions of temperature, concentration, etc., is a long-standing method of elucidating the chemistry. The development of gas chromatography, with its high analytical sensitivity, has allowed the chemistry to be studied through product analysis in well-defined chemical environments, and particularly at small extents of reaction before the behaviour is influenced by reactive intermediate products. As discussed in Chapter 1, it is apparent that the nature of the chemistry changes with the conditions, particularly with temperature and pressure. The primary focus of this book is on the "low-temperature" region, since this is responsible for much of the interesting phenomenology and is of significance in practical applications like knock in spark-ignition engines. Low-temperature chemistry The detailed chemistry is described in Chapter 1 but a brief introduction to the chemistry of alkanes will be given here. Other classes of organic substance, such as aldehydes and ethers have a characteristic chemistry in this region, but alkanes are of special significance, because different structures show a wide range of activity and because they are constituents of practical fuels. For propane and larger alkanes the low-temperature chemistry follows the general scheme shown in Fig. 1 at temperatures between about 500 and 1000 K, depending on the pressure. The parent alkane RH reacts with a radical, usually OH, and loses a hydrogen atom to form an alkyl radical R. In general, R undergoes three types of reaction: decomposition to an alkene and a smaller alkyl radical; reaction with 0 2 to produce the conjugate alkene (with the same number of carbon atoms as the parent) together with an H 0 2 radical; or addition of oxygen to form a peroxy radical. The main fate of peroxy radicals above 500 K (other than to fall apart again) is to undergo an internal hydrogen abstraction to form a hydroperoxide group (—OOH) and a new alkyl radical centre further along the carbon chain. The product of this peroxy

XIII

Introduction RH

.0 — ()• n

/

i

•1 R»

II It It

H 0 2 * +alkene > *

Termination Smaller R* +alkene

R0 2 «

,0 — 0, \ i

H

Peroxy radical isomerisation

>QOOH QOO

Propagation Products + HO •

•0 2 QOOH QC

1

Branching

RO • + HO • ROOH Fig. 1. Simplified outline mechanism for "low-temperature" alkane oxidation. Peroxy radical isomerization, an internal hydrogen abstraction, plays a crucial role at two points in the mechanism and is illustrated on the left.

isomerization is a type of species commonly denoted as QOOH. It can undergo a cyclization, throwing out an OH molecule, to produce a cyclic ether which may have a ring size from 3 to 6 atoms. They are among the major products of alkane oxidation at these temperatures. Since QOOH is an alkyl radical, it can add another 0 2 . Subsequent isomerization of the peroxy radical produces, through an unstable intermediate, a molecular (non-radical) hydroperoxide, shown here as ROOH, and an OH radical. ROOH may have a significant lifetime, particularly at the lower end of the temperature range, and is a degenerate branching agent. Its thermal decomposition to two radicals is the branching step in the mechanism: it provides the multiplication in the number of radicals needed for the runaway phenomenon of autoignition. The termination steps which moderate or prevent this runaway are those early in the mechanism producing inert radicals, H 0 2 and small alkyl radicals, such as CH 3 . The consequent competition in reactions of R and R 0 2 has the important consequence that branching is greatly favoured, and autoignition enhanced, when peroxy isomerizations are facile, since

XIV

Introduction

the route to branching involves two such reactions, while that to termination involves none. As explained in Chapter 1, isomerization is easiest in long straight alkane chains. This is the reason that large n-alkanes autoignite under milder conditions than small or branched alkanes. Another significant and related feature of the mechanism is connected with the reversibility of the formation of R 0 2 : (1)

R + 02 ^ R02

As described above, the subsequent chemistry of R 0 2 radicals leads to branching while that of the termination occurs through the alkyl R. At successively higher temperatures the equilibrium shifts to the left and the termination reactions through R are favoured over the branching reactions through R 0 2 , so that the reaction gets slower. This is the cause of the negative temperature coefficient and cool-flames. Regimes of oxidation chemistry In order to get an overall picture of the nature of the different kinds of chemistry it is helpful to examine the rates of the branching reactions since the final termination reactions (radical recombinations) are less dependent on the temperature. At high temperatures the most important branching reaction is H + 02ÔH + 0 but at lower temperatures and higher pressures this is in competition with the recombination reaction H + 02 + M ^ H 0 2 + M This is not a propagation reaction and usually is effectively a termination because H 0 2 is a much less active radical than H. The upper line in Fig. 2 represents the conditions of temperature and pressure where the rates of these reactions are equal. Broadly, above this line the reaction of H with 0 2 leads to branching and below it to termination. The line represents

XV

Introduction 1400 Small radical chemistry H + 02 ii|_i;„|_"

1200 H

'eroxy chemistry

600 0.1

1

10

100

Pressure / bar

Fig. 2. Regimes of hydrocarbon oxidation chemistry as delineated by the main kinetic chainbranching processes. The upper line connects points where the overall H + 0 2 reaction is neutral: above the line it is net branching; below it is net terminating. The lower lines (applicable to alkane oxidation) are where the peroxy chemistry is neutral: above these lines there is net termination and below net branching, (This is the region of the negative temperature coefficient.) The "low"-, "intermediate"- and "high"-temperature regions are broadly characterized by the types of chemistry indicated.

the lower boundary of what we will call "high-temperature" chemistry. Most flames are controlled by this kind of chemistry. The chemistry which leads to most of the interesting cool-flame, twostage ignition and related phenomena is centred around the reversible formation of peroxy radicals, R 0 2 , by reaction (1). At higher temperatures the equilibrium is over to the left, so that peroxy radicals and their rate enhancing reactions are not important. The equilibrium is of course also

XVI

Introduction

dependent on 0 2 concentration (or pressure in a system with constant composition). The lower lines in Fig. 2 plot the conditions when the rates of branching (via R0 2 ) and of termination (via R) are equal for two alkanes of widely differing autoignition propensities. Below the line for a particular fuel "low-temperature" alkyl peroxy radical chemistry is dominant. Above the line alkyl peroxy radicals are of less significance. Between the lines neither of the previously mentioned branching mechanisms is very effective and consequently rates in the lower part of the region are low. Further into this "intermediate chemistry" regime the reactions of H 0 2 can lead to branching: XH + H 0 2 ^ X + H 2 0 2 H 2 0 2 + M ^ OH + OH + M The XH can be the parent hydrocarbon but is more usually an intermediate oxidation product with weaker C—H bonds, such as an aldehyde or alkene. Even so, the abstraction reaction has a large activation energy, as does the hydrogen peroxide decomposition (which is also pressure dependent), so that the branching mechanism tends to be of greater importance towards the higher temperature and pressure part of the region. It is worth emphasizing the pressure dependence of the boundaries between the different regimes. For instance, at 800 K with air as oxidant, a glass-bulb oxidation system at less than 0.1 bar may be controlled by H + 0 2 branching (high-temperature chemistry), while at the same temperature in an engine, at a pressure of tens of bar, peroxy radical chemistry (low-temperature chemistry) could dominate. The lower boundary is sensitive to the oxygen concentration so that, for instance, turbulent flow reactors which use highly diluted oxidant will need to operate at rather low temperatures to be in the low-temperature regime, even if their total pressure is several bar. Organization of the book The aim of this book is to provide an understanding of both fundamental and applied aspects of low-temperature combustion chemistry and autoignition. The topic is rooted in classical observational science and has grown,

Introduction

XVII

through an increasing understanding of the linkage of the phenomenology to coupled chemical reactions, to quite profound advances in the chemical kinetics of both complex and elementary reactions. The driving force has been both the intrinsic interest of an old and intriguing phenomenon and the centrality of its applications to our economic prosperity. We have attempted to produce a coherent view of the subject while, at the same time, making each chapter self-contained. This has led to some duplication and also to some inconsistencies in style and notation - while we have tried to minimize the latter, the development of the subject has not always been coherent and historic variations in notation, in different but overlapping areas, are difficult to avoid. This Introduction has laid some of the groundwork for the subject. Chapter 1 takes this further and examines the chemical mechanism of lowtemperature combustion and of the component elementary reactions. It is largely based on the use, by Walker and his co-workers at the University of Hull, of well-defined conditions to study the rates and products of specific reactions using gas chromatography. While many of the ideas on the basic mechanism were developed earlier, the work of the Hull group has been central to the determination of much of the quantitative detail in our present understanding and the chapter provides an ideal vehicle for describing the underlying chemistry. It includes a detailed discussion of hydrocarbon oxidation chemistry, providing data on rates of many of the key reactions in the temperature range of interest, and an illustration of how reactivity patterns emerge as reaction types are studied in detail for a range of reactants. These patterns can then be used for related reactants that have not been studied. While end-product analysis experiments of this type can be applied to a wide range of reactions, they only provide comparative measurements of reaction rates and are also only applicable over a narrow range of temperatures. Direct measurements of radical reactions, using techniques such as flash photolysis, which was first developed to study combustion reactions, are needed to provide absolute rate constants; such techniques can also be used over a wider range of temperatures, although they too are more limited than is ideal. Chapter 2 discusses the use of direct methods to study a number of reactions, generally for small radicals, that lie at the heart of autoignition chemistry. As discussed above, often it is not feasible to study the reactions under conditions of practical combustion interest and extrapolations of rate data must be made. This can only be reliably

XVIII

Introduction

undertaken on the basis of sound theoretical models and with a good understanding of the detailed mechanism of the elementary reaction. Chapter 2 includes a limited discussion of the appropriate theory and gives an account, in particular, of pressure effects in radical reactions. The chapter also touches briefly on reaction dynamics, which is a field capable of providing considerable insight into mechanisms of elementary reactions and is a good testing ground for theoretical models. Even for those reactions for which experimental data are available, agreement between parameter values from different laboratories using different (or even similar) techniques is not guaranteed. One of the major challenges in the chemical kinetics of elementary reactions lies in ensuring that the experimental system is not invalidated by the presence of secondary or competing reactions. This problem presents the combustion modeller with a dilemma - which results should be believed? The problem has been solved by the process of evaluation where practising kineticists examine the available data and make recommendations on parameter values. This process is described in Chapter 3, which discusses kinetic databases for combustion. The chapter provides a critical analysis of kinetic data and of the evaluation process and is illustrated with a range of specific examples. Armed with an understanding of the elementary reactions involved in low-temperature oxidation chemistry and with their rate constants, it is now possible to construct quantitative chemical mechanisms to describe processes such as autoignition. Chapter 4 discusses approaches adopted to construct such mechanisms and the relationship between the chemical mechanism and the overall model describing the combustion process of interest. The chapter briefly addresses the issue of numerical integration of the ordinary differential (rate) equations describing the chemical kinetics and delves somewhat more deeply into the topic of sensitivity analysis whereby the species and reactions that most sensitively determine relevant features of the process can be identified. The bulk of the chapter is devoted to methods for reducing the size of and for lumping a chemical mechanism. The former involves identifying those reactions and species which are redundant to the quantitative description of certain specified target features; lumping involves the generation of single variables capable of representing groups of species. Both processes are necessary if detailed mechanisms are to be incorporated into fluid dynamic models of combustion: the important step lies in reducing the number of ordinary differential

Introduction

XIX

equations needed accurately to describe the chemistry and both reduction and lumping provide ways of achieving this goal, although the degree of compression available is, as yet, often insufficient for real systems. The dynamics of complex reactions, and in particular of combustion reactions, are examined in Chapter 5. The repeated cool-flames that can be generated have already been referred to. Much more complex behaviour, including oscillations and even chaos, can be observed if the systems are studied in a continuous stirred tank reactor. The behaviour derives from the non-linearity of the kinetics and to the existence of thermokinetic feedback, i.e., to the type of chemistry leading to ignition and to the negative temperature coefficient in hydrocarbon oxidation. While this kind of "exotic" behaviour has been studied in alkanes, by far the most detailed work has involved the oxidation of hydrogen and CO. The chapter forms an interesting contrast with the earlier chapters where the emphasis has been on detailed mechanisms based on a full, or reasonably full, understanding of the component elementary reactions. Here the emphasis is on the global behaviour and relies on a minimalist approach to the underlying chemistry, using a small number of variables; an underlying aim is to discern the necessary interactions between reactions that can lead to the complex behaviour observed experimentally. The subject is based on dynamical systems theory, but also has firm roots in combustion through the theory of thermal explosions. These topics are covered in detail in the chapter, which provides through its primarily analytical approach, a counterpoint to the numerical emphasis of Chapter 4. The application of the basic ideas to real combustion systems is then taken up in Chapters 6 and 7. In Chapter 6, experimental and modelling studies are described which link the mechanistic observations of Chapter 1 to combustion characteristics of fuels studied under laboratory conditions. The experimental emphasis is initially on global combustion phenomena - ignition and oscillatory cool-flames - for a range of hydrocarbons. Section 6.5 then addresses the distribution of products in hydrocarbon oxidation; this discussion differs from that in Chapter 1 where the conditions were optimized to allow the investigation of specific reactions. The focus is now on studies of oxidation products over a range of isothermal and non-isothermal conditions, the interpretation of the results in terms of elementary reactions and the use of the experimental data as a detailed test of combustion models. The chapter provides an overview of the success of detailed models in describing combustion phenomena and combustion

XX

Introduction

chemistry and has an essentially chemical emphasis - while the experiments refer to complex systems, they are conducted under generally well-defined conditions with the aim of providing well-designed tests of chemical mechanisms. The emphasis in Chapter 7 moves in the direction of engineering applications and the effects of transport. The chapter is concerned specifically with autoignition in spark-ignition engines and with the generation and avoidance of knock. After a discussion of fuels which includes a brief resume of fuel processing, octane numbers and their links to chemical structure, blending and additives, the chapter examines chemical models of autoignition, illustrating the range of lumped mechanisms that have been employed. The importance of concise representations of the chemistry are emphasized through a brief analysis of combustion in engines and the importance of fluid dynamics. Overall, the chapter serves to demonstrate, and illustrate, the importance of chemistry in an important application and to emphasise the need to adapt the description of the chemistry to the physical complexity of the problem. As chemists we always strive for a full understanding of the chemical mechanism. When this knowledge is applied to problems requiring detailed fluid dynamics, we may have to jettison some of the detail and concentrate on a much more concise description that still contains the salient features. The detailed knowledge is still necessary, but an extra dimension, of understanding how the global behaviour is most effectively linked to this detail, is added. The problem becomes even more intriguing. CHRISTOPHER MORLEY and MICHAEL J. PILLING

Contents

Introduction

IX

1 Basic 1.1 1.2 1.3 1.4

1 1 1 4 9 9 9 12

1.5

1.6

1.7

1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15

1.16

chemistry of combustion (R.W. Walker and C. Morley) Historical perspective Characteristics of low-temperature combustion Computer modelling and associated problems Overview of alkane oxidation 1.4.1 Introduction 1.4.2 Autoignition and engine knock 1.4.3 Regimes of oxidation chemistry Review of experimental methods for establishing mechanisms and determining rate constants 1.5.1 Basic difficulties involved 1.5.2 Review of Hull approaches Primary initiation reactions 1.6.1 Experimental determination of kx 1.6.2 Database for the primary initiation process (1) 1.6.3 Experimental determination of k1A 1.6.4 Relative rates of reactions (1) and (1A) Propagation reaction X + R H - * XH + R (2) 1.7.1 Introduction 1.7.2 Kinetic data for X = OH, H, H 0 2 , CH 3 and CH 3 0 2 Homolysis of alkyl radicals The reaction of alkyl radicals with 0 2 1.9.1 Reaction between C 2 H5 and 0 2 Reactions of R 0 2 radicals Reactions of QOOH and QOOH0 2 radicals Oxidation chemistry of CH 3 radicals Reactions of alkoxy radicals Branching reactions Oxidation of cyclic alkanes 1.15.1 Oxidation of cyclohexane and cyclopentane 1.15.2 Formation of benzene 1.15.3 Mechanism of c-C5H9 decomposition 1.15.4 Alkyl-substituted cycloalkanes Oxidation of alkenes 1.16.1 Introduction .

17 17 19 24 25 33 34 35 36 36 38 44 48 53 56 61 64 67 69 73 73 78 80 81 82 82

XXII

Contents

1.16.2 Product formation 1.16.3 Oxidation of isobutene 1.16.4 Oxidation of butene-1 and butene-2 1.16.5 Alkenyl oxidation chemistry 1.16.6 Oxidation of the smaller alkenyl radicals at high temperatures 1.17 Atom and radical addition to alkenes 1.17.1 H atoms 1.17.2 O atoms 1.17.3 H 0 2 radicals 1.17.4 OH radicals 1.17.5 CH 3 radicals 1.18 Oxidation of oxygenated compounds 1.18.1 O-heterocycles 1.18.2 Aldehydes 1.19 Oxidation of aromatic compounds 1.20 Conclusions References

86 89 90 92 95 96 96 97 98 103 107 108 108 Ill 114 119 120

2 Elementary reactions (S.H. Robertson, P.W. Seakins and M.J. Pilling) 2.1 Introduction 2.2 Reaction initiation 2.3 Abstraction reactions 2.3.1 Introduction 2.3.2 Time resolved studies of OH abstraction reactions 2.3.3 Transition state theory (TST) 2.3.4 Isotope effects 2.3.5 Additivity of rate coefficients 2.3.6 Other radical abstractions 2.4 Radical decomposition reactions 2.4.1 Introduction 2.4.2 Real time experimental techniques for radical decompositions . 2.4.3 Modelling radical decomposition 2.4.4 Microcanonical rate coefficients 2.4.5 Unimolecular dynamics 2.5 Radical recombination and association reactions 2.5.1 Introduction 2.5.2 CH 3 + CH 3 -* C 2 H 6 2.5.3 CH 3 + H-> CH 4 2.5.4 Theory and discussion 2.5.5 Addition reactions, equilibria and alkyl radical heats of formation 2.5.6 H 0 2 self reaction 2.6 R + 0 2 ^ R 0 2 2.6.1 Introduction 2.6.2 Experimental

125 125 130 134 134 135 139 143 146 151 153 153 159 162 171 174 176 176 179 188 193 195 197 199 199 201

Contents

XXIII

2.6.3 Discussion 2.6.4 High temperature pathways 2.7 Peroxy radical isomerization 2.7.1 Introduction - Formation of degenerate branching agents . . . . 2.7.2 Direct studies of peroxy radical isomerizations 2.7.3 Discussion 2.8 Theoretical and dynamical studies of the hydrogen/oxygen system . . . . 2.8.1 Introduction 2.8.2 State specific rate coefficients 2.8.3 Theoretical treatment of bimolecular reactions 2.8.4 Reagent excitation References

203 205 214 214 217 219 222 222 224 225 229 230

3 Kinetics databases (D.L. Baulch) 3.1 Data for combustion modelling 3.1.1 Modelling combustion chemistry 3.1.2 Data needs for modelling 3.1.3 Other uses of kinetic data in combustion 3.2 Primary sources of kinetic data: the need for evaluation 3.2.1 Experimentally measured rate constants 3.3.2 The quality of kinetic data 3.2.3 Critical evaluation 3.3 Evaluation of kinetic data 3.3.1 Evaluation groups 3.3.2 Collecting the information 3.3.3 Evaluation procedures 3.4 Interpolation, extrapolation and estimation procedures 3.5 Data sources for modelling 3.5.1 Collections of critically evaluated data References

235 235 235 236 237 238 238 243 254 257 257 258 259 272 283 283 286

4. Mathematical tools for the construction, investigation and reduction of combustion mechanisms (A.S. Tomlin, T. Turanyi and M.J. Pilling) 4.1 Introduction 4.2 Notation 4.3 The construction of combustion mechanisms 4.3.1 Matrix rearrangement 4.3.2 The logical programming approach 4.3.3 Applications to hydrocarbon oxidation 4.3.4 A comprehensive program system 4.3.5 The GRI mechanism 4.3.6 Discussion 4.4 Numerical investigation of complex models 4.4.1 Basic equations 4.4.2 Numerical integration

293 293 300 300 304 305 306 308 310 311 312 312 313

XXIV

Contents

4.5 Sensitivity and uncertainty analysis 4.5.1 Introduction 4.5.2 Local sensitivities 4.5.3 Principal components 4.5.4 Applications of local concentration sensitivities 4.5.5 Uncertainty analysis 4.5.6 Discussion 4.6 Mechanism reduction without time-scale analysis 4.6.1 Finding redundant species 4.6.2 Finding redundant reactions 4.6.3 Sensitivity of temperature rates 4.6.4 Application to hydrogen oxidation in a flow system 4.6.5 Discussion 4.7 Formal lumping procedures 4.7.1 Linear lumping 4.7.2 Non-linear lumping 4.7.3 Chemical lumping 4.7.4 Lumping of continuous mixtures 4.7.5 Discussion 4.8 Reduction based on the investigation of time-scales 4.8.1 Low-dimensional systems 4.8.2 Jacobian analysis 4.8.3 Computational singular perturbation (CSP) theory 4.8.4 Slow/inertial manifolds 4.8.5 The quasi-steady-state approximation (QSSA) 4.8.6 Discussion 4.9 Approximate lumping in systems with time-scale separation 4.9.1 Linear lumping in systems with time-scale separation 4.9.2 Approximate non-linear lumping in systems with time-scale separation 4.9.3 Application of approximate non-linear lumping to the hydrogen/oxygen example 4.9.4 Discussion 4.10 Fitted kinetic models 4.10.1 Fitting a system of odes to experimental data 4.10.2 Fitting a system of odes to detailed kinetic simulations 4.10.3 Fitting algebraic equations to detailed kinetic simulations 4.10.4 A fitted algebraic model for wet CO ignition 4.10.5 Discussion 4.11 Conclusions and future directions 5 Global behaviour in the oxidation of hydrogen, carbon monoxide and simple hydrocarbons (S.K. Scott) 5.1 Introduction 5.2 Notation

314 314 316 318 320 323 325 326 327 329 330 331 342 342 344 348 350 354 357 358 359 359 361 364 373 391 392 392 396 397 401 403 404 409 410 414 416 419

439 439 440

Contents Non-linearity and feedback in chemical kinetics: stoichiometry and elementary steps 5.4 Chemical feedback: branched-chain ignition 5.5 Thermal feedback: ignition, extinction and singularity theory 5.6 Thermokinetic feedback: oscillations and local stability analysis 5.7 The H 2 + 0 2 reaction: p-Ta ignition limits in closed vessels 5.7.1 First limit 5.7.2 Second limit 5.7.3 Reactions involving H 0 2 and the third limit 5.8 Flow reactor studies of the H 2 + 0 2 reaction 5.9 Complexity in the oscillatory ignition region 5.10 Mechanistic modelling of complexity in the H 2 + 0 2 reaction 5.11 The CO + 0 2 reaction 5.11.1 Closed vessel studies 5.11.2 Open systems 5.11.3 Mechanistic interpretation and modelling 5.12 Hydrocarbon oxidation 5.12.1 Oxidation of acetaldehyde 5.12.2 Non-isothermal oxidation of alkanes 5.13 Conclusions and future directions References

XXV

5.3

6 Experimental and numerical studies of oxidation chemistry and spontaneous ignition phenomena (J.F. Griffiths and C. Mohamed) 6.1 Introduction 6.2 Measurements of reaction rate and its dependence on experimental conditions 6.2.1 Introduction 6.2.2 The overall dependence of reaction rate on temperature 6.2.3 Ignition delay times 6.3 Experimental methods 6.3.1 Introduction 6.3.2 Closed, constant volume reaction vessels 6.3.3 Flow systems 6.3.4 Shock tubes 6.3.5 Rapid compression machines 6.3.6 Motored engines 6.3.7 Chemical analysis 6.4 Global combustion phenomena associated with hydrocarbon oxidation 6.4.1 Introduction 6.4.2 Spontaneous ignition and oscillatory cool-flames in closed vessels 6.4.3 Spontaneous ignition in flowing gases 6.4.4 Ignition diagrams for selected hydrocarbons 6.4.5 Ignition delays of alkanes and other hydrocarbons

442 451 460 470 487 489 490 492 494 504 509 514 514 516 526 529 530 536 538 541

545 545 546 546 548 559 562 562 562 563 565 568 572 573 575 575 575 578 579 589

XXVI

Contents

6.4.6 Spontaneous ignition in supercritical fluids Product distributions during hydrocarbon oxidation and its kinetic interpretation 6.5.1 Introduction 6.5.2 Oxidation of propane and propene 6.5.3 Oxidation of hydrocarbons containing four carbon atoms 6.5.4 Oxidation of isomers of C5 and C6 alkanes 6.5.5 Oxidation of n-heptane and 2,2,4-trimethylpentane (/-octane) . 6.5.6 Oxidation of Cn alkanes and Cn alkenes of (n > 8) 6.5 7 Oxidation of aromatics 6.6 Detailed numerical modelling of alkane oxidation and spontaneous ignition 6.6.1 Introduction 6.6.2 Validation of comprehensive kinetic models for lowtemperature oxidation 6.6.3 Application of comprehensive kinetic models 6.7 Conclusions References Appendix

594

6.5

7 Autoignition in spark-ignition engines (D. Bradley and C. Morely) 7.1 Introduction 7.2 Fuels for spark-ignition engines 7.2.1 Composition and manufacture 7.2.2 Knock propensity and octane number 7.2.3 Research and motor octane numbers 7.2.4 Chemical cause of sensitivity 7.2.5 Blending 7.2.6 Anti-knocks 7.3 Chemical modelling of autoignition 7.3.1 Introduction 7.3.2 Comprehensive models 7.3.3 Reduced mechanisms 7.3.4 Simplified models 7.4 Combustion in engines 7.4.1 Laminar and turbulent burning velocities 7.4.2 Engine diagnostics and computational fluid dynamics 7.4.3 Parametric influences on burn rate 7.5 Autoignition in engines: modelling and experiments 7.5.1 Carry-over of combustion products between cycles 7.5.2 Cylinder wall deposits 7.5.3 End gas reactions: basic modelling considerations 7.5.4 Autoignition models applied to the end gas 7.5.5 Complete engine models

594 594 595 607 612 620 626 628 629 629 633 638 640 643 653 661 661 664 664 668 670 673 675 679 684 684 686 688 688 696 696 702 706 709 710 713 716 719 723

Contents 7.6

XXVII

Development of pressure pulses and knock 7.6.1 Inhomogeneity, autoignition centres, and hot spots 7.6.2 Propagation of reaction from a hot spot and modes of autoignition 7.6.3 Propagation of autoignition: basic modelling considerations . . . 7.6.4 Influence of the fuel 7.6.5 Multiple autoignition centres 7.6.6 Knock damage 7.7 Conclusions References

725 725 727 730 740 744 746 747 749

Author Index

761

Subject Index

785

This Page Intentionally Left Blank

Chapter 1

Basic Chemistry of Combustion R.W. WALKER and C. MORLEY

1.1 HISTORICAL PERSPECTIVE

As described in the Introductory Chapter, attention was focused [1] prior to 1961 mainly on the morphology of the cool-flame and ignition regions, rates were followed by pressure change, and essentially chemical techniques were used for product analysis. The acceptance of free radicals, followed by the masterly and elegant Semenov theory [2], which established the principles of branched chain reactions, provided the foundation for modern interpretations of hydrocarbon oxidation. This chapter builds on these early ideas, and pioneering experiments such as those carried out by Knox and Wells [3] and Zeelenberg and Bickel [4], to provide a detailed account of the reactions, thermochemistry and detailed mechanisms involved in the gas-phase chemistry of hydrocarbon oxidation.

1.2 CHARACTERISTICS OF LOW-TEMPERATURE COMBUSTION

The scope of the book is limited to low-temperature combustion (ca 500-1000 K), and excludes high-temperature phenomena such as flames and regions where thermal decomposition, soot formation and production of nitrogen oxides are important. It is, however, the region where combustion chemistry is rich and varied in the complex gas-phase oxidation of organic species, and where the chemistry dictates many of the observed phenomena. Essentially, the oxidation proceeds either in a controlled (basically isothermal) manner in a slow reaction or through an uncontrolled exponential increase in rate which leads to an explosion or ignition. Under some conditions the ignition is apparently curtailed leading to a cool-flame. Here the rate increases sharply for a few seconds, with accompanying

2

Basic chemistry of combustion

Ch. 1

650

1 600

550

500 0

50

100

150

pressure / TonFig. 1.1. Oxidation diagram for stoichiometric pentane + 0 2 mixtures. Figures 1 to 5 refer to the number of cool flames that occur in each region.

pressure and temperature (150-200 K) rises and usually with a pale blue glow, and then falls away rapidly without leading to complete oxidation. The cool-flame may be repeated several times in a periodic fashion and observed with total reproducibility under fixed conditions (Chapter 5). Figure 1.1 shows a phenomenological diagram for pentane oxidation [1] and indicates the areas of pressure and temperature where slow reaction, ignition and cool-flames are observed. Typically, all three types of behaviour are preceded by an induction period during which little reaction occurs. This is more simply observed in pressure change (AP)-time plots during slow reactions, as shown in Fig. 1.2 for the oxidation of butanal [5]. Definition of the length of an induction period is difficult. If designated as the time to the appearance of product or evidence of reaction, then the time depends on the sensitivity of the measurement. For slow reactions,

Characteristics of low-temperature combustion

3

time / s

10

20

30

1

T

r

U

3

1

100

200

300

time / s Fig. 1.2. AP - time plots for two slow oxidations in aged boric-acid-coated Pyrex vessels. (A) C3H6 = 12, 0 2 = 30, N 2 = 18 Torr at 753 K. (B) butanal = 4, 0 2 = 30, N2 = 26 Torr at 593 K.

the time to maximum rate may be used, but it is often difficult to measure accurately. Straight and branched chain reactions almost invariably have complex rate expressions, as shown by -d[H2]/dt = &[H2]0-9[O2]°-6[N2]0-4 for the H 2 + 0 2 reaction in aged boric-acid-coated vessels at 500 Torr and 773 K [6]. Change of pressure can have striking effects even when achieved by addition of an inert gas such as N 2 . Hydrocarbon combustion reactions proceed through the formation of many intermediates, both radical and molecular, prior to formation of the final products C 0 2 and H 2 0 . Another striking feature is the sensitivity of chain reactions to traces of impurities and to changes in surface properties. This is particularly pronounced in the case of some explosion boundaries or limits where parts per million quantities of an impurity may completely subdue the explosion. In an elementary way, explosion limits may be classified into two types: (i) An isothermal limit, where it is the kinetic features of the reaction

4


Ch. 1

that lead to explosion (although since oxidations are invariably exothermic, the explosion is accompanied by large temperature rises). (ii) A thermal limit, where the explosion is brought about by heat release, the major pre-requisites being strong exothermicity and a high overall activation energy. Thermal explosion limits are thus sensitive to factors such as the heat capacity and thermal conductivity of the gases.

1.3 COMPUTER MODELLING AND ASSOCIATED PROBLEMS

For the last 20 to 25 years, computer modelling has been used increasingly to interpret combustion phenomena. When the chemistry is the main interest, for example, in predicting ignition delay times or predicting product profiles, large comprehensive mechanisms can be used which in theory cover every species possible in the total oxidation. In most treatments differential equations are written for the reactants, intermediate species and for the final products. These are solved by standard mathematical treatments and profiles of concentration - time produced for all species. This approach will be discussed in detail in Chapter 4. An obvious problem is that a rate constant value is needed for each reaction in the mechanism in order that the reaction may be included in the computer calculation. Unfortunately there is a serious lack of kinetic data for elementary reactions, many of which have not been studied experimentally. This is particularly true of reactions associated with large alkyl radicals, which are of prime importance in the combustion of "practical" fuels. Three other pertinent difficulties arise. (i) Many dissociation and association reactions are pressure dependent, e.g., H202 + M ^ 2 0 H + M CH 3 + CH 3 + M ^ C 2 H 6 + M Frequently, kinetic data determined at low pressures require lengthy extrapolation to engine conditions (ca 20 atms or above). Fundamental theories are available but very difficult for modellers to use. Fortunately, "userfriendly" methods have been developed by Gilbert et al. [7] and Baulch etal. [8].

Computer modelling and associated problems 1

T

-11

-12 upper and lower uncertainty limits

6

§ -13 o

-u -15 0

1

2

3

4

IO3K/T Fig. 1.3. Plot of logio k vs 1/7 for the reaction OH + CH 4 ^> CH 3 + H 2 0 .

(ii) Although rate constants are classically related to temperature through the Arrhenius equation In k = In A — E/RT, where E is the activation energy, data obtained for most elementary reactions over wide temperature ranges indicate that plots of In k vs IIT are not linear. Figure 1.3 shows that the plot for the reaction between OH radicals and CH 4 is markedly curved, and fits [8,9] the equation k = 2.6 x l O ' ^ T 1 8 3 exp(-1396K/T) cm3 molecule" 1 s" 1 between 250 and 2500 K. Extensive data exist for only a handful of other OH + alkane reactions. The reaction H 0 2 + H 0 2 -> H 2 0 2 + 0 2

6


-11-0

i

i

'to

Ch. 1 r

Pre-1980 line

'o

B

o

~ -12-0

0

1

2

3

4

3

10 K7T Fig. 1.4. Plot of logio k vs 1/T for the reaction H 0 2 + H 0 2 -* H 2 0 2 + 0 2 .

was originally considered an abstraction reaction with AH = -152 kJ m o P 1 and E = 0, as early data gave k = 3.3 x 10" 12 cm3 molecule -1 s" 1 at both 298 and HOOK. Figure 1.4 shows recent data and a complex variation of k with temperature [10,11]. A change of mechanism is undoubtedly the cause with an association intermediate suggested at low temperatures replaced by 2H0 2 - ^ — H 2 0 4 *

^

H202 + 0 2

Mil H204 abstraction at high temperatures. Clearly, extrapolation of data to considerably different conditions is hazardous, but is increasingly aided by theoretical treatments [12]. (hi) For a number of multi-channel elementary reactions, good overall kinetic data are available but there is an absence of detailed information

Computer modelling and associated problems

7

on products. The reaction between OH and CH 3 radicals is an important example with at least four possible channels to products: OH + C H 3 - > H + CH 2 OH -> H + CH 3 0 -> H 2 0 + 1CH2 - t CH3OH Although k = 6.0 x 10 - 1 1 cm3 mol - 1 s _ 1 is known accurately for the overall reaction between 300 and 2000 K, the relative importance of the channels is not clear, which severely restricts rigorous modelling. Fortunately, sensitivity techniques (discussed in Chapter 4) of everincreasing sophistication have been developed to isolate key reactions in a mechanism under particular conditions. For example, of the 500 reactions listed for butane oxidation [13], only about 10 influence the computed ignition time delay by more than about 1% at HOOK and 1-20 atms. None of these include the C 4 H 9 0 2 radical, which is, however, extremely influential at 600 K in determining reaction rate and the length of the induction period. Frequently, sensitivity analyzes are carried out over narrow ranges of pressure and temperature. It cannot be stressed sufficiently that conclusions reached concerning the nature of the key reactions in the mechanism cannot be used thoughtlessly outside the conditions used in the analysis. (iv) The vital factor in discussing combustion chemistry is the thermochemistry of the elementary steps. Table 1.1 lists the enthalpies of formation of important radicals and corresponding bond dissociation energies. The large differences in C—H bond dissociation energy can lead to large activation energy differences even in a family of reactions, so that generalization may be misleading. For the abstraction reactions of H 0 2 with CH 4 , isobutane, and HCHO, the activation energies are about 105, 65 and 50 kJ mol - 1 respectively, with similar A factors (see Table 1.12). At 700 K, the relative rates are about 1:950:12,500, with the consequence that the oxidation of HCHO is very rapid at 700 K, whereas the oxidation of CH 4 is negligibly slow. For an elementary reaction the activation energies of forward and reverse steps are linked to the enthalpy of reaction, so that Ef- Eb = A//.

8


Ch. 1

TABLE 1.1 Enthalpies of formation and bond dissociation energies at 298 K [113] R

A f //(i?) 298 /kJmor 1

D(R-H)298fcJ mo\

QH5 C2H3 C2H5

HCO CH 2 CHCH 2 CH 3 CHC(CH 3 )CH 2 C2H5CO (CH3)2CC(CH3)CH2 CH 2 CHCHCHCH 2

324.4 278.7 116.0 84.0 41.9 206.5 44.7 172.9 103.0 -46.2 61.8 215.1

459.5 444.4 418.5 405.8 395.5 374.5 371.3 370.5 363.3 362.3 348.5 327.5

H02 OH H

12.5 39.3 218

366.9 499.1 436.0

CH3CHCH3

(CH3)3C C6H5CH2

It is clear that the activation energy of an endothermic step must equal or exceed AH. Furthermore, at any particular temperature, the forward and reverse rate constants are linked through Kc = kflkb, so that since Kc can be calculated thermochemically from the free energy change AGe = -RTlnKc, then either kf or kb can be calculated if the other value is known. Unfortunately, at the moment the available thermochemical data, namely, entropies and enthalpies of formation of radicals, are far from reliable. The reaction H + CH4—> CH 3 + H 2 gives a good indication of the problem. Apparently, both rate constants are known accurately [9], and the thermochemical data for the simple species involved are considered very reliable. Nevertheless, the value of Kc calculated from the individual rate constants differs by a factor of 3 to 4 from that calculated thermochemically. For many other reactions, the thermochemically calculated value of Kc may be in error by two orders of magnitude particularly at low combustion temperatures where each 5 kJ mol - 1 error in AG 0 corresponds to an error in Kc of about 300% (600 K). Potentially, a very useful means of obtaining values of rate constants for reactions where experimental data are scarce is thus severely frustrated.

Overview of alkane oxidation

9

1.4 OVERVIEW OF ALKANE OXIDATION

1.4.1 Introduction In view of the complexity and richness of combustion chemistry, a preliminary overview of alkane oxidation is worthwhile building on that developed in the Introductory Chapter. It is impossible to give a simple general mechanism under all conditions. Above ca 1000 K, homolysis of larger alkyl species and radical-radical processes are of particular importance because of the extremely high radical concentrations that exist alongside the very high rates of oxidation. By the late 1960s [14], there was consensus acceptance of a basic mechanism between 500 and 1000 K, as shown in Table 1.2. Figure 1.5 shows a schematic of the oxidation processes centred around the reversible step R + 0 2 ^ R 0 2 , which may be considered the "heart of combustion chemistry" in this temperature region. In qualitative terms, the mechanism explains the observed characteristics of alkane oxidation through slow reaction, changes in product distributions, cool-flames and explosions (autoignition), as well as another remarkable feature involving a region of negative temperature coefficient. 1.4.2 Autoignition and engine knock The autoignition that leads to knock in spark ignition engines is extremely dependent on the chemical composition of the fuel. The knock phenomenon and the practical methods of characterising a fuel in terms of its octane number are described in Chapter 7, but it is pertinent to illustrate here how the understanding of alkane oxidation can help to elucidate the effect of fuel structure on its behaviour in engines. In an engine the fuel/air mixture is compressed both by the piston and the expansion that accompanies the main combustion. If some of the gas autoignites before the flame reaches it there can be a sudden pressure rise called knock, and the occurrence of this limits either the operating conditions and design of the engine or the composition of the fuel. The mixture which may autoignite is exposed to a range of temperature and pressure during its compression. The reactions which are the most important in controlling the chemistry change throughout the compression. Since the overall process is a branched radical chain reaction the most influential steps are those that change the number of radicals, namely the initiation,

10


Ch. 1

TABLE 1.2 Basic mechanism for hydrocarbon oxidation between 500 and 1000 K (1) (1A) (2) (3) (4) (5A) (5B) (5C) (6) (7A,B,C) (8) (9) (10A,B,C) (10D) (11) (12) (13) (14) (15) (16)

RH RH X R R R R R R R02 R02 R02 QOOH QOOH R02 ROOH H202 H02 H H02

+

02

-> R —> R'

+ (+ (+ + + + + + +

RH M) M)

—» —> —> —» —>

o2 o2 o2

02(+M) RH

+ +

o2

+ + + +

M H02

R02

o2

RH

XH R' H AB ABO -> OR -> R 0 2 ( + M ) -* AB, ABO, OR ROOH -» -> QOOH —» AB, ABO, OR —» 0 2 QOOH -> chain reaction —> RO -> 20H -» H 2 0 2 —» OH H202 ->

+ + + + + + + +

H02 R" R AB' (+M) AB (+M) H02 OH OH

+ +

X R

+

X

+ + + + +

OH M

o2 o

R

AB, alkene; ABO, oxygenated compound; OR, oxygen-ring compound. X is a radical usually OH or H 0 2 ; R is the radical formed from RH. R', R" and AB' contain a smaller number of carbon atoms than RH. Reaction numbers for specific radicals are usually given a further letter. For example, (5Ae) refers to the reaction of C2H5 radicals with 0 2 to give the conjugate alkene (AB), C 2 H 4 and the H 0 2 radical. (5Ae)

C2H5 + 0 2 -> C2H4 + H 0 2 .

branching and termination reactions. In an ideal system, if it were only multiplication of radicals that led to runaway and autoignition, only the difference in the rates of branching and termination would determine whether or not the system explodes. However, in reality, thermal runaway is also significant and this is governed by the overall rate of reaction, which is influenced by the initiation rate. Nevertheless, the branching and termination rates appear in an exponential term so that it is the rates of branching and termination that largely decide how fast the oxidation proceeds.

k t cb 1

0-heterocycle + OH

RH HO2 + alkene

oxygenated products

0

dT

C-tl Split

9.

QOOH

C-C split

9

RH a

Products

fn

Branching

lowr R' + alkene

8

E.

E8'

ROOH

I

Branching HIGH TEMPERATURE

0

LOW TEMPERATURE

Fig. 1.5. General oxidation scheme for alkyl radicals at low and high temperatures.

12


Ch. 1

1.4.3 Regimes of oxidation chemistry For clarity, it is worth considering the regimes from the perspective of both products and rates. (/) Product perspective • Below ca 400 K, alkanes are not prone to oxidation but RH compounds such as aldehydes undergo oxidation because the aldehydic C—H bond is relatively weak (D(RC(0)—H) « SeSkJmol" 1 ) compared with primary (420), secondary (400) and tertiary (385) C—H bonds in alkanes, so that the chain reaction may be propagated through H-atom abstraction from RH by peroxy species such as CH3CO3. However, sensitisation or initiation at the surface (lower activation energy) may be necessary because homogeneous primary initiation through a reaction such as (la) is extremely endothermic (ca ÔkJmol" 1 ). (la)

CH3CHO + 0 2 -> CH3CO + H 0 2

• Secondary initiation (and sometimes spectacular acceleration of rate) may well occur through formation and subsequent homolysis of a molecular hydroperoxide such as CH3CO3H, particularly as the temperature is increased above 400 K. Marked autocatalysis and lengthy induction periods may result. Near quantitative yields of CH3CO3H have been observed [15] consistent with the following type of sequence: CH 3 CO + 0 2 - * C H 3 C 0 3 CH3CO3 + CH3CHO -> CH3CO3H + CH 3 CO CH3CO3H -> CH 3 + C 0 2 + OH • A similar mechanism in which the peroxide ROOH formed from the alkane RH provides secondary initiation, occurs above 450 K for alkanes with tertiary C—H bonds and at increasingly higher temperatures where secondary and primary C—H bonds are involved. In the region 500-600 K, where the oxidation rate increases sharply with temperature, the equilibrium R + 0 2 2 is well to the right and

13

Overview of alkane oxidation TABLE 1.3 Data for the equilibrium R + 0 2 ^ R 0 2 A// 298 kJmol

R

H CH 3 C2H5 ;-C 3 H 7 t-(^4rig

CH 2 CHCH 2

-208 -135 -147 -155 -153 -75

log£/atm * 500 K

750 K

1000 K

1250 K

Ceiling temp/K [R] = [R0 2 ] at O 2 = 0.1atm

16.8 7.4 7.6 7.7 7.1 1.6

9.6 2.7 2.45 2.25 1.75 -1.05

6.0 0.35 -0.10 -0.45 -0.95 -2.35

3.7 -1.15 -1.70 -2.1 -2.65 -3.15

1920 930 900 860 820 550

1

Ref.

[91 [91 [al

fbl [al

[b] [a] I.R. Slagle, E. Ratajczak and D. Gutman, J. Phys. Chem., 90 (1986) 402. [b] I.R. Slagle, E. Ratajczak, M.C. Heaven, D. Gutman and A.F. Wagner, J. Am. Chem. Soc. 107 (1985) 1838.

product formation occurs through reactions of R 0 2 leading to high initial yields of oxygenates such as aldehydes, ketones, alcohols and O-heterocyclic compounds. Table 1.3 summarizes values of the equilibrium constant at different temperatures and gives the "ceiling temperatures" where [R] = [R0 2 ] for 0 2 = 0.1 atm. Further increase in temperature produces, quite remarkably, a fall in rate (negative temperature coefficient) usually between ca 570-630 K. Figure 1.6 shows this effect in the oxidations of propanal and propane, where maximum rates (following the induction period) are plotted. In this temperature range, alternative reactions of R 0 2 occur with higher activation energies (by typically 50kJmol _ 1 ) than (8), and become competitive. The equilibrium in (6) lies increasingly to the left, so that the yield of ROOH falls, secondary initiation through (12) is cut drastically, and the oxidation rate decreases. (6) (8) (12)

R + 0 2 ^ R02 R 0 2 + RH -* ROOH + R R O O H ^ R O + OH

With further temperature increase, two features dominate. The ratio [R]/[R0 2 ] rises further as (6) becomes increasingly reversible, so that reactions of R radicals rather R 0 2 influence the oxidation. Of these the most important is the overall reaction (5A) which gives the

14


Ch. 1

H 0

H-0.5

H-1.0

103K/T Fig. 1.6. Variation of rate with temperature for propane and propanal oxidations in aged boric-acid-eoated vessels. Upper curve: C 3 H 8 = 20, 0 2 = 30, N2 = 70Torr; Lower curve: C 2 H 5 CHO = 4, 0 2 = 30, N2 = 26 Torr.

conjugate alkene and H 0 2 . Initial yields of alkenes, well in excess of 50%, are observed for the C2—C5 alkanes [16,17], indeed the yield is 99% for C 2 H 5 radicals [18]. • H 0 2 is thermodynamically stable and unreactive at these temperatures, and is removed to a considerable extent by reaction (14) to form H 2 0 2 , which then through its homolysis in reaction (13) provides increasingly rapid secondary initiation. As shown in Fig. 1.6 the oxidation rate now increases again with temperature, and the major initial products are alkenes and O-heterocyclic compounds. (5A)

R + 0 2 ^ alkene+ H 0 2

(13)

H202 + M ^ 2 0 H + M

(14)

H 0 2 + H 0 2 -> H 2 0 2 + 0 2

As the temperature rises above ca 750 K, particularly at low 0 2 pressures, C—C homolysis of R radicals becomes important, typified by the behaviour of 2-C4H9 radicals.

Overview of alkane oxidation

15

CH 3 CHCH 2 CH 3 -> CH 3 CH=CH 2 + CH 3 Alkenes with a smaller carbon atom number than the parent alkane become major initial products, and the further oxidation chemistry becomes that of the accompanying small alkyl radicals. In the simplest terms, the dominant radicals are CH 3 , C 2 H 5 , /-C3H7 and f-C4H9, all formed from the homolysis of larger alkyl radicals (or oxygenated radicals), and which structurally are unable to undergo further C—C homolysis. Only at higher temperatures (ca 900 K) does C—H homolysis (4) become important. (4tb)

t-C4U9 -* (CH 3 ) 2 C=CH 2 + H

(if) Rate and ignition perspective The rate of the overall oxidation depends on the elementary steps which change the number of radicals. Initiation reactions (section 1.6) are essential under any conditions but have only a linear effect on the number of radicals. The branching and termination reactions, on the other hand, can affect the number of radicals more strongly and it is the exponential increase in radical concentrations caused by an excess of branching over termination which is responsible for an isothermal explosion. As discussed in the introduction, which branching reactions are most important in hydrocarbon oxidation depends on the conditions. This provides a means of classifying the general character of the oxidation, as shown in Fig. 0.1. At high temperatures, and especially in flames, the most important branching reaction in the oxidation of all types of hydrocarbon (and hydrogen) is (15)

H + 02ÔH + 0

At lower temperature and higher pressures this becomes less significant because H atoms react to form much less reactive H 0 2 H + 02 + M ^ H 0 2 + M Then the significant branching reactions become those involving H 0 2 and CH 3 which are rather unreactive radicals (but which are therefore relatively abundant).

16


Ch. 1

CH 3 + 0 2 ^ C H 3 0 + 0 RH + H 0 2 - + R + H 2 0 2 (13)

H 2 0 2 + M ^ OH + OH + M

These branching reactions are relatively slow so that, in this "intermediate" regime of chemistry, thermal, rather than chemical kinetic effects, are especially significant in bringing about ignition. At lower temperatures, as described above, peroxy radical chemistry becomes important. For the alkyl radicals derived from the longer chain alkanes there is then a route to branching through isomerisation, 0 2 addition, and a further isomerisation which is described in section 1.11. R02->QOOH QOOH + 0 2 ^ 0 2 QOOH 0 2 QOOH-> R'OOH + OH R ' O O H ^ R ' O + OH This branching sequence is in competition with reactions of the alkyl radical R to form inert H 0 2 , CH 3 and C H 3 0 0 . These essentially terminating reactions become less important as the temperature decreases and the R + 0 2 ^ R 0 2 equilibrium shifts to the right, causing the overall rate to accelerate as described in the previous section. All these regimes of chemistry play a part in the autoignition leading to knock in engines. At the moment of autoignition, the temperature may be 1000 to 1200 K and the pressure 25 to 60 atm. This corresponds to the "intermediate" temperature range where H 0 2 chemistry is important. The temperature rises, and the high-temperature chemistry takes over to complete the combustion and to provide a large part of the energy release. But for alkane fuels, the relatively long time during the compression when the conditions are in the low-temperature regime is what decides whether the autoignition occurs or not. During this time there is appreciable chemical reaction leading to a temperature rise and the formation of reactive intermediates such as organic peroxides and hydrogen peroxide which help to initiate the autoignition when the temperature has risen further. In essence, the low-temperature chemistry prepares the mixture for ignition,

Review of experimental methods

17

the intermediate chemistry ignites it and the high-temperature chemistry burns it. The different regimes of chemistry are affected by the structure of the alkane to varying extents, with its influence being weaker at high temperatures. For example, the burning velocities of iso-octane and nheptane are very similar. At lower temperature, the autoignition delay times in shock tubes between 1100 and 1600 K are rather different. This arises because all alkyl radicals are unstable and the ease of autoignition depends on the extent to which they decay to reactive H atoms and hence produce branching or to less reactive CH 3 . But knock propensity (octane number) depends on structure to a much greater extent than would be predicted by this mechanism alone, indicating that it is the low-temperature chemistry, with its very structure-dependent peroxy radical isomerizations, which is most influential.

1.5 REVIEW OF EXPERIMENTAL METHODS FOR ESTABLISHING MECHANISMS AND DETERMINING RATE CONSTANTS

1.5.1 Basic difficulties involved For the purpose of establishing mechanisms and determining rate constants, classical direct investigations of alkane oxidation were severely handicapped in at least two ways. (i) Surface effects and the consequent lack of reproducibility. (ii) The high reactivity of the intermediate products which causes increased complexity as the reaction proceeds. For example in studies of the oxidation of propane at 708 K, Falconer and Knox [19] showed that the yield of propene dropped by 30% over the first 10% of reaction. As indicated earlier, autocatalysis can cause massive changes in radical concentration. Lacking computer power and related techniques, early workers failed almost completely to obtain kinetic data for elementary reactions and, furthermore, were unable to establish mechanisms. Even studies [4] in the induction period ( < 1 % reaction) failed to provide kinetic data on the elementary processes. With the advent of computer modelling applied to alkane oxidation, the emphasis has been on fitting a mechanism to experimental combustion data of all kinds (Chapter 6). In many cases it is questionable how reliable

18


Ch. 1

this approach has been, because frequently the mechanism has been tested over very narrow conditions. Furthermore, very few reliable kinetic data for elementary reactions have been obtained from modelling studies. Essentially only two procedures have been successful. (i) Studies of a single elementary reaction through generation of a clean source of the radical, followed by the measurement of the decay of the radical concentration in the presence of excess substrate so that the interpretation involves pseudo-first order kinetics (Chapter 2). Particularly important examples in combustion chemistry include Tully et al.'s [20] work on OH + RH by use of laser-induced fluorescence to monitor [OH] and the study of C2H5 + 0 2 by Sagle et al. [21] who used laser photolysis to produce the radicals and real-time mass spectroscopy to measure [C2H5]. In many cases, including the two cited above, measurements were made over wide temperature ranges, ca 300-1000 K. Frequently, however, although the rate constant determinations were excellent, no information was obtained on the products formed. Two further problems exist: (a) Reliable sources for many radicals, for example the larger alkyl species, are not available. (b) To avoid mechanistic complications, only relatively fast reactions can be investigated. Consequently the approach is ideal for OH + RH reactions where, for most RH compounds, E ~ 0 and for radical-radical reactions where again there is no energy barrier, but fails because of the high activation energies for H 0 2 + RH (60QOkJmor 1 ) and RH + 0 2 (180-240 kJ mol" x) reactions, which under many conditions are key processes in the mechanism. Even for H + RH, where typically E ~ 40-45 kJmol - 1 , values of rate constants too high by a factor of about 10 were obtained below ca 500 K because the H atoms react further with products formed from the R radicals [22]. Improvements in detection sensitivity, permitting variation in and reduction of the [H]/[RH] ratio, coupled with computer simulation and analysis, have led to accurate determinations of rate constants for such reactions, but have not been widely deployed. (ii) The extraction of rate constants for individual elementary reactions from systems which involve a complex mechanism. Of these, probably the consistently most successful studies have been those carried out at Hull using five different approaches. All have a common feature, that despite the overall complexity of the system used, conditions are carefully chosen


19

so that individual elementary reactions are isolated in importance and the resulting kinetic data show very little dependence on any other uncertain parameters. Usually the data obtained are in the form of a ratio of rate constants which can often be isolated with considerable accuracy (ca 5%). Furthermore, in many cases, as part of the experiment design one of the rate constants is known accurately, so that high quality "absolute" rate constants are obtained. 1.5.2 Review of Hull approaches In the later sections, frequent references are made to mechanistic details and kinetic data determined by use of the competitive methods developed at Hull specifically for combustion chemistry. Therefore, it is useful at this stage to outline the basic principles and chemistry which underpin the usefulness of the approaches. (i) Method I. Addition of RH to H2 + 02 mixtures at about 750 K This approach has been remarkably successful over the last 30 years with a wide variety of organic compounds. It is, however, limited in its use to pressures between 200-600 Torr and temperatures between 720800 K. Use of small amounts of the additive (RH) and an aged boric-acidcoated Pyrex vessel permits an investigation of the oxidation of alkanes, alkenes, aromatics and related oxygenated compounds in the total absence of surface effects in a constant and controllable radical environment determined almost entirely by the H 2 + 0 2 mixture. Many different RH compounds may then be oxidized under identical conditions. This is in marked contrast to the direct oxidation method where the radical environment is controlled by the oxidant and changes constantly as the intermediates are formed and then oxidized. Two types of experiment are carried out: (a) Using traces of RH (^0.025% of RH), measurements of the relative rate of consumption of H 2 and RH give kinetic data for OH + RH and H + RH [23, 24], and in some cases H 0 2 + RH [25]. Cohen [12] has commented recently that this method has in general provided the only reliable high-temperature data for H + RH. Essentially, because the H 2 + 0 2 reaction has been analyzed quantitatively [26] under the conditions used, and RH addition causes negligible per-

20


Ch. 1

turbation, the values of [H], [O], [OH] and [H0 2 ] are known accurately and in effect the rate constants for H, OH and H 0 2 attack on RH are determined absolutely, (b) Detailed product analysis (with 1% RH) over wide ranges of mixture composition, (H 2 /0 2 is typically changed from 0.1 to 10), gives mechanistic and kinetic information on the reactions of R radicals in an oxidizing environment [27]. In many cases, the interpretation is based on the relative rates of two competing reactions. For example, for RH = butane, butene-2 and C 3 H 6 are formed uniquely in reactions (5Ab) and (3b), so that the initial relative rate of formation of the products is given by equation (1.1). 4 C 4 H 8 - 2]/d[C3H6] = k5Ah[02]/k3h (5Ab) (3b)

(1.1)

CH 3 CHCH 2 CH 3 + 0 2 -> CH 3 CH=CHCH 3 + H 0 2 CH 3 CHCH 2 CH 3 -> CH 3 CH 2 =CH 2 + CH 3

As k3b is known accurately, A:5Ab can be obtained [23]. The H 2 + 0 2 addition method has been used to obtain mechanistic and kinetic data on most C2—C7 alkane oxidations, and in many cases the data remain unique. (if) Method II. The oxidation of aldehydes (550-800 K) This method offers at least two advantages in terms of ease of interpretation: (a) The aldehydes are usually more reactive than the molecular organic intermediates formed. (b) A specific alkyl radical is produced in the oxidation. The second advantage is simply illustrated by considering propyl radicals. The oxidation of propane produces both 1-C3H7 and 2-C3H7 radicals, and for quantitative interpretation the proportion of each formed must be known. X + C 3 H 8 ^ X H + 1-C3H7 XH + 2-C3H7


21

As X may be OH, H, H 0 2 and CH 3 , precise calculation of the proportion is difficult, as discussed later. However in the oxidations of butanal and ibutanal specifically only 1-C3H7 and 2-C3H7 radicals, respectively, are produced. X + l-C 3 H 7 CHO -> l-C 3 H 7 CO + XH l - C 3 H 7 C 0 ^ 1 - C 3 H 7 + CO X + 2-C3H7CHO -> 2-C3H7CO + XH 2-C3H7CO -> 2-C3H7 + CO Although attack does occur at the alkyl chain in the aldehydes, any correction necessary in the interpretation is small. This approach has been used very successfully in studying the kinetics of 2-C3H7 + 0 2 [28] and C 2 H 5 + 0 2 [29] over wide temperature ranges. (Hi) Method HI. Decomposition of tetramethylbutane (TMB) in the presence of 02 (650-800 K) The system is both unusual and essentially very simple under the conditions used. The central C—C bond in TMB is sterically highly strained (ca 80 kJ mol - 1 ) with the result that homolysis occurs at a convenient rate over the range 650-800 K. Ninety-nine percent of the £-C4H9 radicals formed react with 0 2 to give j'-butene and H 0 2 . The sequence TMB -> 2r-C4H9 t-C4H9 + 0 2 -» /-C4H8 + H 0 2 thus provides a very clean and reliable source of H 0 2 and f-C4H9 [30] radicals. By determining the initial yields of product, and the use of total pressures of ^60 Torr, OH formation in the following sequence is reduced to minor importance [31]. H 0 2 + H 0 2 -> H 2 0 2 + 0 2 H 0 2 + TMB -> H 2 0 2 + (CH 3 ) 3 CC(CH 3 ) 2 CH 2 H202 + M ^ 2 0 H + M

22


Ch. 1

The system has been used to obtain the only reliable data for H 0 2 + alkane. With C 2 H 6 added to TMB + 0 2 mixtures, the experimental measurement involves only the relative rate of formation of C 2 H 4 and J - C 4 H 8 . Effectively, [H0 2 ] is determined directly from d[i-C4H8]/dt and the rate of formation of C 2 H 4 (99% of C2H5 radicals give C2H4) is given by equation (1.2) [32]. d[C2U4]/dt = £ 16e [H0 2 ][C 2 H 6 ] (16e)

(1.2)

H 0 2 + C 2 H 6 -> H 2 0 2 + C 2 H 5

The key to success here (and in each of methods I-V used) is that a simple analytical treatment gives an accurate answer, with computer-based correction ideally amounting to little more than 10%. The method has also given the only kinetic data for reactions involving the addition of H 0 2 to alkenes to give oxiranes [33]. H02 + ^ C = C OOH The TMB studies, and related investigations of the decomposition of trimethylbutane and 2,3-dimethylbutane, have given accurate values for the heats of formation of 2-C3H7 [34] and t-C4H9 radicals [31] in excellent agreement with those given in Table 1.1. (iv) Method TV. Decomposition of specific alkenes in the presence of 02 This system, using 4,4-dimethylpentene-l, has given the first reliable kinetic data for the oxidation of allyl radicals in the range 650-800 K. CH 2 =CHCH 2 C(CH 3 ) 3 -> CH 2 CHCH 2 + t-C4H9 The steric strain arising from the presence of the t-C4H9 group (ca 40kJmol _ 1 ) and the resonance energy in the allyl group (ca 50kJmol _ 1 ) lead to a considerably faster rate than is normal for C—C homolysis in alkenes. As t-C4H9 radicals react uniquely (99%) with 0 2 to give z-C4H8 and H 0 2 (see above), then the chemistry of allyl radicals in the presence of 0 2 and a clean source of H 0 2 can be studied with ease. As expected

23


the resonance-stabilized allyl radicals are unreactive towards RH and 0 2 , and are, hence, consumed mainly in radical-radical processes such as H 0 2 + CH 2 CHCH 2 -> 0 2 + C 3 H 6 and CH 2 CHCH 2 + C H 2 C H C H 2 ^ CH 2 =CHCH 2 CH 2 CH=CH 2 The kinetics of the multi-channel interactions between H 0 2 and allyl radicals have been determined [35] and it has been established that all allyl + 0 2 reactions have high activation energy barriers [36]. Experimentally the system is very simple and requires only precise measurement of reaction products early in reaction ( CH 2 CHCH 2 + H 0 2

Studies with i-butene [39] and HCHO [40] have given similar information (discussed in the next section).

1.6 PRIMARY INITIATION REACTIONS

According to the Semenov theory of chain reactions [2] the rate of oxidation depends strongly (half to first power) on the rate of production of new chain centres. However, the problem that has bedevilled combustion kinetics over many years is the chemical nature of the process. Reactions (1) and (1A) are the primary initiation reactions in hydrocarbon oxidation, to be distinguished from secondary initiation processes such as reaction (13) where radicals are produced from a "stable" intermediate (1) (1A)

RH + 0 2 ^ R + H 0 2 R H ^ R ' + R"

Reactions (1) and (1A) are extremely endothermic as shown in Table 1.4, so that they can be totally dominated by the presence of minute traces of sensitizers, by photo-initiation, surface catalysis and secondary initiation. Surface initiation is particularly pronounced below 600 K, and has been the cause of lack of reproducibility and reported experimentally-

Primary initiation reactions

25

TABLE 1.4 Endothermicities* of RH + 0 2 -> R + H 0 2 RH

AH°298/kJ mof 1

C6H6 C2H4 C2H6

254 239 213 200.5 190 169 166 165 158 156.5 143 122

d " l 3 CH2CH.3

(CH3)3CH C6H5CH3

HCHO CH3CH=CH2

(CH 3 ) 2 C=CHCH 3 C 2 H 5 CHO (CH 3 ) 2 C=C(CH 3 ) 2 CH 2 =CHCH 2 CH=CH 2 *Error limits symbol

5 kJ mol \

determined values of kx higher by a factor of at least 106 than the true homogeneous value [16, 41]. Dixon et al. [42] give values of kl3L increasing from 1.2 x l O " 2 6 to 4.8 x 10" 26 at 350 K and from 1.0 x 10" 25 to 8.5 x 10~25 cm3 molecule -1 s _ 1 at 393 K when the surface/volume ratio is increased from 0.6 to 6.1 cm - 1 . (la)

CH3CHO + 0 2 -> CH3CO + H 0 2

Experimental determination of k1A has usually been through direct homolysis studies. Secondary initiation is absent in the absence of 0 2 , but complex mechanisms require careful interpretation [43]. In some cases, the presence of 0 2 renders the mechanism extremely simple as, for example, in the case of tetramethylbutane [31] (Method III). 1.6.1 Experimental determination of kx Several factors apart from surface effects contribute to the difficulty of determining k 2 0 H + M

Even when a tertiary C—H bond is involved, R p and Rs become equal when 10~2% of /-butane is converted into H 2 0 2 at ca 800 K. Furthermore, several species of peroxide and aldehyde are found in the initial products of even a simple alkane in the first 1% reaction. Direct determination of fci under these conditions is not possible. (iii) Any radical branching through reactions such as H + 0 2 ^ 0 + OH Q O O H 0 2 - > 2 radicals must be taken into account. In practice, this is extremely difficult to do. Recently a very high value determined [44] for klt was almost certainly due to use of a mechanism which did not include important radical branching reactions [11]. (It)

C 6 H 5 CH 3 + 0 2 -> C 6 H 5 CH 2 + H 0 2

Studies of C 3 H 6 oxidation [38] under carefully controlled conditions have shown that the above difficulties can be reduced to a minor role essentially because the allyl radical produced in reaction (lp) is stabilized by electron derealization and is very unreactive towards 0 2 . The allyl radicals are removed mainly by recombination to give hexa-l,5-diene (HDE) and over the temperature range 650-800 K, the initial products are accounted for by a simple mechanism. (lp)

C 3 H 6 + 0 2 -> CH 2 CHCH 2 + H 0 2

(16p)

H 0 2 + C 3 H 6 -> CH 2 CHCH 2 + H 2 0 2

(18)

CH 2 CHCH 2 + H 0 2 - > CO + other products

27


CH 2 CHCH 2 + H 0 2 -> C 3 H 6 + 0 2

(-IP) (19)

CH 2 CHCH 2 + CH 2 CHCH 2 -»• HDE CH 2 CHCH 2 + 0 2 ^ branching

(20) (21) (22)

H 0 2 + C 3 H 6 -»• C 3 H 6 0 + OH OH + C 3 H 6 ^ C 3 H 6 O H

o,

CH 3 CHO + HCHO + OH

Figure 1.7 shows a [product]-time plot for the mixture C 3 H 6 = 12, 0 2 = 30 and N 2 = 18 Torr at 713 K. The single most vital feature is the very high yield of HDE which, on the basis of the mechanism, implies that about 70% of the allyl radicals recombine in reaction (19), which must therefore be the dominant termination step. As a result klp could be determined with only minor (ca 20%) correction by use of the principle that at the steady state the rate of initiation equals the rate of termination. A:lp[C3H6][02] = fc19[CH2CHCH2]2 = R H D E

(1.3)

0.10 h b

OH

0.050 h

0.050 h 0.025 h

600 time / s Fig. 1.7. Product yields in the oxidation of propene at 713 K C 3 H 6 = 12, 0 2 = 30, N 2 = 18Torr. Upper: x, CO; O, HCHO; • , C2H 4 (x2); • , CH 4 (x4); A, CH 2 =CHCHO(x2); Lower: • , CH 3 CHO; • , C 3 H 6 0; A, 1,5-hexadiene.

28


Ch. 1

where RHDE is the initial rate of formation of HDE. At this stage, no other products need to be considered. Over the range 673-793 K, the simple treatment gave Alp = 10~ 116 cm 3 molecule -1 s _ 1 and Elp = 162kJmol _ 1 . More precisely, equation (1.4) should be used, where R p , Rs, R b and Rt are the initial rates of primary initiation, secondary initiation, radical branching and total radical termination. R p + Rs + R b = R t .

(1.4)

Secondary initiation was considered negligible because initial rates were used so that [H 2 0 2 ] = 0, and a small degree of radical branching through (20) was isolated from the variation of the preliminary value of klp with mixture composition. Any radical branching through CH 2 =CHCH 2 02 radicals is negligible because the CH 2 CHCH 2 + 0 2 ^ CH 2 =CHCH 2 0 2 equilibrium is well to the left (Table 1.3). Reactions (-lp) and (14) were included as minor termination processes. Values for [allyl] and [H0 2 ] were obtained from RHDE and the initial rate of formation of propene oxide, and the known values of k19 and k2\ respectively. Very significantly the net corrections were extremely small, giving refined values of Alp = 10-ii.4

(lib)

o.4icm3

molecule

-i

s-i and

£lp

= 164

6kJmol_1.

(CH 3 ) 2 C=CH 2 + 0 2 -> CH 2 C(CH 3 )CH 2 + H 0 2

Using the same approach and interpretation, values of Ahh = 6.4kJmol" 1 were ob1 0 -n.io o.44 c m 3 m o l e c u l e - i s - i a n d Ehh = 1 6 L 2 tained [45] from studies of isobutene oxidation, as predicted by the similar thermochemistry and inert nature of methylallyl radicals due to electron derealization. The agreement is good, and moreover the Arrhenius parameters are entirely consistent with Au = 10~10 47 cm 3 molecule -1 s _ 1 and E1{ = 163 kJ mol~\ which were obtained from studies of HCHO oxidation under conditions where the chain length was reduced virtually to zero. In the initial stages of reaction, the mechanism in KCl-coated vessels, where H 0 2 and H 2 0 2 are efficiently destroyed at the vessel surface, is very simple. (If)

HCHO + 0 2 -* HCO + H 0 2

(23)

HCO + 0 2 ^ H 0 2 + CO

29


(24)

H 0 2 + HCHO -» H 2 0 2 + HCO

(25)

H0 2 ^> surface

(26)

H 2 0 2 ^ surface

Consequently R c o (the initial rate of formation of CO) is given by equation (1.5). Rco = *if[HCHO][0 2 ] + 2 M W ^ ) [ H C H O ] 2 [ 0 2 ] .

(1.5)

Reaction (23) is extremely fast ( f c ~ 5 x 10~12 cm3 molecule -1 s _ 1 ), so that HCO is removed solely in (23) and cannot lead to radical branching or secondary initiation. Furthermore [H 2 0 2 ] is negligible, particularly as initial rates are used. Rearrangement of (1.5) gives (1.6) and from excellent plots of Rco /[HCHO] [0 2 ] against [HCHO], k1{ was obtained directly from the intercept. R CO /[HCHO][0 2 ] = klt + 2M*if/*25)[HCHO].

(1.6)

The homogenous nature of (If) was confirmed by use of different vessel diameters. The data for the three systems are summarized in Table 1.5. A further indication of consistency, given that the activation energies E.lp ~ E_uh ~ E.u ~ 0, is the observation that Elp = Ehh = Eu = A// l p = A// lib = A// l f = 165 5 kJ m o l " \ The "low" values for Alib and Alp (per C—H bond) compared with Au are acceptable because of the extra loss of entropy of activation (ca 18 J K - 1 mol - 1 ) due to derealization in the emerging allyl and methylallyl radicals. With the C 3 H 6 + 0 2 system validated, Ingham, Walker and Woolford [45] used this "ideal" approach to obtain further values for kx. Again the basis was extremely simple. When small amounts of compound RH TABLE 1.5 Arrhenius Parameters for RH + 0 2 —> R + H 0 2 RH

log (^4/cm3 molecule -1 s"1)

E/kJmol - 1

T range

C3H6 /-C4H8 HCHO

-11.49 -11.10 -10.47

163.5 6 161.2 4 163 6

673-793 673-793 673-815

1 4

30


Ch. 1

containing C—H bonds of comparable dissociation energy to the allyl C—H bonds in C 3 H 6 are added to C 3 H 6 + 0 2 mixtures, then additional primary initiation occurs through reaction (1). RH + 0 2 - * R + H 0 2

(1)

Providing R has a facile reaction, usually with 0 2 , such as (27) for butenyl radicals, then [allyl] > [R], and the recombination of allyl radicals (19) remains by far the dominant termination step. With (19) the only termination, and low [0 2 ] so that radical branching reactions are negligible, then the ratio kjklp can be obtained from the initial rates of formation of HDE in the presence (RRDE) and absence (RHDE)O of RH and the use of equation (1.7) R

HDE

(RHDE)O

(27)

=

J+

fci[RH]

^ 7v

ÎP[C3H6]

CH 3 CHCH=CH 2 + 0 2 -> CH 2 =CHCH=CH 2 + H 0 2

Figure 1.8 shows typical plots for the initial yields of HDE for the mixture containing 12, 18 and 30Torr of C 3 H 6 , N 2 and 0 2 respectively at 693 K in the presence and absence of lTorr of methylbutene-2, and Table 1.6 gives values for kjklp obtained from equation (1.7). With the initial yields of propene oxide also known, corrections for the small amount of radical branching (20) and the extra termination due to (14) and (-lp) were also made. As shown in Table 1.6 the corrections were no bigger than ca 25%. (14)

H 0 2 + H 0 2 -> H 2 0 2 + 0 2

(-lp)

CH 2 CHCH 2 + H 0 2 -» C 3 H 6 + 0 2

Absolute values of kx were obtained from the known parameters Alp and Elp (see Table 1.8). Table 1.7 summarizes the recommended Arrhenius parameters for the temperature range 600-800 K for a number of compounds RH used as additives in the C 3 H 6 + 0 2 system. For alkenes, bearing in mind that only allylic C—H bonds undergo reaction (1) under the conditions used, Alp = Ai (per C—H bond) is assumed to obtain the activation energies.


31

0.06

b

0.04

o H

0.02

0 200

400

600

time / s Fig. 1.8. 1,5-Hexadiene yields in the presence (x) and absence (O) of methylbutene-2 at 693 K. C3H6 = 12, N 2 = 18, 0 2 = 30, methylbutene-2 = 1 Torr. TABLE 1.6 Values of kjklp

for (CH3)2C==CHCH3 at 693 K

Mixture/Torr Methylbutene-2

C3H6

N2

o2

fci/fcip (equation (1.7))

1.0 0.4 0.4

12 12 12

18 38 18

30 10 30

23 27.5 26

ki/kxp

(corr) 28.5 31 32

Table 1.8 summarizes the kinetic data for reaction (1) obtained from other studies at Hull; no other reliable results are available. For CH 3 CHO [46], the recombination of CH 3 radicals to give C 2 H 6 was isolated as the "sole" termination reaction, so that kla was determined again from a very simple equation. fcla[CH3CHO][02] = M C H 3 ] 2 = RC 2 H 6 .

(1.8)

A small refinement in the treatment to allow for the branching process

w

F3

TABLE 1.7 Kinetic data for RH + O2+ R + H 0 2 from C3H6addition studies

RH

kilkip kl/kl, (equation (1.7)) (corr)

kl/cm3 molecule-' s-'

4.8 4.0 11.1 12.7

4.9 12.0 15

8.5 x 7.5 x 1.83 x 2.33 X

hexene-1 (CH3)2C--CHCH3 (CH3)2-C(CH3)2 CH2=CHCHzCH*H2

12.3 25.5 117 43

14.5 30.5 145 52.5

C6HsCH3** C6HK2Hs C6HsCH(CHdz C6HsC(CH&

R + H 0 2 obtained by direct methods at Hull RH

it/cm3 molecule - 1 s - 1

T/K

log(A/cm3 molecule - 1 s - 1 )

£/kJmol

/-C 3 H 7 CHO C2H5CHO CH3CHO HCHO C3H6 /-C4H8 (CH 3 )2CHCH(CH 3 )2

1.4 x 10 - 2 2 1.25 x 1(T 22 4.0 x 1 0 - 2 1

713 713 813 673--815 673--793 673--793 813 773 1200

-10.8* -10.8* -10.8* -10.47 -11.49 -11.10

151 152 150 163 6 163.5 6 161.2 4

C6HsCH3

2.2 x 1 0 - 2 2 3.0 x 1 0 - 2 3

-

1 4

-10.1** -10.3

185 173

l

Ref. a [41] b [40] [38] [39] c

[4]

a R.R. Baldwin, C.J. Cleugh, J.C. Plaistowe and R.W. Walker, J. Chem. Soc, Faraday Trans I, 17 (1979) 1433. b. R.R. Baldwin, M.J. Matchan and R.W. Walker, Combustion and Flame 15 (1970) 109. c. R.R. Baldwin, G.R. Drewery and R.W. Walker, J. Chem.Soc, Faraday Trans I, 90 (1984) 2827. *A (per C—H bond) assumed 10 - 1 0 ' 8 cm3 molecule -1 s - 1 . **For (CH 3 ) 2 CHCH(CH 3 ) 2 , AH = E assumed. Ôbtained by Emdee et al. [44], assume AH = E.

(29) gave k29 = 1.8 x 10" 16 cm3 molecule -1 s" 1 at 713 K (28) (29)

CH 3 + CH 3 + M -> C 2 H 6 + M CH 3 0 2 + CH3CHO -> CH3OOH + CH3CO

1.6.2 Database for the primary initiation process (1) Although even careful studies can lead easily to inaccurate (usually too high) rate constants for reaction (1), the consistency, both kinetically and thermochemically, of the data presented above implies high reliability. Walker [45] has pointed out the strong evidence to suggest that the value of klt (toluene + 0 2 ) , obtained by Emdee et al. [44] from modelling studies of toluene oxidation at about 1200 K, is too high by a factor of approximately 100 due to the neglect of radical branching reactions in the mechanism used. Although the data shown in Tables 1.7 and 1.8 are relatively limited and come from only one laboratory, (1) is a key reaction over a wide

34


Ch. 1

TABLE 1.9 Data base for RH + 0 2 -> R + H 0 2 Type of reaction

A/cm3 molecule -1 s _ 1 (per C—H bond)

£/kJmol _ 1

I II III

1.15 x HT 11 1.3 x 1(T12 1.5 x 10" 13

AH A// AH

Key I Localized electron in R (e.g., alkanes, oxygenates etc). II Delocalized electron involving the loss of one rotation (e.g., alkenes, alkylbenzenes). Ill Delocalized electron involving the loss of two rotations (e.g., CH,=CHCH g CH=CH,).

range of oxidation conditions. Data for alkanes will be extremely difficult to obtain experimentally, and the database in Table 1.9 is recommended for use between 600 and 1200 K. They are based on the experimental data available, the principle of bond additivity and that E = AH for the reaction. The decreasing values of A from I to III are consistent with the extra loss of entropy of activation (ca 1 8 J K _ 1 m o l _ 1 per lost rotation) in the emerging electron-delocalized radicals. The data base is probably reliable to a factor of 2 between 600 and 1200 K unless there is a very marked non-Arrhenius behaviour in kx. Below 600 K, (1) is far too slow to influence events and above 1200 K, reaction (1A) will dominate unless the 0 2 pressure is very high (ca 20 atms) (Table 1.11). 1.6.3 Experimental determination of k1A (1A)

R H - > R ' + R"

Reaction (1A) is the other major primary initiation process, particularly above 1000 K. Kinetic data have been available for a number of years, partly because k1A is relatively easy to measure because secondary initiation is effectively absent. Nevertheless, the systems used are invariably complex and care is necessary in abstracting k1A. Tsang [47, 48] has used a single-pulse shock-tube with considerable success by use of an approach whereby relative rate constants are obtained with one known accurately as a reference. Baldwin, Walker and Drewery [34] have shown that E1A can


35

TABLE 1.10 Arrhenius parameters* + for initiation process (1A) Reaction

log(A1A/s *)

£ 1 A /kJmol

C 2 H 6 -»2CH 3 C4Hio —» 2C2H5 (CH3)2CHCH(CH3)2 -> 2/Pr (CH 3 ) 3 CCH(CH 3 ) 2 -> /-Pr + t-bu (CH 3 ) 3 CC(CH 3 ) 3 -> 2^-Bu CH 2 =CHCH 2 CH 2 CH=CH 2 -> 2 allyl C 6 H 5 CH 2 CH 3 -> C 6 H 5 CH 2 + CH 3

16.7 16.5 16.4 16.5 16.8 15.1 15.6 15.5**

372 343 319 305 290 241 306 373**

*High pressure values. Based on data given in Ref. [48]. **Note parameters given for C—H homolysis. +

be estimated accurately where no experimental data exist as the variation in E reflects directly strain and derealization energies in the radicals formed. Table 1.10 gives high-pressure Arrhenius parameters for a selection of hydrocarbons of varying type. Values of A1A = io 1 6 - - 3 s _ 1 for alkanes are reduced to about i o - 3 s " 1 when an electron-delocalized radical is 3 _1 produced and i o - s when two are formed, as in the case of RH = hexa-l,5-diene, consistent with the decrease in entropy of activation arising from the loss of one or two rotations in the transition state. 1.6.4 Relative rates of initiation by 02 attack and dissociation Table 1.11 shows the relative rates of reactions (1) and (1A), expressed as fci[02]/fciA for [0 2 ] = 1 atm at 600, 1000 and 1500 K for a series of hydrocarbons of varying type. In reaction (1), only the weakest C—H bond is attacked to give R + H 0 2 , and the products for the homolysis are shown in parentheses. With linear alkanes, (1) clearly dominates at low temperatures with fci[02]/fciA * 106 at 600 K, but at 1000 K both reactions contribute almost equally. At 1500 K, reaction (1) is of negligible importance unless [0 2 ] > 10 atms. As the alkane structure becomes more branched, E1A falls due to increases in the strain energy in the C—C bond undergoing homolysis, and homolysis becomes relatively more important, despite the presence of tertiary C—H bonds (which increases the value of ki). As an extreme example, A:i[02]/A:1A for tetramethylbutane is only

36


Ch. 1

TABLE 1.11 Relative rates* of (1) and (1A) (&i[02]/A:iA) at 0 2 = 1 atmosphere *I[02]/*IA

RH (products of homolysis) C 2 H 6 (2CH3) C4H10 (2C2H5)

(CH3)2CHCH (CH 3 ) 2 (2/-Pr) (CH 3 ) 3 CCH(CH 3 ) 2 (/-Pr + t-Bu) (CH 3 ) 3 CC(CH 3 ) 3 (2r-Bu) CH 2 =CHCH 2 CH 2 CH=CH 2 (2 allyl) C 6 H 5 CH 2 CH 3 (C6H5CH2 + CH3) C 6 H 5 CH 3 (C6H5CH2 + H)**

600 K 1.8 x 4.7 x 2.0 x 4.4 x 0.11 5.2 1.8 x 1.6 x

6

10 106 103 101

107 107

1000 K

1500 K

1.3 0.3 3.7 2.8 1.7 1.4 0.4 5.4

1.7 7.5 1.4 1.7 6.3 1.9 6.4 8.8

x x x x

10~2 10" 3 10" 4 10" 3

x 102

x x x x x x x x

10" 3 10~4 10" 4 10" 5 10" 6 10" 5 10" 4 10~2

*High pressure values. **Note C—H homolysis.

about 0.1 at 600 K ( 0 2 - 1 atm) and is reduced to 1.7 x 10" 4 at 1000 K and to 6.3 x 10~6 at 1500 K. With this highly strained alkane, reaction (1) may be regarded as of negligible importance at all temperatures unless [0 2 ] is very high. Similar, but reduced, ratios are noted for 2,3-dimethylbutane and 2,2,3-trimethylbutane, where fci[02]/fciA ~ 10~3 at 1000 K. The double electron-delocalization effect in hexa-l,5-diene homolysis to give two allyl radicals gives values of &i[0 2 ]//: 1A below unity above about 650 K. With ethylbenzene the A factors and activation energies for homolysis and reaction with 0 2 are lowered by almost the same factors by electron derealization, so as Table 1.11 shows the values of fci[02]/fciA are very similar to those for C 2 H 6 at all three temperatures, despite the very different natures of the two hydrocarbons. Similar effects will be found for aliphatic alkenes such as hex-1-ene where E1A is lowered only by the derealization energy in the allyl radical produced. However, ki[0?\lk1A will be much lower for branched alkenes, such as 4,4-dimethylpent-l-ene where E1A is lowered by both strain energy and derealization energy. 1.7 PROPAGATION REACTION X + R H ^ XH + R (2)

1.7.1 Introduction Extensive information exists in the literature on the Arrhenius parameters for hydrogen abstraction from hydrocarbons and related com-

Propagation reaction X + RH -+ XH + R (2)

37

pounds by atoms and radicals [49]. Of the radicals mentioned in Table 1.2, relatively few are important in reaction (2) between 600 and 1200 K. Essentially only OH, H, CH 3 , CH 3 0 2 and H 0 2 are influential in removing RH in (2) or in determining reaction rates, ignition limits, flame speeds or ignition time delays. Many other species can, and do, react with RH, but their competing reactions, such as reaction with 0 2 , homolysis, isomerizations and sometimes radical-radical processes are considerably faster. For example, alkyl radicals in general (CH 3 and neopentyl are notable exceptions) react very rapidly with 0 2 to give alkenes between 600 and 1200 K. At the higher temperatures, alkyl radicals increasingly undergo homolysis. For example for 1-propyl radicals where [C3H8] = [0 2 ] = 1 atm the relative rates of reactions (30), 3p (in high pressure region) and (5Ap) are 1:7:6 x 104 at 600 K and 1:10 4 :10 2 at 1200 K.

(2) (5 Ap) (3p) (30)

X + RH^XH + R CH 3 CH 2 CH 2 + 0 2 -> C 3 H 6 + H 0 2 CH 3 CH 2 CH 2 -> CH 3 + C 2 H 4 CH 3 CH 2 CH 2 + C 3 H 8 -> C 3 H 8 + CH 3 CHCH 3

Alkoxy radicals react rapidly with 0 2 or undergo homolysis with relatively low activation energies. (31) (32)

CH 3 CH 2 0 -> CH 3 + HCHO CH 3 CH 2 0 + 0 2 -> CH 3 CHO + H 0 2

R 0 2 radicals (with carbon numbers above 3) can undergo relatively rapid isomerizations involving H atom transfers (discussed later) to form QOOH radicals which cyclize to give the hydroxy radical and oxiranes, oxetanes, tetrahydrofuranes and tetrahydropyranes. Very high yields (—50%) of these O-heterocyclic compounds are observed in the initial products when C5-C10 alkanes are oxidized at 600-800 K [17]. O atoms are only of importance above about 1000 K, along with species such as CH 2 , CH and radicals such as C2H which are derived from the high-temperature product C2H2.

38


Ch. 1

1.7.2 Kinetic data for X = OH, H, H02, CH3 and CH302 OH + RH abstractions are generally very exothermic, (typically 75 to lOOkJmol -1 for alkanes), with low activation energies, so that the reactions are fast even at room temperature and below. (OH radicals are the main remover of organic compounds under atmospheric conditions.) Consequently, although there are major sources of OH radicals, their concentration is extremely small compared with those of other radicals (typically a factor of 104 lower), and OH is only rarely involved in termination processes, particularly radical-radical reactions, unless the temperature exceeds about 1200 K where rates and hence radical concentrations are extremely high. As a result, with the exception of flame speeds, the OH + RH reaction is not a key process in determining combustion behaviour, so that in modelling studies, major changes in the rate constant have little effect on the computed characteristic. It is ironical, therefore, that OH + RH is probably the most well defined reaction kinetically of its type. % over For RH = CH 4 and C 2 H 6 , the rate constant is known within the range 200-2000 K. Baulch et al. [9,50] from a critical analysis of many investigations give £(CH4) = 2.6 x 10" 17 T 1 83 exp(-1400/T) and &(C2H6) = 1.2 x 10~17 T 2 0 exp(-435/T) cm3 molecule" 1 s" 1 as both reactions show marked non-Arrhenius effects. Although fewer in number, sufficient reliable data are available for other alkanes to set-up expressions of the type A:overaii = V * P Q~EP/RT + nsAs t~E-,RT + ntAt e~E^ where np, ns and nt refer to the number of primary, secondary and tertiary C—H bonds, respectively, in RH. This type of expression is of crucial importance in modelling as it permits the calculation of the rate of attack at specific sites in RH and consequently the proportion of each species of alkyl radical produced. For example, in the reaction OH + C 3 H 8 , both 1-C3H7 and 2C 2 H 7 radicals are formed. OH + C 3 H 8 -> H 2 0 + CH 3 CH 2 CH 2 -> H 2 0 + CH 3 CHCH 3 While 1-C3H7 can rapidly undergo C—C homolysis to give C 2 H 4 and CH 3 , the 2-C3H7 radical is limited structurally to C—H homolysis to form H atoms which can undergo the branching reaction (15), which is a key process in determining the rate of the oxidation.

39

Propagation reaction X + RH -> XH + R (2)

CH 3 CH 2 CH 2 -> CH 3 + C 2 H 4 CH 3 CHCH 3 ->C 3 H 6 + H (15)

H + 02ÔH + 0

The OH + RH data are considered so reliable that realistic attempts have been made to allow for the effect of near neighbour groups [51]. For example, for H abstraction from a CH 2 group in an alkane, it is possible to assign specific Arrhenius parameters for the following environments as well as others, based on k = ATz~E,RT A/cm3 molecule" 1 s " 1 ^ 1 4.8 x 10~20 4.8 x 10~20 4.8 x 10" 20

OH + CH 3 CH 2 CH 3 OH + CH 3 CH 2 CH 2 OH + —CH 2 CH 2 CH^

E/Jmol" 1 1080 500 -90

In a similar way, Atkinson [52-54] claims to have fitted rate constants to within a factor of 2 between 250 and 1000 K by consideration of CH3—, CH2—, and >CH— as groups attacked and functioning as near-neighbour groups. For OH + alkane, fc(CH2—X) = £PF(X) k(X— CH2—Y) = £SF(X)F(Y) k(X—CH(X)(Y) = JfctF(X)F(Y)F(Z) where kp, ks and kt are the rate constants per —CH 3 , —CH2—, and >CH— group for X = Y = Z = —CH 3 as the standard substituent group and F(X), F(Y) and F(Z) are the factors for X, Y and Z substituent groups. Atkinson gives the following kinetic data kp = 4.5 x 10" 18 T 2 e - 3 0 3 / T cm3 molecule" 1 s" 1 £s = 4.3 x 10" 18 T 2 e 233/T cm3 molecule -1 s" 1 A:t = 1.9 x 10~18 T 2 e 710/T cm3 molecule" 1 s" 1 with substituent factors of

40


Ch. 1

F(—CH3) = 1.00 F(—CH2—) = F(>CH—) = F(>C < ) = e 76/T Further factors are introduced for the reaction of OH with cycloalkanes [52, 53]. Several critical reviews of the OH + RH data are available for wide temperature ranges [54,55]. At high temperatures, (above ca 900 K) H atoms are formed readily through C—H homolysis of alkyl radicals, (4) (15)

R^R' + H H + 02ÔH + 0

and react rapidly with 0 2 in the branching process (15). Reaction (33) is an important source of H atoms in the later stages of alkane oxidation even at low temperatures when the CO concentration is high. However, at low temperatures, the high activation energy of (15) reduces k15 sufficiently that reactions (15M), particularly at high pressures, and (34) dominate. (33) (15M) (34)

OH + C O ^ C 0 2 + H H + 0 2 + M-* H02 + M H + RH^H2 + R

As H 0 2 is inactive under many conditions, (15M) is effectively a termination process. Cohen [12] has stressed the shortage of data for H + RH, but has used those available very efficiently to extrapolate the rate constants between 250 and 2000 K by use of transition state theory. A large proportion of the data used by Cohen for the larger alkanes was obtained by the Method I approach discussed earlier. He considers that the entropy of activation per attackable H atom in RH increases slightly from CH 4 to C 5 H 12 by about l O J K ^ m o l " 1 at 298 K and then stays virtually constant at 100-103 J K _ 1 m o l _ 1 for high alkanes. Under these conditions, the bond addivity concept can be used and Cohen gives a general expression for the reaction of H atoms with large (C > 4) alkanes, which allows for relatively pronounced non-Arrhenius behaviour.

Propagation reaction X + RH -* XH + R (2)

41

)t H+R H = 9.0 x 10- 1 8 n p T 2 0 exp(-3540/T) + 7.8 x 10~ 1 8 n s T 2 2 exp(-2640/T) + 6.1 x 10~ 18 n t T 20 exp(-970/T)cm 3 molecule"^" 1 . Kinetic data for H + CH 4 and H + C 2 H 6 have been critically analyzed [9,50] and Tsang has reviewed H + C 3 H 8 [56] and H + isobutane [57]. Kerr and Moss [49] list uncritically Arrhenius parameters for a wide range of alkanes and related compounds. Sensitivity analyzes with large comprehensive mechanisms have established the importance of H 0 2 / H 2 0 2 chemistry in determining auto-ignition behaviour between 800 and 1300 K. Modelling studies with methanol [58] propane [59], butane [13, 60] and methane [61] have shown that although alkanes are removed mainly through OH attack, the branching reaction (13) is the principal source of OH, particularly at pressures above 1 atm. (13)

H202 + M-*20H + M

Reactions (14) and (16) are the major sources of H 2 0 2 . (14)

H 0 2 + H 0 2 -» H 2 0 2 + 0 2

(16)

H 0 2 + RH -> H 2 0 2 + R

The ignition delays and induction periods arise simply because the system needs time to build up the H 2 0 2 concentration. Reaction (14) only becomes fast when [H0 2 ] has risen sufficiently and reaction (16) is relatively slow because the reaction is endothermic with high activation energies (65-100 kJmol - 1 ). Pitz and Westbrook's [13] butane oxidation model predicts that at pressures of about 20 atm at 1000 K, the ignition time delay is more sensitive to A:(H02 + butane) than to the value of any other rate constant (in a mechanism containing over 500 elementary reactions), a mere doubling of k(H02 + butane) lowering the ignition time delay by about 20-30%. When the pressure is reduced below 1 atm, the importance of k16 is reduced markedly because the rate of (13) falls, and the branching reaction (15) becomes the key rate determining process

42


Ch. 1

TABLE 1.12 Kinetic data for H 0 2 attack on alkanes Alkane

T/K

^4/cm3 molecule * s *

£/kJ mol

CH 4 C2H6 (CH3)3CC(CH3)3 c-C6H12 (CH3)2CHCH(CH3)2

716 653-773 673-773 673-793 753

-10.83 -10.65 -10.50 -10.55 k = 2.08 x

102.8 85.3 84.2 74.0

3 1 5 5 10" 16

1

2.5 6 5

Table 1.12 shows that very few data are available for H 0 2 + alkane, despite its key role in determining the rate of alkane oxidation between 700 and 1200 K. The reactions are too slow for "real-time" methods and classical techniques suffer from the presence of the highly reactive OH radical. For an alkane, k(OU + RH)/£(H0 2 + RH) = 105-106 at 750 K, so that RH is removed at the same rate by the radicals when [OH]/[H0 2 ] = 10 _5 -10~ 6 . Several studies have started as the investigation of H 0 2 kinetics and produced OH data [62]. All the alkane data in Table 1.12 were determined by Baldwin, Walker and co-workers by use of the decomposition of tetramethylbutane (TMB) in the presence of 0 2 as a source of H 0 2 between 650 and 800 K (Method III, outlined earlier). In each case the kinetics gave k16/k\f and k14 = 3.1 x 10" 12 exp(-775/T) cm3 molecule -1 s _ 1 was used [9] to calculate k16. Preliminary values of &(H0 2 + cyclohexane) = 2.0 x 1 0 " u exp(-8780/T) cm3 molecule -1 s - 1 have been obtained [63] recently, and a simple database is recommended in Table 1.13 for use between 600 and 1200 K. Between these temperatures, the rate constants should be reliable to a factor of about 2, and outside the range H 0 2 + RH reactions are unimportant. Nevertheless, confirmation of the data in Tables 1.12 and

TABLE 1.13 Generic rate data for H 0 2 + alkane Type of C—H

^4/cm3 molecule * s (per C—H)

Primary Secondary Tertiary

-11.5 -11.3 -11.5

x

£/kJmol 85 74 66

4 4 4

1

Propagation reaction X + R H ^ XH + R (2)

43

1.13 by use of a different technique and at higher temperatures is clearly desirable. As is clear from Table 1.3, R 0 2 + RH reactions are important only at low temperatures. No reliable experimental data are available for the reaction where RH = alkane. On the basis that D(R0 2 —H) = D(H0 2 —H), then E ( R 0 2 + RH) « E ( H 0 2 + RH), with A lowered by a factor of about 10 on entropy grounds. In a related way, Walker [11] has shown that A 35 /A 36 = 10, and gives A 36 = 2.5 x 10 - 1 3 cm3 molecule -1 s _ 1 and E 36 = 45 kJ mol - 1 . Comparison with the A factors in Tables 1.12 and 1.13 for H 0 2 + alkanes, supports the view that A ( R 0 2 + RH)/ A ( H 0 2 + RH) « 10. (35)

H 0 2 + C 2 H 5 CHO -> H 2 0 2 + C 2 H 5 CO

(36)

CH 3 0 2 + CH 3 CHO -> CH 3 0 2 H + CH 3 CO

CH 3 radicals are formed in abundance in alkane oxidation at all temperatures. At low temperatures, homolysis of alkoxy radicals with E = 7090kJmol _ 1 competes readily wth RO + 0 2 reactions, and at higher temperatures, C—C homolysis of alkyl radicals becomes important. CH 3 CH 2 0 -> CH 3 + HCHO CH 3 CH(0)CH 2 CH 3 -> CH 3 + C 2 H 5 CHO CH 3 CH 2 0 + 0 2 -* CH 3 CHO + H 0 2 CH 3 CH(0)CH 2 CH 3 + 0 2 ^ CH 3 C(=0)CH 2 CH 3 + H 0 2 CJri3C'H2CH2 —> CH 3 H~ C2H4 (CH 3 ) 2 CHCH 2 -^ CH 3 + C 3 H 6 Homolysis to give CH 3 radicals will occur with any alkyl radical where the CH 3 group is attached to a carbon atom next to that which contains a free valency and with any alkoxy radical of the general structure RiR 2 C(0)CH 3 where either or both Rx and R2 could be H atoms. In both cases, a double bond ( C = C or 0 = 0 ) is formed which provides the energy "pay-back" for a relatively low activation energy (see later). High yields of CH 4 formed through CH 3 + RH —> CH 4 + R are thus formed in many alkane oxidations. The reaction CH 3 + C 2 H 6 has been studied [8, 9] extensively between 500 and 1400 K, by use of a range of techniques. The rate data

44


Ch. 1

show a marked non-Arrhenius effect with k = 2.5 x 10 - 3 1 T 6 0 exp(-3043/T). Although the expression is considered very reliable (AA: = 1 at 300 K rising to 2 at 1500 K), no current theory of kinetics will 6 predict a T term in the rate expression. One possible explanation is that the competing reaction CH 3 + CH 3 —> C 2 H 6 was considered incorrectly to be in its pressure-independent region in the high-temperature studies, so that [CH3] was "modelled" below its true value with a resulting "high" value for k(CH3 + C 2 H 6 ). Tsang has reviewed the kinetic data for CH 3 + propane [56] and CH 3 + isobutane [57], and Kerr and Moss [49] list data for many alkanes and related compounds. Many of the data were obtained at low temperatures ( CH 3 CH=CHCH 3 + H 0 2 CH 3 CHCH 2 CH 3 -> CH 3 CH=CH 2 + CH 3

The initial rate of formation of butene-2 and C 3 H 6 is then given by equation (1.10) and careful measurement of the product ratio over the first 5% reaction gives a very precise value of d[C4H8-2]/d[C3H6]. 4C 4 H 8 -2] _ 4C 3 H 6 ]

fc5Ab[Q2]

( i m

k3b

The expression can be tested over a wide range of mixture composition, and with k3b known, a value for A:5Ab can be obtained. Data for many R + 0 2 reactions have been obtained in this way, as illustrated later, and reliable databases and rate constant-structure patterns established [11,16]. Table 1.14 gives the values of k3 obtained from the H 2 + 0 2 addition studies at 753 K. Although measurements at other temperatures were not possible, activation energies were calculated from estimates of the A facT A B L E 1.14 (a) Kinetic data for homolysis of branched alkyl radicals at 753 K Reaction

kls~l (753 K)

(CH 3 )3CC(CH 3 )2^ (CH 3 )2C=C(CH 3 )2 + CH 3 (CH 3 ) 3 CCH(CH 3 )CH 2 ^ (CH 3 ) 3 CCH=CH 2 + CH 3 (CH 3 ) 3 CCH(CH 3 )CH 2 ^ f-C4H9 + C 3 H 6 (CH 3 ) 2 C(CH 2 )CH(CH 3 ) 2 -> CH 2 =C(CH 3 )CH(CH 3 ) 2 + CH 3 (CH 3 ) 2 C(CH 2 )CH(CH 3 ) 2 -+ z-C3H7 + (CH 3 ) 2 C=CH 2 (CH 3 ) 3 CC(CH 3 ) 2 ^;-C 3 H 7 + (CH 3 ) 2 C=CH 2 (CH 3 ) 3 CC(CH 3 ) 2 CH 2 ^ f-C4H9 + (CH 3 ) 2 C=CH 2 (CH 3 ) 3 CC(CH 3 ) 2 CH 2 ^ (CH 3 ) 3 CC(CH 3 )=CH 2 + CH 3 C H 3 C H 2 C H C H 2 C H 3 ^ CH 3 CH 2 CH=CH 2 + CH 3 CH 3 CHCH 2 CH 2 CH 3 -* C 3 H 6 + C 2 H 5 ( C H 3 ) 3 C C H 2 ^ (CH 3 ) 2 C=CH 2 + CH 3

7.2 6.2 1.3 2.4 2.4 7.4 1.9 1.1 1.6 2.3 3.5

x x x x x x x x x x x

E/kJmor1 (AÎO13-8^0-^"1)

102 104 106 105 106 105 106 104 105 105 103

158 ±7 130 ±7 111 ±7 121 ±7 107 ±7 114 ±7 108 ±7 140 ±7 124 ±7 122 ±7 148 ±7

(b) Experimental kinetic data for lower alkyl radicals [68] £/kJmol-1

Reaction

A/s~x

«-C 3 H 7 -* CH 3 + C 2 H 4

1.6 x 1014

136

2

13

120 137 128

4 5 7

«-C 4 H 9 -* C 2 H 5 + C 2 H 4 5-C4H9 -* CH 3 + C 3 H 6 Z-C4H9 -^ CH 3 + C 3 H 6

2.5 x 10 2.3 x 1014 6.3 x 1013

The reaction of alkyl radicals with 0 2

47

tors. From transition-state theory, the A factor for a unimoleeular process is given by expression (1.11), so that A is effectively determined by the value of AS*, the entropy of activation. A = (ekT/h^xpiAS^R).

(1.11)

Based on Rabinovich and Setser's [66] considerations of the transition state, and Benson and O'Neal's [67] empirical methods for the calculation of AS*, Baldwin, Walker and Drewery [68] concluded that the values of AS* for alkyl homolysis were marginally above zero. This arises mainly from a loss of entropy (ca 20 J K _ 1 mol - 1 ) due to the stiffening of one free rotation in the transition state which is slightly more than compensated by gains due to the lowering of several vibrational frequencies. They took AS* = 4 J K" 1 mol" 1 , so that with ekTIh = 10 136 s _ 1 at 753 K, then A = 10 138 s _ 1 and activation energies were calculated with this value. Table 1.14 also gives Arrhenius parameters for the homolysis of lower alkyl radicals obtained experimentally. Within experimental error, all the A factors are consistent with the value of 10 1 3 ' 8 s _ 1 calculated above. Baldwin, Walker and Drewery [68] showed that both the experimental and the derived activation energies are reasonably consistent with the thermochemistry and the Arrhenius parameters of the reverse reactions. In particular, they showed that for the homolysis of trimethylbutyl radicals there is a very good correlation between logk at 753 K and AU, the internal energy change (Fig. 1.9). Although the data are limited, for the other radicals the rate constant and Arrhenius parameters for homolysis fit a common pattern when allowance is made for the strain energy in all the species involved. Hence, for the homolysis. (CH 3 ) 3 CC(CH 3 ) 2 ^ (CH 3 ) 2 C=C(CH 3 ) 2 + CH 3 the strain in the alkyl radical has been increased in forming the alkene, so that a low rate constant is expected (7.2 x 1 0 2 s - 1 at 753 K), and the reverse reaction should be "normal". However for the homolysis (CH 3 ) 3 CC(CH 3 ) 2 CH 2 -> t-C4U9 + (CH 3 ) 2 C=CH 2 the strain in the radical has been reduced significantly, so that a much higher rate constant (1.9 x 106 s _1 ) is expected together with a very high activation energy for the reverse reaction.

48


Ch. 1

'c/J

50

70

90

110

1

AU / kJmor

Fig. 1.9. Plot of log™ k against AU for the homolysis of trimethylbutyl radicals at 753 K. 1.9 THE REACTION OF ALKYL RADICALS WITH 0 2

Conjugate alkenes are the major initial products in the reaction of lower alkyl radicals [15-17] with 0 2 (where structurally possible) above 550 K (5 A)

R + 02-*AB + H02

As the alkyl chain lengthens, increasing amounts of oxygenated product, particularly O-heterocyclics, are observed in the initial products. At 750 K, the percentage of alkene decreases from about 99% (C 2 H 5 , 2-C3H7, tC4H9) to 85% (I-C3H7), 65% (1-C4H9, 2-C4H9) down to less than 50% for C 5 and C6 alkyls [16,17] with accompanying increases in homolysis products and O-heterocyclics. Figure 1.10 shows the product yields when pentane is added to H 2 + 0 2 mixtures at 753 K (Method I). Structural factors are important, for example, 2-C3H7 and f-C4H9 both give 99% alkene because alternative reactions are very limited. Neopentyl radicals with no H atom on the carbon atom adjacent to the carbon with the free valency are unable to form conjugate alkene. The formation of products through oxidation of alkyl radicals is potentially very complex because at least three routes are available, the direct bimolecular process


20

49

40

% pentane consumed Fig. 1.10. Product yields for a pentane + 0 2 + H 2 mixture at 753 K. C 5 Hi 2 = 5, 0 2 = 70, H 2 = 140, N 2 = 285Torr. (A) O, trans-pentene-2; A, pentene-1; # , 2-methyltetrahydrofurane; V, 2,4-dimethyloxetane; x, cis-pentene-2; + , tetrahydropyrane; (B) O, butane; x, 2-methyl,3-ethyloxirane; D, 2-ethyloxetane; (C) V, C 3 H 6 ; • , CH 4 ; x , C 2 H 4 ; O, CO.

(5A), through R 0 2 decomposition (7) or via reaction (10) of QOOH radicals, formed through H atom transfer in R 0 2 radicals

R + 02 (5A)\

(6)

R0 2 :M - QOOH

1(7) products

/(10)

50


Ch. 1

Baldwin and Walker [14] showed that very complex expressions may rise for the overall rate constant for product formation if all routes contribute, particularly if (6) and (9) are both equilibrated, and R 0 2 can form a number of QOOH radicals. For example, 2-hexylperoxy radicals can undergo six totally different internal H atom transfers to form six QOOH radicals. Mp 1,3t

CH3CHCH2CH2CH2CH3 6-0 2 1

) CH2CH(OOH) CH2CH2CH2CH3 ) CH3C(OOH) CH2CH2CH2CH3

1,4s

) CH3CH(OOH) CHCH2CH2CH3

1,5s

> CH3CH(OOH) CH2CHCH2CH3 > CH3CH(OOH) CH2CH2CHCH3

1,6s 1,7p

> CH3CH(OOH) CH2CH2CH2CH2

The symbols p, s and t refer to the nature, primary, secondary and tertiary, respectively, of the H atom undergoing transfer. Decomposition of the QOOH radical involves homolysis of the weak O—O bond to give a heterocyclic compound (OR) and OH (IOC)

QOOH -> OR + OH

Table 1.15 lists the nature of the O-heterocycle associated with each type of transfer. Baldwin, Walker and co-workers [69] have obtained a number of values of /c5A for C2—C5 alkyl radicals at about 750 K, by use of the H 2 + 0 2 addition studies (Method I). A good correlation between logk5A and the TABLE 1.15 Formation of O-heterocyclic compounds Transfer

Compound

1.3 1.4 1.5 1.6 1.7

Ketone (aldehyde) Oxirane Oxetane tetrahydrofurane tetrahydropyrane

51


1 1

"a

< o

0

50

1

AH/kJ mol"

Fig. 1.11. Plot of log k10 against IIT for the generic reaction R + 0 2 —» alkene + H 0 2 (5A).

enthalpy of reaction is observed as shown in Fig. 1.11 and Table 1.16. The agreement even extends to the analogous reaction of pentadienyl and allyl radicals (AH = +40kJmol~ 1 ). Studies of the separate oxidations of C 2 H 5 CHO [29] and *-C3H7CHO [28] gave small negative activation energies (-6.3 3.5) and (-9.0 5.0) kJ mol - 1 , respectively) over the temperature range 590-750 K for reactions (5Ae) and (5Aip). (5Ae) (5Aip)

C 2 H 5 + 0 2 -> C 2 H 4 + H 0 2 z-C3H7 + 0 2 -> C 3 H 6 + H 0 2

The small negative activation energies are consistent with Gutman's [21] conclusion that A:5Ae is almost temperature independent between 700 and 900 K. Walker [11] has argued that reaction (5A) may have a small positive activation energy above about 900 K, so that k5A is virtually independent of temperature between 700 and 1000 K. A value of £ 5 A = 0 is then recommended between 700 and 1000 K for all the alkyl + 0 2 reactions listed in Table 1.16. The mechanism for reaction (5A) has been the subject of considerable

wl

N

TABLE 1.16 Variation of rate constant for R

R*

+ 02+alkeiie + H 0 2 with AH at 753 K

Alkene

k/cm3 molecule-' s-l

WC-H bond (cm3 molecule-' s-l)

A H / ~ mol-' J

1.10 x 10-l3 2.0s x 10-l~ 8.5 x 10-l~

3.7 x 10-l~ 3.5 x 10-l~ 2.8 x 10-l~

-46 -40 -38

1.1s x 10-l~

-54

2.7 x 10-l~ 1.30 x 10-l~ 7.2 x 10-l~

5.9 x 10-l~ 1.35 x 10-l~ 6.5 x 10-l4 3.6 x 10-l~

-49 1 -45

2.9 x 10-'3

7.3 x 10-l~

-55

1.13 x 10-l~ 1.40 x 10-l~

1.13 x 10-l~ 1.40 x lo-"

-62 -64

3.5 x 10-l~ 4.2 x 10-l~

1.75 x 10-l~ 4.2 x 10-l~

-18

-56

+40

0

Er

(D

E!

L' Y

s

53


controversy. Walker et al. [11,23,70] have provided strong evidence that conjugate alkene from R + 0 2 in the range 600-1000 K does not arise through QOOH radicals, but are unable to distinguish between a direct reaction (6A) and reaction (7) because under the conditions used R and R 0 2 are fully equilibrated in (6). (6)

R + 02

(7)

2

R 0 2 ^ A B +H02 direct

(5 A)

R + 02

> AB + H 0 2

1.9.1 Reaction between C2H5 and 02 The reaction of C 2 H 5 radicals with 0 2 has received particular attention over the last 10 years, has been reviewed recently, [11] and will be discussed further in Chapter 2. Slagle, Feng and Gutman [21] in a very elegant study produced C 2 H 5 radicals photochemically in the presence of 0 2 and monitored their decay in "real time" mass spectrometrically. Between 300 and 1000 K, the overall rate constant decreased monotonically, below 500 K the rate constant increased with pressure between 1 and 15 Torr, and above this temperature C 2 H 4 was the only product observed. Later work by Gutman et al. [71] confirmed the results quantitatively and particularly that k5Ae was effectively independent of temperature between 700 and 1000 K. Gutman and co-workers argued strongly against any contribution from a direct molecular abstraction reaction (5Ae) and proposed a coupled mechanism which accounted for the pressure effect, negative temperature coefficient below 500 K and the unique formation of C 2 H 4 above this temperature.

C2H5 + 0 2 =

C2H502*

— C2H4

+ H02

M C2H5O2 At low temperatures and high pressures, C 2 H 5 0* radicals are mainly

54


Ch. 1

Path Fig. 1.12. Potential energy diagram for C2H5 + 0 2 - > C2H4 + H 0 2 , —- Gutman et al. [21, 71], Walker et al. [11, 70].

stabilized by collision with M to give the final product (in real time) C 2 H 5 0 2 . As the pressure is lowered (15 to lTorr in Gutman's experiments), the C 2 H 5 0 2 radicals increasingly reform C 2 H 5 + 0 2 and the overall rate constant falls. The energy of C 2 H 5 0* increases with temperature, so that for a fixed pressure, the overall rate constant falls, and C 2 H 4 becomes the dominant product. A vital pre-requisite for the mechanism is that there are no positive energy barriers in the sequence from C 2H5 + 0 2 to C 2 H 4 + H 0 2 , so that the whole process is achieved through chemical activation of C 2 H 5 0 2 radicals. The potential energy surface suggested by Gutman et al. is shown in Fig. 1.12, and the energies of both transition states TS1 and TS2 are below the enthalpy of formation of C 2 H 5 radicals. Walker and co-workers have disputed the height of the two barriers and suggest that the full line in Fig. 1.12 is more consistent with other evidence. As indicated later (Table 1.17), the activation energy for the H atom transfer in C 2 H 5 0 2 is reported as 153 kJ moF 1 . C H 3 C H 2 0 2 ^ CH 2 CH 2 OOH Further, studies [Method III] of the addition of C 2 H 4 and other alkenes to tetramethylbutane + 0 2 mixtures over the range 650-800 K have given Arrhenius parameters for the overall reaction,

Reactions of R 0 2 radicals

55

which (with C 2 H 4 as the example) proceeds through the sequence. H 0 2 + C 2 H 4 -> C 2 H 4 OOH -> C 2 H 4 0 + OH There is very strong evidence to support the view that the overall parameters A = 6.3 x 10 - 1 2 cm3 molecule -1 s - 1 and E = 75 kJ mol - 1 refer to the first step, the formation of the adduct C 2 H 4 OOH. If so, the height of TS2 in Fig. 1.12 is considerably higher than suggested by Gutman et al., and the "chemical activation" route through QOOH cannot be the mechanism. Critical evidence in support of this view was obtained by Stothard and Walker [70] through studies on the addition of H 0 2 radicals to trans-butene-2. They showed that the initial product ratio [2,3,dimethyloxirane]/[cis-butene-2] was at least 5 between 700 and 750 K, implying that alkene formation from QOOH is not important and that QOOH radicals predominantly decompose to give oxiranes. As Walker [11] has pointed out, there are many other problems associated with the Gutman mechanism, and it is very unfortunate that so much of the experimental work has been focused on the C 2 H 5 + 0 2 reaction. A number of modelling studies by Dean and co-workers [72, 73] have been interpreted as support for the Gutman mechanism, but they are usually based on uncertain thermochemical data. Walker [11] has concluded that the answer may lie in the assumption that only one potential energy surface is involved. Two recent theoretical treatments for C 2 H 5 + 0 2 [74] and CH 3 + 0 2 [75] indicate the need to consider two potential energy surfaces (see Chapter 2).

1.10 REACTIONS OF R 0 2 RADICALS

Reaction (8) has been discussed earlier. At low temperatures, ca. 600 K, it is the main source of the secondary initiation product ROOH which on homolysis gives the alkoxy and OH radicals, both of which are very reactive. The reaction is relatively slow (see earlier) and the build up to maximum concentration may take considerable time so that lengthy indue-

56


Ch. 1

tion periods are observed. As the temperature is raised, two reactions with higher activation energies become very important, namely the decomposition of R 0 2 into R + 0 2 and the isomerization by H atom transfer of R 0 2 into QOOH radicals which then undergo homolysis at the weak O—O bond to give O-heterocyclic compounds and OH radicals. (6) (8)

R + 02 ^ R02 R 0 2 + RH -* R 0 2 H + R R02-*QOOH

(9) (IOC)

QOOH ^ OR + OH

As E-6 and E9 do not differ greatly, the relative importance of (-6) and (9) is not markedly influenced by temperature, and the relative yields of O-heterocyclic compounds (from QOOH homolysis) and conjugate alkenes are similarly little affected between 600 and 800 K. This is particularly noticeable [17] with the larger alkanes where O-heterocyclic compounds and conjugate alkenes are formed in similar quantities. Fish [76] was the first to develop the extensive peroxy radical isomerization and decomposition (PRID) theory which was used to explain the many oxygenated compounds in alkane oxidation. Fish also considered the possibility of group transfer in R 0 2 radicals, for example R—CH 2 CH 2 CHR' -> CH 2 CH 2 CHR'

I O—O

I ROO

Energetically less favourable, no real evidence has been found for its occurrence. Usually it has been introduced into mechanisms to explain the formation of lower oxygenated products, but it is far more likely that they are formed in secondary processes from the primary products. Fish made estimates of the rate constants for the various types of isomerization by equating the activation energy to the sum of the strain energy in the ring transition state and the normal activation energy of the bimolecular R 0 2 + RH reaction. Ring transition state energies were taken as those for cyclic alkanes, first suggested by Benson [77], namely 42, 27, 3, 27, 109 and 117kJmol _ 1 for 8, 7, 6, 5, 4 and 3-membered rings,

57

Reactions of R0 2 radicals

respectively. Fish suggested activation energies of 32, 44 and 59kJmol~ 1 for bimolecular R 0 2 attack at tertiary, secondary and primary C—H bonds, respectively. He argued that the entropy of activation AS* = —62JK~ 1 mol~ 1 is effectively unchanged for all ring transition states involved in intramolecular H atom transfer reactions, so that A ~ 1011 s _ 1 . These data give Jfc(l, 4s) = 1011 exp(-8540/T) s" 1 and fc(l, 7p) = 1011 exp(-12150/T) s _ 1 for the two transfers involving 2-hexylperoxy radicals. Later it will be shown that these calculated parameters are seriously in error. CH3CHCH2CH2CH2CH3 -}&-+ CH3CH-CHCH2CH2CH3

I

I

O-O

O-OH

I*

CH3CHCH2CH2CH2CH2

I

OOH The internal H atom transfer (9) is extremely important in the chemistry of engine efficiency and knock, as illustrated later, and in particular the competition between reactions (IOC) and (37) is the key to the amount of branching that occurs in low-temperature ignition. (IOC)

QOOH -* OR + OH QOOH + 0 2 -» branching

(37)

Baldwin and Walker have obtained quantitative experimental data for reaction (9), the kinetically significant reaction (along with reaction (6)) in the formation of oxiranes, oxetanes, tetrahydrofuranes and tetrahydropyranes in the generalized sequence. (6)

R + 02 ^ =

(IOC)

(9)

R02

> QOOH

• OR + OH

With R and R 0 2 fully equilibrated in the temperature region 600-800 K, and considerable evidence that k10c > k_9 [23,78], the rate of formation of O-heterocycle is given by equation (1.12).

58


40-heterocycle] at

=

^

^

=

^

^

Ch. 1 { i n )

By use of the H 2 + 0 2 addition approach (Method I), and measurement of products from competing homolyses and oxidation reactions, kohs has been measured for most types of common H atom transfers in R 0 2 . K6 was of necessity at the time calculated from Benson's additivity rules, so that k9 could be determined [23, 78, 79]. The system most carefully examined by Baldwin and Walker [78, 80, 81], and more recently by Pilling et al. [82], is the chemistry associated with neopentylperoxy radicals, which leads to the formation of isobutene, acetone and 3,3-dimethyloxetane (DMO) as the major initial products. The data were obtained from a study of the addition of neopentane to slowly reacting mixtures of H 2 + 0 2 between 673 and 773 K. Neopentane is a particularly attractive alkane for such a study because structurally only one alkyl radical is produced in the initial attack, and secondly reactions leading to oxygenated products are magnified in importance because the "normal" conjugate alkene-forming reaction 5A is not available to the neopentyl radical. Very importantly, therefore, a simple, clean system is available for quantitative interpretation. Two key observations concerning the variation of initial rates of product formation with mixture composition (the 0 2 concentration was varied by a factor of 50) are summarized in equations (1.13) and (1.14).

d([DMO + [CH3COCH3]) = ka[02] d[i-C4H8]

(1.13)

4CH3COCH3] = kh[02]. d[DMO]

(1.14)

These relationships provided conclusive evidence for the following mechanism, where (38) represents an overall reaction:

59

Reactions of R0 2 radicals

[i-Butene] / TonFig. 1.13. Initial product yields from neopentane +H 2 + 0 2 mixtures at 753 K. H 2 = 140, N2 variable to total pressure of 500Torr. V, 0 2 = 35; • , 0 2 = 70; • , 0 2 = 140; x, 0 2 = 210, A, 0 2 = 280; O, 0 2 = 355Torr.

CH2 (6n)

(CH 3 )3CCH 2 + O2 ~ ^ = - (CH 3 )3CCH 2 02 - ^ U On)

(CH 3 )2CCH 2 OOH

C 38 )/^) (lOCn)

(CH3)2C=CH2 +CH3

( C H 3 ) 2 C = 0 + HCHO + OH (CH 3 ) 2 C

I

CH 2

I

+ OH

CH2-0 As an example of this type of work, Fig. 1.13 indicates the care with which the initial values of d[(CH 3 COCH 3 ) + (DMO)]W[/-C4H8] were obtained for a range of the mixtures used at 753 K. Typically the ratio was determined over the first 5-10% consumption. As can be seen, the change in the ratio is significant and arises from a complex mechanism often involving secondary formation of products and widely differing rates of removal of the intermediate products. Here, for example, acetone is

60


Ch. 1

relatively stable, but DMO and isobutene are removed quite quickly. Extrapolation to zero consumption gives reliable initial values. However, in the past, many studies involved sampling as late as 30% consumption where the product ratios might be factor of 3 or 4 different from the initial values. Under these circumstances, validation of mechanism is frequently difficult and the determination of reliable rate constants and particularly Arrhenius parameters impossible. From the mechanism, ka = K6n k9Jk3n and kh = k38/k10cn. Using a literature value for k3n and k6n calculated from Benson's [83] bond additivity data, k9n = 1.20 x 1013 exp(-14430/T) s _ 1 was obtained for the l,5p transfer involved. Table 1.17 lists kinetic data and Arrhenius parameters for other H atom transfers in R 0 2 radicals obtained in essentially the same way. The consistency of the values is compelling in relation to the variation of the A factors with ring size and of the activation energies with ring strain and type of H atom transferred (i.e. primary, secondary or tertiary). Pilling [82] has enhanced the investigations by direct measurement of [OH] and most importantly confirmed the accuracy of ka for neopentylperoxy radicals. Further his technique permits the direct determination of K6n and his results show that it is a factor of about 10 greater than the value calculated from Benson's additivity rules, so that the Baldwin and Walker values of k9n are reduced by the same factor. Pilling's direct determination of k9n is of paramount importance because of the large set of self-consistent data obtained by Walker and co-workers [78] for a whole series of 1,4, 1,5, 1,6 and 1,7 H atom transfers in R 0 2 radicals, all based on calculated values of K6 for R + 0 2 ^ R 0 2 . In effect, Pilling's value for k9n for the l,5p transfer in neopentylperoxy radicals calibrates and validates the complete set of data which have been used to model many practical combustion problems. The "new" set of values are shown in Table 1.17. Essentially at about 750 K, the new rate constants are a factor of 10 lower than the "old" values.

1.11 REACTIONS OF QOOH AND 0 2 QOOH RADICALS

Although still somewhat equivocal, there is strong evidence to suggest that reaction (9) [16,23], R02—> QOOH, is effectively irreversible in the region 600-1000 K. Values of rate constants for the homolysis of QOOH radicals remain rare. Baldwin, Hisham and Walker [78] from the

TABLE 1.17 Arrhenius parameters* for H-atom transfers in R 0 2 radicals [a]

US-'

Reaction RO2

QOOH

(753 K)

Ah-' EkJmol-' (per C-H) (original)

Type

Ah-' E M mol-' (per C-H) (modified [82])

CH3CH2CH(00H)CHz CHzCH200H

2.2 x 10' 1.95 x I d

8 x 10'2 8 x 10'2

144.8 145.5

1,4p 1,4p

1.4 X 10"

(CH3)2C(CH200H)CHZ CHjCH(CHzOOH)CH2 CHZCH~CH~CHZOOH CH;LCHzCHzCHzCH200H

7.5 x 104 4.5 x lo4

1.0 x 10'2 1.0 x 10'2 1.25 X 10" 1.55 X 10"

116.7 117.2 98.0 82.1

1sp

1.75 X 10"

1,6p 197~

2.2 x 10'0 2.75 x lo9

8 x 10'2 8 x 10'2

127.2 123.0

(1,3s) 1,4s

1.0 x 10'2 1.0 x 1oI2 1.25 x 10"

100.9 98.4 80.4

1,5s

6.0 x lo4 9.3 x lo4

2.4 x lo4 CHjCH(0OH)CHCHs CH~CH(OOH)CHCHZCH~ 4.7 x lo4 CHjCHCHzCHzOOH CHsCH(OOH)CH2CHCH3 CHjCHCHpCHzCH200H

2.0 x lo5 3.0 x lo5 6.6 X Id

1,6s (197s) CHjCH(OO)CH2CH3 1.5 x lo3 6.4 x 10'3 153.3 1,3t (CH3)3CHCH2OO 1.8 x 105 8 x 10'2 110.2 1,4t (1,5t) (1,6t) (1,7t) *Values in parenthesis are calculated on the basis that the differences in E for the transfer of a particular differences in strain energy.

(1.15 x 1 0 ~ ~ ) 1.4 X 10" 1.75 X 10"

2.2 x 10'0 (2.75 x lo9) 1.15 x 1013 1.4 X 10l2 (1.75 X 10") (2.2 X 10'9 (75) (2.75 X lo9) (62) type of H atom are due to

62

Ch. 1


H 2 + 0 2 + neopentane addition studies give A10Cn —

1.5 x 1 0 n s _ 1 and £iocn = 69 5 kJ mol for the formation of 3:3 dimethyloxetane (DMO) from neopentylhydroperoxide radicals. They are similar to Mill's [84] experimentally determined values of A = 1011'5 s _ 1 and E = 59 8 k J m o l - 1 for the liquid phase decomposition of the 2,4-dimethylpentylhydroperoxide radical into 2,2,4,4-tetramethyloxetane. -1

(CH 3 )2C(OOH)CH 2 C(CH 3 ) 2 -* (CH3)2C

O

I I

+OH

CH 2 -C(CH 3 ) 2 No other experimental data exist for any QOOH homolysis. Benson estimated A ~ 1011 s _ 1 and E = 63 kJ mol - 1 for the decomposition of QOOH (1,5 formation) radicals into oxetanes. As solvent effects are unlikely to influence the liquid-phase homolysis, the kinetic parameters for the formation of oxetanes are taken to be consistent and relatively reliable. Fish [76] suggested A = 1011 s" 1 and E = 59, 12.5 and 0 kJ m o P 1 for the decomposition of QOOH radicals into oxiranes, tetrahydrofuranes, and tetrahydropyranes, respectively, on the basis of the strain energy in the ring transition state. If this is so, the A factors should decrease by a factor of about 10 as the ring size is increased by each extra atom because a further rotation is "frozen". Kojima [85] has also suggested a fall in the activation energies (with A constant at 1.3 x 10 1 0 s - 1 ) from 66 to 63 to 52 kJ mol - 1 for the formation of the oxirane, oxetane and tetrahydrofurane from the three conjugate butylhydroperoxide radicals. CH2(OOH)CHCH2CH3

/°\

> CHr—CHCH2CH3

CH2(OOH)CH2CHCH3 CH 2 (OOH)CH 2 CH 2 CH 2

• CH2—CH 2 —CH 2 —CH 2

Baldwin et al. [78] also found clear evidence for O—O and C—C homolysis in the neopentylhydroperoxide radical and gave A39 = 3.5 x lO^s" -1 and £39 = H 3 k J m o r 1 .

Reactions of QOOH and 02QOOH radicals

63

TABLE 1.18 Variation of k10CJk39 with temperature [84] kiocJk39

3.2

600 800

0.86 0.36

1000 1200

29.2

(39)

T/K

(CH 3 ) 2 C(CH 2 OOH)CH 2 -> (CH3)2C—CH2 + HCHO + OH

Table 1.18 shows the calculated values of k^cJ^g at temperatures between 600 K and 1200 K. Clearly formation of the oxetane, DMO, predominates below 800 K, and although k10Cn/k39 is well below unity at 1200 K, R 0 2 and QOOH chemistry is not important at this temperature as the equilibrium R + 0 2 2 greatly favours R radicals. Cyclization and C—C homolysis are also apparent in the chemistry of CH 3 CH(OOH)CH 2 CH 2 radicals (lOCb)

CH3CH(OOH)CH2CH2

> CH 3 CH-CH 3 CH 2 + OH

(40)

CH3CH(OOH)CH2CH2

• CH3CHO + C 2 H 4 + OH

as shown by studies [23] of the addition of butane to H 2 + 0 2 mixtures at 753 K, where A;10cb/^4o ~ 5. Cyclization of QOOH radicals to give oxiranes is even more relatively favourable with (likely) A = 10 1 2 s _ 1 and £ — 75kJmol _ 1 and (by inspection) less energetically favourable C—C homolysis reactions. Although Fish's estimates of the activation energies for cyclization to tetrahydrofuranes and tetrahydropyranes are too small, values significantly lower than 69kJmol _ 1 (for oxetanes) are likely, so that formation of the O-heterocycles will be the major path. The addition of 0 2 to QOOH is clearly important, as shown by the high yields of acetone found in the neopentane + H 2 + 0 2 studies [80, 81]. The mechanism involves the two stage sequence (41) (CH 3 ) 2 C(CH 2 OOH)CH 2 + 0 2 ^ (CH 3 ) 2 C(CH 2 OOH)CH 2 0 2 (42)

(CH 3 ) 2 C(CH 2 OOH)CH 2 0 2 -> CH 3 COCH 3 + 2HCHO + 0 2

64


Ch. 1

where (41) is equilibrated. By assuming K41 = K6n, Baldwin et al. [78] report A42 = 1.7 x 1011 s" 1 and E42 = 99 kJ mol" 1 . For a 1,5s H atom transfer in the (CH 3 ) 2 C(CH200H)CH 2 00 radicals, A43 = 3.5 x K ^ s " 1 (Table 1.17) and E43 = 110 kJ mol" 1 , and k42/k43 = 4.5 at 600 K falling to 1.8 at 1000 K. Although, on the basis of the figures used, the rate of formation of the dihydroperoxide radical is lower than the rate of homolysis to give acetone, even small amounts of the dihydroperoxide radical can strongly influence the overall rate of oxidation through a radical initiation sequence such as (44)-(46). (43)

(CH 3 ) 2 C(CH 2 OOH)CH 2 00 -> (CH 3 ) 2 C(CHOOH)CH 2 OOH

(44)

(CH 3 ) 2 C(CHOOH)CH 2 OOH -> (CH 3 ) 2 C(CHO)CH 2 OOH + OH

(45)

(CH 3 ) 2 C(CHO)CH 2 OOH -> (CH 3 ) 2 C(CHO)CH 2 0 + OH

(46)

(CH 3 ) 2 C(CHO)CH 2 0 -* (CH 3 ) 2 CCHO + HCHO

1.12 OXIDATION CHEMISTRY OF CH3 RADICALS

CH 3 is a unique alkyl radical, first because it is present in virtually all alkane and alkene oxidations, particularly at high temperatures, and secondly because its range of reactions is very limited. Above 700 K, the main source of CH 3 radicals is through homolysis of alkyl radicals (3), for example (3p). (3p)

CH 3 CH 2 CH 2 -* CH 3 + C 2 H 4

At low temperatures, the CH 3 4- 0 2 ^ CH 3 0 2 equilibrium favours CH 3 0 2 and this radical is consumed mainly in radical-radical processes such as (47), (48) and (49), although (50) is very important because above about 400 K it can lead to secondary initiation through the formation and decomposition of CH 3 OOH (51), which can be influential up to 800 K. (47a) (47b) (48)

CH 3 0 2 + CH 3 0 2 -> 2CH 3 0 + 0 2 -> HCHO +• CH 3 OH + 0 2 CH 3 0 2 + H 0 2 -> CH 3 OOH + 0 2

Oxidation chemistry of CH3 radicals

(49) (50) (51)

65

CH 3 0 2 + CH 3 -» 2CH 3 0 CH 3 0 2 + RH -> CH3OOH + R CH3OOH + M -> CH3O + OH + M

Lesclaux et al. [86] used flash-photolysis u.v. absorption to follow reactions (47) and (48) between 373 and 573 K. In particular, the presence of polar gases did not affect the derived rate constants and no pressure effect was apparent in contrast to behaviour observed for the H 0 2 + H 0 2 reaction at these temperatures. They reported (k47a + k47h) = 1.3 x 10 - 1 3 exp(365/T) cm3 molecule -1 s _ 1 with k47a/k47h = 45 exp(-1465/T), so that k47a dominates above about 500 K. They also reported k48 = 4.3 x 10" 13 exp(780/T) cm3 molecule -1 s - 1 . Keiffer, Pilling and Smith [87] have made excellent direct measurements of the rate constant for CH 3 + 0 2 + M—>CH 3 0 2 + M, and most importantly have provided information for the calculation of the rate constant in the fall-off region. The unique behaviour of CH 3 radicals is brought out starkly by comparison of the products of the separate oxidations of propanal and ethanal at 813 K using a mixture containing 2, 30, and 28Torr of aldehyde, 0 2 and N 2 , respectively [41, 46]. Mechanisms for the initial stages of reaction may be written as follows. C 2 H 5 CHO + 0

2

^ C 2 H 5 CO + H 0 2

C 2 H 5 CO + M -> C 2 H 5 + CO + M (5 Ae)

C 2 H 5 + 0 2 -> C 2 H 4 + H 0 2 H 0 2 + C 2 H 5 CHO -* H 2 0 2 + C 2 H 5 CO H 0 2 + H 0 2 -> H 2 0 2 + 0 2 CH3CHO + 0 2 -> CH3CO + H 0 2 CH3CO + M ^ CH 3 + CO + M CH 3 + CH3CHO -> CH 3 CO + CH 4 CH 3 + 0 2 -> oxidation products (CH 3 OH + HCHO) CH 3 + CH 3 + M ^ C 2 H 6 + M

66


Ch. 1

Whereas the product ratio [C2H4]/[C2H6] ~ 600 from C2H5 radicals, [oxidation products]/[CH4] — 0.2 from CH 3 radicals, and moreover C 2 H 6 was formed in yields of about 10%. Despite repeated efforts, no trace of butane could be found in the products from propanal oxidation. The underlying features are worth stressing. C 2H5 radicals have a very facile reaction with 0 2 (/c5Ae ~ 1 x 10 - 1 3 cm3 molecule -1 s _ 1 ), so that H atom abstraction (E~ 50-70 kJmol - 1 ) is not an attractive alternative, and moreover [C2H5] is extremely low so that radical-radical reactions are of negligible importance. With CH 3 radicals, the rate constant for any bimolecular oxidation reaction is at least a factor of 103 lower, so that H abstraction is competitive and [CH3] a factor of ca. 103 higher, resulting in the conditions for radical-radical reactions of major importance. As a consquence, whereas in general CH 4 and C 2 H 6 are important initial products in alkane oxidation between 700 and 1000 K, other alkanes are only produced in negligible amounts. The high temperature CH 3 + 0 2 reaction has constantly attracted controversy [16,89]. Reaction (52) has in recent years been considered the major process above 1000 K. However, Imamura et al. [8,89] reported that (53) dominates below 2500 K because of its low activation energy (52) (53)

CH 3 + 0 2 -> CH 3 0 + O -> HCHO + OH

They obtained, by use of O and OH resonance absorption spectroscopy, k53 = 5.3 x 10" 13 exp(-4530/T) compared with k52 = 3.7 x 10" 10 exp(-16960/T). Ewig et al. [88] have shown that the latter parameters are consistent with a QOOH-type mechanism for formaldehyde formation. CH 3 + 0 2 -> CH 3 0 2 -> CH 2 OOH -> HCHO + OH At high temperature, CH 3 + radical reactions become more prevalent, but existing kinetic descriptions are often inadequate. k54 = 3.0 x 10 - 1 1 cm3 molecule -1 s" 1 between 300 and 2500 K has been recommended [9], 7 (see Chapter 3). No measurebut with an uncertainty of A log k54 = ment of k_lm has been made, although under some circumstances, the reaction is a key termination process. k55 has a recommended [9] value of 6 x 10" 11 between 300 and 2500 K, but it is uncertain to a factor of 3. Moreover, the relative importance of the products is unknown, which is a serious problem for modellers. In contrast, k56 = 1.4 x 10" 10 cm3 mole-

Reactions of alkoxy radicals

67

cule - 1 s _ 1 between 300 and 2500 K is probably known to a factor of better than 1.5. (54)

CH 3 + H 0 2 -* CH 3 0 + OH

(-lm)

CH 3 + H 0 2 -> CH 4 + 0 2 CH 3 + O H - * H + CH 2 OH

(55)

-> H + CH3O -> H 2 0 + 1 CH 2

(56)

O + CH 3 -> HCHO + H

1.13 REACTIONS OF ALKOXY RADICALS

As alkoxy radicals are only formed at low temperatures from R 0 2 + R 0 2 reactions and the decomposition of peroxides, they are only of general importance below 650 K R02 + R02->2RO + 0 2 R O O H - + R O + OH ROOR' -> RO + R'O As is well known, they play a key role in atmospheric chemistry through sequences involving NO. OH + R H - > H 2 0 + R R + 0 2 -> R 0 2 R02 + N O ^ R O + N02 RO radicals undergo only two reactions: C—C homolysis; reaction with 0 2 to give an aldehyde or ketone and an H 0 2 radical; isomerization to give a hydroxyalkyl radical when the ring transition state involves little strain energy. These possibilities are illustrated below for 2-pentoxy radicals.

68


Ch. 1

T A B L E 1.19 Kinetic parameters for alkoxy radicals [50, 54,95] Reaction C H 3 0 + M ^ HCHO + H + M C 2 H 5 0 - + C H 3 + HCHO C 6 H 5 0 ^ c - C 5 H 5 + CO CH3O + 0 2 -> HCHO + H 0 2 C2H5O + 0 2 -* CH3CHO + H 0 2 (CH 3 ) 2 CHO + 0 2 - ^ CH3COCH3 + H 0 2 CH 3 CH(0)CH 2 CH 2 CH 3 -* CH 3 CH(OH)CH2CH 2 CH2 OCH2CH2CH2CH3 -» HOCH2CH2CH2CH2

:3.16xl02T" 8 x 1013 a , b 2.5 x 1 0 l l a 6.7 x 10" 14 1.0 x 10" 13 1.5 x 10214 8.2 x 10 1 0 c ' d 5.5xl01Oc'a

exp(- 15400/T) 90.0 a ' b 183.7a 8.9 6.9 1.7 34.7C 21.8C

a

High pressure limit, units s _ 1 . Estimate [54]. c Estimate [95], units s - 1 . d Units s" 1 . b

o CH3CHCH2CH2CH3

lsom

7lzation> CH3CH(OH)CH2CH2CH2

decomposition J>2

/

CH3CHO + CH2CH2CH3 and CH3+CH3CH2CH2CHO

H0 2 + CH3C(=0)CH2CH2CH3

Atkinson [54] has reviewed these reactions in the context of atmospheric chemistry, and all are important in combustion under specific conditions up to about 600 K. Arrhenius parameters for a selection of reactions are given in Table 1.19. It is clear that 1,5 isomerization is the dominant process where structurally possible. Increased ring strain in 1,4 and 1,6 isomerizations will render them much less likely. Methoxy radicals are important up to 800 K, where moderate yields of CH3OH can be observed in the initial products. CH.O + RH -> CHUOH + R As homolysis in CH3O is restricted to the considerably slower reaction

Branching reactions

69

(57) and the reaction with 0 2 is somewhat slower than for other alkoxy radicals (ca. factor of 10), then formation of CH 3 OH becomes competitive. With other alkoxy radicals little conjugate alcohol is observed unless the pressure of 0 2 is very low. (57)

CH 3 0 + M -> HCHO + H + M

Rather surprisingly, reliable rate data for reaction (57) and reactions such as (58) are not available at combustion temperatures; Atkinson [54] reviews those obtained at low temperature. (58)

RCH 2 0 -* R + HCHO

1.14 BRANCHING REACTIONS

Although branching reactions have been mentioned earlier, more focussed attention is necessary. In the oxidation of propanal, the rate of primary initiation = 2&ipa[C2H5CHO][02] and the weakness of the aldehydic bond promotes a sufficiently high value of klpa that a measureable initial rate of oxidation is observed. Figure 1.14 shows a plot of the pressure change (AP) against time for the oxidation of propanal at 713 K. The dissociation of the H 2 0 2 formed in the reaction leads to AP - time curves which show autocatalysis. Baldwin, Walker and Langford [41] have modelled the oxidation and through sensitivity analyses have shown that the maximum rate (wmax) is determined almost completely by k13 and k2p/k\f and the initial rate (= ^2p(îpa/î4)1/2[C2H5CHO]3/2[02]1/2) by klpa and k2/k\/4. wmax is totally insensitive to the value of klpa used unless the value exceeds 103 times the correct figure. n>max is given by expression (1.15). HVax = M2* 1 3 [H 2 0 2 ][M]/M 1 / 2 [C 2 H 5 CHO] (lpa)

C 2 H 5 CHO + 0 2 -» C 2 H 5 CO + H 0 2

(2p)

H 0 2 + C 2 H 5 CHO -* H 2 0 2 + C 2 H 5 CO

(13)

H202 + M ^ 2 0 H + M

(1-15)

70


Ch. 1

time / s Fig. 1.14. AP - time plot for propanal oxidation in an aged boric-acid-coated vessel at 713 K. O, experimental; computed.

(14)

H02 + H02-»H202 + 02 OH + C 2 H 5 CHO -> C 2 H 5 CO + H 2 0

Reaction (13) represents secondary initiation, in that the initiation arises from a product of the reaction, and at any time (assuming no other initiation) then the effective overall rate of forming new centres is (2£ lpa [C 2 H 5 CHO][0 2 ] + 2* 13 [H 2 0 2 ][M]). Secondary initiation at low temperatures involves alkylhydroperoxides through sequences such as s-C 4 H 9 0 2 + R H - ^ C 4 H 9 O O H + R 5-C 4 H 9 OOH-^5-C 4 H 9 0 + OH Dialkylperoxides can equally provide secondary initiation, but their concentration will be much lower because they are generally formed only in termination reactions, although under many conditions the kinetic chain length may be low

Branching reactions

71

R 0 2 + R 0 2 -> ROOR + 0 2 ROOR->2RO Above 650 K, because of their weak C—H bonds, the reaction of aldehydes and alkenes with 0 2 can become important. (1) (lp)

RCHO + 0 2 -> RCO + H 0 2 C3H6 + 0 2 - + C 3 H 5 + H 0 2

In the oxidation of C 3 H 6 itself, the marked autocatalysis is mainly caused by reactions (lh) and (lAh), and the results suggest that klh/klp ~ 1500 at 753 K [91]. (lh)

CH 2 =CHCH 2 CH 2 CH=CH 2 + 0 2 -> CH 2 =CHCHCH 2 CH=CH 2 + H 0 2

(1 Ah)

CH 2 =CHCH 2 CH 2 CH=CH 2 -> 2CH 2 CHCH 2

At low pressures, the isomerization of oxiranes will cause initiation, as release of the ring strain energy combined with other rearrangements produces chemically activated intermediates which decompose into radicals unless the intermediate is stablized [90]. C 2 H 4 0 ^ CH 3 CHO* -> CH 3 + HCO |M CH 3 CHO Table 1.20 lists Arrhenius parameters for a number of secondary initiators. When acting as a secondary initiator in an autocatalytic reaction, the time to maximum rate is comparable with the half-life of the initiator. Thus the half-life of H 2 0 2 at 60 Torr (mostly N 2 and 0 2 ) and 753 K is about 60 sec, consistent with the induction period shown in the oxidation of butene-2 (see later). The half-life of alkyl peroxides is much shorter, but in the high pressure rapid combustion observed in gasoline engines this rate of initiation is too slow and only radical branching is sufficiently fast to explain the observed phenomena. The formation of dihydroperoxide

72


Ch. 1

TABLE 1.20 Kinetic parameters for secondary initiation reactions E/kJmoF 1

Aa

Reaction b

H 2 0 2 + M -*20H + M CH 3 OOH c ^ CH 3 0 + OH CH3(CH2)5OOHc-* CH3(CH2)50 + OH HCHO + 0 2 -> HCO + H0 2 (CH3)3CC(CH3)3C -* 2t-Bu CH2=CHCH2CH2CH==CH2C -> 2CH2CHCH2

-7

2.0 x 10 4.0 x 1015 1.1 x 1016 3.4 x 10"10 6.3 x 1016 1.25 x 1015

202.9 179.6 180.8 163 290 241

f 1/2/s at 750 K

5.5 2.5 4.7 1.7

48 x 10"4 x 10"4 x 102 x 103 33

"Units cm3 molecule"1 s _1 or s - 1 . Low pressure, second order values. c Limiting high pressure values. b

radicals with subsequent homolysis to give OH radicals is the favoured route at about 700 K. CH3CHCH2CH2CHCH3 -2?_ CH3CHCH2CH2CHCH3 OOH OOH 00

• 2CH3CHO + C2H3 + 20H —

CH3CHCH2CHCHCH3 OOH OOH

Here there is no induction period, other than that to build up the radical concentration, typically, ca. 10~4 s. In the oxidation of trans-butene-2 at 673 to 773 K, an unacceptable rate constant ratio of fclb/fclp ~ 150 is required unless a radical branching reaction is introduced [92]. The sequence below is consistent with the kinetic results. H0 2 + CH3CH = C H C H 3 — CH3CH—CHCH3 OOH 02 * 3 radicals — —

CH3CH^CHCH3 OOH02

Oxidation of cyclic alkanes

(lb)

CH 3 CH=CHCH 3 + 0 2 -> CH 2 CH=CHCH 3 + H 0 2

(lp)

CH 3 CH=CH 2 + 0 2 -» CH 2 CH=CH 2 + H 0 2

73

At high temperatures, reaction (15) is the most important radical branching process in combustion at pressures below 1 atm. Emdee et al. [44] missed out many of the reactions that produce H atoms in modelling toluene oxidation at 1200 K, and as a result obtained a value for klt a factor of about 100 too high. (15)

H + 02->OH + 0

(It)

C 6 H 5 CH 3 + 0 2 -> C 6 H 5 CH 2 + H 0 2

Finally it should be realized that quadratic branching may occur where there is no net increase in radicals. Reaction (59) is a key step in determining the second limit of the H 2 + 0 2 reaction under conditions where H 0 2 is inert, so that effectively one active radical gives two active radicals. Similarly reaction (54) is a very important branching reaction in alkane oxidation between 700 and 1000 K. (59) ((54)

H + H02->20H CH 3 + H 0 2 -» CH 3 0 + OH

1.15 OXIDATION OF CYCLIC ALKANES

1.15.1 Oxidation of Cyclohexane and Cyclopentane Some reference to the oxidation of cycloalkanes is merited in view of their significant presence in conventional fuels, particularly as relatively little attention has been paid to the chemistry involved. Recently Walker and co-workers [93, 94], by separate addition to slowly reacting mixtures of H 2 4- 0 2 , have made a quantitative analysis of the oxidation products of cyclohexane (753 K) and of cyclopentane (673-783 K). They also determined rate constant ratios k61/k62 and k60lk15 for OH and H attack

74

Ch. 1


TABLE 1.21 Kinetic data for H and OH attack on cycloalkanes at 753 K [97, 98] Alkane

k61lke2

c-pentane c-hexane CH 2 group* in alkanes

24.6 31.5

4 5

ke\lk62 (per CH2)

W&15

keo'k\5 (per CH2)

4.9 5.3 4.2

390 555

78 93 89

30 50

*Mean value for CH2 in C 3 H 8 , C 4 H 10 and C 5 Hi 2 .

on the cycloalkanes. Table 1.21 summarizes the data obtained and compares them with the mean values obtained for a —CH2— group in propane, butane and pentane on a bond additivity basis. Walker [51] has discussed the effect of near-neighbour groups on the rate of attack at the methylene group in alkanes and quotes k61/k62 = 5.0 at 753 K in excellent agreement with the value of 4.9 and 5.3 for OH H (CH2—) in cyclopentane and cyclohexane, respectively. It is likely, therefore, that —CH2— groups in the two cycloalkanes are indistinguishable from those in alkanes when near-neighbour effects are taken into account. Table 1.22 gives the recommended Arrhenius parameters for H and OH attack on cyclopentane and cyclohexane. (60cp)

H + c-C5H10 -» H 2 + c-C5H9

(61cp)

OH + c-C5H10 -> H 2 0 + c-C5H9

(15)

H + 02-+OH + 0

(62)

OH + H 2 - > H 2 0 + H TABLE 1.22 Temperature coefficients for H and OH attack on cyclo-pentane and cyclo-hexane based on k = A Tn t~B/T [97, 98] Reaction

T/K

A* 15

n

B/K

H + c-pentane H + c-hexane

300-1200 300-1000

4.0 x K T 5.7 x KT 15

1.5 1.5

2440 2440

OH + c-pentane OH + c-hexane

250-1500 300-1000

6.1 x UT 19 1.10 x KT 15

2.5 1.47

-519 -125

*Units, cm3 molecule

1

s

1

K


20

U0

75

60

Consumption of c-hexane Fig. 1.15. Product yields from a cyclohexane + 0 2 + H 2 mixture at 753 K. (A) A, 1,4-chexaneoxide; x, 1,2-c-hexaneoxide; • , C 2 H 4 ; O, 1,3-butadiene. (B) A, c-hexene; V, hex-5-en-l-al; # , CO; D, 1,3-onexadiene; O, benzene.

(60ch)

H + c-C6H12 -> H 2 + c-C 6 H n

(61ch)

OH + c-C6H12 -> H 2 0 + c-C 6 H n

Figure 1.15 shows the product profiles for the mixture containing 2.5, 140 and 357.5 Torr of cyclohexane, H 2 and 0 2 at 753 K. The results are typical of the quality of data obtained not only from cyclohexane but alkanes and related compounds in general. Primary and secondary products may be characterized from the shape of the profiles, Cyclohexane, 7-oxabicyclo(2,2,l)heptane (1,4 cyclohexane oxide), 1,2 epoxycyclohexane, CH 2 =CHCH 2 CH 2 CH 2 CHO (hex-5-en-l-al), buta-l,3-diene and possible

76


Ch. 1

TABLE 1.23 Initial yields* of products from c-pentane and c-hexane oxidation at 753 K [97,98] c-pentane

%

c-hexane

%

c-pentene 1,2-epoxycyclopentane C2H4 c-pentadiene

87 9.1 1.9 4.0

c-hexene 1,2-epoxycyclohexane C2H4 c-hexa-l,3-diene hex-5-en-l-al 1,4-epoxycyclohexane

67 4.6 0.7 3.5 24.0 7.5

*5% consumption, mole product per mole consumed. H 2 = 140, 0 2 = 70, c-alkane = 2.5, N2 = 287.5 Torr

TABLE 1.24 Initial yield of conjugate alkene for alkyl radicals at 753 K and 70 Torr 0 2 Radical

%

C2H5

99 99 80 60 50 90 70

£-C4Al9

/-C 3 H 7 /-C4H9

/-C 5 H n C-C5H9

c-QHn

ethene are primary products. Cyclohexa-l,3-diene and benzene are formed in very rapid secondary processes. Table 1.23 compares the initial yields of products (ca. 5% consumption) for the same mixture and that containing separately cyclopentane. There are clear features that mark the oxidations as following the same general mechanism as that for alkanes. (i) In general, even at 753 K, the ring is preserved during the formation of the initial products. With cyclohexane, hex-5-en-l-al is the only important non-ring compound. With cyclopentane, C 2 H 4 is only observed at very low 0 2 pressures. (ii) Conjugate alkene is the dominant initial product from both cycloalkanes in all mixtures studied. Table 1.24 compares the initial yields of alkene with those from alkyl radicals under the same conditions


77

TABLE 1.25 Relative initial yields of products from c-hexane at 753 K [97] Mixture

[1,2 epoxycyclohexane]/[c-hexene] [hex-5-en-l-al]/[c-hexene] [1,4 cyclohexane oxide]/[c-hexene] I II III IV

c-hexane c-hexane c-hexane c-hexane

= = = =

2.5, 2.5, 2.5, 2.5,

02 02 02 02

= = = =

I

II

III

IV

0.072 0.38 0.116

0.062 0.38 0.127

0.078 0.39 0.117

0.069 0.36 0.121

70, H 2 = 140, N2 = 287.5 Torr. 357.5, H 2 = 140 Torr. 70, H 2 = 427.5 Torr. 10, H 2 = 140, N 2 = 347.5.

c-C6Hn +

02

-0

— [ ^1

+ H0 2

(iii) Peroxy radical isomerization and decomposition is of importance, particularly for cyclohexane. 1,2-cyclohexane oxide and 1,4-cyclohexane oxide are formed via 1,4s and 1,6s H atom transfers as shown, but the 1,5s product (1,3-cyclohexane oxide) is unstable and decomposes to hex5-en-l-al. OOH

OO

OOH

ô^tf-d-

I

?-®-

OH

OH

CH2 = CHCH2CH2CH2CHO

Table 1.25 gives the relative rates of formation of the three products obtained from various mixtures. Gulati and Walker [93] argue that the relative yields of 1,2-cylohexane oxide and hex-5-en-l-al are consistent with the values of kiAJki,5s = 0.19 0.02 at 753 K obtained directly from studies with pentane as additive [79].

78

Ch. 1

Basic chemistry of combustion 14

CH 3 CHCH2CH2CH3^

CH3CHCHCH2CH3 0 0 H

V

OO

l,5sV

CH3CHCH2CHCH3 OOH

It must be emphasized that the ratio is independent of the value of the equilibrium constant for C 5 H n + 0 2 ^ C 5 H n 0 2 . The relative amount of 1,2- to 1,4 isomer is, however, much larger than expected (Table 1.17) for alkyl radicals, k(l,4s)/k(l,6s) « 0.07, compared with the value of [1,2 cyclohexane oxide]/[l,4 cyclohexane oxide] = 0.60 0.10. Walker and Gulati point out that although about 99.5% of the cyclohexane molecules adopt the chair form at 753 K, the O—O group at either an axial or an equatorial position in the chair form is not favourably structured for a low strain energy in the transition state, as shown by H %

^

-

H

H

H

H ^ ^ "

H

°0 However, the boat form can offer a very facile low energy route for the transfer. O O H „

OOH

n

The relatively low value of k(l,6s) is then a compromise arising from a low percentage of the boat form and a relatively strain-free H atom transfer in this species. The nature of the cyclohexane structure also indicates why the 1,3cyclohexane oxide is not observed in the products. As seen, there is relatively little strain in the six-membered ring transition state for a 1,5s transfer, but the rigidity of the chair imposes too much strain for the

79


formation of the 1,3 oxide, and rupture occurs to produce the energetically more favourable hex-5-en-l-al.

A H'AÂ

^

H'VÂ

^ CH2 = CHCH2CH2CH2CHO

In the case of cyclopentylperoxy radicals, the C—O bond and the C—H bonds in the 3 position are configured even further away from each other, so that formation of the QOOH radical itself is highly strained and no pent-5-en-l-al is observed. 1.15.2 Formation of benzene As shown in Fig. 1.15, benzene is formed as a secondary (probably tertiary) product when cyclohexane is added to H 2 + 0 2 mixtures at 753 K. Use of cyclohexene [93] as additive shows secondary formation of benzene, and when cyclohexadiene is the additive, benzene is formed as a primary product in yields of 60-70%. Clearly then cyclohexane and substituted cyclohexanes will be major sources of aromatics in the temperature region 600-1000 K. The formation is eased by the production of delocalized cyclohexenyl and cyclohexadienyl radicals which hastens the sequence through to benzene as shown below where X = OH, H 0 2 usually.

-Of* & * «

XH

-O + XH

- o* 0 0*°2

H0 2

+ XH

+ H0 2

80


Ch. 1

TABLE 1.26 Competing reactions of hex-5-en-l-yl radicals at [0 2 ] = 1 atmosphere A/cm3 molecule-1 s _1 or A/s'1

Products

10

> c-hexyl

1.0 x 10

£/kJ mol"1

(kor ^[O.D/s"1 600 K

1400 K

35

6

9.6 x 10

4.9 x 108

-%> CH2=CHCH2CH2CH=CH2

1.67 x 10"13

0

2.0 x 106

8.7 x 105

- ^ CH2=CHCH2CH2CH2CH202 > CH2=CHCH2CH2 + C2H4 > CH2=CHCHCH2CH2CH3 > CH2=CCH2CH2CH2CH3

1.67 x 10"12 1.0 x 1013 1.0 x 1012 1.0 x 1011

0 125 125 85

2.0 x 107 1.3 x 102 13 4.0 x 103

8.7 x 106 2.2 x 108 2.2 x 107 6.7 x 107

Production of benzene could be important in terms of pollution from exhaust emissions in gasoline engines. In the cyclohexane chemistry above, increased ring homolysis above 750 K might well be expected with the formation of C 2 H 4 and butadiene through the sequence c-C 6 H n -> CH 2 =CHCH 2 CH 2 CH 2 CH 2 -> C 2 H 4 + CH 2 =CHCH 2 CH 2 CH 2 =CHCH 2 CH 2 + 0 2 ^ CH 2 =CH—CH=CH 2 + H 0 2 However, Walker and Handford-Styring [94] have shown that because there is no strain energy in the ring transition state, the fastest reaction of CH 2 = CHCH 2 CH 2 CH 2 CH 2 radicals is cyclization back to c-C6 Hn with A = 1.0 x 1010 s" 1 and E = 35kJmol _ 1 . Table 1.26 lists the competing reactions of the hex-5-en-l-yl radicals, together with their Arrhenius parameters. A s R + 0 2 ^ R 0 2 for hex-5-en-l-yl radicals will be fully equilibrated at low temperatures, it is clear that below 600 K oxidation will be the main reaction, and only above 1400 K will homolysis to give butenyl and C 2 H 4 dominate. Between these temperatures, the major process will be cyclization to c-C 6 H n . A similar set of kinetics will apply to any hexenyl radical of the type RCH=CHCH 2 CH 2 CH 2 CHR (the three CH 2 groups could also be substituted further), which will rapidly cyclize even under oxidising conditions to give substituted benzenes as indicated above for the cyclohexyl radical.

81


300 C2H4 + C4H7

_

C2H4 + C3H5

200

I

J

" ^ i-c5u9 c-C5H9//

100 c-C 6 H n

reaction path Fig. 1.16. Potential energy diagrams for c-hexyl homolysis and c-pentyl homolysis.

1.15.3 Mechanism of cyclopentyl radical decomposition Homolysis of c-C5H9 radicals involves the sequence [95] c-C5H9 - ^ > CH 2 =CHCH 2 CH 2 CH 2 -> CH 2 CHCH 2 + C 2 H 4 From measurements of the relative yields of c-pentene and C 2 H 4 at low pressures of oxygen, Walker and Handford-Styring show that reaction (63) is effectively non-reversible. The contrasting behaviour of the c-C5H9 and c-C 6 H n systems arises from two features involving the thermochemistry, which is shown in Fig. 1.16. In the case of CH 2 =CHCH 2 CH 2 CH 2 CH 2 radicals, cyclization is favoured because there is zero strain energy in the ring transition state and a "normal" activation energy ( l ^ k J i n o r 1 ) for homolysis into butenyl and C 2 H 4 . For CH 2 =CHCH 2 CH 2 CH 2 radicals, the

82


Ch. 1

energy barrier to cyclization is increased by about 30 kJ mol - 1 and that for homolysis is reduced by about 22 kJ mol - 1 because the electron-delocalized allyl radical is formed [94]. Even allowing for a lower A factor due to the extra loss of entropy of activation in the formation of a six-membered ring, the sharp contrast in behaviour is easily understood. Walker and Handford5 exp(-17260/T) s" 1 . Styring obtained k63 = i o 1.15.4 Alkyl-substituted cycloalkanes Little is known of the chemistry of alkyl-substituted cycloalkanes such as C 6 H n CH 3 . The presence of one tertiary C—H bond will not noticeably increase the overall rate of radical attack because of the abundance of secondary C—H bonds. Attack at a ring C—H position will tend to give a similar product distribution to that observed for cyclohexane with high yields of the various isomeric cyclohexenes at most temperatures between 600 and 1000 K. Loss of a hydrogen atom at the (3 position to the R group would induce homolysis to give cyclohexene and R radicals, particularly at temperatures above 800 K. R

R

H

Rapid formation of benzene would then follow. Interesting QOOH chemistry would also be favourable and include not only the compounds analogous to those observed for cyclohexane, but also two oxabicyclooctanes which would have some associated stability arising from the boat and chair forms of cyclohexanes. H. CHyO

W —

CH20

CH 3 + C 2 H 4 (+ M) 1-C3H7 + 0 2 -> C 3 H 6 + H 0 2 Consequently, in modelling alkane oxidation through to the final products, a sub-set of elementary reactions are required to account for the oxidation of any alkenes formed. Two prominent properties of alkenes result in a number of distinguishing features in their oxidation chemistry. (i) Addition of atoms and radicals can occur across the double bond. X + R ' C H = C H 2 ^ R'CH—CH 2 X -> R'CH—CH 2 X For asymmetric alkenes, two adducts are formed for each radical X undergoing addition. As at least 4 species of radical, H, OH, H 0 2 and R 0 2 will be mechanistically important, a minimum of 8 different adducts are formed. At higher temperatures (ca. 1000 K), the addition of O atoms must also be considered. When coupled with abstraction, then the number of possible radicals formed from the alkene is very large. Even for propene, three C 3 H 5 radicals may be formed, completing a total of 11 species if only 4 radicals are involved in addition.

84


Ch. 1

TABLE 1.27 Activation energies used to calculate the relative rate of attack on the CH 3 group in C3H6 and C 3 H 8

E/kJmor1 Radical OH H02

o2

C3H6

C3H8

0 52 163

12 85 218

X + C3H6^CH2CHCH2 -> CH 3 CH=CH + XH -* CH 3 C=CH 2 (ii) When H abstraction occurs at the carbon atom in a /3 position to the double bond, electron derealization occurs in the emerging alkenyl radical with several significant consequences. The j8 C—H bond strength is considerably weakened (ca. 52 kJ mol - 1 ) [96] relative to that of a "normal" C—H bond in alkanes, so that abstraction may occur considerably faster than expected. With k = Ae~E,RT and equal A factors, then at 750 K, the relative rates of H abstraction by 0 2 , H 0 2 and OH are 6.8 x 103, 200 and 7 from the CH 3 groups in C 3 H 6 and C 3 H 8 (one CH 3 ), using the activation energies given in Table 1.27. Compensation occurs, however, through the reduction AS in the entropy of activation (by ca. 18 J K _ 1 mol - 1 ) in forming the delocalized radical due to the loss of one rotation in the radical. The lowering of the A factor is given by eAS/R = 8.7, and taken as 10 here. Thus although abstraction by 0 2 and H 0 2 remains considerably faster for propene, this is not the case for attack by OH radicals. The "low" A factor for H abstraction from the jS(allyl) position has interesting consequences. Considering C 3 H 6 , with bond dissociation energies of 364, 435 and 452 kJ mol - 1 for the three C—H positions, abstraction by selective species such as 0 2 , H 0 2 , R 0 2 and CH 3 will occur almost solely at the allyl position except at very high temperatures where activation energy differences are less important. For H 0 2 , allowing for the "low" allyl A factor, abstraction at the double bond C—H atoms only becomes important ( A £ ~ 50kJmol _ 1 ) above 1200 K. However, for OH

85

Oxidation of alkenes

radicals A£ ~ 20 kJ mol - 1 , so that above about 750 K, attack at the double bond positions in propene becomes more important and will become dominant above 1000 K. This analysis probably accounts for Tully's [97] observation on the temperature coefficient for the OH + C 3 H 6 reaction. The rate constant determined by direct monitoring of [OH] displayed a small negative temperature coefficient between 300 and 500 K, consistent with C 3 H 6 OH adduct formation. The measured rate constant then fell markedly, almost certainly due to the instability of C 3 H 6 OH which reformed OH radicals. However, above about 750 K, the rate constant increased again, the rise corresponding to an activation energy of about 20kJmol _ 1 . Although ascribed by Tully to formation of the allyl radical through abstraction at the /3 position, E « 0 would be expected, and the explanation is almost certainly that above about 750 K, abstraction at a double bond C—H becomes faster. As OH is usually the main propagating species at all temperatures between 300 and 2000 K, then the chemistry of C 3 H 6 oxidation should increasingly involve the localized radicals CH 3 CH=CH and CH 3 C=CH 2 as the temperature approaches and exceeds 1000 K, whereas below 800 K the allyl radical should dominate. Figure 1.17 shows the AP - time plots for the separate oxidations of

500

1000

1500

time/s Fig. 1.17. AP - time plots for alkene oxidation in aged boric-acid-coated vessels at 753 K. alkene = 4, 0 2 = 10, N 2 = 46Torr.

86


u

10

20

30

0

50

100

Ch. 1

150

200

Time/s

Fig. 1.18. Oxidation products from trans-butene-2 (Sections A and C) and from C3H6 (B and D) at 753 K. alkene = 4, 0 2 = 30, N 2 = 26Torr. (A) O, 1,3-butadiene; A, CH 3 CHO; V, cis-butene-2; • , C3H6; D, CO; O, acrolein; x, butene-l(x2); (C) A, trans-2,3-butaneoxide; V, cis-2,3-butaneoxide; D, CH 4 ; # , C2H4 (x4); (O) C 2 H 5 CHO; x, methylethylketone; (B) x, acrolein(x2); O, C 2 H 4 (xl.5); # , CO; D, HCHO; V, CH 4 (xlO); (D) V, 1,5-hexadiene (x4); x, propanal (x3); O, C 3 H 6 0; • , CH 3 CHO.

butene-1, trans-butene-2, propene and isobutene under the same initial conditions at 753 K. Effectively the rates of oxidation of both butene-1 and trans-butene-2 are the same both initially and at the maximum values, and are spectacularly faster initially than those for /-butene and propene which differ by a factor of about 2. Another very striking feature is the difference in the initial product distribution, which is illustrated in Fig. 1.18 for propene and trans-butene-2, and in Table 1.28. The key difference centres on the presence of high yields (up to 30%) of hexa-l,5-diene (HDE) from propene and 2,5-dimethylhexa-l,5-diene (DMHDE) from isobutene. Despite repeated efforts, no trace of the equivalent compounds from butene-1 and butene-2 could be found. However, with butene-1 and butene-2, high yields (up to 50%) of buta-l,3-diene were obtained, together with up to 40% yields of 1,2-butaneoxide and 2,3-butaneoxide,

87

Oxidation of alkenes TABLE 1.28 Initial yields3 of products from propene and trans-butene-2 oxidations at 753 K [36, 92] Propene products

%

t-butene-2 products

%

CO HCHO CH 4 C2H4 CH 3 CHO CH 2 =CHCHO CH 3 CH 2 CHO C3H60 CH 2 =CHCH 2 CH 2 CH=CH 2

30.4 65.1 7.3 12.2 15.3 2.9 1.4 11.3 16.4

CO HCHO CH 4 C2H4 CH3CHO CH 2 =CHCHO CH 3 CH 2 CHO

7.1 5.4 6.0 0.3 23.2 1.8 1.2 0 0 6.0 0.8 15.8 29.7 18.1 8.7 0.6

C3H6 /-C4H8 cis-butene-2 butadiene-1,3 r-2,3 butaneoxide c-2,3 butaneoxide methylethylketone

a

About 1% reaction, mole product per mole consumed, alkene = 4, 0 2 = 30, N2 = 26 Torr

respectively. In contrast, the yields of epoxides from propene and isobutene were relatively low. 1.16.2 Product formation Walker and co-workers [35,36,38] considered that recombination of allyl radicals is the only plausible route to HDE, which logically emphasizes the inertness of allyl radicals towards 0 2 and hydrocarbons. They show that CO, HCHO and acrolein are formed through allyl + H 0 2 reactions, and that reaction (-lp) is an important termination reaction in propene oxidation.

2CH 2 CHCH 2 ^ CH 2 =CHCH 2 CH 2 CH=CH 2

88


Ch. 1

TABLE 1.29 Arrhenius parameters for allyl + 0 2 reactions [36]

V 1 ) E/kJmoT1

Reaction

log(v4/cm3 molecule

CH2CHCH2 + 0 2

-12.6 6 -11.12 5 -11.08 5 -11.78 Jfc=1.7xl0~ 17 at 753 K

-> 3 radicals + products —> CO + products -> 2HCHO + CO + OH -> CH 2 =C=CH 2 + H 0 2 CH2C(CH3)CH2 + 0 2 ^ CH2=C(CH3)CHO + OH

72.5 78.6 80.3 95

8.3 4.5 60

*Defined [36] as a radical breaching step, alkene = 4, 0 2 = 30, N2 = 26Torr

CH3CHCH2 + H0 2

[CH2 = CHCH2OOH]

CH2 = CH + HCHO — —- CH2 = CHCH20 + 0H

Jo 2

|o2 1

HCHO + HCO

CH2 = CHCH0 + H0 2

I02 1

H0 2 + CO (-lp) (17)

CH2CHCH2 + H 0 2 ->C 3 H 6 + 0 2 H 0 2 + C3H6 -»• C 3 H 6 0 + OH

Although propene oxide is not a major product, very importantly from measurements of its rate of formation, [H0 2 ] may be calculated reliably as k17 is known accurately [37]. As [CH2CHCH2] can similarly be calculated from the rate of formation of HDE, Walker and co-workers have made a comprehensive analysis of allyl radical reactions in both propene oxidation and the decomposition of 4,4-dimethylpentene-l (DMP) in the presence of 0 2 . Table 1.29 summarizes the Arrhenius parameters for various reactions of allyl radicals with 0 2 . Kinetic data for a variety of other allyl reactions has also been critically examined recently [9]. Table 1.28 shows that CH 3 CHO is an important primary product from

89


C 3 H 6 . Although the evidence is not secure, the general consensus supports formation through an OH addition sequence [98, 99]. OH + C3H6

CH 3 CH—CH 2 O

O

i

i

H

O

CH3CHO + HCHO + OH

As OH radicals are re-formed and are not involved in any termination sequence, formation of CH 3 CHO is not kinetically of importance at least in the initial stages of reaction. 1.16.3 Oxidation of isobutene Below 800 K, H abstraction from isobutene results almost uniquely in methylallyl (MA) radicals which undergo homolysis to a small degree to give allene and CH 3 radicals, but otherwise are as unreactive as allyl radicals. With their resulting high radical concentration, significant yields (up to 30%) of 2,5-dimethylhexa-l,5-diene (DMHDE) are observed in the initial products of isobutene oxidation. X + (CH 3 ) 2 C=CH 2 2CH 2 C(CH 3 )CH 2

XH + CH 2 C(CH 3 )CH 2 CH 2 =C(CH 3 )CH 2 CH 2 C(CH 3 )=CH 2

Because fcubAip ~ 5, then [H0 2 ] is increased in isobutene oxidation, and significant yields of methacrolein are found through the sequence, CH 2 C(CH 3 )CH 2 + H 0 2

CH 2 =C(CH 3 )CH 2 OOH

CH 2 =C(CH 3 )CH 2 OOH •

CH 2 =C(CH 3 )CH 2 0 + OH

CH 2 =C(CH 3 )CH 2 0 + 0 2

CH 2 =C(CH 3 )CHO + H 0 2

90


Ch. 1

Isobutene oxide is formed through H 0 2 addition, and acetone through OH addition in an analogous way to CH 3 CHO from propene. Measurement of the initial rates of formation of DMHDE and isobutene oxide coupled to relevant rate constants from the literature gave accurate values for [MA] and [H0 2 ]. As indicated earlier, the basic mechanism is so simple that reliable values have been obtained for khh from the fundamental equation, ^lib[*-C4H 8 ][02j

=

RDMHDE

where RDMHDE is the initial rate of formation of DMHDE. Corrections have been made for small contributions (—20%) from branching processes and for the termination reactions (-lib) and (14). (-lib) (14)

CH 2 C(CH 3 )CH 2 + H 0 2 -> (CH 3 ) 2 C=CH 2 + 0 2 H02 + H 0 2 ^ H 2 0 2 + 02

The basic oxidation mechanism for propene and isobutene appears almost unique and apart from OH and H 0 2 addition involves the formation, through abstraction, of highly inert radicals which are almost completely consumed through radical-radical processes. The tremendous acceleration (over a factor of 100) shown between the initial and maximum rates for both alkenes (Fig. 1.17) can be explained through secondary initiation involving three reactions for each alkene. They are given below for propene HDE^2CH2CHCH2 (lh)

HDE + 0 2 -> CH 2 =CHCHCH 2 CH=CH 2 + H 0 2

(13)

H202 + M-+20H + M

1.16.4 Oxidation of butene-1 and butene-2 Turning to butene-1 and trans-butene-2, a totally different abstraction mechanism becomes apparent. The high yields of butadiene (Table 1.28) indicate that the delocalized butenyl radicals formed via H abstraction


91

from the j8 positions in the two butenes react rapidly with 0 2 in reaction (63).

(63)

X + CH 3 CH=CHCH 3 v ^CH 3 CHCHCH 2 + XH X + CH3CH2CH=CH/ CH 3 CHCHCH 2 + 0 2 -> CH 2 =CHCH=CH 2 + H 0 2

Although k63^2x 10" 15 cm 3 molecule" 1 s" 1 at about 750 K is a factor of 50-100 lower than k5A for alkyl + 0 2 reactions, it is about 104 higher than k64 for allyl + 0 2 , and so provides a good "sink" for butenyl radicals. (64)

CH 2 CHCH 2 + 0 2 -* C H 2 = C = C H 2 + H 0 2

As a result, [butenyl] is very low and [H0 2 ] is typically a factor of 103 higher than in propene oxidation under the same initial conditions. The high yield of H 0 2 has several consequences. (i) Termination in butene-1 and butene-2 oxidation is almost entirely due to reaction (14), with virtually no contribution from reaction (65). No evidence is obtained for a mutual reaction of butenyl radicals (66). (14)

H 0 2 + H 0 2 -* H 2 0 2 + 0 2

(65)

CH 3 CHCHCH 2 + H 0 2 -* C 4 H 8 + 0 2

(66)

2CH 3 CHCHCH 2 -» CH 3 CH=CHCH 2 CH 2 CH=CHCH 3

(ii) H abstraction by H 0 2 radicals gives H 2 0 2 which then undergoes homolysis as the main source of secondary initiation. H 0 2 + CH 3 CH=CHCH 3 -> H 2 0 2 + CH 3 CHCHCH 2 H202 + M->20H + M Figure 1.17 shows that the degree of autocatalysis is considerably less marked than with propene and isobutene oxidations. (hi) Modelling studies [92] of the initial stages of trans-butene-2 oxidation have been carried out over a wide range of mixture composition and temperature, and product distributions can be predicted very accurately.

92


Ch. 1

However, as indicated earlier, unless a linear branching reaction is included in the mechanism, a value of klhlklp = 100-120 is required to explain the very high initial rate of oxidation, as shown in Fig. 1.17. Independent studies, however, show that klh/klp = 5 2 between 700 and 800 K (Table 1.6). Any attempt to model the oxidation with quadratic branching produces impossibly high autocatalysis at about 10% consumption of butene-2 because of the high radical concentrations at this point.

(lb) (lp)

C4H8-2 + 0 2 -* CH 3 CHCHCH 2 + H 0 2 C 3 H 6 + 0 2 -> CH 2 CHCH 2 + H 0 2

The kinetics are consistent with QOOH0 2 branching through the sequence

H 0 2 + C 4 H 8 -> C 4 H 8 OOH

> branching

As mentioned earlier, [H0 2 ] is typically a factor of 103 lower in C 3 H 6 oxidation, and further, H 0 2 addition to C 3 H 6 is significantly slower, so that the equivalent sequence in C 3 H 6 oxidation is of negligible importance. Indeed, Walker and Stothard [38] have shown that radical branching in this system is explained by the reaction of allyl radicals with 0 2 (see Table 1.29). CH 2 CHCH 2 + 0 2 ^ branching Summarising the essentials, two sub-mechanisms appear to operate in the oxidation of the four alkenes just discussed. With butene-1 and butene-2, the mechanism is sharply dependent on H 0 2 / H 2 0 2 chemistry, whereas with the "unique" cases of C 3 H 6 and isobutene, [H0 2 ] is relatively low, and the kinetics of allyl and methylallyl reactions through radical-radical processes are of key importance. The recombination products HDE and DMHDE are responsible through homolysis and reaction with 0 2 for the striking autocatalysis observed (Fig. 1.17).

93


1.16.5 Alkenyl oxidation chemistry It is relevant to comment here on four types of alkenyl radical, mainly with respect to their reactions with 0 2 . (i) Alkenyl radicals where the free valency occurs at a carbon atom in the double bond. The simplest case occurs with vinyl radicals, where between 600 and 800 K the initial products are HCHO and CO. Gutman et al. [100] and others [98,101] have suggested formation via a cyclic intermediate. C2H3 + 0 2

— CH—CH2

Li

— HCHO + HCO

|*

H0 2 + CO

No evidence has been found for acetylene formation in a reaction equivalent to (5A) at low temperatures, although at higher temperatures (>1000 K), it is considered important [9] as is reaction (67) because of the high O atom concentrations that exist. C 2 H 3 + 0 2 -> H C = C H + H 0 2 (67)

C 2 H 3 + O -* H C = C H + OH

Whilst the evidence is not completely firm, it is likely that CH 3 G=CH 2 and CH 3 CH=CH radicals themselves undergo reactions (68) and (69) in the region 600 - 900 K, although the radicals are probably only important near 1000 K.

?-? (68)

CH3C = CH2

— CH3C—CH2

— CH3CO +HCHO

?-? (69)

CH3CH = CH

— CH3CH— CH

— CH3CHO + HCO

(ii) Alkenyl radicals typified by CH 2 CHCH 2 and CH 2 C(CH 3 )CH 2 , but generally CH 2 C(R)CH 2 . These radicals react extremely slowly with 0 2

94


Ch. 1

(Table 1.29) and with organic species in either abstraction or addition mode. Further, only below 500 K does the equilibrium position for R + 02 2 favour R 0 2 radicals, so that R 0 2 reactions are generally unimportant. Apart from C—C homolysis (with the exception of allyl) which is relatively slow below 800 K, the radicals will be consumed mainly in radical-radical processes, as illustrated by the strikingly high yields of hexadiene (HDE) and dimethylhexadiene (DMHDE) observed in the initial oxidation products of C 3 H 6 and /-C4H8, respectively. (hi) Electron-delocalized radicals which react relatively quickly with 0 2 to give a conjugate diene and H 0 2 . Structurally, H abstraction from the C atom j8 to the double bond coupled to the availability of a C—H bond in the y position permits the reaction to occur. CH 2 CHCHCH 3 + 0 2 ^ CH 2 =CHCH=CH 2 + H 0 2 CH 2 =CHCHCH 2 R + 0 2 - > CH 2 =CHCH=CHR + H 0 2 At 750 K , f c ~ 2 x 10~15 cm3 molecule -1 s _ 1 for these reactions, compared with k5A ~ 2 x 10 - 1 3 for the analogous reaction of alkyl radicals, but is a factor of about 104 greater than for allene formation from CH 2 CHCH 2 . (5 A)

R + 0 2 -> alkene + H 0 2 CH 2 CHCH 2 + 0 2 -> C H 2 = C = C H 2 + H 0 2

H 0 2 / H 2 0 2 chemistry will therefore be very important for alkenyl radicals in this class, and product-time curves will show similarities with those for butene-1 and butene-2 in Fig. 1.17. As the temperature rises above 700 K and the pressure of 0 2 falls, homolysis of the radicals will increasingly dominate, as shown for CH 2 CHCHCH 2 CH 3 radicals. CH 2 CHCHCH 2 CH 3 -> CH 2 =CHCH=CH 2 + CH 3 (iv) Alkenyl radicals where the free valency is remote from the double bond, for example in reaction (70). Here the rate constant (/c70) would be that expected for a "normal" alkyl radical, (1-2) x 10 - 1 3 cm3 molecule" 1 s" 1 between 600 and 900 K. K(R + 0 2 ^ R0 2 ) should not be disimilar from that for 2-C3H7 radicals. However, for the radical shown, homoly-


95

sis with k71 = 106 s 1 will be favoured above 700 K because the "stable" allyl radical would be produced. (70)

CH 2 =CHCH 2 CH 2 CHCH 3 + 0 2 -»• CH 2 =CHCH 2 CH 2 CH=CH 2 + H 0 2 4(71) CH 2 CHCH 2 + CH 2 =CHCH 3

Large alkenyl radicals with certain structural characteristics may cyclize or isomerize by H atom transfer. As indicated earlier, CH 2 =CHCH 2 CH 2 and CH 2 =CHCH 2 CH 2 CH 2 will cyclize to cyclobutyl and cyclopentyl radicals, but with considerable ring strain energy barriers, whereas C H 2 = CHCH 2 CH 2 CH 2 CH 2 will cyclize extremely rapidly (A = 1.0 x 1010 s" 1 and £ = 35kJmol~ 1 ) arising from zero strain in the six-membered carbon ring formed. H atom transfers would only become competitive at temperatures exceeding 1200 K. However, a small change in structure would induce a very rapid 1,6 H atom transfer involving abstraction from an allyl position. Arrhenius parameters of 1.0 x 1010 s - 1 and E = 35 kJ mol - 1 for isomerisation (72) arise from zero strain and the size of the ring. No other reaction could compete between 600 and 1200 K unless [0 2 ] was extremely high. (72) CH 3 CH 2 CH=CHCH 2 CH 2 CH 2 -> CH 3 CHCHCHCH 2 CH 2 CH 3 Without doubt, structure plays a major part in determining the oxidation chemistry of alkenyl radicals. 1.16.6 Oxidation of the smaller alkenyl radicals at high temperatures Further consideration of the behaviour of the smaller alkenyl radicals at temperatures near to 1000 K is merited. As indicated earlier, methylallyl radicals undergo C—C homolysis to some extent at 750 K. CH 2 C(CH 3 )CH 2 -* C H 2 = C = C H 2 + CH 3 Homolysis of allyl radicals may become important, but the energy barrier is very high due to electron derealization; A// 73a = 200 and E73a = 240 kJ mol - 1 , whilst A# 7 3 b = 240 and E73b = 260 kJ mol" 1 .

96 (73a) (73b)


Ch. 1

CH 2 CHCH 2 -> CH 3 + C 2 H 2 -> C H 2 = C = C H 2 + H

With A73a = 5 x 1013 s" 1 , k73a = 20 s" 1 at 1000 K, assuming first-order kinetics which is unlikely below 1 atm. k73a rises to 107 at 2000 K, so that homolysis of the allyl radical could be a major process unless the pressure is low. Very similar values of k73h are obtained from Tsang's [102] recommendation of 1.5 x i o n T 0 - 8 4 exp(-30050/T)s~\ However, the importance of allyl and related (e.g., methylallyl) radicals at very high temperatures is questionable following the discussion earlier on the "low" A factor (down by a factor of 10) for H abstraction from the allylic position in alkenes. As the OH radical remains a very important abstracting radical at high temperatures, then with the activation energy only about 20kJmol _ 1 for abstraction from the double bond C—H position, formation of CH 3 CH=CH and CH 3 C=CH 2 from propene will dominate above 1200 K. The high temperature chemistry of the propenyl radicals is largely unknown, but reactions such as (74)-(77) would be relatively fast. (74)

CH 3 CH=CH + M -* CH 3 + C 2 H 2 + M

(75)

CH 3 C=CH 2 + M -> CH 2 CHCH 2 + M

(76)

CH 3 CH=CH + O -> products

(77)

CH 3 C=CH 2 + O -> products

Further study of the chemistry of alkenyl radicals in the range 1000-2000 K and up to pressures of 30 atms is required, with particular emphasis on the smaller electron-delocalized radicals. Gutman etal. [103] have investigated the reactions of highly unsaturated alkenyl radicals with 0 2 , but at temperatures well below 1000 K.

1.17 ATOM AND RADICAL ADDITION TO ALKENES

1.17.1 H Atoms Extensive kinetic data exist for the low temperature addition of H atoms to alkenes [65]. Activation energies vary from 17 (C 2 H 4 , high pressure) to

Atom and radical addition to alkenes

97

about 5 k J m o l _ 1 for larger alkenes. Here interest lies mainly in the influence of the reaction in the oxidation process. With C 2 H 4 (D(C2H3—H ~ 452kJmol _ 1 ), H abstraction competes effectively only at high temperatures unless the pressure is low, and below 800 K C 2 H 4 acts catalytically to convert H atoms (potentially a branching species) to inactive H 0 2 radicals M

H + C2H4

> C2H5

o2

> C2H4 + H 0 2

However, H atom addition to alkenes is ca. 170kJmol _ 1 exothermic, so that C—C homolysis reactions are enhanced through the "chemically activated" alkyl radicals formed. Relative yields of products are thus very dependent on pressure. CH3CHCH3* - M _ c 3 H 7 - ^ ^ C3H6 + H 0 2 H + C3H6 = ^

CH3CH2CH2*

— CH3 + C2H4

IM C 3 H 6 + H 0 2 — ^ - CH 3 CH 2 CH 2 Studies [23] of the separate addition of butane, butene-1 and butene-2 to H 2 + 0 2 mixtures (Method I), where H atoms are important chain carrying intermediates, have confirmed that the yield of homolysis products such as C 2 H 4 and C 3 H 6 increase significantly with the butenes even at 500 Torr total pressure. Formation of 2-C3H? (from H + C3H6) and *-C 4 H| (from H + *-C4H8) again induces efficient catalytic conversion of H to H 0 2 , as even in their "chemically activated" state the radicals are unable to undergo C—C homolysis. It is clearly apparent from the H + C 3 H 6 scheme written above why C 3 H 6 is a good inhibitor of alkane oxidation. Even if 1-C3H* rather than 2-C3H* radicals are formed, either C—C homolysis gives the unreactive CH 3 radical or stabilization occurs resulting in either more CH 3 radicals or the production of H 0 2 radicals. When coupled with H abstraction from C 3 H 6 to give the "stable" allyl radical, which is almost inert towards 0 2 , the main role of the alkene is to convert highly reactive H and OH species into unreactive H 0 2 and allyl radicals.

98


Ch. 1

1.17.2 O Atoms The addition of O atoms to alkenes has been studied extensively, particularly at low temperatures [104], although Klemm et al. [105] (244-1052 K) and Mahmud et al. [106] (290-1510 K) studied the kinetics of O + C 2 H 4 over wide temperature ranges. In their (78a)

O + C 2 H 4 -> CH 2 CHO + H

(78b)

^ C H O + CH 3

(78c)

-> HCHO + CH 2

(78d)

-> CH2CO + H 2

critical review, Baulch et al. [8,9] recommend k78 = 5.75 x 10~18 T 2 0 8 cm3 molecule -1 s _ 1 between 300 and 2000 K with an accuracy of better than 20% below 1000 K. No pressure effects are observed above 300 K. The product channels are less well defined [50], but k78Jk78 = 0.35 and k78h/k78 = 0.6 are recommended. Very little detailed product information is available for the higher alkenes [104]. Fortunately, O atom reactions are not of importance in combustion below 1000 K. 1.17.3 HO2 radicals Addition of H 0 2 radicals to alkenes to give oxiranes is kinetically important between 600 and 1000 K because of the formation of the very reactive OH radical in the sequence

H02 +> = < — ) f - < — X-/ C ( + °H OOH ° Below 700 K, addition of R 0 2 to alkenes contributes to the formation of oxiranes and also produces alkoxy radicals.

Baldwin, Walker and co-workers have added a number of alkenes separately to tetramethylbutane + 0 2 mixtures (Method III) to obtain extensive data on the addition of H 0 2 radicals to alkenes by measurement of the oxirane produced.


99

TABLE 1.30 Arrhenius parameters for H 0 2 and R 0 2 addition to alkenes Alkene H 0 2 [33, 70] C2H4 C3H6 pentene-1 hexene-1

Ionization potential/ kJ mol - 1

log(A/cm3 molecule -1 s _1 )

E/kJ mol

1

T/K

1013 939 917 912

11.77 11.77 11.90 11.87

5 0 4 4

71.9 62.6 59.7 58.6

5 5 5 5

673-773 673-773 673-793 673-773

/-butene cis-hexene-2 trans-hexene-2 trans-butene-2

891 884 884 881

11.98 12.37 12.37 12.17

4 0.35 0.35 0.30

53.1 53.4 53.4 50.0

5 5 5 5

673-793 673-773 673-773 673-793

c-hexene (CH 3 ) 2 C=CHCH 3 (CH 3 ) 2 C=C(CH 3 ) 2

841 836 801

12.31 12.44 12.20

0.25 0.32 0.35

45.5 43.0 35.4

5 5 5

673-773 673-773 653-773

CH 3 0 2 [107] (CH 3 ) 2 CHCH=CH 2 (CH 3 ) 2 C=CHCH 3 (CH 3 ) 2 C=C(CH 3 ) 2

880 836 801

12.18 12.62 12.64

2 0.49 0.32

52.8 42.4 36.4

4 3.6 2.8

373-403 373-403 373-403

/-(CH 3 ) 2 CH0 2 [108] C3H6 / C4H8 (CH 3 ) 2 CHCH=CH 2

939 891 880

11.86 12.19 12.48

6 9 0.72

67.7 62.7 54.9

2.5 2.2 5.4

373-408 373-408 363-408

(CH3)2C==CHCH3 (CH 3 ) 2 C=C(CH 3 ) 2

836 801

12.75 12.82

0.35 0.50

48.2 40.9

9 8

303-363 303-363

Table 1.30 summarizes the data available and Fig. 1.19 shows a plot of the activation energies against the ionization potential of the alkene. Walker et al. [70] have argued that the excellent correlation indicated in Fig. 1.19 provides clear evidence that the parameters in the table refer to the initial addition step. Kinetic data for R 0 2 + alkene have also been obtained by Waddington and co-workers [107,108]. Reference has already been made to the very high activation energies for H 0 2 addition (72 kJ mol - 1 for C 2 H 4 ), which if true raises problems for the mechanism suggested by Gutman etal. [71] for the C 2 H 5 + 0 2 reaction. They argue that C 2 H 4 is produced through C 2 H 4 OOH formation as indicated in Fig. 1.12 but require an activation energy of only 20-25 kJ mol - 1 for H 0 2 addition to C 2 H 4 . Walker et al. consider that the main reaction

100

Ch. 1


o

s? c c

>

800

900

1000

Ionisation potential/kJ mol"1 Fig. 1.19. Plot of activation energy against ionization energy of alkene for the addition of H 0 2 to alkene.

of QOOH radicals is O—O homolysis and cyclization to give oxiranes. Emphatic evidence that QOOH radicals do not predominantly decompose to give conjugate alkene + H 0 2 has been obtained recently by Stothard and Walker [70] from studies of the separate addition of cis- and transbutene-2 to slowly reacting mixtures of H 2 + 0 2 (Method I) and to TMB + 0 2 mixtures at 713 and 753 K. The relevant chemistry is shown below for trans-butene-2. H 0 2 + t-C4H8-2 - s r — CH3CHCHCH3

— cis-C4H8-2 + H 0 2

OOH

CH3CHCHCH3 + OH


101

TABLE 1.31 Oxidation of cis-butene-2 [70] Mixture/Torr* trans-butene-2 02

H

[2,3 DMO]/[cis-butene-2]

753 K

2.5 2.5 2.5 2.5 10 10 4

70 210 358 70 70 50 56

140 140 140 428 140

3.8 3.9 4.2 4.6 3.5 3.9 4.1

20 70 310 20 70 50

140 140 140 140 140

5.5 5.6 5.8 4.6 5.0 4.5

713 K

2.5 2.5 2.5 6.3 10 10 cis butene-2

O,

H2

753 K

10 5 2 10 4

50 30 50 70 30

140 0

7.3 7.1 7.7 6.7 7.5

*Mixtures with H 2 , up to 500Torr with N 2 ; mixtures without H 2 , up to 60Torr with N2.

If CH 3 CH(OOH)CHCH 3 radicals predominantly decompose to give alkene, then a high proportion should reform trans-C4H8-2, but as the trans form is only marginally favoured [23] then the initial product ratio [2,3dimethyloxirane]/[cis-C4H8-2] should be very low. In marked contrast, Stothard and Walker report average values of the ratio of about 4 at 753 K and 5 at 713 K over a wide range of mixture composition. The results of more recent experiments with cis-C4H8-2 are reported in Table 1.31, together with the r-C4H8-2 values. In reality the values of the product ratios should probably be considerably higher than indicated but OH addition is reversible at the temperatures used.

102


OH + trans-CH3CH = CHCH3 = = :

Ch. 1

CH3CHCHCH3 OH

cis-CH3CH = CHCH3 + OH These experiments appear to indicate that QOOH radicals predominantly decompose to oxiranes, and as the thermochemistry of the participating reactions is essentially independent of the alkene system, this conclusion can be extended to C 2 H 4 OOH radicals, to the C2H5 + 0 2 reaction and R + 0 2 systems in general. Batt [109] carried out a number of interesting experiments with isobutylhydroperoxide radicals, produced in the absence of 0 2 in the range 350-500 K by radical attack on r-butylhydroperoxide where H abstraction occurred mainly on one of the CH 3 groups. He measured the relative yields of 2,2-dimethyloxirane and isobutene, considered to be formed in competing reactions of the f-butylhydroperoxide radical. CH3

CH3

I

I

X + CH 3 —C—OOH

— CH2 — C — OOH

I

+ XH

I

CH3

^

(CH3)2C = CH2 + H0 2

CH3 (CH3)2C — CH2 + OH

V

A He reported a value of logkjk h = 1 0 exp(-2620/T) which corresponds to a value of (iso-butene]/[2,2-dimethyloxirane] of about 0.06 at 300 K and 10 at 750 K. Thus oxirane formation is dominant at low temperature, but QOOH decomposition to conjugate alkene appears more important at 750 K, which appears to contradict the conclusions above. However, Batt's results give Ea- Eh = 22 kJ mol - 1 and if this number is used for the H 0 2 + C 2 H 4 sequence,

H 0 2 + C2H4 ^

C 2 H 4 OOH - A C 2 H 4 0 + OH ka


103

the energy barrier for H 0 2 addition is even higher than suggested by Walker et al. If, as they suggest, kb > fca, then (-a) is rate determining and £-a = £obs = 72kJmol~ 1 , where Eohs is the overall activation energy for oxirane formation (Table 1.30). With Gutman's view that C 2 H 4 OOH decomposes to give mainly C 2 H 4 + H 0 2 , then ka> kh, so that the addition step is equilibrated and JEobs = E.a + Eh - Ea. Assuming E.a - Eb = 22kJmol _ 1 (as for isobutylhydroperoxide radicals), then with Eobs = 72, E.a = 94kJmol _ 1 , compared with the maximum value of 25 k J m o l - 1 required by the Gutman model (see Fig. 1.12). Further discussion of the problem appears elsewhere [11], and the topic is reconsidered in Chapter 2. 1.17.4 OH radicals Few kinetic data are available for the addition of OH radicals to alkenes. In the absence of 0 2 , the addition becomes reversible above about 550 K, but 0 2 addition to the adduct appears to provide an important route to aldehydes and ketones at relatively elevated temperatures with initial yields of 20-30%. OH + CH3CH=CHCH3

— CH3CHCHCH3 O H

|o 2 2CH3CHO + OH—

CH3CHCHCH3 OHOO

OH + (CH 3 ) 2 C=CH 2 —— (CH 3 ) 2 C—CH 2 O H

I jo 2

O |

°

CH3COCH3 + OH + HCHO

104


Ch. 1

Detailed modelling studies of the oxidations of propene, trans-butene-2 and isobutene have been carried out [111] for the temperature range 650800 K, and the yields of aldehydes and ketones formed as above are consistent with mechanism proposed. A more intriguing problem concerns the formation of lower alkenes, namely, in particular, C 2 H 4 from C 3 H 6 and C 3 H 6 from isobutene. When C 3 H 6 is added to mixtures of H 2 + 0 2 at 753 K and to TMB + 0 2 mixtures between 673 and 773 K, C 2 H 4 , CH 3 CHO and HCHO are major initial products. CH 3 CHO and HCHO may be formed through OH and then 0 2 addition as indicated above for f-butene-2 and /-butene. For the H 2 + 0 2 mixtures [98], H addition to give chemically activated CH 3 CH 2 CH* radicals, followed by C—C homolysis, is a plausible route for C 2 H 4 formation. H + C 3 H 6 -* CH 3 CH 2 CH^ -> CH 3 + C 2 H 4 However, the concentration of H atoms is very low in the C 3 H 6 + TMB + 0 2 mixtures and, as an alternative, formation of C 2 H 4 through OH addition and a 4-centre transition state may be possible. OH + C3H6

— CH3CH — <j:H2 H — O

— C2H5 + HCHO 99%|o 2

C2H4 + H0 2 Given the tightness of the transition state, and the ensuing strain energy, it is very unlikely that the sequence can compete with dissociation back to OH and C 3 H 6 , or with 0 2 addition to the C 3 H 6 OH adduct. Even more emphatic evidence against simple OH addition is shown by the total absence of C2H4 [23, 98, 111] in the initial products when trans- or cis-butene-2 is added to H 2 + 0 2 mixtures or to TMB + 0 2 mixtures under the same conditions. Given that OH addition to either carbon atom in the double bond would result in C 2 H 4 through the analogous 4-centre transition state, the marked contrast in behaviour of the two alkenes is very significant. Major yields of propene (ca. 10%) are found in the initial products from isobutene oxidation between 673 and 773 K, but effectively no propene is observed initially from the oxidations of butene-1, 2-methylbutene-2 and 2,3-dimethylbutene-2. Structurally, propene formation is possible via C 3 H 7 radicals in all cases through the hydroxy adduct.

105


CH3—C —CH 2

CH3^ i

I

H—O

O

(A) CH3

CH3— CHCH2CH3 H

(B) /CH3

CH3/ j I ^CH3 H—O

(Q

CH3CH—C

/

CH3 CH3

0--H

(D)

If C 3 H 6 from isobutene is formed from the adduct A, it is difficult to understand why adducts B to D should not react similarly. However, for example, studies [110] of 2,3-dimethylbutene-2 oxidation between 650 and 790 K show that whilst acetone is a major product (ca. 25% yield), the ratio [CH3COCH3]/[C3H6] is at least 30, which appears to exclude the sequence below as an important step CH3

. CH3

CH3^| ; ^CH3 O—H

CH3COCH3 + CH3CHCH3

lo 2 CsH* + H 0 2

A sequence initiated through H 0 2 addition to isobutene could account for C 3 H 6 formation. H0 2 + (CH3)2C = CH2

(CH3)2CCH2OOH H atom transfer

(CH3)2CHCH2 + 0 2

(CH3)2CHCH200

CH3 + C3H6 A similar sequence could account for the formation of C 2 H 4 from C 3 H 6 . Unfortunately modelling studies [111] indicate that the predicted variations

106

Ch. 1


of the yields of the lower alkene with mixture composition are inconsistent with experimental data. Furthermore, QOOH—» R 0 2 isomerization is generally considered too slow to compete with alternative reactions of QOOH radicals at the temperatures used. As a final comment here, it is pertinent to point out that the problem over lower alkene formation arises solely with C 3 H 6 and isobutene which are the only alkenes to give "stable" electron delocalized radicals, allyl and methylallyl, which do not have a facile reaction with 0 2 and are consumed (at least in the range 600-1000 K) mainly through radical-radical processes. Together with the mutual reactions of the two species, their reactions with H 0 2 will be the major processes involved. It is conceivable that C 2 H 4 is formed in one of the reaction channels from the CH 2 CHCH 2 + H 0 2 reaction. CH2CHCH2 + H0 2 —~- C3H6 + 0 2

HCHO .+ CO + H0 2

I

S

[CH2 = CHCH2OOH] —— OH + CH 2 =CHCH 2 0

— C2H3 + HCHO - ^ -

CH2=CHCHO + H 0 2

m- C2H4 + HCO

A similar scheme may be written for methylallyl radicals. CH2C(CH3)CH2 + H0 2

— i-(C4H8) + 0 2

I [CH2=C(CH3)CH2OOH]

— OH + CH2=C(CH3)CH20

— CH3C=CH2 + HCHO - ^ - CH2 = C(CH3)CHO + H0 2 — C3H6 + HCO

A key set of experiments have been carried out by Walker and co-workers [111] in connection with the CH 2 CHCH 2 + H 0 2 reaction. Allyl and H 0 2 radicals were produced in three separate systems.


107

(i) The oxidation of C 3 H 6 C3H6 + 0 2 - > C 3 H 5 + H 0 2 X + C 3 H 6 ->C 3 H 5 + XH (iii) Decomposition of 4,4-dimethylpentene-l in the presence of 0 2 (Method IV, Section 1.5.2). (CH 3 ) 3 CCH 2 CH=CH 2 -^ (CH 3 ) 3 C + CH 2 CHCH 2 (CH 3 ) 3 C + 0 2

> /-C4H8 + H 0 2 99%

(iii) Decomposition of 1,5-hexadiene in the presence of 0 2 CH 2 =CHCH 2 CH 2 CH=CH 2 -> 2CH 2 CHCH 2 CH 2 =CHCH 2 CH 2 CH=CH 2 + 0 2 _> CH 2 =CHCHCH 2 CH=CH 2 + H 0 2 X + CH 2 =CHCH 2 CH 2 CH=CH 2 _> CH 2 =CHCHCH 2 CH=CH 2 + XH CH 2 =CHCHCH 2 CH=CH 2 + 0 2 -> C H 2 = C H C H = C H C H = C H 2 + H 0 2 The three systems have been examined [35, 38] in some detail at the same temperature and pressure, but the essential point here is that the initial ratio of CO: C 2 H 4 : CH 2 =CHCHO is the same within 5% for the three sources of H 0 2 and CH 2 CHCH 2 . The formation of CO and CH 2 =CHCHO has been well characterized and it seems very likely that C 2 H 4 is also an important product of the CH 2 CHCH 2 + H 0 2 reaction. Further investigations are obviously necessary, particularly in relation to possible pressure effects on the relative yields of products.

108


Ch. 1

1.17.5 CH3 radicals In general, the addition of alkyl radicals to double bonds is not important in combustion because competing processes such as homolysis and reaction with 0 2 are too fast. The only real exception to this guideline is the CH 3 radical which, as indicated earlier, has a restricted range of reactions particularly at low 0 2 pressures. In consequence, minor amounts of alkenes with a carbon number one greater than the parent alkane are found as secondary products. For example with propane, the three butenes are observed. X + CH3CH2CH3

— CH3CHCH3 + XH

CH3CHCH3 + 0 2

— CH 3 CH=CH 2 + H0 2 CH3CHCH2CH3 - ^ * - C4H8-1 + H 0 2

CH3CH = CH2 + CH3

- 2 ^

^ CH3CHCH2 CH3

^

( C

C4H8-2 + H 0 2 H

3

)

2

C = CH2

+ H02

Extensive data are available, particularly on CH 3 addition, between 300 and 600 K. In general, for CH 3 , Arrhenius parameters lie [65] in the range A = 1 0 s _ 1 and E = 30-35 klmol" 1 . Pressure effects will be particularly important for the smaller systems below 1 atm.

1.18 OXIDATION OF OXYGENATED COMPOUNDS

1.18.1 O-heterocycles A brief introduction to the oxidation of oxygenated compounds is appropriate as they are important intermediates in the total oxidation of an alkane through to C 0 2 + H 2 0 . In the temperature region below 1000 K, the major oxygenated species are aldehydes (ketones from highly branched alkanes) and O-heterocyclic compounds formed via R 0 2 —> QOOH isomerizations. An impression of the relative activity of oxiranes and oxetanes

109

Oxidation of oxygenated compounds

Ql

1

1

1

1

1

25

50

75

100

125

1

time/s Fig. 1.20. Reactivities of additives to H 2 + 0 2 mixtures at 753 K. Additive = 5, H 2 = 140, 0 2 = 70, N 2 = 285Torr. V, z-butanal; A, /-butaneoxide; x, 3,3-dimethyloxetane; O, neopentane; • , acetone.

in an oxidation environment is given in Fig. 1.20. Results are shown for various compounds when 1% is added separately to an H 2 + 0 2 mixture at 753 K [112]. An important advantage here is that each compound is effectively added to a fixed environment determined by the H 2 + 0 2 mixture. Clearly 3,3-dimethyloxetane and isobutene oxide are relatively reactive with half-lives of about 20 and 15 seconds, respectively, compared with 50, 80 and over 500 seconds for isobutene, propene and acetone, respectively. Relatively little detailed information is available on the oxidation of oxygenates. Baldwin, Walker and Keen [113] have made a comprehensive study of both the decomposition and the oxidation of oxirane at 753 K and list rate constants for a number of the elementary steps. For oxiranes, isomerization occurs exothermically due mainly to the strain energy in the ring (ca 120kJmol - 1 ). It is probable that oxirane isomerization proceeds via "chemically activated" CH 3 CHO.

110


QjRtO

— [CH3CHO]*

^ ^ . _ ~--^

Ch. 1

CH3 + HCO CH3CHO CH4 + CO

The major products from methyloxirane (propene oxide) are C 2 H 5 CHO, CH3COCH3 and CH 2 =CHCH 2 OH [90]. Similarly, isobutyraldehyde is the dominant product from isobutene oxide. Oxetanes are known to undergo homolysis through ring splitting, so that isobutene and HCHO are major products from 3,3-dimethyloxetane. CH 2 (CH3)2C I I CH2—O

— (CH 3 ) 2 C = CH 2 + HCHO

In a combustion environment, removal by free radical attack will also occur, although the C—H bond strength in oxirane is surprisingly high [113], D(CH2 —O—CH—H) = D(C 2 H 5 —H) = 420 kJ moH The resulting oxiranyl radicals isomerize 120 kJ mol - 1 ) to a variety of products.

/°\

exothermically

CH2—CH

_ CH3 + CO — CH2CHO

/ ° \ CH 3 CH—CH

— CH3CHCHO — CH3COCH2 — CH 3 C=CHOH

A

CH 3 C—CH 2

— CH3COCH2 — CH3CHCHO

(AH

Oxidation of oxygenated compounds

111

Oxetanyl radicals will readily yield alkenes. — (CH3)2C = CH2 + HCO (CH3)2C—CH2 CH-0 CH 2 —CH 2 I I — C2H4 + HCO CH — O Tetrahydropyranes and tetrahydrofuranes, with their low strain energies will tend to retain the ring more in the initial products. For example, a tetrahydrofuranyl radical will form an unsaturated ring compound in reacting with 0 2 although homolysis to give unsaturated aldehydes will also be competitive. O

CH 2

O

CH2

CH2CH2CHCH2 - ^ — CH 2 CH=CH

I

CH2 = CHCH 2 CH 2 0

lo2 CH2 = CHCH2CHO + H0 2

1.18.2 Aldehydes Inspection of most oxidation mechanisms and product profiles shows that aldehydes are primary products, particularly HCHO. Outline oxidation mechanisms for aldehydes have been discussed earlier. Essentially they are good sources of alkyl radicals through the sequence X + RCHO -* RCO + XH RCO(+M) -* R + CO(+M) At the higher temperatures, attack will also occur in the R group (particularly when X is OH).

112


Ch. 1

X + CH 3 CH 2 CH 2 CHO -* CH 3 CHCH 2 CHO + XH The CH 3 CHCH 2 CHO radical could then undergo the usual gamut of reactions available to an alkyl radical viz. CH3CHCH2CHO ° 2 , , CH3CH = CHCHO + H0 2 O'2 CH3CH = CH2 +HCO

CH3CHCH2CHO

I

00 11,5 H transfer CH3CHCH2CO

I

OOH

CH3CHO + CH 2 C=0 + OH The 1,5 H atom transfer is particularly attractive because of the weak aldehydic C—H bond involved. Baldwin, Walker and Langford [41] have explained in this way the significant yields of CH 3 CHO found in the initial products from C 2 H 5 CHO oxidation X + CH 3 CH 2 CHO -* XH + CH 3 CHCHO CH3CHCHO + 0 2 ^ CH3CHCHO -> CH3CHCO

I OO

I I

OOH

CH3CHO + CO + OH Formaldehyde is formed in a variety of ways, and together with HCO and RCO radicals is the major source of CO. X + HCHO -» HX + HCO (79)

HCO + 0 2 - » H 0 2 + CO

113

Oxidation of oxygenated compounds TABLE 1.32 Kinetic parametersa for X + HCHO -» XH + HCO based on k = AT" e~B/T [9] A/(cm3 molecule 2.1 5.7 5.0 6.9 1.3 a

x x x x x

x

s _ 1 K"")

16

10" 10" 15 10" 12 10" 13 10" 31

n

B/K

1.62 1.18 0 0.57 6.1

1090 -225 6580 1390 990

T/K 300-1700 300-2000 600-1000 300-2200 300-2000

Rate constants accurate to better than a factor of 2 over the range shown.

TABLE 1.33 Kinetic parameters3 for CO and HCO reactions based on k = AT" e v4/(cm3 molecule

Reaction OH + C O - * C 0 2 + H H 0 2 + C O - + C 0 2 + OH HCO + 0 2 - + H 0 2 + CO HCO + Ar -» H + CO + Ar a

l

s

-l

K

-n)

17

1.05 x 10~ 9.7 x 10" 11 5.0 x 10~12 2.6 x 10" 10

n

B/K

1.5 0 0 0

-250 11550 0 7930

T/K 300-2000 500-1000 300-2500 600-2000

Rate constants accurate to better than a factor of 2 over temperature range given.

(80)

HCO + M - ^ H + CO + M RCO(+M) - • R + CO(+M)

Table 1.32 summarizes the data for these reactions, where X = H, OH, O, H 0 2 and CH 3 . It is important to note (Table 1.33) the high rate constant for reaction (79) which effectively means that all HCO radicals are converted very rapidly into CO. Reaction (80) becomes more important at elevated temperature, low [0 2 ] and high total pressures. Apart from possible surface formation, above 500 K all C 0 2 is formed via CO in reactions (81) to (84). Reactions (81) and (79) act catalytically in converting H to H 0 2 radicals. (81)

H + CO(+M) -+ HCO(+M) (79)| 0 2 CO + H 0 2

(82)

OH + CO ^ C 0 2 + H

114 (83) (84)


Ch. 1

H 0 2 + CO -+ C 0 2 + OH O + CO(+M) -> C 0 2 + M

Rate constants for these key reactions have been studied extensively and are given in Table 1.33. Reaction (82) is of particular interest [50] because, first, flame-speed simulations are very sensitive to the rate of this reaction and secondly, the establishment of its non-linear temperature coefficient in the late 60s first raised doubts about the validity of the simple Arrhenius equation over wide temperature ranges.

1.19 OXIDATION OF AROMATIC COMPOUNDS

Beyond doubt, relatively little is known about the oxidation chemistry of aromatics, despite the increasing use of BTX (benzene, toluene and xylene) aromatics for improving the anti-knock behaviour of internal combustion engines. Further, the problem of polyaromatic hydrocarbons (PAH) associated with soot particles which originate particularly in diesel engines focuses attention on combustion-generated pollution. At low temperatures, radical attack involves addition to the ring, but above about 600 K the process is reversed, with the exception of O atom attack [114] so that phenol is an important product for benzene. 0 + C6H6^C6H5OH -> C 6 H 6 0 + H Arising from the high C—H bond dissociation energy in the ring (465kJmol _ 1 ), abstraction occurs preferentially on the alkyl side chain, mainly at the site which gives rise to electron-delocalized radicals, for example, benzyl, C 6 H 5 CH 2 , from toluene and C 6 H 5 CHCH 3 from ethylbenzene. Even for the non-selective OH radical, only a small percentage (C6H60

H 4- C6H5C2H5 —> C6H5C2H4

473

300-2000

0

2280

300-1400

5.3 x 10" 2.6 x 10 -11 2.8 x 10 -11

1.21 0 0

1200 1630 1840

300-2500 300-600 300-600

6.6 x 10"22 12 k- = 2.4 x 10"

3.44

1570

600-2500 773

2.7 x 10"16 2.2 x 10" U 8.6 x 10"15

1.42 0 1.0

730 400-1500 5330 1000-1200 440 400-1500

6.6 x 10"13 9.1 x 10"12

0 0

OH + C 6 H 6 -»H 2 0 + C6H5 ^ H + C6H5OH OH + C6H5CH3 -* H 2 0 4- C6H5CH2 H0 2 + QH5CH3 -> H 2 0 2 4- C6H5CH2 -»H 2 02 + e6H4CH3

C2H5

T/K

15

O 4- QH5CH3 -> products O + p-C6H4(CH3)2 -> products O 4- C6H5C2H5 -» products H 4" C6H5CH3 —> C6H5CH2 4" H2

3.8

B/K

7080 14500

600-1000 600-1000

H C2H5

••O-O

+ C2H5

0

D J — CO?Below 1000 K, the phenyl (C6H5) radical is likely to undergo oxidation to CO, probably through a sequence such as

Oxidation of aromatic compounds C6H5 + 0 2

— C6H502

117

^ 0 + C6H50

CH—CH CH2

=

CH — CH = CH "^ + CO

pjT \

pij

/ CHO

+ OH

In experiments involving the oxidation of benzene at 750 K, Baldwin et al. [115] were unable to detect acetylene in the products. Two features of the sequence should be emphasized. First, the cyclopentadienyl radical (C5H5) is stabilized through electron derealization and will predominantly react with other radicals, of which (below 1000 K) the most important will usually be H 0 2 . Secondly, the overall step (85) represents (85)

C6H5 + 0 2 ^ C 6 H 5 0 + 0

a major source of radical branching, as the majority of phenyl radicals will normally be consumed in this way. It is probably the neglect of radical attack at the ring in toluene oxidation and hence the non-inclusion of the equivalent reaction CH3QH4 + 0 2 -> CH 3 C 6 H 4 0 + O in the toluene mechanism that results in a value for klt obtained by Emdee et al. [44] which is a factor of about 100 higher than expected (see earlier) (It)

C 6 H 5 CH 3 + 0 2 ^ C 6 H 5 CH 2 + H 0 2

Above 1000 K, the C 6 H 5 + 0 2 ^ C 6 H 5 0 2 equilibrium will favour C 6 H 5 radicals which will then disappear in radical-radical processes or decompose. Stock tube studies [118] using atomic resonance absorption spectrometry to measure [H] provided strong evidence that the phenyl radical

118


Ch. 1

undergoes intramolecular rearrangement to give linear C 6 H 5 , followed by rapid decomposition to C 6 H 4 + H and other highly unsaturated species. C6H5 -> C H = C H C H = C H C = C H -> H C = C C H = C H C = C H + H -> C 2 H 2 + C H = C H C = C H -+ C 2 H 3 + C H = C C = C H C2H + C H 2 = C H C = C H Attempts to model the [H]-time profiles were frustrated by the lack of kinetic and thermodynamic data on the species involved. Stock-tube studies of toluene pyrolysis [118,119] also revealed interesting aspects of the thermochemistry. The activation energy of the initiation process was 355 10 kJ m o l " \ consistent with reaction (lAt), but not with (lAt 1 ) for which A/J « 430 kJ mol" 1 . (1 At) 1

(1 At )

C 6 H 5 C H 3 -• C 6 H 5 C H 2 + H C 6 H 5 C H 3 -> C 6 H 5 + C H 3

The high dissociation energy of the C—C bond linking the alkyl group to the ring is characteristic of alkylbenzenes and is close to D ( C H 2 = CH—CH3) = 420 lOkJmol" 1 in propene. In alkanes, D(C—C) does not exceed 370 kJ mol - 1 , so that C—C homolysis is much more rapid. Consequently for aromatics with the structure C 6 H 5 CH 2 R, for example ethylbenzene, the initiation process in the absence of 0 2 will always produce benzyl. Electron derealization is, of course, again the cause. (1 A)

C 6 H 5 CH 2 R -> C 6 H 5 CH 2 + R

Below 1000 K, in the presence of 0 2 , the initiation process will be (1) (see section 1.6.4). (1)

C 6 H 5 CH 2 R + 0 2 -> C 6 H 5 CHR + H 0 2

The reactions of benzyl are of considerable interest at all temperatures.

119

Summarizing remarks

Below 1000 K, benzyl will be consumed largely through radical-radical processes as discussed earlier. However, at higher temperatures, particularly at low 0 2 concentration, some form of homolysis should be important. Bearing in mind the stability of the radical (electron derealization), benzyl should be consumed relatively slowly. However, it is clear from Just's [118] experiments that it reacts very rapidly which rules out a number of the more obvious homolyses such as C 6 H 5 CH 2 —> C 5 H 5 + C 2 H 2 ^ c - C 5 H 5 + C2H2 —> C4H4 ~h C3H3

because each has an activation energy in excess of 350 kJ mol - 1 , compared with Just's observed value of only 190 kJ mol - 1 . He has suggested a rapid isomerization to the bicyclo-hexadienyl radical, followed by rapid fragmentation to account for the low activation energy.

o

E=190 E=190kJ.moH

CH 2 | \ | / 1

fragmentation

1.20 SUMMARIZING REMARKS

In conclusion it is worth noting that the general oxidation chemistry of alkanes and of alkyl radicals is reasonably well understood, although many of the detailed aspects are still speculative. For example, as pointed out in Section 1.9.1, there is still extensive controversy over the mechanism of the apparently simple reaction (5Ae). (5 Ae)

C 2 H 5 + 0 2 -* C 2 H 4 + H 0 2

As indicated in the Chapter, alkane oxidation proceeds through many intermediate compounds prior to the formation of the final products C 0 2 and H 2 0 . These intermediates which include O-heterocyclic compounds, aldehydes, ketones, other oxygenated species, alkenes, peroxides and CO

120


Ch. 1

can play an important part in determining the path of the oxidation, and particularly in influencing the extent of pollution from practical combustion units. However, for many of these compounds both mechanistic information and kinetic data on the elementary reactions involved are very limited. As has been pointed out, there are a number of key reactions which, under some circumstances, have enormous influence on oxidation phenomena. For a significant proportion of these reactions, vital information is still not available. In order to interpret a wide range of combustion problems over a wide range of conditions, advances in understanding have of necessity had to be made in the theories of kinetics and dynamics, modelling and sensitivity analysis, the interplay of chemical and physical effects, and the interactions of both of these with thermal factors. The later chapters will focus on these and other topics.

REFERENCES [1] V.Ya. Shtern, The Gas-Phase Oxidation of Hydrocarbons (Pergamon, London, 1964). [2] N.N. Semenov, Chemical Kinetics and Chain Reactions (Oxford University Press, Oxford, 1935). [3] J.H. Knox and C.H.J. Wells, Trans. Faraday Soc. 59, 2786 (1963) 2801. [4] A.P. Zeelenberg and A.F. Bickel, J. Chem. Soc. (1961) 4014. [5] R.W. Walker and D.A. Yorke, unpublished work. [6] R.R. Baldwin and R.W. Walker, Essays in Chemistry, eds J.N. Bradley, R.D. Gillard and R.F. Hudson, 3 (1972) 1. [7] R.G. Gilbert, K. Luther and J. Troe, Ber. Bunsenges, Phys. Chem. 87 (1983) 169. [8] D.L. Baulch, C.J. Cobos, R.A. Cox, C. Esser, P. Frank, Th. Just, J.A. Kerr, M.J. Pilling, J. Troe, R.W. Walker and J. Warnatz, J. Phys. Chem. Ref. Data 21 (1992) 411. [9] D.L. Baulch, C.J. Cobos, R.A. Cox, P. Frank, G. Hayman, Th. Just, J.A. Kerr, T. Murrells, M.J. Pilling, J, Troe, R.W. Walker and J. Warnatz, J. Phys. Chem. Ref. Data 23 (1994) 847. [10] H. Hippler, J. Troe and J. Willner, J. Chem. Phys. 93 (1990) 1755. [11] R.W. Walker, Research in Chemical Kinetics, Vol. 3, eds R.G. Compton and G. Hancock (Elsevier, Amsterdam, 1995) p. 1. [12] N. Cohen, Int. J. Chem. Kinet. 23 (1991) 683. [13] W.J. Pitz and C.K. Westbrook, Comb, and Flame 63 (1986) 113. [14] R.R. Baldwin and R.W. Walker, 14th Int. Symposium on Combustion (Combustion Institute, Pittsburgh, 1973) p. 241.

References

121

[15] G.J. Minkoff and C.F.H. Tipper, Chemistry of Combustion Reactions (Butterworths, London, 1962). [16] R.W. Walker, "A Critical Survey of Rate Constants for Reactions in Gas-Phase Hydrocarbon Oxidation, Specialist Periodical Reports (The Chemical Society, London (1975) p. 161. [17] R.T. Pollard, Comprehensive Chemical Kinetics, Vol. 17, eds C.H. Bamford and C.F.H. Tipper (Elsevier, Amsterdam, 1977) p. 249. [18] R.R. Baldwin, LA. Pickering and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 76 (1980) 2374. [19] J.W. Falconer and J.H. Knox, Proc. Roy. Soc. A250 (1959) 493. [20] F.P. Tully, A.T. Droege, M.L. Koszykowski and C.F. Melius, J. Phys. Chem. 90 (1986) 691. [21] I.R. Slagle, Q. Feng and D. Gutman, J. Phys. Chem. 88 (1984) 3648. [22] D. Jones, P. A. Morgan and J.H. Purnell, J. Chem. Soc, Faraday Trans. 1, 73 (1977) 1311. [23] R.R. Baker, R.R. Baldwin, A.R. Fuller and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 71 (1975) 736. [24] R.R. Baker, R.R. Baldwin and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 71 (1975) 756. [25] R.R. Baldwin, G.R. Drewery and R.W. Walker, J. Chem. Soc, Faraday Trans. 2, 82 (1986) 251. [26] R.R. Baldwin and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 75 (1979) 140. [27] R.R. Baldwin, J.P. Bennett and R.W. Walker, 16th Int. Symposium on Combustion (Combustion Institute, Pittsburgh, 1977) p. 1041. [28] S.K. Gulati, and R.W. Walker, J. Chem. Soc, Faraday Trans. 2, 84 (1988) 401. [29] K.G. McAdam and R.W. Walker, J. Chem. Soc, Faraday Trans. 2, 83 (1987) 1509. [30] G.M. Atri, R.R. Baldwin, G.A. Evans and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 74 (1978) 366. [31] R.R. Baldwin, H.W.M. Hisham, A. Keen and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 78 (1982) 1165. [32] R.R. Baldwin, C.E. Dean, M.R. Honeyman and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 82 (1986) 89. [33] R.R. Baldwin, D.R. Stout and R.W. Walker, J. Chem. Soc, Faraday Trans. 87 (1991) 2147. [34] R.R. Baldwin, G.R. Drewery and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 80 (1984) 2827. [35] Z.H. Lodhi and R.W. Walker, J. Chem. Soc, Faraday Trans. 87 (1991) 2361. [36] N.D. Stothard and R.W. Walker, J. Chem. Soc, Faraday Trans. 88 (1992) 2621. [37] R.R. Baldwin, C.E. Dean and R.W. Walker, J. Chem. Soc, Faraday Trans. 2, 82, (1986) 1445. [38] N.D. Stothard and R.W. Walker, J. Chem. Soc, Faraday Trans. 87 (1991) 2361. [39] T. Ingham, The Oxidation of Isobutene, PhD thesis (University of Hull, U.K., 1995). [40] R.R. Baldwin, A.R. Fuller, D. Longthorn, and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 70 (1974) 1257.

122


Ch. 1

[41] R.R. Baldwin, R.W. Walker and D.H. Langford, Trans. Faraday Soc. 65 (1969) 792, 806. [42] D.J. Dixon, G. Skirrow and C.F.H. Tipper, Combustion Institute European Symposium (Academic Press, London, 1973) p. 94. [43] D.J. Allera and R. Shaw, J. Phys. Chem. Ref. Data 9 (1980) 523. [44] J.L. Emdee, K. Brezinsky and I. Glassman, J. Phys. Chem. 96 (1992) 2151. [45] T. Ingham, R.W. Walker and R.E. Woolford, 25th Int. Symposium on Combustion (Combustion Institute, Pittsburgh, 1994) p. 767. [46] R.R. Baldwin, M.J. Matchan and R.W. Walker, Comb, and Flame 15 (1970) 109. [47] W. Tsang, Int. J. Chem. Kinet. 1 (1969) 245. [48] W. Tsang and J.A. Walker, Int. J. Chem. Kinet. 11 (1979) 867. [49] J.A. Kerr and S.J. Moss (Ed), Handbook of Bimolecular and Termolecular Gas Reactions, Vol. I (CRC Press, Boca Raton, Florida, 1981). [50] D.L. Baulch, C.J. Cobos, R.A. Cox, P. Frank, G. Hayman, Th. Just, J.A. Kerr, T. Murrells, M.J. Pilling, J. Troe, R.W. Walker and J. Warnatz, Comb, and Flame 98 (1994) 59. [51] R.W. Walker, Int. J. Chem. Kinet. 17 (1985) 573. [52] R. Atkinson and S.M. Aschmann, Int. J. Chem. Kinet. 20 (1988) 339. [53] R. Atkinson and S.M. Aschmann, Int. J. Chem. Kinet. 24 (1992) 983. [54] R. Atkinson, J. Phys. Chem. Ref. Data, Monograph 2, (1994) p. 1. [55] D.L. Baulch, M. Bowers, D.G. Malcolm and Tuckerman, J. Phys. Chem. Ref. Data 15 (1986) 465. [56] W. Tsang, J. Phys. Chem. Ref. Data 17 (1988) 887. [57] W. Tsang, J. Phys. Chem. Ref. Data 19 (1990) 1. [58] C.K. Westbrook and F.L. Dryer, Comb. Sci. Tech. 20 (1979) 125. [59] C.J. Jachimowski, Comb, and Flame 55 (1984) 213. [60] W.J. Pitz, C.K. Westbrook, W. Proscia and F.L. Dryer, 20th Int. Symposium on Combustion (Combustion Institute, 1985) p. 831. [61] D.B. Smith, Paper presented at the Autumn Meeting of the Chemical Society (University of Hull, U.K., September, 1984). [62] D.J. Hucknall, D. Booth and R.J. Sampson, Int. J. Chem. Kinet. Symp. Ed. 1, (1975) 301. [63] S. Handford-Styring and R.W. Walker, to be published. [64] H.M. Frey and R. Walsh, Chem. Rev. 69 (1969) 103. [65] J.A. Kerr and S.J. Moss (ed), Handbook of Bimolecular and Termolecular Gas Reactions, Vol. 2 (CRC Press, Bocca Raton, Florida, 1981). [66] B.S. Rabinovitch and D.W. Setser, Advances in Photochem., Vol.3 (Wiley, New York, 1964) p. 1. [67] S.W. Benson and H.E. O'Neal, Kinetic Data on Gas Phase Unimolecular Reactions, NSRDS-NBS-21 (U.S. Government Printing Office, Washington D.C., 1970). [68] R.R. Baldwin, R.W. Walker and Robert W. Walker, J. Chem. Soc, Faraday Trans. 1, 77 (1981) 2157. [69] R.R. Baldwin, J.P. Bennett and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 76 (1980) 2396.

References

123

[70] N.D. Stothard and R.W. Walker, J. Chem. Soc, Faraday Trans. 2, 86 (1990) 2115. [71] A.F. Wagner, I.R. Slagle, D. Sarzynski and D. Gutman, J. Phys. Chem. 94 (1990) 1853. [72] A.M. Dean, J. Phys. Chem. 89, (1985) 4600. [73] J.W. Bozzelli and A.M. Dean, J. Phys. Chem. 94 (1990) 3313. [74] G.E. Quelch, M.M. Gallo and H.F. Schaeffer, J. Am. Chem. Soc. 114 (1992) 8239. [75] S.P. Walch, Chem. Phys. Lett. 215 (1993) 81. [76] A. Fish, Organic Peroxides, Vol. 1, ed D. Swern, (Wiley, New York, 1970) p. 141. [77] S.W. Benson, F.R. Cruickshank, D.M. Golden, G.R. Haugen, H.E. O'Neal, A.S. Rodgers, R. Shaw and R. Walsh, Chem. Rev. 69 (1969) 279. [78] R.R. Baldwin, M.W.M. Hisham and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 78 (1982) 1615. [79] R.R. Baldwin, J.P. Bennett and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 76 (1980) 1075. [80] R.R. Baker, R.R. Baldwin, C.J. Everett and R.W. Walker, Comb, and Flame 25 (1975) 285. [81] R.R. Baker, R.R. Baldwin and R.W. Walker, Comb, and Flame 27 (1976) 147. [82] K.J. Hughes, P.A. Halford-Maw, M.J. Pilling and T. Turanyi, 24th Int. Symposium on Combustion (Combustion Institute, Pittsburg, 1992) p. 645. [83] S.W. Benson and R. Shaw, Organic Peroxides, Vol. 1, ed D. Swern (Wiley, New York, 1970) p. 105. [84] T. Mill, 13th Int. Symposium on Combustion, Vol. 1 (Combustion Institue, 1971) p. 237. [85] S. Kojima, Comb, and Flame 99 (1994) 87. [86] P.D. Lightfoot, R. Lesclaux and B. Veyret, J. Phys. Chem. 94 (1990) 700. [87] M. Keiffer, M.J. Pilling and M.J.C. Smith, J. Phys. Chem. 91 (1987) 6028. [88] F. Ewig, D. Rhasa and R. Zellner, Ber. Bunsenges, Phys. Chem. 91 (1987) 708. [89] K. Saito, R. Ito, T. Kakumoto and A. Imamura, J. Phys. Chem. 90 (1986) 1422. [90] M.C. Flowers, J. Chem. Soc, Faraday Trans. 1, 73 (1977) 1927. [91] N.D. Stothard and R.W. Walker, J. Chem. Soc, Faraday Trans. 88 (1992) 2621. [92] N.D. Stothard and R.W. Walker, to be published. [93] S.K. Gulati and R.W. Walker, J. Chem. Soc, Faraday Trans. 2, 85 (1989) 1799. [94] S. Handford-Styring and R.W. Walker, J. Chem. Soc, Faraday Trans. 91 (1995) 1431. [95] A.S. Gordon, Can. J. Chem. 43 (1965) 570. [96] D.F. McMillen and D.M. Golden, Ann. Rev. Phys. Chem. 33 (1982) 493. [97] F.P. Tully and J.E.M. Goldsmith, Chem. Phys. Lett. 116 (1985) 345. [98] R.R. Baldwin and R.W. Walker, 18th Int. Symposium on Combustion (Combustion Institute, Pittsburg, 1981) p. 819. [99] D.J.M. Ray and D.J. Waddington, Comb, and Flame 20 (1973) 327. [100] I.R. Slagle, J.Y. Park, M.C. Heaven and D. Gutman, J. Am. Chem. Soc. 106 (1984) 4356. [101] J.W. Bozzelli and A.M. Dean, J. Phys. Chem. 97 (1993) 4427. [102] W. Tsang and J.A. Walker, private communication.

124


Ch. 1

[103] I.R. Slagle, A. Bencsura, S-B. Xing, and D. Gutman, 24th Int. Symposium on Combustion (Combustion Institute, Pittsburg, 1992) p. 653. [104] R.J. Cvetanovic, J. Phys. Chem. Ref. Data 16 (1987) 261. [105] R.B. Klemm, J.W. Sutherland, Wickramaaratchi and G. Yarwood, J. Phys. Chem. 94 (1990) 3354. [106] K. Mahmud, P. Marshall and A. Fontijn, J. Phys. Chem. 91 (1987) 1568. [107] D.A. Osbourne and D.J. Waddington, J. Chem. Soc, Perkin Trans. 2, 65 (1980) 925. [108] M.I. Sway and D.J. Waddington, J. Chem. Soc., Perkin Trans. 2, 139 (1983). [109] L. Batt, private communication. [110] R.W. Walker and R.E. Woolford, to be published. [Ill] R.R. Baldwin and R.W. Walker, to be published. [112] R.R. Baker, PhD thesis (University of Hull, 1969). [113] R.R. Baldwin, A. Keen and R.W. Walker, J. Chem. Soc, Faraday Trans. 1, 80 (1984) 435. [114] J.M. Nicovich, C.A. Gump and A.R. Ravishankara, J. Phys. Chem. 86 (1982) 1684. [115] R.R. Baldwin, M. Scott and R.W. Walker, 21st Int. Symposium on Combustion (Combustion Institute, Pittsburg, 1986) p. 991. [116] J.L. Emdee, K. Brezinsky and I. Glassman, J. Phys. Chem. 95 (1991) 1626. [117] T.A. Litzinger, K. Brezinsky and I. Glassman, Comb, and Flame 63 (1986) 251. [118] M. Braun-Unkhoff, P. Frank and Th. Just, 22nd Int. Symposium on Combustion (Combustion Institute, Pittsburg, 1989) p. 1053. [119] C. Hippler, C. Reihs and J. Troe, J. Phys. Chem. Neue Folge 167 (1990) 1.

Chapter 2

Elementary Reactions STRUAN H. ROBERTSON, PAUL W. SEAKINS and MICHAEL J. PILLING

2.1 INTRODUCTION

Chapter 1 emphasized the relationship between the component elementary reactions and the macroscopic behaviour of hydrocarbon oxidation. Any detailed description of autoignition, for example, must be based on a chemical mechanism comprising these elementary reactions, including specification of both reactants and products, and the associated rate constants. For many of the reactions involved in hydrocarbon oxidation, the products have been established over many years through end-product analysis, especially by Baldwin, Walker and coworkers (see Chapter 1). Their techniques have also been a major source of rate data via comparative methods, relating the target-rate constants to established values using observed product ratios. This approach has been widely used throughout chemical kinetics, because the direct methods for measuring rate constants, such as flash photolysis, found comparatively limited application. This situation has changed dramatically over the past twenty years as the range of techniques for monitoring transient species has expanded. Thus a wide range of intermediates can now be studied and, since the sensitivity is often very high, their concentrations can be kept low, minimising interference from unwanted reactions. The precision of the techniques has also improved, through the use of repetitively pulsed lasers and better methods of recording and analysing data. This has led to the generation of more reliable rate data which have been validated over a wide range of conditions. A consequence is the ability to address more complex reaction systems in which the target reaction is one component of a more complex reaction chain. Often the skill of the experimentalist lies in devising conditions in such systems under which the required rate data can be extracted with optimal accuracy.

126

Elementary reactions

Ch. 2

RH + X ; X = 0 2 , O H , . . .

Decomposition

Fig. 2.1. Schematic diagram of the prototypical reactions occurring during low temperature combustion of a hydrocarbon RH.

In this chapter we examine the use of direct methods, largely based around laser flash photolysis and discharge flow, to determine rate constants for reactions central to hydrocarbon oxidation (Fig. 2.1). This chapter also demonstrates how theory can be used to interpret the results of rate measurements and act as a framework for extending their use to conditions which may not be accessible to experiment. The alkyl radical lies at the centre of the oxidation scheme shown in Fig. 2.1 and the first two sections discuss its method of generation both in the initiation and chain propagation processes. The former has already been discussed in some detail in Chapter 1 and is not amenable to study by direct techniques. It is included here to produce a more complete picture of the overall methodology needed to quantify a reaction mechanism and to emphasize that direct techniques cannot provide all of the answers. The hydroxyl radical (OH) is the main chain carrier and the bulk of Section 2.3, is devoted to the measurement of rate constants for its hydrogen atom abstraction reactions with hydrocarbons. Much of the data and the methodology described in this section derive from studies of tropospheric chemistry, where the oxidative chain, at least in its early stages, shows a close relationship to the higher temperature processes which are central to this book. Indeed, it is fair to say that many of the developments in

Introduction

127

technique, referred to in the previous paragraph, derive from the need for the development of a better understanding of atmospheric chemistry. Abstraction by other radicals is also discussed, together with the increasing appreciation of the curvature present in many Arrhenius plots. This observation derives from the improved experimental precision and the increased temperature range over which radical reactions can be studied. It illustrates the need for experiment rather than simple extrapolation, or for extrapolation firmly based on a sound theoretical understanding of the reaction in question. Happily for us, at the temperatures of significance in the processes central to this book, curvature is of minor importance. Radical decomposition is examined in Section 2.4. These reactions provide the main competitive route to reaction of the radical with 0 2 and lead to a reduction in the size of the alkyl radical centres. Until recently, they were studied primarily by end product analysis, usually in competition with recombination. The application of laser-flash photolysis, coupled with photoionization mass spectrometry to monitor the alkyl radicals, has provided a major contribution to the quantitative characterization of radical decomposition. The reactions are pressure dependent and this characteristic raises an added level of complexity. It is often necessary to use rate data at pressures significantly higher than those used in the laboratory (e.g. for modelling autoignition in engines). Therefore, it is important that the rate data can be reliably extrapolated to the conditions required for modelling. Section 2.4 also includes a discussion of pressure-dependent reactions, of the methods for fitting and extrapolating the laboratory rate data and for parameterizing the data for incorporation in numerical models. Termination of the radical chains occurs through radical recombination, primarily involving small radicals. The experimental and theoretical study of radical recombination processes has been an area of intense activity. The reactions present experimental challenges because their second-order nature requires the determination of the radical concentration. The reactions are also pressure dependent, especially at the temperatures of interest here and particularly for the small radicals that contribute significantly to termination in hydrocarbon oxidation. They also provide an interesting challenge to the theoretician because they occur on so-called Type II potential energy surfaces, which have no potential energy maximum. The transition state, in consequence, has to be located variationally, a process which requires an understanding of the interaction between the angular

128


Ch. 2

vibrational modes of the molecule, e.g., the torsional and rocking vibrations. A full discussion is beyond the scope of this book, but reference is made to the major determinants of the recombination rate constants through the medium of the simple, but incompletely understood, CH 3 + CH 3 and CH 3 + H reactions. Section 2.5 examines addition reactions which are the reverse of the radical decomposition reactions considered in Section 2.4. These reactions in themselves are comparatively unimportant in hydrocarbon oxidation, but they have provided a good source of thermodynamic data on radicals. Thermodynamic parameters are central to the modelling of autoignition because of the importance of heat release, but also because of their use in determining the rate parameters for the reverse of well characterized reactions. Section 2.5 includes a brief review of the currently accepted alkyl radical heats of formation. This field has been in turmoil in recent years because of disagreements on the values, which largely derive from kinetic measurements. Consensus is emerging but controversy still remains. The reactions which lie at the heart of hydrocarbon oxidation are discussed in Sections 2.6 and 2.7. The former addresses the reaction between alkyl radicals and oxygen, both the formation (and the decomposition) of the alkyperoxy radical and the formation of the alkene + H 0 2 , which predominates at high temperatures. The formation of the alkyl peroxy radical is now well characterized; several rate constants have been determined over a range of temperature and pressure, and correlations developed to assist the estimation of parameters for uncharacterized reactions. The reactions of smaller alkyl radicals with 0 2 are pressure dependent. The thermodynamic parameters for the formation of the alkylperoxy radical are also well defined. These reactions were the first to have their approach to equilibrium monitored directly by laser-flash photolysis. Such an investigation permits both the equilibrium constant and the forwardand reverse-rate constants to be measured. The onset of the decomposition of the peroxy radical, and therefore of the negative temperature coefficient in the overall oxidation is determined by the thermodynamics of the reaction. While considerable progress was made over a number of years, using thermodynamic parameters estimated using the group additivity methods developed by Benson [1] and coworkers, the direct measurements revealed significant errors in this approach for this particular application, related partly to the alkyl radical values referred to above, but also to the peroxy

Introduction

129

radical group value itself. This revision has changed substantially the magnitude of the equilibrium constants, especially for the larger radicals, and influences the temperature at which the negative temperature region is calculated to commence. Also, there has been some controversy over the mechanism of the reaction leading to the alkene and H 0 2 . The current status of this reaction is reviewed and an attempt made to resolve the controversy. Interestingly, this reaction provides a good example of the contributions that can be made to our qualitative and quantitative understanding of reaction mechanisms by ab initio calculations. The proposed mechanism and the radical thermodynamics have an impact on the peroxy/hydroperoxy radical isomerization that provides the route out of the peroxy system and both continues the propagation, and leads to the degenerate branching agent (Fig. 2.1). The isomerization, which is discussed in Section 2.7, constitutes the component of the mechanism that is most sensitive to the structure of the parent alkane and so provides the main link between ignition characteristics and molecular structure. Most of our understanding of these reactions derives from the end-product work of Walker and coworkers (Chapter 1), but it has been recognized that there are problems in interpreting their data deriving from the uncertainties in radical thermodynamics. Direct measurements of the rate constant for the neopentyl system, that provide a means of setting the whole of the Walker data on a quantitative basis that is totally consistent with the new thermodynamic parameters, is described in Section 2.7 and links are made with the mechanism proposed for the R + 0 2 reactions. Section 2.8 briefly discusses reactions central to the H 2 / 0 2 system. There are three reasons for this discussion. First, the hydrogen/oxygen reaction in a continuous stirred tank reactor is used widely in Chapters 1, 4 and 5 as a model system of relevance to reactions involving thermokinetic processes. Second, the elementary reactions contained within the complex mechanism become increasingly important as the temperature of a combustion process increases. As discussed in Chapters 1 and 7, while low-temperature oxidation chemistry determines the main features of the onset of autoignition, it is the high-temperature chemistry that drives the damage associated with knock. So, although the high-temperature reactions are not of central concern to the subject of this book, they are relevant. Finally, elementary reactions of small radicals are most amenable to detailed study from the viewpoint of both kinetics and dynamics and using both experimental and theoretical techniques. Therefore, such reactions

130


Ch. 2

provide a detailed framework within which our understanding of reactions of larger species can be developed. The emphasis throughout this chapter is on the measurement and calculation of rate constants for elementary reactions of central importance in hydrocarbon oxidation. The major aim of the activity is to provide data for modellers and so the temperature and pressure of real systems is, as discussed earlier, paramount. Thus, the mere provision of data is not sufficient; it is necessary to extrapolate and, therefore, to understand the processes involved. In consequence, the study of reaction dynamics and the development of theoretical models are also key components of the strategy and of this chapter. The study of elementary reactions for a specific requirement such as hydrocarbon oxidation occupies an interesting position in the overall process. At a simplistic level, it could be argued that it lies at one extreme. Once the basic mechanism has been formulated as in Chapter 1, then the rate data are measured, evaluated and incorporated in a data base (Chapter 3), embedded in numerical models (Chapter 4) and finally used in the study of hydrocarbon oxidation from a range of viewpoints (Chapters 5 7). Such a mode of operation would fail to benefit from what is ideally an intensely cooperative and collaborative activity. Feedback is as central to research as it is to hydrocarbon oxidation! Laboratory measurements must be informed by the sensitivity analysis performed on numerical models (Chapter 4), so that the key reactions to be studied in the laboratory can be identified, together with the appropriate conditions. A realistic assessment of the error associated with a particular rate parameter should be supplied to enable the overall uncertainty to be estimated in the simulation of a combustion process. Finally, the model must be validated against data for real systems. Such a validation, especially if combined with sensitivity analysis, provides a test of both the chemical mechanism and the rate parameters on which it is based. Therefore, it is important that laboratory determinations of rate parameters are performed collaboratively with both modelling and validation experiments.

2.2 REACTION INITIATION

The gas phase initiation of hydrocarbon oxidation is thought to occur by the following slow reaction:

Reaction initiation

(1)

131

RH + 0 2 ^ H 0 2 + R

Isolated study of this type of reaction is difficult for the following reasons: (a) Primary initiation is rapidly masked by secondary initiation from molecules generated in the oxidation of the hydrocarbon. For example, the rates of primary and secondary initiation, from hydrogen peroxide decomposition, in a mixture of 100 Torr of methane and oxygen at 773 K, will become equal when only 10" 4 % of methane has been converted to H 2 0 2 [2]; (b) the consumption of reagents and production of products is very quickly dominated by rapid chain processes rather than by the initiation process; and (c) surface catalysed initiation may be the dominant process, proceeding with a significantly lower activation energy than its gas phase counterpart. Slow reactions such as (1) are difficult to study by direct-time resolved spectroscopic methods. They can be investigated by end-product analysis, but the number of systems that can be examined is limited because of the need to suppress the rapid chain oxidation processes and careful experimental procedures are required to eliminate the effects of surface or secondary initiation. Walker and coworkers have used gas chromatographic (GC) end-product analysis to investigate initiation in propene [2] and formaldehyde [3] oxidation, systems in which, for different reasons, rapid chain oxidation can be prevented. The experimental technique is deceptively simple, consisting of a heated pyrex reaction vessel, coated with aged boric acid, from which samples can be taken for GC analysis. For the initiation of propene/oxygen mixtures in the temperature range 673-793 K, the major hydrocarbon product is hexa-l,5-diene (HDE) and the reaction scheme can simply be written as, (2) (3)

C3H6 + 0 2 ^ C 3 H 5 + H 0 2 2C3H5^C6H10

The simplicity of the scheme arises from the fact that, even at these relatively low temperatures, the equilibrium for the addition of oxygen to form the allylperoxy radical (from which secondary radicals could be formed by processes outlined in Fig. 2.1) lies predominantly to the left [4]

132


50

100

Ch. 2

150

200

f/s

Fig. 2.2. Pressure vs time profiles for products from C3H6 + 0 2 at 480°C, showing the autocatalytic nature of the reaction. Initial conditions C3H6 = 12, 0 2 = 30, N2 = 18 Torr. O, CO; x, HCHO; • , C2H4; _, CH 3 CHO; V, CH 2 =CHCHO; • , hexadiene (x5); O, propene oxide (x5).

(4-4)

C3H5 + 0 2 - C 3 H 5 0 2

and the only significant reaction of the allyl radical is recombination. Under these conditions the initial rate of formation of HDE, (R H DE)O> is equal to the rate of initiation. (RHDE)O

= £i[C 3 H 6 ][0 2 ]

(2.1)

Figure 2.2 shows that there is some autocatalysis present in the system and, hence, (R H DE)O is obtained by plotting the tangents of the HDE curve vs time and extrapolating to zero time. In general, Table 2.1 shows that there is relatively good agreement for kx over a range of oxygen and propene pressures; however, at low propene: oxygen ratios alternative termination routes such as reactions (5) and (6) should be considered,

(5)

H0 2 + C3H5 -* C3H60 + OH

(6)

H0 2 + H0 2 -> H 2 0 2 + 0 2

133

Reaction initiation TABLE 2.1 Summary of results for the determination of kx at 753 K Mixture composition3 /Torr C3H6

o2

2 4 4 4 12 12 12 16

30 5 10 30 5 10 30 30

(RHDE)O

» *-*

15 i—

jo

o

0

4

8

TIME (milliseconds)

Fig. 2.4. (a) [OH]t profile (O) and exponential fit (—) for OH + C 2 H 6 experiment, T = 478 K, P = 600 Torr of helium, [C2H6] = 1.26 x 1014 molecule cm" 3 ; (b) and (c) diffusion) for OH + C 2 D 6 experiment, corrected and uncorrected [OH]t profiles ( • , _) and fits ( T = 705 K, P = 600 Torr helium, [C2D6] = 5.91 x 1013 molecule cm - 3 .

Pseudo-first-order rate coefficients are obtained from the gradient of a plot of ln[OH] t vs t, e.g., Fig. 2.4. If the pseudo-first-order rate coefficient is determined for a number of different ethane concentrations, then the bimolecular rate coefficient for reaction (10) will be the gradient of a plot of k' vs[C 2 H 6 ]. OH radicals can be generated from a number of different precursors. In one of their studies Tully et al. [7] used an indirect method, first producing excited oxygen (0( 1 D)) atoms from the photolysis of nitrous oxide with 193 nm photons from a nanosecond pulse from an excimer laser

en)

N 2 0 + h»'-»N2 + 0( 1 D)

followed by a rapid reaction with water to generate two hydroxyl radicals (12)

0( 1 D) + H 2 0 ^ 2 0 H

138


Ch. 2

Substrate concentrations are chosen such that OH is generated on a timescale that is short in comparison to its decay via reaction (10). Alternative methods include direct photolysis of suitable precursors, e.g., (13)

H N 0 3 + hi OH + N 0 2

For all production methods and especially for direct photolytic methods, care needs to be taken so that the radicals are generated, or rapidly converted, into a thermal distribution as relaxation of electronic or vibrational states into the monitored ground state would complicate the kinetics. The advantages of pulsed lasers as light sources are well documented [9]. In the experiments of Tully et al. [7] the time dependence of the concentration of OH radicals is monitored by laser-induced fluorescence. Light at 308 nm, corresponding to a transition from the ground vibrational state of OH to the ground vibrational state of the first electronically excited state (A c a n be varied t 0 build up a time profile of m e OH decay

me

Fig. 2.5. Schematic diagram showing the method of LIF data collection for a pulsed dye laser.

recorded from 292-705 K. The concentrations of N 2 0 , water and ethane and the total pressure in the cell were monitored by calibrated flow controllers and a capacitance manometer respectively. Figure 2.6 shows an Arrhenius representation of the data of Tully and a number of other workers. There is generally good agreement ) between this work and that of previous authors, the variation being close to the experimental uncertainty involved with each determination. The graph also shows significant non-linearity, even over the limited range of the experimental determinations (240-830 K), and k10 can be represented by the three parameter expression k10 = (8.51 x i o - i y ) r Z U 6 e x p ( - 4 3 0 / r ) cm 3 molecule" 1 s" (2.7) 2.3.3 Transition state theory (TST) It is often found that the canonical rate coefficient can be expressed in terms of simple formulae with a small number of parameters, the most

140

Ch. 2


2.2

3.2

4.2

1000/ T(K) Fig. 2.6. Temperature dependence of reaction (10). Full details of the experimental measurements can be found in reference (7).

well known being the Arrhenius expression

k(T)=Aexp(-E/RT).

(2.8)

Moreover, the parameters themselves can often be related to certain features of the potential energy surface (PES) on which the reaction occurs, the most obvious being E of equation (2.8) which is clearly related to the height of the barrier which the system must cross to react. The classical approach to TST, in terms of a quasi-equilibrium between reactants and transition state, can be summarized as follows: consider the reaction (14)

A+B^C+D

Abstraction reactions

141

A line may be drawn on the PES of the reaction which connects reactants to products such that each point along this line the energy is a minimum with respect to motion normal to the line. Such a curve is called the reaction coordinate. The profile of the energy along such a curve is shown in Fig. 2.7, and as can be seen, the profile exhibits a maximum which acts as a barrier to reaction. The configuration of the reactants at the top of the barrier is referred to as the transition state (TS). The mechanism of reaction (14), can be represented as ^C+D

A+B

(2.9)

where {AB}* represents the TS. A quasi-equilibrium is assumed to exist between the reactants and the TS and it can be shown that the rate coefficient for this mechanism is given by (2.10)

k(T) = kBT/h.K*

where K* is the equilibrium constant for the quasi-equilibrium. Application of standard statistical mechanical techniques for the calculation of equilibrium constants to equation (2.10) yields k

^

=

^TTrt h

eM-Eo'RT)

(2.11)

QAQB

where QA and QB are the partition functions, per unit volume of the reactants, QTS is the partition function, per unit volume, of the TS and E0 is the reaction threshold energy (difference in zero point energies between the reactants and the TS). The calculation of QA and QB can be performed using standard statistical mechanical techniques. The calculation of QTS is a more difficult problem because there is usually little information about the structure of the TS. An approximation of the TS properties can be determined by examining the properties of the reactants and products as it is often possible to correlate vibrational frequencies between the two. The change over from reactant vibrational frequencies to those of the products can be modelled by using a suitable interpolation function which has adjustable parameters that can be altered in a fitting exercise. More recently, much use has

c P h)

(A + BC) 1

Potential encrgy

I

Transition stale \

\

Reaction co-ordinate

I,

n

Completely dissociated state

u

A+BtC

(4 Fig. 2.7. (a) Potential energy surface for an A

(b) + BC-

AB

+ C reaction;

(b) minimum energy pathway across the PES. 0 3 N


143

been made of ab initio quantum mechanical techniques in locating and determining the properties of the TS. This highlights one of the main advantages of TST, that information about one point only is required, rather than the whole PES. The partition functions can be calculated either classically or quantum mechanically and consequently there are both quantum and classical versions of TST. The cost of a quantum calculation is comparable to that of a classical calculation and as a consequence the quantum mechanical theory is predominately used. 2.3.4 Isotope effects One of the simplest ways of obtaining mechanistic information about Hatom abstractions is to investigate the effects on the rate coefficient of replacing hydrogen atoms with deuterium. Deuterium substitution will affect both the pre-exponential term of equation (2.11) via the partition functions of the reagents and TS and will also increase the effective activation energy by lowering the zero-point energy of the reagents to a greater extent than for the TS (Fig. 2.8). A plot of the natural logarithm of the ratio of the rate coefficients vs 1/Twill yield AE/R as the gradient of the graph, where A£ is the difference in activation energies for the two systems. Pre-exponential contributions will be revealed by curvature of the plot. Hess and Tully [8] have looked at the effect of deuterium substitution on reactions (15) and (16). (15)

OH + CH 3 OH -> H 2 0 + CH3O H 2 0 + CH 2 OH

(16)

OH + CD3OH -» H 2 0 + CD3O HDO + CD 2 OH

Figure 2.9 shows the isotope effect, the ratio of k15 to &16, plotted against IIT (on a log scale). The graph breaks down into two regions. At lower temperatures the weaker C—H or C—D bond is broken, the activation energy of the two reactions is different and hence the ratio of rate coefficients is temperature dependent. A linear least squares fit to the lower temperature data gives an isotopic change in activation energies of

144


Ch. 2

£?

i

|:/2hvH\ XJl/2hv \ l D Reactant

Reaction co-ordinate Fig. 2.8. Potential energy profile for an isotopic reaction showing the increase in activation energy which arises on deuterium substitution.

1.01 kJmol" 1 , a value characteristic of the C—H bond strength in methanol. At higher temperatures the isotope ratio deviates from the extrapolation of the best fit line indicating the onset, and increasing dominance, of the hydroxyl abstraction channel. Hess and Tully [8] conclude that the room temperature channel to form methoxy is insignificant, in contrast to earlier workers who determined minor, but significant, channel efficiencies of 0.11-0.25. All studies show an increasing contribution of the hydroxyl abstraction with temperature but there is still significant scatter in the values of the temperature at which the two channels are of equal importance. In the experiments on reaction (10), described in detail in Section 2.3.2, Tully et al. [7] investigated the reaction of OH with deuterated and partially deuterated ethanes (10)

OH + C 2 H 6 -> H 2 0 + C 2 H 5

145

Abstraction reactions 2.4 •

2.2-

2-

2"

L8

'

J-

]= 1.6-

I 1.4-

1 1.2-

1 -

1

1

1

1.5

2

2.5

1

1 3.5

1000/T(K)

Fig. 2.9. Ratio of rate coefficients for the isotopic reactions of OH + CH 3 OH and OH + CD 3 OH as a function of temperature.

OH + CD 3 CH 3 -+ HDO + CD 2 CH 3

(17a)

H 2 0 + CD 3 CH 2

(17b)

OH + C 2 D 6 -+ HDO + C 2 D 5

(18)

It was found that the temperature data for reactions (10) and (18) could be represented by modified Arrhenius equations with identical pre-exponential factors. In its thermodynamic formulation TST defines a bimolecular rate coefficient as fc=M:.wAct exp(AS*AR)

h

exp(-AHVRT).

(2.12)

The neglect of pre-exponential factors in isotope analyses of activation energies is only valid if the entropy changes on forming the TS or the temperature dependence of the entropy change is identical for the two

146


Ch. 2

isotopomers, i.e., the intercept on the y axis of an isotopic analysis is 1 or there is no curvature in the plot. Tully et al. performed a theoretical calculation on reactions (10) and (18), determining the structure of the TS from ab initio calculations. Vibrational frequencies and rotational constants were then determined and used to calculate the entropy change on forming the TS for the reactions. While there were significant variations in the entropy changes associated with the translational, rotational and vibrational contributions, the net difference and the variation with temperature was very small, confirming the experimental observations and validating the use of identical pre-exponential factors. If the rate coefficients for abstraction at a particular site are unaffected by isotopic substitution of neighbouring groups then simple geometric arguments would predict that k17(T) = 0.5klo(T) + 0.5kls(T).

(2.13)

Tully et al. found this equality to be correct within an average deviation of 1% over the entire range of their studies. This separability of reactivity by site is of great significance and is discussed in detail in the next section. 2.3.5 Additivity of rate coefficients From the viewpoint of the combustion chemist, mechanistic and theoretical studies of abstraction reactions serve two purposes. First, they can determine the overall rate coefficient for an abstraction over a range of temperatures, especially when there are limited experimental data. Second, the combustion modeller wishes to know the rate of abstraction at any particular site on a hydrocarbon molecule. For reaction (10) this is trivial as there is only one type of site; a primary C—H bond. However, for more complex fuels there will be a variety of different sites which to a first-order approximation can be considered as primary, secondary and tertiary C—H bonds. As mentioned in the introduction to this section, Atkinson et al. [10] and Walker [11] have attempted to describe radical/ alkane kinetics with the following simple model based on equation (2.4) ^(radical + alkane)

=

^p^p

+ ^s^s

+ ntKt

(2-4)

i.e., the overall rate coefficient is the sum of the primary, secondary and

147

Abstraction reactions TABLE 2.3 Comparison of room temperature rate coefficients for OH abstractions Alkane

1012 x A:/cm3 molecule -S"

1

A%

Observed3

Calculated5 4.1 3.9 -1.2 -6.5

n-alkanes propane n-butane «-hexane /i-nonane Branched alkanes Isobutane Neopentane 2,2,3-Trimethylbutane 2,2,4-Trimethylpentane

1.22 2.58 5.70 10.7

0.09 0.4

1.27 2.68 5.63 10.0

2.29 0.77 4.21 3.66

0.06 0.05 0.08 6

2.23 0.71 4.33 6.03

-2.6 -7.8 2.9 64.8

Unstrained cycloalkanes Cyclohexane Methylcyclohexane

7.57 10.6

0.05 0.3

8.84 11.0

16.8 3.8

0.05

Experimental determinations from references [10, 12, 13] using relative rate techniques with rc-butane as the reference compound. b Values calculated including neighbouring group corrections from Ref. [10].

tertiary rates where np is the number of primary C—H bonds and kp is the primary rate coefficient per primary C—H bond etc. Measurements of the rates of radical reactions with neopentane and ethane allow a determination of kp which can then, in turn, be used to calculate A:s and kt from reactions with say propane and isobutane respectively. The temperature dependence of kp, ks and kt can be expressed in simple Arrhenius form or as an ATn exp(EIRT) expression [11]. More complex formulations, taking into account neighbouring group effects (or the nonequivalence of each primary, secondary or tertiary site), have been introduced by Atkinson et al. [10]; however, it is debateable whether such detail is justified by the precision of the experimental data. The validity of the procedure can be verified by comparisons of calculated and experimental overall rate coefficients [10]; examples are shown in Table 2.3. The excellent agreement between calculated and experimental determinations indicates that unknown rate coefficients for alkane abstractions can be predicted by this method with a high degree of confidence. Some of the branched chain alkanes may be an exception to this rule. The large deviations between experiment and calculations are attributed by Atkinson et al. [10] to possible steric hindrance.

148

Ch. 2


TABLE 2.4 Calculated percentages of radicals formed in the overall reaction OH + alkane Alkane

propane ^-butane

Radical

n-C3H7 Z-C3H7 A1-U4XI9

5-C4H9

n-pentane

l-CsHu 2-C5Hn 3-C5Hn

Temperature/K 300

500

750

1000

30 70 14.5 85.5 9.5 55.5 35.0

46.5 53.5 27 73 19 51.5 29.5

55 45 36 64 26.5 47.5 26

59.5 40.5 40.5 59.5 31 45 24

The importance of such calculations not only lies with the prediction of the absolute values, but also in determining the relative site specific rate coefficients, and hence, the fractions of the different isomeric radicals which can be formed (Table 2.4). Experimental determinations of site specific rate coefficients are scarce. As seen in subsequent sections, the final products of combustion depend strongly on the nature of the radical intermediates. An alternative method of predicting site specific rate coefficients based on TST has been devised by Cohen [14, 15, 16]. The thermodynamic formulation of TST (equation (2.12)) is based on examining the free energy change on forming the TS and can be divided into two components, (i) an entropy factor relating to the change in entropy on forming the TS and, (ii) an enthalpy factor (related to the activation energy). Molecular constants such as moments of inertia and vibrational frequencies are required to calculate entropies and, in general, these are not available for the TS. Cohen has developed a method whereby the entropy of the TS is calculated from known values of stable model compounds (generally the corresponding alcohol) with suitable corrections from group additivity considerations. Barrier heights and activation energies are found by fitting the room temperature TS prediction of the rate coefficient to known experimental values with the barrier height as a variable parameter. Figure 2.10 shows that, in general, the agreement with experimental data is good and hence the method can be used to extrapolate experimental data. The earlier work by Cohen [14] has recently been updated [15, 16] in the light of arguments about the non-equivalence of all primary, secondary

149


•p-io

a a

-11

O

0.0

1.0

2.0

3.0

1000 (K)/T Fig. 2.10. Temperature dependence of the OH + /i-C4Hi0 reaction. Solid line = calculated rate coefficient using TST; points = experimental determinations. Full details of the calculations and reference to the experiments can be found in Ref. [14].

and tertiary sites. Cohen concludes that for primary C—H bonds in small alkanes (and possibly secondary C—H bonds) such effects are measurable and should be accounted for. However, for larger molecules such effects become negligible in comparison to experimental uncertainties. Cohen concludes 'The database is too small, and the experimental uncertainties too large to establish unambiguously whether simple additivity, the formulation of Atkinson et al. [10], or the TST model described here is more accurate." Similar conclusions were drawn for H-atom reactions [17] and the resulting general expression for the reaction with large (more than 4 carbon atoms) was obtained WH+RH

= 5.4 x 10 3 « p r 2 0 exp(-3450/r) + 4.7 x 103nsT22 x exp(-2640/r) + 3.7 x 10 3 n t r 2 0 exp(-970/r). (2.14)

Table 2.5 contains a comparison of the experimental data [18] for the

150


Ch. 2

TABLE 2.5 Site specific and total rate coefficients for OH + propane. A comparison of experiment with various calculations Experiment3

Transition state theoryb

Atkinson0

Walkerd

3.09 7.76 2.51 10.9

2.95 7.59 2.57 10.5 -3.6%

3.55 7.25 2.04 10.8 -1%

3.81 8.89 2.33 12.7 + 16.5%

14.8 18.6 1.26 33.4 -1.5%

14.1 15.1 1.07 29.2 -13.8%

16.0 18.4 1.15 34.4 + 1.5%

38.9 36.3 0.93 75.2 +8.4%

34.7 27.6 0.80 63.2 -8.9%

37.5 26.1 0.70 63.6 -8.4%

295 K kp /c

e

ks Its' Kp f^tot

A%f 503 K

-

K

15.9 17.8 1.12 33.9

ks Its' Itp K-tot

A% 732 K

-

K ^tot

36.3 33.1 0.91 69.4

A%

-

Ks Its' ftp

a

From Tully and Droege [18]. From Cohen [16]. c From Atkinson et al. [10]. d From Walker [11]. e Units cm3 molecule -1 s~\ Percentage difference in total rate coefficient from experimental value. b

reaction of OH with propane with the calculated values using the methods of Atkinson et al. [10], Walker [11] and Cohen [16]. In general, the agreement between the calculated and experimental rate coefficients is good, at least for the total rates, predominantly within the % error that one would expect for the experimental determinations. It should also be noted that the predictions of Walker and Atkinson were made before the publication of the comprehensive study of Tully et al. [7] and, therefore, the data for this reaction were not incorporated into the models of these authors.


151

2.3.6 Other radical abstractions (/) H02 abstractions As will be seen in subsequent sections, the H 0 2 radical is an important combustion intermediate, however, due to its low reactivity, it plays little part in propagating the combustion process via abstractions from fuel molecules. As discussed in Chapter 1, abstraction by H 0 2 from intermediates such as aldehydes, generates H 2 0 2 and provides a route to branching in the intermediate temperature region. The reaction with formaldehyde has been studied directly as a function of temperature by Jemi-Alade et al. [19]. H 0 2 radicals were generated by the photolysis of H 2 CO/0 2 mixtures (19)

H 2 CO + hv^

(20)

HCO + 0 2 -> H 0 2 + CO

(21)

HCO + H

H + 02^H02

and [H0 2 ] and [H 2 0 2 ] were monitored in real time by UV absorption spectroscopy at 210 nm. At low temperatures ( H 2 0 2 + HCO

(20)

HCO + 0 2 -> H 0 2 + CO

An unweighted Arrhenius plot for A:22, incorporating the work of JemiAlade et al. [19] and other 'low' temperature measurements yields the following expression fc22 =

x 10" 13 x exp[-(38.4

2.9) kJ mol"1/RT] cm3 molecule" 1 s - 1 . (2.15a)

High-temperature shock tube studies [20] show significant curvature in the

152


Ch. 2

2.000

Fig. 2.11. Modified Arrhenius plot for k22. (A) Ref. 19, (O) Ref. [20], (A) R.R. Baldwin, D.H. Langford, M.J. Matchan, R.W. Walker and D.A. Yorke, 13th Int. Symp. Comb. 251 (1971), (O) R.R. Baldwin and R.W. Walker, 17th Int. Symp. Comb. 525 (1979), (*) S. Hochgreb, R.A. Yetter and F.L. Dryer, 23rd Int. Symp. Comb. 171 (1990). Adapted from a figure by D.W. Stocker.

Arrhenius plot and the resulting modified expression as shown in Fig. 2.11 is k22 = 9.66 x 1(T27 r 4 5 3 e x p ( - 3 3 0 0 / r ) .

(2.15b)

(ii) CH3 abstractions CH 3 abstraction reactions can play a significant role in combustion chemistry, for although there is no net change in either hydrocarbon or alkyl

Radical decomposition reactions

153

radical concentration following a reaction such as (23)

CH 3 + C 3 H 8 -> CH 4 + i-C3H7

we shall see that the subsequent chemistry of the alkyl radicals CH 3 and j-C 3 H 7 is very different. In contrast to their reaction with oxygen (Section 2.6) there is very little direct time resolved experimental data on the reactions of methyl radicals with hydrocarbons, most of the studies having been performed by gas chromatographic end-product analysis. Interestingly, hydrogen abstraction by methyl radicals is characterized by a strong pre-exponential temperature dependence. A thorough review of a majority of the literature is given by Warnatz [21]. More recent work on methyl reactions with propane, hydrogen and formaldehyde can be found in references [22-24] respectively.

2.4 RADICAL DECOMPOSITION REACTIONS

2.4.1 Introduction Figure 2.1 shows that, once formed, the hydrocarbon radical can undergo a number of reactions, one of which is decomposition; e.g., the decomposition of the isopropyl radical into H atoms, which can propagate the chain reaction, and propene. (24)

Z-C 3 H 7 ^H + C 3 H 6

By their nature, radical decompositions involve bond cleavage and hence have considerable activation energies, 150kJmol - 1 for reaction (24). Decomposition only becomes competitive with alternative reactions, such as oxygen addition, at higher temperatures or for large radicals which have weaker bonds. Under these conditions radical decomposition can be of great importance, converting larger less reactive (in terms of abstraction reactions) radicals into H or smaller radicals and generating alkenes. Decomposition can tend to simplify the combustion process. Instead of considering the complex reaction pathways of a very large number of different radicals produced from a complex fuel mix, we need only to follow the reactions of smaller radicals which are more stable with respect

154


Ch. 2

to decomposition. At the same time, the routes to the degenerate branching agent, ROOH, can be significantly slowed, because the R02—> QOOH isomerization is generally faster for larger radicals. Information on the rates and products of radical decompositions is therefore of vital importance. C—C fission is favoured for most radicals at low temperatures, ethyl, /-propyl, and £-butyl being the main exceptions. However the study of these radicals is important as their relatively small size allows for detailed modelling of the experimental data. Radical decompositions are unimolecular reactions and show complex temperature and pressure dependence. Section 2.4.l(i) introduces the framework (the Lindemann mechanism) with which unimolecular reactions can be understood. Models of unimolecular reactions are vital to provide rate data under conditions where no experimental data exist and also to interpret and compare experimental results. We briefly examine one empirical method of modelling unimolecular reactions which is based on the Lindemann mechanism. We shall return to more detailed models which provide more physically realistic parameters (but may be unrealistically large for incorporation into combustion models) in Section 2.4.3. Direct experimental measurements of radical decompositions are relatively rare but recently a number of techniques have been applied and are producing new and interesting information. In Section 2.4.2. we shall mainly focus on the technique of laser flash photolysis coupled with photoionization mass spectrometry as a method of monitoring radical decompositions although other techniques will be briefly mentioned. In the final subsection we shall look in more detail at one of the crucial aspects of decomposition reactions, the true unimolecular and energy dependent dissociation of an activated molecule. Exciting new developments, both theoretical and experimental, are taking place in this area confirming and interpreting some of the proposed theories of unimolecular reactions. (/) The Lindemann mechanism Figure 2.12 illustrates the potential energy profile for a unimolecular radical decomposition. Bimolecular collisions with a bath gas can activate the molecule so that it contains enough energy to react. However, for reaction to occur this energy must be located in the bond to be broken and hence the dissociation is not instantaneous. During the intramolecular energy redistribution further collisions can occur with the bath gas which will deactivate the molecule. The simple treatment proposed by Linde-


155

A+M Reaction coordinate Fig. 2.12. Schematic diagram of a 2-D potential energy curve for a unimolecular reaction. ka and A^ are the bimolecular rate coefficients for activation and deactivation respectively by a bath gas M. kT is the rate coefficient for the unimolecular reaction of an energized molecule, A*, to give product, P.

mann assumes that the energized molecule, A*, is in a steady-state, i.e., that its rate of formation is equal its rate of removal. ka[A][M] = kJLA*][M] + kT[A%

(2.16)

[A*] = ka[A][M]/(kd[M] + kr).

(2.17)

The rate of product formation dP/dt is dP/dt = kr[A*] = Mk[A][M]/(Jfcd[M] + kT).

(2.18)

Two possible extreme conditions exist (a) kd[M] > kT The rate of collisional deactivation is much greater than that of reaction

156


Ch. 2

(at high pressures). Equation (2.18) reduces to 6P/dt = krka[A]/kd9

(2.19)

and the rate coefficient is independent of the bath gas pressure. This is known as the high pressure limit and the associated unimolecular rate coefficient as k°°. (b) kd[M]0).

(2.37)

Davies et al. [39] have shown that if the general form of kZ{(5) can be written as,

C(/3)=^°Q)

exp(-/3Zf°)

(2.38)

for n > — 1.5 and E> E™ + AH% then the equation (2.38) becomes,

k(E) =

A T N(E)T(n° +1.5) E-E^-AHft X

Jo

Np(x)[(E -ET- Aflg) - x]nOC+05dx (2.39)

where N(E) is the density of states for the unimolecular reactant, Np(x)

168


Ch. 2

is the convoluted densities of states of the products, AH% is the zero-point energy difference between the reactants and products and T is the Gamma function. C is given by, r.

C

, .

, .

2TTMAMB

-.3/2

]

(2.40)

\_h\MA + MB)\ where MA and MB are the masses of the dissociation products. The feature that makes the use of equation (2.39) particularly attractive in data analysis is that it is a convolution and as such can be calculated very rapidly by the use of the Fast Fourier Transform (FFT) method. This technique clearly depends on the parameters A°°, ri*\ and E™, which may be determined from a least squares fit to the fall-off data. It is noted that this form of ILT produces association and dissociation data that are thermodynamically consistent, based as it is on the equilibrium constant. Equation (2.30) can be solved analytically only for a few simple systems, for most other cases numerical solution is required. Solution is usually effected by using the graining technique whereby the energy is partitioned into a series of contiguous intervals or grains, each of which is characterized by a number of states, a mean energy and a mean microcanonical rate coefficient. Equation (2.30) is thus approximated by, = co 2 Piff>,(t) - toPi{i) - kiPl{t),

^ at

(2.41)

j=i

which can be written more concisely using matrix notation as,

:T = M P (O,

(2.42)

dt where M is given by, M = o>(P - I) - K, (2.43) the elements of P being the discretized form, Pih of the probability densities, P(E'\E) in equation (2.30), I is the identity matrix and K is a


169

diagonal matrix containing the microcanonical rate coefficients. The solution of equation (2.42) is, p{t) = Ue A 'lTV(0),

(2.44)

where A is a diagonal matrix of the eigenvalues of M, and U is the corresponding eigenvector matrix. The eigenvalues of M are all negative and for reasonably large threshold energies (>10,000cm -1 ) there will be one value, Ai, whose magnitude will be orders less than the others. At long times, which correspond to experimental time scales, the decay will be dominated by Ax and will be exponential. Equation (2.44) can then, to a very good approximation, be written as, p(0-u1exp(A10,

(2.45)

and the unimolecular rate coefficient kd can be identified with — Ai. Furthermore, the vector Ui is the steady state normalized population density for the reacting system. Rate coefficient calculation thus becomes a question of determining the eigenvalue of smallest magnitude, for the "spatial" operator in equation (2.42) i.e., the matrix M. If the grain size is 100 c m - 1 and the energy range to be considered is 50 000 c m - 1 then M will be of order 500. For data analysis purposes the calculation of Ax of M has to be as efficient as possible. It is inefficient to calculate all of the eigenvalues of such a matrix. There are available a number of routines that will calculate only certain eigenvalues. Gilbert, Smith and co-workers [40-42] have made much use of the Nesbet algorithm for single eigenvalue calculation. Work is currently in progress to determine other ways of enhancing the speed of eigenvalue determination. One method that shows promise is the diffusion equation approximation. The basis of this method is that it can be shown that under certain circumstances the integral operator on the right hand side of equation (2.30) can be replaced by a differential operator that is similar in form (and solution) to a diffusion equation. Such equations can sometimes yield analytic results and even when this is not the case they are much more amenable to numerical solution often with considerable savings in CPU time. The decomposition of the iso-propyl [33] is an example of a system for which the parameters A°°, n°°, and ET are already well known, Harris and

170


Ch. 2

Pitts [43] having measured the association reaction rate coefficient as a function of temperature and close to the high pressure limit. The objectives of the decomposition data analysis in this case were to obtain estimates of the enthalpy of reaction, A#2, and (A£) d for He from the kinetic data. The general strategy for parameter determinations is: (i) Use equation (2.39) to obtain k{E). For the present example this involves making an initial guess for AH%. (ii) Insert the k(E) obtained into the master equation (or diffusion equation) and calculate the rate coefficients for the experimental pressure and temperature points for a specific value of (AE)d. (iii) Compare the calculated and experimental rate coefficients and adjust the parameters as indicated by the data in accordance with a prescribed best-fit criterion, (iv) Repeat this cycle until the best fit criterion is met. As with most fitting procedures the criterion used was that of least squares, that is the best values of the parameters were those that minimized ^ 2 , m

2

{k?v ~ kfc)2.

X =^

(2.46)

1=1

X2 is thus a function of A/fo and (A£")d and the best fit corresponds to its global minimum. The nature of the calculation of the rate coefficients means that x2 *s a non-linear function of the parameters, and so a nonlinear least squares method is required, for example the Levenberg-Marquadt algorithm, in which the gradient vector is followed to the minimum. The difficulty with this method for the present system is that analytic derivatives are not available and have to be calculated using finite difference which significantly increases the processor time. Further difficulties with this method arise if the nature of the surface is such that it has either local minima in which a search might get trapped, or the surface is flat so that the derivatives obtained from finite difference have large errors. This latter difficulty was encountered for the present system and the method was abandoned in favour a simple grid search of parameter space. For the grid search of the x2 surface, values for 500 grid points were calculated. The full experimental data set of 71 points was used. The x2


12000

12250

AH>m-

171

12500 1

Fig. 2.18. Contour plot of \2 surface showing the strong correlation between A//J5 and d.

surface obtained is shown in Fig. 2.18; the contour lines are at intervals of 15000 s - 2 . This surface shows a trench with a flat bottom which is the sort of minimum that Levenberg-Marquadt algorithms have difficulties in searching. It is also apparent that the best value of AH% is very strongly correlated to the value of (A2?)d. Even so, good definition of both parameters, and particularly of AH% is obtained. The most suitable choice of parameters is Mf0 = 12300 c m - 1 and d = 230 cm" 1 . Figure 2.17 shows a plot of the experimental data together with calculated fall-off curves and illustrates the quality of the fit. An estimate of the enthalpy of formation of the iso-propyl radical of 87.7 kJ mol - 1 is obtained from A//$.

2.4.4 Microcanonical rate coefficients In the above analysis of the iso-propyl radical decomposition the microcanonical rate coefficients were obtained from an experimentally

172


Ch. 2

determined Arrhenius form of the association reaction and as a consequence can be said to be experimentally determined. There is, as stated above, considerable interest in calculating k(E) from purely theoretical considerations. The reasons for this are twofold: firstly, a theoretical calculation would reduce the number of parameters that would need to be adjusted in any fitting/analysis of experimental data, and secondly, it gives a greater insight into the processes occurring at the molecular level. Modern unimolecular theory has its origins in the work of Rice, Ramsberger and Kassel [44] who investigated the rate of dissociation of a molecule as a function of energy. Marcus and Rice [44] subsequently extended the theory to take account of quantum mechanical features. This extended theory, referred to as RRKM theory, is currently the most widely used approach and is usually the point of departure for more sophisticated treatments of unimolecular reactions. The key result of RRKM theory is that the microcanonical rate coefficient can be expressed as

hN where G(E) is the sum states at the transition states for those degrees of freedom that are orthogonal to the reaction coordinate, h is Planck's constant and N(E) is the density of states of the stable molecule. Below the threshold energy, Zs0, for the reaction, G{E) is zero. To be able to utilize this formula a great deal of information concerning molecular parameters is required. To calculate N(E) rotational constants and vibrational frequencies of internal motion are required and in many case these are available from spectroscopic studies of the stable molecule. Unfortunately the same cannot be said for the parameters required to calculate G(E) because, by definition, the transition state is a very short lived species and is therefore not amenable to spectroscopic analysis. The situation is aggravated still further by the fact that many unimolecular dissociation processes do not have a well defined transition state on the reaction coordinate. It is precisely these difficulties that make ILT an attractive alternative as it does not require a detailed knowledge of transition state properties. A number of workers have tackled the problem of transition state prop-


173

erties by constructing them from the known properties of the reactant and products. In this way a set of transition state vibrational frequencies and rotational constants are obtained which usually depend on some interpolation parameter which can be altered to fit the experimentally observed data. The interpolation procedure is not appropriate for reactions proceeding on Type I surfaces, with a pronounced maximum which constrains the position of the transition state. In this case, the most reliable approach to estimating transition state parameters is through quantum mechanical ab initio calculations. Such calculations are also of value for Type II surfaces. Molecular parameters such as vibrational frequencies, rotational constants and threshold energies depend on the details of the potential energy surface on which the reaction takes place. Recent advances in both computer hardware and software have meant that quantum chemistry methods can be used to investigate these surfaces and in particular the properties of the system as a function of the reaction coordinate. A complete description of these techniques is beyond the scope of the current volume. However, in summary, the starting point is the Schrodinger equation for the electronic wavefunction, for a given nuclear geometry. The wavefunction is expressed as an anti-symmeterized product of molecular orbitals. The molecular orbitals are themselves expressed as a linear combination of atomic orbitals, often referred to as the basis set. The values of the linear combination coefficients are determined by variational minimization of the energy of the system for a given geometry. The resultant wave function can be further refined by application of techniques which account for electron correlation etc. In this way an estimate of the energy for the specified system may be obtained. The precision of these estimates depends very much on the size and type of the basis set used and the method used to refine the initial wavefunction; currently the best techniques can give values _1 . Estimates of the vibrational frequencies can also to within be obtained from the wave function. It is often found that these tend to about 10% too high owing to electron correlation and anharmonicity effects. The great advantage of the ab initio approach is that it is not restricted to stable molecules in their equilibrium configuration - it can also be used to examine radicals that would be hard to analyse spectroscopically and, perhaps more importantly, it can probe the PES in the region of the transition state, giving information about the vibrational frequencies at the

174


Ch. 2

transition state that can be substituted into equation (2.47) to yield the microcanonical rate coefficient. 2.4.5 Unimolecular dynamics State specific experiments can now test unimolecular rate theories by probing microcanonical rate coefficients. Moore and coworkers [45] have studied the dissociation of ketene close to the reaction threshold in an attempt to test RRKM theory. (29)

CH.CO -> CH9 + CO

Using a short tunable laser pulse they excited ketene into its first electronically excited state from which internal conversion occurs to give a highly

isc

T-

",5C

CH 9 C0

REACTION COORDINATE

Fig. 2.19. The lowest three potential energy surfaces of ketene along the reaction coordinate. The ketene molecule is excited by a UV laser pulse to the first excited singlet state (Si), undergoes internal conversion to S0 and intersystem crossing to Ti, and dissociates into 1 CH 2 + CO (singlet channel) or 3 CH 2 + CO (triplet channel) fragments.

175


i CO

o 3

5

UJ

2

0 250

350 Energy -

450

550

28,000

650

cm"1

Fig. 2.20. Rate coefficient for deuterated ketene dissociation as a function of the photolysis energy. The error bars on selected data points represent 2 SD from three independent rate coefficient measurements. The solid line is an RRKM fit.

vibrationally excited ground state molecule with a defined energy content (Fig. 2.19). They then monitored, by laser induced fluorescence, the rate at which the product CO was formed (Fig. 2.20). Close to reaction threshold RRMM theory predicts that k(E) should be a stepwise function of energy (eqn 2.47), each step occurring when there is enough reagent energy to access a new vibrational level in the transition state. Figure 2.20 shows that this is exactly what is observed and the energy gap between steps allows the determination of the vibrational frequencies in the transition state. The figure also shows the excellent agreement with ab initio calculations performed using the methods outlined in the previous subsection. In an elegant four laser experiment Temps [46] has also investigated the variation in k(E) for the decomposition of the methoxy radical: (30)

CH3O^CH20 + H

176


Ch. 2

The barrier for dissociation is quite low and hence the vibrational levels of the methoxy radical can still, to a good approximation, be considered as harmonic oscillators, even at several thousand wavenumbers excitation. Following photodissociation of the methylnitrite, a specific rovibrational state of the methoxy radical was produced by stimulated emission pumping (SEP). Information on the lifetime of the state produced is obtained by linewidth measurements or by laser induced fluorescence (Fig. 2.21(a)). The observed state can be removed by intramolecular energy transfer or unimolecular dissociation. Temps was able to differentiate between these two processes and hence obtain state specific unimolecular rate coefficients, kWy Basic RRKM theory predicts that the dissociation rate is dependent only on the total amount of energy in the reagent molecule; however, these experiments show considerable variations in /cvj even for states which have very similar total energies. This might appear to be a blow for basic RRKM theory; however, whilst the reagent molecule still has a discrete rovibrational spectrum the density of states is high and averaging the kyj over a small energy region (or grain) reproduces the RRKM estimate (Fig. 2.21(b)). Current developments in RRKM theory by Marcus, Wardlaw [47-49], Smith [50-52] and Klippenstein [53-56], have rigorously extended the theory to include the effects of / dependence, the result leading to microcanonical rate coefficients which are functions of both energy, E, and the magnitude of the angular momentum, /.

2.5 RADICAL RECOMBINATION AND ASSOCIATION REACTIONS

2.5.1 Introduction Figure 2.1 shows that an alternative fate for radicals generated during the oxidation process is recombination, e.g., (31) CH 3 + H + M ^ C H 4 + M CH 3 + CH 3 + M - * C 2 H 6 + M

10” E=7449cm

..

-.l

1O’O

U

3

G o y , a

Pg

r\

,

r

I

E= 7444crn

2 10’

c

\

83

1

7-

i

a .-

0

E

loe

”_ 0.0

0.2 Time

/

0.4 (ps)

0.6

0.8

Ba

-J

G e

10’

D

I

8.z 8’

Y

1o6

(D 1

E5’ c1

v1

10’

6000

8000 Energy

/

10000

(ern-')

Fig. 2.21. (a) Time-resolved LIF decay profiles for two closely spaced rotational levels of vibrationally excited CH30 (X). The solid line is an exponential fit for the decay convoluted with the dump laser pulse shape function. (b) Measured state specific unimolecular dissociation rate constants for CH30 (X) compared to calculated k ( E , J ) curves without and with tunneling corrections.

178


Ch. 2

or in general (33)

R + R' + M - ^ R R ' + M

and one may also need to consider the disproportionation reaction (33a)

R + R' -> RH + R"CH=CH 2

which may be an activated process. Recombination of small alkyl radicals is a terminating step removing chain carrying radicals from the combustion system and hence is an important determinant for global features such as flame propagation and ignition. For larger radicals, decomposition and reaction with 0 2 occur too rapidly for recombination to be significant under autoignition conditions. Recombination reactions are also major routes for forming higher hydrocarbons; for example in methane combustion, reaction (32) is the primary source of C2 hydrocarbons. Rate coefficients for recombination reactions are related to those for dissociation via the equilibrium constant, which can generally be calculated from thermodynamical information with a high degree of precision, although the accuracy depends on the quality of the thermodynamic data. The rate coefficients are pressure dependent and the theoretical framework of unimolecular reactions can therefore be used to describe them. Because there is little or no activation energy for the recombination process, rates of radical association reactions can be measured over a wide range of temperatures and can be used, in combination with thermodynamic information, to calculate rate coefficients for unimolecular dissociations. The availability of data for a number of radical recombination reactions over a wide range of pressures and temperatures makes these reactions excellent test beds for theoretical models of pressure dependent reactions. In the following subsections we shall look in detail at reactions (31) and (32), from both experimental and theoretical viewpoints. Association reactions of the type (34)

H + C 2 H 4 + M -> C2H5 + M

(35)

CH 3 + C 2 H 4 + M -> n-C3H7 + M

are the reverse of the radical decomposition reactions considered in Section

Radical recombination and association reactions

179

2.4. If the forward and reverse reactions can be measured, then the relationship between rate coefficients and the equilibrium constant allows the thermodynamics of the reaction to be determined from kinetic measurements. This has been an extremely successful method of determining radical heats of formation and the results of a number of recent experiments are discussed in Section 2.5.5. We will conclude our discussion on recombination reactions with the self reaction of the H 0 2 radical. This reaction is of interest as it is an intermediate step in the conversion of the comparatively unreactive H 0 2 radical into OH radicals, via H 2 0 2 . (6)

H02 + H02->H202 + 02

(36)

H202-^20H

Although not strictly an association or recombination reaction the H 0 2 self reaction is best understood via an intermediate complex mechanism and the reaction shows some of the properties of an association reaction. 2.5.2 CH3 + CH3 -* C2H6 (i) Experimental studies Reaction (32) is probably the most well studied radical recombination reaction with data existing from 200-1700 K and 30-2 x 107 Pa. In contrast to the experiments described in previous sections, second order reactions, such as radical recombinations require a knowledge of absolute concentrations in order to determine rate coefficients. The integrated rate expression for a second order decay is: kt = 1/[A]0 - 1/[A].

(2.48)

The major techniques that have been used to monitor [CH3] in real time are mass spectrometry [57] and UV absorption spectroscopy [58]. At higher pressures and temperatures determinations have been made by shock tube studies. In the following we briefly consider each of the experimental techniques (as a function of increasing pressure).

180


Ch. 2

(a) Discharge flow IMass spectrometry Low pressure (0.25-5.5 Torr) studies of reaction (32) have been performed at 300 and 408 K using the technique of discharge flow coupled to mass spectrometric determination of methyl radical concentrations [59]. Methyl radicals were generated by the fast reaction (37)

F + C H 4 ^ H F + CH 3

with [CH4] in considerable excess (—100:1) to prevent further reaction of F atoms with methyl radicals. F atoms were produced by passing a dilute mixture of F 2 in argon through a microwave discharge. Absolute methyl radical concentrations were obtained by the titration: (38)

CH 3 + N 0 2 -> CH 3 0 + NO

calibrating the observed NO signal at the mass spectrometer with values from known flows of NO. Care was taken to account for diffusional loss, pressure drops and heat transfer effects. Details of these procedures may be found in Ref. [59]. The results of this study are summarized in Fig. 2.22. (b) Flash photolysis!Photoionization mass spectrometry The details of this technique have already been discussed in the previous section. Methyl radicals were produced by pulsed excimer dissociation of acetone at 193 nm, which has been shown to give over 95% yield of the methyl radical channel [60]. (39)

CH3COCH3 + M 1 9 3 n m ) -> 2 C H 3 + CO

The time dependence of the methyl radical concentration in the flow tube was monitored mass spectrometrically, methyl ions being generated by photoionization with light from a hydrogen discharge lamp and put onto an absolute measure by monitoring the decrease in the acetone signal following the photolysis. Using this technique Gutman and coworkers [57] were able to cover the range 1-5 Torr and 296-906 K.

181


o

£ J? O T—(

o

-0.4

1

1.5

2 l6

2.5

3

3.5

3

Log (lAr]10 /molee enf ) Fig. 2.22. Temperature and pressure dependence of k32 using a combination of discharge flow/mass spectrometry, flash photolysis/PIMS, and flash photolysis/UV absorption techniques. The solid lines represent the global best fits of all the data as outlined in subsequent text.

(c) Flash photolysis/UV absorption The major limitation of the PIMS technique (Section 2.4.2) is the low pressures required to operate the mass spectrometer and ensure plug flow in the flow tube. In a collaborative paper, Pilling and coworkers [57] were able to extend the pressure range to -600 Torr of argon using UV absorption to monitor the transient behaviour of the methyl radicals. Figure 2.23 shows a schematic diagram of the apparatus. Further work [59] extended the temperature range down to 200 K using a modified spectrosil reaction cell, suspended in an acetone/C0 2 ice bath. In both studies the photolysis of acetone was used as the methyl radical source and the methyl radical concentration monitored by measuring the absorption of radiation from a xenon arc lamp at the maximum of the methyl radical absorption spectrum at 216.4 nm. To put these absorption measurements onto an absolute scale requires a knowledge of the absorption cross-section of CH 3 as a function

182

Elementary reactions beam dump

Ch. 2

To exhaust

Valve to control pressure in reaction vessel

Heatabie Stainless Steel Reaction Vessel

Monochromator

Premixed gases flow slowly through reaction vessel

PMT]

Flowcontroller Transient Digitiser Signal Averager

ArF excimer laser

Flowcontroller

Flowcontroller

Microcomputer to control experiment, gather and analyse data

Fig. 2.23. Schematic diagram of the laser flash photolysis/UV absorption spectroscopy apparatus employed by Slagle et al. [57] to study reaction (32) at higher pressures. B, Baratron pressure transducer, PS, flow sensor; I, iris; L, lens; NV, needle valve to control total pressure; PD, photodiode; PM, photomultiplier; V, solenoid valve; Zn, zinc lamp for calibration.

of temperature. Experimental measurements of the cross-section (end product analysis and repetitive photolysis experiments [58]) were made at temperatures between 296 and 537 K and validated theoretical calculations allow their extension to the full range of the CH 3 + CH 3 experimental study (200-906 K). Very high pressure (1-200 bar) room temperature data have been obtained by Hippler et al. [61] by following the [CH3] produced by the photolysis of azomethane with UV absorption spectroscopy. The agreement with the data of McPherson et al. [58] is excellent and shows no pressure dependence between 1 and 10 bar. Thereafter there is a slight decrease in the rate due to the onset of diffusion limited kinetics.


183

(d) Shock tube determinations Temperatures of around 1000 K are the upper limit for conventional flash photolysis experiments, higher temperatures require specially designed apparatus or shock tubes. There have been three shock tube studies of reaction (32). Glanzer et al. [62] determined k32 at 1350 K, between 1 and 25 atm, initiated by the rapid thermal decomposition of azomethane with the methyl radical concentrations being monitored by UV absorption. Direct measurements of the absorption coefficient at 1400 K were used to determine absolute methyl radical concentrations. Similar measurements were performed by Hwang et al. [63] More recently Hessler and coworkers have measured k32 in the temperature and pressure ranges 1175-1750 K and 1.13-2.27 atm. Once again the rapid decomposition of azomethane was used to generate methyl radicals which were then monitored by the laser flash absorption technique [64] at 215.94 nm. A/32, the rate of azomethane dissociation and the absorption cross section were determined simultaneously by non-linear least squares fits to the methyl absorption profiles. (if) Modelling methyl radical recombination The recombination of methyl radicals is an important reaction not only because of its participation in combustion processes but also as a test bed for theoretical treatments. It is also the subject of some controversy regarding the form of the temperature dependence of the limiting high pressure rate coefficient. The theoretical treatment of Wardlaw and Marcus [47-49] using Flexible Transition State Theory suggests that k32 should have a negative temperature dependence. A detailed analysis of the experimental data obtained by Slagle et al. [57] by Wardlaw and Wagner [65] also requires a negative temperature dependence. However, Troe et al. [61] argue that the data are compatible with a rate coefficient which is temperature independent. In this section we describe methods, based on the master equation, that can be used to analyse the data obtained for this reaction and show results obtained for an expanded data set that includes new experimental points. The analysis of kinetic data from recombination systems can be tackled using the same techniques that where introduced in Section 2.3.3. Again the most widely employed technique is that due to Troe but the use of master equations for examining these systems has advantages for reasons similar to those discussed above.

184


600

I

1 11 1 1

500

o

i

i

I

I

H coFji

.....*a

u

400 . 1

1 — 1

a

§

i"

1

Ch. 2

1

300

Fc

200

J

100 1

\ y

1

1.5

i

I

i

1

2.5

3 R/k

3.5

. , 1

1

4.5

Fig. 2.24. Schematic diagram showing the potential energy of the methyl + methyl reaction as a function of separation. Ec is the position of the absorbing barrier at a temperature of 1000 K.

Methyl radical recombination is a special example of the more general recombination, (40,-40) (41)

A + B ^ C* C* + M-~»C + M

considered by Smith et al. [42] who demonstrated that the overall recombination rate coefficient is related to the dissociation rate coefficient by the equilibrium constant regardless of the pressure of the system. The equilibrium constant is, of course, independent of pressure and so it is possible to calculate the dissociation rate coefficients from the data, analyse these using the Troe method [25-27] and then calculate the desired association rate using the equilibrium constant. A similar approach (Fig. 2.24) is possible using MEs following the procedure of Section 2.4.3 to determine the eigenvalue of smallest magni-


185

tude. An alternative route, which is more intuitively appealing, is to modify the ME given in Section 2.4.3 to take account of the reactive input:

= co 2 PijPj(i) - coPi{t) - kiPi(t) + Rt{t).

^ at

(2.49)

y=i

The term Rt represents reactive gain due to bimolecular association and is constrained by detailed balance at equilibrium. If A and B remain in thermal equilibrium throughout the course of reaction (a good approximation) then Ri is given by,

Rlt) = k:[A]eq[B]eq^-r 'Ztkifi

(2.50)

Ri(t) = k:[A]eg[B]eqcf>i

(2.51)

Rt{t) = R'(t)4>i

(2.52)

where CH4 (/) Introduction This reaction is an important terminator in methane combustion but its main relevance for this work is as a test bed for theoretical models of unimolecular and recombination reactions. The most thorough experimental results to date (to be described below), covering the temperature and pressure ranges 296-604 K and 50-600 Torr, are still far from the high pressure limit even at the lowest temperature of the study. Whilst the experimental data can be extrapolated using phenomenological or RRKM procedures, Brouard et al. [66] used the isotopic reaction (31a) (31b)

CH 3 + D -* CH 2 D + H -> CH 3 D

in order to measure k^x directly. Monitoring the disappearance of D atoms directly, under conditions where [CH3] ^> [D], gives the rate of formation of the activated methane molecule (fc£>) because zero-point energy effects make the ejection of the D atom from the activated molecule, reforming reagents, extremely unlikely. Once k& has been corrected for the isotopic substitution the resulting &31 should agree with the value extrapolated from the CH 3 + H fall off data. However, the agreement between the two values of &31 is poor whichever method of extrapolation is used; the discrepancy possibly arising from a number of areas:


• • • •

189

An error in the CH 3 + H experimental data. An error in the CH 3 + D experimental data. Invalid extrapolation of the fall off data. An anomalous isotope effect for reaction (31).

(ii) Experimental determinations The experiments were carried out in a conventional multiport flash photolysis cell. CH 3 and H radicals were both generated from the flash photolysis of acetone at 193 nm, H atoms being produced from a very minor photolysis channel. Methyl radicals were monitored by UV absorption spectroscopy with H atom detection by resonance fluorescence. The experimental requirements of the two detection systems are somewhat incompatible and hence neither system is operated under ideal conditions. This brief description may give the impression that these are relatively simple experiments; however, like all radical-radical cross reactions a great deal of care is required to determine valid experimental conditions and extract the target rate coefficient. The methyl radicals were generated in considerable excess over hydrogen atoms and are removed predominantly by self recombination. The expression for the [H] decay is given by

[H] = [H] 0 exp - J k31[CU3] dt - kdt

(2.56)

Jo

where kd accounts for diffusional loss which was measured in separate experiments. CH 3 decays by a purely second order mechanism so that the integral in equation (2.56) can be evaluated to give: [H] = [H]0[l + 2k32[CH3]0t]~k3l/2k32 e x p ( - M ,

(2-57)

where k32 and [CH 3 ] 0 are determined from fits to the methyl radical decay profiles. Figures 2.26 and 2.27 show the satisfactory fits of the methyl and H atom decay traces and the fall off data over the entire experimental range. For the isotope experiments D atoms were generated by the photolysis

190


Ch. 2

CO

* 03 CO

'

^

^

^

&?

5.0

7.5

CO D "D

"(75

Q)

a:

0.0

2.5 Time / ms

0.3

B

o) 5 0.2 CD * . =

c 9c5 -Q 0.1 a.
N 2 + 0( 1 D) 0( 1 D) + D 2 - ^ O D + D OD + D 2 -> D 2 0 + D

Unfortunately at the lower temperatures of the isotopic study reaction (42) generates D atoms on the timescale of the target reaction. Brouard and Pilling accounted for this by a numerical analysis and produced consistent values for k31 over a 7 fold variation in D 2 concentrations showing that reaction (42) could successfully be accounted for. Figure 2.28 shows that as expected kZ is pressure independent (50-600 Torr) and over the limited range of the experiments (300-400 K) temperature independent with a 10 cm3 molecule" mean value of 1.75 x 10

192


Ch. 2

z.s T

2-

T

1.5 -

x

J

T

t

l

k

T

I

^

1 100

1 200

x

T

?

I

T

T

¥

i

1

I

1 o

0.5 -

0-

1 300

1 400

1 500

1 600

He Pressure /Torr Fig. 2.28. Variation of k^

with pressure.

(Hi) Comparison of the isotopic reactions The application of transition state theory to the isotopic reactions yields the following equation for the ratio of the rate coefficients t OO

J OO

QU

Q?

GcH3D QT

exp

f(£ H - £ D ) kT

(2.58)

where Qx is the rovibrational partition function of the activated complex (with the reaction coordinate factored out), QTi$ the translational partition function of the atomic reagent and E is the activation energy for recombination for the two reactions CH 3 + H and CH 3 + D. To a first approximation the exponential term and the ratio of the vibrational partition functions can be approximated to unity and hence the isotope ratio is merely the product of the ratio of the rotational partition functions of the activated complex and the translational partition functions of the atomic reagents. The latter term is simply 2 3/2 . Deuteration of the atomic reagent results in a doubling of the rotational partition function for the CH 3 D activated complex and hence the final ratio of the rate coefficients /CR/^D is V2. The predicted value of k3i at the high pressure limit for the CH 3 + H


193

reaction is V2 x 1.75 x 10" 10 cm3 molecule" 1 s" 1 = 2.47 x 10" 10 cm3 molecule -1 s - 1 . A variety of different fitting procedures were used in an attempt to determine ku from the experimental data. Whatever the fitting proceedure used it was difficult to obtain good fits to the data with k^ constrained to the value determined by the deuterium atom experiments. Unconstrained fits return a value of 4.68 x 10 - 1 0 cm 3 molecule -1 s - 1 for k^ using the master equation techniques outlined in the previous section. One is left with the conclusion that either the experimental data are in error or some aspect of the modelling procedure, either the isotopic correction or the extrapolation is at fault. Independent evidence for the validity of the CH 3 + H data and extrapolation comes from a calculation by Quack and Troe [67] who calculated a value of 4.7 x 10 - 1 0 cm 3 molecule -1 s - 1 for &H- However, more detailed ab initio calculations [68] predict k°^ close to the experimental values and with a V2 isotope dependence. The two sections of experimental evidence would therefore seem to be incompatible unless there is an anomalous isotope effect. Further work, both experimental and theoretical, is required as the reaction is of considerable significance in combustion. 2.5.4 Theory and discussion The two association reactions have been examined theoretically by Marcus, Wardlaw and co-workers [47-49, 69]. They treated these reactions using Flexible Transition State Theory (FTST), a variational derivative of transition state theory. The difficulty with association reactions such as reactions (31) and (32) is that there is no barrier to association and so there is no obvious location on the reaction coordinate for the transition state. Recent developments of TST place more emphasis in locating the molecular geometry for which the reactive flux is a minimum, and the transition state is associated with this geometry. These remarks apply equally to the complementary unimolecular reaction and it is helpful to look at the unimolecular reaction to begin with, always bearing in mind that association and dissociation are connected via the equilibrium constant. In Section 2.4.4 it was shown that for the RRKM model, the microcanonical rate coefficient is proportional to the sum of states, G, at the transition state, which is a function of the energy, E. Application of the minimum flux criterion means that G must be altered

194


Ch. 2

so that it is now also a function of the reaction coordinate, R. The microcanonical rate coefficient is determined from equation (2.47), using the value G(E, R) which is a minimum with respect to i?, i.e., k{E) = G(£, Rx)lhN{E),

(2.59)

where dR The high pressure canonical rate coefficient for dissociation is given by *5(T) = — — G(£, R*) t-^dE. hQ Jo

(2.61)

The association rate coefficient is obtained from equation (2.61) by application of the equilibrium constant and gives ka(T) =

1 f°° G(E, /?*) exp(-jSE) d£, PV F hQA(T)QB(T)J0

(2.62)

where G A ( T ) , QB(T) are the partition functions of the associating fragments at infinite separation. Marcus, Wardlaw and coworkers [47-49, 69] calculated G(£, /?*) using FTST for both reactions (31) and (32) and showed that the methyl recombination reaction did indeed exhibit a negative temperature dependence and that the methyl/hydrogen atom association reaction exhibited a positive temperature dependence. The potential energy surfaces on which these two reactions take place are quantitatively similar so the obvious question is why do these reactions have such different temperature dependences? The answer to this question lies in the functional dependence of the sum of states, G, at the transition state. The dependence of G on R is complex, but can be approximately separated in to two competing factors: (i) As 7? decreases and the reactants go over to form products, the degrees of freedom that were free rotors in the reactants become progres-


195

sively more hindered and ultimately become vibrations, which consequently leads to a decrease in the sum of states at a given energy. (ii) Counteracting this effect is that of the reaction exothermicity. As the reaction coordinate decreases the reaction exothermicity is released initially into the reaction coordinate. This excess energy is available to the other degrees of freedom and has the net effect of increasing the number of states that are available at the transition state. The relative magnitude of these two competing factors determines the observed temperature dependence. For the methyl radical/hydrogen atom association reaction there are two degrees of freedom that become vibrations as reactants go over to the product. For the methyl recombination there are five such degrees of freedom. It is thus expected that the first factor, the decrease in the density of states as R approaches the product will be greater for methyl radical recombination, and as a consequence G(£, i?*) will be a weaker function of E. 2.5.5 Addition reactions, equilibria and alkyl radical heats of formation Rate coefficients for addition reactions such as H + C 2 H 4 + M -> C 2 H 5 + M

(34)

show similar pressure dependencies to recombination reactions although, as the reaction requires the breakup of the rr system in the carbon double bond of the alkene, there is usually a positive activation energy and a well defined transition state. Addition reactions are linked to the alkyl radical decomposition reactions via the equilibrium constant. The equilibrium constant is in turn related to the thermodynamics of the reaction via equation (2.63) ArGe=

-RTlnK^

(2.63)

Kinetic studies of the forward and reverse reactions (either under isolated conditions or during direct equilibration) can therefore be used to determine ArGf£8 a n d fr°m this the heat of formation of the alkyl radical, AfHf98(R). Alkyl radical heats of formation are of considerable importance because they are linked to the C—H bond strength of the corresponding

196


Ch. 2

hydrocarbon. In this sub-section we shall briefly review addition reactions, their use in determining radical heats of formation and the current consensus on the values of A f //^ 8 (R). Hydrogen atom addition reactions can be studied under isolated conditions using the techniques outlined in Section 2.3.2, examples may be found in the studies of Pilling and coworkers [70-72], Harris and Pitts [43] or Stief and coworkers [73]. Forward and reverse rate coefficients are related via the equilibrium constant and hence thermodynamic data for the reaction can be obtained, A r G e (T). If A r 5 e (T) is known or can be calculated then A r / / e may also be determined by the Third Law Method. Heats of formation of the alkene and H are well known and hence the heat of formation of the radical can be determined. Alternatively, if both reactions are measured over a significant, overlapping region of temperature, subtraction of the forward and reverse activation energies directly yields the enthalpy of reaction for the midpoint of the temperature range (Second Law Method). As the temperature of the system rises it is noted that the decays of the atomic species in the presence of excess alkene are no longer exponential, but show a biexponential decay. At high temperatures the rate of alkyl radical dissociation occurs on the same timescale as the addition reaction and equilibrium is established. The two portions of the biexponential decay are related to the rapid approach to equilibrium, which depends on the sum (&34[C2H4] + /c-34), followed by the slower diffusive loss of the equilibrated species from the observation region. Analysis of such data leads directly to third law determinations with the advantage of reduced systematic errors as only one experimental technique is used [71, 72]. Other chemical systems can also be utilized to determine A f // e (R), probably the most well documented is the hydrogen halide equilibrium: (43, -43)

R + HX ^ RH + X

(X = Br, I)

Up until the late 1980's measured activation energies for the reverse reaction were combined with assumed values of the forward reaction to give Second Law heats of formation for R [74]. However, the values of A f // e (R) were found to be incompatible with those measured by addition (or recombination)/dissociation equilibria [71, 75]. The reasons for this were rationalized by a series of experiments performed by Gutman and coworkers [76, 77] and others [78, 79] who directly measured the rate


197

TABLE 2.6 Radical heats of formation based on kinetic studies Radical

AfH298(R) kJ m o F 1

CH 3 C2H5

145.6 120.9 90.0 67.4 51.5 -12.1 -2.4

1-C3H7 5-C4H9 £-C4H9

CH 2 OH CH3CO

3 7 7 1 7 2 3

D 298 (R-H)/ kJ mol" 1

Reference

438.5 423.0 412.5 410.9 403.8 406.7 374.0

82,83 71, 72, 77, 79 77 77 77-79 84-86 87

1.3 7 1.7 2.1 7 4.2 3

coefficients for the R + HX reactions and found activation energies significantly lower than those assumed by earlier workers. The currently accepted values for various radical heats of formation and the related C—H bond energies are given in Table 2.6 and reviews on methods of determining radical heats of formation may be found in two articles by Gutman [80, 81]. 2.5.6 HO2 self reaction In low temperature combustion of H 2 / 0 2 H 0 2 radicals are formed from the pressure dependent combination of hydrogen atoms and oxygen. The relative unreactivity of the H 0 2 radical means that this reaction is effectively a termination step and determines the onset of the second explosion limit in the hydrogen oxygen system. One of the most important H 0 2 reactions is the self reaction to generate hydrogen peroxide (6)

H 0 2 + H 0 2 -* H 2 0 2 + 0 2

which via reaction (36) converts H 0 2 radicals into reactive OH radicals. The H 0 2 self reaction shows many of the characteristics of a recombination reaction exhibiting a negative temperature dependence and a positive pressure dependence. Closer inspection of the variation of k6 with pressure shows a nonzero intercept and at higher temperatures the reaction begins to show a positive temperature dependence. Reaction (6) has been a source of controversy compounded by the combined experimental difficulties of the lack of a simple photolytic

198


Ch. 2

method of generating H 0 2 and a sensitive detection method. Besides the method of H 0 2 generation described in Section 2.3.6 an alternative indirect method photolyzing chlorine in the presence of methanol and oxygen has also been used for real time studies [88]. (44)

Cl2 + hi/->2Cl

(45)

CI + CH 3 OH -> HC1 + CH 2 OH

(46)

CH 2 OH + 0 2 -> HCHO + H 0 2

A corresponding F 2 /CH 3 OH/0 2 system has been used in discharge flow studies or alternatively the F 2 /H 2 0 2 system. Detection methods include absorption [88], and i.r. diode laser spectroscopy [89] for time resolved studies and laser magnetic resonance for flow systems [90]. The overall rate coefficient is given by equation (2.64) k6 = k'6 + AS[M]

(2.64)

where k'6 describes the pressure independent component of the reaction and k^ refers to the pressure dependent contribution. The pressure dependence of reaction (6) is best described by the schematic potential energy diagram shown in Fig. 2.29 [91]. The intermediate complex can either redissociate or isomerize directly to products giving a pressure independent term. Alternatively, at higher pressures the H 2 0 4 complex can be stabilized increasing the rate of H 0 2 removal. As the exit barrier to products lies below that of the entrance channel, reactivation of the stabilized H 2 0 4 intermediate will preferentially form products rather than regenerate the reactants. Figure 2.30, which illustrates the temperature dependence of the overall rate coefficient, indicates that at higher temperatures a direct channel with a significant activation energy opens up. The data of Lightfoot et al. [92] neatly covers the temperature range including the minimum in k6 and illustrates the onset of the direct abstraction. A comprehensive shock tube study by Hippler et al. [93] has characterized the Arrhenius parameters for the high temperature reaction and confirmed an earlier single temperature determination of Troe [94] which had appeared to contradict the lower temperature data.

R + 02

199

2

2H02

Energy

H202 + 0 2

H02 + H02 ^ = ^

H204

*,[M]

• H202 + 0 2

*-s(M]

H204 Fig. 2.29. Potential energy surface for reaction (6) adapted from Ref. [91]. 2.6 R + 0 2

2

2.6.1 Introduction The equilibrium (47, -47)

R + Q2 ^ R 0 2

is central to the nature of the oxidation process. At low temperatures a classical pressure dependent association reaction occurs and the peroxy radical formed can undergo a number of reactions, some of which lead to the formation of degenerate branching agents crucial to the ignition process. At higher temperatures the peroxy radical becomes increasingly unstable, the equilibrium shifts to the left and alkyl radicals react by an alternative mechanism discussed in Section 2.6.4. The reverse peroxy rad-

200


Ch. 2

o E E

KT K/T Fig. 2.30. Temperature dependence of k6. (X) Ref. [93], (O) R. Patrick and M.J. Pilling, Chem. Phys. Lett. 1982, 91, 343. ( • ) Ref. [94], (A) B.Y. Andersson, R.A. Cox and M.E. Jenkins, Int. J. Chem. Kinet. 1988, 29, 283. ( • ) Ref. [92]. Solid line is fitted expression from Ref. [93].

ical decomposition involves breaking the C—O bond and hence is a highly activated process, whereas the formation of the peroxy complex has no activation energy. The equilibrium constant and hence the position of equilibrium is therefore highly sensitive to temperature changes and this temperature sensitivity is the key to the oscillatory behaviour observed in some cool flames (Chapter 4) and to the negative temperature regime in hydrocarbon oxidation. In this section we shall briefly review the experimental data on the equilibrium (47, -47) and then move on to discuss the strong correlation between the rate coefficients for reaction (47) and the ionization potential of the alkyl radical. The direct determination of the enthalpies for reactions (47, -47) during the mid and late 1980's produced values significantly larger than those estimated by group additivity methods [95]. The reasons for these discrepancies are discussed in the final part of this section.

R + 0 2 ^ R0 2

201

2.6.2 Experimental A majority of the studies of reactions (47, -47) have been carried out using flash photolysis/UV detection of hydrocarbon radicals [96-98] or flash photolysis coupled to photoionization mass spectrometry [99, 100]. Both techniques have been discussed in previous sections. The experiments fall into two categories, those measuring only the forward reaction (which may be pressure dependent) and secondly direct studies of the equilibrium (47, -47). The details of some studies are tabulated in Table 2.7. At low temperatures the hydrocarbon radicals are removed by reaction (47) (or for some studies by reaction (47) and the recombination reaction requiring the use of a mixed order analysis programme [96]) but as the

TABLE 2.7 Experimental studies of the R + 0 2 reaction Radical

Method

Temperature /K

CH 3

LFP/UV

298-530

20--600

CH 3 CH 3 C2H5 C2H5 C2H5 C2H5 «-C3H7

LFP/PIMS LFP/UV LFP/PIMS RR DF/MS PR/UV LFP/PIMS LFP/PIMS LFP/PIMS LFP/PIMS

298 298 296-904 298 298 298 298 298 298 266-374

0.5--6.7 1.5 X105 0.3--6.0 3-1500 ? 760 4 4 3 3

LFP/PIMS LFP/PIMS

298 298

3 3

/"Z-C^rig

/-C4H9

neo-C 5 Hn C-C5H9

c-C 6 H n

Pressure /Torr

r/cm3 molecule -1 s _ 1

Ref.

(1.2 0.2) x 10~12 (T/300)1-2*0-4 na 2.2 x 10" 12 na (9.2 0.9) x 10" 12 na 12 -12

(7.5 1.4) x 10" 12 (2.9 0.7) x 10~12 2.1 x 10" 12 ) (T/300)" (1.7 0.3) x 10" 11 n

96 99 97 100 101 102 98 103 103 104 105 104 104

LFP/UV, Excimer laser photolysis, UV absorption detection of hydrocarbon. LFP/PIMS, Excimer laser photolysis, photoionization mass spectrometric monitoring of hydrocarbon. RR, Relative rate determination. DF/MS Discharge flow production of radicals, mass spectrometric monitoring of hydrocarbon radicals. PR/UV, Pulsed radiolysis in 760 Torr H 2 , UV absorption detection of ethyl and ethyl peroxy.

202


Ch. 2

temperature increases the hydrocarbon radicals are reformed by decomposition of the peroxy radical. For small radicals (e.g., methyl and ethyl) the association reaction is in the fall off region under normal laboratory conditions ( 2CH 3 0

which will be present in the high radical density experiments of Cobos et al. There is still some considerable scatter in the values for the high pressure limiting rate coefficient for reaction (49) (49)

C 2 H 5 + 0 2 + M -> C 2 H 5 0 2 + M

The most recent determination, made using relative rate methods [101], gives a value approximately a factor of two higher than either the extrapolation of the low pressure data of Plumb and Ryan [102] or the atmospheric pressure measurement of Munk et al. [98]. However, it is close to the extrapolation of the low pressure data of Wagner et al. [108] made using variational RRKM methods. Figure 2.31 illustrates the strong correlation between the reaction crosssections for oxygen addition and the ionization potentials of linear alkyl radicals and the physical reasons for this are discussed in the following section. It can be seen that the data of Cobos et al. [97] (CH3) and Kaiser et al. [101] (C2H5) lie someway off the best fit line to the remaining data points which have been determined by a number of different experimental techniques, indicating that these two determinations may be subject to some systematic errors. Figure 2.32 shows a typical resultant biexponential decay of the hydrocarbon radical observed for equilibrating systems when the rate of decomposition of the peroxy radical becomes comparable with that of the addition reaction. The initial rapid decay is due to the establishment of the equilibrium (47, -47) and the rate coefficient which characterizes this

203

R + Q2 ^ R 0 2

10

~"

^Vbutyl 5-butyl | ft c-hexyl c-pentyl ' * \ f t /-propyl

/2-butyl ""•

p p L

p

O ethyl

n -propyl ft \ ft.ethyl /-butyl O neo-pentyl O

\ \ O methyl

\ ft methyl

o.i

_j

6

.

10 (D?-EA)/eV

Fig. 2.31. Variation of rate coefficient of R + 0 2 with ionization potential of the radical species. Full details of the experimental measurements can be found in references 99 and 103-5.

portion of the decay is the sum of k'47 (the pseudo-first-order rate coefficient, k47 [0 2 ]) and k-47 [4]. On a longer timescale the radicals are gradually removed by radical-radical reactions and diffusive loss from the observation region. Such analyses allow a direct determination of the equilibrium constant which can be compared to those calculated by group additivity rules. 2.6.3 Discussion Bayes and coworkers [99, 103-5] have carried a systematic survey of hydrocarbon radicals and found that there is a very strong correlation between k41 and the ionization potential for the hydrocarbon radical (Fig.

204


Ch. 2

TABLE 2.8 Experimental studies of the equilibrium (47, -47) Radical

Temperature/K

Method

Reference

AH R +o2

/kJ mol" 1

IRMPD/PIMS LFP/PIMS LFP/PIMS LFP/UV LFP/PIMS LFP/PIMS

CH 3 C2H5 *-C3H7 C3H5 C3H5 t-K^4rig

694-811 609-654 592-692 382-453 352-413 550-580

135.4 147.1 157.6 76.1 75.2 153.4

2.9 6.3 7.5 1 8 7.9

109 110 111 4 111 110

IRMPD/PIMS, Infrared multiphoton dissociation/photoionization mass spectrometry. LFP/UV, Excimer laser photolysis, UV absorption detection of hydrocarbon. LFP/PIMS, Excimer laser photolysis, photoionization mass spectrometric monitoring of hydrocarbon.

3.0 2.5 2.0 1.5 *

1

1.0 0.5 0.0

• A , -.••rf'V^residuais

U-U"

0.0

.0

2.0

3.0

4.0

Fig. 2.32. Time dependence of the allyl % absorption signal showing the approach to, and establishment of equilibrium in reaction (4,-4). T= 440 K, oxygen pressure = 5.44 Torr, hexadiene pressure = 166 mTorr, total pressure (argon) = 50 Torr.

2.31). Bayes suggested that the relationship is due to long range attractions between the two radical species R and 0 2 . A lower ionization potential implies that the outer electrons of the hydrocarbon radical are less tightly bound and hence more polarizable increasing the dipole interactions be-

R + 0 2 ^ R0 2

205

tween the two radicals and hence the cross-section of the reaction. The effects of the changing masses of the hydrocarbon radicals (and hence the collision frequency) can be removed by dividing by the mean relative velocity of the reactive systems. Branched chain radicals such as neopentyl and isobutyl seem to fall someway below the best fit line implying a steric or possibly a hyperconjugation effect which decreases the reactivity of these radicals. A number of the R + 0 2 reactions show a negative temperature dependence. It is not possible to explain this by classical transition state theory (where the transition state is of a defined structure and located at a constant position on the reaction coordinate) and the reaction needs to be modelled using the variational techniques discussed earlier for methyl + methyl and methyl + H (Section 2.5.4). Direct measurements of the equilibria (47, -47) brought into question previous estimates of the thermochemistry of the R + 0 2 system which had been based on group additivity (Chapter 1) calculations. These calculations were subject to considerable uncertainty as prior to the mid 80's there was little direct experimental evidence on Af//(R) or A r // R + 0 2 Group additivity calculations predicted an almost constant bond energy (or A r // R + 0 2 ) with increasing complexity of the alkyl radical. In contrast the equilibrium studies of Gutman, Slagle and coworkers [108-111] showed a tendency for -A r /f R + 0 2 to increase with increasing complexity of the alkyl radical. A qualitative agreement in the trend of A r / / R + 0 2 with R could be achieved if the values of Af//(R) used in the group additivity calculations were replaced with the higher recommendations of Tsang [75]. Direct determinations of alkyl radical heats of formation [81] have confirmed the values of Af//(R) proposed by Tsang. The calculated values of A f H(R0 2 ) were now consistently 7.9kJmol _ 1 too low suggesting that the heat of formation of the O—(C)(0) group should be adjusted downwards by a corresponding amount. Slagle et al. [110] suggested that such a small adjustment is consistent with the errors of the group additivity calculations which are based on a number of complex thermochemical cycles. 2.6.4 High temperature pathways At high temperatures the peroxy-radical becomes increasingly unstable with respect to redissociation to reactants and an alternative reaction

206


Ch. 2

pathway is required. The observed products are predominantly alkenes and H 0 2 but epoxides and OH can also be formed, usually in lower amounts. The overall reaction for ethyl + 0 2 occurs via three channels (49a)

C2H5 + 0 2 ^ C 2 H 5 0 2

(49b)

-> C 2 H 4 + H 0 2

(49c)

-> C 2 H 4 0 + OH

with channels (49a) and (49b) predominating. Two models have been developed for the reaction. One, [112] based primarily on the detailed chemistry of the peroxy radical and of subsequent steps in the oxidation chain, proposes independence of the channels with (49b) occurring by direct H-atom abstraction. The other model [108], which was developed to explain the pressure and temperature dependence of the overall rate coefficient and the channel efficiencies, proposes a composite mechanism in which channel (49b) occurs via the peroxy radical and involves intermediate isomerization to give an unstable hydroperoxy radical. Neither mechanism is able fully to explain all the experimental observations; indeed the validity of some observations has been questioned by proponents of the composite model, Bozzelli and Dean [113]. In the following we summarize the experimental observations and the apparently contradictory conclusions which arise if the results are interpreted with either of the models outlined above. An alternative model is presented which invokes involvement of two electronic states of differing symmetry for the peroxy radical. The model is compatible with all the available experimental data on ethyl + 0 2 . (/) Survey of experimental results Detailed discussions of some of the experimental data can be found elsewhere [108]. In this subsection major results are summarized to provide a basis for the subsequent mechanistic proposals. (a) Measurements of the overall rate constant (k49) and the channel efficiencies (49a, 49b) • k49 is pressure dependent [108], as would be expected for an association reaction, but there is a zero pressure intercept, compatible with a non-associative channel, whose magnitude increases with temperature.

R + 0 2 ^ R0 2

207

• The pressure dependence of k49 becomes less pronounced at higher temperatures [108]. • 049b? the channel efficiency for reaction (49b), decreases with pressure and increases with temperature. • At high pressures, k49b shows a weak and possibly negative temperature dependence. (b) Detailed chemistry of reactions subsequent to peroxy radical formation • At temperatures of approximately 700 K, oxirane is formed from channel (49c), in competition with channel (49b), with a yield which is much lower than that of ethene. The mechanism is presumed to involve the the formation of the peroxy radical and subsequent isomerization to the QOOH radical. • There have been extensive measurements of other alkyl radical systems. All are compatible with a mechanism in which the peroxy radical isomerizes to form a QOOH radical which then dissociates in a variety of ways. • Time resolved measurements of the equivalent reaction for the neopentyl radical, using LIF of OH as a probe, are fully compatible with gas chromatographic analysis of the products (see Section 2.7). For this radical, the analogue of channel (lb), to form the alkene, is not possible and the reaction proceeds exclusively via the peroxy/hydroperoxy route. • The hydroperoxy radical (QOOH) is hypothesized from the end product analysis experiments of Walker and co-workers [6] as a distinct intermediate. The radical is also important in proposed mechanisms of the low temperature (^700 K) oxidation of alkanes. It is considered sufficiently long lived to form hydroperoxy-peroxy radicals. (50)

QOOH + 0 2 -> 0 2 QOOH

• It is also relevant to look at the reverse reaction of the products of (-49b). Baldwin et al. [114] have measured the rate of formation of the oxirane: (51)

C 2 H 4 + H 0 2 -> C 2 H 4 0 + OH

208


Ch. 2

They proposed that the initial stage involves addition of H 0 2 to form the hydroperoxy radical which represents the rate determining step. They obtained an activation energy of 70kJmol _ 1 for reaction (51) which they ascribed to the addition step of H 0 2 to ethene. (ii) Possible reaction mechanisms (a) Independent channel mechanism Benson [112] proposed a mechanism in which the 0 2 directly abstracts a hydrogen atom from C 2 H 5 , so that reactions 49a,b occur via independent, parallel channels. Such a mechanism could reproduce many of the observed experimental features, giving a non-zero intercept in the k/[M] plots and an increased efficiency for channel 49b at low pressures and high temperatures, simply because of the reduction in k49a. The direct mechanism, though, cannot explain the observed coupling in k49a and k49h (i.e., the pressure dependence of both channels a and b) or the observed negative activation energy for reaction 49b. Benson originally proposed an activation energy of 20-40 kJ mol - 1 for channel b. It is clear that a realistic mechanism must incorporate coupling and a common peroxy intermediate. (b) Composite mechanism In this brief resume we concentrate on the mechanism proposed by Wagner et al. [108], noting any significant differences in the Bozzelli and Dean [113] mechanism at the end of this sub-section. The mechanism seeks primarily to explain the kinetic observations enumerated above. It addresses, in particular, the pressure dependence of k49h and its zero or negative activation energy. These observations are compatible only with a mechanism in which C 2 H 4 and H 0 2 are formed from the peroxy radical (bold curve in Fig. 2.33). Wagner et al. constructed the potential energy surface using thermodynamic data and results of ab initio calculations. They carried out RRKM/modified strong collision calculations, fitting the wide range of experimental data by varying a number of surface parameters. Thefitsare excellent and require an energy for the peroxy/hydroperoxy radical transition state that lies below the C 2 H 5 + 0 2 asymptote by approximately lOkJmol - 1 . This corresponds to a threshold energy for reaction (49b) of approximately 125 kJ mol - 1 , somewhat lower than the activation energy observed in the formation of oxirane from the peroxide (145 kJmol - 1 ). The barrier for dissociation of QOOH to C 2 H 4 + H 0 2 was placed at a still lower energy based on estimates by Benson [112] of 25kJmol _ 1 for the activation energy for

R + O2

209

2

1 Energy

C2H5 + Q2

C 2 H 4 + H0 2

C2H4OOH

Reaction Coordinate

Fig. 2.33. Possible potential energy surfaces for reaction (49). Solid line from Ref. [108]. Dashed line from ref [114].

C 2 H 4 + H 0 2 . This leads to an extremely short lifetime for the hydroperoxy radical formed in reaction 2 so that their model becomes totally insensitive to the properties of this radical. Baldwin et al. [114] obtained a much higher experimental activation energy (assumed to represent the barrier to hydroperoxy formation, the dashed surface of Fig. 2.33) for reaction (51) (70kJmol - 1 ), which, as pointed out by Wagner et al. [108], places this transition state above the C 2 H 5 + 0 2 asymptote, which is totally incompatible with the requirements that the kinetic data for reaction (49) place on the mechanism. Note that the Wagner mechanism does not permit a lifetime for C 2 H 4 OOH, as formed from C 2 H 5 + 0 2 , which is sufficiently long to enable it to undergo the reactions proposed in alkane oxidation chemistry. The Bozzelli and Dean [113] mechanism used QRRK theory and placed the transition state for isomerization at even lower energies. An alternative, high energy four membered ring transition state was also proposed which leads to CH 3 CHO + OH.

210

Ch. 2


2

A'

2

A

2

A"

2

A'

Fig. 2.34. Orbital arrangements of the two lowest electronic states of the ethyl peroxy radical.

(Hi) Proposed mechanism for the C2H5 + 02 reaction The above system has attracted attention from a number of workers using ab initio methods. In a recent series of papers Quelch et al. [115, 116] have examined the properties of the ethylperoxy radical using various basis sets. Quelch et al. make two pertinent observations as a consequence of their calculations. The first is that the ethyl peroxy radical has an excited state (2Af) that is close in energy to the ground (2A") state (72.8 kJ mol - 1 ) and, which they believe is derived from the *Ag state of oxygen. Examination of the molecular orbitals for the cis conformer of both states suggests that in the 2A" state the unpaired electron on the terminal oxygen occupies a p-like orbital that is perpendicular to the plane containing the carbon and oxygen atoms, whereas in the 2 A' state the unpaired electron occupies an orbital that lies in the carbon-oxygen plane. These orbitals arrangements are shown in Fig. 2.34. The second observation derives from

R + 0 2 ^ R0 2

211

these orbital arrangements: examination of the cyclic transition states of the two electronic states shows that each yield different products. (52)

( 2 A")C 2 H 5 0 2 _> C 2 H 4 . . . H 0 2

(53)

( 2 A')C 2 H 5 0 2 -> C 2 H 4 0 2 H

In the (2A") case a concerted reaction occurs resulting in the elimination of H 0 2 and possible formation of a van der Waals complex. The ( 2 A') state, on the other hand, yields QOOH, the intermediate required for epoxide formation. Recently Walch [117] has reported calculations on a related system, (54)

CH 3 + 0 2 -> Products

using a large atomic natural orbital basis at the CASSCF level. As with the calculations of Quelch et al. a low lying excited state was found. The ground 2A" state correlates with CH 3 + 0 2 ( 3 2~) and leads only to the products CH 3 0 + O. The excited 2 A' state correlates with CH 3 + 0 2 ( 1 A g ) and leads to the products CH 2 0 + OH. The unpaired electron on the terminal O-atom occupies a p-like molecular orbital and, as with the ethylperoxy radical, this orbital is perpendicular to the C—O—O plane for the 2A" state and in the plane for the 2 A' state. Furthermore, Walch argues that 1,3-hydrogen migration is not favourable for the 2A" state, but it is for 2 A' because in this state the unpaired electron is in the C—O—O plane. The consequence of this is that only the 2 A' state can go on to form CH 2 OOH which subsequently decomposes to CH 2 0 and OH. The 2A" state on the other hand, because 1,3-hydrogen migration is not possible, gives the products CH 3 0 and O. Given the close similarities between the work of Quelch et al. and Walch analogous arguments should explain some of the observations mentioned previously. Starting with C 2 H 5 + 0 2 ( 3 2~) the initial product will be the 2 A" ethylperoxy radical whose orbital structure has been shown by Quelch et al. to be such that the unpaired electron on the terminal O-atom is in a p type orbital perpendicular to the C—C—O—O plane. By analogy with the 2A" state of CH 3 0 2 ,l,4-hydrogen migration is expected to be inhibited and the formation of the hydroperoxy radical is improbable. Instead,

212

Elementary reactions H' \

Ch. 2

O /

c-c

Energy

C 2 H 5 +0 2 C ,H, + H0 2

C2H40 + OH C2H4OOH

C2H502

Reaction Coordinate Fig. 2.35. Potential energy surface for the ethyl + 0 2 reaction involving the first electronically excited level of the ethyl peroxy radical.

reaction occurs to form C 2 H 4 + H 0 2 via a concerted mechanism (Fig. 2.35). On the other hand if the initial reactants were C 2 H 5 + 0 2 ( 1 A g ) then the 2 A' state is formed initially, the orbital structure of which does permit 1,4-hydrogen migration, and the hydroperoxy radical can be formed, and can decompose to give OH and C 2 H 4 0. Since the direct formation of the ethylhydroperoxy radical is improbable for the 2A" state, the question still remains as to how the ethylperoxy radical forms oxirane, which can be produced from C 2 H 5 + 0 2 ( 3 2~) albeit in smaller yield than from C 2 H 4 + H 0 2 , as observed by Baldwin et al. [114]. Reaction (49b) must involve internal conversion from the 2A" to the 2 A' state. Such a process will occur with low probability because the density of states in the latter is smaller than that in the former, because of the difference in zero point energies. Nevertheless, internal conversion allows access from ground state reactants to the excited electronic state

R + 0 2 ^ R0 2

213

with the correct orbital symmetry for formation of the hydroperoxy radical. Since the mechanism proposes distinct surfaces leading to C 2 H 4 + H 0 2 and to C 2 H 4 0 + OH, it allows the former to occur with a lower activation energy and the latter to occur via a hydroperoxy radical that is sufficiently long-lived to have its own identifiable chemistry. It remains to explain reaction (51) and the experimental results of Baldwin et al. [114], which require a large activation energy. It is helpful to consider the following composite mechanism: (55, -55)

C 2 H 4 + H 0 2 ^± C 2 H 5 0 2

(-49a, 49a)

C 2 H 5 0 2 ^± C 2 H 5 + 0 2 C 2 H 5 0 2 -> C 2 H 4 0 + O H

(56)

where k56 incorporates both internal conversion and reaction. McAdam and Walker used a large concentration of 0 2 , so that reaction (49a) is significant. Applying the steady state approximation to C 2 H 5 0 2 and C 2 H 5 , recognising that k-55 > k56 and equating the rate of formation of C 2 H 4 0 to both fc55[C2H4][H02] and £ 56 [C 2 H 5 0 2 ], we find k51 = k56(k55/k.55)

(2.65)

i.e., k51 is a composite rate constant containing the equilibrium constant for reaction (55) and k56. The effective activation energy thus approximates to the difference in energy between the C 2 H 4 + H 0 2 asymptote and the 2 A' transition state, which is clearly larger than E55, as required by both Wagner et al. and Baldwin et al. This analysis is necessarily limited because it assumes complete thermal relaxation within the C 2 H 5 0 2 well, which may not be valid. Nevertheless, it provides qualitative agreement with the experimental results. Further evidence to support for this mechanism comes from the work of Salem [118]. Salem examined a number of photochemical reactions with particular attention to potential energy surface symmetry for both ground and excited states. One such reaction considered was hydrogen abstraction by ketones: (57)

RR'CO + HR" + hi/-> R R ' C O H + R"-

214

Ch. 2


1.6

1 1-2 fc: b 0.8 o

*

0.4

700

650

600

550

Temperature (K) Fig. 2.36. The rate of propene oxidation as a function of temperature showing the region of negative temperature dependence.

The main products of this reaction are two radicals. Examination of the orbital symmetry of the excited state by which this reaction proceeds shows that the electron density on the O-atom is similar to that of the 2Ar state mentioned above. Furthermore, the ground state reactants are shown to correlate with ionic products. This latter abstraction is not seen for the reactants in reaction (55) as the products are of higher energy, however the cyclic transition state of reaction of the ethylperoxy radical allows a concerted reaction which does not involve the explicit formation of charged intermediates. 2.7 PEROXY RADICAL ISOMERIZATION

2.7.1 Introduction - Formation of degenerate branching agents A characteristic of alkane oxidation is the negative temperature dependence of ignition delays (Fig. 2.36). This occurs in the temperature region where peroxy radicals are becoming unstable and hence low temperature ignitions were associated with reactions of peroxy species. Once formed, peroxy radicals can only undergo a limited number of propagation reactions and some of these will be dependent on the nature of the fuel. If the fuel contains labile hydrogens (such as aldehydes) then the peroxy radical can abstract a hydrogen to form the hydroperoxy species (58)

R02 + RH-^ROOH + R

Peroxy radical isomerization

215

which at high temperatures acts as a degenerate branching agent (DBA) yielding RO and OH radicals (59)

R O O H - ^ R O + OH

which in turn can react with the fuel to generate more radical species. However, for alkanes the strength of the C—H bond is such that we must look elsewhere for the source of the DBA that leads to cool flames and ignitions characteristic of alkane oxidation. In 1968 Fish [119] proposed an alkylperoxy isomerization theory that explains the temperature dependence of the ignition delays and the complex spectrum of products that can be obtained in hydrocarbon oxidation. Once formed the ROO radical can under go an internal isomerization. Cleavage of the resulting R'OOH (or QOOH) radical is a chain propagating step leading to generation of a number of different products and an OH radical. Alternatively at low temperatures the addition of 0 2 can compete with unimolecular dissociation. (50)

QOOH + 0 2 -> 0 2 QOOH

The resulting radical species can undergo a number of reactions including the formation of the hydroperoxide degenerate branching agent. For larger hydrocarbons a number of different isomerization processes can occur, the efficiency of each process being controlled by the free energy change on formation of the cyclic transition state. This in turn is made up of two components, an entropy term (reflecting the probability of the end oxygen atom colliding with a particular site) which decreases with increasing ring size and an enthalpy term favouring the formation of 6 and 7 membered rings which have minimum ring strain energy. Baldwin, Walker and coworkers [120, 121] have determined Arrhenius parameters for a number of isomerization reactions using steady-state, end product analysis techniques, adding alkanes to slowly reacting H 2 / 0 2 systems. A steady-state analysis of the kinetic scheme shown in Fig. 2.37a gives the product ratio ([DMO] + [CH3COCH3])/[C4H8]) = K61k62[02]/k60

(2.66)

provided that k-61 > k62. From the measured product yields, literature values for k60, and thermochemical estimates of K61 they were able to determine rate coefficients for a wide range of radicals and postulate generalized structure-related Arrhenius expressions.

216

Ch. 2

Elementary reactions (CH3)3CCH2

UOj| CH4

H2

%

CH3 + (CH3)2CCH2

HCHO

(CH 3 ) 3 CCH 2 0 2

°2

(CH3)2C - CH 2 «-

(CH 3 ) 2 (CH 2 )CCH 2 0 : H

I I

H2C-0

02

+OH

CH3COCH3 + 2HCHO + OH

(a)

(CH3)3CCH2I hv (CH3)3CCH2

*«[OJ CH 3 + (CH3)2CCH2

other products +

«

>*£ TZL

*-61

(CH 3 ) 3 CCH 2 0 2

(CH^CH^CCH.O^

OH

diffusive loss of OH

(b)

Fig. 2.37. (a) Scheme for study of reaction (62) by end product analysis, (b) Scheme for time resolved study of reaction (62).

Peroxy radical isomerization

217

However, direct studies of the R + 0 2 equilibrium, of the type described in Section 2.6.2, produce values for the equilibrium constant over an order of magnitude different from the thermochemical estimates affecting the absolute values of the isomerization rates. Pilling and coworkers [122, 123] devised a time resolved equivalent of Baldwin and Walker's experiment (Fig. 2.37b) to directly measure the isomerization rate for the neopentyl radical. This technique is described in Section 2.7.2. The results of the direct studies are shown to be compatible with the earlier indirect work and can be used to scale the results of Baldwin and Walker. The relationship between isomerization rate and structure is discussed in Section 2.7.3. 2.7.2 Direct studies of peroxy radical isomerizations The laser flash photolysis/laser induced fluorescence technique used to study reaction (62) is the time resolved equivalent of scheme (2.37a). Neopentyl was once again the radical of choice as the reaction with oxygen cannot lead to the formation of an alkene species. The hydrocarbon radical was generated by excimer laser (248 nm) photolysis of neopentyl iodide and the production and decay of OH radicals monitored by laser induced fluorescence (Fig. 2.38). Originally it was thought that steps 60, 61, - 6 1 and 63 would be much faster than the isomerization process. In that case the rising portion of the OH temporal profile could be associated with k62 and hence would be independent of oxygen concentration. The slower decay is due to diffusive loss of the OH radical and reaction with the precursor. The success of the technique depends on the use of low concentrations of both radicals and radical precursors to prevent complications from radical-radical and radical-precursor reactions. Due to the sensitivity of the laser induced fluorescence technique these conditions can easily be achieved. However, Fig. 2.39 shows that the rising rate coefficient (A+) depends sensitively on [0 2 ], tending to a limiting value at high [0 2 ]. If one also includes reactions (60-62) and (65) in the analysis [122] then the expression for A+ depends on [0 2 ], tending to -k62 at high concentrations and -(A:_61 + k62) as [0 2 ] approaches zero. The full expression for A+ is: A+ = l / 2 [ - i 8 + ( j 8 2 - 4 y ) 1 / 2 ] ,

(2.67)

P = k60 + k61[02] + k-61 + k62,

(2.68)

where

218


Ch. 2

T—i—i—i—(—i—i—i—i—|—i—i—i—i—|—i—i—i—i—f—i—i—i—i—|—i—r—r—i—j—T

0,0 *—J ' ' ' '—'—'—'—'—' '—'—' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' L 0.0000.001 0.001 0.002 0.002 0.003 0.003

time/s Fig. 2.38. Time dependence of the OH signal following the photolysis of neopentyliodide/0 2 mixture.

y = k60k-61 + k60k62 + k61[02]k62.

(2.69)

k60 has been measured by Slagle et al. [34] and an extrapolation of the lower temperature determination of k6l by Xi et al. [105] allows both k62 and k-61 to be determined from a fit to variation of A+ with [0 2 ] (the solid line in Fig. 2.39). The determinations were repeated over a range of temperatures from 660-750 K to obtain the following Arrhenius parameters for k62 k62 = 10

- 77 exp(-(123

10) k J m o r V / ? r ) s" 1 .

(2.70)

The variation in A+ with [0 2 ] arises from the complex interplay between reactions (60-62). At high [0 2 ], decomposition of the alkyl radical cannot compete with the addition of oxygen and peroxy radical formation is fast and irreversible (k61[02]> k-61). The concentration of oxygen required to achieve this condition will increase with temperature as both k60 and k-61 depend sensitively on temperature. Under these conditions k62 is the

Peroxy radical isomerization 1.2xl04

219

,

1.0

0.8

H2 + CI reaction were eventually shown to be in error due to reactions on the walls of the vessels used [35]. When these sources of error were eliminated the results of the rate measurements were compatible with the thermodynamic data. As well as leading to accurate rate data for the reactions involved these studies also prompted more awareness of the potential importance of wall reactions in fast-flow discharge studies. The limited accuracy of much thermodynamic data can sometimes present problems. The best kinetics data have an accuracy approaching %

264

Kinetics databases

Ch. 3

at ambient temperatures and often % over a limited temperature range. If equilibrium constants are to be used at this level then, for an error in the rate constant to be within , free energies need to be known to 1 kJ at ambient temperatures. For most radicals this is not the case and the best semi-empirical method for estimating thermodynamic properties do not approach this level of accuracy. Conversely, in principle, the measurement of k{ and kT offers a particularly accurate route to enthalpies of formation of radical species. The extent to which k{ and kr can be combined or compared via thermodynamic data is sometimes affected by the fact that the kinetics data on k{ and kr have often been measured over different temperature ranges. For example, there are numerous measurements of the rate constant for the reaction OH + H 2 -> H 2 0 + H over the temperature range 2002500 K, but for the reverse there are only a few studies at T> 1000 K [36]. This situation usually arises because the activation energy of the forward and reverse reactions are very different. Thus it would be expected to be particularly pronounced for decomposition/combination reactions where the combination step will normally have an activation energy close to zero whereas the energy barrier to decomposition will be of the order of bond strengths. In cases where a reaction leads to molecular products the reverse reaction will almost certainly be too slow to measure with existing techniques e.g., OH + OH —> H 2 0 + 0 2 , and the use of thermodynamic data to check the rate data will be inapplicable. On the other hand, in such a case, knowledge of kf and Kc provided an estimate of kr which is not accessible experimentally. Another limitation in its use may occur when reactions have more than one channel open to them. For example, the following reaction (7) (8)

0 + C H 3 0 ^ 0 2 + CH 3 -> OH + HCHO

has been studied at 298 K, the rate constant has been measured, and a value for the branching ratio determined [37] but, since there is only one study, it would be useful to be able to check the result through measurements of the rate constants of the reverse reactions and equilibrium data. Although there are data for the reverse reaction between 0 2 and CH 3 , in the case of channel 8 the reverse reaction preferentially yields different products

Evaluation of kinetic data

265

OH + HCHO -* H 2 0 + CHO

(9)

and the channel leading to CH 3 0 + O is too slow to be detected. The temperature coefficients of k{ and kr can be combined to yield data on the enthalpy changes in the reaction and through that procedure give values of bond strengths. Conversely, thermodynamic data on bond strengths can be used in the assessment of data on the temperature dependence of kf and kr. In the case of decomposition reactions, values of bond strengths may suggest the most likely bond to be broken in the molecule i.e., the most likely channel for the reaction, and may also provide an estimate of the activation energy. Established thermodynamic data can also be used as a check on the measured temperature coefficients of the rate constants for forward and reverse directions of reactions. (iv) Comparison with theory Theories of chemical kinetics are not yet at a stage where they can provide values of rate constants as reliable as experimental measurements. Nevertheless simple theory can be useful to the evaluator as a guide to "reasonable" values of rate parameters to be expected for particular types of reaction. Transition state theory has been particularly useful in providing a framework for assessing the data on metathetical reactions. According to this theory the rate constant can be expressed as

h

QAQB

where C 0 2 + H.

This reaction plays a major role in determining the amount of CO produced by combustion systems but, despite its importance, until 1971 there was considerable controversy over the apparent discrepancy between rate constant studies at high and low temperatures. The general feeling prior to that time was that the reaction involved an atom transfer and that the rate data should conform to the equation k = ATn cxp(BIT)

(3.2)

where A, n and B are constants. In 1971, Dryer et al. [39] presented a theoretical model based on the formation of a COOH intermediate. A simple transition state treatment of the model could not provide absolute values of k but, nevertheless, showed that the temperature dependence was not as anticipated (equation (3.2)). This is illustrated in Fig. 3.10 where the experimental data up to 1967 are plotted together with an evaluation at that time (solid line). Subsequent measurements soon confirmed that the general trend was as Dryer et al. [39] suggested (dashed line) and recent studies have been able to characterize the pressure as well as the temperature dependence of the rate constant [40]. At the time this was a striking example of the way in which theory may be used to guide evaluation. The most highly developed theories in chemical kinetics are those describing unimolecular reactions. Reference has already been made to them in some detail in Chapter 2 where their use in analysing data on radical

267

Evaluation of kinetic data T/K 2000

1000 1

1

ri

500 |

400 i

300 1

I

0 •

CD

o 'O -12.0

E

-

\"|" +

jv

: \ • \

n

>^\

o

i

O -4

^

-

D

^^^

T

a

I 1.0

I

_J

1

lO 3 ?" 1 /^* Fig. 3.10. Arrhenius plot of experimental data (up to 1967) for the reaction CO + O H - * C 0 2 + H: ( • ) [124]; ( • ) [125]; ( • ) [126]; ( • ) [127]; (O) [128]; (|-|) [129]; (|-|) [130]; (T) [131]; (+) [132]; ( ) [133]; ( • ) [134]; (O) [135]; (O) [136]; ( - ) evaluation, 1967 [137]; ( ) evaluation based on subsequent experimental data [42].

decomposition/combination processes has been described. These theories provide a very sound framework for interpreting experimental data on such reactions and for extrapolating the results to different pressures and temperatures (Section 3.4). For reactions such as bond dissociation and those proceeding by a well defined energy maximum, e.g., isomerization, successful methods based on Transition State Theory have been developed to calculate k°°. The computation of k(T, [M]) has been performed traditionally using integral expressions but more detailed models based on the Master Equation are now also used (Chapter 2). Irrespective of the model used, for the evaluator and modeller the results are most usefully

268

Kinetics databases

Ch. 3

expressed in terms of the parameters k00, k0, and Fc introduced by Troe and described in Chapter 2 and Section 3.3.4. Ab initio methods can also be useful; they have now reached the stage where readily available computer programs can produce information on the energetics of chemical reactions which is sufficiently accurate to be used as a guide, or to support arguments, where choices are made between alternative experimental findings. They may also be useful to the modeller where no experimental results are available. Thus for the reactions (11)

OH + C 2 H 2 -> H 2 0 + C2H

(12)

-» CH 2 CO + H

(13)

-* HOC 2 H + H

which are the main channels for the reaction at temperatures above HOOK calculations [41] show that reaction (11) is a factor of 5-10 faster than other channels and that the next most important channel leads to HOC 2 H + H. There are no experimental results to guide the evaluator but, in their absence, these calculations can be recommended as a rough guide to the modeller and provide the experimentalist with targets for future study. 3.3.4 Recommendations The principle aim in evaluating the kinetic data on a particular reaction is to provide for the modeller an expression giving the rate constant as a function of temperature and, where relevant, pressure. As well it is usual to indicate the temperature range over which the expression is expected to be valid and to assign error limits to any values of the rate constant derived from the expression. (J) The rate constant expression The format used in presenting the rate constant is governed by the need for modellers to have a simple but general analytical expression convenient for use in computer codes. For bimolecular reactions the form chosen is equation (3.2). Variations occur with n = 0 or B = 0 in some cases, and occasionally it


269

becomes necessary to use different sets of A, n and B for different temperature ranges. On rare occasions a complete departure from this form is required. For example, the data on the reaction CO + OH—> C 0 2 + H referred to earlier (equation (3.2.2)) is best expressed in the form ln(A:/cm3 molecule" 1 s _1 ) = -12.95 + 3.94 x 10~ 4 T at pressures close to atmospheric [42]. This form arises from an interplay between the temperature and pressure dependence of the rate constant. Now that recent measurements have been able to characterize these pressure effects in more detail [40] this expression for k may become redundant and may eventually be replaced by expressions based on the Troe formulae (Chapter 2). For reactions with pressure-dependent rate constants this formalism developed by Troe and his co-workers is now most widely used. The rate constant k(T, [M]) is written as,

where F is given by the expression

°gF

l + [{\ogk°[M]lk~)INf

where, at not too high temperatures, N= 0.75 - 1 . 2 7 log Fc Thus the rate constant can be expressed in terms of the three quantities k°, k™, and Fc all of which are temperature dependent. These expressions, developed from Troe's adiabatic channel model, have the great virtue of sufficient complexity to express adequately the variation of k with T and /?, but simple enough for ready programming, and hence, of being convenient for modelling purposes. It is quite possible to use other ways of expressing evaluated data for decomposition/combination reactions but none are so useful. For example, Tsang and Hampson have adopted a rather different approach, described in Section 3.4, but their methods do not lead to a simple analytical expression for k(T, [M]).

270

Kinetics databases

Ch. 3

(ii) Temperature limits Evaluators usually designate together with the expression for the rate constant a temperature range (and occasionally pressure limits) over which the expression is expected to be valid. In many instances this corresponds to the temperature range over which there are experimental data. However, mindful of the need of the modeller for data at combustion temperatures, there are times when the evaluator will extend the range beyond that covered by measurements. This extrapolation carries with it greater uncertainty (wider error limits) in the values of k derived for the extrapolation region, and even the possibility that some feature of the reaction mechanism may cause a quite unexpected change in the behaviour of k with change in temperature, to produce corresponding order-of-magnitude errors in k. Users of the recommended expression are at liberty to extrapolate if they wish but the temperature limits are meant to serve as a warning that using it outside the recommended temperature range may lead to large errors. (Hi) Error limits Most evaluations include an estimate of the reliability of the expression proposed for k. This usually is given in the form A log k = x or its equivalent statement that k is uncertain by a factor , where x = log F. Thus, if k has a value Y, then it is expected to lie in the range given by FY. log k = (log or Y/F^k^ It is important to appreciate that these error limits are not based on any objective statistical treatment of the data. They are subjective assessments drawing on the experience of the evaluator as well as the extent to which the data scatter. For example, in the extreme case where there is only one measurement of the rate constant of a reaction, the evaluator will assess the quality of the experimental work, the difficulties in determining the rate constant, and any other pertinent factors, and on that basis will assign error limits to any recommendation for k. These will be well outside the reported precision of the results, recognizing that experience shows all measurements are subject to undetected systematic errors, and that many chemical systems reveal unexpected complexity when studied in more depth, or by other techniques. Recognizing this, in their evaluation of data for atmospheric modelling, the IUPAC Panel have a policy of using a factor of 2 as the error factor for a rate constant measured in a single


271

study, irrespective of the quality of the work, until confirmatory independent measurements are made [43]. Thus, error assignments by evaluators tend to be conservative, reflecting their experience of the way in which measured values of rate constants can change as techniques improve and knowledge advances. If further measurements by different techniques lead to better definition of the rate constant, then later evaluations should lead to a reduction in the error limits even if the recommended value of k does not change. The error limits on k may vary with temperature (and pressure) depending on how well the temperature coefficient of k has been determined. There are different ways of expressing this. Evaluations of data for atmospheric modelling which, generally are only concerned with a relatively small temperature range (200-400 K, although occasionally recommendations are extended to higher temperatures) sometimes give the change in error limits with temperature in the form

A l o g £ ( r ) = Alog£(298K)

[*=*> (?-£)]

C 3 H 6 + 0 2 a value of k = 4.4 x 10~ 12 cm 3 molecule -1 s _ 1 has been recommended [17] 3 over the over the range 300-1000 K with error limits of A log k = range 600-800 K rising to 5 at other temperatures. A number of other variations on these types of changes in error limits with temperature are possible, and can be expressed in similar ways.

3.4 INTERPOLATION, EXTRAPOLATION AND ESTIMATION PROCEDURES

Values of rate constants are often required for modelling at temperatures and pressures well outside the ranges covered by experimental measurements. Therefore, part of the evaluation process is to interpolate between experimental data, or extrapolate from them, to provide information over as wide a range of conditions as possible. To do so evaluators make use of a number of estimation techniques based on their own experience and existing knowledge, as well as theories of chemical kinetics. The extent to which evaluators are willing to depart from experimental data and trust to theory and estimation varies a great deal from one evaluator to another. This becomes most obvious when there are no experimental data for a reaction and recourse must be made entirely to empirical or theoretical methods. Some evaluators would be unwilling to make recommendations in these circumstances arguing that it is better for the modeller to realize that such gaps in our knowledge exist, and to avoid using estimated values which may introduce large errors into the model. But others would argue that, of necessity, the modeller will use a value from some source and that one estimated by an evaluator is likely to be more accurate than one estimated by the modeller. As knowledge increases and theories improve the gap between these two extremes can be expected to diminish, but there is still sufficient uncertainty concerning the mechanisms of many elementary reaction to make estimation an extremely uncertain procedure. On the other hand, estimation can be used with some confidence in many cases where the kinetics of the process are thoroughly understood. The interpolation and extrapolation procedures described in this section make use of existing experimental data and are less controversial than estimation techniques but also not without problems.

Interpolation, extrapolation and estimation procedures

273

(i) The variation of rate constants with temperature: interpolation and extrapolation So long as there are no indications that the rate constant may be pressure dependent it is usually assumed that it conforms to equation (3.2). When n = 0 equation (3.2) reduces to the Arrhenius equation and will give a linear plot of In A: vs. IIT. In practice, n takes on small positive values leading to a degree of curvature of the Arrhenius plot which becomes more pronounced as 1/T becomes smaller. At temperatures less than approximately 1000 K the curvature is usually difficult to detect experimentally at the current precision of measurement. As indicated in Section 3.3.3, where there are data over the whole temperature range the curvature on the Arrhenius plot i.e., non-zero values of n, presents no problem. Equation (3.2) can be fitted to the data "by eye" or statistically to yield values of n, A and B. However, it is often the case that there are data at high temperatures and data at low temperatures requiring interpolation between these two relatively isolated sets if an expression for k covering the whole temperature range is to be derived. To perform the interpolation on the basis of equation (3.2), a value of n must be known, or assumed, leaving A and B to be fixed by forcing the fitted expression to pass through the data. In the earliest evaluations it was usual to assume that n = 0 and to use a simple linear interpolation on the Arrhenius diagram. This assumption reflected the fact that, at the time, there was insufficient confidence in the data to make it worthwhile to try to define n more closely. That situation has changed and the evaluator will now look for evidence for a more realistic value of n. The evidence may come from the data itself or from theory. As an example of the way in which the experimental data may be used to indicate the curvature, the data for the reaction H + CH 3 CHO -> H 2 + CH3CO are shown in Fig. 3.11. There are measurements over a temperature range at low temperatures and corresponding measurements at high temperatures. A significant difference between the two sets of data is the larger temperature coefficient of k (higher activation energy) observed for the high-temperature results, reflecting the dependence of activation energy on n. Therefore, the evaluation line has been fitted to be compatible with the temperature coefficients of the measurements as well as the absolute values. There are times when the temperature coefficient has not been measured in this way and, under those circumstances, the evaluator will either use

274

Kinetics databases

Ch. 3

T/K 2000 i

1000

500

400

300

.

|

.

|

-10.5

A A

\ \ \ \

\

-11.0

\ \ \ \ \ \ 7

_iw

\\ \\\ \\\ \\ \\ \ \\

""-5 -

o

3o~~ o o

£

"I -12.0

so

\ ^CA

\V \ \ ^^

o

*j

-12.5

1

>> -13.0

\N

\

X

-13.5

-

|

1

1

1

1

lOV/K"

1

X

X

X

•

1

1

1

Fig. 3.11. Arrhenius plot of experimental data for the reaction H + CH 3 CHO-» products: ( ) [138]; ( • ) [139]; ( - ) [140]; (•••) [141]; (O) [142]; ( ) evaluation [17].


275

n = 0 and make a linear interpolation between the data sets, or use a value of n derived from theory. An example of the former, H + H 2 0 2 —> H 2 0 + OH is shown in Fig. 3.3. Most evaluators adopt this pragmatic approach to fixing a value of n. Either n = 0 is assumed, or where the data are sufficiently numerous, the line of best fit is obtained statistically or 'by eye'. An exception to this approach is to be found in the work of Cohen [44] who has used the transition state methods developed by Benson and his colleagues [38] as a framework for evaluating experimental data for bimolecular metathetical reactions such as those of H, O and OH with alkanes. Using a model of the transition state, a theoretical value of the pre-exponential factor in the rate expression is derived which may be combined with experimental data at one temperature to give the exponential term. The rate expression so derived may be used to calculate values at other temperatures. By treating families of reactions adjustments can be made to the transition states to make them compatible with the experimental data on the whole range of reactions considered. Where there are data available over a range of temperatures there is little to choose between the pragmatic approach and the more theory oriented methods of Cohen. However, Cohen's method probably provides a more satisfactory basis for extrapolation and it also provides a useful method for estimation of rate constants (see later in this section). Extrapolation of recommended rate expressions to temperatures outside the range of the experimental data is inevitably accompanied by an increase in the uncertainty associated with the values of k so obtained. Furthermore, gross error may result if the extrapolation is taken into a region where the reaction mechanism changes. Extrapolation can most confidently be performed when there is little curvature on the Arrhenius plot which will usually be the case at temperatures less then 1000 K. At higher temperatures, unless n is very small or accurately known, extrapolation is accompanied by considerable uncertainty. (if) Multichannel reactions The discussion of interpolation and extrapolation methods has so far assumed that the reaction involved proceeds to one set of products over the whole temperature range. If more than one channel is open to the reaction, the branching ratio between the channels may also vary with temperature and this variation will lead to curvature of the Arrhenius plot

276

Kinetics databases

Ch. 3

irrespective of any curvature due to the rate parameters of the individual channels. A simple example is provided by the reaction of atomic hydrogen with propane which can proceed by two channels (14)

H + CH 3 CH 2 CH 3 -* H 2 CH 2 CH 3 + H 2

(15)

-> H3CHCH3 + H 2

The rate constant for the overall reaction will be the sum of the rate constants of the two channels (3.4)

k = k14 + k15

If k14 and k15 each obey a simple Arrhenius relationship without a temperature dependent pre-exponential factor, then k(T) = A14 exp(B14/T) + A15 cxp(B15/T)

(3.5)

This will give rise to curved Arrhenius plot unless B14 = B15, which, in this instance, will not be the case because of the difference in bond strengths of the primary and secondary C—H bonds involved. Thus curvature of the Arrhenius plot arises purely because of the occurrence of more than one reaction channel, even if each of the reactions concerned individually would give rise to a linear plot. Interpolation between data sets, or extrapolation, can only be carried out if the rate parameters associated with each of the channels is known or the branching ration (k14/k15) is known as a function of temperature. In some cases the problem is not so acute. The reaction of H atoms with acetaldehyde can also proceed by two channels, (16) (17)

H + CH3CHO -> H 2 CHO + H 2 -> H3CO + H 2

but the aldehydic H is so much more labile than the alkyl H that channel (17) dominates the reaction until sufficiently high temperatures are reached. There is no experimental evidence for the occurrence of channel (16) but it is possible to estimate by methods outlined later in this section


277

that it is unlikely to contribute significantly (k16/(k16 + k17) ~ 0.1) below 1000 K. A more extreme case of the problems of dealing with multichannel reactions is provided by reactions which may occur by an addition channel and an abstraction channel. Reactions of this kind are extremely common in combustion systems. An example is the reaction of OH radicals with but-1-ene. The kinetic data associated with that reaction are shown in Fig. 3.12, to demonstrate the difficulty of extrapolating or interpolating between the available data at high temperatures and those at low temperaT/K

-10.0

c

—li-i

1

1

1

IOVVK" Fig. 3.12. Arrhenius plot of experimental data for the reaction OH + but-1-ene —> products: (€) [143]; (+) [144]; (O) [145]; (O) [146]; (x) [147]; ( • ) [148]; (V) [149]; (B) [150]; (O) [151]; ( • ) [152]; ( • ) [153]; ( • ) [154]; ( ) evaluation [65]. The filled symbols ( • , • , • ) represent data obained in low-pressure studies. All other data was obtained at, or close to, 1 atm.

278

Kinetics databases

Ch. 3

tures. At low temperatures the values of k demonstrate a negative temperature coefficient and are pressure dependent characteristic of an addition mechanism. At higher temperatures the temperature coefficient is substantial and positive exhibiting no pressure dependence, typical of say a bimolecular atom transfer reaction. In this case very little interpolation is possible; the value of k needs to be established experimentally as a function of pressure as well as temperature over the two regimes where the different mechanisms operate. (Hi) Estimation methods The evaluator may be called on to provide values of rate constants for reactions which have not been studied experimentally. The methods that are used are numerous and varied sometimes being based on little more that the 'chemical intuition' of the evaluator but in other cases having a sound theoretical foundation. This section reviews the most well established. (a) Use of the equilibrium constant The relationship between the rate constants for forward and reverse steps and the equilibrium constant has already been discussed in describing its use in the evaluation process. Where the thermodynamic data are accurately known the relationship can be used with confidence to estimate the rate constant of a reaction from a knowledge of data on its reverse. The limitations are the accuracy of the thermodynamic data and the possibility that the reaction may proceed by a different channel to that defined by the equilibrium constant. (b) Analogy It is often possible to identify a reaction which has many features similar to the one for which a rate constant value is required: the mechanism is expected to be the same, similar chemical groups are involved, and the energetics may be similar. In these circumstances it is plausible to assume that the reaction kinetics are analogous. Examples are to be found in all of the major compilations discussed in Section 3.5. For example, in evaluating data for modelling the combustion of methane Tsang and Hampson [45] have assigned to the reaction C 2H5 + HCHO -> C 2 H 6 + CHO a rate constant identical with the analogous reaction CH 3 + H C H O ^ H 4 CHO.


279

Taking the estimation process a step further they have used the equilibrium constant for the reaction C 2 H 5 + HCHO -* C 2 H 6 + CHO together with the estimated rate constant to estimate the values for the reverse reaction of CHO with C 2 H 6 . (c) Correlations Where rate constants have been measured for the identical type of reaction of an atom or radical with a range of compounds it may be possible to correlate the values of the rate constants with structural features in the compounds concerned. The structure-reactivity relationship so developed may then be applied to predict values of the rate constants for the reactions with other compounds where experimental data are lacking. The reactions of H, O and OH with alkanes have been treated in this way. In the case of the alkanes the reaction involved is abstraction of an H atom and the structural feature of importance is the strength of the C—H bond being attacked. For alkanes with C > 4 this can be generalized further since primary, secondary and tertiary hydrogens can each be assumed, to a first approximation, to have the same bond strength in different alkanes. Thus a rate constant kp, ks, kt can be assigned to the reaction in which a primary, secondary, or tertiary hydrogen atom is abstracted and, to a first approximation, the overall rate constant can be written as k = npkp + nsks + ntkt

(3.6)

where n p , ns and nt are the numbers of primary secondary and tertiary hydrogen atoms in the molecule. Values of the rate constants may then be obtained by fitting this formula to the experimental data on the alkanes, the only requirement being that the data must cover examples of abstraction of all three types measured in circumstances where the total rate constants show some sensitivity to each of the /c's. The use of this approach to estimating rate constants has been discussed in some detail in Chapter 2. Formulae have been developed for H abstraction from alkanes by OH [46, 47], H [47, 48] and CH 3 [49]. The database for H 0 2 and 0 2 attack on alkanes is too limited to proceed in the same way to obtain an empirical expression for the rate constants of such H abstraction reactions. However, extensive measurements of k(02 + RH) and k(H02 + RH) have been made for a number of alkanes at 753 K in Baldwin and Walker's laboratory. By estimating values of the

280

Kinetics databases

Ch. 3

pre-exponential factors involved they are able to derive corresponding activation energies for reactions involving primary, secondary and tertiary hydrogens as described in Chapter 1. These estimates have been used in evaluations for estimating rate constants for 0 2 and H 0 2 abstraction reactions e.g., see Tsang's evaluation of the reactions for modelling methane combustion [45]. Ranzi et al. [50] have adopted a rather different approach to estimating the kinetic parameters for H-abstraction from hydrocarbon compounds. For the reaction of a radical, R, with a compound, R'H, the rate constant is expressed as k ~ A: R C R 'H

where A:R represents the intrinsic reactivity of the R-radicals and C R H is related to the reactivity of the H atom abstracted. Both A:R and C R H are expressed in Arrhenius form and values of the Arrhenius parameters are derived from the available experimental data. The method is applied to a wide range of species but the differences between predicted and experimental values, where available, are fairly large. (c) Activation energies In the estimation methods discussed so far the quantities estimated have either been the rate constant or the pre-exponential factor in the Arrhenius expression. Methods for estimating the activation energy of bimolecular reactions are much less developed. Theoretical prediction, at the level required, is beyond current computational techniques except in some exceptional, simple cases. However, there have been empirical attempts to relate the activation energy for a series of related reactions e.g., H abstraction by methyl radicals from hydrocarbons, to the thermodynamics of the process. The most widely used correlation of this type is the Evans-Polanyi relationship E act = a(BDE-jS)

(3.7)

where a and j8 are empirical constants and BDE is the bond dissociation energy of the bond under attack. Kerr and Parsonage have obtained a good correlation for the reactions of the methyl radical with


281

0.494 and j8CH3 = 351.0 kJ mol - 1 [49]. Similarly Cohen obtains «OH = (0.36 0.04) and j8 OH = 425 4kJmol" 1 for OH abstracting H atoms from alkanes [44]. However, Cohen comes to the conclusion that "The problem with using the Evans-Polanyi relationship as a predictive tool, however, lies in the large uncertainties in most of the BDE's. Recall that an error of 4 kJ mol - 1 in the BDE means an error in k (298 K) of more than a factor of 5". The correlation of k or E act with bond strengths are the most commonly used for estimation purposes but correlations with other properties of the molecules are possible. For example, it is possible to correlate the rate constant for OH + alkane reactions with the ionization potential of the alkane [10] and, for other types of reaction, it has been suggested that both electron affinities and ionization potentials of the species involved may correlate with the kinetics parameters [51]. However, because of the great sensitivity of the rate constant to small errors in E act , correlations aimed at predicting E act are very rarely used. «CH 3 =

(iv) Pressure dependent rate constants Reactions having rate constants which are both pressure and temperature dependent, quite commonly found in combustion processes, originate from reactions proceeding by initial formation of an adduct which may then have open to it a number of possible reaction channels. These may include decomposition back to the original reagents or to new products, collisional quenching to give a stabilized adduct, isomerization followed by decomposition or collisional stabilization. The competition between the collisional quenching of the energized adduct and the other channels produces the pressure dependence of the rate parameters. The reverse processes will, of course, also have pressure-dependent rate constants. As indicated earlier (Section 3.2.2) at high and low pressures the rate constants take on an extremely simple pressure dependence and interpolation and extrapolation with respect to temperature can be carried by the methods outlined in the previous parts of this section. Between these limiting conditions the pressure and temperature dependence is complex and can only be handled satisfactorily within a framework of theory. It is useful at this stage to distinguish between two types of combination/ decomposition reactions. The first is the relatively straight forward process of atoms or radicals coming together to form an adduct which may then be collisionally stabilized or decompose back to the original reacting spe-

282

Kinetics databases

Ch. 3

cies. The second type involves an adduct where collisional stabilization must compete with other channels of isomerization and/or decomposition to products other than the original reactants. The former is the classical unimolecular reaction mechanism which has been the subject of continuous theoretical scrutiny since it was first proposed in 1922. The way in which evaluators handle these reactions has been described in Sections 3.2.2 and 3.3.4. The interpolation or extrapolation or, indeed, estimation procedures that may be used for k0 and koo are analogous to those described earlier in this section. F c is usually based on fitting the theory to existing experimental data, but theoretical estimation methods are also available [18] and, in some cases, for small molecules at temperatures close to 300 K, a standard value of 0.6 has been used by some evaluators [52] where no experimental data are available. In Tsang's evaluations the pressure-dependent factors in the expression for the rate constant are handled differently [45]. Using the RRKM model a factor governing the pressure dependence is calculated and presented in tables as a function of the energy transferred per collision. The user may then select a typical value for this energy for the collision partners involved and read the pressure-dependent factor from the tables. Satisfactory application depends on the ability to select the correct energy transferred which may be derived from experimental data on the particular reaction concerned but, more often, may have to be estimated from data on analogous reactions which, usually, are quite limited. In the case of formation of an energized adduct capable of isomerization or decomposition via a channel different from the entrance channel, there is only very limited assistance to be obtained from theory or analogous reactions. The interpolation, extrapolation and estimation of the pressure and temperature dependence of such rate constants is extremely difficult. The overall rate constant for the reaction can often be handled (interpolated, extrapolated) over a limited range of temperatures by the empirical methods outlined earlier. However, also as indicated earlier, a great difficulty presented by these reactions is lack of information on the products. Only experimental studies can establish with confidence the product channels and the ways in which branching ratios vary with temperature and pressure, but experiments to quantitatively characterize product channels are usually extremely exacting and, for the moment, theory often offers the only, very limited guidance. For this type of reaction, progress

Data sources for modelling

283

in computation of branching ratios may well offer the best hope for future progress.

3.5 DATA SOURCES FOR MODELLING

In this section collections of kinetic data for modelling combustion systems are listed and described. The emphasis is on critically evaluated data because of its greater reliability but other sources are also given since only a proportion of the data necessary for modelling have been thoroughly assessed. The present volume is concerned with low-temperature oxidation of hydrocarbons but the data sources cited in this section cover both higher and lower temperature regimes. As indicated earlier most of the direct measurements of rate constants of elementary reactions have been carried out at high temperatures (>1000 K) or temperatures close to ambient. It is only relatively recently that experimental techniques have been modified to produce substantial quantities of data for the intermediate temperatures pertinent to low-temperature oxidation of hydrocarbons, and much of the data for modelling low-temperature oxidation of hydrocarbons must be obtained from extrapolation of low-temperature data, interpolation between high- and low-temperature data, or by estimation methods. Consequently both the evaluations produced for modelling flames and those for atmospheric modelling are relevant. 3.5.1 Collections of critically evaluated data (i) Data for modelling combustion The two major sources of critically evaluated data for combustion modelling are the publications prepared by the CEC group [17, 36] and from Tsang and his collaborators at NIST [45, 53-56]. As well, there are a number of excellent reviews and collections of data sheets prepared by Cohen and Westberg [29, 57] in which data for a selection of combustion reactions are evaluated. The CEC Group on Data for Combustion Modelling was established as part of the European Commission Energy Research and Development Programme. In its first publication the data on some 200 reactions relevant

284

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to combustion were evaluated [36]. The reactions relate to the combustion in air of methane, ethane and some aromatic compounds. Some reactions of nitrogen containing species are included, particularly those relevant to the chemistry of pollutant formation and abatement strategies. This material has been updated and data on a number of additional reactions evaluated in subsequent publication [17]. Of particular relevance is the inclusion in this latter publication of the data on ethyl, /-propyl, £-butyl and allyl radicals, which are species prominent in the low-temperature combustion processes considered in this present volume. The cut-off point for the literature collection for these evaluations was mid-1992. A summary of the recommended rate constant expressions from that volume has also been published [58]. Tsang has prepared a series of publications on data required for modelling the combustion of methane [45], methanol [53], propane [54], isobutane [55] and propene [56]. The treatment is less detailed than in the case of the CEC publications; there is less evaluation of the primary data, often previous evaluations are accepted, and there is no graphical presentation of results. However, data on more than 500 reactions are considered and, where there are no experimental data, estimates of the rate constants are given. Unimolecular reactions are also treated slightly differently from the CEC evaluations as described in Section 3.3. The nitrogen chemistry associated with combustion in air is not dealt with in the set of papers by Tsang just cited, but in two similar publications on the high-temperature data for modelling propellant combustion, data on some of the relevant nitrogen reactions are evaluated. The reactions covered are those of the species NO, N 0 2 , HNO, HN0 2 , HCN, N 2 0 [59] and CN, NCO, HNCO [60]. The evaluations of Cohen and Westberg [28, 29] deal with a small number of reactions relevant to combustion (some reactions from the H 2 /0 2 system, reactions of atomic oxygen and of OH with alkanes, NH 2 and NH). The other reactions covered are mainly relevant to atmospheric chemistry. Their data sheets provide a much more detailed assessment of the primary data than most; in some cases the original data are reanalysed in the light of advances since the original publication. They also make extensive use of transition state theory to interpolate between sets of data over a temperature range. The review of high-temperature nitrogen chemistry by Hanson and Salimian [61] contains a number of evaluations of kinetic data and estimates

Data sources for modelling

285

of rate constant values. The kinetics data on gas phase reactions of species containing halogens and the cyanide group have been evaluated by Baulch et al. [62]. Although the other early evaluations of Baulch et al. [63] have now largely been superseded they are still of occasional value where no alternative evaluations exist. (ii) Data for modelling atmospheric chemistry There are two major evaluation groups (IUPAC, NASA) which regularly compile and evaluate kinetic data for atmospheric chemistry [43, 52]. These collections are updated and expanded at 2-4 year intervals. They are concentrated on reactions relevant to stratospheric chemistry but more recently the tropospheric content has been expanded, particularly in the IUPAC compilations, bringing more coverage of reactions of organic species and hence more relevance to low-temperature oxidation processes. Atkinson has also produced evaluations for tropospheric modelling but of the species dealt with (OH, N 0 3 and 0 3 reactions with organic compounds) only the reactions of OH are directly relevant to combustion [64, 65]. (Hi) Evaluations and reviews of data for specific species There are a number of reviews in which the kinetic data are compiled and often evaluated for the kinetics of reactions of a specific atom or radical with a range of compounds. In most cases the available data over the whole temperature range are considered. The following reviews on H, O, OH, CH 3 reactions and those of organic peroxides are the most relevant for low-temperature oxidation of alkanes. Because of its importance in both combustion and atmospheric chemistry, the OH radical has received most attention. Atkinson [65] has produced an extremely comprehensive collection and evaluation of data on OH reactions aimed mainly at atmospheric modelling but evaluating data at higher temperatures as well. Other evaluations of the reactions of OH with alkanes are those of Cohen and Westberg [29], Baulch et al. [66], and, more recently, Cohen's revision of his earlier evaluations [44]. As indicated in Section 3.3 a number of semi-empirical formulae have been derived to predict the rate constants of such reactions. Herron [67] has evaluated the data for reactions of 0( 3 P) with saturated organic compounds, data for reactions of 0( 3 P) with methane, ethane and

286

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neopentane have been evaluated by Cohen [30], and Cvetanovic [68] has evaluated data for the addition of 0( 3 P) to unsaturated hydrocarbons. The experimental data for H atoms reacting with alkanes are limited and lacking in accuracy at low temperatures. The data have been evaluated by Cohen [48]. There are extensive relative rate measurements at temperatures close to ambient for hydrogen transfer reactions of methyl radicals. Their data have been compiled and evaluated by Kerr and Parsonage [49]. The same authors have also evaluated the data on addition reactions of atoms and radicals with alkenes, alkynes and aromatic compounds [69]. Alkyl peroxy radicals are important intermediates in low-temperature oxidation of alkanes but there is only limited information on their reactions, particularly at temperatures much above ambient. Their properties have been recently reviewed and the kinetics data for their mutual reactions have been evaluated [70]. (iv) Other reviews Warnatz [71] has produced an extensive compilation of data for modelling combustion systems. Although there is little direct evaluation of the material, recommendations are made, usually based on previous evaluations, where those are available. Reactions of small alkanes are covered. Mention has already been made of the publications of Kerr and Moss [23] as a source of bibliography. The same works list information on the kinetics of the reactions surveyed. Although the information is not critically evaluated and is now slightly dated, the coverage is broad, and where no evaluation of data on a particular reaction is available, the Kerr and Moss collection provides a valuable source for the existing data as does the NIST database [22], also discussed in Section 3.3. This too lists the kinetics data and also provides a graphical presentation. The older data on a wide range of unimolecular reactions has been compiled and evaluated by Benson and O'Neal [72].

References [1] G. Barbieri, F.P. Di Maio and P.G. Lignola, Comb, and Flame, 98 (1994) 95. [2] Gutheil, E., Comb, and Flame 93 (1993) 239. [3] J.A. Miller and C.T. Bowman, Prog. Energy Combust. Sci. 15 (1989) 287.

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C.K. Westbrook and F.L. Dryer, Comb. Sci. Technol. 20 (1979) 125. C.K. Westbrook, F.L. Dryer and K.P. Schug, 10th Symp. (Int.) Comb. (1982) 153. C.K. Westbrook, J. Warnatz and W.J. Pitz, 22nd Symp. (Int.) Comb. (1988) 893. W. Tsang, J. Chem. Phys. 40 (1964) 1171; ibid 43 (1965) 352; ibid 44 (1966) 4283, Int. J. Chem. Kinet. 1 (1969) 245. B. Atakan and J. Wolfrum, Chem. Phys. Lett. 178 (1991) 157; K. Mahmud and A. Fontijn, J. Phys. Chem. 91 (1987) 1918. T. Ko, G.Y. Adusei and A. Fontijn, J. Phys. Chem. 95 (1991) 8745. J.P.D. Abbott, K.L. Demerjian and J.G. Anderson, J. Phys. Chem. 94 (1990) 4566. I.R. Slagle and D. Gutman, J. Am. Chem. Soc. 107 (1985) 5342; I.R. Slagle, E. Ratajczak, M.C. Heavan, D. Gutman and A.F. Wagner, J. Am. Chem. Soc. 107 (1985) 1838; I.R. Slagle, E. Ratajczak and D. Gutman, J. Phys. Chem. 90 (1986) 402. V.A. Nadtochenko, O.M. Sarkisov, V.I. Vedeneev, Izv. Akad. Nauk SSR, Ser. Khim. 3 (1979) 651. C.J. Hochanadel, T.J. Sworsky and P.J. Ogren, J. Phys. Chem. 84 (1980) 231. B. Veyret, R. Lesclaux and P. Roussel, Chem. Phys. Lett. 103 (1984) 389. F. Temps and H.Gg. Wagner, Ber. Bunsenges. Phys. Chem. 88 (1984) 410. J.E. Baggott, H.M. Frey, P.D. Lightfoot and R. Walsh, Chem. Phys. Lett. 132 (1986) 225. D.L. Baulch, C.J. Cobos, R.A. Cox, P. Frank, Th. Just, J.A. Kerr, T. Murrells, M.J. Pilling, J. Troe, R.W. Walker and J. Warnatz, J. Phys. Chem. Ref. Data 23 (1994) 847. J. Troe, J. Phys. Chem. 83 (1979) 114; J. Troe, Ber. Bunsenges. Phys. Chem. 87 (1983) 161; R.G. Gilbert, K. Luther and J. Troe, Ber. Bunsenges. Phys. Chem. 87 (1983) 169. K.J. Hsu, S.M. Anderson, J.L. Durant and F. Kaufman, J. Phys. Chem. 93 (1989) 1018. L.F. Keyser, J. Phys. Chem. 90 (1986) 2994. U.C. Sridhanan, L.X. Qiu and F. Kaufman, J. Phys. Chem. 86 (1982) 4569. G. Mallard, NIST Chemical Kinetics Database, available from Standard Reference Data, NIST, 221/A320 (Gaithersburg, Maryland, USA). J.A. Kerr and S.J. Moss, Handbook of Bimolecular and Termolecular Gas Reactions, Vols I and II (CRC Press, Boca Raton, Florida 1981); ibid Vol. Ill (CRC Press, Boca Raton, Florida 1987). F. Westley, Tables of Recommended Rate Constants Occurring in Combustion, NSRDS-NBS (1980) 67. COD ATA Task Group on Chemical Kinetics, The Presentation of Chemical Kinetics Data in the Primary Literature, CODATA Bull., No. 13 (1974). R.F. Hampson and D. Garvin, J. Phys. Chem. 81 (1977) 2317. D.L. Baulch and D.C. Montague, J. Phys. Chem. 83 (1979) 42. N. Cohen and K. Westberg, J. Phys. Chem. 83 (1979) 46. N. Cohen and K.R. Westberg, J. Phys. Chem. Ref. Data 12 (1983) 531. N. Cohen, Int. J. Chem. Kinet. 18 (1986) 59.

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[44] [45] [46] [47] [48] [49] [50] [51] [52]

[53] [54] [55] [56] [57] [58]

[59] [60]

Kinetics databases

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J. Blauwens, B. Smets, and J. Peeters, 16th Symp. (Int.) Comb. (1977) 1055. Y. Matsui and A. Yuuki, J. Appl. Phys. 24 (1985) 598. D.L. Lindackers, M. Burmeister and P. Roth, 23rd Symp. (Int.) Comb. (1990) 251. A.J. Dean, R.K. Hanson and C.T. Bowman, 23rd Symp. (Int.) Comb. (1990) 259. J.E. Spencer and G.P. Glass, J. Phys. Chem. 79 (1975) 2329; P.F. Ambidge, J.N. Bradley and D.A. Whytock, J. Chem. Soc. Faraday Trans. 1, 72 (1976) 2143. D.L. Baulch, C.J. Cobos, R.A. Cox, C. Esser, P. Frank, Th. Just, J.A. Kerr, M.J. Pilling, J. Troe, R.W. Walker and J. Warnatz, J., Phys. Chem. Ref. Data 21 (1992) 451. F. Ewig, D.D. Rhasa and R. Zellner, Ber. Bunsenges. Phys. Chem. 91 (1987) 708. S.W. Benson, Thermochemical Kinetics, 2nd edn. (Wiley, New York, 1976). F. Dryer, D. Naegeli and I. Glassman, Comb, and Flame 17 (1971) 270. R. Foster, M. Frost, D. Fulle, H.F. Hamann, H. Hippler, A. Schlepegrell and J. Troe, J. Chem. Phys. 103 (1995) 2949. J.A. Miller and C.F. Melius, 22nd Symp. (Int.) Comb. (1988) 1031. D.L. Baulch and D.D. Drysdale, Comb, and Flame 23 (1974) 215. IUPAC Subcommittee on Gas Kinetic Data Evaluation for Atmospheric Chemistry, R. Atkinson, D.L. Baulch, R.A. Cox, R.F. Hampson Jr., J.A. Kerr and J. Troe, J. Phys. Chem. Ref. Data 21 (1992) 1125. N. Cohen, Int. J. Chem. Kinet. 23 (1991) 397. W. Tsang and R.F. Hampson, J. Phys. Chem. Ref. Data 15 (1986) 1087. R. Atkinson, W.P.L. Carter, S.M. Aschmann, A.M. Winer and J.N. Pitts Jr., Int. J. Chem. Kinet. 16 (1984) 469. R.R. Baldwin and R.W. Walker, J. Chem. Soc. Faraday Trans. 1, 75 (1979) 140. N. Cohen, Int. J. Chem. Kinet. 23 (1991) 683. J.A. Kerr and M.J. Parsonage, Evaluated Kinetic Data on Gas Phase Hydrogen Transfer Reactions of Methyl Radicals (Butterworths, London, 1976). E. Ranzi, M. Dente, T. Faravelli and G. Pennati, Comb. Sci. Technol. 95 (1994) 1. J.P.D. Abbatt, D.W. Toohey, F.F. Fenter, P.S. Stevens, Wm.H Brune and J.G. Anderson, J. Phys. Chem. 93 (1989) 1022. NASA panel for Data Evaluation, Chemical Kinetics and Photochemical Data for Use in Stratospheric Modeling, Evaluation Number 11, W.B. DeMore, S.P. Sander, D.M. Golden, R.F. Hampson, M.J. Kurylo, C.J. Howard, A.R. Ravishankara, C.E. Kolb and M.J. Molina, JPL Publication 94-26. W. Tsang, J. Phys. Chem. Ref. Data 16 (1987) 471. W. Tsang, J. Phys. Chem. Ref. Data 17 (1988) 887. W. Tsang, J. Phys. Chem. Ref. Data 19 (1990) 1. W. Tsang, J. Phys. Chem. Ref. Data 20 (1991) 221. N. Cohen and K.R. Westberg, J. Phys. Chem. Ref. Data 20 (1991) 1211. D.L. Baulch, C.J. Cobos, R.A. Cox, P. Frank, G. Hayman, Th. Just, J.A. Kerr, T. Murrells, M.J. Pilling, J. Troe, R.W. Walker and J. Warnatz, Comb, and Flame 98 (1994) 59. W. Tsang and J.T. Herron, J. Phys. Chem. Ref. Data 20 (1991) 609. W. Tsang, J. Phys. Chem. Ref. Data 21 (1992) 753.

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[61] R.K. Hanson and S. Salimian, in: Combustion Chemistry, ed. W.C. Gardiner (Springer-Verlag, New York, 1984). [62] D.L. Baulch, J. Duxbury, S.J. Grant and D.C. Montague, Evaluated Kinetic Data for High Temperature Reactions, Vol. 4: Homogeneous Gas Phase Reactions of Halogen- and Cyanide-Containing Species, J. Phys. Chem. Ref. Data 10 (1981) Supplement No. 1. [63] D.L. Baulch, D.D. Drysdale, D.G. Home and A.C. Lloyd, Evaluated Kinetic Data for High Temperature Reactions, Vol. 1: Homogeneous Gas Phase Reactions of the H 2 -0 2 System (Butterworths, London, 1972); D.L. Baulch, D. Drysdale and D.G. Home, Evaluated Kinetic Data for High Temperature Reactions, Vol. 2: Homogeneous Gas Phase Reactions of the H 2 -N 2 -0 2 System (Butterworths, London, 1972); D.L. Baulch, D.D. Drysdale, J. Duxbury and S. Grant, Evaluated Kinetic Data for High Temperature Reactions, Vol. 3 (Butterworths, London, 1976). [64] R. Atkinson, Gas Phase Tropospheric Chemistry of Organic Compounds, J. Phys. Chem. Ref. Data (1994) Monograph No. 2. [65] R. Atkinson, Kinetics and Mechanisms of the Gas Phase Reactions of the Hydroxyl Radical with Organic Compounds, J. Phys. Chem. Ref. Data (1989) Monograph No. 1. [66] D.L. Baulch, M. Bowers, D.G. Malcolm and R.T. Tuckerman, J. Phys. Chem. Ref. Data 15 (1986) 465. [67] J.T. Herron, J. Phys. Chem. Ref. Data 17 (1988) 967. [68] R.J. Cvetanovic, J. Phys. Chem. Ref. Data 16, 261 (1987). [69] J.A. Kerr, Parsonage, Evaluated Kinetic Data on Gas Phase Addition Reactions. Reactions of Atoms and Radicals wtih Alkanes, Alkynes and Aromatic Compounds (Butterworths, London, 1972). [70] T.J. Wallington, P. Dagaut and M.J. Kurylo, Chem. Rev. 92 (1992) 667; P.D. Lightfoot, R.A. Cox, J.N. Crowley, M. Destriau, G.D. Hayman, M.E. Jenkin, G.K. Moortgat and F. Zabel, Atmos. Environ. 26A (1992) 1805. [71] J. Warnatz, in: Combustion Chemistry, ed. W.C. Gardiner (Springer-Verlag, New York, 1984). [72] S.W. Benson and H.E. O'Neal, Kinetic Data on Gas Phase Unimolecular Reactions (NSRDS-NBS21, Washington, 1970). [73] C.P. Fenimore and G.W. Jones, 9th Symp (Int.) Comb. (1963) 597. [74] D.E. Hoare and M. Patel, Trans. Faraday Soc. 65 (1969) 1325. [75] R.R. Baldwin and R.W. Walker, J. Chem. Soc. Faraday Trans. 1, 75 (1979) 140. [76] N.R. Greiner, J. Chem. Phys. 53 (1970) 1070. [77] S. Gordon and W.A. Mulac, Int. J. Chem. Kinet., Symp. 1 (1975) 289. [78] D.J. Hucknall, D. Booth and R.J. Sampson, Int. J. Chem. Kinet., Symp. 1 (1975) 301. [79] J.N. Bradley, W.D. Capey, R.W. Fair and D.K. Pritchard, Int. J. Chem. Kinet. 8 (1976) 549. [80] J.J. Margitan and R.T. Watson, J. Phys. Chem. 86 (1982) 3819. [81] F.P. Tully, A.R. Ravishankara and K. Carr, Int. J. Chem. Kinet. 15 (1983) 1111. [82] D.L. Baulch, R.J.B. Craven, M. Din, D.D. Drysdale, S. Grant, D.J. Richardson, A. Walker and G. Watling, J. Chem. Soc. Faraday Trans. 1, 79 (1983) 689.

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[83] K.M. Jeong, K J . Hsu, J.B. Jeffries and F. Kaufman, J. Phys. Chem. 88 (1984) 1222. [84] C.A. Smith, L.T. Molina, J.J. Lamb and M.J. Molina, Int. J. Chem. Kinet. 16 (1984) 41. [85] F.P. Tully, A.T. Droege, M.L. Koszykowski and C.F. Melius, J. Phys. Chem. 90 (1986) 691. [86] R.A. Stachnik, L.T. Molina and M.J. Molina, J. Phys. Chem. 90 (1986) 2777. [87] C. Lafage, J.-F. Pauwels, M. Carlier and P. Devolder, J. Chem. Soc. Faraday Trans. 2, 83 (1987) 731. [88] J.F. Bott and N. Cohen, Int. J. Chem. Kinet. 23 (1991) 1017. [89] H. Niki, E.E. Daby and B. Weinstock, J. Chem. Phys. 48 (1968) 5729. [90] J.T. Herron and R.D. Penzhorn, J. Phys. Chem. 73 (1969) 191; G.P.R. Mack and B.A. Thrush, J. Chem. Soc. Faraday Trans. 1, 69 (1973) 208. [91] A.M. Dean and G.B. Kistiakowsky, J. Chem. Phys. 54 (1971) 1718. [92] R.B. Klemm, J. Chem. Phys. 71 (1979) 1987. [93] J.S. Chang and J.R. Barker, J. Phys. Chem. 83 (1979) 3059. [94] R.B. Klemm, E.G. Skolnik and J.V. Michael, J. Chem. Phys. 72 (1980) 1256. [95] J. Peeters and G. Mahnen, 14th Symp. (Int.) Comb. (1973) 133. [96] S. Toby and K. Kutske, Can. J. Chem. 37 (1959) 672. [97] A.M. Held, K.C. Manthorne, P.D. Pacey and H.P. Reinholt, Can. J. Chem. 55 (1977) 4128. [98] K.C. Manthorne and P.D. Pacey, Can. J. Chem. 56 (1978) 1307. [99] K.R. Selby, Ph.D. Thesis (University of York, 1978). [100] C. Anastasi, J. Chem. Soc. Faraday Trans. 1, 79 (1983) 749. [101] V.Ya. Basevich, S.M. Kogarko and G.A. Furman, Izv. Akad. Nauk. SSSR, Ser. Khim. (1971) 2191. [102] N. Washida, R.I. Martinez and K.D. Bayes, Z. Naturforsch. A29 (1974) 251. [103] T. Tsuboi, Jpn. J. Appl. Phys. 15 (1976) 159. [104] K. Shibuya, T. Ebata, K. Oki and I. Tanaka, J. Phys. Chem. 81 (1977) 2292. [105] J.P. Reilly, J.H. Clark, C.B. Morre and G.C. Pimentel, J. Chem. Phys. 69 (1978) 4381. [106] J.H. Clark, C.B. Moore and J.P. Reilly, Int. J. Chem. Kinet. 10 (1978) 427. [107] V.A. Nadtochenko, O.M. Sarkisov and V.I.Vedeneev, Dokl. Akad. Nauk. SSSR 224 (1979) 152. [108] B. Veyret and R. Lesclaux, J. Phys. Chem. 85 (1981) 1918. [109] M.A. Cherian, P. Rhodes, R.J. Simpson and G. Dixon-Lewis, 18th Symp. (Int.) Comb. (1981) 385. [110] F. Temps and H.Gg. Wagner, Ber. Bunsenges. Phys. Chem. 88 (1984) 410. [Ill] A.O. Langford and C.B. Moore, J. Chem. Phys. 80 (1984) 4211. [112] J. Vandooren, L. Oldenhove de Guertchin and P.J. van Tiggelen, Comb, and Flame 64 (1986) 127. [113] R.S. Timonen, E. Ratajczak and D. Gutman, J. Phys. Chem. 92 (1988) 651. [114] K. Glanzer, M. Quack and J. Troe, Chem. Phys. Lett. 39 (1976) 304; 16th Symp. (Int.) Comb.(1977) 949. [115] H. Hippler, K. Luther, A.R. Ravishankara and J. Troe, Z. Phys. Chem. NF 142 (1984) 1.

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Chapter 4

Mathematical Tools for the Construction, Investigation and Reduction of Combustion Mechanisms ALISON S. TOMLIN, TAMAS TURANYI and MICHAEL J. PILLING

4.1 INTRODUCTION

Chemical mechanisms have been employed for many years in hydrocarbon combustion [1]. Initially they were used as a means of understanding the underlying phenomenology of the combustion process in terms of the elementary reactions of individual species. This could be at a very schematic level, such as in thermal explosions where the process was modelled by a single reaction [2], or at a more complex level, exemplified by the peroxy radical mechanism of autoignition [3]. As understanding developed, mechanisms were required to play a more quantitative role. Greater demands were made on their agreement with experiment both at the macroscopic level, and in the simulation of minor products and, more recently, radical intermediates [4]. These requirements have coincided with the existence of expanding databases of rate parameters for elementary reactions (see Chapter 3), so that the construction of ever more detailed "complete" mechanisms is becoming both increasingly feasible and complex. To this development has been added the need to embed chemical mechanisms in computational fluid dynamic codes, so that the understanding developed in studying homogeneous chemical kinetics can be employed in the reactive flow conditions found in real combustion devices (see Chapter 7). The present limitations of computer hardware mean that complete chemical mechanisms cannot be incorporated in computational fluid dynamics (CFD) codes, and for turbulent combustion systems these limitations are especially pronounced. In general, some approximations have to be made which allow the size of the mechanism to be drastically reduced.

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Studies of the dynamics of complex chemical reactions, exemplified by the work described in Chapter 5, also make extensive use of reduced or lumped mechanisms. Such studies have shown that a range of rich dynamic behaviour observed in experimental systems can be found in models of lumped mechanisms. The interesting question is whether, or not, most systems can be represented by such simplified reaction schemes and, if so, how can the relationship between a full mechanism and a reduced one be defined. If such a relationship can be deduced then the use of simplified schemes would make both the mathematical analysis and the numerical modelling of hydrocarbon systems more tractable. Frequently, the lumping is achieved heuristically, with skeleton mechanisms being proposed which generate the correct dynamics and are chemically sensible [5]. Such mechanisms, although proving useful in many engineering applications, cannot be mapped uniquely onto a set of elementary reactions and their rate parameters. The links between these lumped and the detailed mechanisms are at best implicit, and in some ways this empirical approach prevents the modeller from reaching any deep understanding of the underlying processes involved. In this chapter we discuss the construction, analysis and reduction of chemical mechanisms. The rationale is that the mechanisms should be comprehensive and make full use of the available rate data for elementary reactions. At the same time, since their major uses are to understand the underlying chemical processes, and to model real combustion systems, possibly in turbulent environments, there is a need to generate concise mechanisms from the comprehensive ones. A schematic diagram illustrating both the structure of this chapter and the strategy employed for mechanism construction and reduction, is shown in Fig. 4.1. The basis of the mechanism of hydrocarbon oxidation has been discussed in Chapters 1 and 2 and clearly forms the kernel of the construction of a comprehensive chemical mechanism. Traditionally, detailed mechanisms have been constructed "manually", with chemical experts examining the species likely to be present in the system, and assessing which reactions they are likely to undergo under the appropriate conditions. In the precomputer era, when the steady-state approximation formed a primary tool and analytic solutions were necessary, there was a need to limit the size of the mechanism, and this was achieved in an a priori manner with the expert selecting the principal reactions on the basis of experience. The

295

Introduction Quantum Chemistry

Direct Measurements

Thermochemistry / Thermodynamics Measured values

L

*** Mass Balance Equilibration Data Reconciliation Noise Form and Magnitude Functional Analysis Linear Dependencies Identification

Kinetic Informations

KINETIC MODEL STRUCTURE DEVELOP/ THEORETICAL ANALYSIS

T

T

I Stoichiometry consistency checks |

| Reaction steps/intermediates identification |

Transport Model Assumptions Transport Parameters Identification

T PE criteria choice

PARAMETER ESTIMATION

Initial Guess Generation

SOLUTION ANALYSIS (Statistical Inference) (model adequacy, estimate quality)

Sensitivity Analysis I MODEL REDUCTION I Lumping Analysis Reaction Path Analysis I MODEL CHANGEMENTS I Restart experimental Design

MODEL IMPROVEMENTS

Restart theoretical Analysis

Fig. 4.1. A schematic diagram illustrating the strategy employed for mechanism construction and reduction.

advent of numerical integration and codes that deal with stiff systems, relaxed this condition and it became feasible to construct and employ comprehensive mechanisms. The development of these mechanisms is time-consuming and iterative. Validation against experimental data is necessary, and usually several independent research groups are involved in the process. Rate data are needed for each elementary reaction and, preferably, taken from the evaluated databases described in Chapter 3.

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These parameters are dependent on the experimental and, increasingly, theoretical rate determinations described in Chapter 2. Thermodynamic data constitute an additional requirement. It is generally accepted that these data sets are inviolate and that the tuning of rate parameters to obtain better agreement with the validating experiments is not acceptable. If this practice is followed then the lack of agreement can usually be attributed to shortcomings of the mechanism or to deficiencies of the parameters of certain highly sensitive reactions. Even so, the temptation to tune the rate constants has to be resisted, because the experiments provide a comparatively limited information base and the problem is often highly over-parameterized. A recent development, described in Section 4.3.7, exploits tuning of the rate constants but only over an accepted parameter uncertainty range. Clearly, mechanism construction is highly labour intensive, and this characteristic is leading increasingly to a recognition of the need for community "ownership" and the development of "master" mechanisms. The use of computer-based expert systems provides an alternative to manual mechanism construction and is discussed in Section 4.3. Such an approach draws from the experience gained in the manual mechanism development, which is used to construct rules that can be converted into algorithms for assembling mechanisms. The advantage of the computerbased approach is that total mechanisms are easily constructed within the confines of the rules. The procedures can then be readily modified if changes in our understanding lead to a modification of these rules. Expert systems perform at their best for complex molecules, such as alkanes, that are degraded via a relatively small set of generic reactions. Hydrocarbon oxidation and pyrolysis provide excellent examples. Once the mechanism has been constructed and the rate parameters incorporated, then it is necessary to integrate the resulting coupled differential rate equations. A full discussion of the numerical solution of differential equations is beyond the scope of this book, but a brief reference is made in Section 4.4 to integration methods and to complete packages specially designed for chemical systems. Potentially, the results from numerical integration can provide a mass of information which can be compared with experiment, but, in itself, is fairly indigestible. Mechanisms such as that for hydrocarbon oxidation are complex, and there is a need to understand their structure and validity. In order to gain such understand-

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ing, information is required about the major routes and cycles, the key reactions and groups of reactions, the determinants of the major features such as the temperature, the dominant products and the key intermediates. It is difficult to achieve these aims without mathematical tools for the analysis of the data generated by integration methods. Sensitivity analysis, which is described in Section 4.5, offers such tools by providing a quantitative measure of the dependence of species concentrations, rates, or reaction features on the rate parameters or on the initial concentrations of other species. In addition, principal component analysis can be used to combine the sensitivities, to allow an understanding of the interactions between reactions and their grouped behaviour. The next stage in the logical process is the use of methods for mechanism reduction, to obtain a subset of the detailed mechanism by removing species and reactions that are redundant under the conditions required. One procedure, based on rate and sensitivity analyses, is described in Section 4.6. The approach is exemplified using the H 2 /0 2 reaction in a continuous stirred tank reactor. This system shows rich dynamical behaviour as described in detail in Chapter 5. It is employed as an exemplar throughout the chapter, first, because of its rich dynamics, but second, since it is really the only system where a large proportion of the techniques we will introduce have been applied. The application of these techniques leads to quite substantial reductions; typically the number of reactions is halved and the number of species reduced to one- to two-thirds of the original value. These figures could possibly be higher for mechanisms generated by an expert system, since for manually constructed mechanisms, some reactions have usually been left out by the experienced modeller. For some applications this degree of reduction would be enough, however, for CFD models, the number of species remaining will still be too large. In a properly reduced mechanism all the remaining species and reactions are necessary and any further progress is generally achieved by representing the mechanism in a more compact way. One way forward is by lumping the species. This can involve either expressing the concentrations of some species in terms of others via algebraic equations, or constructing a new set of lumped variables from linear or non-linear combinations of the chemical species, thereby reducing the dimension of the problem. Formal aspects of both linear and nonlinear lumping are briefly reviewed in Section 4.7. Chemical lumping, a

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less formal approach linked to the structure of the reaction network, is illustrated by reference to the formation of polycyclic aromatic hydrocarbons in flames. For the incorporation of reduced chemical models in CFD codes, lumping techniques can take advantage of the wide range of chemical timescales, and the very limited range of transport time-scales present. The wide range of chemical time-scales presents problems in that it leads to stiffness in the resulting differential equations, but it also provides a possibility for reduction using time-scale separation. There are a number of different approaches based on this idea, including the well known quasisteady-state approximation. A comparison of these approaches is presented in Section 4.8 and provides an opportunity to discuss the dynamic evolution of chemical systems through a hierarchy of manifolds each of a successively reduced dimension. The mathematical description of the chemical system on the final slow manifold forms the basis of reduced models appropriate for incorporation in transport codes. One approach is to employ this manifold directly, using a look-up table, as proposed in the seminal work of Maas and Pope [6, 7]. Slow manifolds are implicitly utilized in the repro-modelling approach of mechanism reduction. Another way to use time-scale separation is to develop approximate lumping techniques, as discussed in Section 4.9, and illustrated, once again, by reference to hydrogen oxidation. Section 4.10 includes a brief survey of global reaction mechanisms, which have generally been obtained in a less formal way and with no direct link to the comprehensive mechanism. These skeleton mechanisms have played a major role in the development of our understanding of combustion. In the present context, the Shell model of autoignition [8] provides an excellent example, based on the underlying chemistry and providing a minimal basis for a description of the overall dynamics. An ultimate aim of reduction and lumping techniques is to provide models of this type, but with well-defined links to the comprehensive mechanism. The techniques described in the present chapter follow on logically from Chapters 1 to 3 which described the tools needed for the generation of comprehensive mechanisms. In the future, they are intended to provide a link between these mechanisms and the sort of tools introduced in Chapter 5 for the analysis of the global behaviour and the rich dynamics found in combustion systems. The beauty of the methods described in Chapter 5 is

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that they can give incredible insight into combustion processes, but the reality is that the mathematical analysis often requires concise mechanisms with very few variables. Conversely, the understanding of detailed mechanisms, and of the complexity required to generate specific dynamical features, should be guided by the advances described in Chapter 5. The need for concise chemical representation also applies to the work described in Chapters 6 and 7, so that the links between these chapters and the present one are crucial if chemical mechanisms, based on the type of understanding of elementary reactions discussed in Chapters 1 to 3, are ever to be incorporated in models of real combustion devices. At present, the generation of such concise mechanisms directly from full mechanisms has not been widely achieved. Most of the objective techniques described here are in their infancy, and this is evident from the need to use the hydrogen/oxygen example as an illustration, rather than one more appropriate to a volume on hydrocarbon oxidation. Therefore, one aim of the chapter is to give a comprehensive account of the mathematical techniques available, with the expectation that they will indeed be applied, before long, to the problems that lie at the heart of this book. What we have tried to achieve in the text is to give some information about the various methods available, their principle of application and their scope in complex combustion systems. Because of limited space we cannot give enough details for the methods to be applied from this review, and detailed information should be found from the references. However, even our simplified description might be too technical in many cases, and for readers interested primarily in applications, we recommend the example and discussion sections only. The chapter tries to give an up-to-date account of all the main mathematical tools which have been used for the construction, investigation and reduction of complex reaction mechanisms. However, there are reviews which discuss some of these methods in more detail. Frenklach [9] has reviewed several techniques, developed until 1989, for the reduction of combustion mechanisms. Application of reduced chemical mechanisms in turbulent combustion has been reviewed by Seshadri and Williams [10]. Warnatz [4] has reviewed the relation of combustion modelling to detailed reaction mechanisms. The recent review of Griffiths [11] concentrates mainly on global mechanisms for the description of low-temperature alkane combustion. The review of Turanyi [12] gives an almost full account until

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1989 on the application sensitivity analysis methods in chemical kinetics. Formal mathematical models of chemical mechanisms have been discussed byErdiandT6th[13].

4.2 NOTATION ki

A E R T Ta

cP 1i

cu Q *res

p Vij

RJ

p V t X

s

J M €

Q 4H 9i

mth order rate constant (molecule cm 3 ) 1 pre-exponential factor (molecule cm - 3 ) 1 J mol" 1 activation energy gas constant J K" 1 mol" 1 K gas temperature K ambient temperature specific heat capacity per unit volume J K - ' k g " 1 typically kJ mol~ exothermicity of yth reaction step molecule c m - 3 species concentration s residence time of reactor Torr pressure stoichiometric number rate of reaction ; molecule c m - 3 s~ kg cm - 3 density cm3 volume s time W cm" 2 K" 1 heat transfer coefficient cm2 surface area Jacobian matrix lumping matrix small parameter Schur matrix specific mole number of species / Schur vector

4.3 THE CONSTRUCTION OF COMBUSION MECHANISMS

Most mechanisms used nowadays are the result of slow and painful development work. Preliminary mechanisms are published by certain

The construction of combustion mechanisms

301

groups and then criticized by others, suggesting the addition of new reaction steps and the elimination of previously proposed steps. As a result of hard work over several decades, detailed reaction mechanisms are now available for the high-temperature combustion of alkanes up to iso-octane. Reaction mechanisms are also available for the combustion of other simple fuels like H 2 , wet CO, and simple alcohols. However, these reaction mechanisms are not widely accepted, although it is mainly the rate parameters rather than the main reaction routes which are debated. Recent combustion research requires many more new reaction mechanisms. Such requirements include, for example, the generation of mechanisms for the combustion of ethers, which can be important as blending agents of petrol. The search for new fire extinguishers also requires the generation of new detailed mechanisms. The manual generation of reaction mechanisms starts with the assessment of necessary species. Reactants and main products clearly have to be included into the mechanism, but their interconversion routes also introduce a large number of intermediate species. Such a procedure means that a huge number of highly coupled reaction steps have to be assessed and there is always a possibility for human error. For this reason, several groups have proposed the application of computers for the generation of reaction mechanisms. Superficially it may seem likely that this procedure can be well automated, but producing a general program is technically very difficult. In spite of efforts over several decades, there are still no widely used programs for mechanism generation. A mechanism generation program consists of the following parts: 1. The structures of chemical species have to be stored in a form which can be manipulated easily by the reaction generator that will predict the products of each elementary reaction. The computer representation must be unique and non-ambiguous so that reaction steps are not misinterpreted or repeated. As an example, for the neopentane + H atom reaction: C(CH 3 ) 4 + H-> C(CH 3 ) 3 CH 2 + H 2 the reaction generator produces 12 reactions (one for each hydrogen) with apparently different products. Once the structure of the product species are rearranged, comparison reveals that the 12 reactions give identical products. If the starting hydrocarbon was iso-butane (CH(CH 3 ) 3 ), a similar rearrangement would identify two different products.

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2. For a reaction generator to produce a viable set of elementary reactions given a number of chemical species, it must consider all combinations of species and never produce the same reaction twice. The input of the reaction generator is the structure of one or two reacting species (for unimolecular and bimolecular reactions, respectively) and the output is the list of the possible reactions together with the structure of the corresponding possible products. On a purely combinatorial basis, reactions of large organic molecules may result in a huge number of possible products. To avoid unlikely reactions, the reaction generator must know the types of likely reactions for given types of reactant structures. Thus, expert chemical knowledge has to be encoded into this module. For example, it would be possible to produce a large number of products from radicals CH 3 and C 2 H 5 . However, if the reaction generator considers only recombination and disproportionation reactions, it will produce CH 3 CH 2 CH 3 and CH 4 + C 2 H 4 , respectively. 3. Kinetic parameters of the generated reactions are searched for in a database. However, usually parameters for only a small fraction of reactions generated are likely to be found in the connected database. Therefore, the program must be capable of parameterizing all reactions, using empirical rules associated with the type of reaction and the size and structure of reactants. As an example, Arrhenius parameters for the abstraction of a primary hydrogen atom from a large alkane by radical OH can be predicted well for any hydrocarbon. Such prediction rules have been discussed in Chapters 2 and 3. Also, reaction enthalpies can be calculated from thermodynamic data stored in a database, or from approximated data calculated from Benson's additivity rules. 4. The reaction generator will produce so many possible reactions that the program has to filter out those which are obviously unimportant. The two basic techniques are the elimination of very endothermic reactions, and the elimination of reactions which are too slow. All the steps described above have to be carried out iteratively by the mechanism generation program. First, a primary mechanism is obtained by allowing the initial species to react to give their primary products. These primary products react similarly to reproduce a secondary reaction scheme, introducing secondary products and so on. Continuing this process generates a series of reaction sets. The nth term of the series contains the permitted reactions of species evolved in the (n - l)th term. The procedure is terminated at a predefined n, and the sum of the series up to the nth


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TABLE 4.1 Reaction types for alkane pyrolysis Decomposition Radical decomposition Atom abstraction Radical addition to unsaturated molecule Radical recombination Radical disproportionation Molecular elimination Alkene conversion Radical conversion Radical isomerization

M—> .R + .R' .R' —> .R' + U .R + M—> .R' + M' .R + U—> .R' .R + .R' —> M .R + .R' —> M + U M—» M' + U U + .A—• .R + U' .R + .A—> .R' + .R" .R—».R'

Key: M = molecule; .R = radical; U = unsaturated molecule; .A = atom.

term gives the final mechanism. The efficiency of mechanism generation programs can be tested on tasks which have been solved previously by "human" mechanism builders. Another benchmark is the numerical testing of the mechanism against experimental data. As an example, we consider in outline the mechanism generated by Chinnick et al. [14] for methane pyrolysis. The reaction types are defined in Table 4.1 and a set of rules is associated with each of them. For example, decomposition corresponds to the rupture of every unique single bond in a molecule M, taking care to identify only unique reactions. Limitations can be placed on the reactions, essentially through a generalization of the associated rate constants. Radical isomerization, for example, is only permitted in the Chinnick system for 1-4, 1-5 and 1-6 H shifts, and since it does not occur for hydrocarbons with carbon chains less than C4, it is absent for methane. All possible reactions are then constructed, allowing all possible pairs of species to react by all the appropriate reaction types, including unimolecular processes. Species attributes are used to assign which of the reaction types are feasible. For example, for an addition reaction, one reactant must be an unsaturated molecule and the other a radical. Table 4.2 shows the first three orders or generations of reactions for methane pyrolysis. The primary mechanism includes only the unimolecular reaction, methane decomposition, since only one initial reactant is present with only one bond type. The reaction generates two radical species, and the secondary mechanism includes all possible reactions of these species and methane, a total of four reactions generating C 2 H 6 and H 2 . The tertiary mechanism

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TABLE 4.2 Primary, secondary and tertiary mechanisms for methane pyrolysis determined by the Leeds Expert System Reaction

Reaction type

Primary (first-order) mechanism PI CH4-+.CH3+.H

Decomposition

Secondary (second-order) mechanism SI .CH3+.CH3^CH3CH3 S2 C H 4 + . H ^ H 2 + .CH3 S3 .CH3 + . H ^ C H 4 S4 .H+.H-^H2

Recombination Abstraction Recombination Recombination

Tertiary (third-order) mechanism Tl CH 3 CH 3 -> .CH3 + .CH3 T2 CH 3 CH 3 -> .CH2CH3 + .H T3 CH 3 CH 3 -+ H 2 + CH 2 = CH 2 T4 CH 3 CH 3 + .CH3 -> .CH 2 CH 3 + CH 4 T5 CH 3 CH 3 + .H -* .CH 2 CH 3 + H 2 T6 H2^.H+.H T7 H2+.CH3-*CH4+.H

Decomposition Decomposition Molecular elimination Abstraction Abstraction Decomposition Abstraction

contains seven reactions and generates two new species, the fourth-order mechanism (not shown) contains 17 reactions and five new species and so on. The number of reactions increases rapidly with mechanism order and needs to be limited in a rational way, related to the percentage conversion of the reactant and some cut-off value for the products. The limitation can be checked only by subsequent numerical integration. The choice of reaction types also needs to be assessed by comparison of simulations with experiment. In the following sections, a brief historical review of expert systems is given. 4.3.1 Matrix rearrangement In his pioneering work, Yoneda [15] used a technique which is different from the logical scheme given above. Yoneda's approach was to represent the structure of each chemical species as a square matrix. The reaction generator joined these matrices, and searched for elementary reactions by rearranging the matrix elements. An elementary reaction was defined as


305

a simple reaction that cannot be decomposed further into simpler reactions in the sense of the transformation of bonding. Changes in the joint matrix of reactants resulted in a matrix description of reaction products. No reaction types were defined, but limitations were enforced for the number of bonds changed and the structure of reaction products. The approach of Yoneda has advantages and drawbacks. Definition of reaction types brings human arbitrariness into the process of mechanism generation and may result in errors. However, restricting the types of reactions is the best way to limit the number of reactions generated. Yoneda demonstrated the applicability of his program, called GRACE, on a simple example of catalytic hydrogenation of ethylene. In the case of more complex reactions GRACE probably would have generated too many meaningless reactions. Unfortunately, the development of GRACE seems to have been prematurely stopped. A group of researchers in Budapest continued the line of Yoneda [1621] but avoided the combinatorial explosion of the number of products by the preliminary definition of acceptable reaction products. Thus, the species to be included in the mechanism were fixed a priori, and the program provided the list of reactions. They used the matrix technique of Yoneda for the representation of reactions and species structures, but the number of generated reactions was limited by applying certain restrictions. The most important restriction was that bimolecular reactions were considered only with a maximum of three products. The number of generated reactions was kept low based on reaction complexity and thermochemical considerations. The mechanism obtained was reduced by qualitative and quantitative comparisons with experimental results, including contributions of elementary reactions to measured rates. The method proposed 538 reactions for the liquid phase oxidation of ethylbenzene. The reactioncomplexity investigation approved only 272 reactions and the reaction heats were feasible in the cases of 168 reactions. This mechanism was reduced to a 31-step final mechanism. 4.3.2 The logical programming approach Chinnick et al. [14,22] seem to be the first to have developed an expert system for mechanism generation based on logical programming. The program was written in POPLOG, which is an integrated combination of programming languages POP11, PROLOG and LISP. Most of the expert

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system was encoded in PROLOG. The outline of the program was constructed according to the steps described in the general discussion. The canonical representation of the structure of chemical species was based on the DENDRAL project [23]. The molecule was interpreted as a tree, rooted at a unique point in its structure called the centroid atom. The DENDRAL rules had to be extended for a unique representation of acyclic hydrocarbons. This coding is not applicable for cyclic molecules. Ten types of elementary processes were defined, which included all the reaction types characteristic for the pyrolysis of alkanes. To speed up the process of reaction generation, five basic structural attributes were attached to each species. These attributes described the type of the species (molecule, radical), the length, the saturation, etc. The application of attributes enabled the sensible application of reaction types, since, for example, molecules do not take part in radical recombination reactions, and short radicals cannot take part in 1,4-isomerization. Where possible, rate parameters were taken from databases and others were estimated on the basis of ten rules which were applied according to the type of reaction and reaction partners. Most of the generated reactions were eliminated during the construction of the mechanism. The primary tool was the comparison of reaction rates for parallel reaction channels. This approach required careful planning of the order of generation of the reaction types. The program was used for the pyrolysis of Q-C4 hydrocarbons. Mechanisms for ethane, propane and butane contained 15, 49 and 76 species and 18, 115 and 179 reactions, respectively. These mechanisms were compared with schemes which had been proposed by human experts. Most of the reactions were identical, and, in general, the program proposed a superset of those presented in the literature. 4.3.3 Applications to hydrocarbon oxidation (i) Chevalier et al. [24, 25] at Stuttgart University, created a program for the automatic generation of reaction mechanisms for the description of the oxidation of higher hydrocarbons. Unfortunately, their two publications do not reveal many technical details. The main ideas for the unique representation of species and the generation of reactions seem to be similar to those of Chinnick et al. Their program was written in LISP and uses the CEC kinetic data evaluation [26] for the Ci-C 4 chemistry submechan-


307

ism. Most of the rate parameters are estimated based on reaction type and reactant structure using simple rules. Chevalier et al. went further in terms of applications than the Chinnick work. Their reaction mechanism for the ignition of heptane contains 620 species and 2400 reactions, while the mechanism for the hexadecane (cetane) consists of 1200 species and 7000 reactions. Ignition delay times were calculated by the automatically generated n-heptane mechanism and were compared to both experimentally measured delay times, and values calculated by a human-generated mechanism. The agreement with the experimental data was excellent, slightly better than in the case of the manually generated mechanism. The Chevalier program was developed further by Nehse et al. [27] in Heidelberg. (ii) Ranzi et al. [28] have developed a FORTRAN program called MAMOX for the automatic generation of the reaction steps for the lowtemperature oxidation of normal alkanes. The logical scheme is similar to earlier work. Mechanisms were generated for the oxidation of pentane and butane, and the simulation results were in good agreement with experimental data. The novelty in the work of Ranzi et al. is the automatic simplification of the large detailed reaction mechanism obtained by lumping both the species and the reactions. Isomers with similar kinetic behaviour were considered as single-lumped species (see Section 4.7.3 for a discussion of chemical lumping). Parallel reaction routes were lumped together based on kinetic assumptions. Finally, the model parameters were fitted to the predictions of the complete scheme. (iii) Blurock [29-31] wrote an X-window based program, called Reaction, for the generation of reaction mechanisms. The logical scheme of the program is identical again to those by the other authors. The program was applied to the generation of the low-temperature oxidation mechanisms of 21 different alkane molecules. The reaction patterns considered in this case were hydrogen abstraction, addition of 0 2 to a radical, isomerization, addition of a second 0 2 and internal rearrangement providing two OH radicals. The mechanisms were used for the calculation of the time-dependent concentration profile of OH whose rate of production was compared to research and blend octane numbers of the alkanes. (iv) Broadbelt et al. [32-34] developed an integrated system for the computer generation of kinetic models. The required input consists of the structure of the reactants, the rules by which the reactant and product

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species react and the parameters of a structure/property kinetics correlation. The algorithm transforms this information into the reaction network, species properties and rate constants. 4.3.4 A comprehensive program system Come et al. [35] and Vogin [36] have dealt with the topic of automatic mechanism generation for a number of years. As a result of a long research program they have developed a comprehensive working system. The structure of the program is given in Fig. 4.2. This system includes a kinetic database (KERGAS), software for automatic reaction generation (EXGAS), and programs for the estimation of kinetic parameters and thermodynamic data (THERGAS, KINBEN and KINCOR). EXGAS and KINCOR are written in PASCAL, while THERGAS and KINBEN are written in FORTRAN. As an indication of the size of this program system, EXGAS is about 23,000 lines long and THERGAS and KINBEN consists of 41,000 lines. KERGAS is a kernel of reaction steps involving C 0 , Cx and C 2 molecules and free radicals. KERGAS includes data for all the important reactions of methane and ethane combustion and data for 850 reactions of 42 species are included [37]. EXGAS generates comprehensive primary reaction mechanisms, and produces lumped secondary mechanisms of lumped primary products and secondary species. The method for the unambiguous coding of species is documented in [38-40]. EXGAS is connected to the programme THERGAS [41,42] which computes the coefficients of thermodynamic NASA polynomials for the species by means of Benson's estimation techniques. Kinetic data for molecular initiation, isomerization and combination, which cannot be found in the literature, are estimated using Benson's techniques in the program KINBEN. For other types of reactions kinetic data are estimated on the basis of structural correlations in KINCOR. Subroutines KERGAS and EXGAS write reaction mechanisms, thermodynamic and kinetic data to a CHEMKIN-format data file, so that the mechanism obtained can be directly used in the CHEMKIN [43] simulation package (see Section 4.4). The capabilities of the system have been demonstrated by the generation of a reaction mechanism of 142 species and 269 reactions for the oxidation of butane [44]. A mechanism of 1200 reactions was generated for the oxidation of iso-octane. In both cases the results calculated by the auto-

primary mechanism

EXGAS

COMPREHENSIVE PRIMARY MECHANISM

8 2 Rg.

v

lumped secondary mechanism

3

9,

V CHEMKIN II

z

P

I '

B Fig. 4.2. Flowchart of the NANCY program system for mechanism generation.

w

0 W

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matically-generated mechanism were in good agreement with experimental data. The Nancy program system [45-49] seems to be the most advanced of all the similar programs for the generation of reaction mechanisms. It also includes lumping and the estimation of thermodynamical data, making the package self-sufficient in terms of generating all the information needed to carry out simulations. 4.3.5 The GRI mechanism The main advantage of mechanistic kinetic models is their modularity. The reaction steps of hydrogen oxidation form the core of most combustion mechanisms. By adding further steps, a mechanism describing the combustion of wet carbon-monoxide can be formulated. By adding yet more elementary reactions the mechanism describes the combustion of methane, and so on. If mechanisms are generated in this way, using measured rate data for each elementary step, the previous steps do not have to be altered or eliminated. Of course, some of them might become unimportant and may be eliminated, but it is not necessary to do so. The rate parameters and the product channels of these reaction steps are determined by experiments, where reactions can be studied individually or in small groups. However, the error of the determination is often rather high, and many rate coefficients are known with an uncertainty as high as a factor of three. Therefore, when large reaction mechanisms are assembled, they do not satisfactorily reproduce the experimental data in many cases. Frenklach [50] has shown that by fitting the parameters of some reaction steps to bulk experimental data, large reaction mechanisms can be optimized. This method is recommended for the systematic improvement of reaction mechanisms, and has been demonstrated for the improvement of a large mechanism of methane combustion having 149 reversible reactions [51]. The project was funded by the Gas Research Institute and several US workgroups have contributed to the GRI mechanism [52]. The mechanism and the results of its testing are available through the World Wide Web [53]. An earlier version (ver. 1.2) of the GRI mechanism described the high-temperature oxidation of methane and the current version (ver. 2.11) has been extended to include NO x reactions present in methane flames. The basis of the GRI mechanism is an elementary reaction mechanism, where experimentally determined values are assigned to the rate para-


311

meters. In the next step, the simulation results from the initial mechanism were compared to bulk experimental data such as ignition delay times, laminar flame speeds, species profiles in shock-tube ignition experiments, laminar flames and flow reactors. An extensive sensitivity test revealed which parameters needed to be tuned to minimize the difference between the experimental data and the simulation results. The parameters were then optimized automatically by a computer program, but their values were always kept within the uncertainty limits of the original experimentallydetermined parameters. This optimization process was done simultaneously for all the rate parameters in the mechanism. There are several significant advantages and drawbacks associated with this approach. The obvious advantage is that the resulting mechanism describes the experimental data more accurately than other untuned mechanisms. The technique provides a huge advantage so long as the conditions of practical application are close to the conditions of one of the experiments used for tuning, and if the corresponding experiment provided accurate data. The drawbacks of the approach are associated with the loss of the modularity of elementary mechanisms. The parameters are no longer independent and it is not advisable to substitute new values for the parameters if more accurate experimental data become available for some reactions. Another consequence of lost modularity is that this mechanism may not be suitable as a starting module for building combustion mechanisms of other hydrocarbon fuels. If the product channels of some reactions are improper, or if one of the fixed parameters or some of the experimental data are erratic, then the whole mechanism can be mistuned. The GRI mechanism lies somewhere between an elementary and empirical mechanism. The latter type of mechanisms will be described in detail in Section 4.4.9. Given the above precautions, the approach used in the GRI mechanism is an important and novel advance of the mathematically-assisted generation of reaction mechanisms. 4.3.6 Discussion The principles for the automatic generation of reaction mechanisms seem to be well elaborated. Independently, several research groups have written successful programs for the generation of pyrolysis and low-temperature alkane combustion mechanisms. However, while simulation programs are generally available, mechanism generation programs are not

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either on a commercial basis or in the public domain. A possible reason for this might be that making such a program "fool-proof", i.e., suitable for use by an inexperienced user, is as time-consuming as the creation of the program itself. In spite of this, it is expected that such programs will be more widely available in the future. The concept used for the GRI methane mechanism is generally applicable for the improvement of the agreement of an existing detailed mechanism with bulk experimental data. Care should be taken in the use of such an application because of the limitations discussed above.

4.4 NUMERICAL INVESTIGATION OF COMPLEX MODELS

Once a mechanism has been generated using the best available rate information, it has to be validated against a range of experimental data. Mechanistic information has to be converted in some way into species profiles and/or reaction features for a range of different conditions, so that comparisons can be made with experimental measurements. Therefore, the mechanism becomes part of a larger model which may seek to represent many different situations, ranging from a simple stirred reactor to a complex engine environment. The usual way to include the chemical component in any larger model is by constructing differential equations describing the rate of change of species concentrations. 4.4.1 Basic equations The rate of change of concentration in a homogeneous reaction system can be described by the following system of ordinary differential equations (odes):

«f(c,k), at

c(0) = c°,

(4.1)

where c is an n-dimensional concentration vector. For a homogeneous reaction taking place isothermally in a closed vessel f (c, k) can be represented as a function of the reaction rates

Numerical investigation of complex models

/,(c,k) = 2 ^ i ? / .

313

(4.2)

J

Here i is the species number, vtj is the stoichiometric coefficient of the fth species in the ;th reaction and Rj is the yth reaction rate. In a homogeneous isothermal flow reactor, the rate of change of the species concentrations at a constant volume can simply be represented as ^ =/,(c, k) = ^ ^ dt tres

+X

(4.3)

ViJRj9

J

where ct is the concentration of species i and c° is the inflow concentration or the steady-state concentration existing in the vessel when no reaction takes place. For non-isothermal reaction schemes then we should also include energy balance, (Cp) ^

= (Cp) ^

dt

^ tres

+ 2 qjRj ~^(T-Ta). j

(4.4)

V

The heat-loss terms describe both the loss via the flow of gases leaving the reactor and via Newtonian cooling through the walls where tres is taken to be the average residence time of the reactor. Ta is the ambient temperature, V the volume, S the reactor surface area and x the heat transfer coefficient. Cp is the heat capacity per unit volume which is assumed to be independent of temperature, and g7 the exothermicity of the yth reaction step. 4.4.2 Numerical integration In order to find the evolution of species concentration or temperature with time, the above equations must be integrated. For complex reaction mechanisms this usually means integration by numerical methods. There are a large number of schemes for the numerical integration of coupled sets of differential equations, but not all will be suitable for the types of mechanisms we are discussing. Chemical systems form a difficult problem because of the differences in reaction time-scale between each of the

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species. The existence of fast time-scales associated with intermediate species, within an overall time-scale which is quite long, makes the computational problem expensive. This problem is called stiffness and is a feature of the majority of combustion models. The consequence of stiffness is that the use of standard integration methods, such as the explicit Runge-Kutta methods, becomes inefficient, because too small a time-step is required in order to achieve stability. Special integration packages for stiff systems can be found, such as LSODE [54-56] which is based on a predictor corrector method, and DASSL [57] which uses the backward differentiation method and can be used to solve coupled sets of differential and algebraic equations. Comparisons between these methods and standard Gear [58, 59] algorithms have been made by Radhakrishnan [56]. A range of different integration schemes to suit chemical, and particularly combustion problems, have been included in the program packages LSENS [56] and SPRINT [60]. Such codes form the basic tools for integrating sets of ordinary differential equations describing the time evolution of chemical species. However, in order to use one of these packages the rate equations must be expressed explicitly within the driving program for the integration routine. For users wishing to address a particular chemical application the task is made simpler by using specifically designed codes such as CHEMKIN [43], RUN1DL [61] or FACSIMILE [62], where the mechanistic information can be expressed easily in kinetic form. CHEMKIN and RUN1DL also contains a thermodynamic database so that non-isothermal systems can easily be interpreted.

4.5 SENSITIVITY AND UNCERTAINTY ANALYSIS

4.5.1 Introduction Possession of a comprehensive mechanism and a suitable integration scheme does not mean that a chemical process is well understood. Particularly for large mechanisms with many species, it is not easy to understand the competition between reaction steps or the coupling between species. We shall see that sensitivity analysis and components of the mechanism reduction techniques described below, provide a good method of under-

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standing the fine details and behaviour of complex chemical kinetics processes. Many rate parameters in combustion are known only within fairly large error bands. For example, rate constants are often known only within a factor of 3-5, which is defined as the ratio of the best estimate and the assumed extreme values. These uncertainties are listed in collections of evaluated reactions (see Chapter 3 or Refs [26,63]). At a first glance it seems incomprehensible that kinetic mechanisms can accurately predict experimental data with such poorly known parameters. However, quite an accurate prediction is possible in many cases. The reason is that reaction mechanisms include both stoichiometric information (about the interconversion routes of species), and parametric information (rate parameters and thermodynamic data). For some reactions only the stoichiometric information is important. These reactions have to be present in the mechanism, but the value of the rate constant is almost arbitrary provided that it is in a given region (in most cases it has to be large enough). In the case of other reactions, the rate coefficient has to be known with some precision. For some reactions a small deviation in the parameter value causes a significant change in the model output, while for other reactions, a much larger deviation will cause a similar change. There are no direct methods for the inspection of the stoichiometric information. If a reaction proves to be redundant by one of the methods described in the next section, then obviously the stoichiometric information of the reaction is not important. Methods for the analysis of time-scales (see Section 4.8) can also provide hints for the transformation of reaction systems without losing stoichiometric information. A good global reaction scheme is the minimal reaction system that contains all the stoichiometric information of the detailed mechanism. In Sections 4.8 to 4.10, a series of methods will be presented for the generation of global reaction schemes. The family of methods for the study of parametric information in mathematical models is called sensitivity analysis. Sensitivity analysis investigates the relationship between the parameters and the output of any model. It is usually used for two purposes: first, for uncertainty analysis and, second, for gaining insight into the model. Sensitivity methods in chemical kinetics have been reviewed by Rabitz et al. [64], who concentrated mainly on the interpretation of sensitivity coefficients in reaction-diffusion systems. Turanyi [12] considered sensitivity methods as tools for studying reaction kinetics problems and reviewed several applications. Recently, Radha-

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krishnan published several articles on the numerical comparison of methods for the calculation of local sensitivities (see e.g., [65, 66]). There are several reviews on the different fields of sensitivity analysis. For an almost complete list of publications on the application of sensitivity analysis in reaction kinetics up to 1989 see [12]. 4.5.2 Local sensitivities Sensitivity analysis investigates the output of models as a function of parameters. In local sensitivity analysis, this relation is given as the partial derivative of the output of the model with respect to the parameters. These partial derivatives can be simply introduced via a Taylor series expansion: m

d(t, k + Ak) = c,-(r, k) + 2 — A*y + • • •,

(4.5)

y=i dkj

where the partial derivatives dcjdkj are called the first-order local concentration sensitivity coefficients. The parameters are perturbed at time tu and the change in concentrations is studied at a later time t2. In most cases time ti is set to be equal to the starting time of simulation. The simplest method for the calculation of local sensitivities is by changing each individual parameter by a small amount, rerunning the model and approximating the sensitivities by differences. This brute-force method is only appropriate if better methods are not available, since it is relatively slow and inaccurate. If the original system of kinetic differential equations (4.1) is differentiated with respect to kj, the following set of sensitivity differential equations is obtained: d dc

x/x

dc

df(t)

- — = J(') — + ^M> at dkj

dkj

.

,

; = l,...,m

/A

^

(4.6)

dkj

where J(t) = 8(1'dc and the initial condition for 3d dkj is a zero vector. Equations (4.1) and (4.6) are coupled through the matrices df/dc and df/dk, that is equation (4.6) can only be solved if the concentration values, calculated in equation (4.1), are available at times where the above matrices are calculated during the numerical solution of the sensitivity equation (4.6).


317

A possible way to achieve this is to solve equation (4.1), together with all the sensitivity equations belonging to different parameters. In this case, the stiff ode solver has to decompose a large (m + l)n x (m + l)n Jacobian at each time-step, which is very inefficient. If the kinetic ode (4.1) is coupled with a single sensitivity equation at one time, the joint Jacobian is smaller, but the kinetic system of odes has to be solved m times. The most efficient algorithm for the solution of the sensitivity differential equations is called the decoupled direct method (ddm), which was first applied in chemical kinetics by Dunker [67, 68]. He drew attention to the fact that equations (4.1) and (4.6) have the same Jacobian, so that a stiff ode solver will use the same step size and order of approximation in the solution of both odes. The ddm method first takes a step for the solution of equation (4.1) and then performs steps for the solution of equation (4.6) for j = 1,. . . , m. The procedure is repeated in the subsequent steps. Since the Jacobian of the equations is the same, it has to be triangularized only once for each time interval. This method is applied in the program SENKIN [69]. The sensitivity coefficient dcjdkj is of limited applicability in its original form. The parameters and the various output quantities of a model may have different units, for example, rate coefficients belonging to reactions of different orders have different units. In such cases, the elements of the sensitivity matrix are incomparable. The usual solution to this problem is the introduction of normalized sensitivity coefficients. These coefficients form the normalized sensitivity matrix,

g

/ ^ \

/ajncA

\CidkjJ

\d In kjj

(4.7)

The normalized sensitivity coefficients represent the fractional change in concentration ct caused by a fractional change of parameter kj. Differentiating equation (4.1) with respect to initial concentrations c° gives d dc ^ = j

_ , . dc ^

( , )

. -

1

- - -

(4 8>

-

The matrix valued function dc/dc°, is called the initial concentration sensiti-

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vity matrix, and its initial value is a unit matrix. This matrix gives the effect of the change of initial concentrations on the solution of the model. The above comments on the effective solution, and on the normalization of the sensitivity matrices, are also applicable for the initial concentration sensitivity matrix. In flames, the ratio of concentration sensitivity vs. space variable (distance) functions, where the sensitivities are calculated with respect to any two parameters, are found to be the similar for all concentrations and temperatures of the model. This phenomenon is called the self-similarity of sensitivity functions [70, 71]. The reason is that in flames the temperature is the dominant variable and any perturbation in the system affects the concentration-distance functions mainly through the changes induced in the temperature [72]. The solution of combustion models can not only represent concentration vs. time or concentration vs. distance functions, but also some features of the model. Such features could include the time to ignition or the flame velocity of freely propagating premixed flames. The sensitivity of these features can also be investigated. Flame velocity sensitivities (see Fig. 4.3) are frequently calculated and used for investigating reaction mechanisms. Representing sensitivity results with bars, also used in Fig. 4.3, was probably first introduced by Warnatz (see e.g. [73]). This kind of representation is very transparent and has become a kind of standard. It is well known that good flame velocity results do not mean that the chemical mechanism is correct. Similarly, lists of reactions with high flame velocity sensitivity do not show all the important parameters of a flame mechanism. Tuning of rate parameters having high-flame velocity sensitivities would alter significantly the flame propagation speed, but there are other parameters in the mechanism which have to be known precisely for the calculation of realistic concentration vs. distance profiles.

4.5.3 Principal components In many cases, one is interested in the effect of a parameter change on the concentrations of several species. This effect can be interpreted as the sensitivity of the value of an objective function, where the objective function is defined as the sum of squares of normalized differences in the original and perturbed concentrations. This sensitivity value, called the

319

Sensitivity and uncertainty analysis -0.2

0.0

0.2

0.4

0.6

0 2 + H -> OH + O H + CH 3 + M -> CH 4 + M CO + O H - » C 0 2 + H C H 3 + O H - * 1 C H 2 + H20 OH + 0 - > 0 2 + H C 0 2 + H -» CO + OH CH 3 + O -» CH 2 0 + H H 2 + OH -> H 2 0 + H HCO + M - > H + CO + M CH 4 + H -> CH 3 + H 2 O + H 2 0 -> 20H H 2 0 + H -> H2 + OH 20H -> O + HzO CH 4 + OH -> CH 3 + H 2 0 H2 + O -> OH + H H + OH + M -> H 2 0 + M 1

CH 2 + H 2 0 - > C H 3 + OH 0 2 + H + M -» H0 2 + M OH + H -> H 2 + O

0 2 + 1 C H 2 - > C O + OH + H

Fig. 4.3. Flame velocity sensitivities of stoichiometric, atmospheric methane flame. The sensitivities were calculated with program PREMIX [248] using the Leeds methane oxidation mechanism (version 1.3) [239].

overall sensitivity [74], can easily be calculated from the normalized local sensitivities,

J3> = S

d In ct

d\xvk,

(4.9)

where the summation runs over the indices of species present in the group investigated. The sensitivity matrix shows the effect of individual parameter changes on the calculated concentrations. In most applications the parameters may change simultaneously. Principal component analysis is a mathematical method that assesses the effect of simultaneously changing parameters on several outputs of a model [74]. The objective function of the principal component analysis is the square

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of the normalized deviation of concentrations, summed overall concentrations, and for all time points:

h=i i=i \ Ci(th) J

where A Q ( ^ ) is the deviation of concentration at time th as a result of parameter change Aa, Aa = a - a 0 , and a7 = In kj. This function can be approximated using the sensitivity matrix, e(a)«(Aa) r S r S(Aa),

(4.11)

where S = [S1? S2, . . . S^ . . . S/] r and S^ is the normalized sensitivity matrix for time th. Let U denote the matrix of normalized eigenvectors U7 of S r S such that U r U = I. The new set of parameters ty = U r a are called principal components. The objective function can also be expressed in terms of the principal components, m

e(a)~I\j(A%)\

(4.12)

7=1

where A^P = \JTAa and A7 are the eigenvalues of S r S. It is apparent from the last equation, that the eigenvectors of matrix STS reveal the related parameters, and the corresponding eigenvalues express the weight of these parameter groups. Principal component analysis (PCA) provides a way of summarizing the information contained in the sensitivity matrix, and at the same time revealing connections between parameters which would not be apparent when studying the raw sensitivity matrix. 4.5.4 Applications of local concentration sensitivities Local sensitivity information has numerous applications in uncertainty analysis, parameter estimation, experimental design, mechanism investigation and mechanism reduction. Uncertainty analysis, a quantitative study of the effect of parameter uncertainties on the solution of models, is


321

discussed in the next section. Although application of real uncertainty analysis have been rare in combustion, sensitivity analysis has been frequently applied to a qualitative assessment of which parameters have to be known more precisely to reproduce the experimental results more satisfactorily. Sensitivity analysis can be a useful ingredient of any parameter estimation procedure [74, 75]. Since parameter estimations are often based on the method of least-squares, principal component analysis is easily applied in conjunction with these techniques. If PC A indicates that a parameter is not influential, then this parameter should not be fitted if ill-defined parameter-estimation problems are to be avoided. In many cases, however, PC A reveals that only the quotient or product of some parameters can be determined. In such cases, one of the coupled parameters has to be fixed, and one must remember that the parameterestimation procedure provides the ratio of parameters rather than the real values. In a similar way, sensitivities and principal component analysis can be used for the design of experiments. Several ways of collecting experimental data (various selections of measured species and measurement times) can be tested on the computer and checks made on the information content regarding the parameters to be studied. The best variant can then be used in the real experiment. Sensitivity analysis can also be applied for the reduction of mechanisms. A reaction is redundant if its sensitivity is small with respect to each species of the reaction system at all times in the interval considered [76]. As discussed in the next section, it is sufficient if not all species are considered, but only the necessary species. Principal component analysis has also been applied to mechanism reduction [74, 77-79]. Although the application of sensitivity analysis (and principal component analysis) for mechanism reduction is entirely correct, mechanisms can be reduced in an easier way via the inspection of reaction rates. Methods using reaction rates are based on the comparison of rates of reactions having a similar role in the mechanism. For example, rates of reactions producing the same species can be compared. This approach requires attention because a reaction with a large rate can be unimportant, and a reaction with a small rate can be important, depending on their role in the mechanism. This problem does not emerge in mechanism reduction based on sensitivity analysis, because the concentration sensitivity methods inspect the effect of parameter changes rather than the origin of production rates of species. Thus,

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mechanism-reduction methods based on concentration sensitivities offer a substantially different way for mechanism reduction, than the usual methods based on the study of reaction rates. Sensitivity analysis can also be applied to reveal the relation of reaction steps to each other even in large and complex mechanisms. Reactions with high sensitivities have been considered to be identical to the rate-limiting steps [80]. Based on the definition of rate-limiting steps, the significance of these high sensitivities can be understood, and a more formal method can be elaborated. According to a proposed definition [81], a reaction step is rate-limiting if an increase in its rate coefficient causes a significant change in the overall reaction rate. This overall reaction rate is usually considered to be equal to the production (or consumption) rate of an important species. In many systems, a single overall reaction rate is not interpretable, and several rate-limiting steps, corresponding to several important species, can be found, for example, in the combustion of fuel mixtures. The sensitivity of production rates to the rate coefficients is given by the following equation: at d(dddkf) ¥ / J c " ^ ^ ( 0 - +- , at

dkj

dkj

. , y = l,...,m.

,„ _ (4.13)

This equation is identical to equation (4.6) which has been applied to the calculation of local sensitivities, but it can also be used for finding ratelimiting steps. The rate-limiting steps of the ith species can be identified (if they exist) by searching for very large elements in the ith row of the matrix d d In c dt dink

Interactions among parameters, as revealed by the principal component analysis, always have kinetic reasons. Such interacting parameter groups are formed by fast equilibrium conditions and the existence of quasisteady-state species. Thus, sensitivity and principal component analyses can be used for the detection of such kinetic features, although further methods for this purpose will be listed in following sections.


323

4.5.5 Uncertainty analysis The local concentration sensitivity matrix shows the effect of equal perturbations of parameters. This means equal unit perturbation in the units of each parameter for the case of the original sensitivity matrix, and unit fractional (percentage) perturbation in the case of the normed sensitivity matrix. The local sensitivity matrix itself does not carry any information on the uncertainty of parameters. The local sensitivities can be used for a rough assessment of the parameter uncertainties on model output. The variance of the model output (T2(Ci) can be estimated from the variance of the parameters (T2(kj) and the local sensitivities, m

2

*\cd=l^y\ki),

(4.14)

if the covariance of parameters is neglected. For a complete analysis see [82]. Local sensitivities have been used in a different way by Warnatz [4] for assessing the uncertainty of rate coefficients on the results of combustion modelling. The problem with the uncertainty analysis based on local sensitivities is that the first-order local sensitivities are linear approximations and valid only near the nominal values of parameters. The uncertainty of parameters in combustion models is usually very large and, therefore, the above calculated variances are first approximations only. The global sensitivity analysis methods address the problem of the precise calculation of the uncertainty of the model output as a result of uncertainties in parameters. These methods can handle any large uncertainty in the input parameters. More refined techniques include the determination of the extent of the uncertainty in output as a result of the uncertainty of each parameter. The first widely used global method was the Fourier Amplitude Sensitivity Test (FAST) (for a review see [83]). In the FAST method, all rate parameters were simultaneously perturbed by sine functions with incommensurate frequencies. Fourier analysis of the solution of the model provided the variance aj(t) of concentration /, and also the variance ojj{i) of ct arising from the uncertainty in the yth parameter. Their ratio

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a At)

was called partial variance and it was the basic measure of sensitivity in the FAST method. The FAST method proved to be very slow. It requires 1.2 x m2*5 simulations of the kinetic model, where m is the number of parameters. In case of a 50-parameter model this means 21,200 runs. This technique seems to be too expensive for detailed chemical models containing several hundred parameters. Therefore, usage of the FAST method has gradually vanished. Similar methods can be and have been elaborated based on other orthonormal perturbation functions, but all these methods require a huge number of simulations of the original model. Another family of global methods is based on selecting a large number of parameter combinations within their domain of uncertainty. The model is then simulated using these parameters and the output is processed. These methods differ from each other in the selection of parameter combinations and in the way of processing the model output. In the Monte Carlo method, a random number generator is used to select values of parameters according to their probability density function. The model is solved for each of the parameter combinations and the computed results are analyzed using standard statistical methods. The Monte Carlo method is conceptually and computationally very simple, but requires large numbers of simulations. There are other methods for the selection of the values of parameters which are more economical. In the Latin hypercube method [84, 85] the input parameter sets have to correspond to their probability density functions and are designed according to a Latin hypercube. The Latin hypercube arrangement ensures that the sample parameter values cover the whole uncertainty range and there are no chance correlations between any of the uncertain parameters. Following this method, an acceptable solution is obtained using only a few dozen simulations, but this method provides biased estimates of variances. Recently, Saltelli et al. [86-88] and Homma and Saltelli [89], have developed a new global sensitivity analysis method based on the sensitivity indices of Sobol [90]. This is a Quasi-Monte Carlo approach which, unlike the original Monte Carlo method, is effective because it uses quasi-random numbers [91]. The number of required simulations is about two orders of magnitudes greater


325

than the number of parameters. Saltelli and Hjorth [92] have applied this method to an atmospheric chemical system. All global methods require considerable computer time and these methods have not been applied to combustion problems. It is expected in the near future that they will be applied to the evaluation of spatially homogeneous combustion models. 4.5.6 Discussion For a long time the main topic of research in the area of sensitivity analysis was to find an accurate and effective method for the calculation of local concentration sensitivities. This question now seems to be settled, and the decoupled direct method (ddm) is generally considered the best numerical method. All the main combustion simulation packages such as CHEMKIN, LSENS, RUN1DL and FACSIMILE calculate sensitivities as well as the simulation results and, therefore, many publications contain sensitivity calculations. However, usually very little information is actually deduced from the sensitivity results. It is surprising that the application of principal component analysis is not widespread, since it is a simple postprocessing method which can be used to extract a lot of information from the sensitivities about the structure of the kinetic mechanism. Also, methods for parameter estimation should always be preceded by the principal component analysis of the concentration sensitivity matrix. Local concentration sensitivities can be used for mechanism investigation. As an example, it has been shown that rate-limiting steps can be identified on the basis of the time derivative of the sensitivity matrix. This approach to rate-limiting steps is in complete accordance with the classical definition and yet allows their identification for mechanisms of any size. The question remains open as to whether, or not, the application of concentration sensitivity analysis for mechanism reduction is feasible. Various methods for the study of reaction rates, to be discussed in the next section, are applicable for mechanism reduction, with the advantage that they use much less computer time. However, with the increasing power of computers, this point of view may become less significant, and the application of concentration sensitivities might become important as a principally different way of finding unimportant reactions. Application of formal uncertainty analysis to combustion systems has been very rare so far, and restricted to the utilization of local sensitivities.

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Possible reasons are the limited knowledge of the uncertainty of parameters, and the fact that global sensitivity methods are computationally very intensive. In the future, it is expected that both these limitations will be lifted and detailed uncertainty analyses will appear for combustion calculations. On the other hand, one of the main applications of sensitivity analysis has been to form a qualitative picture about which parameters should be known precisely in order to reproduce accurately a set of experimental observations.

4.6 MECHANISM REDUCTION WITHOUT TIME-SCALE ANALYSIS

The wording "full mechanism" and "reduced mechanism" suggests that there are full mechanisms which include all species present in the reactor, and that the model includes all significant reactions. It is easy to see that this is not true for several reasons. The quantum state of molecules and radicals is almost never considered in complex chemical models. Examples of rare exceptions are the distinctions between CH 2 (X 3 Bi) and CH 2 (a 1 A 1 ) in combustion and between 0( 3 P) and 0( 1 D) in atmospheric chemistry. All species in complex mechanisms are effectively lumped, since molecules are treated as single species with a Boltzman distribution of quantum states. In many cases isomers are also considered as identical species. Detailed kinetic models almost never include all species that are known to be present in the reactor. As an example, it is well known to everyone who has used a gas chromatograph with a flame-ionization detector, that ions are present in hydrocarbon flames. However, mechanisms for methane flames do not, in general, include the reactions of ions. The fact is that implicitly reduced mechanisms are used more often than not in modelling work; understanding how objectively reduced mechanisms can be generated is, therefore, of primary importance. Also, it is not true that full mechanisms are universally applicable. All detailed mechanisms have a domain of applicability in temperature, pressure and initial concentration space. They are constructed in such a way as to fulfil certain requirements. For example, they may be required to reproduce some experimental concentration or temperature profiles, or experimental features such as ignition times. Although the mechanisms claimed to be "general" or "full" are already tailored to a domain of assumed applications, they still usually contain many species and reactions,

Mechanism reduction without time-scale analysis

327

which are not necessary for a well-working mechanism within this domain. Therefore, a first step to mechanism reduction is to find a subset of the detailed mechanism containing fewer reactions and perhaps fewer species, which is applicable over the same range of conditions and reproduces the same experimental data. In other cases, the reduced mechanism can be a "tailored" mechanism according to special requirements. 4.6.1 Finding redundant species The primary stage in finding an appropriate submechanism is the determination of redundant species. Species of chemical mechanisms can be classified into three categories. The reproduction of the concentration profiles of important species is the aim of the modelling process. Important species might, for example, include reaction products or initial reactants. Other species, termed necessary species, have to be present in the model to enable the accurate reproduction of the concentration profiles of important species, temperature profiles or other important reaction features. The remaining species are redundant species. If redundant species are on the lefthand side of a reaction, this reaction can then be eliminated from the mechanism without any effect on the output of the model. If such a species is on the righthand side, then the reaction may or may not be deleted. Even if the reaction has to be retained, the redundant species can be deleted from the list of products of the reaction. Of course the latter can only be done if preservation of atoms or mass is not a formal requirement for the mechanism. Two methods [88] have been proposed for the identification of redundant species. The first one is based on the fact that if a species has no consuming reactions, a change in its concentration has no influence on the concentration of other species. Therefore, a species is certainly redundant if the elimination of all of its consuming reactions does not cause significant deviation from the solution of the full model on inspecting the important species and features. However, there are situations where a species can be redundant even if the elimination of its consuming reactions causes significant changes, for example when it is formed by fast reversible reactions. In this case, the species, not found to be redundant in the first test, has to be reinvestigated by eliminating simultaneously its fast producing and consuming reactions. This second test may reveal further redundant species.

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An alternative method is based on the investigation of the Jacobian of the kinetic system of odes, J = df/dc. A species may be considered redundant if its concentration change has no significant effect on the rate of production of important species. An element of the normed Jacobian (dln/;)/(dlnc ; ) shows the fractional change of the rate of production of species i caused by the fractional change of the concentration of species ;'. The influence of the change of the concentration of species i on the rate of production of an 7V-membered group of important species can be taken into account by the sum of squares of normalized Jacobian elements,

Bi

= £ (iMA 2 . n=l \d In C;/

(4.i6)

The higher the Bt value of a species, the greater is its direct effect on important species. However, there are necessary species which influence the concentration of important species, not through direct coupling, but by influencing another necessary species which is so coupled. Thus, the group of necessary species has to be identified by an iterative procedure. The highest ranked species on the basis of the Bt values are also considered in the summation for each subsequent iteration, providing new Bt values. This procedure is repeated in a cycle until no new species are admitted into the Af-membered group. The values of the vector B converge and the redundant species are those which are not included in the summation after the final iteration. This procedure is then repeated at several reaction times. These calculations provide a hint for the selection of a group of redundant species which has to be validated via comparison of the full mechanism and the obtained truncated mechanism. The first method requires several simulations of different reduced models as each species is eliminated. The number of simulations is of the order of the number of species. Using this method the effect of species elimination on important features can be obtained directly in a quantitative way. The second method requires a single simulation of the original model where the Jacobian is calculated from the concentrations at several reaction times. Therefore, this method is more efficient and can be applied automatically, but cannot be used to investigate the effects on important features such as ignition time.


329

4.6.2 Finding redundant reactions Once the necessary species have been found, the second step in reducing a mechanism is the elimination of its non-important reactions. A classical and reliable method is the comparison of the contribution of reaction steps to the production rate of necessary species. A description of this method - without the pre-selection of redundant species - is given by Warnatz [4, 93]. A more recent application of this technique for methane flames is presented in [94]. According to this method a reaction is redundant if its contribution to the production rate of each necessary species is small. This rule sounds simple and obvious but there are, however, several drawbacks and pitfalls. First, the reaction contributions have to be considered at several reaction times (or at several heights in the case of steady flames). Second, all reaction contributions to each necessary species have to be considered, and it is not easy to analyze such huge matrices. The threshold of unimportance will vary from time to time, and from species to species. Applying a uniform threshold for each time and species (e.g., minimum 5% contribution) can either result in redundant reactions being left in the scheme, or an over-simplified mechanism. An alternative technique for the reduction of mechanisms, using reaction rates, is based on the sensitivity of production rates to changes in rate parameters [95]. If the parameters are the rate constants and the reactions are considered irreversible, then the normed rate sensitivities have the following form:

This equation shows that an element of the normed rate sensitivity matrix is the ratio of the rate of formation or consumption of species i in reaction y, to the production rate of species i. It is possible to consider the effect of each parameter on several production rates simultaneously using a leastsquares objective function. This approach leads to the application of the following overall sensitivity type measure:

330

Mathematical tools

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Here, the summation is over the indices of all necessary species. This measure gives a rank order of reaction importances in the system at each of the reaction times considered. A more sophisticated way for the study of the rate sensitivity matrix F is based on the principal component analysis of the matrix. In a very similar way to the case of principal component analysis of the concentration sensitivity matrix, the eigenvalue-eigenvector analysis of the matrix F r F quantifies the importance of reaction groups for the set of important and necessary species [95]. The parameter groups are defined by the eigenvectors, with significant members having high eigenvector elements. The importance of parameter groups is given by the relative size of the eigenvalues. Reactions, which are significant members of a parameter group having high importance, are the important reactions in the mechanism. The method of the principal component analysis of the rate sensitivity matrix with a previous preselection of necessary species is a relatively simple and effective way for finding a subset of a large reaction mechanism that produces very similar simulation results for the important concentration profiles and reaction features. This method has an advantage over concentration sensitivity methods, in that the log-normalized rate sensitivity matrix depends algebraically on reaction rates and can be easily computed. For large mechanisms this could provide considerable time savings for the reduction process. This method has been applied for mechanism reduction to several reaction schemes [96-102]. 4.6.3 Sensitivity of temperature rates Often, in combustion modelling, the important feature is not a species concentration but the time to ignition, the maximum temperature rise or simply the critical ignition temperature. Frequently, temperature is a measured quantity in an experiment since measuring species concentrations is more difficult, especially for radicals. Therefore, the sensitivity of the rate of change of temperature to a change in the rate parameters is of importance. If the reduced model is required only to produce accurate temperature profiles then the temperature sensitivities become useful. This does not mean, however, that in this case mechanism reduction can be based on temperature rate sensitivities alone, and overall rate sensitivities should also be considered.


331

The normalized temperature rate sensitivity is given by ain(dr/dQ= -qfRf d In kj Cp(dT/dt)' In the example which follows it is calculated separately from the usual rate sensitivities and is a simple one-dimensional array. Another approach would be to treat temperature as though it were a concentration, and to include it in the eigenvector/eigenvalue analysis. 4.6.4 Application to hydrogen oxidation in a flow system We now introduce an example mechanism which will serve to illustrate some of the techniques discussed in Section 4.6. We have chosen for this purpose a hydrogen oxidation system in continuously stirred batch and flow reactors [99]. Other similar examples can be found in the literature such as the application of the above techniques to a propane pyrolysis mechanism [103]. Although one of the simplest of combustion mechanisms, the hydrogen system contains all the main types of elementary steps and dynamic behaviour present in larger combustion mechanisms, and the elementary reactions describing the process and their rate data are well established. The H 2 + 0 2 mechanism shown in Table 4.3, comprising 47 reactions involving 9 species [104], will be studied in the region of the second explosion limit, where interesting oscillatory behaviour is found in a CSTR. An example of such a computed oscillatory ignition trace is shown in Fig. 4.4. The dynamics of the H 2 + 0 2 reaction in a CSTR are discussed in detail in Chapter 5. Specifically, a stoichiometric reaction mixture was studied at a pressure of 20 Torr and a residence time of 8 s. Both isothermal and non-isothermal models were considered for a range of temperatures between 500 and 2311 K. The third body M is assumed to be made up from the molecular species H 2 , 0 2 and H 2 0 with relative efficiencies of 1:0.4:6, respectively, for all third-body reactions [105]. Although the chemistry is derived from the original Dougherty and Rabitz scheme, the rate data were updated and, where possible, obtained from the CEC evaluation tables [26]. The sources for other reaction rate data are shown in Table 4.3. Oscillating ignition reactions constitute a particularly stringent test of mechanism

TABLE 4.3 Full reaction mechanism for hydrogen oxidation including rate data and sources

Reaction no

Reaction

~/(cm molecule-')'-" ~

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0 + HOz -+ 0 2 + OH Hz+Oz-+H+HOz HZ + OH-H + HzO 0 2 + H-+ OH + 0 Hz + O-+H + OH 0 2 + H + M + M + HOz H + WALL 0- WALL OH -+ WALL H02 + WALL HOZ + HOZ -+ HzOz + 0 2 0 2 + OH + 0 + HOZ H+HOz+20H H + HOz-+HzO + 0 HOz + Hz+HzOz + H H02+H+Hz+Oz Hz02 + M + 2 0 H + M H202+H-+HZ+HOZ HZ02 + H + H 2 0 + OH HZ02 + OH -+ HZO + HOz HzOz + 0 + OH + HOz

2.90 x 2.40 X 1.70 x 3.30 x 8.50 X 1.37 x 75 75 75 75 3.10 x 3.70 X 2.80 X 5.00 x 5.00 X 7.10 X 8.00 x 2.80 X 1.70 x 1.30 x 1.10 x

lo-" lo-'' lo-'' lo-''

lo-''

lo-'' lo-'' lo-" lo-'' 10-l' lo-' lo-'' lo-" lo-''

'-'

(E/R)/ K

n

0 0 1.60 0 2.67 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

-200 28500 1660 8460 3160 -500 0 0 0 0 755 26500 440 866 13100 710 22900 1890 1800 670 2000

Reference

22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

HZO + 0 + 20H H+H,O+OH+H, OH + O+ H + 0 2 OH + H + 0 + H2 20H+O+H20 H + OH + M+H20 + M H + 0 + M-+ OH + M H + H + M+ H2 + M OH+OH+M+Hz02+M 0 + 0 + M +0 2 + M HOz + OH+H2O + 0 2 H20 + M + H + OH + M H2 + M+ H + H + M 0 2 + M+ 0 + 0 + M O2 + 0 + M+ O3 + M 0 3 + M+02 f 0 + M O3 + 0 + 2 0 2 O3 + H + O2 + OH 03+OH+H02+02 03+HO2+202+OH HOz+ M + O + OH + M H20 + HO2 + Hz02 + OH HOz + M + 0 2 + H + M 20H-+HO,+H OH + M+ 0 + H + M H202 + 0 2 + 2H02

7.60 x 7.50 X 7.50 x 8.10 x 2.50 x 3.90 x 1.30 x 8.30 x 1.60 x 2.20 x 4.80 X 3.70 x 1.46 x 3.01 X 2.15 x 5.80 X 8.00 X 2.15 x 1.60 X 1.10 x 5.30 X 4.70 x 2.00 x 2.00 x 4.00 x 9.00 x

10-lS

lo-'' lop2' lo-''

lo-" lo-' 10-~

lo-'* 10-l~

lo-''

lo-" 10-~ lo-" 10-~ lo-''

1.30 1.6 -0.5 2.8 1.14 -2.0 0 0 -3.0 -1.5 0 0 0 -1.0 0 0 0 0 0 0 0 0 -1.18 0 0 0

8605 9270 30 1950 50 0 0 0 0 0 -250 52900 48300 60643 345 0 2060 345 940 500 0 16500 24363 20200 5 m 2 m

w w w

334

Mathematical tools 2000-4 1800

Ch. 4

I

I

1600-J T/K

1400 H

1200 - j 1000 H

800 J 0

' 5

1 10

' 15

20

25

30

t/S

Fig. 4.4. An example temperature trace from the full 47-step scheme showing a maximum temperature rise of around 1400 K. Pressure = 20 Torr, Ta = 790 K.

reduction procedures and, therefore, provide an ideal vehicle for the illustration of the systematic approaches outlined above. (/) Redundant species The first stage of any reduction procedure should always be to establish which are the necessary species in the reaction mechanism over the range of conditions to be considered. In order to carry out the redundant species analysis, decisions must first be made about the important species and features which the reduced model must be able to reproduce accurately. In this example the important species were chosen as the primary reactants H 2 and 0 2 , and the product H 2 0 . The analysis described in Section 4.6.1 was then applied over a range of ambient temperatures and reaction times for both the isothermal and non-isothermal model. Calculation of the Bt values revealed H 2 0 2 and 0 3 to be redundant species for both models and at all reaction conditions tested. Table 4.4 shows examples of redundant species calculations from the full non-isothermal scheme at differing parts of the oscillatory trace

335


TABLE 4.4 Estimated effect of species for the rate of change of necessary species: T = 2311 K, T = 818 K. A " - " indicates species below the chosen threshold which are therefore redundant 2311 K

818 K

species 1 2 3 4 5 6 7 8 9

H H2 H20 OH O

o2

H02

o3

H202

+ + + + + + +

-

lOg B;

species

9.46 9.45 9.32 9.25 8.50 6.64 3.16 -3.66 -4.65

OH H2 H H20 O

o2

H02

o3

H202

log Bi + + + + + +

+ -

4.43 4.25 4.21 3.69 3.36 3.27 2.89 -3.87 -4.34

and at two different temperatures. They show the effect, on a logarithmic scale, of each species on the rate of production of important and necessary species for the final iteration (i.e., when convergence has been achieved) based on equation (4.16). H 2 0 2 and 0 3 have by far the lowest Bt values at all conditions. Their consuming reactions have the smallest effect on the rates of production of H 2 , 0 2 and H 2 0 both directly and via the necessary species H, O, H 0 2 and OH at these conditions. It is also worth noting that H 0 2 has a Bt value below the other remaining radicals at all conditions tested. (ii) Redundant reactions From the redundant species analysis it is clear that all reactions which consume H 2 0 2 and 0 3 are redundant and can be removed automatically from the mechanism. In order to identify other redundant reactions the techniques of rate sensitivity analysis coupled with a principal component analysis of the resulting matrix can be used. The principal component analysis of the rate sensitivity matrix containing only the remaining important and necessary species will reveal the important reactions leading to reduced mechanisms applicable at various ambient temperatures. In principle it may be possible to produce a reduced scheme which models non-isothermal behaviour from analysis carried out on an isothermal model. An isothermal system is easier to model since thermodynamic and heat-transfer properties can be excluded from the calculations. However,

336

Mathematical tools

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it is important to make sure that a simulation of the full non-isothermal scheme is carried out first in order to establish the temperature extremes under which the isothermal scheme must be studied. Both isothermal and non-isothermal systems will be illustrated here in order to compare their reduced mechanisms. The isothermal system is examined first, at a series of temperatures between the extremes found in the non-isothermal model. From Fig. 4.4 we can see that the extremes of temperature found during an ignition are approximately 700 and 2000 K. The maximum temperature rise is actually slightly higher than that shown in Fig. 4.4 but can only be captured using very small time-steps due to the steepness of the ignition peak. Isothermal model for low temperatures At a low temperature of 700 K, where the overall rate of reaction remains small, a subset of only 6 reactions involving the primary branching and termination routes are selected by the principal components. Selecting only the necessary species as part of the objective function, reactions 2, 3, 4, 7, 8 and 9 are chosen as important. However, if all species are included in the objective function then reactions 11, 36 and 37 are also selected by the principal component analysis, even though their removal from the scheme has little effect on the concentrations of the important species. This illustrates the importance of using the redundant species analysis prior to calculating rate sensitivities in choosing the optimum reduced scheme. Reactions 36 and 37 are fast-reversible reactions of 0 3 which has a low concentration. The present example demonstrates that in many cases such coupled reaction sets can be automatically removed from the model via the identification of redundant species. Isothermal model for high temperatures and oscillatory region At a higher temperature of 1000 K further reactions become significant. Considering the overall sensitivities at reaction time t = 10~2 s, Table 4.5 shows that reaction 13, H + H02—> 2 0 H becomes of major significance at high temperatures in terms of OH production, and has a higher sensitivity coefficient than both reactions 4 and 5. The OH radical now becomes more significant as is illustrated by the inclusion of reactions 9 and 22-27 in the high-temperature mechanism, all of which involve OH. Although the third-body reaction H + 0 2 + M —> H 0 2 + M (reaction 6) remains the main termination route, other termination reactions become important at


337

TABLE 4.5 Ordering of overall rate sensitivities for isothermal scheme at t= 1.0 x 10 - 2 s and 1000, 2000 K 1000 K

2000 K

Order

Reaction number

Overall sensitivity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

6 13 39 16 14 42 19 11 3 1 23 24 4 6 43

1.37 7.73 2.31 2.90 1.05 2.56 2.14 1.48 1.26 1.14 9.25 6.86 6.16 5.51 5.08

x x x x x x x x x x x x x x x

108 107 107 106 106 104 104 104 104 104 103 103 103 103 103

Reaction number

Overall sensitivity

23 3 26 22 6 13 5 25 24 4 16 45 14 2 42

1.13 1.08 8.74 7.72 3.14 2.21 7.64 4.14 1.85 1.85 1.08 8.68 4.60 1.49 5.38

x x x x x x x x x x x x x x x

108 108 106 106 106 106 105 105 105 105 105 104 104 104 103

high temperatures, e.g., reaction 16, as well as other radical-radical reactions, such as OH + O —> H + 0 2 (reaction 24), which leads to a reduction in the concentration of the radical pool. At 2000 K, reactions 14, 28, 29, 35 and 45 were also selected, leading to a total of 20 reactions over all ambient temperature conditions. Non-isothermal behaviour If the rate sensitivity analysis is carried out over a non-isothermal oscillatory trace, then it is possible to see how certain reactions, in particular radical termination reactions and the reactions of water, become important at high temperatures and high conversions. The highest number of reactions is selected at the peak maximum where the temperature rises above 2000 K. All reactions required by the non-isothermal analysis, reactions 2-10, 13, 14, 16, 22-29, 35 and 45, were featured in one of the isothermal mechanisms. Therefore, it is possible to produce a reduced scheme for a non-isothermal simulation from an isothermal analysis provided the full

338

Mathematical tools

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temperature range is covered. Figure 4.5 compares the 47-step full model with the 22-step reduced model, illustrating the excellent agreement. Further reduction can be be achieved, however, by studying the overall and temperature sensitivities for the non-isothermal scheme. The temperature sensitivities calculated from equation (4.19) are presented for t = 3.5 x 10~2 s and T = 2311 K, in Table 4.6 along side the overall rate sensitivities (equation (4.18)). The results, as expected, are not identical to the rate sensitivities. However, the reactions selected by the overall rate sensitivities remain at the top of the temperature sensitivity order. There are exceptions of course, such as the wall reactions, which in our model have no direct effect on the temperature and so such an analysis cannot be useful in assessing their significance. By choosing a suitable threshold and taking the sum of all reactions which have a temperature sensitivity higher than this threshold at the chosen time points, we should arrive at a scheme close to that obtained from the principal component analysis. In practice, however, the threshold value is not easily chosen since no natural cut appears in the temperature sensitivity order. Therefore, temperature rate sensitivities are more useful in assessing the general importance of reactions for temperature changes rather than providing a systematic reduction method, and are recommended as a useful tool but not as a single method of model reduction. By excluding reactions with overall and temperature sensitivities, which are significant only over small reaction periods, a final mechanism of 16 reactions involving 7 species is achieved which reproduces the oscillatory behaviour over a large temperature range fairly accurately. The reduced mechanism is as follows:

(2)

H2 + 0 2 ^ H + H 0 2

(3)

H2 + O H ^ H + H 2 0

(4)

02 + H ^ O H + 0

(5)

H 2 + O -> H + OH

(6)

02 + H + M ^ M + H02

(7)

H-^WALL

(8)

O -> WALL


339

TABLE 4.6 Comparison of overall rate sensitivity and temperature sensitivity order for t = 3.5 x 10~2 s, 7=2311 K Order

Reaction number

Overall sensitivity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

3 5 4 23 22 37 6 10 9 7 8 17 2 35 43 39 19 36 20 14

6.94 1.28 4.71 1.72 2.02 9.66 9.81 7.75 1.52 1.09 9.30 5.22 1.18 2.64 1.15 2.64 2.34 2.80 3.48 1.02

x x x x x x x x x x x x

107 107 106 106 105 104 103 103 103 103 102 102

x x x x x x x

10" 1 10" 1 10" 2 10~2 10~4 10" 5 10 - 6

Reaction number

Temperature sensitivity

6 3 13 23 16 4 14 22 5 42 1 24 27 32 39 26 29 11 19 28

0.43 0.41 0.22 0.11 0.61 0.48 0.34 0.16 0.42 0.41 0.29 0.90 0.58 0.40 0.34 0.19 0.15 0.52 0.21 0.94

(9)

OH -> WALL

(10)

H02^WALL

(13)

H + H02-^20H

(16)

H + H 0 2 ^ H 2 + 02

(22)

H20 + 0 ^ 2 0 H

(23)

H + H 2 0 ^ 0 + H + H2

(24)

OH + O ^ H + 0 2

(25)

OH + H - ^ 0 + H 2

(26)

2 0 H ^ O + H20

x x x x x x x x x x x x x x x x

10" 1 10" 1 10" 1 10" 1 10" 2 10" 2 10" 2 10" 3 10" 3 10" 3 10" 3 10" 3 10 - 3 10~4 10" 4 10 - 5

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Mathematical tools

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15.5 - i

i 6

r 8

- 1

10

I

1

1

I

I

12

14

16

18

20

t/S 17.5

^ E

17

if 3

16.5 H

g

16.0

X S

15.5

Y >Z in Fig. 4.10 illustrates this case, where kx = 1 s and k2 = 10 s _ 1 for the solid line, and k2 = 20 s - 1 for the dashed line. Y can be considered a QSSA species whose real concentration decays with time. The QSSA concentration, therefore, never reaches the real concentration but remains above it since dy/dt is negative. The final distance depends on the size of k2 and, hence, on the lifetime of Y. This distance

377

Reduction based on the investigation of time-scales

1.4-

1.2H

>

1.0-

0.8 H

0.6 H i

I

0.0

0.1

0.3

0.2

0.4

0.5

t/S

Fig. 4.9. Relaxation to a steady-state: X^ Y, kf= kb = 10 s x: (a) A steady-state solution X=Y=1; ( b ) Z o = 0.5, Y0 = 1.5; (c) X0 = 1.5, Y0 = 0.5.

is the error induced by the application of the QSSA and is governed by the speed of the quasi-stationary point. (ii) Errors for groups of QSSA species In reality the QSSA is rarely applied to a single species, and error calculations must take into account the interactions between QSSA species. It is important, therefore, to use equation (4.91) only as a first guide to the selection of QSSA species and then to calculate the error resulting from the application of the QSSA to a group of species with small single errors using the following equation in matrix form: dc (2) = J(22)Ac*, At

(4.95)

where Acg represents the group error. This can be rewritten as a system of linear equations,

378

Mathematical tools

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1.4-J \b

1.2H

\\

1.0 J V

\ N

X

o.a-

\ \

/ S~ N^> ! s \ v

// if

0.6-

\ V X

1C 0.4-

u.c

i

0.0

0.2

i 0.4

l

I

l 0.6

0.8

t.o

i T.2

l 1.4

l T.6

t/S

Y x - for &i = 1 s 1 and k2 = 10 Fig. 4.10. Relaxation to a quasi-steady-state: X—^->. Y—^Z _1 _1 ) and k2 = 20 s ( — ) : (a) QSSA solution, X0 = 10 ( ), 20 ( — ) , Y0 = 1.0; s ( (b) Z o = 10 ( ), 20 ( — ) , y 0 = 1.5; (c) X0 = 10 ( -), 20 ( — ) , Y0 = 0.5. The perturbations of a shorter lifetime species relax more quickly and its trajectory becomes closer to the QSSA solution.

J;; CU

J,; kyî

(4.96)

where / and k run over all QSSA species in the group. Therefore, the calculation of the group errors requires the solution of a coupled set of linear algebraic equations in Acf. If the group errors remain small, then the chosen group of species can be considered as QSSA species. The error of the quasi steady-state approximation in spatially distributed systems has recently been studied by Yannacopoulos et al. [160]. It has been shown qualitatively that QSSA errors, which might decay quickly in homogeneous systems, can readily propagate in reactive flow systems so that the careful selection of QSSA species is very important. A quantitative analysis of QSSA errors has not yet been carried out for spatially distributed systems but would be a useful development.


379

(Hi) Practical application of the QSSA The solution of equation (4.90) is the crucial part of the application of the QSSA and can be achieved in a number of ways. Direct application. If a QSSA species has only one main consuming reaction, then it can be easily removed from the scheme by lumping its forward reactions with its consuming reaction. For example, the radical (CH 3 ) 3 CCH 2 can be removed from the following reaction sequence at high temperatures since all other consuming reactions, apart from its decomposition, can be neglected. (CH3)4C + O H ^ (CH 3 ) 3 CCH 2 + H 2 0 (CH 3 ) 3 CCH 2 ^ (CH 3 ) 2 CCH 2 + CH 3 The two reactions can be combined to give a single reaction, (CH3)4C + O H ^ (CH 3 ) 2 CCH 2 + CH 3 + H 2 0 Numerical application. The second way of applying the approximation is the numerical solution of the differential + algebraic equation (DAE) system given in equations (4.89) and (4.90). Many numerical codes can solve general coupled sets of equations (e.g., DASSL [57]), and this is an ideal way to test the reduced system to confirm the choice of QSSA species made using the calculation of instantaneous errors and lifetimes. In terms of reduction in computer simulation time, these general methods are not the best approach, although in certain cases they may lead to savings because of the reduction in stiffness of the model. Special classes of integrators based on the QSSA approach have been developed in the field of atmospheric chemistry [161, 162]. These provide more significant savings than general DAE solvers because they specifically use the form of chemical rate equations in the design of the integration algorithm. Such methods could have applications in combustion chemistry. More substantial savings can be achieved through the explicit algebraic solution of the QSSA equations, since this leads to the removal of species from the differential equation system. Generation of explicit algebraic expressions. In many cases explicit algebraic

380

Mathematical tools

Ch. 4

QSSA expressions can be generated by hand. This is often difficult for large mechanisms since the original implicit equations can be too complex to allow the analytical solution of the QSSA equations. However, the use of algebraic manipulation packages such as REDUCE, MAPLE and MATHEMATICA may allow the solution of equation (4.90) to form explicit expressions for the concentrations of the QSSA species in terms of the other species and rate parameters. These expressions can be then substituted back into the differential equations (4.89) for the non-QSSA species to produce a reduced set of odes. Finding such expressions involves the algebraic solution of coupled sets of equations. Some systems may involve radical-radical type reactions and so non-linear equations can arise, which leads to certain limitations. The algebraic solution of sets of nonlinear equations is not always possible and so, unless linear approximations can be made, QSSA species which are coupled in this way cannot be included. Future work could involve finding systematic ways of truncating the steady-state expressions to a linear form so that the full number of species could be included in the approximation. Partial equilibrium assumptions. At present, the only systematic method for the truncation of QSSA expressions is the use of partial equilibrium assumptions which may be applied in the case of fast reversible reactions. If a steady-state species has many reactions, but two of these are fast and reversible with rates much higher than the other reactions involved, e.g.,

then the concentration of the steady-state species can be described by the expression [B] = Keq[A], or for a second-order reaction of the type, A +

+ D.

the expression is of the form

(4.97)

Reduction based on the investigation of time-scales [C] =

KÂ][B]

381 (49g)

These partial equilibrium assumptions are equivalent to truncating the QSSA expression after the largest term. The study of relative reaction rates is, therefore, important in order to find such fast reversible reactions. Inner iteration. In some cases explicit expressions can be produced for most QSSA species, but for some other species the QSSA equations are still implicit coupled non-linear expressions. These equations can be solved separately and the concentration of QSSA species calculated by an iteration cycle. This so-called inner iteration method has proved to be a successful technique for this purpose. See Chapter 6 in [163] for an example of its application in methane and ethylene flames. Peters' method for producing global reaction steps. The work of Peters and coworkers illustrates a method for producing global mechanisms using the QSSA. They have produced a number of mechanisms for the description of the combustion of several fuels. For example, the global mechanism for methane flames, deduced by Peters [164,165], consists of the following steps: CH 4 + 2H + H 2 0 ^ CO + 4H 2 CO + H 2 0 ^ C 0 2 + H 2 H+H+

2

+M

0 2 + 3H2 ^ 2H + 2H 2 0 The method used by Peters and followers starts from an already simplified reaction mechanism. The concentrations of most radicals are determined by assuming quasi-steady-state approximations and partial equilibrium. Main reaction chains are identified and reactions of the chains are summed-up forming global reactions. The rates of the global reactions are derived by rate-limiting step assumptions and using the algebraic expressions for radical concentrations. As illustrated by the above example, these global reactions may not have mass action form as do elementary reaction steps, and in some cases can exhibit non-integer stoichiometrics.

382

Mathematical tools

Ch. 4

In the original procedure of Peters, the starting simplified mechanism was obtained by chemical experience, and the QSSA and partial equilibrium assumptions were verified only a posteriori by comparing simulation results of the starting and reduced mechanisms. This method of Peters has been criticized by Warnatz [4] (see, e.g., p. 567), who stated that the starting mechanisms are over simplified (leaving out C2-chemistry for methane combustion for example). Warnatz objected that the quasi-steady-state and partial equilibrium assumptions are employed mainly on an intuitive basis. He also drew attention to the fact that it is not necessary to use the same reduced mechanism for the description of different systems. Clearly, it is easy to improve this method by using a systematically reduced starting mechanism and verifying a priori the QSSA species and equilibrium relations. Recently, several works have been published, where Peters' method is used together with a mathematically backed selection of QSSA species [131, 166, 167]. In spite of criticism, the Peters method has proved to be very successful, and in a series of articles, Peters and others have produced reduced mechanisms for a range of practically important fuels [163, 168-186]. These mechanisms are applicable in premixed and diffusion laminar flames at both fuel-rich and fuel-lean conditions and there are some applications in turbulent flames as well. Most of the applications of this method are related to flames; in the present context it is particularly interesting to examine cases where the method has been applied to the reduction of ignition mechanisms. In a recent paper by Mixller et al. [185], a kinetic mechanism of 1011 elementary reactions and 171 chemical species for n-heptane ignition was analyzed and reduced to four global steps with adjusted rate coefficients to describe ignition at pressures around 40 atm. Two of these steps account for the high-temperature branch and the other two for the low-temperature branch of the ignition mechanism. The model results were compared to the adiabatic ignition delay times obtained in a shock tube. This model was criticized by Griffiths [187], who pointed out that the global mechanism is incapable of reproducing ignition delays which exhibit a negative temperature dependence. The reason is that the proposed chemical structure is not appropriate for the reproduction of such qualitative features, since the prediction of ignition rests exclusively on thermal feedback. He also suggested an extension of the chemical model to overcome this difficulty. Trevino and Mendez [131] have applied the global reaction method for


383

the reduction of the ignition mechanism of methane. The starting elementary mechanism contained 231 reactions while the reduced mechanism contained six global steps. The QSSA species were selected on the basis of a CSP analysis. There was a good agreement between the results obtained from the starting and the reduced mechanisms for initial temperatures higher than 1300 K. Mulholland et al. [188] derived reduced ignition mechanisms for four simple hydrocarbons based on perfectly stirred reactor (PSR) calculations. A minimum reaction mechanism was produced by successive elimination of minor reaction pathways. The global models were generated on the basis of quasi-steady-state and rate-limiting step assumptions. Very good agreement was found between the ignition times calculated by the detailed mechanisms and the global mechanisms over the temperature range 800 to 1600 K. In this temperature range, the ignition times span six orders of magnitude. (iv) Application of the QSSA to the hydrogen oxidation example The method of rate sensitivities has enabled the removal of all redundant species and reactions from the original mechanism (Section 4.6.4). We now turn to time-scale methods [99], in particular the application of the QSSA, to condense the remaining equations to the smallest number of species. The most important aspect of applying the QSSA is the choice of QSSA species. Calculations of QSSA errors and species lifetimes (equation (4.92)) will reveal steady-state species. In an oscillatory reaction these will obviously change in time through the ignition cycle and Fig. 4.11 shows some examples of species lifetimes for the isothermal system in the oscillatory region at T = 790 K. As expected, the lifetime varies periodically with time in the same way as the species concentrations. However, for some species, the lifetime and instantaneous errors always remain low. For example, OH and O have lifetimes which remain below 10 - 4 s throughout the oscillation. Since these times are significantly less than the characteristic times of the overall reaction, the application of the QSSA should be possible for these species. Previously, several authors thought that the QSSA can be applied to all radical intermediates in a model partly because of their low concentrations. Therefore, H and H 0 2 are also possible candidates and have fairly short lifetimes. Results using the 16-step scheme generated in Section 4.6.5 show that the QSSA is applicable to species O, OH and H 0 2 with no significant

384

Mathematical tools 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 -3.0 -35 -40

-

Ch. 4

(a) r

i

i

j

i

l

20

30

40

50

60

70

"I 10

I

l 80

90

I 100

-us •5 0 -5 5 -60

-S5 CD

'o

O Q.

o

1 0 0500-05-1 0 -1 5 -2.0•2.5 -3.0 -3.5 -40 • -4 5 • -5.0•5 5 -6 0 -

i o 0.5 00 -0 5 -1 0 •1 5 •2 0

-

(b) 10

30

40

50

60

70

80

90

100

(c) '.0

1 0 — 05 -

20

(d)

20

30

40

50

60

70

SO

90

100

t/s

-05 — -10 — • 1 5 •2 0 — .•3 0 •3 5 40 J 5 50 •5 5 50

-

Fig. 4.11. Log-species lifetime for radicals showing OH to be the most likely QSSA species: (a) H; (b) OH; (c) O; and (d) H 0 2 .


385

18-,

E

if 3 O

o E

o

Fig. 4.12. Example of oscillatory traces of OH and H 2 , for the 16-step reduced scheme with non-QSSA variables vs schemes where the QSSA has been applied to OH, O and H 0 2 .

change in the behaviour of the system. Figure 4.12 compares the concentrations of H 2 , and one of the QSSA species OH, with and without the application of the QSSA to OH, O and H 0 2 . Note that the periodically changing OH can still be described by an algebraic function of the nonQSSA species. The QSSA does not apply to all radicals, however, since its application to H leads to negative concentrations in H and to concentrations of H 2 and 0 2 which exceed their initial concentrations. It is, surprising, perhaps, that H 0 2 is a successful QSSA species and H is not, given that H has a lower average lifetime. The reason is that the errors inherent in the application of the quasi-steady-state to H spread quickly to the nonQSSA species causing drastic changes in the dynamics of the system. One measure of the spread of such errors is the initial concentration sensitivity, i.e., the effect on species concentration Ct at a time t2 of a change in species concentration C7 at a time t\ (see equation (4.8) in Section 4.5). Figure 4.13 shows examples of such initial concentration sensitivities for the radical species H, OH and H 0 2 during the ignition period of the

386

Ch. 4

Mathematical tools 20-i

15-1 CO

0)

7""

c

OH CO

O) O

H02 10

~i

15 J i

i

-4.0

-3.5

;

-3.0

.

-2.5

i

-2.0

:

-1.5

i

-1.0

l

-0.5

0.0

Log (t/s) Fig. 4.13. Initial concentration sensitivities [dCi(t2)]/[dCi(ti)]. for radicals H, OH and H 0 2 during the first ignition stages of the reaction showing that the system is far more sensitive to initial errors in H concentration than to errors in the concentration of H 0 2 .

reaction. The effect can be positive or negative (a perturbation can result in an increase or decrease of the concentration) but for clarity the log of the absolute sensitivity is used. The sensitivity to both H and OH is orders of magnitude higher than to H 0 2 showing that any spread of errors will be much faster for these species. Because OH has a shorter lifetime, the QSSA errors propagated are much smaller than in the case of H. H 0 2 has a lifetime comparable to that of H but has a low sensitivity which limits the propagation of errors. Therefore, we have some indication as to why H is the only unsuccessful radical QSSA species. Producing a skeleton scheme Although the QSSA can be applied to the 16-step scheme, its application leads to fairly complicated algebraic expressions and, therefore, it is difficult to generate a much reduced global mechanism. In this section we examine how far the scheme can be further reduced while retaining the


387

qualitative features of the full model. We do not seek to reproduce the period of the oscillations but to simulate oscillatory behaviour and ignition delay times. The resulting reduced scheme requires the competition of a set of overall branching reactions 3-5 with termination reactions 6 and 7 as follows: (2)

H2 + 0 2 - * H + H 0 2

(3)

H2 + O H ^ H + H 2 0

(4)

02 + H-*OH + 0

(5)

H 2 + O ^ H + OH

(6)

0 2 + H + M-*M + H02

(7)

H^WALL

and is identical to the low-temperature scheme. Note that the wall termination reaction has to be present in order to produce oscillatory behaviour for the chosen conditions. The application of the QSSA to the above scheme leads to a simple set of differential and algebraic equations describing the system and to an algebraic relationship between the QSSA and the non-QSSA species. As above we choose O and OH as the QSSA species ( H 0 2 is no longer in the scheme; it is considered a stable product) and set d[OJ/dr = d[OH]/df = 0. This results in the following algebraic equations:

[0] q s s = ^ Jqss

fOH1

i

L

,

(4.99)

A:5[H2] + l/? r e s =

k4[Q2][H] + (fc5*4[H2][Q2][H])/(fcs[H2] + 1/fre.) MH 2 ] + l/*res (4.100)

Substituting for the QSSA species in the original odes, the governing set of differential equations for this system becomes

388

Mathematical tools

Ch. 4

d[p = [H2]o ~ [H2] _ ^ [ H 2 ] [ 0 2 ] _ ^ ^ ^ Qf

fres

2fc4fc5[H2][Q2][H] k5[H2] + ties ' d^]

=

[Ojo

Clf

[O2] _

(4101)

_M

fc2[H2][Q2]

o d [ H ]

_M H ] [ o 2 ] [ M ] ,

fres

(4.102)

aaoi. _ [m +

+

fres

dr

famwrojM

(4.103)

^ [ H 2 ] + 1/^res

jW,,[H]+MH2][02]+2^5[Hj[Q;][H] df

fres

£5[H2] + l/fres

- A:6[H][02][M] - A:7[H].

(4.104)

Since k3 is large it is assumed that k3 [H2] > l/£res a n d so we have neglected the tres term in the case of [OH] qss . k5 is smaller and a similar assumption cannot be made for [0] q s s . Figure 4.14 shows a comparison of the temperature profile obtained from the solution of the above system along with the energy balance equation (4.4), with that from the original 47-step scheme. The period from the minimum scheme is less than half that for the full scheme and the maximum temperature rise is much less. Therefore, this scheme could not be used as an accurate representation of experimental results but is of interest because it provides the minimal oscillating mechanism that we are able to generate for the present conditions. In closed systems, or even in systems with large residence times, it is possible to make further simplifications to the above scheme. In such cases it can be assumed that &5[H2] > l/tres and the steady-state concentrations for O and OH reduce to

389

Reduction based on the investigation of time-scales 1000 -,.

T/K

790

5

10

t/S

Fig. 4.14. Temperature profiles for full scheme and minimum four-variable scheme for Ta 790 K. The maximum temperature rise for the minimum scheme is only about 30 K.

[OH]qss =

2A:4[Q2][H] *3[H2] '

(4.106)

and the set of odes becomes, d[H2]

[H 2 ] 0 -[H 2 ]

dt

*res

d[0 2 ] dt

[O 2 ] 0 - [0 2 ]

- ^2[H2][02] - 3*4[02][H],

(4.107)

A:2[H2][02] - *4[02][H] - fc6[H][02][M],

*res

(4.108) d[H 2 0]

[H2Q]

dt d[H] = _[H] dt

tm

+ 2k4[02][H],

(4.109)

+ *2[H2][02] + 2fc4[02][H]

k6[H][02][M]-k7[U].

(4.110)

390

Mathematical tools

Ch. 4

From this set of equations the following set of four global reactions can be derived:

H 2 + 0 2 -> H + products 3H2 + 0 2 -> 2H 2 0 + 2H H + 0 2 + M - * M + products H^WALL

rate = fc2[H2][02], rate = Jfc4[02][H], rate = Jfc6[H][02][M], rate = fc7,

which describe the overall stoichiometry for slow flow or for a closed system. Mass balance could provide a further reduction of the above equations to only three variables. This scheme provides a good estimate for times to ignition in both closed and flow systems since, for high H 2 concentrations, the flow is insignificant and the rate of formation of water is well approximated in equation (4.109). At low H 2 concentrations, the rate /c4[02][H] is an over-estimation for a flow system and the faster production of H 2 0 switches off the ignition more quickly. The four-step scheme does not reproduce sustained oscillatory behaviour. In order to reproduce oscillations successfully, we cannot neglect the outflow of QSSA species. The important point to note here is that a significant number of assumptions have to be made in order to arrive at a global reaction model with a minimal number of steps. If we want the reduced scheme to reproduce almost exactly the important results of the full model, most of the original terms must be retained in some form. Even if the number of species and model stiffness is reduced, the remaining lumped equations will be more complex. Achieving a global model in kinetic form usually involves some simplification of the remaining terms as we have done above. Undoubtedly, this will lead to a loss of accuracy of the reduced model but, by following an objective procedure, the effects of these assumptions can be seen at each stage. When using global mechanisms generated by other methods it is important to remember that in such cases similar assumptions are implicit, and that these global mechanisms will have similar limitations as the ones described above.


391

4.8.6 Discussion Methods based on the separation of time-scales between chemical species have shown a high degree of success in reducing chemical systems. Such methods may also reduce the stiffness of the equation system by removing fast equilibrating processes, thus allowing the use of less expensive integration methods. The QSSA is the oldest method in use and perhaps has the best proven success rate. The selection of QSSA species can now be automated using recently developed methods such as CSP and techniques to evaluate QSSA errors. Once identified, the QSSA species can be eliminated from the reaction scheme, either by solving the QSSA expressions explicitly or by using methods for the generation of global reactions such as used by Peters etc. There are two remaining problems in the application of the QSSA to general systems. The first is the algebraic solution of the non-linear equation systems or their truncation to some linear form. Some methods have been used, such as the application of partial equilibrium assumptions, but the criteria for their application often rely on trial and error. The technique of inner iteration provides a solution to the remaining non-linear QSSA expressions, although these will not be as computationally efficient as an explicit solution would be. The second problem is the limitation of the approach associated with the fact that it identifies species directly via the time-scales of the associated linear system. If there are more fast time-scales (as identified by an investigation of the eigenvalues) than QSSA species, then the optimum reduced model may not be produced by the application of the QSSA. In some cases the fast variables of the model will not be individual species, but will be functions of the original species, as shown by the application of CSP. The ILDM technique proposed by Maas and Pope overcomes this problem by describing geometrically the optimum slow manifold of a system. The criterion for reduction is based on the time-scales of linear combinations of variables and not on species themselves. The main advantage of the technique is that it requires no information concerning which reactions are to be assumed in equilibrium or which species in quasi-steadystate. The only inputs to the system are the detailed chemical mechanism and the number of degrees of freedom required for the simplified scheme. The ILDM method then tabulates quantities such as rates of production on the lower-dimensional manifold. For this reason, it is necessarily better suited to numerical problems since it does not result in sets of rate

392

Mathematical tools

Ch. 4

equations which are directly related to the inputs of the model. The CSP method can formulate such a set of equations but, because it relies on a linear approach, these equations change with time and, therefore, expressions valid for any reaction time cannot be written down explicitly for nonlinear systems. In the next section we describe techniques under development which address these problems based on the application of lumping to systems with time-scale separation. These methods will be based on the idea of transforming the system to reveal purely fast variables, each of which will be associated with a single time-scale.

4.9 APPROXIMATE LUMPING IN SYSTEMS WITH TIME-SCALES SEPARATION

4.9.1 Linear lumping in systems with time-scale separation The lumping techniques described in Section 4.7 are exact techniques, so that the reduced model is an exact representation of the features of the full model. In systems with time-scale separation it is possible to carry out approximate lumping by selecting only the slow subspaces without a significant loss of information about the system. Exactly as in the QSSA and slow manifold techniques, we then arrive at a reduced model which is perfectly good for describing long-time behaviour. The advantage is that, since we are looking for only approximate conditions, a higher degree of lumping can be achieved than for the exact case. In this way a new set of lumped equations can be defined which describe the motion along the slow manifold, i.e., the inertial form of the equations. Following the same approach as for the ILDM and CSP techniques, the simplest way to achieve this is to define a new orthogonal basis where the slow and fast modes become decoupled. (/) The use of Schur vectors in lumping In Section 4.8.4 we introduced the Schur vector. We will now formalize this concept and show how the Schur matrix can be used as a lumping matrix for systems with time-scale separation. If A C ( # nX " then there exists a unitary transformation matrix Q C 4 = [H], * = [OH], y6 = [0],

y 7 = [H0 2 ].

The mass balance equations then become — 1 = -ktftf2 dt

~ k2yxy5 ~ k4yxy6,

dy2 —- = ~kxyxy2 ~ k3y2y4 ~ k5y2y4(y1 + 0Ay2 + 6y3), dt —~ = k2yxy5, dt d^4

dt

~k3y2y4 ~ k5y2y4(y1 + 0Ay2 + 6y3) - k6y4 + kxyxy2 + k2y y5 + A ^ ^ ,

(4.128)

dy5 —- = - ^ 1 ^ 5 + k3y2y4 + fc4yij65 dr dy d* dt

- ^ 1 ^ 6 + k3y2y4,

dy r- = k1y1y2 + k5y2y4(y1 + 0.4y2 + 6y3). dt Strictly, the application of the algebraic method in non-linear perturbation theory requires the existence of a small parameter e in the equations, and this can be revealed by a non-dimensionalization procedure. However, even for such a simple set of equations, this is a complicated process which can be avoided by carrying out a numerical investigation of the time-scales present in the problem. By examining the eigenvalues of a linear approximation to the system as described in Section 4.7, it becomes clear that there are two negative fast modes in the above equations over all conditions tested. These are indicated by the presence of two large

Approximate lumping in systems with time-scale separation

399

negative eigenvalues in the spectrum over the whole of each trajectory. In fact, there are two purely fast variables related to y5 and y6 which can be eliminated. These correspond to species O and OH which were identified as potential quasi steady-state species in the previous section. For some combustion systems the fast modes may not have an obvious relationship to particular species, and a more detailed comparison of reaction fluxes would have to be carried out in order to select an appropriate A0. Such fast modes could also be identified using computational singular perturbation theory (CSP). For the above system we set e = 1 and choose A to be the following: A = A0 + Al9 where A0 = (-ktfiys

+ k3y2y4 + k4y1y6) — + (~k4y1y6 + k3y2y4) — ,

Ax = {-kly1y2 - k2yiy5 ~ k4y^y6) — + (-kiyiy2

~ k3y2y4 - k5y2y4(y1 + 0Ay2 + 6y3)) —

Sy2 »\ + Ktfiys — + {~k3y2yA ~ ksy?yA(yi + 0Ay2 + 6y3) dy3 - k6y4 + kxyxy2 + k2y1y5 + k4y^-—.

(4.129) dy4

The first stage in the process is to convert AQ to a suitable canonical form. This is achieved by using the transformation from the original fast variables to some new, purely fast, variables q>x and cp2 where,

k2yi and

{k4 - k2)y1

k4 - k2

400

Mathematical tools

4C0 2 + 5H 2 0

In each case, these stoichiometric equations are appropriate to the complete combustion of the fuel and, hence, assume the availability of sufficient oxidant. If there is less than a stoichiometric proportion of 0 2 , then other partial combustion products such as CO or HCHO etc., may be important in the stoichiometric equation. Thus, there is some sense of pragmatism in specifying whether a particular chemical species is best defined as a product and, hence, featured in the stoichiometric equation or as an intermediate formed perhaps in the early stages but effectively completely consumed again in later stages. Such intermediates do not feature in the stoichiometric equation and do not contribute significantly to the overall thermodynamics of the reaction. The intermediate species may, however, be significant (perhaps even dominant) in determining the

Stoichiometry and elementary steps

443

reaction kinetics, i.e., how fast the conversion of reactants to products is proceeding at any given instant. In general, the reaction stoichiometry tells us nothing about the kinetics. The reaction does not proceed in a single concerted chemical process in which all the molecular rearrangements are achieved in one single stroke. Instead, the chemistry occurs through a sequence of elementary steps with which a variety of other chapters in this volume are primarily concerned. Even for the stoichiometrically-simple oxidation of hydrogen, governed by equation (i) above, it is possible to identify something of the order of 90 different elementary steps that contribute to the overall reaction over a sufficiently wide range of pressure and temperature. The phenomena discussed later in this particular chapter arise over a limited range of experimental conditions, but even so involve approximately 30 elementary steps. These elementary steps involve the intermediate species as well as the initial reactants: in some cases intermediate species are produced, in others they are transformed and in yet others they may combine to form the final products. In the H 2 + 0 2 reaction, even under the limited range of conditions of interest in this chapter, the intermediate species H, O, OH, H 0 2 and H 2 0 2 can be invoked along with the final product H 2 0 . (Under some conditions sufficient H 2 0 2 survives for this to be appropriately classified as a minor product.) Examples of elementary steps from this reaction are (iv)

H2 + 0 2 - ^ H 0 2 + H

(v)

OH + H 2 ^ H 2 0 + H

(vi)

H + 02ÔH + 0

(vii)

H + OH + M-» H 2 0 + M

In step (iv) two intermediate species are produced from the initial reactants. At least one of these species, the H atom, is a reactive radical intermediate (it possess an unpaired electron) that plays a major role in carrying the chain of the subsequent chemical transformations. Thus, reaction (iv) is classified as an initiation step (see Chapter 1). Step (v) sees one reactive chain-carrying intermediate, the OH radical, transformed into one other, the H atom, and so it is a propagation step. Step (vi) sees the production of two chain carriers from one: such a step, which brings about

444

Global behaviour in simple oxidations

Ch. 5

an increase in the number of chain carriers, is termed a branching step and we will see that such steps underpin the chemical feedback processes in combustion reactions. Step (vii) is a termination step as the number of chain carriers decreases. The species M in such reaction steps is termed a third body. Its role is to extract some of the energy released in the bondformation process. This acts to stabilize the nascent product molecule, in this case H 2 0 , which would otherwise re-dissociate on the first vibration of the newly-formed bond (see Chapter 2). Any molecule present in the reaction mixture can play the role of a third body, although some are more efficient at participating in particular energy-transfer processes than others and, hence, different species have different third-body efficiencies, and the rate of such steps thus depends on the total concentration of molecules present, i.e., on the pressure and composition of the reacting gas mixture. In the case of elementary steps (only), the reaction kinetics can be inferred from the stoichiometric equations such as (iv)-(vii). Thus, the rate at which step (v) proceeds will be proportional to the local instantaneous concentration of OH radicals and to the local instantaneous concentration of H 2 molecules. We can write, therefore, rv = * v [OH][H 2 ],

(5.1)

where rv is used to denote the rate at which OH and H 2 are converted to H and H 2 0 through this particular step. This term will contribute to governing reaction rate equations for each of these species, i.e., to the differential equations specifying the overall (local) rates of change of these particular concentrations. The proportionality coefficient kv is the reaction rate coefficient (sometimes called "rate constant") for this step. This is likely to be a temperature-dependent quantity (and hence not appropriately called a constant) as described elsewhere in Chapters 1-3 of this volume and to which we return later in this section. The form of equation (5.1) suggests that any bimolecular elementary step will naturally give rise to a term in the reaction rate equations that involves the product of two concentrations. Such a quadratic term in a differential equation provides for a non-linearity and so we see that chemical kinetics naturally produces non-linear terms and equations. Steps (iv) and (vi) are also bimolecular (involving two molecules) and, hence, give rise to quadratic terms: step (vii) gives rise to a cubic term as the total concentration [M] is the sum of the instantaneous individual concentrations, al-


445

though this is frequently more conveniently expressed in terms of the local pressure and temperature [M] = p/RT. The elementary steps contributing to a chemical mechanism explicitly specify the (local) rates at which individual species concentrations are varying by summing all the terms of the form of equation (5.1) that contribute to the production, or removal, of the individual species of interest. However, the overall reaction rate is a quantity that must first be defined. In this case, it is appropriate to return to the overall stoichiometric equation (i), for the H 2 + 0 2 reaction. A generalized version of a stoichiometric equations is (viiia) aA + bB -* cC + eE Here the numbers a, b, c and e are related to the stoichiometric coefficients for this overall reaction. These important quantities are obtained by rewriting the stoichiometric equation in the form (viiib)

0 = cC + eE - aA - bB = 2 vjJ J

The final form indicates a summation over all species J appropriate to the particular reaction under consideration. Thus, the stoichiometric coefficient for species C in reaction (viii) is given by vc = c while the stoichiometric coefficient for species A is vA = -a. The stoichiometric coefficients for the reactants and products in the overall reaction (i) are similarly, for the reactants, J^H2=-2,

ô2=-l

for the product ^H2O

= +2

The stoichiometric coefficients for reactants are typically negative while those for products are positive. We can use the stoichiometric coefficients to define the overall rate based equivalently on the rate of consumption of reactants or the rate of production of products. In terms of the general

446


Ch. 5

stoichiometric equation (viii), the equivalent definitions of the "reaction rate" R have the form reaction

rate,

« - ifiS , I M , - !4U c dt e dt a dt

=

_ Iffil, b dt (5.2)

or, in general

Vj dt

where vj and [J] are the stoichiometric coefficient and concentration of species J. In some cases, it is appropriate to divide such a rate expressed in terms of a product species such as C by the initial concentration of one of the reactants such as A divided, in turn by its stoichiometric coefficient. The term (û/^c)([C,]/[A]0) represents the extent of reaction £, which is a dimensionless measure of the progress of the reaction varying from 0 at the beginning of the reaction to 1 at the end (complete conversion). The form of equation (5.2) is useful in that it relates the rates of removal or production of the species in the stoichiometric equation. To specify how this rate depends on the concentrations of the various species involved in the reaction, we must return to the individual elementary steps that comprise the mechanism for this reaction. When dealing with a set of elementary steps, we can use stoichiometric coefficients of the form v{J that specify the coefficient for species ; in step i. The overall rate equation will then involve the rates of the individual elementary steps summed, as they involve the stoichiometric coefficients of the species in terms of which the overall rate has been expressed. As an explicit example, we can consider a truncated version of the mechanism for the H 2 + 0 2 reaction that may be of relevance at very low pressures:

(0)

H2 + 0 2 - ^ 2 0 H

rate = A:0[H2][O2]

(1)

OH + H 2 ^ H 2 0 + H

rate = ^![OH][H2]

(2)

H + 02ÔH + 0

rate = fc2[H][02]


(3)

O + H 2 -> OH + H

(4)

H-*§H2

447

rate = * 3 [0][H 2 ] rate = *4[H]

The last step here is apparently a unimolecular process and depends only on the concentration of H atoms. This is a step that occurs on the surface of the reaction vessel and the above is a simplification of the rate expression for such a process. We can apparently define the reaction rate R in three different ways: R = - \ ^ 2 at

= ô[H 2 ][0 2 ] + ^ 1 [ O H ] [ H 2 ] 2 2 (5.3a)

+ h3[0][H2]--k4[H], 2 4 ^

R=~

= MH 2 ][0 2 ] + k2[U][02],

(5.3b)

at or R=ld[H£l_l

2

At

2

L

JL

J

(5.3c)

It is not apparent that these are the same and, in fact, in general this will not be the case. The reason for this is that the stoichiometric equation does not consider the intermediate species. If any of these are being formed or removed at significant rates, then the above three choices are not equivalent. If, however, the intermediate species manage to establish a quasi-steady-state in which their rates of formation and removal effectively balance, then these three different definitions of the overall rate become equivalent. This point has been discussed as an example in Chapter 4. Even if the quasi-steady-state approximation is valid, all of the forms of the reaction rate equation (5.3) given above involve the concentrations of at least some of the intermediate species. If we are to be concerned with the intermediate species, and particularly when we wish to employ the methods of non-linear mathematics (or even direct numerical computation) then we must find ways of including such quantities. The appropriate rate equations for the concentration of H, O and OH are readily constructed

448


Ch. 5

in a similar manner. There are various ways forward. The six rate equations (one for each different chemical species) can be integrated numerically forwards in time from a given set of initial concentrations and for a particular set of the experimental parameters. Such computations could then be repeated for different parameter values or initial concentrations and a map of the global behaviour in the parameter space be constructed. Experimentally, this is equivalent to determining the pressure-temperature ignition diagram for a particular mixture and vessel size. We may note that it is not, in theory, necessary to integrate all six equations: there are two constraints that arise from the conservation of atoms as an H atom is never converted to an O atom in a chemical process. Thus we know that there is a total atom balance of the form 2[H2] + [H] + [OH] + 2[H 2 0] = 2[H 2 ] 0 ,

(5.4a)

2[0 2 ] + [O] + [OH] + [H 2 0] = 2[O 2 ] 0 ,

(5.4b)

relating the instantaneous concentration of various intermediates to the initial concentrations of the reactants. For complex situations, such as for spatially-inhomogeneous systems, this atom balance must be expressed as an integral over all space and may not hold locally and in many cases, even for well stirred systems, it is often not particularly helpful invoking these constraints. The quasi-steady-state approximation works by replacing the differential equations for the rates of change of the intermediate species by algebraic conditions obtained by setting d[H]/df = 0 etc. (see Section 4.8.5). In some cases, the resulting equations can be solved and manipulated algebraically allowing substitution into the overall rate equation to obtain a form that only involves explicitly the concentrations of the reactants and (perhaps) products. Such rate equations can then be compared with the empirical rate equations determined from experiment to test the validity of the assumed mechanism and to obtain quantitative values for the rate coefficients involved. The non-linear mathematics approach begins with the set of governing differential equations but puts great emphasis in the initial stages of analysis on the true steady-states of such systems, i.e., the combinations of concentrations for which all the rate equations, including those for the reactants and products, become zero. This is only appropriate in so-called


449

open systems in which there are inflows of fresh reactants. In closed vessels, the system "simply" has a single steady-state corresponding the state of chemical equilibrium: such states are basically "uninteresting" in the present context, although the way in which the system evolves "early on" in its approach to the equilibrium state may be "interesting". Other approximate approaches can be developed to analyze such situations. The simplest type of open system of interest in combustion is the continuous-flow, well-stirred tank reactor or CSTR, which is an idealization of tank reactors used widely in industry. In essence, this is simply a tank into which reactants flow continuously at some known volumetric flow-rate and the reactant-intermediate-product mixture is efficiently stirred so that there are no spatial concentration or temperature gradients. In order to maintain a constant reaction volume, there is a matching volumetric outflow of the mixture from the CSTR so that molecules spend only a finite time in the reactor. This is known as the mean residence time tTes and is determined by the volumetric flow-rates and the reactor volume. In terms of the governing mass balance equations which again determine the rates of change of individual species concentrations then, under certain mild assumptions, there are simply additional flow terms added to the kinetic terms. For the reactant species there will be both inflow and outflow terms while for intermediates and products there are likely to be only additional outflow terms as these species do not have significant concentrations in the inflow mixture. The nature of these extra flow terms can be exemplified with the simplified H 2 + 0 2 mechanism discussed above. For the fuel H 2 , the mass balance equation appropriate to reaction in a CSTR can be written in the form

ffl= at

H!l0il tres

- ^[OH][H 2 ] - * 3 [0][H 2 ] + ik 4 \H\.

(5.5)

The first term on the righthand side of this equation gives the net inflow rate of H 2 : here [H 2 ] 0 is the "inflow concentration" of the fuel. This must be specified with some care if, as is likely, there are separate inflows of fuel and oxidant (and perhaps also of other diluents) to the reactor. The appropriate definition of the "inflow concentration" in such equations is,

450


Ch. 5

in fact, the concentration of the fuel (or any other species) that would be established in the reactor under the prevailing flow conditions but in the absence of chemical reaction. In other words, we must allow for the dilution effects that arise when more than one inflow stream is mixed in the reactor. For intermediate or product species, such as H 2 0 , there is typically no inflow term so the mass balance equation has the form

ffioi.,i[OHIH2]-a°i,

(5,6) *res

ill

with an outflow term appended to the kinetics. We can write a general form (cf. equation (4.3)) for the mass balance equation of any species J

afl.SBkzJffi.s^, Qt

tres

(5.7)

i

where [J]0 is the inflow concentration as defined above (which may be zero for intermediates and products) and the final term is the sum of the kinetic terms with vx j being the stoichiometric coefficient for species J in reaction step i which has a rate rt that will depend in some (typically non-linear) way on the concentrations of the various species in the reactor. If the mechanism is not known in detail, the kinetic terms may be replaced by empirically-determined rate laws, i.e., by approximations to the reaction rate term that typically will be some (non-linear) polynomial fit of the observed rate to the concentrations of the major species in the reaction (reactants and products). Such empirical rate laws have limited ranges of validity in terms of the experimental operating conditions over which they are appropriate. Like other polynomial fitting procedures, these representations can rapidly go spectacularly wrong outside their range of validity, so that they must be used with great care. If this care is taken, however, empirical rate equations are of great value. In many respects, it is surprising that complex chemical mechanisms often underlie relatively simple empirical rate expressions. This is a consequence of the way that the individual elementary steps, of which there may be several hundred, frequently appear to group themselves into a much smaller number of "timescale bands". Then, the kinetic tune may be called, to a great extent, by only one or two such groups, each of which

Chemical feedback: Branched-chain ignition

451

introduces only a simple term to the approximate rate equation. This is discussed more formally in Section 4.8. Examples of such empirical rate laws for the H 2 + 0 2 reaction are found in the literature [1-4]: rate = fc[H2]2[02], rate =

fc[H2]12[02f8,

rate = fc[H2][02]1/2, rate =

k[H2][02]2,

each for different pressure, temperature, mixture composition and vessel surface preparations. For our general reaction (viii) we might be able to provide an empirical rate equation of the form ld[C] c At

=

ld[£]= _ld[A]= e At a At

ld[£] b At

= kvÂr[BY[CY[EY.

(5.8)

Here the coefficients a-e are the individual (empirical) reaction orders with respect to the species A-E, respectively. The overall reaction order would then be given by a + fi + \ + e> but this may not be a particularly significant quantity. Other possible (and indeed quite common) forms for the empirical rate law involve rational polynomial terms such as [A]a/(1 + ^[C]*) in which case the concept of an overall order, or of an order with respect to C, is not appropriate.

5.4 CHEMICAL FEEDBACK: BRANCHED-CHAIN IGNITION

Empirical rate laws are particularly useful for introducing the idea of feedback. For this, we can proceed by plotting a graph of how the overall reaction rate varies with the degree of advancement of the reaction. The latter can be simply represented by the extent of reaction described above, i.e., by the ratio of how much of a selected reactant has been used up with respect to its initial concentration. In terms of species A in our general reaction, which for convenience we can simplify here to

452 (ix)


A + B -» C + D

Ch. 5

rate = f([A], [B])

where we have restricted the rate law to be some function of the reactant concentrations only, we could sensibly choose the extent of conversion as

f-W£4

(5.9)

The form of the dependence of R on £ will be determined by the order of the reaction with respect to A and B. Provided the reaction does indeed follow the overall stoichiometry in the above equation (in this particular case we have vA = vB = - 1 ) then, once we know the initial concentrations of A and 5 , the concentration of B at any time can be determined uniquely if the concentration of A is known, i.e., these concentrations are not independent, and so the rate can be formally expressed as a function solely of one concentration or extent of reaction variable. For example, if the reaction is first order in A (a = 1) and zero order in all other species (/3, *, e = 0) then the rate law R = k[A] = k[A]0(l - £) gives rise to a linear relationship as indicated in Fig. 5.1(a). This is a rather special case and, in the context of the previous section, we have a reaction that shows linear dynamics rather than non-linear dynamics. For any other rate law, the plot of R versus £ will give rise to a curve. Figure 5.1(b) shows the appropriate form for a second-order reaction, e.g., with a = j3 = 1, [A]0 = [B]0, or a = 2 /3 = 0, and also shows the curves for a number of other overall reaction orders of the general form R = kix[A]o (1 - £)", with n not necessarily specified as being an integer. The curves in Fig. 5.1(b) with n + 1 are "non-linear" but do share the feature that the rate is highest at the beginning of the reaction (£ = 0) and the rate decreases monotonically in each case as the extent of reaction increases. This feature is characteristic of deceleratory reactions. As the number of elementary steps and intermediate species (particularly if these involve reactive species such as radicals) increases, so the possibility of more "interesting" shapes for the reaction rate curve increases. Two such "interesting shapes" are illustrated in Fig. 5.1(c). These are characteristic of reactions that display an acceleratory phase at low extents of reaction before attaining a maximum prior to a final deceleratory phase at high extents at the end of the reaction as the state of chemical equilibrium is

453


k[A]0 11=1/2

06

3

2

o

-a o

Fig. 5.1. Variation of reaction rate R with extent of reaction £: (a) linear relationship for first-order reaction; (b) non-linear deceleratory reactions of overall order n; (c) reactions showing chemical feedback in the form of autocatalysis; (d) comparison of chemical and thermal feedback curves.

approached. These two curves have approximately "quadratic" (parabolic) and "cubic" character, and many observed rate curves for reactions that have some form of chemical feedback mechanism can be approximated quite closely by one or other, or by some linear combination of these two curves. In order to give these idealized feedback curves a "chemical face", they are frequently represented in terms of autocatalysis [5, 6]. A purely quadratic autocatalytic curve would arise if the A + B reaction does not give rise to products C and Z), but instead, produces two molecules of species B quadratic autocatalysis

A + B —>2B

rate = A:^[A][5]

We can think of this as a reaction in which the conversion of a reactant species A to an intermediate or product B is assisted or "catalyzed" by B

454


Ch. 5

itself. An example of how such a rate law arises in the H 2 + 0 2 will be presented below, but it should be emphasized that the above representation is simply a convenient shorthand rather than a suggestion that any species B directly combines with a reactant A to reproduce. If we start with pure A so that [B]0 « 0, then the reaction rate will be very low (R would actually be equal to 0 if [B]0 is actually equal to zero and there is no other route from A to B). As A is converted (we might initiate the reaction by adding a small trace of B), then [A] decreases as [B] increases with [A] oc (l - £) and [B] 3B

rate = fcc[y4][l?]2

We still have [A] « ( l - £) and [B] « £, so now cubic autocatalysis

R = kc[A]l^2(l — £)

The higher order, with respect to the autocatalyst, skews the rate curve so that the maximum lies at higher extents of conversion and there is a longer induction phase during which the reaction rate is close to zero at low extents of conversion. Cubic autocatalysis is apparently less significant than quadratic which is relatively common as chemical feedback in combustion systems, although cubic-type curves have been reported and exploited in the oxidation of H 2 for which a rate expression of the form d[H20]/d£ = &[H2][H20]2 was observed [7] and also the oxidation of CS2 in heavilydiluted air mixtures [8]. An important example of chemical feedback in combustion reactions is provided by the so-called branching cycle in the H 2 + 0 2 reaction [9] which


455

is comprised of steps (l)-(3) given earlier. Under typical conditions, the slowest or rate determining step in this sequence is step (2) involving the H atom and 0 2 molecule. The net stoichiometry of the branching cycle is obtained by taking step (2) + step (3) + 2 x step (1) which may be written as, 3H2 + 0 2 + H -» 3H + 2H 2 0

rate = fc2[H][02]

The H atom is retained on the lefthand side of this stoichiometric equation to emphasis its role (because its concentration occurs explicitly in the rate law). Thus, the overall reaction can be regarded as the conversion of H 2 to 2H, "catalyzed" by H and with the simultaneous conversion of 2H2 + 0 2 to 2H 2 0. The latter provides the free energy driving-force for the process. In terms of the reaction stoichiometry, the stoichiometric coefficient for H atoms is now +2 (i.e., +3 - 1) so that H is a "product" but also appears with a positive exponent (reaction order) in the rate expression. Thus, it clearly plays a role that parallels B in the simplified representation with A corresponding to 0 2 as the latter appears as the "reactant" in the rate law. In the simple mechanism for the H 2 + 0 2 reaction given earlier, the branching cycle which increases the H-atom concentration competes with the termination step (4) which removes H atoms from the system. This competition appears more clearly if we make steady-state assumptions on O and OH (see Section 4.8 for a justification of this procedure) and then substitute for these into the rate equation for d[H]/d£, to give ^ 1 = 2* 2 [H][0 2 ] - k4[U] = [H], at

(5.12)

where = 2k2[02] - k4 is termed the net branching factor. If > 0, i.e., if the coefficient for branching exceeds that for termination, then d[H]/dr is positive and there is (exponential) growth in the H-atom (and other radical) concentration. The condition = 0 is important in determining the classic p-Ta ignition limits as we discuss later in this chapter. A simplified and more generalized representation of the chain-branching chain-termination competition involving a single reactant A, an intermediate X and a final product P is the three-step scheme

456


initiation

A—>X

rate = k{ [A]

branching

A + X-+2X

rate = fc*[i4][Z]

termination

X—>P

rate = kt[X]

Ch. 5

The concentration of the reactant [A] is frequently assumed to be constant (this is known as the "pool chemical approximation"). Only one reaction rate equation, for the chain carrier X, need be considered. This has the form M = r{ + kb[A][X] - kt[X] = r{ + 4{X]9 at

(5.13)

where r{ = k{[A] is the initiation rate and is the net branching factor as before. Provided (/> + 0, this integrates to give [X] = ^(e+-l). 9

(5.14)

If kt > kb[A], then < 0 and the chain carrier concentration evolves to a steady-state value given by [*]ss=-ri/0.

(5.15)

If is not small, this will be a low steady-state concentration, but as cf) tends to zero so [X]ss increases. For 0, [X] grows exponentially with time again until the consumption of A must be taken into account. The condition for an acceptable steady-state then is clearly (/> < 0, with the condition 0 = 0 being the "critical" case separating steady-state from explosive runaway. The "critical" concentration of the reactant obtained from the condition = 0 is then [A]cr = kt/kb.

(5.16)

Although this mathematical manipulation leads to an explicit criterion


457

for the "ignition limit", and has contributed immensely to the understanding of branched chain reactions, this approach has been criticized in one respect. As is increased towards zero from some negative value, so [X]ss increases smoothly towards infinity, as indicated in Fig. 5.2. This seems to contrast the "expected" behaviour in chemical systems that show a "steady-state rate" that generally increases as the system parameters are brought close to the critical values, but then show a sharp, discontinuous change in behaviour as the limit is crossed. We will see examples of such discontinuous bifurcation structures with other models below but should also note that in closed systems there is, in fact, no such discontinuous change, rather the rate simply increases, usually very rapidly, over a typically very narrow range of experimental conditions (so narrow and with such a dramatic increase that it appears to be discontinuous). The interpretation of "ignition" in closed systems is still a subject of academic study. The initiation step is necessary in the above system to maintain a nonzero steady-state chain carrier concentration if < 0. The problem can be re-specified by setting r{ = 0 and examining how [X] varies if we have a non-zero initial chain-carrier concentration [X]0. If c/> < 0, the [X] decreases exponentially to zero; if >0, then [X] increases exponentially. So again, we see that <j> = 0 is a "critical" value separating qualitatively different responses.

|[ IX].

net branching factor,

ty

0

Fig. 5.2. Variation of the quasi-steady-state radical concentration with net branching factor for the simple model of chain branching and chain termination showing [X]ss -» °° as -* 0 (note: the steady-state only exists for (f> < 0).

458

Ch. 5


If the same model is examined in a CSTR, then the governing equations are

d[A]_([A]0-[A]) Qf

d\X]

(5.17a)

kÂ] - kb[A][X],

*res

=

_[X]

(5.17b)

+ k-M] + kb[A\[X\ - k,[X\.

In this case we can incorporate the consumption of the reactant and still study (true) steady-state behaviour. Setting d[A]/df = d[X]/df = 0 we obtain [A]0 - [A]ss = (i + A^ res )[;r] ss ,

(5.18)

and, hence, the steady-state condition become

ki[A]0 + { kb[A]0 - ki(l + kttrcs)

k, l res

in

- kb(l + kttres)[Xfss = 0.

(5.19)

The first term in this quadratic equation is the initiation reaction rate based on the inflow concentration of the reactant. The coefficient for the term in [X]ss has something of the character of the previous net branching factor. The above equation has a single positive solution for any set of rate constants, residence time and inflow concentration: a typical variation of [X]ss with [A]0 is shown in Fig. 5.3(b) and shows a rapid increase in the vicinity of some "critical" concentration [^4]o,crThe behaviour can be quantified if we make the approximation of ignoring the (probably small) initiation terms, setting k{ = 0. The steady-state condition can then be written in the form fiow is given by

(5.20)

459


X

(a)

(b)

[A]„,r

[A] 0

Fig. 5.3. Variation of steady-state radical concentration as a function of reactant concentration appropriate to chain branching and chain termination reactions in a flow reactor (a) ignoring initiation reaction, there is a root [X]ss = 0 for all [A]0 and a non-zero root that is positive for [A]0 > [A]0,cr; (b) including initiation step provides a non-zero positive radical concentration for all [A]0 but the radical concentration increases rapidly over a relatively small range of [A]0.

0f low = kb[A]0 - kt

1 ^res

= 0

1

.

(5.21)

'res

The flow term adds an extra chain-carrier loss term with a pseudo-firstorder rate constant given by the inverse of the residence time. Equation

460


Ch. 5

(5.20) has a root [X]ss = 0 for all parameter values, with a non-zero root whose sign is determined by the sign of f low . For $ f low < 0, the non-zero root is negative and, hence, physically unacceptable. For fiow > 0, the non-zero root is positive and physically acceptable. The character of the two roots in this case is such that [X] will evolve to the non-zero root from any non-zero (positive) initial value. The variation of the two roots [X]ss with [A]0 is shown in Fig. 5.3(a). There is a transcritical bifurcation where the two roots become equal and their loci cross at [A]0 = [A]0,cr which corresponds to the condition for f low = 0 [A]o,a = (kt + —)/kb. \

(5.22)

^ res' '

Provided rt is small, then the critical inflow concentration for this branching-termination model under CSTR conditions differs slightly from the so-called "pool chemical" result which is obtained by assuming [A] = constant. For typical chemical systems the residence time will be such that kt > VtTes, so the two results are not significantly different but the extra influence of the flow is clearly evident in the above forms. In neither of the analyses above, however, is there a discontinuous jump in the steadystate response as the parameters are varied.

5.5 THERMAL FEEDBACK: IGNITION, EXTINCTION AND SINGULARITY THEORY

Very few chemical processes are precisely thermoneutral and combustion reactions are characteristically highly exothermic processes. If the reaction rate is not negligible, therefore, there will be a significant nonzero rate of chemical heat evolution. Unless this energy is transferred equally rapidly from the system, it will give rise to local increases in the temperature of the reacting mixture above that of the surrounding fluid or ambient heat reservoir. This thermal effect is highly significant because of the other characteristic feature of combustion reactions, i.e., the high sensitivity of the reaction rate coefficients to temperature. A common form of representing the temperature dependence of reaction rate "constants" is the Arrhenius equation k = Ac'E/RT.

(5.23)

Thermal feedback

461

Here A is the pre-exponential factor, E is an effective activation energy or temperature coefficient for the overall reaction, R = 8.314 J K _ 1 m o l _ 1 is the Gas Constant and T is the local absolute temperature. The quotient EIR has units of temperature and is sometimes known as the Arrhenius temperature TArr. It is a characteristic feature of combustion processes that if Ta is the ambient temperature, then TArr > Ta so that the group RTJE and, more generally, the scaled or "dimensionless" temperature RTIE are typically (very) small quantities in the systems of interest. The evolution of the local temperature in a reacting system is governed by the heat balance equation. This may become quite complex in unstirred systems, especially if convective heat transfer processes develop as a consequence of local heating. For the simple CSTR described earlier we can proceed with an ordinary differential equation of the form ( c ^ ^ ^ C c ^ ^ ^ ^ +I^^-^Cr-Ta). at tTes t V

(5.24)

Here cpp is the heat capacity per unit volume and is assumed to be independent of temperature in this simplified formulation, T0 is the temperature of the inflowing reactants. The second term on the righthand side is the sum of the rate of chemical heat evolution due to the individual elementary steps: q{ is the exothermicity (-A//: typical units = kJmol - 1 ) of the ith step and plays a similar role to that of the stoichiometric coefficients in the mass balance equations and rx is, as before, the rate of the ith step and, hence, depends on various species concentrations and, now, also on the temperature T of the reacting mixture. The final term is a Newtonian heat transfer term indicating heat transfer from the reacting mixture at temperature T to the surrounding heat bath at the ambient temperature Ta subject to a heat transfer coefficient \ a n d a surface-tovolume ratio S/V. For a well-lagged reactor we may approach the adiabatic case ^ = 0, for which the only heat transfer process is the inflow of reactants that are cool relative to the outflowing fluid. For the simplest chemical case imaginable, that of a single first-order reaction A—>B

rate = ka

exothermicity = q

where we use a for the concentration of A and the rate constant k has the

462


Ch. 5

Arrhenius form given above, the two governing equations for reaction in a CSTR have the form ^ =^ ^ - ^ K At tres

(5.25a)

(cpp) ^ = (cpP) H^H at tr^

+ qk(T)a

- & (T - r a ) . V

(5.25b)

This provides a pair of coupled, non-linear (through the Arrhenius temperature dependence) ordinary differential equation for the two variables a and T. If the temperature increases, the reaction rate increases through the increase in k. The consequent increase in T will lead to increases in the heat transfer rates and also to a decrease in the concentration of A, which in turn tends to decrease the reaction rate term ka. To quantify this effect, we can examine the adiabatic case x= 0. In this situation, the temperature rise above the inflow is uniquely linked to the extent of reaction £ = (a0 - a)/a0 through the condition a0-a

T-T0 ad

?0

=

^

(5 26)

00

Here, Tad - T0 = qa0/cpp is the temperature rise that occurs under adiabatic conditions accompanying complete consumption of the reactant A. Because of this relationship, the temperature and the reactant concentration are not independent (this relationship is only true under adiabatic conditions), and so the instantaneous reaction rate can be expressed in terms of one of these quantities alone, e.g., we can express R as R(€) as in the previous section. A typical form for the reaction rate curve is shown in Fig. 5.1(d). This shows the nature of thermal feedback even in this very simple chemical example. At low extents of reaction, the increase in k as T increases dominates and the rate, which is relatively low when £ = 0 so that T = T0, increases as the reaction proceeds. Only at very high extents of reaction, i.e., close to complete consumption of the reactant, does the rate fall, approaching zero as £ —> 1. The curve has some similarities with the cubic autocatalytic curve but thermal feedback tends to have a shallower initial development and its maximum at higher £.

Thermal feedback

463

The governing mass and heat balance equations can be usefully recast in the following dimensionless forms: da dr

.,. 1— a a/(0), T res

^ = eadaf(d) " ( — + - ) 0. ar \r r e s TNJ

/e. ^ x (5.27a)

(5.27b)

Here 6= (T - T0)E/RTl is a dimensionless measure of the rise in the temperature above the inflow value (for simplicity T0 = Ta has been assumed). The unspecified function f(0) then reflects the increase in the rate coefficient k above its value when T= T0 that arises for a particular dimensionless temperature excess 0, i.e., k(T) = k(T0)xf(d).

(5.28)

The Arrhenius temperature dependence can be represented exactly in this way with /(0) = exp — — . U + eflJ

(5.29)

For the purposes of qualitative discussion, however, this form can be usefully approximated by the simple exponential dependence f(0) - e e ,

(5.30)

which holds as the quantity e = RT0/E = T0/TArT is typically very small compared with unity as discussed earlier. The quantity 0ad = (T ad - T0)E/ RTl is the dimensionless adiabatic temperature excess and the dimensionless concentration used here a = a/a0 = 1 - £. The remaining dimensionless quantities Tres and r N are the dimensionless residence time tres/tch and Newtonian cooling time tN/tch, where tN = cppV/xS and the chemical timescale tch = llk(T0) is the inverse of the rate constant evaluated at the inflow temperature T0. Using the relationship between the temperature rise and extent of con-

464


Ch. 5

version determined earlier we have for the adiabatic case (i.e., with TNôo)

^ - = l - « = £.

(5.31)

The reaction rate term R — k{T)a can thus be written as R = i? 0 (l " €) e"ad*,

(5.32)

where R0 = k(T0)a0 and gives rise to the form in Fig. 5.1(d). The behaviour of this model has been widely studied in the chemical reactor engineering literature, where it is a standard form. Under non-adiabatic conditions it gives rise to complex steady-state responses and to sustained oscillations in the concentration and temperature. A detailed description can be found elsewhere (see e.g., Chapters 6 and 7 in reference [5] and references therein). In the "pool chemical" formulation of this model, the consumption of the reactant A is ignored and so there is only one variable, the temperature T. The governing heat balance equation has the form (cPp) ^ = qk(T)a0 ~^(Tat V

Ta),

(5.33)

there being no flow terms and the reactant concentration is now constant and equal to its initial concentration a0. The dimensionless form of this equation can be written as

f = m~e-r

(5-34)

where ifj is known as the Semenov number and is given by 0adTN- This model is the basis for the theory of thermal explosion or thermal runaway because of chemical self-heating [10]. The neglect of reactant consumption is generally justifiable due to the large magnitude of the dimensionless adiabatic temperature excess 0ad, which indicates that only small extents

Thermal feedback

465

of reaction are need to achieve temperature rises that make 0 of order unity. Criticality for ignition in either of the two formulations of this simple model of thermal feedback can be interpreted by means of a thermal diagram. The variation of the rate of heat evolution with temperature for the simple Semenov model above is, effectively, the graph of the function f{6). For relatively low-temperature excesses, the curve is well approximated by the exponential form/(0) = t6 and so has the appearance shown in Fig. 5.4. The heat transfer rate corresponding to the term dlifjin equation (5.34) is simply a straight line on the thermal diagram. For low values of the Semenov number, corresponding to low reaction exothermicities or to high heat transfer rates, the loss line is steep and intersects the heat release line twice on the diagram as shown. The system starting with 0 = 0, i.e., with T= Ta initially, will evolve to the lower of these intersection points at which d0/dr = 0 and, hence, a steady-state is attained. The corresponding steady-state temperature excess 0SS will have a value less than unity, typically indicating a temperature excess of less than 10 K. As if/ is increased, so the slope of the heat loss line decreases and for high if/, there are no

\|/ small

m

o

e

Fig. 5.4. Thermal diagram for Semenov model of thermal explosion: the rate of chemical heat release varies with the dimensionless temperature excess 0 according to/(0) « e e ; the rate of heat transfer is given by the straight line with a gradient of 1/I/J. For small ifj the loss line is steep and makes two intersections corresponding to two steady-states; for large ip the loss line has a low gradient and does not allow steady-state intersection points; the critical case corresponds to tangency of the heat release and heat loss lines.

466

Ch. 5


intersections between the two curves on the diagram. In this case, the heat release rate always exceeds the heat loss rate so the extent of self-heating increases continuously until a high temperature rise is attained (at which point the assumption that the reactant concentration remains constant becomes totally invalid). This latter behaviour corresponds to thermal runaway. The "critical" case separating these two different responses (low 6 steady-state for low i/> and thermal runaway for large if/) arises when the heat release and heat loss curves just touch tangentially. The two intersection points at low i/> approach each other as \\f is increased and merge at the tangency condition. These steady-state intersections vanish for larger \\f. If we plot the variation of the steady-state intersection points with the parameter \\f, to give the bifurcation diagram as shown in Fig. 5.5, then we see two branches that meet at a vertical turning point at the point of tangency. (There is an additional intersection point giving rise to a third branch at large 0: for the Semenov model in which reactant consumption is ignored, this branch relies on the "saturation" of the Arrhenius function at large T>TArr and is totally physically unrealistic. For the CSTR equations, consumption is included explicitly and the third branch of intersection points corresponds to a high, but physically acceptable, temperature and an "ignited" steady-state.) Following the steady-state behaviour then, if we begin with some low value for I/J, the system will evolve to the lowest branch in Fig. 5.5. If if/ is increased slowly, the system effectively moves along the lowest branch, with the steady-state temperature excess increasing slowly and smoothly

9s*cx

Vex

V

Fig. 5.5. Variation of the steady-state temperature excess 0SS with the Semenov parameter \ft indicating the turning point at the critical condition.

Thermal feedback

467

with the parameter. At the turning point, however, the system must jump away from the low 6 steady-state solution since this solution vanishes as tangency occurs. Thus, there is a discontinuous jump in the steady-state of this model, corresponding mathematically to a genuine saddle-node bifurcation point. Such a genuine discontinuity does not arise in the simple quadratic chain-branching model, and this absence leads to the earlier (but probably unwarranted) criticism alluded to in the previous section. (In fact, if we allow for reactant consumption, which must clearly be a feature of the real situation, the distinct bifurcation structure vanishes from the Semenov model too as the only steady-state for any i/> is that with all the reactant consumed and the temperature returned to the ambient. Only open systems have true bifurcations between non-(chemical) equilibrium steady-states.) The critical condition can be located in a straightforward manner using the simultaneous conditions for a steady-state and for tangency of the two curves steady-state condition

f{&) = - , = —,

tangency condition d0

(5.35a) (5.35b)

\fj

with the latter being the condition for equal slopes for the heat release and heat loss curves on the thermal diagram. If we take f{6) = e 0 , then these give 0ss,cr = l,

iAcr = e _ 1 .

(5.36)

For the CSTR model, the highest intersection point is physically realistic and, indeed, important. It corresponds to an intersection point on the deceleratory part of the reaction rate curve as indicated in Fig. 5.6(a, b). There are now two possible tangencies of the heat loss line with the heat release curve as some parameter such as the residence time is varied, as indicated in the figure. The first has a similar implication as in the Semenov case. It corresponds to the merger of two low-lying steady-states and to an ignition point on the steady-state locus and, in this model, arises typically as the residence time is increased. The system now jumps to a high steady-

468


Fig. 5.6. Bistability for an exothermic reaction in a flow reactor: (a) flow diagram for system with large adiabatic temperature excess, 0ad > 4 showing heat release curve and indicating the position of the flow line corresponding to ignition and extinction events; (b) corresponding variation in steady-state temperature excess with mean residence time showing region of bistability: the "jumps" associated with the tangencies in (a) are indicated by vertical arrows at the turning points; (c) and (d) flow diagram and steady-state locus for a system with 0ad < 4 for which multistability has been unfolded.

state temperature excess, corresponding to the uppermost branch in the figure. If the experimental conditions are now changed, e.g., by decreasing the residence time, we move back to the left on the steady-state locus, but remain on the upper branch. Only at some lower residence time do the upper and middle branches coalesce at an extinction turning point. This is another tangency (saddle-node) bifurcation between the heat release and heat loss curves, as indicated, and sees the system jump back to a low temperature steady-state. Thus, this model shows both ignition and extinction and accounts for a range of hysteresis in the parameter space over

Thermal feedback

469

which different steady-states coexist. The system can sit at either steadystate, depending on its previous history. If some other parameter of the system, such as the adiabatic temperature excess 0ad is varied, so the shape of the steady-state locus may deform. For low values of 0ad in this model, corresponding to weakly exothermic processes, then the hysteresis loop is unfolded, as indicated in Fig. 5.6(c, d), and a simple smooth variation of the steady-state temperature excess with the residence time is observed. Thus, systems can lose criticality as other experimental parameters are changed. The recipe of locating the critical (ignition or extinction) conditions through tangency in a thermal or flow diagram of the form shown in the previous figures or equivalently as a vertical turning point in the steadystate locus can be written as a general prescription [5]. The steady-state condition can be written in the form F(xss, A; p) = 0,

(5.37)

where F is some set of functions, x is potentially a vector of the variables in the system, A is the primary bifurcation parameter, i.e. the parameter chosen as that most likely to be varied in a given set of experiments and p is a vector of the remaining unfolding parameters. For criticality, this condition will be satisfied along with the condition for a vertical turning point in the xss-X curve, which occurs when dF(xss, A; p)/dx = 0,

(5.38)

where the form dF/dx indicates the matrix obtained by differentiating the vector of functions F with respect to the variables. If there is more than one independent variable in the model, then equation (5.38) represents a condition on the determinant of the Jacobian matrix, det(J) = 0. To illustrate this, we can apply this prescription to the two variable CSTR model for which x = (a, 6)T. The steady-state condition F is simply obtained by setting da/dr = 0

Fx = — - a / ( 0 ) = O, ''"res

(5.39a)

470

Ch. 5


F2 = eadaf(6) - ( — + — ) 0 = 0.

(5.39b)

The Jacobian matrix has the form

3a

30

Tres

§

§

da

30,

L 0ade.«e

.

/ 1 +. J-) i\

0 a d afle « - ( J -

(5.40) taking f(6) = e e , for simplicity. If we also restrict ourselves to the adiabatic case so that l/r N = 0, then the tangency/determinant condition becomes det(J) = I — + te

1

dadace

*

+ 0 a d ae z " = O.

(5.41)

Using the steady-state equations, this can be reduced to a quadratic equation in 0ss?cr with roots 0ss,cr = koad

V0 ad (0 ad - 4)}.

(5.42)

This indicates that we need 0ad > 4 for there to be real roots. The roots correspond to the critical temperature excess at the points of ignition (lower root) and extinction (upper root) respectively. As 0ad is decreased towards the transitional value 0ad,trans = 4, so the two turning points approach each other and merge as the hysteresis loop unfolds (Fig. 5.6).

5.6 THERMOKINETIC FEEDBACK: OSCILLATIONS AND LOCAL STABILITY ANALYSIS

In many combustion systems there will be the possibility of both thermal and chemical feedback, with the two processes coupled together through the heat and mass balance equations just for added fun. A simple model

Thermokinetic feedback

471

involving both effects would be that of an exothermic, quadratic autocatalytic reaction with an Arrhenius temperature dependence A + B -* IB

rate = k{T)ab

for which the governing equations would have the form da _ (a0 - a) k(T)ab, dt tre$ db = (fro - b)

(5.43a)

+ k(T)ab,

(Cpp) ^ = (cpP) at

(T

T)

°

tres

(5.43b) + qk(T)ab - ^ (T - T a ), V

(5.43c)

or, in dimensionless forms noting that a0 + b0 = a + b from the reaction stoichiometry, ^ = — - - «(1 + A, - * ) / ( * ) , dr r res

(5.44a)

^ = 0 ad a(1 + ft " a)/(fl) - ( — - — ) 0, dr

\Tres

(5.44b)

TN/

where a, 0, etc. are as defined before and j80 = bja0. This system has been investigated using singularity theory [11,12]. The condition for thermal runaway under the assumption a = a0 in a closed system was determined by Frank-Kamenetskii [13] as i/fcr = 4e _ 1 in terms of the Semenov number described earlier. Another model studied under the name of thermokinetic feedback is due to Salnikov [14,15]. This has two first-order reaction steps which, in general, could both be exothermic and both have an Arrhenius temperature dependence: (51)

A^X

rate = M

(52)

X-*P

Tate = k2x

472


Ch. 5

The most interesting behaviour can be obtained, however, by taking a slightly simplified version in which only step (S2) is exothermic and has a temperature-dependent rate law, so that the governing equations have the form: dx — = k1a-k2(T)x, dt (cpp) ^ = qk2(T)x - & (T - Ta ), at V

(5.45a) (5.45b)

and the concentration of the reactant a is assumed to be constant. These equations can again be recast in dimensionless terms, with the following form being particularly convenient [16a]: ^ = fi-Kyf(d), dr

(5.46a)

An

(5.46b)

— =yf(B)-e. dr

Here, /x is a dimensionless measure of the initial reactant concentration a0, y is the dimensionless concentration of X and K is the ratio of the Newtonian cooling time to the chemical reaction time at ambient temperature. Equations (5.46a, b) have a steady-state solution which, taking f(8) = ee for convenience, is given by 6ss = fi'K,

yss = (fi/K)e-^/K\

(5.47)

The variation of the steady-state with the parameter jx is shown in Fig. 5.7 for a particular choice of K. There is no "criticality" in this case as the steady-state solution varies smoothly with the parameter \x across the whole range, so there are no discontinuous jumps. There is, however, a different type of bifurcation that arises in this model. Provided the second parameter K is sufficiently small, there is a range of the parameter /JL over which the steady-state solution becomes unstable. To understand this statement we can examine how a system sitting at

473


0ss

/

s

H1

&

H

Fig. 5.7. Variation of quasi-steady-state temperature excess 0SS with the dimensionless reactant concentration JJL for the Salnikov model. The steady-state is stable at high and low fi but is unstable over a region fxf < \x < /JL* as indicated by the broken section of the locus.

the above steady-state will respond to a (very small) perturbation. Imagine that the perturbation takes y and 6 to the following values: y = y ss + Ay,

6 = 0SS + A0,

(5.48)

where Ay and A0 are small quantities. The equations governing the evolution of the perturbations can be expressed in terms of the original governing equation in which the functional forms of the righthand sides can be expanded about their steady-state values in a Taylor series in the perturbations. Thus if, in general, dy/dr = f(y, 6) and dd/dr = g(y, 6) then we can write

^ =/(7ss, 0SS) + 7^ (7ss, 0ss)Ay + ^(yss, dr dy 30

6ss)A0 + (5.49a)

474


dA0 de dz — = g(7ss, ess) + -*- (y ss , 0ss)Ay + -* (y ss , 0SS)A0 + dr dy ad

Ch. 5

(5.49b)

By the definition of the steady-state condition, the first terms on the righthand side are zero and provided the first-order partial derivative terms do not all vanish, we can ignore the additional terms which are second-order in the perturbations. Thus, we obtain a pair of linear equations for the evolution of these perturbations in the vicinity of the steady-state point V * = ? ( 7 s s , 0ss)Ay + ^ ( y s s , 0SS)A0, dr dy dd

(5.50a)

^

(5.50b)

= — (7ss, 0ss)Ay + ^ ( y s s , 0ss)A0,

dr

dy

36

or, in the most general form, dT d(Ax)

= J(Ax),

(5.51)

where (Ax) is the vector of perturbations and J is, as before, the Jacobian matrix of first partial derivatives which is evaluated with the steady-state values for the variables y and 0. The evolution of the perturbations is then given by the sum of exponential terms in the form Ay = ax eAlT + a2 eA2T,

A0 = bx eAlT + b2 eA2T,

(5.52)

where the coefficients arb2 depend on the initial perturbation. The qualitative nature of the time dependence is determined by the exponents Ala. These are obtained as the eigenvalues of the Jacobian matrix, i.e., from the equation J - A I = 0, where I is the identity matrix.

(5.53)

475


For the Salnikov model, the partial derivatives can be evaluated and the steady-state solution substituted to obtain the Jacobian matrix for this twovariable system in the form

dg \dy

dg\ ae/

\ ee

yee-l)

I t^ K

^-1 K

(5.54) The characteristic equation for this matrix has the form of a quadratic equation with the eigenvalues then given by the roots A2 - tr(J)A + det(J) = 0,

(5.55)

where tr(J) is the trace (the sum of the terms on the leading diagonal) and det(J) is the determinant. For the present model, we have, therefore, A2 + (1 + K^/K

- ^) A + KS*K = 0.

(5.56)

Four possibilities exist for the roots depending on the sign and magnitude of the trace and determinant: (i)

tr(J) < 0, tr(J) 2 - 4 det(J) > 0 two negative real roots

(ii)

tr(J) < 0, tr(J) 2 - 4 det(J) < 0 two complex conjugate roots with negative real parts

(hi)

tr(J) > 0, tr(J) 2 - 4 det(J) < 0 two complex conjugate roots with positive real parts

(iv)

tr(J) > 0, tr(J) 2 - 4 det(J) > 0 two positive real roots

In case (i), the two exponential terms in the series for the evolution of the perturbations decay monotonically to zero, so the system decays

476


Ch. 5

monotonically back to its original steady-state. In this case the system is stable and termed as stable node. In case (ii) the steady-state is again stable as the perturbations decay but now, through the imaginary parts of the exponential terms the decay has a damped oscillatory character and the steady-state is termed a stable focus. In case (hi) there is again oscillatory behaviour, but now the positive real parts indicate that the perturbations grow. This growth will not continue in an unbounded manner as higherorder terms in the Taylor expansion of the full non-linear equations will become important once the perturbation is not infinitesimally small. Nevertheless, the system departs from its initial steady-state which is termed an unstable focus. In case (iv) there is again divergence from the vicinity of the steady-state, but in a direct manner characteristic of an unstable node. The terms node and focus are most easily understood if instead of plotting the perturbations as a function of time, we plot one variable against the other. This gives rise to a trajectory in the y-d phase plane. The steady-state corresponds to a point on this plane and the trajectory indicates the direction in which the system evolves in the vicinity of this singular point (it is termed singular as the slope of the trajectory which is given by d0/dy = (d0/dT)/(dy/dr) = 0/0 at this point). In the case of a stable steady-state, the local trajectory is directed towards the steady-state point either approaching directly (node) or as an inward spiral (focus): for unstable points the flows are in the opposite direction. The phase portraits associated with the four cases above, and also for a fifth case to be discussed below but not present in the Salnikov model, as shown in Fig. 5.8. Returning to the specific case of the Salnikov model, the major qualitative change in behaviour occurs when damped oscillatory decay of the perturbation gives way to oscillatory growth. The condition for this change from case (ii) to case (hi) which is known as a Hopf bifurcation is, in general terms, Hopf bifurcation

tr(J) = 0.

(5.57)

In this model this becomes l + K e M / K - - = 0.

(5.58)

K

Provided K < e~2 there are two values of ^, which we may denote by /A* and ^ 1 ? at which this occurs, and these mark the ends of a range of steady-


477

Fig. 5.8. Phase plane portraits of different possible steady-state singularities: (i) stable node, trajectories approach singular point without overshoot; (ii) stable focus showing damped oscillatory approach; (iii) unstable focus showing divergent oscillatory departure; (iv) unstable node showing direct departure; (v) saddle point x showing insets and outsets and typical trajectory paths.

state instability as indicted by the broken portion of the steady-state loci in Fig. 5.7. If K increases towards the value e~2, the ends of range approach each other and instability is lost from the system. If the steady-state is unstable over the range between the Hopf bifurcation points, what happens to the concentration and temperature excess in the region /i* < \x < ju,*? We can answer this in a straightforward manner by integrating the full equations for a particular value of K < e~2 and with JJL then chosen in the above range and with initial conditions close

478


Ch. 5

to, but not exactly equal to, the steady-state values. In this particular case, we will then find that the concentration and temperature settle into a regular, periodic oscillation about their steady-state values as shown in Fig. 5.9(a, b). The amplitude and period of this oscillatory motion vary with the parameter JJL across the region of instability, with the amplitude tending to zero at the two ends as the steady-state regains stability. If we plot the oscillatory behaviour on the y-0 phase plane, it draws out a closed loop or limit cycle around the steady-state singular point, as indicated in Fig. 5.9(c). It is a characteristic feature that a limit cycle is born in the phase plane at a Hopf bifurcation point. In the Salnikov model with the exponential approximation we have a particularly simple scenario. At the lower Hopf point, the steady-state becomes unstable as the parameter /x is increased through /i*. As this occurs, a stable limit cycle is born off the steady-state point at the Hopf point and grows in size around the singular point as /JL is increased further. This is characteristic of a supercritical Hopf bifurcation. What we will see in the reaction is a steady-state give way to oscillations in a smooth manner, a so-called soft excitation of the oscillations that grow in size from zero. If we reduce /JL again, the oscillations shrink back to zero amplitude an the steady-state is regained at exactly the same parameter value without any hysteresis. Such a bifurcation is often represented by Fig. 5.10(a) which shows the amplitude of a stable limit cycle grow from zero as the parameter is varied so that the steadystate becomes unstable. The behaviour at the upper Hopf point is also that of a supercritical Hopf bifurcation although the loss of stability of the steady-state and the smooth growth of the stable limit cycle now occurs as the parameter is reduced. This is sketched in Fig. 5.10(b). We can "join up" the two ends of the limit cycle amplitude curve in the case of this simple Salnikov model to show that the amplitude of the limit cycle varies smoothly across the range of steady-state instability, as indicated in Fig. 5.11(a). The limit cycle born at one Hopf point survives across the whole range and dies at the other. Although this is the simplest possibility, it is not the only one. Under some conditions, even for only very minor elaboration on the Salnikov model [16b], we encounter a subcritical Hopf bifurcation. At such an event, the limit cycle that is born is not stable but is unstable. It still has the form of a closed loop in the phase plane but the trajectories wind away from it, perhaps back in towards the steady-state as indicated in Fig.


479

•MM Fig. 5.9. Variation of (a) concentration y and (b) temperature excess 6 in time for a value of fx in region of steady-state instability showing sustained oscillations; (c) typical limit cycle lying around unstable steady-state o produced by plotting y against 0; (d) for some parameter values there are two limit cycles surrounding a stable steady-state, one unstable (broken curve) the other stable (solid curve). In (c) and (d) example trajectories are shown (thin curves) that wind either onto the stable limit cycle or, in (d) onto the stable steady-state point.

5.9(d). Such a limit cycle grows so that it surrounds a stable steady-state, as sketched in Fig. 5.10(c, d) and so would grow out of the upper Hopf point /x* as the parameter /JL increases. This then leaves the question as to what happens if we start outside the unstable limit cycle. The system does not wind onto an unstable cycle, nor can the trajectory cross it so it cannot approach the stable steady-state inside. The system must move further away across the phase plane and either find another steady-state (of which there is not one in the Salnikov model) or perhaps to find another, but

480


(a)

sic

sss f _uss_

Ch. 5

(b)

I^ J c Luss_ ji sss

^ (c) - - jilc sss

\ uss

n

(d) ulc.- uss

/

\

sss \

^

Fig. 5.10. The four possible types of Hopf bifurcation: (a) a stable steady-state (sss) becomes unstable (uss) as a parameter /JL is increased through the bifurcation point (/A*) and a stable limit cycle (sic) emerges - the growth of the limit cycle is indicated by plotting the maximum and minimum of the variable as it undergoes the oscillatory motion around the limit cycle; (b) the scenario is reversed, with the steady-state losing stability and a stable limit cycle emerging as the parameter is reduced; (a) and (b) are termed supercritical Hopf bifurcations. In (c) and (d) there is an unstable limit cycle emerging to surround the stable part of the steady-state branch: this is characteristic of a subcritical Hopf bifurcation.

stable, limit cycle. The latter occurs in the Salnikov case and the modified bifurcation diagram is shown in Fig. 5.11(b). The stable limit cycle born at the lower Hopf point "overshoots" the upper Hopf point but is extinguished by colliding with the unstable limit cycle born at the upper Hopf point which also grows in amplitude as /x is increased. Over a, typically narrow range, then there are two limit cycles, one unstable and one stable around the (stable) steady-state point. If we start with the system at some large value of JJL, so we settle onto the steady-state locus, and then decrease the parameter, we will first swap to oscillations at the Hopf point juf- At this point there is a stable limit cycle available as the system departs from the now unstable steady-state, but this stable limit cycle is not born at this point and so already has a relatively large amplitude. We would expect to


Hi

m

£

Hi*

H»*

p.

Fig. 5.11. Variation of the oscillatory (limit cycle) solution with /x for the simple Salnikov model showing that the stable limit cycle born at one supercritical Hopf bifurcation exists over the whole range of the unstable steady-state, shrinking to zero amplitude at the other Hopf point; (b) in this case, each Hopf point gives rise to a different limit cycle, with a stable limit cycle born at /if growing as /JL increases and an unstable limit cycle born at /x* also increasing in size as \x increases. At some /JL > /xj the two limit cycles collide and are extinguished.

see the system suddenly jump from steady-state to large amplitude oscillations in a hard excitation process. Furthermore, if we now increase the parameter again, we stay on the stable limit cycle locus until the later bifurcation point where this coalesces with the unstable cycle born at /**• Thus, there is a region of hysteresis between steady-state and oscillatory behaviour characteristic of a subcritical Hopf bifurcation, although in many real systems this is too small to be clearly identifiable in practice. There

482


Ch. 5

are other important ways in which limit cycles disappear or lose stability that we will discuss later. Experimental confirmation of the existence of oscillations in a system with Salnikov-type kinetics has been obtained [17] by Griffiths and co-workers. We saw earlier that the simple model for a single exothermic reaction in a CSTR gives rise to multiple steady-state solutions and also that a mathematical recipe for locating the critical turning points in the steadystate locus involved, in general terms, the determinant of the Jacobian matrix becoming equal to zero. Referring back to the eigenvalue equation above, we can see that under such a circumstance one of the eigenvalues A will become equal to zero. If the determinant changes sign from positive to negative, then we will have a situation where one root will be positive and the other negative, irrespective of the sign or magnitude of tr(J). The evolution of any perturbation will then be governed by one exponentially decreasing term but also by one exponentially increasing term in equation (5.50). Eventually, the growth term will dominate and the system will move away from the vicinity of the steady-state even though there may some initial "fast motion" in its direction. The steady-state point is known as a saddle point and its phase portrait was given earlier in Fig. 5.8(e). There is one special path that passes through the steady-state along which the coefficient of the term with the positive eigenvalue are exactly equal to zero. If the system is perturbed exactly onto this pair of saddle insets then the system will actually travel along them to the steady-state. The saddle outsets correspond to the case in which the coefficients of the terms with the negative exponent are zero. In general, a trajectory from an arbitrary point may evolve towards the steady-state parallel to the inset but will then begin to move away, ultimately parallel to the outset. The insets and outsets play important "organizing" roles in the phase plane when there are multiple steady-states or other attractors such as stable limit cycles. Saddle point steady-states are those on the middle branches in regions of hysteresis. We never encounter an isolated saddle point in chemical systems: they always occur with a node or focus partner. At a vertical turning point in a steady-state locus, where the zero eigenvalue arising because det(J) = 0 occurs, the steady-state locus splits into two branches (or two branches merge): one branch of saddles and the other which is initially a node although it may gain focal character an perhaps change stability at a Hopf point later.


483

If we consider a typical phase plane for a two-variable model in which three steady-state solutions lie (i.e., we have parameter values so that we are in a region of hysteresis) and if the steady-states corresponding to the uppermost and lowermost branches are stable, then the insets to the saddle lie across the plane so as to divide it in to two distinct regions. The inset is then known as a separatrix. If we examine the trajectories in the phase plane then we will notice that none cross the separatrix: those that start on one side end up approaching one of the stable steady-states and those that start on the other approach the other steady-state. Thus, the separatrix defines the edges of the two basins of attraction of the individual stable steady-state attractors on the phase plane as indicated in Fig. 5.12. In the simplest cases, the separatrix lies fairly simply across the phase plane, but it is not unusual for the insets to weave complex dances across the plane, although they avoid intersecting each other except under very special (but also very interesting) circumstances. If the insets are wound in a complex manner so as to divide the plane into interlocked thin strips, the different initial conditions that lie quite close but in different bands can end up evolving to different steady-states. An important event in the phase plane occurs if the inset to a saddle manages to join up with an outset from the same saddle point. This then gives rise to a closed loop with the saddle point lying on it as a "corner". Such a loop is known as a homoclinic orbit as it forms a path connecting

Fig. 5.12. Typical phase plane arrangement of two co-existing stable-steady-states separated by a saddle point. The insets to the saddle divide the phase plane into two parts: trajectories starting above this separatrix tend to the steady-state point in the upper righthand part of the plane; those starting below tend to the lower steady-state.

484


Ch. 5

the steady-state to itself. (There can also arise heteroclinic orbits which are closed paths connecting two different steady-states.) A homoclinic orbit only occurs for a particular value of the parameter being varied. If the parameter is varied a bit further, then the orbit may break up so that it is no longer closed, or it may shed off from the saddle point to form a separate closed loop in the phase plane independent from any steady-state point. The latter, of course, is simply a limit cycle of the type seen earlier, so homoclinic orbit formation is important as another mechanism for limit cycle (and hence oscillation) formation or, in reverse, extinction in systems that exhibit multiple steady-states as sketched in Fig. 5.13. One more model scheme is of interest: the Gray-Yang model for some aspects of the low temperature oxidation of hydrocarbons [18-21]. This involves the features of chemical and thermal feedback described previously with a chain-carrier X coupled to the temperature T. Four reaction steps are required: (GY1)

initiation

A->X

rate = kxa

(GY2)

branching

A + X —> 2X

(GY3)

termination I

X—>P

rate = ktXx

(GY4)

termination II

X->Q

rate = kt2x

rate = kbax

In the original formulation, the reactant concentration a is assumed to be constant, so this is not being used to model full ignition problems. The

(a)

(b)

(c)

1 £r \cr Fig. 5.13. Formation of a stable limit cycle about an unstable steady-state through a homoclinic orbit as the inset and outset of a saddle point merge.

485


branching reaction (GY2) and the termination step (GY4) are taken to be exothermic, with the termination step being the more strongly exothermic of the two. The termination step (GY3) is taken to be thermoneutral (in fact, it is only necessary that this be less exothermic than the branching step) as is the initiation step (GY1). Steps (GY1), (GY2) and (GY4) are taken to have rate constants following the Arrhenius temperature dependence, with Et2 > Eb. The activation energy Etl for the termination step (GY3) is taken to be effectively zero, so that the rate constant is independent of temperature, but again the requirement is actually less strict as Etl < Eb. The mass and heat balance equation for this scheme have the form dx — = ki(T)a + cl>(T)x, dt (cPp)^= at

qMT)a

+ © ( 7 > ~^(TV

(5.59a) 7 a ),

(5.59b)

where = kb - kn - kt2 is the net branching factor and @ is a related quantity @ = qjtb + qnKi + qtiK2,

(5.60)

involving the reaction exothermcities. Both of these factors are temperature dependent via the rate coefficients. The inequalities described above for the exothermicities and activation energies mean that at low temperatures there is an overall exothermic reaction involving steps (GY1) and (GY3). As the temperature increases, so the initiation rate and the branching rate increase, leading to an increase in the number of chain carriers and to a greater rate of heat evolution. On further increasing the temperature of the reacting mixture, the second termination step becomes significant and, although this is also exothermic, the consequent decrease in the net branching factor sees a fall in the chain carrier concentration and, hence, an overall reduction in the rate of heat release. At very high temperatures, the rate of heat release increases again as the rate of the initiation step increases. The resulting dependence of

486


Ch. 5

gas temperature, T Fig. 5.14. Thermal diagram (compare Fig. 5.4) for the Gray-Yang model showing a maximum and minimum and a region of negative temperature coefficient. Also shown are three heat loss lines: Li intersects R four times, with states I and III being potential stable steady-states; L2 is a critical case with I and II merging at a point of tangency, so the system would have to jump to the ntc region (steady glow or cool-flame); L3 has a different tangency corresponding to the critical condition for ignition.

the total rate of heat release on the reacting mixture temperature, i.e., the thermal diagram for this model, may have the form shown in Fig. 5.14, with a maximum and a minimum superimposed on the general exponential increases seen earlier for the single one-step exothermic reaction. In between the extrema is a region in which the rate of heat evolution decreases with increasing temperature, a phenomenon known as the negative temperature coefficient or ntc. This region is also intimately connected with the limit cycle oscillation behaviour in this model scheme. The basic Gray-Yang scheme has been extended by Wang and Mou [22] who wrote the proper forms of the governing equations appropriate to a CSTR and allowed for the consumption of the reactant A and an additional branching step (GY5)

branching II

A + X-+2X

rate = kb2ax

This creates a model with three independent variables, a, x and T and allows for some more complex responses and for behaviour that might be related to the full ignition of hydrocarbons.

487

The H2 + 0 2 reaction 5.7 THE H 2 + 0 2 REACTION: p-Ta IGNITION LIMITS IN CLOSED VESSELS

The reaction between hydrogen and oxygen has been thoroughly reviewed elsewhere on several occasions [2, 4, 9, 23] and so the account of the classical behaviour in closed vessels will be restricted here to the basic features necessary for setting the background and interpreting the "new" behaviour from studies in open (flow) reactors. Figure 5.15 shows the classic form of the three pressure-temperature (p-Ta) ignition limits for the H 2 + 0 2 reaction in a static, closed reaction vessel. The experimentalist has control over the mixture composition, the total pressure and the ambient temperature (i.e., the temperature of the oven in which the reactor is enclosed) as well as other less obvious factors such as the coating on the inside of the vessel walls. Depending on the values for these various parameters, the reaction at reduced pressure typically has one of two qualitative forms. Either the reaction is very slow, perhaps even undetectable, or it occurs rapidly on a millisecond timescale. In either case, the final product is essentially complete conversion to water vapour, but the route from reactants to products is obviously different to the observer. The locus of pressure and ambient temperature conditions that separate

third limit

°: 400 ignition

GO

00 CL,

5

700 800 ambient temperature, Ta/K Fig. 5.15. Schematic representation of the p-Ta ignition limits for the hydrogen + oxygen reaction in a closed reactor.

488


Ch. 5

P2

pressure, p Fig. 5.16. Schematic variation of "reaction rate" as a function of pressure at fixed ambient temperature showing dramatic increase in rate approaching the three limits.

these two types of behaviour for a given mixture composition and vessel preparation forms an approximately Z-shaped curve comprised of the first, second and third "explosion limits" in order of increasing pressure. At any pressure, the system moves from "slow reaction" to "ignition" as the ambient temperature is increased, but the response to changes in pressure is more interesting. Starting at very low pressures, the reaction rate is extremely low, but increases with increasing pressure until the first limit is reached, at which point ignition appears with a high instantaneous rate. Ignition is the response across a range of pressures, but as we cross the second limit, so the "rate" falls again. The first and second limits form a peninsula on the ignition diagram. If the pressure is increased further, the rate increases again and becomes high as the third limit is approached. This description is frequently illustrated with a diagram of the form of Fig. 5.16. Here the term "rate" needs a rather careful interpretation as the system does not really attain a steady-state at any point: rather the maximum rate is generally much lower outside the ignition region than inside where it becomes high (although always remains finite). The reaction in the region "above" the second limit (i.e., at higher pressure in the slow reaction zone) is sufficiently rapid to be measured with conventional methods: the exothermic reaction can support transient temperature excesses (gas temperature being heated above the ambient temperature) of several kelvin [24] as indicated in Fig. 5.17.

489

The H2 + 0 2 reaction

(a) 250

(b) S

s

\

s

(c)

, AT = 3K NAT = 5K

\

NÂT = 3 K

^

4T = 2 K ^ ^

AL

N A T = 2K

AT =

AT = 3K k s

^

iks

WIKV

AT = 0 K V > - ^ îgnition

780

ignition

860

780

AW = OK

860780

ignition

860

ambient temperature, T./K Fig. 5.17. Self-heating in the slow reaction zone above the second limit in the H 2 + 0 2 system: (a) large vessel, equimolar mixture; (b) small vessel equimolar mixture; (c) small vessel stoichiometric mixture. The numbers indicate the maximum instantaneous temperature excess observed at a given p-Ta location and the "isotherms" connect points at which the same AT is observed. (Reprinted with permission from reference [24], © Royal Society of Chemistry.)

5.7.1 First limit The location of the first limit is sensitive to a number of experimental parameters including the vessel diameter, surface coating, 0 2 concentration and also to packing the vessel with glass rods (another way of varying the surface: volume ratio). These features are explained by invoking the competition between the branching cycle (steps 1-3) and the termination step (4) occurring on the vessel surface. As described earlier, the net branching factor for this system would have the form $ = kb-

kt = 2k2[02] - k4.

(5.61)

The rate constant k2 shows a typical Arrhenius form, at least over a modest range of temperature, while k4 involves a combination of diffusion to the walls and a subsequent surface-phase reaction.

490


Ch. 5

The condition for criticality, i.e., for the first limit, will then be parametrized by cf> = 0, i.e., by the condition (5.62)

2k2[02] = k4.

In unstirred systems, the effective rate of diffusion will be decreased by increasing the total pressure, allowing inert gases to influence the explosion pressure. The concentration of 0 2 will be directly related to the partial pressure, (5.63)

[O2]=p02/RT.

If we ignore the effect of p on k4, which may be much smaller in wellstirred systems, the ignition condition can be written as Po2,.r = W t c c r = ( ^ ) RT = ( ^ )

e+*'*

(5.64)

where ptot,cr is the limit pressure for a given mole fraction x02 of 0 2 , indicating that the limit pressure decreases as T increases. Different surface coatings affect the value of k4, the surface termination coefficient. For some particularly inert or "reflective" surfaces, k4 ~ 0 and so the first limit disappears to zero pressure. 5.7.2 Second limit The important part of the mechanism applicable to pressures and temperatures along the major part of the second limit involves the competition between the branching cycle (l)-(3) and a gas-phase "termination" step (5)

H + 02 + M ^ H 0 2 + M

This is effectively a termination step if the species H 0 2 does not continue the reaction chain. The latter situation arises if, say, the radical is efficiently removed in a surface reaction of the form (6)

H 0 2 —> products at wall

Provided step (5) is rate determining in this termination step then the termination rate rt = /c5[H][02][M] and the net branching factor will have

The H2 + 0 2 reaction

491

the form cf> = 2k2[02]-k5[02][M],

(5.65)

and the condition = 0 for the second limit will be equivalent to (5.66)

2k2 = k5[M].

In this expression, the term [M] is a representation of the concentration of "third body" species M. The role of M can be played by any molecule or radical in the vessel and is essentially to stabilize the H 0 2 by energy transfer. In general, only species present in significant concentrations will be important here, but different species have different effectivenesses at facilitating the energy transfer. This is accommodated by assigning different values of third body efficiencies a{ to different species /. The termination rate can then be written as rt = kf2 (xUl + a02Xo2 + 2 a{x{ J -^z [0 2 ],

(5.67)

where kf2 is the rate coefficient for step (5) with H 2 acting as the third body M and the summation is over all species present in addition to H 2 and 0 2 . The relative third-body efficiency ax any species / is the ratio kl5/k™2 and, thus, by definition aU2 = 1. For 0 2 , a0l ~ 0.3, i.e., 0 2 is only 30% as effective as H 2 at stabilizing the H 0 2 species, so as H 2 is replaced by 0 2 in a mixture, the net efficiency of the termination process decreases. The ignition limit criterion $ = 0 can now be written as 2k2 = kf2 (xU2 + a02x02 + E axxx \ - ^ f , or 2k2RT kf2 (aU2 + x02x02 + 2 axx{

(5.68)

492


Ch. 5

This indicates that the limit pressure will increase as the temperature increases mainly through the Arrhenius temperature dependence of the branching step rate coefficient k2. The termination step relies on collisional energy transfer and its net rate is likely to decreases as the temperature (and, hence, average energy of the reacting species) increases, equivalent to a negative temperature coefficient (activation energy). The influence of the mixture composition on the limit pressure arises from the term in the denominator relating to the relative third body efficiencies. If H 2 is replaced by a less efficient third body, so ptot,cr increases, i.e., ignition arises at a lower temperature for a given fixed pressure. Conversely, if H 2 is replaced by a more efficient third-body, the termination rate is enhanced and the limit pressure decreases or, equivalently, the limit moves to higher temperatures. An important case of this arises if H 2 is replaced by H 2 0 , the product of the reaction. Water vapour is particularly efficient as a third body in step (5), with aU20 typically being regarded as having a value of approximately 6.3 (although a recent direct measurement has suggested that this may need to be reduced by a factor of 3: see Section 3.2.2(iii)). If a mixture of H 2 and 0 2 is maintained at some pressure above the corresponding second limit for a significant period of time, the slow reaction will cause the production of some water. This in turn will cause the limit for the instantaneous mixture to change its location. Baldwin and Walker [25] describe this for a series of experiments in which the location of the limit is determined by premixing the reactants at a pressure well above the limit and then withdrawing gas through a capillary. They noted that the observed limit pressure depends on the rate of withdrawal of gas and that, with a sufficiently low withdrawal rate the ignition, could be completely suppressed by the formation of the "inhibitor" product. The inhibiting role of the product on the reaction is also of great importance in interpreting flow-reactor phenomena and we return to this in a later section. 5.7.3 Reactions involving H02 and the third limit This simple interpretation of the second-limit mechanism is appropriate provided the reaction vessel surface is "efficient" with respect to removal of the species H 0 2 . This situation arises with many salt-coated vessels, with KC1 being widely exploited. Provided this is arranged, the rate determining part of the termination process is the gas-phase step (5) and the

The H2 + 0 2 reaction TABLE 5.1 The Baldwin-Walker mechanism H2 + 0 2 H2 + 0 2 H 2 + OH H + 02 0 + H2 H + 02 + M H O OH H02 + H02 H02 + H H02 + H H02 + H H 0 2 + H2 H202 + M H202 + H H202 + H H 2 0 2 + OH H202 + O H20 + 0 H20 + H OH + O OH + H OH + OH H + OH + M H+H+M OH + OH + M O+O+M H02 + M H2 + M 02 + M

-* -* -> -» -» —> -> -> -> -* -» —» -» -> -> -» —> —> -* -> -» —» —> -* -> -> -> -» -> -> ->

20H H02 + H H20 + H OH + O OH + H H02 + M wall wall wall H202 + 0 2 20H H20 + 0 H2 + 0 2 H202 + H 20H + M H2 + H 0 2 H 2 0 + OH H20 + H02 OH + H 0 2 20H H 2 + OH 02 + H H2 + 0 H20 + 0 H20 + M H2 + M H202 + M 02 + M H + OH + M 2H + M 20+ M

limit is relatively insensitive to vessel size. With other coatings, and in particular with "aged boric acid" which has been extensively exploited by the Baldwin-Walker group [26-28] for the determination of reaction rate constants, the removal of H 0 2 at the walls is not significant. In such systems, the H 0 2 concentration will increase with time and further reactions of this species may become important. Of these, the reaction (7)

H 0 2 + H 0 2 -* H 2 0 2 + 0 2

494


Ch. 5

sees the formation of hydrogen peroxide. This reaction will also become important even for vessels with efficient surfaces at sufficiently high pressure. Hydrogen peroxide can dissociate thermally (8)

H202 + M->20H + M

returning active chain carriers to the radical pool. This decreases the effectiveness of the termination process and, eventually, will provide for a less negative net branching factor. The latter effect, along with the increasing rate of heat generation, brings the possibility of a combined chain-thermal ignition which is believed to be the character of the third limit. Additional reactions that need to be considered when modelling the H 2 + 0 2 reaction in the vicinity of the second limit are given in the Baldwin-Walker scheme in Table 5.1.

5.8 FLOW REACTOR STUDIES OF THE H2 + 0 2 REACTION

The H 2 + 0 2 reaction has been studied in the vicinity of second explosion limit pressures in well-stirred, continuous-flow reactors over the last 15 years [29-38]. The typical arrangement of the apparatus is shown in Fig. 5.18. Experiments are commonly performed by setting the inflow rates with electronic flow controllers to achieve the desired mixture composition and residence time at a specified total pressure. The inflow channels of fuel and oxidant are preheated separately before reactor entry which is achieved through jet nozzles to enhance mixing. In some experiments, additional mechanical mixing is provided. During an experiment it is generally most convenient to maintain a constant total pressure (controlled by a needle valve in the outflow line) and to vary the experimental conditions through the oven temperature Ta as required. In between small step changes in Ta, the system will typically be allowed to adjust to a steadystate appropriate to the operating conditions, i.e., the system is left for sufficient time for transient features to die out. The p-Ta ignition diagram obtained in the above manner shows an explosion limit, Fig. 5.19, similar in both form and location to that observed for identical mixture compositions in closed vessels and also to similar slow reaction behaviour for pressures above the limit. Use of a flow reactor, however, also allows the behaviour on the "ignition" side of the

495

Flow reactor studies of the The H2 + 0 2 reaction Oven

Preheating Coils

Reaction Vessel

V

Liquid Nitrogen Trap and Vacuum Pump Chart Recorder

Mass Flow Controllers

\ Thermocouple

Photomultiplier

Jet Nozzle

Personal Computer

Fig. 5.18. Diagrammatic representation of flow-reactor apparatus for studying combustion reactions. (Reprinted with permission from reference [33], © Royal Society of Chemistry.)

50 r13 o

40

slow reaction

steady ignited state

3.0

20

10 0 650

700

750

800

ambient temperature Ta/K Fig. 5.19. Thep-T a ignition diagram for an equimolar H 2 + 0 2 mixture with mean residence time tres = 5.2 0.7 s showing region of slow reaction separated by second limit from regions of oscillatory ignition and steady-ignited state. (Reprinted with permission from reference [33], © Royal Society of Chemistry.)

496


Ch. 5

limit to be investigated experimentally. In a closed system, the ignition encountered on crossing the limit sees the rapid completion of all reaction with no further kinetic developments. In a flow system, the products of the ignition will subsequently be replaced by an inflow of fresh reactants. Thus, we can imagine at least one of two scenarios: either the inflowing reactants will support a continuation of the ignition process giving rise to a "steady flame" in which there is a steady combustion of the fuel + oxidant stream at a rate matching its inflow; or, we might expect a periodic sequence of ignition event separated by periods of relatively little chemistry but during which the mixture composition is changing under the influence of the inflow and outflow. For an equimolar mixture ( H 2 : 0 2 = 1:1) and if the total pressure in the reactor exceeds approximately 30Torr, then crossing the ignition limit by increasing the ambient temperature causes the reaction to jump to a steadystate corresponding to a steady, high consumption of the incoming reactants. That is, the steady-state concentration of H 2 in the reactor and outlet is approximately zero, and the 0 2 concentration is one-half of its inflow concentration. The reaction rate is not particularly high (compared with those associated with a normal burning or flame state) as it is limited by the inflow rates which are relatively low in such systems (fres is of the order of seconds and is long compared to the Newtonian cooling timescale). If the ambient temperature is subsequently decreased, the system may stay in this high reaction state to temperatures well below the ignition limit temperature located on the upward sweep, with an extinction event at some lower Ta. This corresponds to the type of steady-state hysteresis loop described in Section 5.3 and we can sketch the bifurcation diagram for this system as in Fig. 5.20(a) with a branch of unstable steady-states (saddle points) in between. Pursuing the dynamical systems theory description of this response, we see that the ignition limit can now be identified with the turning point or "saddle-node bifurcation point" in the steadystate locus where the branch of low reaction rate (high [H2]ss) states meets the middle, saddle point branch. Similarly, the extinction point, which has no analogy in the closed system, is also a turning point in the steady-state locus. If the reaction is perturbed while the system is in the ignited state, e.g., by a momentary disturbance to the inflow, there is typically a damped oscillatory return indicating stable focal character. If the system is in the region of hysteresis and the perturbation is sufficiently large, it can cause a transition to the low reaction rate (steady slow reaction) state. The slow

Flow reactor studies of the The H2 + 0 2 reaction

T x

497

T a,cr

x

a,Hopf

Fig. 5.20. Schematic representations of bifurcation diagram appropriate to H 2 + 0 2 system in flow reactor, (a) A low conversion (high [H2]) steady-state exists at low ambient temperature but terminates in an ignition turning point at r a , cr at which the system must jump to the lower branch of high conversion steady-state. If Ta is subsequently reduced, the system can stay on the lower branch for Ta < Ta,CT until the extinction turning point. There is experimental evidence for hysteresis at the limit between steady slow reaction and steady ignited states, (b) Variation in experimental conditions allows high reaction (low [H2]ss) branch to lose stability over range of ambient temperature (for Ta < Ta,Hop{). System evolves along high [H2]ss (low reaction) branch until Ta = Ta,CT at which point it evolves to large amplitude limit cycle (sic). As Ta is increased, amplitude of oscillations decreases, and a steady-state emerges for Ta > Ta,Hop{. If the ambient temperature is reduced again, there is no hysteresis, with oscillatory ignition re-appearing at r a , Hop f and the oscillations terminating at (the saddle-node) bifurcation at r a , cr . (c) As in (b) except that the limit cycle locus is now folded: two separate branches of stable limit cycles (sic) corresponding to different oscillatory solution now exist, separated by a branch of unstable limit cycles (ulc). Hysteresis is observed between the large- and small-amplitude oscillations over a (typically narrow) range of ambient temperature.

498


Ch. 5

reaction branch typically has stable nodal character with perturbations decaying monotonically. At a lower total pressure, the reaction again exhibits a low reaction steady-state at low Ta, equivalent to the slow reaction in a closed vessel. This lies on a branch of steady-states that terminates with a saddle-node bifurcation point corresponding to the ignition limit, as indicated in Fig. 5.20(b). In this case, however, the steady-state corresponding to the ignited state is not stable and so the system does not jump to a steady-flame state. Instead it oscillates around the ignited state, showing an oscillatory sequence of ignitions. The typical waveform for a set of experimental conditions lying just beyond the ignition limit is shown in Fig. 5.21, in which the H 2 , H 2 0 and a nominal record of the gas temperature (which is different from the ambient or oven temperature) and the emitted visible light intensity are plotted as a function of time. The temperature is recorded in such experiments using fine-wire thermocouple junctions (Pt/Pt + 13%Rh, coated with silica to avoid catalysis), but these have significant response times compared with the short timescale of the sharp ignition phase in these "relaxation-type" oscillations. (The response-time distortion of the signal is even more pronounced if the voltage is output directly to a traditional chart recorder, but this is reduced with an oscilloscope or A/D converters coupled to computer-based data acquisition systems.) A more realistic record of the temperature excursion is obtained indirectly from the rotational level populations from laser absorption studies on the OH radical concentration, Fig. 5.22, indicating that temperature rises approaching the adiabatic temperature excess are achieved. The H atom concentration has been followed using resonance enhanced multiphoton ionization (REMPI) [34-38]. For conditions close to the limit, there is virtually complete H 2 consumption during the ignition stage of the oscillation, followed by a flow-based recovery during which [H2] approaches its inflow concentration. The closer the conditions are held to the limit condition, the closer [H2] approaches [H 2 ] 0 and also the longer the period between ignition events. In a careful measurement of the variation of the oscillatory period with the ambient temperature in this region, Griffiths et al. [32] showed that the period varies approximately according to the relationship

499

Flow reactor studies of the The H2 + 0 2 reaction Light Output ( arbitary units ) AT\ K 200-1

100

100 Time \ seconds

200

100 Time \ seconds

200

[H:0]

50 100 Time \ seconds

10,]

mm 50 100 Time \ seconds

Fig. 5.21. Typical instrumental records for oscillatory ignition; (a) AT as measured by finewire thermocouple; (b) light emission intensity as measured by photomultiplier; (c) [H 2 0] and (d) [0 2 ] as measured by mass spectrometer. (Reprinted with permission from reference [33], © Royal Society of Chemistry.)

500


Ch. 5

8.0-

-2500

6.0-

h2000

4.0-

-1500 ^

2.0-

-1000

-500

t/ms

Fig. 5.22. Records of [OH], [H] and T from rovibrational spectrum and REMPI studies. (Reprinted with permission from reference [37], © Royal Society of London.)

where Ta,CT is the value of the ambient temperature exactly on the ignition limit, implying an infinite period as Ta —> Ta^CT. Experimentally, a period of 158 s was recorded for a system with tres = 2 s by holding the ambient temperature 0.05 K above r a?cr . The lengthening of the period close to the limit is illustrated in Fig. 5.23. If the ambient temperature is raised so that the system is taken further into the ignition region, the amplitude of the oscillations decreases and their frequency increases. For Ta sufficiently high, the amplitude decreases to zero and a steady-state (steady flame) emerges. The period of the oscillations remains finite as the ambient temperature approaches this oscillatory extinction. Figure 5.24 shows how the waveform changes with Ta close to this upper limit to the oscillatory range. The waveforms become noticeably more sinusoidal as evidenced by the fourier transforms with the waveform for Ta = 800.5 K having only a single frequency component. The limit cycles corresponding to these oscillations can be obtained from a single experimental record (most conveniently, the thermocouple signal) using the "method of delays". In this, the value of the signal at some time t is plotted against the value of the same signal at a later time t + td where

501


T./K

727.80

727.00

724.00

723.70

723.50

8

1

T a (t0.05)/K

I

I

I

723.20

I

L

extinction

$723.15

Fig. 5.23. Variation of oscillatory period in vicinity of ignition limit (saddle-node bifurcation point) showing extreme lengthening as limit is approached. (Reprinted with permission from reference [32], © Manchester University Press.)

td is known as the "delay time". This plot is performed for all the points in a time series and gives rise to the limit cycles shown in Fig. 5.25 for the present system. The choice of td is not particularly important but is typically taken to be approximately one-half of the oscillatory period. (The additional "chaining" on the trajectories in this figure corresponds to a frequency of 50 Hz and is due to electrical interference picked up through the thermocouple leads.) As Ta increases, so the limit cycle shrinks to a point (corresponding to the stable steady-flame state. If the ambient temperature is reduced again, oscillations return at the same temperature at which they were lost, indicating a lack of hysteresis. These features are, therefore, characteristic of a supercritical Hopf bifurcation (see Section 5.5.d). We can now sketch the corresponding bifurcation diagram for this pressure and mixture composition, Fig. 5.20(b). This again shows a folded steady-state curve but with the additional feature of the Hopf point along the high reaction rate (low [H2]ss) branch at some temperature r a?Hopf . The high reaction state is unstable for Ta < ra?HoPf a n d stable for higher ambient temperatures. For Ta < Taôp{, the high reaction state is surrounded by a stable limit cycle whose amplitude is indicated on the bifurcation diagram by plotting the maximum and minimum values of [H2]

502


Ch. 5

784 OK

AT \ K 100 AT\ K

T = 799.8K

60H I/VVVVWSA/VN/VVVVNAA/VVVNA/VVVWNA'VVS/WVVW^

Ta = 797.1 K

30

60 10

0

30

AT \ K 0 AT \ K

TQ = 800.5K

™ 60798.5K

30 J

60 A 10

30 J Al

10 60 "

0 AT \ K

\ = 800.6K

T = 799.3K

30 -

60 J

0 -

i

'

•' r

10

30 JWWVV\AAAAAAAAAAAA^^

Time \ seconds

0 10 Time \ seconds

Fig. 5.24. Variation of oscillatory waveform in vicinity of boundary between oscillatory and steady ignition showing characteristic nature of a supercritical Hopf bifurcation. (Reprinted with permission from reference [33], © Royal Society of Chemistry.)

attained during the oscillation. This limit cycle increases as Ta is decreased away from r a?Hopf but undergoes an extinction at lower ambient temperatures. Experimentally, it is observed that this extinction typically occurs at approximately the same ambient temperature as that at which the saddle-node bifurcation (ignition limit point) is located as Ta is increased

503


800.5K

3

799.3K

o O

o

o

T3

798.5K

797.1K

digitised thermocouple output Fig. 5.25. Reconstructed limit cycles for oscillations shown in Fig. 5.24. (Reprinted with permission from reference [33], © Royal Society of Chemistry.)

from the low reaction steady-state. This, along with the observed scaling law for the oscillatory period given above, suggests that the limit cycle is lost by collision with the turning point that marks the ignition point on the diagram, as illustrated in Fig. 5.20(b). Technically, this is known as the formation of a homoclinic orbit to a saddle-node point (earlier we saw the extinction of a limit cycle by formation of a saddle point for which the inset and outset connected into a loop). The inverse square-root scaling is characteristic of such as structure. This would account for the lack of hysteresis between the slow reaction state and the oscillatory ignition at the limit. Thus, the ambient temperature range between r a c r and ra?Hopf bounds the range of oscillatory ignition in this instance. Both of these points move as we vary the total pressure between experiments: the variation of TacT corresponds to the ignition limit itself while the variation of r a H o p f describes the second boundary on Fig. 5.19 giving the upper temperature limit to the region of oscillatory ignition. The latter curve appears to approach the ignition limit at a definite angle at the upper end of the pressure range. Arguing from the shape of the bifurcation diagrams

504


Ch. 5

sketched previously, and noting that there must be smooth transitions from one type to another, leads us to conclude that as ptot increases, the homoclinic orbit formation must move away from the saddle-node (ignition) point and up along the saddle point branch as TÛopf approached r a c r . In such cases, the oscillatory ignition response will survive to ambient temperatures below that corresponding to the ignition limit, as indicated in Fig. 5.19. We should then see hysteresis between the low reaction state and the oscillatory ignition. Furthermore, at some pressure, the Hopf point may move along the high reaction branch so that it lies at a lower ambient temperature than the ignition limit. The system should then jump to a high reaction steady-state for Ta > r a>cr on the initial upward sweep in ambient temperature, but on the way down will show hysteresis in which the system loses stability of the steady flame to oscillatory ignition. In practice the region of the experimental conditions under which such relatively complex behaviours are exhibited are very small and so these are unlikely to be observed by accident. However, armed with the above argument it has been possible to search successfully for such behaviour in a narrow range of pressure above the crossing point of the two loci on the bifurcation diagram [33,39].

5.9 COMPLEXITY IN THE OSCILLATORY IGNITION REGION

Returning to the variation of the limit cycle amplitude across the region of oscillatory ignition, for the equimolar mixture this is relatively simple, with the amplitude decreasing continuously. Typically, there is a narrow range of Ta over which the amplitude decreases very rapidly as the waveform changes from the relaxation type observed close to the ignition limit, to the sinusoidal character observed near to r a?Hopf . Such a region of rapid change is known rather misleadingly as a "canard" (i.e., false) bifurcation (as there is no qualitative change). Experimentally, it is often difficult to distinguish between a very rapid change and a genuine discontinuity, so care is needed in these instances. With different mixture compositions, however, the range of behaviour supported becomes richer. The variation of the oscillatory period with the ambient temperature for a stoichiometric mixture (2H2 + 0 2 ) at an operating pressure of 16 Torr and a residence time of 4 s is compared with

Complexity in the oscillatory ignition region

505

8

T3

a. 4

0

725

750 775" TJK Fig. 5.26. Variation of oscillatory period with ambient temperature for different H 2 + 0 2 systems: (a) p = 14 Torr, tres = 4 s; (b) p = 14Torr, tres = 2 s, showing region of birhythmicity. (Reprinted with permission from reference [40], © Combustion Institute.)

that of the equimolar mixture in Fig. 5.26. Whereas the period simply decreases across the range for the latter, there is a region of oscillatory hysteresis for the stoichiometric case. Over a range of approximately 730 K < Ta < 745 K, the reaction exhibits either relatively large amplitude ignition events or small amplitude, more sinusoidal oscillations, depending on whether the ambient temperature is being increased from low values or decreased from the steady-flame state at high ambient temperature [40]. The coexistence of two different oscillatory states for the same experimental conditions is analogous to the coexistence of steady-states (multistability) seen earlier, and is known as birhythmicity. We can imagine this arising from a folding of the limit cycle locus in the bifurcation diagram, as indicated in Fig. 5.20(c), just as steady-state multi-stability arises from a folding of the steady-state locus. This also indicates that there will be a third oscillatory state, corresponding to a branch of unstable limit cycle solutions between the two different stable oscillatory states. Thus, there are three limit cycles arranged concentrically around the (unstable) steadystate point in the phase plane, as indicated in Fig. 5.27. The fold in the

506


Ch. 5

Fig. 5.27. Schematic representation of three co-existing limit cycles surrounding an unstable steady-state. The middle limit cycle is unstable: trajectories that start outside this evolve to the outer stable cycle; those that start inside the unstable limit cycle evolve to the smallest, stable cycle.

limit cycle locus must arise as the mixture composition, which plays the role of an unfolding parameter in this system, is varied from equimolar to stoichiometric. The unfolding of the birhythmicity feature can also be accomplished by varying other parameters, such as the residence time (with birhythmicity favoured by low fres) or the total pressure. Another type of complexity is a feature for stoichiometric mixtures at certain pressures and residence times [33]. Thep-T a ignition limit diagram s shown in Fig. 5.28. Within the for such a system with tTes = region of oscillatory ignition lies a subregion denoted "complex oscillations". The complex oscillations in this system have a mixed mode waveform. These comprise of a single large ignition followed by a number of small amplitude oscillations before the process repeats. A selection of observed mixed-mode states are shown in Fig. 5.29. Also displayed are the associated limit cycles obtained by the method of delays described previously. These show an apparent crossing corresponding to the small amplitude events. In fact, as stated earlier, trajectories in the phase plane cannot cross, except at the singular points of the system. In this case, then, the trajectory must be embedded in a phase space of more than two dimensions, so the 2-D portraits shown are just projections of a trajectory winding around in at least 3 dimensions. The crossings are, therefore, nothing more than one part of the cycle passing above or below another

Complexity in the oscillatory ignition region

507

50 40

slow reaction steady ignited state

20 10 650

region of complex oscillations

700 750 800 ambient temperature, Ta/K

Fig. 5.28. The p-Ta ignition diagram for a stoichiometric 2H2 + 0 2 mixture with mean residence time tTes = 2.0 0.2 s showing additional region of complex oscillatory ignition. (Reprinted with permission from reference [33], © Royal Society of Chemistry.)

part but not actually intersecting. (We should remember that there is really one independent variable representing each chemical species involved in the reaction mechanism + one more for the reacting mixture temperature, so the H 2 + 0 2 system is genuinely multidimensional from the point-ofview of dynamical systems analysis. It is striking how for much of its behaviour this, and most other chemical reactions, confine themselves to behaviour that lies in low-dimensional phase spaces, e.g., a 0-D steadystate or a 1-D limit cycle embedded in a 2- or 3-D phase space.) As the ambient temperature is increased across the region of mixedmode states, so the number of small amplitude oscillations in the repeating unit increases. A particular waveform can be denoted by the symbol 1", where n is the number of small events, as n increases with Ta. As we approach the righthand limit (high Ta) so that n effectively tends to infinity, i.e., the large amplitude ignition disappears and we have simply repetitive, small amplitude oscillations which undergo a Hopf bifurcation to yield the stable steady-state at some higher ambient temperature. Mixed-mode oscillations are also favoured by shorter residence times and by mixtures

508


T

.•

200-

Ch. 5

734.5K

100-

l

00

jjLiL

I 10

20

200

-600

-1400

-2200

DIGITALISED THERMOCOUPLE OUTPUT Xlt)

30

Time \ seconds

AT \ K 1r a = 7 4 4.1K

200-

K

0.

1 j\

u1_J 10

ID

o o o

) THERM

100-

0-

O

>

1|

-MOO

X

-600

200 •

,

r

J>^~\

///

\

\\1 \

\

(

,

i

1

|

DIGITALIZED THERMOCOUPLE OUTPUT Xlll

20

Time \ seconds

E OUTPUT )

AT \ K

o3 oo

1 '\\

cc X

-1400 •

-600

, / _ _

o

>-

200 J 200

,

-600

, ,

-1400

-2200

DIGITALIZED THERMOCOUPLE OUTPUT X|t) 10

20

30

Time \ seconds

Fig. 5.29. Typical mixed-mode oscillatory ignition waveforms for stoichiometric 2H2 + 0 2 mixtures with corresponding reconstructed limit cycles. (Reprinted with permission from reference [33], © Royal Society of Chemistry.)

Mechanistic modelling of complexity in H2 + 0 2 reaction

509

of composition close to stoichiometric. Some experimental evidence now exists for so-called concatenated states. These are waveforms apparently comprising two different mixed-mode "parents", e.g., a mixing of l 1 and l 2 states to form a repeating unit of the form 1*12 lying in a region of the parameter plane between the l 1 and l 2 states. Between the 1*12 and l 2 , it is also possible to find 1*1212 oscillations in which the repeating unit involves eight events, three of which are of large amplitude and five small ignitions. Some very complicated bifurcation sequences involving hysteresis (birhythmicity) between small amplitude oscillations and mixedmode state have also been observed but not characterized in any quantitative sense to date.

5.10 MECHANISTIC MODELLING OF COMPLEXITY IN H2 + 0 2 REACTION

The first feature modelled for the flow system [29, 30] was thep-T a ignition limit (corresponding to the locus of saddle-node points in terms of the bifurcation diagram). This would correspond to an application of singularity theory as outlined in Section 5.3. In fact, these studies were predated by a singularity theory approach to computing the ignition limits for H 2 + 0 2 mixtures directly by such techniques for closed systems with reactant consumption ignored [41]. Formally, the approach would involve setting up the mass balance equations for the chemistry and flow rates for each species involved in any postulated mechanism and then solving, simultaneously, the steady-state condition and the vanishing of the determinant of the corresponding jacobian matrix. Such computations are now feasible but for the present system represent taking a large sledgehammer to a fairly innocuous nut. We can proceed with sufficient accuracy by building on the earlier analyses involving the net branching factor and how it becomes modified appropriate to flow systems. The ignition limit lies in the region of the p-Ta plane, corresponding to the second limit in classical closed vessels, and so we may surmise that the dominant features of the mechanism will be the competition between the branching cycle (1-3) and the gas-phase termination step producing H 0 2 , step (5). A full steady-state analysis on the intermediates OH and O would introduce (out)flow terms for each species and a fairly complex polynomial in terms of tTes. The full analysis appears in Chapter 4 of this volume. For now, we can note that the typical residence times of interest, 1 to 10 s,

510


Ch. 5

mean that the flow terms are relatively insignificant compared with the fast kinetic terms for these reactive species and, hence, can be neglected. Thus, we can use a similar approach to that given earlier in terms of a net branching factor governing the H atom concentration, (5.71)

4>=(2k2-k5[M])[02l

as before. To allow for the loss of H by the outflow, we can modify this to a form appropriate to a flow reactor as in Section 5.2, to yield fciow = (2fc2 " * 5 [M])[0 2 ] -

fcfiow,

(5.72)

where kf low = l/fres- Implicit in this is the assumption that the steady-state concentrations of the major reactant species do not differ significantly from their inflow concentrations along the branch of low reaction steady-states. This is unlikely to fail by more than 5%. The condition for the limit then reduces to 4>{iow = 0. In fact, for the experimental studies described in the previous sections, the influence of the flow term on the (p-Ta) location of the limit is almost insignificant. The individual values of 2k2 and, hence, A:5[M] are many orders of magnitude greater than the flow term at the typical pressure and ambient temperatures of interest, for instance 2£;2[02] ~ 103 s _ 1 for an equimolar mixture at 16 Torr and 750 K compared with /cflow ~ 0.1 s" 1 . Furthermore, the relatively high activation energy for step (2) ensures that only very small changes in Ta will be needed to allow for the -/cfiow term in equation (5.72). Thus, the problem of predicting the p-Ta limit, and the influence of mixture composition etc., is effectively the same as for equivalent closed vessels. In order to model the behaviour on the "ignition side" of the limit, however, it is clearly necessary to be more sophisticated, at least during the transient ignition stages, as there will be high radical concentrations, temperature excursions and significant temperature rises. There has been an interest in discovering the "minimal" mechanism that will predict at least some form of oscillatory event. A scheme involving the branching cycle (1-3), the termination step (5) and one extra step (9)

H + H02^H20 + 0

gives oscillatory solution for some parameter values provided certain cm-


511

cial features pertaining to reactant consumption and, in particular, product formation were also included [42-44]. For some ambient temperature Ta slightly in excess of the limit condition r a c r , the basic nature of the oscillatory ignition process involves a branched chain ignition that leads to consumption of H 2 and 0 2 and formation of H 2 0 . Remembering that the product species has a significantly higher third-body efficiency for the termination step (5) than the fuel and oxidant mixture it was formed from, the ignition serves to increase the effective value of the term k5[M] in the net branching factor. (During the actual ignition event, the temperature Twill increase dramatically, increasing the instantaneous value of /c2, but this is short-lived as the temperature falls rapidly back to the vicinity of Ta.) Prior to the ignition, fiow will have been marginally positive (as we have just crossed the limit), but the change in mixture composition will now arrange that 0 flow (based on the instantaneous, product-dominated mixture composition) will now be negative. This causes the reaction rate to fall to a low value which we can treat as zero to a first approximation. Thus, the system thus enters a quiescent phase in which changes in composition within the reactor arise solely from the flow processes. The product concentration decreases while those of the original (inflowing) reactants recover. If these are truly flow-controlled processes with no chemical contribution, then the concentrations will vary exponentially with the time elapsed since the ignition event, i.e., they will have the form [H2] = [H 2 ] 0 (l - e - ' H , [0 2 ] = ([O 2 ] 0 - [0 2 ] resid )(l - e - " - ) + [0 2 ] r e s i d ,

(5.73)

[H2O] = [ H 2 ] 0 e - ^ s ,

(5.74)

and

for a fuel-lean mixture, where [0 2 ] r e s i d is the "residual" concentration of 0 2 after the ignition event ([02]resid = 2[O2]0 for an equimolar mixture) and the concentration of H 2 0 immediately after the ignition event is equal to the concentration of the fuel before the ignition. Thus, the concentration of the inhibitor will fall to e _ 1 of its post-ignition value and the reactant

512


Ch. 5

concentrations recover to the same factor of their initial concentration within one residence time and e _ n within ^-residence times. During this period, the effective (mixture-composition dependent) value of the net branching factor will be increasing as H 2 0 is replaced by the less efficient reactant mixture. Eventually, flow will pass through zero again and another ignition will develop. From the above argument we can see why the period between successive ignition events increases the closer we are to the limit, r a?cr . The smaller the "degree of supercriticality", the smaller c/>flow based on the initial reactant concentrations will be (it is exactly equal to zero at the limit and slightly positive for Ta marginally in excess of T^cr). This means that the mixture will need to recover virtually to its original composition through the exponential flow processes above, requiring more residence times to elapse before the next ignition, the observation of Griffiths et al. [32] of a period equivalent to 60 x tres indicates such a marginally supercritical system that only once the H 2 0 concentration has decreased to e~60 of its post-ignition value can another ignition develop. As the difference between Ta and r a?cr increases, so the branching term involving the rate coefficient k2 increases and 4>f\ow based on the inflow concentrations is more positive. This means that following an ignition (/>fiow will become zero before the initial composition is regained, so ignition develops "earlier". The reactant concentrations do not recover to their inflow values at any stage, so the amplitude of the ignition will be slightly smaller. The above argument can be used to predict the oscillatory period reasonably accurately, provided Ta is neither too far above T^cr (such that there is significant chemistry occurring in the period before the actual ignition event) nor too close to r a?cr . The latter case must involve additional features as the exponential variation of the mixture composition with time would lead to a logarithmic lengthening of the period as Ta approaches â,cr? whereas, the experimentally determined form involves the inverse square root of the difference as given earlier. The latter scaling arises because once the flow has returned the reactant concentrations to their original values, the reaction almost manages to achieve a steady-state and so "hovers" in the vicinity of a state in which mass and heat balance equations almost vanish. The same scaling is observed in the vicinity of any saddle-node point, even in the simple theory of thermal runaway (see Section 5.3). The loss of oscillatory ignition can also be rationalized on the above


513

arguments. If the ambient temperature is increased sufficiently, then the branching term 2k2 will become so large that $fiow remains positive even for the system in which the H 2 0 concentration is at its highest value, i.e., [H 2 0] = [H2]o just after the ignition event. In this case, the "ignition" will continue rather than be quenched and the inflowing reactant streams will simply support a steady burning state. In order to model the oscillatory waveform and to predict the p-Ta locus for the (Hopf) bifurcation from oscillatory ignition to steady flame accurately, it is in fact necessary to include more reaction steps. Johnson et al. [45] examined the 35 reaction Baldwin-Walker scheme and obtained a number of reduced mechanisms from this in order to identify a minimal model capable of semi-quantitative p-Ta limit prediction and also of producing the complex, mixed-mode waveforms observed experimentally. The minimal scheme depends on the rate coefficient data used, with an updated set beyond that used by Chinnick et al. allowing reduction to a 10-step scheme. It is of particular interest, however, that not even the 35 reaction mechanism can predict complex oscillations unless the non-isothermal character of the reaction is included explicitly. (In computer integrations it is easy to examine the "isothermal" system by setting the reaction enthalpies equal to zero: this allows us, in effect, to examine the behaviour supported by the chemical feedback processes in this system in isolation TABLE 5.2 The "minimal complex oscillator" model for the H 2 + 0 2 reaction Rate constant, k = Ae~ E/RT

H2 + 0 2 - > 2 0 H H2 + O H - + H 2 0 + H H + 02ÔH + 0 0 + H2ÔH + H H + 02 + M->H02 + M H-^wall OH -> wall H02 + H ^ 2 0 H H02 + H-*H2 + 02 OH + 0 - > 0 2 + H

A*

(E/R)/K

g/kJmol - 1

1 x 108 2.2 x 107 2.2 x 108 1.37 x 107 5.1 x 103 12 12 2.5 x 108 2.5 x 107 1.3 x 107

35194 2590 8450 4480 -500 0 0 950 350 0

-77.748 64.0 -70.66 -8.247 199.451 0 0 159.707 238.651 8.247

*Units for A are (m3 mol *)" 1 s step in the mechanism.

1

where n is the order of each elementary

514


Ch. 5

from the thermal feedback routes. The isothermal system supports ignition and simple oscillations but not complex oscillations.) Chapter 4 of this volume discusses the formal aspects of such mechanism reduction further. The minimal complex oscillator mechanism of Johnson et al. is given in Table 5.2.

5.11 THE CO + 0 2 REACTION

The reaction between carbon monoxide and oxygen is the other "stoichiometrically simple" combustion process and has been studied for about as long as the H 2 + 0 2 reaction [46,47]. Unlike the latter, however, the progress towards a full mechanistic understanding - or even agreement amongst different workers as exactly how to interpret each other's observations in a consistent manner - has only emerged in the last two decades. The main obstacle to studying this reaction is its extreme sensitivity to traces of any species containing H atoms [48-50]: H 2 0 and CH 4 being the primary source of such contaminants. This was recognized relatively early and there were extensive programmes of research dedicated to "drying" the reactants by storage in liquid 0 2 etc. for long periods prior to mixing the gases [51-54]. In these studies the p-Ta location of the "ignition limit" shifted to higher ambient temperature as the reactants were dried further. The reaction also shows phenomena not exhibited by the H 2 + 0 2 system, including trains of oscillatory light emission that may continue over several hours even in a closed system. The reaction is sensitive to the state of preparation of the vessel surface to a degree beyond that found for the H 2 + 0 2 reaction. Relatively detailed review of the classical ignition limit phenomena can be found elsewhere [4, 5, 23], and so we concentrate here on features related to the oscillatory and steady glow in this system in closed vessels and on the behaviour in flow reactors. 5.11.1 Closed vessel studies In addition to showing branched-chain ignition, with (first and second) p-Ta limits similar to those of the H 2 + 0 2 system, the CO + 0 2 reaction supports a response known as glow. This is a spontaneous chemiluminscent state, with a weak, pale-blue emission arising from electronically-excited

The CO + 0 2 reaction

515

CO* formed from the three-body recombination reaction (10)

CO + O + M ^ C O | + M

with the triplet O atom giving rise to a triplet C 0 2 molecule. This step is a termination process (there have been occasional suggestions that excited C 0 2 can support a process of "energy-branching" with its excess enthalpy being sufficient to allow the reaction (11)

COf + 0 2 - > C 0 2 + 2 0

but there is no actual evidence in support of this type of process in this reaction). The glow can take one of two forms. The earliest reports of steady glow are due to Prettre and Laffitte [55] who observed a long-lived chemiluminescence from a mixture of the reactants in a closed vessel. Ashmore and Norrish [56] were the first to report observations (made in 1939) of oscillatory glow, which they termed the "lighthouse effect". In this mode, "bursts" of chemiluminscent emission are separated by periods of relative "darkness". Depending on the total pressure and oven temperature, short or long trains of emission were observed. Even in a closed system, they were able to record trains consisting of over 100 pulses. The reactor used by Ashmore and Norrish had been treated by exposure to chloropicrin (CC13N02) a known inhibitor of radical chains. If the surface was washed with acid, the oscillatory glow response was lost, but reappeared on further pre-treatment with the inhibitor. Linnett and various co-workers [57, 58] also observed oscillatory glow, somewhat intermittently, in their studies of the reaction with extensivelydried reactants (see above) in vessels with untreated surfaces. These authors unfortunately used the phrase "multiple explosions" to describe the phenomenon which they observed only by eye. Later, meticulous work exploiting instrumental monitoring of the reaction by Bond et al. [59, 60] revealed that the "ignition limit" reported by the Linnett group on the basis of a detectable emission actually corresponds to the limit for steady glow. The work by Bond has provided a firm framework for the interpretation of the isolated reports by Ashmore and Norrish, Linnett and co-workers and those of McCaffery and Berlad [61] and Aleksandrov and Azatyan [62], by paying particular attention to controlling the H 2 content of the

516


Ch. 5

mixture and by following light emission intensity (photomultiplier), selfheating (fine-wire thermocouple) and stable species concentrations (mass spectrometry) during the reaction. The extent of reaction accompanying a single oscillatory excursion is typically less than 0.5% if the reaction mixture is "dry" ( C 0 2 + OH

is relatively unimportant, but has the potential significance that it provides a reaction channel for the relatively unreactive radical H 0 2 and thus would help sustain the radical chain. CO and C 0 2 can also play the third-body role in the reaction (5)

H + 02 + M ^ H 0 2 + M

(a) Modelling the ignition limit The ignition limit is a feature of systems containing approximately 0.5% H 2 or more. If we begin the discussion by imagining a pure H 2 + 0 2 system in which the H 2 is then systematically replaced by CO, we find initially that the (second) limit moves to lower ambient temperatures and higher pressures. This feature can be modelled quantitatively simply by including the relative third body effect of CO, which is less than 1 (i.e., CO is a less

527

The CO + 0 2 reaction

55

(ii) (i)

fa o

15

700

ambient temperature, Ta/K

780

Fig. 5.38. Variation of CO + H 2 + 0 2 ignition limit with fuel mixture composition, all mixtures have 50% 0 2 : (i) 50% H 2 ; (ii) 10% H 2 + 40% CO; (iii) 2% H 2 + 48% CO; (iv) 1.33% H 2 ; (v) 0.5% H 2 ; (vi) 0.33% H 2 , (v) 0.167% H 2 . (Reprinted with permission from reference [67], © American Institute of Physics.)

efficient third body than H 2 in the primary termination process in the H 2 + 0 2 second limit mechanism) so replacing H 2 by CO decreases the effective termination rate and favours ignition. The ignition limit reaches its "minimum" position, furthest to the left on the p-Ta diagram as indicated in Fig. 5.38, for a mixture composition with approximately 10% H 2 . A further replacement of H 2 with CO sees the limit moving back to higher ambient temperature, as indicated in the figure. This "inhibitory" effect arises for systems with low [H2] for which the extra termination step (10) involving CO becomes able to compete with the branching process (3). Using the reactions

(i)

OH + H 2 ^ H 2 0 + H

(2)

H + 02ÔH + 0

(3)

0 + H2ÔH + H

(13)

CO + O H - * C 0 2 + H

(5)

H + 02 + M ^ H 0 2 + M

(10)

CO + O + M -> C 0 2 + M

528


Ch. 5

a quasi-steady-state analysis on the radical intermediates yields the following expression for the net branching factor [67]

* = 1 |

ir r r n i r ^ i ~ ^ [ M ] ' fc10[CO][M]

(5 75)

'

k3[H2] where k5 = k™2{xU2 + a02x02 + acoXco + K}, (5.76) with xU2 + xCo = 0.5 for an equimolar fuel + 0 2 mixture. At relatively high H 2 , the influence of CO is felt mainly through the final term in equation (5.75) and with aco < 1, increasing x c o , and, hence, decreasing jtH2> has the effect of decreasing the effective value of k4 and, hence, increasing the value of products), but also by the same change throughout the course of reaction and throughout the temperature range under investigation. The sensitivity is highest in fuel + oxygen mixtures when the considerable dilution by atmospheric nitrogen is absent.

1.0-,

8.

/

0.8.

If

/o

0.6.

i

^

°l

° \\

0.4_

0.2_

o\.°

§*r

^**

0

550

650

600

700

T/K

Fig. 6.3. First evidence for a region a negative temperature dependence of reaction rate in a closed vessel. The ordinate refers to the rate of change of reactant concentration, deduced from the fractional rate of pressure change. (After Pease [4].)

552

Experimental and numerical studies of oxidation chemistry

Ch. 6

Consider the stoichiometric equation F + y02 = zP.

(6.2)

In the absence of diluting gases and for an initial fuel: oxygen ratio = 1 :y, the governing equations are \ = z([F]o - [F]) + (1 + y)[F] = z[F]0 + (l + y- z)[F],

(6.3)

and V\dn V/ dt

0+/-z)(£),

100 K). The quantitative evidence for changes of mechanism cannot be distinguished if no temperature record is obtained simultaneously. Chemical analyses made in non-isothermal conditions must also be interpreted with caution if the system is not wellstirred so that a uniform temperature is maintained throughout the reactants [39, 40, 42, 52]. Strictly speaking the results cannot be used as a fully quantitative basis to test kinetic schemes when a zero-dimensional model is adopted. Nevertheless, there is a useful qualitative, or semi-quantitative function for the validation of models, as exemplified in Section 6.6. Many results that can serve this purpose are documented in an Appendix. 6.3.3 Flow systems Flow systems in use may be classified as heated laminar tubes, or plug flow tube reactors, (PFTR) and burners, or heated turbulent flow reactors and well-stirred reactors, or continuous stirred-tank reactors, (CSTR). The simplest flow systems, namely laminar flow tubes operated at atmospheric pressure, were used in some of the earliest chemical studies of hydrocarbon oxidation [4, 28, 53]. In this type of application, pre-mixed gaseous fuel and air flowed through a heated tube and the products were collected at the outflow under stationary-state conditions. Product compositions were analyzed and extents of reaction were deduced. Even in these earliest studies, the possibility of temperature changes owing to exothermic oxidation were noted and in some cases "mildly explosive reaction", probably cool-flames, were detected [4]. CSTRs and turbulent flow reactors have been operated mainly at pressures from 0.1 to 1 MPa, with residence times varying from many seconds to less than 100 ms at the highest operating pressures. Studies of the combustion of hydrogen [54], carbon monoxide [55] and acetaldehyde [56, 57] at sub-atmospheric pressure also have been reported. In most cases the reactants are mixed after metering and pre-heating, and then flow through the reactor at constant pressure and flow rate. The reactor temperature may be held constant, or it may be varied continuously or in a stepped manner, in order to probe the modes of behaviour for a given fuel + oxidant composition over the temperature range. A well-mixed condition can be established in the CSTR by either a

564


Ch. 6

mechanical stirrer [17, 45] or jet mixing of the gases [30, 32, 58]. The former is required when mean residence times in the reactor exceed a few seconds, since jet velocities are then too low to sustain reasonable spatial uniformity. Surface effects in a metal vessel have been minimized by Dagaut et aL [30], in their high pressure kinetic studies in a jet-stirred flow system, by surrounding a silica vessel with a stainless steel pressure chamber. Different modes of operation of CSTRs are reported. The studies by Dagaut et aL [30] in a 35 cm3 CSTR, some of which have been discussed in Section 6.2, were designed for isothermal chemical measurements at high pressures. Very high dilutions with N2 were employed so that temperature changes within the reactants during exothermic reaction were minimized. The investigations of non-isothermal phenomena at high pressures, by Lignola et aL [32], were performed in stoichiometric mixtures of fuels in air in a larger CSTR (100 cm3). A turbulent flow reactor, at Princeton University, has been operated at atmospheric pressure over many years to give information on the detailed chemistry of hydrocarbon oxidation in the temperature range 1000-1200 K. Aromatic compounds as well as alkanes and alkenes have been investigated, as summarized in the Appendix [59-67]. Constant molar proportions, RH:0 2 :N 2 , are established at the inlet, usually with a fuel:oxygen ratio close to 95% inert gas. Representation of how some of the events unfold in time is shown in Fig. 6.9. The contact surface is at the interface between the driver gas and the test gas. It is moves rapidly along the tube but only at a subsonic velocity as opposed to the supersonic velocity of the shock front. The driver gas at the contact surface is cold, and so when it envelopes the test gas, any high-temperature reactions are quenched instantly. The reaction time is represented by the interval between the arrival of the incident shock front and the arrival of the contact surface. Its duration is determined by the point in the tube at which the kinetic measurements are being made. The rarefaction fan, which is driven back into the driver gas, is an expansion wave and it has a divergent character because the velocity of its "tail" is governed by gas that has already been cooled by the rarefaction head. These are the principles, but to obtain meaningful kinetic information many corrections have to be made to the calculations through which the conditions within the shocked gas are derived. The corrections relate to shock tube dimensions and gas viscosity, both of which affect the build up of a boundary layer. Movement of the test gas along the tube affects its reaction time. The correction methods and procedures for deriving the

567

Experimental methods Rarefaction wave />

tk

ff

Fig. 6.9. Representation of events in a shock tube. The locations of the shock front, the contact surface and the rarefaction fan in the driven and driver sections of the tube after an interval h following the bursting of the diaphragm are marked.

shocked gas temperature and the shock velocity are very well established [80-82]. The simplest form of shock tube is closed at both ends. The shock would be reflected from the test gas end, so that the reactants may experience a double heating as the reflected shock passes through them. This creates exceedingly high temperatures before the contact surface arrives. Unless care is taken to tailor the gas mixtures, so that the reflected shock passes through the contact surface, it will undergo a second reflection, which may destroy the control of heating and quenching that is being sought for kinetic measurements. Another effect is that the rarefaction fan would be reflected from the end wall of the driver section so that it also follows the shock and the contact surface. In fact, the expansion wave moves at the local sound speed, which is faster than the contact surface velocity so, with a sufficiently long test section and sufficiently short driver section, the rarefaction may overtake the contact surface and become the major gas cooling factor. The gaseous reactants behind a reflected shock are stationary. Some tubes, termed chemical shock tubes, have been designed so that conventional chemical analysis can be performed on the reaction products, by use of gas chromatography, or related, techniques. One example of the application of the chemical shock tube, perhaps unique in the lowtemperature oxidation of hydrocarbons, is the study by Dahm and Voerhook [83] of n-pentane oxidation. They measured the organic peroxide yields over the shocked gas temperature range 570-630 K at reaction times of 1-3 s. In most cases kinetic data are derived from spectroscopic studies of the

568


Ch. 6

reactants or products via windows in the shock tube walls. Spectroscopic methods also identify the occurrence of ignition when ignition delays are being measured (such as the time to the maximum emission from C*, CH* or OH*). Typically the shocked gas temperatures for kinetic studies [84] or ignition delay times [85, 86] fall in the range 1500-2500 K. Lower temperatures are not often used for kinetic studies because non-idealities become a problem and also because other experimental techniques become accessible, but they are of interest for ignition delay measurements of hydrocarbons [29]. The near discontinuity in temperature that is achieved when the shock passes through the reactants gives a very well defined start to the reaction, from which an ignition delay can be well characterized. There can be complications in other types of experimental systems because gas admission or heating rates can occur on a time-scale that is comparable with the chemical timescale. Extremely high pressures are attainable in shock tubes and conditions are close to those in engines. However, if the development of reaction is relatively slow, with ignition delays of several milliseconds, there can be appreciable heat losses during the induction period, which complicates the interpretation of the behaviour [29]. 6.3.5 Rapid compression machines In their most familiar role, rapid compression machines (RCM) comprise a mechanically driven piston which is used to compress a gaseous charge in a combustion chamber to a closed, constant volume following the end of compression. That is, there is no reciprocation as in a piston engine. The compression occurs sufficiently rapidly to induce heating of the gaseous charge, commonly in the range 500-1000 K according to the mechanical compression ratio and the ratio of heat capacities of the gaseous mixture. Very high pressures can easily be achieved (0.5-5 MPa) with a typical compression ratio of about 10:1. Such machines enable measurements to be made under conditions close to those of the end gas in reciprocating engines, but without a number of the experimental complications that may arise in an engine as a result of the opening and closing of valves or the expansion caused by the piston in its downstroke. The charge is fuel + air, the fuel being either premixed as a vapour in oxygen and other diluting gases, which is a further simplification, or injected as liquid droplets. Rapid compression machines have been used also for other types of kinetic studies [87] or to investigate the gas motion and turbulence that

Experimental methods

569

are set up by the piston motion [88], and to measure the heat transfer rates from moving gases in a combustion chamber [89]. The operating temperature range of the RCM is lower than that typical of shocked gas conditions and reaction times are several orders of magnitude greater than in shock tubes, < 100 ms. These conditions also complement those of closed and flow reactors, especially with respect to the higher operating pressures and shorter reaction times in a rapid compression machine. Rapid compression machines are not without technical difficulty, nor are the data obtained easy to interpret quantitatively. Amongst the major technical problems associated with the operation of an RCM is the ability to attain a sufficiently fast compression, yet arrest the motion of the solid piston without vibration or bounce. The early development and applications of rapid compression machines have been reviewed by Jost [90], Sokolik [91] and Martinengo [92]. The earliest recorded application of an RCM to spontaneous ignition studies appears to be that of Falk [93] in 1906, which was a gravity driven machine. Rapid compression machines have been used over many years in the former Soviet Union [91, 94]. No single design or mode of operation has dominated the field. A compressed air-driven piston is generally favoured to achieve high piston speeds, ranging from about 5 to 15 m s _ 1 . Park and Keck [95] have shown that piston speeds of about 10 m s - 1 are optimal for near adiabatic compression without inducing vigorous gas motion, and the creation of a rollup vortex in the corner between the moving piston crown and cylinder wall. A hydraulic control for both compression and expansion of the charge, at piston speeds below 5 m s _ 1 , has been adopted by workers at the Toyota Research Laboratories [96]. At Shell Research Centre, Affleck and Thomas [97] adopted the innovative, double opposed piston action in order to minimize the piston stroke while maintaining a high compression ratio. Additional advantages of this design were believed to be the creation of a plane of aerodynamic symmetry of the compressed charge and reduced gas motion in the centre of the chamber. However, there was restricted optical access in a machine of this design. This apparatus was applied extensively to the study of alkane combustion in the 1970s, by Fish and co-workers, as discussed in Section 6.5. The design of the RCM at Leeds University, used by Griffiths and co-workers, was derived from that of the Shell machine but incorporating only a single piston [50]. Most modern RCM systems operate with the driven piston directly in

570


Ch. 6

line with the driving piston. There are two pistons in the RCM at Lille which are connected by a cam [21, 22, 98]. The driving piston is perpendicular to the driven piston in the combustion chamber. Compression within the combustion chamber results from motion of the driving piston away from the axis of the connection. The speed of the driving piston and the profile of the connecting cam control the rate of compression. The cam profile also ensures the anchoring of the compression piston at the end of its travel. There is scope to control the "profile" of the piston motion with this design, and also measurement of the piston speed is very easy. However, the piston speeds that can be attained during compression are limited by the mechanical stresses imposed. Amongst the main parameters of interest in rapid compression studies are the temperature and pressure that are reached at the end of compression. Pressure measurements are made by fast response pressure transducers (>10kHz), and ignition delay times are measured from the pressure-time profiles. The measurements of pressure may be supplemented by the detection of light output through windows, and by chemical analysis at intermediate stages of reaction by rapid expansion and quenching methods [22,99-101]. Although piston motion may be rapid, a perfect adiabatic compression does not occur in an RCM. Heat losses to the chamber wall and boundary layer development as a result of the gas motion generated by the piston are the main causes of departures from ideality. Nevertheless, gas at the core of the compressed charge may be regarded to have experienced an adiabatic isentropic compression, assuming heat losses are confined to the boundary layer. In most, but not all circumstances, the core gas temperature, Tc, is the natural reference temperature for the compressed gas because the highest temperature at the end of compression is responsible for the development of spontaneous ignition in the shortest time [88, 95]. Exceptionally, when the compression heats the reactants to temperatures that correspond to the region of ntc for that particular mixture, combustion may be initiated in the cooler boundary layer region. That is, gases which, at the end of compression, are colder than those in the adiabatic core control the duration of the ignition delay. This was demonstrated by Schreiber and coworkers by the simulation of alkane combustion, using various reduced kinetic schemes, in computational fluid dynamic calculations [102-104].


571

The core gas temperature is derived from the measured pressure ratio. For a change TX—*T2, produced by the small pressure change P\-*P2, ,

Ay-Dly

(6.15)

T2=T1[^j

where y is the ratio of the heat capacities (Cp/Cv) of the reactant mixture. The expression for an isentropic compression over the range Tt -^ Tc and Pi —>Pc, which takes into account the temperature dependence of the heat capacities, is

I ^ ^ d l n T = I Cdin/?.

(6.16)

If heat losses during compression are only slight then the bulk of the compressed gas will be at the core gas temperature. However, if heat losses during compression are very significant, as in slow compression, then a rather smaller fraction of the compressed charge will be at the core temperature. The extent to which heat losses during compression cause departures from the adiabatic ideal may be assessed from a comparison of (6.16) with the temperature (T ad ) which is predicted on an ideal volumetric basis from knowledge of the dimensions of the RCM [50]. That is rTad

rvc

i

-dlnr= JTt

(7-1)

din 7,

(6.17)

JVi

where Vt and Vc are the initial and final volumes of the combustion chamber respectively [50]. The compressed gas temperature calculated by (6.16) does not represent an appropriate reference temperature when the fuel + air mixture is so reactive that exothermic oxidation begins during the compression stroke. The pressure reached at the end of compression does not then arise solely form p-V work, and application of a thermodynamic relationship (6.16) based on the measured pressure is not valid. The ideal gas equation may be applied to the system in these circumstances, the compressibility factor

572


Ch. 6

of the gas being assumed to be unity, but it gives only a spatially averaged compressed gas temperature, r a v , where

Ti

PiVt

If there is little or no change in the number of moles of material as a result of reaction an average gas temperature may also be interpreted during the post-compression period from the instantaneous pressure by use of equation (6.18). Experiments are normally performed under relatively dilute conditions (—80% inert gas) and, in general, the number of moles of product and reactant are approximately equal during the "slow oxidation" of hydrocarbons. Equation (6.16) is the most satisfactory reference temperature for the compressed gas but it is not valid in all circumstances. The application of equations (6.16)—(6.18) was tested by Griffiths et al. [50]. Since the state of the gas at the end of compression is a function of the RCM and its operating conditions, the quantitative characteristics of ignition, such as the duration of the ignition delay under given conditions, are specific to a particular machine. The qualitative structure, such as the occurrence of two-stage ignition and the existence of a region of negative temperature dependence of ignition delay, are common features of corresponding experiments performed in different machines. 6.3.6 Motored engines Many combustion scientists have exploited motored engines for the study of partial combustion of fuel-air mixtures. Invariably a single cylinder CFR (Co-operative Fuel Research) engine is used, driven by an electric motor but there is no spark ignition. Thus, although the fuel-air mixture does not experience the additional compression of the flame "piston" (see Chapter 7) or consumption by it, the charge does undergo compression and heating, which may lead to spontaneous combustion. This is one way in which end-gas autoignition can be investigated and the technique can be used to study the unburnt hydrocarbons. Supplementary sampling devices are often used in conjunction with motored engines so that the intermediate stages of the processes may be examined. These could involve


573

a rapid sampling system [105, 106], which snatches a small amount of the charge while the overall process continues, probe sampling [107, 108], or exhaust gas analyses [109-112]. Chemical analyses made in the exhaust gas of a motored engine lead to a different technique from that normally employed for the low-temperature combustion processes of alkanes. The motored engine is a chemical reactor in which the period for reaction is governed by the engine speed. The conditions under which reaction occurs are governed by the compression ratio which, linked to the inlet temperature and pressure, controls the highest temperature and pressure. Whilst there may be some difficulty in defining very precisely the temperature history of the reactants, comparative chemical measurements can be made by variations of engine speed, inlet temperature and pressure. Leppard [111, 112] has adopted this approach to useful effect. The exhaust gas is analyzed from a number of motored cycles so that cycle-to-cycle variations are averaged. Examples of the application of motored engines are discussed in Section 6.4 and 6.5. These have contributed significantly not only to the better understanding of the mechanisms of hydrocarbon oxidation but also to a bridging of the gap between the interpretation of reactions investigated in more conventional chemical systems, and the reactions of hydrocarbon fuels in spark-ignition engines. 6.3.7 Chemical analysis Continuous sampling from closed or flow systems is possible via a probe, but, ideally, neither its presence in the vessel nor its action should perturb the reaction. If these conditions are satisfied, it is possible to obtain data throughout the course of a single experiment. The products must be extracted in such a way that reaction ceases at the probe orifice. A pumping system may be used and a probe designed to quench adiabatically by a suitably large pressure drop across the probe orifice. Fine-drawn silica probes are often used in laboratory scale experiments. The analytical technique must also be capable of a continuous response if events are varying in time. Sampling to a mass spectrometer for analysis is an obvious possibility. The instrument must be capable of repetitive scanning and signal storage throughout a suitable mass range in very short cycles (10-50 ms). Alternatively, it may be possible to focus on a specific mass spectral peak which characterizes a particular component, and to

574


Ch. 6

follow its variation in time. Mass spectra for many hydrocarbons and other organic compounds have such similarity that it requires particular ingenuity to extract quantitative records pertaining to specific components of the system. The most successful mass spectrometric applications are confined to relatively low molecular weight fuels. Flow systems do not require analysis on the same response time-scale, provided that the reaction is maintained in a stable stationary state. Batch sampling from closed vessels requires that many reproducible measurements over different time intervals are made in order to build up a profile of the chemistry of the reaction. The reward is that a much wider range of analytical methods may be brought into use. Foremost, from about 1960, has been gas chromatography, and its modern derivative, capillary column gas chromatography, in which very fine, internally-treated capillary columns are used for the chromatographic separation. The present trend is to couple mass spectrometry with chromatography so that the mass spectral analysis is restricted to an individual component of the mixture as it is eluted from the chromatography column. Computer libraries of data files are available for efficient and accurate identification of the mass spectra obtained. Much of the remaining difficulty is associated with the similarity of the mass spectra of compounds formed in hydrocarbon oxidation, especially if a distinction is to be made between isomeric structures of individual compounds. Spectroscopic methods are required for free radical intermediates. Laser induced fluorescence of hydroxyl radicals has been used successfully to determine elementary rate parameters associated with the isomerization reaction R 0 2 ^ QOOH [113]. Laser perturbation of hydroxyl radical concentrations in stabilized cool-flames has been used to obtain "global" kinetic data for chain-branching rates at temperatures of importance to the low-temperature region [79]. These methods appear to be most suited at present to combustion studies in flow systems. There are also several studies of the relative intensity from OH radical fluorescence during oscillatory cool-flames [58,114]. Laboratories in Armenia [115] and France [21,116] have a well-established esr technique for the detection of alkylperoxy and hydroperoxy radicals. The procedure involves the freeze-trapping of free radicals on a liquid nitrogen cooled trap located within the cavity of an esr machine. The reaction products, including relatively long-lived R 0 2 radicals, are pumped continuously at low pressure from a flow reactor to the collection

Global combustion phenomena

575

point. The accumulated signal is scanned repeatedly across the appropriate microwave frequency range, during which time the concentration of radicals builds up at the cold trap.

6.4 GLOBAL COMBUSTION PHENOMENA ASSOCIATED WITH HYDROCARBON OXIDATION

6.4.1 Introduction Spontaneous ignition and associated features of organic gases and vapours are a consequence of the exothermic oxidation chemistry discussed in Chapter 1, but the way in which events unfold is determined by the physical environment within which reaction takes place. The heat transfer characteristics are probably most important, as may be illustrated with respect to the different consequences of adiabatic and non-adiabatic operation in a CSTR (Section 5) [117]. The notion of adiabatic operation may seem remote from any practical application, but this idealized condition may be approached if the chemical time-scale is considerably shorter than the time-scale for heat losses. 6.4.2 Spontaneous ignition and oscillatory cool-flames in closed vessels The reactants pressure-vessel temperature (p—Ta) ignition diagram for thermal ignition is characterized by the two regions of "slow combustion" and "ignition", which are separated by a critical boundary. Even in very early investigations of hydrocarbon oxidation in closed vessels (1935), it was found that the ignition diagram for hydrocarbons was considerably more complicated (Fig. 6.10), as discussed in Chapter 5. This type of ignition diagram is constructed from the results of many experiments performed at different reactant pressures in a vessel which is maintained at constant temperature in an oven. There is a sharp boundary beyond which ignition occurs, but the shape of the ignition limit curve is complex: it comprises three branches, the central one of which shows an initial increase in critical pressure as the temperature is raised. This is a consequence of the negative temperature coefficient of reaction rate, discussed in Section 6.2. The non-ignition (or subcritical) region of the ignition diagram is also subdivided into different types of behaviour in which there is a boundary

576


Ch. 6

Ignition

Slow reaction

Fig. 6.10. Idealized (p-Ta) ignition diagram which is typical of that obtained for the combustion of many hydrocarbons in oxygen or air in closed vessels and well-stirred flow systems. Ignition, oscillatory cool-flame and slow reaction regions are shown.

between the slow combustion and the multiple (oscillatory) cool-flame regions. Such oscillations can be observed by a faint blue luminescence which originates from an electronically excited state of formaldehyde formed in reactions (2) and (3), [118,119], (2)

CH 3 0 + OH -» CH 2 0* + H 2 0 ; AH%8 = -296.4 kJ mol"-1

(3)

CH3O + CH3O -> CH 2 0* + CH3OH; AH^s = -278.5 kJ mol" 1

where the enthalpies refer to the production of ground-state formaldehyde. Similar emissions are associated with much of the slow combustion region at conditions close to those for the occurrence cool-flames but, in general, they are too weak to be seen even by a dark-accustomed eye. A small,

Global combustion phenomena

577

but distinct, light pulse may also be associated with the final stage of slow oxidation. This is called the "pic d'arret" [120, 121]. On average only one photon is emitted per 103 molecules of fuel consumed by reaction [122]. It is possible to see this very weak, pale blue light by eye in a darkened laboratory when it is isolated in a cool-flame reaction, as opposed to a two-stage ignition (see Chapter 5). In two-stage ignition the emission is indicative of the free radical concentrations in the first stage of the ignition. In some combustion systems there is a further complexity of behaviour at conditions between the cool-flame region and the ignition boundary, termed multiple stage ignitions [124]. The light output which accompanies the final stage of two-stage ignition is considerably more intense. It is characteristic of the emission from premixed flames of hydrocarbons and arises predominantly from chemiluminescent reactions, mainly giving electronically excited CH*, C*, C O | and OH* [123]. Although Fig. 6.10 is presented in a general form, very many specific examples are published in the literature because, often, it is important to characterize the conditions in which different reaction modes are observed before any other detailed investigations are made [125-127]. The reactant pressures and vessel temperatures at which the different features occur are dependent not only on the particular fuel and its proportions with oxygen or air but also on the size and shape of the vessel [121]. Surface reactions may play some part also, especially in the initiation processes, so the quantitative details are usually specific to the particular system employed [128]. Nevertheless, the qualitative structure of this ignition diagram is characteristic of that for alkanes and alkenes containing three or more carbon atoms, of acetaldehyde (ethanal) and higher molecular mass aldehydes, higher alcohols, ethers other than dimethyl ether, and a variety of other organic compounds which contain fairly large aliphatic groups. Although there are some resemblances of the behaviour, the features of Fig. 6.10 are not exhibited fully by methane, ethane, ethene or formaldehyde. Nor are they shown by benzene, toluene or xylene, but coolflame characteristics are shown by propyl- and n-butyl-substituted benzenes [129]. For the most reactive substances (such as acetaldehyde or diethyl ether in equimolar proportions with oxygen) the minimum ignition temperature is at about 480-500 K as the pressure approaches 100 kPa in normal laboratory experiments (typically vessels of up to 0.5 dm 3 ). The apex of the

578


Ch. 6

ignition boundary occurs at about 550-600 K. The minimum pressure within the ignition peninsula is about 13kPa (~100torr) and cool-flames may be detected at pressures as low as 4 kPa. The normal alkanes are more reactive than their isomeric, branched chain structures and the reactivity increases considerably in the range C3-C6. As an approximate guide, equimolar mixtures of alkanes with oxygen exhibit ignition at temperatures above about 520 K, and within the ignition peninsula, at pressures above about 30 kPa. Most organic fuels mixed with air do not undergo spontaneous ignition in the low-temperature range at pressures much below 100 kPa. The lowest autoignition temperature at a given pressure is generally observed in mixtures that are in the range of molar proportions RH: 0 2 = 1:2 to 2:1 (>10% by volume of the vapour in air, O > 3). This is considerably richer than the optimum conditions for the propagation of pre-mixed flames, which occurs in stoichiometric mixtures, for which 10 1 8 cm - 3 ) at temperatures below 750 K.

598


Ch. 6

Alkylperoxy radical isomerization has a rather higher probability when 1-propyl radicals are involved in a corresponding competition because the activation energy for the primary H atom abstraction is higher than that for a secondary H atom abstraction. Even if it does not occur very readily, alkylperoxy radical isomerization may play an important part because the decomposition products, which include aldehydes, lead to propylhydroperoxide via H atom abstraction reactions of appreciably lower activation energy, for example, (14)

C H 3 C H ( 0 0 ) C H 3 + CH 2 0 -> CH 3 CH(OOH)CH 3 + CHO; E = 45 kJ mol - 1

A prolonged debate prevailed in the 1950s and 1960s regarding the role of aldehydes versus organic peroxides as the "important" branching intermediates in alkane oxidation at low-temperatures [125]. It would appear that there is a synergy of the two, and both classes of compounds play an important part in the low-temperature autocatalytic process. Degenerate chain branching may occur by the decomposition of 2-propyl hydroperoxide, as (15)

CH 3 CH(OOH)CH 3 -> CH 3 CH(0)CH 3 + OH

The OH radical is able to continue propagation of the chain, whereas the most likely fate of the propoxyl radical (see also Section 1.13) is its decomposition [151] (16)

CH 3 CH(0)CH 3 -> CH 3 CHO + CH 3

Thus, acetaldehyde (ethanal) may be expected as a prominent molecular intermediate, which also readily undergoes further oxidation. An additional source of CH 3 CHO is from the oxidation of propene, formed as an intermediate by OH radicals (see Chapter 1). The oxidation of acetaldehyde is particularly well documented over a wide temperature range [152156], an important feature being the generation of methyl radicals and carbon monoxide at temperatures in excess of 650 K, in the sequence of

Product distributions during hydrocarbon oxidation

599

reactions (17) (18)

CH3CHO + X -> CH3CO + HX CH3CO + M -> CH 3 + CO + M

where X represents any propagating free radical. Below 1000 K the oxidation of methyl radicals yields formaldehyde (methanal), methanol and hydrogen peroxide (Sections 1.12 and 1.18). Carbon monoxide is also a product of the further oxidation of formaldehyde, mainly by (19)

CH 2 0 + X -> CHO + HX

(20)

CHO + 0 2 -> H 0 2 + CO

The acetylperoxy radical, CH3CO3, is formed by oxidation of CH 3 CO at temperatures below 650 K, which also leads to degenerate branching via peracetic acid, and is a source of C 0 2 [157, 158]. Supplementary reactions during propane oxidation may also be proposed through the formation of diperoxy species following alkylhydroperoxy radical isomerization, taking the form (21)

HOOCH 2 CHCH 3 + 0 2 ^ H O O C H 2 C H ( 0 0 ) C H 3

(22)

HOOCH 2 CH 2 CH 2 - + 0 2 ^ HOOCH 2 CH 2 CH 2 00-

(23)

CH 2 CH(OOH)CH 3 + 0 2 ^

OOCH 2 CH(OOH)CH 3

These species would be expected to undergo a further internal isomerization either to form a dihydroxyalkyl radical, which would readily decompose to give chain branching, or to lead to a relatively unreactive propagation via H 0 2 , e.g., (24)

HOOCH 2 CH(00)Cm ^ fCH3CHO + CH20 + H0 2

(25)

OOCH2CH(OOH)CH3J "~ 1HOOCCH(OOH)CH 3 -^CH 3 CHO + CHO + 20H

(26)

HOOCH 2 CH 2 CH 2 00- ^ HOOCHCH 2 CH 2 OOH ^ CH 2 CHO + CH 2 0 + 2 0 H

600


Ch. 6

The abstraction of an H atom from another molecule to form an alkyldihydroperoxide, followed by decomposition, might give, (27) HOOCH 2 CH(00)CH 3 + RH N

HOOCH2CH(OOH)CH3 -» CH3CHO + CH 2 0 + 20H •OOCH2CH(OOH)CH3 + R H 7

(28)

HOOCH 2 CH 2 CH 2 00- + RH -> HOOCH 2 CH 2 CH 2 OOH -> CH 2 (0)CH 2 + CH 2 0 + 2 0 H

As discussed in Section 1.16, the competing reaction to alkylperoxy radical formation, (8) and (9), is the H atom abstraction reaction (29)

C3H7 + 0 2 ^ C 3 H 6 + H 0 2

The outcome is the same regardless of whether C 3 H 7 is 1- or 2-propyl, although there may be some variation of the reaction rate. Reaction (29) is expected to be dominant at higher temperatures. Propene is also susceptible to further oxidation, but the overall conclusion must be that there is a shift towards its formation from that of partially oxygenated molecular intermediates as the temperature is raised through the range 650-800 K. Commensurate with this is the transition from the vigorous reaction which involves degenerate branching via alkylhydroperoxides and OH propagation, and quite substantial exothermicity, to the relative unreactive regime governed by H 0 2 propagation with a virtually thermoneutral overall stoichiometry given by (30)

C 3 H 8 + 0 2 = C 3 H 6 + H 2 0 2 ; AH%98 = 12.2 kJ mol"*

The bond strength of the allylic hydrogen in propene D(H—CH 2 CHCH 2 ) is 351 kJ mol - 1 , which confers an unusual initial reactivity as a result of the susceptibility to H atom abstraction by propagating free radicals, (31)

CH 3 CHCH 2 + X -> CH 2 CHCH 2 + HX

By contrast (see Section 1.5d), the resonance stabilized allyl radical formed


601

as the product is extremely unreactive to further oxidation [159], and tends to react mainly through radical + radical reactions, such as (32)

CH 2 CHCH 2 + H 0 2 -» CO + products

Reactions with molecular oxygen are also possible, which are believed to have high activation energies [159]

(33)

CH 2 CHCH 2 + 0 2

2 radicals + product; CO + product; 2CH 2 0 + CO + OH; CH 2 CHCH 2 + H 0 2 ;

72.5 78.6 80.3 95 kJ

8.3 kJ mol" 1 4.5 kJ mol" 1 6.0 kJ mol" 1 mol" 1

There is a more detailed discussion of propyl radical and propene oxidation in Chapter 1. Experimental observations A shift in the overall mechanism is certainly reflected in a global way from studies by Jones et al. [160] of propane: oxygen mixtures at an equivalence ratio 4> = 6 at 0.5 MPa in a tubular flow reactor (Fig. 6.23), although a negative temperature dependence of rate is not strongly developed in the extent of conversion of the fuel. In a recent study, Koert et al. [14] investigated in more detail the molecular products formed during the oxidation of propane over the temperature range 600-900 K at an equivalence ratio of 0.4. They used a turbulent flow, tubular reactor (a "pressurized flow reactor", or PFR, volume 140 cm3) operated at 1.0 and 1.5 MPa with a mean residence time of 200 ms. Gases were sampled at the reactor outlet at a fixed residence time and constant pressure, while the temperature was reduced in a controlled manner. An ntc region was observed in the reactor temperature range, approximately 700-740 K. The reactant temperature at the sampling point did not exceed 740 K over this range of tube temperature. The carbon oxide yields and oxygen consumption were continuously monitored by gas analyzers and other products were detected and measured by gas chromatography. The major products at the tube temperatures 700, 750 and 800 K, expressed as a fractional yield of the fuel consumed, are given in Table 6.2. Approximately 25% of the fuel was consumed at its maximum, just prior to the onset of the negative temperature

602

Experimental and numerical studies of oxidation chemistry a 70 - i

60-J

50J

u

40-4

30H

20-

10' Conversion 673

633

T— 793

753

713

1 833

T/K

Fig. 6.23. Variation of the yields of alkenes and partially oxygenated products formed during the combustion of propane in a tubular flow reactor as a function of the tube temperature. (After Jones et al. [160].)

TABLE 6.2 Major products of propane oxidation in air at 1 MPa expressed as fractional yield of fuel consumption at three reactor temperatures in a plug flow reactor ( = 0.4) [14] Major products

700 K

750 K

800 K

CO CH 3 CH=CH 2 HCHO

0.5 0.075 0.125 0.225 * 0.05

0.325 0.660 0.001 0.001 * 0.05

0.25 0.60 * 0.04 0.425 *

co2

CH 3 (CH 2 ) 2 CH 3 CH3CHO *Minor products.


603

dependent range and, within the residence time in this reactor, the extent of reaction decreased virtually to zero at the highest temperature of the ntc range. There were no major distinctions in the results at the two pressures investigated. Although the carbon oxides were detected in strikingly high proportion at the lowest temperatures, almost certainly as products of the secondary oxidation of acetaldehyde and formaldehyde, acetaldehyde and formaldehyde were themselves both important molecular products. Propene became the most important product at temperatures beyond 750 K, but n-butane was found to exceed the propene yield at temperatures beyond 800 K. Various other oxygenated, molecular intermediates, detected in minor yield below 750 K were virtually undetectable at higher temperatures. Ethene was also prominent among this group of compounds. With the exception of the n-butane yields detected in this high-pressure study, there are resemblances of the product yields from the sub-atmospheric pressure studies under appreciably richer conditions (typically O = 5) by Shtern and co-workers [161] and by Falconer and Knox [162]. Shtern and Polyak [163] paid particular attention to the formation of peroxides by polarographic detection, and concluded that the only peroxide formed in detectable proportions during propane oxidation was hydrogen peroxide. However, Bonner and Tipper [164] reported the formation of per acetic and perpropionic acids, and also trace amounts of propylhydroperoxide during slow oxidation and preceding a cool-flame during propane oxidation. Direct evidence for the existence of alkylperoxy radicals at 600 K has been reported by Gukasyan et al. [165]. The mole fraction versus time profiles from the sub-atmospheric pressure studies by Wilk et al. [13] in a closed vessel, shown in Figs 6.24 and 6.25, are rather typical of the records obtained by many different workers subsequent to the seminal work by Newitt and Thornes [8,134] in 1937. The results in Fig. 6.24 were obtained by Wilk et al. [13] during the occurrence of periodic cool-flames. The associated temperature changes were measured by a thermocouple. Although these chemical analyses of propane oxidation [13] are among the best that have been reported, it is rather difficult to make a clear judgement from them on the applicability of the "alkylperoxy radical" mechanism of oxidation. There is reasonable evidence in support of enhanced yields of propene at higher temperatures, especially if allowance is made for the non-isothermal effects in cool-flames. The susceptibility of

604


(a)

5n

^

Ch. 6

r-748

2 o

s 0

130

260

390

520

650

780

t/s

(b)

0.4—i

O A X

-

CH3CHO C2H3CHO + C 3 H 6 0 C H 2 0 * 1/2

Fig. 6 24. Mole fraction - time profiles for propane/air mixtures at <j>= 1.0; p0 = 80kPa; To = 723 K. (After Wilk et al. [13].)

605


(a)

1.2-,

130

260

390

520

650

780

t/s (b)

io n

Fig. 6.25. Mole fraction - time profiles for propene/air mixture. (After Wilk et al. [168].)

606


Ch. 6

propene to oxidation can be seen at both the low and high vessel temperatures. There is a marked reduction in yields of acetaldehyde and methanol at the higher vessel temperature. The greater yields of propene oxide at the higher vessel temperatures probably arise from further oxidation of propene by (34)

CH 3 CHCH 2 + H 0 2 -> CH 3 CH(0)CH 2 + OH

The underlying difficulty in the interpretation of propane oxidation at lowtemperatures appears to be that the intermediate molecular products are so much more reactive than the alkane that much of the relevant detail is masked. An indication of this lies in the very high proportions of CO and C 0 2 that are formed, probably through acetaldehyde and formaldehyde oxidation (reactions (18-20)). The concentration profiles of molecular intermediates give no clue regarding the subtlety of the mechanism involving excited states of C 3 H 7 0 2 [166]. Such information has to be obtained from studies of the elementary reaction processes themselves, as discussed in Chapters 1 and 2. By contrast to the oxidation of higher alkanes, there is no qualitative way of distinguishing evidence for a diperoxy radical mechanism during the low-temperature oxidation of propane. It is possible that the considerably lower reactivity of propane than that of higher alkanes, as demonstrated by phenomenological aspects (Section 6.4.4), may be explained by the absence of these types of processes. The matter may not be resolved until more insight is obtained from numerical modelling and simulation of experimental results. Propene is the principal intermediate species in the oxidation of many higher alkanes, as well as that of propane, so an understanding of the mechanism of propene oxidation is important to the interpretation of the oxidation of these fuels. Following previous studies [125,167], Wilk et al. [168] investigated the oxidation of propene over the temperature range 580-715 K in a closed vessel at sub-atmospheric pressure (Fig. 6.24). The procedures were similar to those discussed above. A region of ntc that was observed in the temperature range (560-620 K) may be related to the reversible reaction of allyl radicals with oxygen oxidation, (35)

C3H5 + 0 2 ^ C 3 H 5 0 2


607

However, this equilibrium has an abnormally low temperature coefficient which ensures that the position of equilibrium is displaced to the left even at temperatures below those at which the ntc is observed, as discussed in Chapter 1. Wilk et al. [13] have proposed that the chain branching reactions during the low-temperature oxidation of propene originate from reactions of methylperoxy radicals. Although the origins of methyl radicals themselves are not disclosed, there is a possibility that the underlying branching/non-branching interactions could arise from (36)

CH 3 + 0 2 ^ CH 3 0 2

because propene is able to offer a very labile H atom for the subsequent formation of methyl hydroperoxide, (37)

CH 3 0 2 + C 3 H 6 -> CH 3 0 2 H + C 3 H 5

which is able to decompose readily at the prevailing temperatures. 6.5.3 Oxidation of hydrocarbons containing four carbon atoms Experimental investigations of the low-temperature oxidation of n-butane and /-butane are important because they are the simplest compounds to give insight into the consequences of different isomeric structures on the overall reactivity of alkanes. There have also been analytical studies of the low-temperature oxidation of the butenes in closed vessels [169171] and in a motored engine [172]. Representative results from analytical studies of the oxidation of n-butane and /-butane at low pressures and temperatures in closed vessels, obtained at great extents of reactant consumption, are shown in Table 6.3. The results of analyses from the butanes during the initial stages of reaction at both low and high temperatures, are shown in Table 6.4. There is considerable evidence for the formation of partially oxidized products from alkylperoxy radical isomerization in these chemical analyses, and there is supplementary support for a transition to alkenes as the major primary products at higher temperatures. Mechanisms for alkylperoxy radical isomerization and decomposition that involve each

608


Ch. 6

TABLE 6.3 Relative product yields for the extensive slow oxidation of n-butane and /-butane Reactants

C4H10 + 3.50 2

1-C4H10 + 0 2

Reference

137

173

Vessel temperature/K

588

579

Initial pressure/kPa

21.3

26.7-53.3

% Product Yield, x (moles per mole reactant) JC>30

but-1-ene

/-butene carbon monoxide water

30 > JC> 10

but-2-ene

10 > JC> 2

propene ethene

/-butyraldehyde propene carbon dioxide methyloxiran r-butanol methanol formaldehyde

2>JC>0.1

tetrahydrofuran 2-methyloxetan 2,3-dimethyloxiran 2-ethyloxiran methyl ethyl ketone propionaldehyde propene oxide acrolein acetone ethene oxide acetaldehyde

2,2-dimethyloxiran 5-butanol methacrolein /-propanol propionaldehyde ethene methane

of the four butyl radical structures which may be formed from n-butane and /-butane oxidation are shown in Table 6.5. Berry et al. [137] detected a range of C 3 and C4 O-heterocyclic compounds amongst other partially oxygenated molecular products during the early stages (

H 3 C.

CH3CHO + C2H4

0

0—OH i-bumne oxidation

r

CH3CHCH2 O—O

CH, CH3CCH2

CH3CCH3 0-0

None

0

HO—0

r r

CH2CHCH2

r

H3C

H3C\7

HO—0

CH3CCH2

0-OH

"'b

CH2CHCH3 + CH 2 0

«— 0

H3

V,

H 3 CT7 0

CH3CCH2CH3

II O


611

TABLE 6.6 Effect of vessel temperature on the early product yields during non-isothermal oxidation of n-butane (pf- = 21 kPa; n-C4H10: 0 2 = 0.29) [137] Products

588 K

603 K

643 K

700 K

Total butenes (mole/mole rc-butane consumed) Ratio but-1-ene but-2-ene Total C 4 heterocycles (mole/mole n-butane consumed) Ratio 2,3-dimethyloxiran + 2-ethyloxiran tetrahydrofuran + 2-methyloxetan Ratio Total butenes Total C 4 heterocycles

0.68

0.70

0.71

0.74

1.7

1.9

2.1

2.4

0.072

0.067

0.050

0.040

0.5

0.8

1.2

1.9

9.4

10.5

14.2

18.5

of 700 K (Table 6.6). The extent to which temperature change accompanied the reaction was not noted. Carlier et al. [21] studied the combustion of n-butane + air at 0.18 MPa under stationary-state conditions on a heated flat-flame burner. Chemical analyses by probe sampling from different regions of the spatially-resolved, cool-flame and second-stage hot flame structure provided sufficient information to enable a quantitative species profile to be constructed for the following species: butane, hydrogen, methane, carbon monoxide, carbon dioxide, ethylene, ethane, water, formaldehyde, propene, propane, methanol, acetaldehyde, but-1-ene, hydrogen peroxide and alkylperoxy radicals. The products of oxidation during the induction period leading to twostage, spontaneous ignition of normal butane have been studied at a pressure exceeding 0.8 MPa and an initial compressed gas temperature of 730 K in a rapid compression machine [21,22]. In all, twenty-five intermediate and final products were identified, the proportions of the most important of which are given in Table 6.7. The concentration - time profiles for the primary fuel and several intermediate and final products are shown in Fig. 6.26. The solid lines shown in this figure represent numerical modelling studies which are discussed in Section 6.6.3. There are considerable temperature changes associated with the reaction that occurred during the development of ignition, and the temperature probably exceeded 800 K in the interval following the first stage. There is a sharp increase in the yield

612


Ch. 6

TABLE 6.7 Major molecular products identified during the ignition delay of two-stage ignition of rt-butane (Tc = 730K;p c >0.8MPa) [21] Product

Yield (% total carbon)

Butane 1-butene Methanol carbon monoxide 2-butene Ethene Propene propanal 2,3-dimethylbutane Tetrahydrofuran 2-ethyloxiran 2-methyloxetan Propanone

80 3.3 0.25 1.6 1.3 0.79 0.66 0.19 0.19 0.18 0.12 0.12 0.11

of but-1-ene at the first stage of two-stage ignition and it continues to rise thereafter. By contrast, the yield of tetrahydrofuran, which typifies the products of alkylperoxy radical isomerization and decomposition, rises quite sharply at the first stage, commensurate with the increased extent of consumption of the fuel, but its concentration hardly varies during the development of the second stage in which higher gas temperatures prevail. Butyl hydroperoxides have been identified from the oxidation of each of the butanes, especially at T < 5 2 0 K [177,178], but there seems to be no unequivocal experimental evidence that diperoxyalkyl species are formed, such as would be obtained from the identification of dicarbonyl products. Many of the products identified by Minetti et al. [22] could be derived from a diperoxyalkyl oxidation route, as described in their paper, but since alternative mechanisms based on butylperoxy radical isomerization and decomposition are feasible, this evidence is not conclusive. 6.5.4 The oxidation of isomers of C5 and C6 alkanes Dimethylpropane (neo-pentane) is unique amongst the isomers of pentane since it contains no secondary or tertiary C—H bonds. This unusual structure of the dimethylpropyl radical has been exploited in the study of


613

100-1

t/tign

t/tign

Fig. 6.26. Concentration - time profiles for intermediate products measured during the two-stage ignition of rc-butane in a rapid compression machine. The concentrations are based on the carbon balance. For the purposes of numerical modelling the abscissa is normalized to the duration of the ignition delay. Experimental data are marked as points, numerical results are shown as solid lines. (After Minetti et al. [22]).

alkylperoxy isomerization reactions, as described in Section (1.10) [113,179]. The low-temperature oxidation of dimethyl propane has also been studied under isothermal [180] and non-isothermal [27] conditions. A considerable range of partially oxygenated products were detected, the major primary products being /-butene, 3,3-dimethyloxetan, j-butyraldehyde and formaldehyde. Acetone was also detected in high yield [180], but this was thought to be a secondary oxidation product. There is the prospect of only one possible alkylperoxy isomerization process involving an H atom, namely, (38)

(CH 3 ) 3 CCH 2 00- ? (CH 3 ) 2 (CH 2 )CCH 2 OOH,

which accounts for the formation of acetone, 3,3-dimethyloxetan, and also /-butene and formaldehyde at low-temperatures, as discussed in Section

614


Ch. 6

1.10. The activation energy of this isomerization (38) would be regarded to be the same as that for (12). The formation of i-butyraldehyde as an important product requires not only the loss of a carbon atom from the structure, but also an internal H atom transfer from an adjacent carbon atom. Zeelenberg [181] and Fish [27] proposed a (1,4) methyl radical transfer, (CH 3 ) 3 CCH 2 00- ^ (CH 3 ) 2 C()CH 2 OOCH 3

(40) followed by (41)

(CH 3 ) 2 C()CH 2 OOCH 3 -» (CH 3 ) 2 C()CH 2 0 + OH

and (42)

(CH 3 ) 2 C()CH 2 0 -* (CH 3 ) 2 CHCHO

Clearly, this aspect of dimethylpropane oxidation requires further investigation. Studies of the low-temperature oxidation of normal pentane, by Hughes and Simmons [182], showed that acetone was the principal partially oxygenated product in the early stages of reaction. This was corroborated by Kinnear and Knox [183], A major aspect of concern at the time was the way in which vessel surface treatments could affect the product distributions, as well as the initial reaction rate [128, 183-185]. This remains true and may influence the product distribution, especially in the early stages of alkane oxidation. However, the formation of acetone may also be interpreted as evidence for the formation of dialkylperoxy species, via the sequence of reactions (43)

CH 3 CHCH 2 CH 2 CH 3 + 0 2 ^ CH 3 CH(00)CH 2 CH 2 CH 3

(44)

CH 3 CH(00)CH 2 CH 2 CH 3 ^ CH 3 CH(OOH)CH 2 CHCH 3

(45)

CH 3 CH(OOH)CH 2 CHCH 3 + 0 2 ^ CH 3 CH(OOH)CH 2 CH(00)CH 3

(46)

CH 3 CH(OOH)CH 2 CH(00)CH 3 ^ CH 3 C(OOH)CH 2 CH(OOH)CH 3


(47)

CH 3 C(OOH)CH 2 CH(OOH)CH 3 -> CH3COCH2- + CH3CHO + 2 0 H

(48)

CH 3 COCH 2 + XH -> CH 3 COCH 3 + X

615

A mechanism for acetone formation which involves only the alkylperoxy species requires a rather unlikely intramolecular H atom transfer (51), as follows: (49)

CH 3 CH(00)CH 2 CH 2 CH 3

(50)

CH 3 CH(OOH)CH 2 CH 2 CH 2 -

3 CH(OOH)CH 2 CH 2 CH 2 -

-> CH 3 CH(0)CH 2 - + C 2 H 4 + OH (51)

CH 3 CH(0-)CH 2 . -> CH 3 COCH 3

Propene oxide would appear to be a far more likely product of the isomerisation (50). However, an alternative mode for the dihydroxy radical decomposition (47) also gives rise to propene oxide. That is, (52)

CH 3 C(OOH)CH 2 CH(OOH)CH 3 -» CH 3 CH(0)CH 2 + CH 3 CO + 2 0 H

and is a preferred mechanism, since analyses by Cernansky et al. [186] do not show an equivalent yield of ethene, which would be required if (50) is a major route to propene oxide formation (Table 6.9). The investigations by Cernansky et al. [110,186] were made at much higher pressures in a motored engine. An interesting application of a finewire thermocouple to measure the gas temperature is reported, which showed that temperatures of around 750 K were reached prior to the onset of the second stage of two-stage ignition. This was comparable to the temperature predicted from the pressure record. Analysis of samples taken at "top dead centre" during the compression stroke confirmed that acetone and propene oxide were major products of alkylperoxy radical isomerization (Table 6.8). Propanal and 2-butanone were also found to be predominant among the oxygenated products. Blin-Simand et al. [187] have also reported evidence, from motored engine studies, of a ketoalkylhydroperoxide species formed by diperoxyalkyl radical isomerization.

616


Ch. 6

TABLE 6.8 Results of analysis of samples taken at 'top centre" (tc) of motored engine cycle when charged with w-pentane fuel [110,186] Temperature (inlet)/K Temperature2 (tc)/K

412 724

416 731

418 735

418 x 735

Product (at tc)

%

%

%

%

Carbon monoxide Carbon dioxide Methane Ethene Formaldehyde Propene Methanol Acetaldehyde 1-butene c/s-2-butene frans-2-butene Ethanol Propene Oxide Acrolein Propanal Acetone Butanal Unknown C4 2-butanone 1-pentene 2-methyltetrahydrofuran Tetrahydropyran Unknown C5 oxygenate Unknown C5 oxygenate Unreacted «-pentane Total C5 carbon, %

5.19 2.94 0.02 0 0 0 0 0.02 0 0.01 0 0 0.03 0 0 0.08 0.01 0 0 0.04 0 0 0 0 98.24

6.77 3.36 0.02 0 0 0 0 0.03 0 0.01 0 0.02 0.11 0 0.12 0.1 0.03 0.02 0.12 0.1 0.01 0 0.04 0.02 97.5

100.00

100.06

6.44 3.61 0.03 0.06 0 0.07 0.05 0.3 0.02 0.02 0.01 0.04 0.13 0 0.22 0.18 0.03 0.06 0.15

10.95 4.12 0.09 0.48 8.66 0.42 0.48 1.5 0.14 0.02 0.01 0.03 0.11 0.05 0.49 0.19 0.05 0.06 0.31

**

**

0.06 0.01 0.03 0.04 96.88

0.19 0.03 0.05 0.07 94.2

99.76

101.18

Engine speed - 900 rpm; n-pentane-air, <j> = 1.0; Inlet manifold pressure - 1.34 atm; Engine - motored. ^ h i s data set is at 30° ATDC. 2 Using n = 1.34, CR = 5.25. **Peak coincided with fuel peak.

The earliest mechanistic proposal related to the formation of diperoxyalkyl species appears to be that made in 1951 by Neu et al. [188]. It was based on the formation of dicarbonyl products, detected by absorption spectroscopy, during the slow oxidation of pentane and that of other organic compounds.


617

Normal pentane oxidation was also the subject of an investigation by Cullis and Hirschler [189] with respect to the formation of 1- and 2pentene, and the effect of their addition to pentane + oxygen mixtures. Mechanisms of reaction were interpreted from radioactive isotope tracer studies by addition of 14C substituted pentenes to the reactants [189], albeit under conditions involving appreciable temperature change. In an earlier study, Chung and Sandler [26] showed that the proportion of pentenes formed during pentane oxidation in a flow system reached a maximum at a reactor temperature of about 750 K. Much of the outstanding chemical investigation of the low-temperature combustion of alkanes was performed on the C5 and C6 alkanes, in the late 1960s and early 1970s, mainly by Cullis and co-workers [137, 189-193] and Fish and co-workers [99, 194-199]. The quality and extent of the chemical analyses in these studies has rarely been equalled in subsequent work, and the data obtained provide very strong evidence, not only for the importance of alkylperoxy radical formation and isomerization in the low-temperature chemistry, but also the role of these reactions in the development of cool-flames and two-stage ignitions. However, one constraint on quantitative application is that much of the information was obtained under markedly non-isothermal conditions in the absence of any record of the reactant temperature. Moreover, the effects of convection on the movement of cool-flame combustion waves within an unstirred reaction vessel were not appreciated at the time [52], which casts doubt on some of the mechanistic interpretations of the evolution of multiple cool-flames that were made [195]. The low pressure, non-isothermal oxidation of 2-methylpentane, which contains primary, secondary and tertiary H atoms, was investigated by Fish [195,199] under closed vessel conditions ([RH]: [0 2 ] = 1:2). Pressure changes were obtained throughout reaction, but temperature changes were not measured. Multiple cool-flame behaviour and two-stage ignitions were observed. Extensive and detailed chemical analyses were made throughout the course of the non-isothermal reaction under a wide range of reactant pressures (20-33 kPa) and vessel temperatures (515-725 K). The concentrations of partially oxygenated molecular products, particularly carbonyls and peroxides, increased with temperature through the slow combustion region. O-heterocyclic intermediates, such as methyl substituted tetrahydrofurans and oxirans, became particularly prominent. Most oxygenates ceased to be formed in substantial amounts above a

618


Ch. 6

vessel temperature of about 620 K, but the concentration of acetaldehyde continued to increase until the temperature approached 670 K. The major products were alkenes, also with some lower molecular mass alkanes (Table 6.9). A product of ignition in these fuel-rich mixtures was benzene, which was formed in surprisingly high yield. This seems to have received no comment in the literature. Organic peroxides were detected during the slow oxidation of 2-methylpentane at a vessel temperature of about 525 K, although their structures were not identified in the analysis. Total peroxides were also measured during cool-flame propagation, but the extent to which hydrogen peroxide contributed to the overall yield was not determined. Radioactive tracer studies of 2-methylpentane oxidation at 515 K enabled Cullis et al. [200] to show that the overall mechanism was more strongly affected by abstraction from the four secondary C—H sites than from the single tertiary C—H site, despite its lower bond strength. The temperature at which this study was performed would suggest that abstraction by OH radicals might be rather more important than abstraction by H 0 2 or R 0 2 radicals. The chemistry of the non-isothermal oxidation of 2-methylpentane also TABLE 6.9 Comparison of the major products for non-isothermal oxidation of 2methylpentane in oxygen (l:2mol) identified at low pressure (2033.3 kPa) in a closed vessel during slow combustion at 530 and 620 K [195,199]

Products

Ta = 530 K

Ta = 620 K

Peroxides Ethylene Butanone Propionaldehyde Pentan-2-one 3-methyloxetan Acetaldehyde Methacrolein /-Butyraldehyde Acetone 3-methylbutan-2-one 2,2-dimethyltetrahydrofuran C5 alkenes

Acetaldehyde C5 alkenes Propylene Z-butene C6 alkenes 2-methylpentan-3-one


619

has been studied at high pressures, ~2MPa, in stoichiometric mixtures with air in a rapid compression machine [99, 194, 196]. Rapid sampling during the ignition delay and chromatographic analysis yielded profiles of the reactant consumption and change of intermediate molecular product concentrations throughout the evolution of two-stage ignition. The aim of this study was to compare the chemical mechanism of the non-isothermal oxidation of 2-methylpentane in the region of cool-flame at sub-atmospheric pressures with that at 2 MPa. A comparison of the product distributions revealed a broad similarity at low and high pressures, in similar temperature ranges. One main difference was that large concentrations of CO were formed at high pressures, but this may also be a consequence of the much higher proportion of oxygen present than that in the low-pressure studies, which may promote the further oxidation of primary products. Among other significant features was that at low pressures fewer oxirans were formed, and the distribution of unsaturated carbonyl compounds differed. Amongst the products from the high-pressure studies were several ketoaldehydes [103]. These compounds could be formed only via a dihydroperoxy oxidation mechanism. The distributions of products from the low-temperature chemistry following alkylperoxy radical isomerization were classified as: O—O homolytic bond fission and ring closure —> O-heterocycle + OH C—O homolysis —> alkene + H 0 2 O—O homolysis and intramolecular H-transfer —> carbonyl + OH O—O and C—C homolytic fission —> OH + alkene + carbonyl/O-heterocycle O—O and C—C homolysis and group transfer —> carbonyl + alkene (one rearranged) + OH O—O and two other bonds undergo homolytic bond fission —»unsaturated carbonyl + alkyl radical + water These chemical links between low- and high-pressure studies also support common features, notably between multiple cool-flames and two-stage ignitions at low pressures in closed vessels, and two-stage ignitions at high pressures in rapid compression machines and in the combustion chambers of engines. The relevant chemical context is that, as reactant temperatures during the evolution of two-stage ignition are raised in the first, "coolflame" stage, to beyond 800 K, the overall mechanism is that of the higher

620


Ch. 6

temperature regime. It embraces not only the oxidation of the remaining primary fuel, largely in H 0 2 driven chemistry, but also the oxidation of partially oxygenated intermediates and lower molecular mass hydrocarbons. Amongst these also are hydrogen peroxide and formaldehyde, which readily oxidizes via H 0 2 to H 2 0 2 . Hydrogen peroxide decomposition leads to chain branching and the rate of decomposition increases dramatically as thermal feedback causes an increase of the reactant temperature. The rate of development in the second stage can be quite slow initially because the driving reactions are not necessarily very exothermic, but the later, more rapid, phase can generate strong pressure pulses with the consequences for engine knock discussed in Chapter 7. As discussed in Section 1.14, the decomposition of hydrogen peroxide in the reaction, (53)

H 2 0 2 + M -> 2 0 H + M ; L = 3 x 1014 exp ( - 24400/r) s" 1

is usually the principal chain-branching reaction characteristic of the temperature range above about 750 K, but there are two important qualifications. First, the rate of decomposition of hydrogen peroxide is exceedingly slow at 750 K (t1/2 = 0.3 s), and becomes effective on the time-scale of the development of the second stage of two-stage ignition only at temperatures beyond 850 K (f 1 / 2 H 2 0 2 + R

The decomposition preceded by the reaction (55)

H 0 2 + H 0 2 -> H 2 0 2 + 0 2

does not lead to a net gain of free radicals. 6.5.5 Oxidation of n-heptane and 2,2,4-trimethylpentane (i-octane) Although there had been earlier attention [201], towards the end of the 1980s interests became focused more strongly on the study of the behaviour of the primary reference fuels for gasoline (petrol), i.e., n-heptane and i-


621

octane (2,2,4-trimethylpentane). Studies of the oxidation at high pressures have begun to emerge, the main work having been performed in jet-stirred flow reactors. Luck et al. [105] compared the product compositions from n-heptane combustion in the end gas of a fired single-cylinder engine (—3.5 MPa) with those from low pressure, closed and flowing systems used in earlier work. Some of the results reported in [105] were obtained in the flow reactor described in Section 6.3.3 [72]. Non-isothermal combustion of stoichiometric fuel: air mixtures ( = 1 are shown in Table 6.11. Greater extents of reaction of /-

622


£ ° •G

£

ai 8, hydroperoxides were detected in studies by Brown and Fish [209] during the liquid phase oxidation of 2-methylhexadecane at temperatures below 500 K. They showed that monohydroperoxides were the principal primary product of the low-temperature oxidation. An autocatalytic reaction developed as a consequence of the peroxide decomposition. This pattern is consistent with other low-temperature, gas phase and liquid phase studies of hydrocarbons and other organic compounds, such as acetaldehyde [152-156]. Brown and Fish [209] found very little evidence that alkylperoxy radical isomerization occurred and no evidence for alkyl dihydroperoxide formation at temperatures below 430 K. The most likely mode of alkylperoxy radical isomerization arose with the possibility of an internal abstraction at the tertiary C—H bond to create a transition state of minimum strain energy, namely,


(56)

627

R02^Q(l,6t)OOH

The primary propagating and branching steps involve intermolecular H atom abstraction from another fuel molecule in the sequence (57) (58)

RH + 0 2 - * R + H 0 2 R + 02-^R02

(59)

R 0 2 + RH -» R 0 2 H + R

(60)

R 0 2 H ^ R O + OH

Further H atom abstraction by the alkoxy radical gave rise to C17 alcohols, but decomposition of the alkoxy radical also occurred to form products containing fewer carbon atoms. Whereas alkyl hydroperoxides predominated at about 450 K during the gaseous oxidation of n-decane [210], dihydroperoxides became increasingly prominent and were measured in equal amounts to the alkyl hydroperoxides up to about 500 K. Neither was present in measurable yield at higher temperatures. Partially oxygenated molecular intermediates of C < 10 and lower alkenes were also detected at these low-temperatures, but O-heterocycles containing 10 carbon atoms were the principal intermediate products throughout the vessel temperature range of 500-650 K (Table 6.13). A single cool-flame occurred during reaction throughout this temperature range at the prevailing composition and pressure. The high-pressure oxidation of n-decane and of kerosene from low to high temperatures has also been studied by Dagaut et al. [211, 212] in their jet-stirred flow reactor. Reactant consumption and intermediate and final product formation have been measured in similar detail to that of nheptane and /-octane oxidations. The relative proportions of a number of O-heterocyclic compounds, partially oxygenated products and alkenes formed from n-decane-oxygen in the stoichiometric ratios <E> = 0.1, 1.0 and 1.5 with excess nitrogen, were studied at 1.0 MPa over the temperature range 550-1150 K. Limited chemical data have been obtained during the spontaneous ignition of n-tetradecane in a high-pressure flow system (3 MPa) by Cavaliere et al. [147]. Liquid n-tetradecane fuel was injected perpendicular to the pre-heated air flow ( 1 m s - 1 ) over the temperature range 500-900 K. Products were withdrawn by a rapid sampling valve

628


Ch. 6

TABLE 6.13 Variation in product distribution with change in initial temperature during n-decane oxidation in a liquid-injected closed vessel (P, = 33.3 kPa; decane: 0 2 : N 2 = 2:1:7) [210] TJK Products O-heterocycles Cio carbonyls Cio alcohols Cio alkenes C9 alkenes C5_8 alkenes C7_9 carbonyls C3_6 carbonyls

103 Product yield (mol per initial mol /i-decane) 483 503 533 563 593

623

673

140.0

260.1

368.6

380.7

381.3

402.9

341.0

59.7

52.3

61.1

58.1

58.8

58.2

50.2

3.8 137.3 291.0

76.7 1.3 87.8 82.3 163.0

106.0 14.3 123.8 38.0

130.3 14.9 132.3 36.5

149.7 15.3 145.4

164.9 18.1 177.6

25.0

30.1

153.1 33.2 379.0 36.5

located 25 mm from the spray nozzle on the axis of the spray. Reaction products were removed at a fixed time after injection and the air inlet temperature was varied in order to give a chemical species profile for the oxidation of rc-tetradecane over the operating temperature range. 6.5.7 Oxidation of aromatics Studies of benzene and toluene oxidation in the turbulent flow reactor at Princeton University have provided valuable information on the mechanisms of oxidation at temperatures in excess of 1000 K [213]. The first extensive study of benzene and toluene at temperatures of about 750 K was made by Burgoyne [35]. Apart from CO and C 0 2 , the major products of benzene oxidation that were detected were phenols and acids. An autocatalytic reaction was observed by Burgoyne [35] presumably driven by H 2 0 2 formation and decomposition. Amongst the main products of toluene slow oxidation were benzyl alcohol, benzaldehyde and benzoic acid. Phenolic compounds were also reported. This reaction also showed an autocatalytic development. An equilibrium constant for the equilibrium between benzyl and benzylperoxy radicals has been measured by Fenter et al. [214], but this cannot be followed by an isomerization in the way that is possible in alkanes. Increases of the length of the alkyl side-chain open the possibility of alkylperoxy radical formation and isomerization, so there is then some

Detailed numerical modelling of alkane oxidation

629

scope for reactions similar to those of the alkanes. Similar reaction mechanisms involving intramolecular H atom transfer are also possible when there is more than one substituted alkyl group, as in ortho- and meta-xylene. Barnard and Sankey [36, 215] confirmed the formation of oxygenated products during the slow oxidation of oxylene at 785 K, including o-xylene oxide, and proposed that its formation resulted from the reactions, (61) (62) (63)

C H 3 C 6 H 4 C H 2 + 0 2 -> C H 3 C 6 H 4 C H 2 0 2 C H 3 C 6 H 4 C H 2 0 2 -> C H 2 C 6 H 4 C H 2 0 2 H C H 2 C 6 H 4 C H 2 0 2 H -> C 6 H 4 (CH 2 )(0) + O H

Reaction (61) would now be expected to be an equilibrium. The R 0 2 radicals involved in these types of reactions are likely to have a considerably enhanced stability as a result of the delocalized character of the aromatic rings. The rate of oxidation of p-xylene was found, by Barnard and Ibbison [216], to be very similar to that of toluene. The slow oxidation of all of the xylene isomers and of toluene were found to be autocatalytic [36, 215], and a mechanism through formaldehyde oxidation was proposed, since it was found to be one of the intermediate products. Hydrogen peroxide must also be formed from the predominantly H 0 2 propagation, and this is likely to be an important contributor to the autocatalysis.

6.6 DETAILED NUMERICAL MODELLING OF ALKANE OXIDATION AND SPONTANEOUS IGNITION

6.6.1 Introduction The construction of a comprehensive kinetic model to represent the oxidation of a hydrocarbon, incorporating the "best" available kinetic parameters, permits a quantitative link to be forged by numerical computation between detailed chemical measurements and the interpretation of the underlying kinetics and mechanism of the combustion system. The first step is the simulation of composition-time profiles for intermediate and final products under conditions resembling the experimental study, as a validation of the model itself. Further insight may then be gained into the

630


Ch. 6

behaviour of free radicals during the course of the reaction from their predicted profiles [217]. If the model yields a satisfactory quantitative representation of the observed chemistry, which requires exhaustive tests over as wide ranges of experimental conditions as possible, then the model is ready for application in a number of different ways. These may have essentially chemical purposes, such as the prediction of a product yield in a chemical process, but often the aim is to predict other sorts of behaviour, such as the possibility of spontaneous ignition, ignition delay times, or more complex phenomena such as engine knock. Measurements involving spontaneous ignition under well-controlled conditions, such as the location of p-Ta ignition boundaries or the duration of ignition delays, constitute a useful step towards an understanding of spontaneous ignition phenomena but, on their own, they have limited value as a diagnostic of the chemical authenticity of a comprehensive kinetic scheme. In order to simulate (p-Ta) ignition diagrams of the hydrocarbons (or other organic compounds) and the associated oscillatory cool-flame phenomena, additional information is required about heat transport properties for the modelling of these non-isothermal processes. Its inclusion creates an opportunity to investigate how well the chain-thermal interaction is represented in the kinetic structure. So far there seems to have been no rigorous application of detailed kinetic models to represent alkane oxidation, or those of any other hydrocarbons at temperatures below 850 K. The only reported simulations of (p-Ta) ignition diagrams using detailed thermokinetic models pertain to acetaldehyde combustion [153-156]. The establishment of a detailed kinetic model provides an opportunity for the numerical prediction of the behaviour of a chemical system under conditions that may not be accessible by experimental means. However, large-scale models with many variables may require considerable computer resource for their implementation, especially under non-isothermal conditions, for which stiffness of the system of differential equations for mass and energy to be integrated is a problem. Computation in a spatial domain, for which partial differential expressions are appropriate, becomes considerably more demanding. There are also many important fluid mechanical problems in reactive systems, the detailed kinetic representation of the chemistry for which would be highly desirable, but cannot yet be computed economically. In such circumstances there is a place for the use of reduced or simplified kinetic models, as discussed in Chapter 7. Thus,


631

not only do comprehensive kinetic models constitute a starting point of the formal mathematical methods for reduction to simplified schemes (Chapter 4), but also they may be used as analytical tools against which the performance of reduced models may be tested. There has been very substantial progress in the United States and Western Europe to construct comprehensive kinetic models to represent hydrocarbon combustion over wide temperature and pressure ranges [218-220]. There have also been advances in large-scale numerical modelling in Russia [221]. The present position is that comprehensive kinetic models exist for the combustion of alkanes, certainly up to /-octane with some exploration towards the components of kerosene and diesel fuels [212]. The temperature range for the intended application of existing models is approximately 600-2600 K. The largest kinetic schemes now run to many thousands of elementary steps, but their origins were more modest. The first successes constituted the modelling of flame properties of the simplest fuels such as hydrogen, carbon monoxide, methane or methanol. Distinguished studies of flame structure of the simpler fuels continue today [222]. Progress to predicting detailed flame properties of more complex fuels by numerical methods is frustrated by the need for fundamental data for heat and mass transport properties (thermal conductivity and diffusion coefficients in laminar flames) in one and two dimensions, in addition to the chemical kinetic parameters. Progression via zero-dimensional, numerical analysis was made through the development of models to represent ethane, ethene, and acetylene at temperatures in excess of 1000 K, followed by additional reactions to represent the oxidation of propane and propene, and n-butane and ibutane combustion. An important underlying feature, particularly with regard to the choice of appropriate numerical data, was the similar structure of the chemistry across a range of fuels, engendered by the predominance of unimolecular decomposition reactions. Moreover, the reactions of the much smaller C/H/O containing species, encapsulated in the mechanisms for the lower molecular mass fuels, proved to be of considerable importance in the flame chemistry of, say, the C 4 or higher hydrocarbons. The interactions at high temperatures, which involve the lower molecular mass species, are admirably illustrated by Warnatz [223]. By 1984, the model of Westbrook and co-workers to represent the hightemperature combustion of hydrocarbons, up to and including the C4 alkanes and (to some extent) alkene isomers, incorporated 255 reversible

632


Ch. 6

reactions. Sub-mechanisms to represent flame retardancy or inhibition were also included. Validation was made by the simulation of flame propagation and also against ignition delays measured in shock tubes [224]. Warnatz [223] had also set up similar models, which give insights into the structure of laminar alkane, alkene and acetylene flames up to C 4 , and the development of a kinetic model for the high-temperature combustion of alkanes up to octane had begun. Until about 1985 the emphasis remained on high-temperature combustion and flame properties. That is that detailed models were confined to temperatures above about 1000 K. There has been some progress towards an understanding of the reaction mechanisms of benzene and toluene at temperatures above 1000 K, supported by experimental studies [213]. The inclusion of reactions to represent the "low-temperature" chemistry in a detailed model for n-butane oxidation at high pressures, that is appropriate to temperatures down to about 600 K began in 1986 [225]. At the present time, models which include around 500 species and more than 2000 reversible reactions to represent alkane isomers up to heptane, are in use [219] and still larger schemes are under development [220]. Progress in the validation and application of these models, and kinetic representations for propane and propene oxidation, are discussed in the next subsection. Modelling of the low-temperature combustion of ethene has also been undertaken more recently [20]. There has not been the same progress in the numerical modelling of the kinetics and mechanisms of combustion of other classes of hydrocarbons at temperatures below 1000 K. The lack of detailed knowledge of mechanisms and quantitative data for the rate constants of elementary reactions from experimental measurements is the main difficulty. The alkenes are important because they are so intimately involved as intermediates in the combustion of alkanes. No less important are the aromatics, toluene and the xylene isomers in particular, because they are components of gasoline. Knowledge of the oxidation of other organic compounds is also relevant to the detailed kinetics of the low-temperature combustion of hydrocarbons, such as alcohols, ethers (linear and cyclic) and aldehydes. Of these the chemistry of the lower aldehydes (formaldehyde and acetaldehyde) are probably best understood [153-156]. There have been recent studies of ethers, such as methyls-butyl ether [226], prompted by the growing importance of this class of compounds as octane boosters in unleaded gasoline and as contributors to "clean technology" gasolines.


633

6.6.2 Validation of comprehensive kinetic models for low-temperature oxidation The experimental study by Wilk et al. [168] of the low-temperature oxidation of propene, discussed in Section 6.5.2, was the subject of numerical interpretation [227] on the basis of a detailed mechanism comprising 296 reversible reactions. Concentration-time profiles for C 3 H 6 , and for the intermediate and final product concentrations (CO, C 0 2 , CH 2 0, CH 4 , CH 3 OH, C 2 H 4 , CH3CHO and C 2 H 3 CHO + C 3 H 6 0) simulated under nonisothermal closed vessel conditions, were compared directly with experimental measurements over a range of vessel temperatures with due allowance for some incompatibility between the zero dimensional (spatiallyuniform) model and the spatially non-uniform experimental conditions, satisfactory agreement was achieved between the numerical analysis and the experimental observations. A number of useful points emerged from this exercise, the main kinetic conclusions of which are discussed in detail elsewhere [227]. For present purposes we may note that a negative temperature dependence of reaction rate, simulated as a rate of pressure change in a closed vessel, was predicted to exist in a similar temperature range to that observed experimentally. Moreover, multiple cool-flames, in satisfactory accord with the experimental observations, were also predicted. The underlying kinetic structure which gave rise to the ntc of rate was of the form, (64) (65) (66)

R + 02 ^ R02 R 0 2 + XH ^ R 0 2 H + X R 0 2 H ^ RO + OH

in which the moiety R may be identified as CH 3 , CH 3 CO, C 2 H 3 , C 2 H 5 , C 2 H 3 CO, rc-C3H7 or /-C3H7. However, in the absence of a sensitivity analysis (Chapter 4), it was not possible to distinguish if there was a dominant organic peroxide which governs the chain branching rate, if indeed any particular species was dominant. Some support was given for the view that the equilibrium involving allyl radicals, (35)

C3H5 + 0 2 ^ C 3 H 5 0 2

634


Ch. 6

is unlikely to play an important part in control of the negative temperature dependent reaction (Section 1.16). In similar modelling studies of n-butane oxidation in a closed vessel, Pitz et al. [228] paid particular attention to the changes in yields of oxygenated intermediates (aldehydes, alcohols, cyclic ethers) and also of the hydrocarbon intermediates, as the reaction vessel temperature was varied over the range of investigation. Non-isothermal reaction occurred under the prevailing experimental conditions, and the accompanying temperature changes were measured by a sensitive thermocouple. As in the propene analysis, although the numerical model itself did not accommodate spatial variations of temperature, due allowance was made for self-heating by a simultaneous integration of the energy conservation equation, with heat loss. The predicted temperature change was checked against the experimental measurement. The kinetic scheme comprised more than 800 reversible reactions, more than 70% of which would be regarded to belong to the "low-temperature" class of reactions. The important concentration-time profiles include those for 1,2- and 2,3-epoxybutane, 2-methyloxetan, tetrahydrofuran, butyraldehyde and 2butanone, the relative concentrations of these products being determined by the relative rates of 1- and 2-butylperoxy radical isomerization and decomposition. The kinetic data used to represent these reactions were the parameters recommended by Walker [229] in 1961. The formation of butyraldehyde and 2-butanone require internal isomerizations of the form (67)

R p 0 2 ^ Q(l,3p)OOH and R s 0 2 ^ Q(l,3s)OOH,

neither of which is well documented. Following this study, Wilk et al. [230] simulated the composition-time profiles for selected alkenes and oxygenated products that were formed from rc-butane and /-butane combustion, and also mixtures of these fuels, in a motored engine. An engine cycle was simulated within a spatially uniform zone of varying volume. The volume history was specified in such a way that the predicted pressure history matched the measured polytropic pressure history in non-reactive conditions. Composition profiles were compared with those measured experimentally. Some of the kinetic features that distinguish the reactivities of the two fuels and their modes of reaction involving alkylperoxy and dialkylperoxy radicals were elucidated in this work. The n-butane oxidation model had also been applied to the


635

simulation of results obtained by Leppard and co-workers in a motored engine [231] and an extension of it was applied to the oxidation of butene isomers under the same experimental conditions [232]. The same variable volume technique was adopted in these simulations as in those for the simulation of the experimental studies of the butane isomers by Wilk et al. [230]. However, the end-of-cycle yields of products were obtained at a range of compression ratios in these cases in order to represent the "averaged" exhaust gas compositions measured by Leppard. The numerical model for n-butane oxidation, by Pitz et al. [228], was used also by Carlier et al. [21] to simulate experimental studies of the twostage combustion of n-butane at 0.18 MPa on a flat-flame burner and, following this validation, to simulate the ignition delays of n-butane in a rapid compression machine. The numerical studies of the burner experiments were extended by Corre et al. [233]. For simulations of the behaviour on a flat-flame burner the chemical model was computed in an isothermal mode, the experimental one-dimensional temperature profile being introduced as an input parameter. Among the important aims of the tests by Corre et al. [233] was the rationalization of the predicted extent of nbutane consumption throughout the development of the first (cool-flame) and second stages of combustion, with that observed experimentally. The experimental study by Minetti et al. [22, 116] included the detection and measurement of R 0 2 and H 0 2 radicals by esr, the one-dimensional spatial profiles of which were simulated by Corre et al. [233]. Carlier et al. [21] established that the kinetic model for butane combustion developed by Pitz et al. [228] gave rise to a prediction of 50% consumption of n-butane in the first reaction stage, whereas only 15% consumption was measured experimentally. Amendments to the kinetic model were then made involving a number of R 0 2 + R 0 2 interactions. The relative importance of reactions involved in the consumption of n-butane and in the formation of OH throughout the two-stage reaction, which were derived from the modified model, are shown in Fig. 6.26. It is important that the extent of fuel consumption during the development of two-stage ignition is satisfactorily represented, since the removal of too much fuel early in the reaction must lead to serious quantitative discrepancies in the simulation of the second stage of a two-stage ignition or, at worst, a qualitative failure to predict its existence. Such problems may be exacerbated in a full thermokinetic simulation, when exothermicity of reaction and thermal feedback are taken into account, because the self-attenuating

636


Ch. 6

properties within the ntc region can accentuate the extent of reactant consumption required to sustain reaction. However, this aspect could not be explored within the more limited scope of the simulations by Minetti etal. [116]. There are some discrepancies in the relative yields of some products shown in Fig. 6.26 but, in general, the agreement is encouraging. This study is among the few tests of comprehensive models that relates to the molecular products of alkylperoxy isomerization. In this study also, attention was paid to the rate of OH formation during the two-stage ignition. The mechanism shows that the OH production rate is controlled by a series of reactions whose relative importance changes with temperature. The contribution of the six individual reaction rates of OH formation corresponded to 95% of the overall rate of OH formation, as follows: (68) (69)

0 2 QOOH -* aldehyde + RO + 2 0 H QOOH -» heterocycle + OH

(70)

H 2 0 2 + M -» 2 0 H + M

(71)

R O O H - * R O + OH

(72)

CH 3 0 2 + H 0 2 -» CH 3 0 + OH + 0 2

(73)

R + H 0 2 ^ R O + OH

A detailed kinetic model for n-butane combustion has also been reported by Kojima [234] comprising 700 reversible reactions. Although this may prove to be a useful development, it has not been the subject of chemical validation, having been used only to simulate ignition delays for n-butane in a shock tube and in a rapid compression machine. In the absence of complementary chemical tests, the prediction of ignition delay really constitutes an application rather than a test of a comprehensive kinetic scheme. Although studies have emerged from a number of different research groups, there are fewer examples of the validation of kinetic models for higher alkanes at present. Minetti et al. [101] have adapted a kinetic model developed by Warnatz to represent n-heptane oxidation. It comprises 168 species in 904 reversible reactions. There is a strong emphasis on the


637

R + 0 2 / R 0 2 equilibria and on the modes of alkylperoxy radical isomerization and further oxidation in this scheme. The model was tested by comparison of the predictions of the concentration-time profiles for a number of unsaturated and oxygenated intermediates with experimental measurements during the development of two-stage ignition in a rapid compression machine. This is a rigorous test because it includes the effect of thermal feedback on the kinetic evolution as the reactant temperatures rise. The predictions of overall ignition delay and the temperature range for the existence of a negative temperature dependence of ignition delay were simulated satisfactorily, but the concentrations of molecular intermediates were not well reproduced. One fundamental difficulty of this kinetic model appears to be that a far higher proportion of ^-heptane (80%) was predicted to be consumed in the first stage of twostage ignition than was measured experimentally (25%). Automatic methods for the generation of reaction schemes have also been applied to alkane oxidation (see Chapter 4 and [217]). A model comprising 640 species in 2400 reactions to represent the oxidation of alkane isomers up to and including octane over the temperature range 500-2500 K has been developed by Chevalier et al. [220]. This model was used to predict ignition delays in constant volume enclosures at high pressures under spatially uniform, adiabatic conditions. For the purpose of test of this kinetic model, these conditions were deemed to represent combustion in shock heated gases, and the results were compared with the experimental ignition delays for rc-heptane + air mixtures measured by Ciezki and Adomeit [29]. A similar procedure had been adopted previously by Warnatz [219]. Subsequent to their experimental studies of kerosene oxidation at up to 4MPa in the high pressure CSTR, Dagaut et al. [202, 212] modelled ndecane oxidation in the temperature range 750-1150 K. The mechanism comprised 90 species in 573 reversible elementary reactions. The principal aim was to reproduce the stationary state concentrations of C 0 2 , CO, CH 2 0, CH 4 and C 2 H 4 that were measured experimentally as a function of the reactor temperature at various equivalence ratios, reactant pressures, and mean residence times. This emphasis on low molecular mass species arose because there was substantial experimental evidence that the high molecular mass alkyl radicals, derived from the predominantly alkanecontaining aviation fuel, decomposed readily at temperatures above 750 K to smaller species, as discussed in Chapter 1. Dagaut et al. [235-238] had

638


Ch. 6

previously developed and validated models for C 2 H 4 , C 2 H 6 and C 3 H 8 and 1-C4H8 oxidation at temperatures in the range 800-1200K and at pressures up to 1 MPa. 6.6.3 Applications of comprehensive kinetic models The only specific application of a comprehensive kinetic model that appears to have been reported so far is the comparison of predicted times of ignition for selected alkanes with the research octane number (RON) measured in a CFR engine (Table 6.14) [220, 239]. The calculations were based on the variable volume numerical model, described in Section 6.6.2, matched to a motored engine pressure history from a primary reference fuel mixture corresponding to RON 90 in a stoichiometric proportion with air (4> = 1.0). In the simulation the compression ratio was set to the critical compression ratio of a RON 90 fuel mixture. The time to ignition was determined from bottom dead centre (bdc.) of the compression stroke in the simulated engine cycle to the maximum rate of temperature change in autoignition. Cowart et al. [143] used a similar procedure as a basis for the validation of a reduced kinetic model to represent /-octane end-gas autoignition, as is discussed in Chapter 7. The duration of the compression stroke in the calculated times (Table 6.14) corresponded to approximately 50 ms, signifying that autoignition occurred after top dead centre, as observed experimentally. The choice of bdc as the initial condition is essential to permit the development of spontaneous combustion during the course of the compression stroke as the gas pressure and temperature increase. This definition of the time for ignition differs from that given by, (i) the admission of reactants to a hot, closed reaction vessel, (ii) the end of compression in an RCM, or (iii) the passage of a shock through the reactants in a shock tube. It is not easy to rationalize these times for ignition shown in Table 6.14 with the research octane number for each fuel, even in the context of a broad correlation based on RON = 60-70, 70-80, 80-90 or 90-100. Moreover, isomers of similar structure that exhibit a similar RON do not show a similar degree of correspondence in the calculated ignition delay. Before any further assessment of these data is made, it would be desirable also to have experimentally measured ignition delays for these single component fuels under high pressure conditions, as in an RCM or shock tube. Most of the numerical modelling discussed in this section has been


639

TABLE 6.14 Research octane numbers for selected alkane fuels and predicted times of ignition [239] Fuel

RON

Time of ignition/ms

n-heptane n-hexane 2-methylhexane tt-pentane 3-ethylpentane 2-methylpentane 3-methylpentane 3,3-dimethylpentane 2,4-dimethylpentane 2,2-dimethylpropane 2-methylbutane 2,2-dimethylbutane 2,2-dimethylpentane ^-butane 2,3-dimethylbutane 2,2,4-trimethylpentane 2-methylpropane Propane 2,2,3-trimethylbutane Ethane

0 25 42 62 65 73 74 81 83 86 92 92 93 94 100 100 102 112 112 115

55.0 55.0 54.5 55.9 57.4 55.5 57.2 55.7 56.2 56.4 59.1 59.9

* 57.6 58.0 58.5

* * * *

*No ignition.

concerned with the simulation of the chemical development of reaction under non-isothermal conditions, usually leading to ignition or cool-flame phenomena. Exceptionally, the work by Dagaut et al. [202, 212] pertained to isothermal reaction under well-stirred conditions. Despite some incompatibility between the zero-dimensional numerical modelling and the spatial non-uniformity of the experiments to which most simulations have been matched, the interaction between modelling and experiment has led to improvements of existing models for alkane and alkene oxidation. Numerical studies have yielded a clearer insight into the kinetic structures that are involved in the complex thermokinetic interactions leading to spontaneous ignition. There is no doubt that numerical modelling has played a very important part in combustion research in the last twenty years [240]. Numerical approaches have enhanced the link between the

640


Ch. 6

formal understanding of the elementary foundations (Chapters 1 and 2) and the interpretation of hydrocarbon ignition.

6.7 CONCLUSIONS

As discussed in Section 6.6, numerical analyses can be used with some success to simulate the chemical features of low-temperature hydrocarbon oxidation. However, there is progress yet to be made on a number of fronts by both experimentalists and numerical modellers. The kinetics and mechanism of certain classes of compounds, such as the alkenes and aromatics, have not been investigated fully in the low-temperature region, the information on aromatic compounds being particularly sparse (Chapter 1). By contrast, there is much better understanding of the low-temperature oxidation of alkanes. Following Section 1.4.1, the reactivity of alkanes arises within the mechanistic structure (74)

cyclic ether + OH alkene + aldehyde + OH ((3-scission) 02

T

°

2

alkene + HO2 branching

i branching The extent of knowledge of each stage of this generalized overview is discussed fully in Chapters 1 and 2. Although the ability to cope with complex mixtures of fuels may be some way off, dealing with alkane mixtures is certainly within our grasp. However, there appears to be one key aspect that requires clarification. From the evidence in Section 6.4, it is clear that the longer chain nalkanes, such as n-heptane, can undergo spontaneous ignition at very lowtemperatures. This is in marked contrast to the lower reactivity of the shorter n-alkane chains such as ^-butane, or of highly branched isomers

Conclusions

641

such as 2,2,4-trimethylpentane (/-octane). Lignola et al. [32] proposed that differences in reactivity between n-heptane and /-octane were due to the amount of acetaldehyde that was formed during alkylperoxy radical isomerization and decomposition. Alternatively, for the alkanes of low carbon number (Cn < 4) the inability to undergo spontaneous ignition at very lowtemperatures may be attributed to the existence of very high energy barriers associated with alkylperoxy radical isomerization and the failure of diperoxy species to be formed, whereas the differences in reactivity of higher carbon number alkanes may lie in constraints of the alkyldiperoxy radical rearrangement and decomposition (0 2 QOOH in (74)). Beyond this point neither the measured product compositions from slow oxidation (Section 6.5), nor the results of numerical modelling (Section 6.6), yet give a proper account of the sensitivity of reactivity to structure. There is ample experimental evidence for the formation of alkyldihydroperoxides, especially from the longer chain n-alkanes (Section 6.5) but, as noted in Chapter 1, the rate of chain branching which arises from the homolysis of molecular hydroperoxides, regardless of whether they are alkyl hydroperoxides or dihydroperoxides, is too slow to account for the short duration of ignition delays in high-pressure gases at low-temperature. There is also experimental evidence for the formation of alkylketohydroperoxy radicals of the form (75)

—CH 2 CH(OOH)CH 2 —CH(00)CH 2 — ^ —CH 2 C(OOH)CH—CH(OOH)CH 2 — —CH 2 C(0)CH 2 —CH(OOH)CH 2 — + OH

Similar rearrangements may also occur involving an H atom, other than that at the hydroperoxy site, but in these circumstances the product is not a molecular species, but a free radical. (76)

— CH 2 CH(OOH)CH 2 —CH(00)CH 2 — ^ —CH 2 CH(OOH)C()H—CH(OOH)CH 2 —

As discussed in Section 1.14, the decomposition of such a species is a branching reaction, e.g.,

642


(77)

—CH 2 CH(OOH)C()H---CH(OOH)CH 2 — -> —CH 2 CH(0)CH 2 + OCHCH 2 + 2 0 H

Ch. 6

and is likely to occur with an activation energy which is appreciably lower than that for molecular hydroperoxide decomposition, and typically that of the energies involved with the decomposition of QOOH (Section 1.11). These particular mechanisms (76 + 77) may satisfactorily account for the high reactivity of long chain n-alkanes. The reactivity of highly branched isomeric structures may be suppressed by alternatives to (77). One example may be seen in the proposed nonbranching mode of decomposition of the alkyldiperoxy species formed from neo-pentane (Section 1.11), viz., (78)

(CH 3 ) 2 C(CH 2 OOH)CH 2 0 2 - -> CH 3 COCH 3 + 2CH 2 0 + OH

Another non-branching mode of alkyldiperoxy radical decomposition that also suppresses OH radical propagation has been proposed for /-butane [175], (79)

(CH 3 ) 2 C(OOH)CH 2 0 2 - -> CH 3 COCH 3 + CH 2 0 + H 0 2

These examples suggest four alkane structures that may be able to undergo non-branching modes of alkyldihydroperoxy radical decomposition, as follows: (80) 02Q(l,4p)OOH: —C—CH200- -> aldehyde/ketone + CH20 + H0 2 (81)

02Q(l,4s)OOH: —CH—C— -> aldehyde + ketone + H0 2 OO- OOH

(82)

02Q(l,4t)OOH: —C—C— -> ketone + ketone + H0 2 OOOOH

(83)

02Q(l,5p)OOH: CH2—C— -> aldehyde/ketone + 2CH20 + OH OOH CH 2 00-

References

643

Apart from the studies of neo-pentane and /-butane, experimental evidence in support of these mechanisms and the determination of appropriate rate constants has yet to be forthcoming. Undoubtedly, improved numerical simulations of the variation in the low-temperature reactivity amongst alkanes and their isomers will emerge. Perhaps, more importantly there will be reduced numerical models, obtained by the methods discussed in Chapter 4, which encapsulate the appropriate kinetic dependences in a fundamental way. The difficulty with a number of current approaches to reduced kinetic modelling [217] is that, although some of the elements of (74) may be identified in the reduced schemes, the implementation of the schemes relies too heavily on empirically derived kinetic parameters. The ability to predict, by numerical methods, the spontaneous ignition of single component fuels or mixtures of hydrocarbons and the dependences on experimental conditions is extremely important with regard to ignition hazards or the performance of fuels in spark-ignition and diesel engines. Aspects of combustion in spark-ignition engines are addressed in Chapter 7. The models employed for such purposes must have fundamental kinetic origins if the response of practical fuels to changes in conditions, such as variations in fuel + oxidant mixtures or reactant pressure, is to be accurately captured.

ACKNOWLEDGEMENTS

The authors wish to thank EPSRC for a research studentship awarded (CM) and for research grants in support of the work related to this review.

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APPENDIX Chronological review since 1976 of a range of experimental studies of hydrocarbon oxidation at temperatures u p to 1300 K Year

Ref.

Hydrocarbon

Conditions

Results

1976-1980

[241]

Ethane

90 cm3 closed vessel; 775-875 K; 1627 kPa;

8 product yields as f ( t ) .

~421

Propane

30 cm3 closed vessel; 700 K; 28 kPa.

12 product yields as f ( f ) .

P31

Propane

Closed vessel; 623 K; isothermal; 33 kPa.


P441

n-butane

Closed vessel; 753 K; isothermal; 67 kPa.

8 product yields as f (butane consumed).

P451

n-butane

Mechanically stirred flow; 120 cm3; tres = 5 and 10 s; 420-520 K; non-isothermal; 100 kPa.

f(T.7).

375 cm closed vessel; 550/640 K; 0.4 MPa.

Relative proportions of 27 products and profiles for 19 products as f ( t ) .

Rates of formation of 7 oxygenates as

4

[I731

1981-1985

i-butane

~2461

n-pentane

350 cm3 flow reactor; 750-870K; 100 kPa; [RH]:[02]:[Nz] = 1:8:30; tre,=0.3s.

Selectivity of 17 partially oxygenated or hydrocarbon products. Profiles for selected products as f ( T ) .

12471

n-pentane

Closed vessel; 80 kPa; 753 K; n[ C S H I ~ ] : [ H ~ ] : [ ~ ~ ] :=[ N 5 :Z140:70:285. ]

9 oxygenates, 1-pentene, 1-butene, propene and methane as f(n-C5H12 consumption).

[248]

Methane

803 cm3 closed vessel; 728 K; 55-75 kPa.

Measured CH4 consumption rate versus alkylperoxy radical concentration.

[451

Ethane

500 cm3 mechanically stirred flow reactor; f,,, = 15-120 s; 500-800 K; 100 kPa;

Consumption of ethane and O2as f( To). Relative yields of CH,, C2H4, CHzO and CH3CH0 asf(T,).

~4x1

Ethene

803 cm3 closed vessel; 570-670 K; 50 kPa.

CzH4 consumption and CzH40 formation VS.[ROz]

(D

3

E

APPENDIX cont. Chronological review since 1976 of a range of experimental studies of hydrocarbon oxidation at temperatures up to 1300 K Year 1981-1985 (cont)

Ref. [249]

~ 9 1

Hydrocarbon Ethene Ethene

Conditions

Results

Tubular reactor; 980-1073 K; 0.11.0 MPa; Q, = 0.25-4.0.

6 product yields as f ( t r e s ) .

CSTR; 1053-1253 K; 1 atm; = 1.8-2.4 S .

9 products and reactants as f ( T )

Q, = 0.36;

I,,,

Propene

% P

600 cm3 mechanically stirred flow; 550745 K; 100 kPa; ires= 10-20 s.

12 product yields and temperature rise measurements over wide range of conditions as f ( t r e s ) .

Propene

Closed vessel; 8 kPa; 713 K; [ C & , ]: [ 0 2 ] : [N,] = 1 :30: 29.

C3Hb0, C2H4, CH,CHO, CH2CHCH0, CH4 as f ( t ) .

Propene and tetramethylbutene

Closed vessel; 8 kPa; 713 K; [c&] : [TMB] : [Oz] [N,] = 1 :5 30: 24.

C3Hh0. C2H4, CH3CH0, CHzCHCHO, CH4 as f ( t ) .

Propane and butane

Flow tube (high diluent); -1000K; 0.10.6 MPa; t,,, = 0-100 ms; Q, = 2.8 (C3H8); 1.6 (C4Hio).

6 product profiles as f ( t r e s )

1,3-butadiene

Adiabatic turbulent flow reactor; T = 1100K; 100 kPa; Q, = 0.55, 1.18, 1.65; t,,, = 0-125 ms.

i butene

100 cm3 closed vessel, unstirred; 587 K; non-isothermal; 45 kPa.

!?

x

2 E Ba 3

c

w2.

Cool flame region; t-C4H90ZHand AT as f(t).

i-butene



1-butene

JSFR; 900-1200 K; 1-10 atm; Q, = 0.154.0 (high dilution); t,,, = 0-0.25 s.

Concentration vs. t,,, for C4H8 isomers and 8 hydrocarbon products at various conditions.

Y

APPENDIX cont. Chronological review since 1976 of a range' of experimental studies of hydrocarbon oxidation at temperatures up to 1300 K Year 1981-1985 (cont)

Ref.

Conditions

Results

[255]

2,3-dimethylbutane

Decomposition in 02;Closed vessel; 773 K; ratio of partial pressures, [RH]:[O2] = (i) 2:13 (ii) 2:118.

7 products as f ( t ) .

~561

2-methylpentane

500 cm3 closed vessel; 600 K; nonisothermal; 0-20 kPa.

Yields of 11 partially oxygenated products.

12571

i-octane

Combustion in a flat flame; 100 kPa.

Product yields as f (distance from burner).

[2101

n-decane

500 cm3 closed vessel; non-isothermal; 450-650 K; 70 kPa; [RH] : [O,] : [Nz] = 2: 1:21.

Yields of 30 products at different T,. Profiles for total peroxide, 0-heterocycle, carbonyls, alkenes as f ( r ) . 9

Benzene, toluene and ethylbenzene

Adiabatic turbulent flow reactor; T = -1200 K; 100 kPa; fres = 0-125 ms; = 0.3-1.5.

Benzene and phenol and 7 other aliphatic hydrocarbon products identified (for benzene and ethylbenzene); 6 aromatics and 8 other aliphatic hydrocarbons identified (for toluene) as f(t,,,).

~581

Kerosene

JSFR; 870-1030 K; 100 kPa;

[259]

Methane

3.3 cm3 flow tube; [CH,]: [O,] = 10: 130: 1; 2.4-6.5 MPa; 690-730 K; isothermal.

CH30H, CO and COz and % conversion as f ( t d

Ethene

JSFR; @ = 1.0 with diluting gases; 10001250 K; 100 kPa; I,,, = 2 ms.

9 alkane and oxygenated product concentrations as f( To).

Ethene

1395 cm3 closed vessel; 700 K and 720 K; isothermal


[601

1986-1990

Hydrocarbon

~ 9 1 ~271

I,,, = 0-0.3

s.

6 major hydrocarbon products plus 11 others in minor yield.

LEI 'FI

9


Ref.

Hydrocarbon

Conditions

Results

[260]

Ethene

JSFR; 900-1200 K; isothermal; 0.11.0 MPa.

7 product yields as f(tr,,).

1131

Propane


11 product yields and AT as f ( t ) over wide range of conditions

~381

Propane

JSR; 800-1200K; 0.1-1.0MPa; @ = 0.150.4; t,,, = 0-0.35 ms.

8 products as f(tres).

[168,227]

Propene

1395 cm3 closed vessel; 580-715 K; 80 kPa; C3H,Jair, @ = 2.0.

10 product yields and AT as f ( t ) over wide range of conditions.

a

Closed vessel; 750 K; 8 kPa: boric acidcoated vessel; [C3H6]: [02] : [Nz] = 1 : 6.5 : 7.8.

Partial pressures of C3H6, propene oxide, acrolein, CH3CH0, C2H4, CO, CHzO and hexadiene as f ( t ) .

0

[2281

n-butane

1395 cm3 closed vessel; 550-740 K; nonisothermal; 73 kPa; [RH] : [Oz]: "21 = 1:2 : 20.

15 product yields as f(r), AT noted.

[391

n-butane

500 cm3 closed, mechanically stirred vessel; 560-600 K; non-isothermal; [RH]:[O2]:[NZ]= 5 : 1 : 4 ; 66kPa.

C4HI0 consumption as f ( t ) .

[211

n-butane

Low P(0.18 MPa; 2-stage flame; 6701070 K) and 2-stage ignitions (RCM; T, = 600-900 K); RH:air, @ = 1.0.

13 hydrocarbon and 0-heterocycle concentrations as f ( t ) .

n-butane and i-butane

Motored engine with fast action, incylinder sampling.

Product yields as f ( t ) .

~301

C

B3. c

s

0

5 m

APPENDIX cont. Chronological review since 1976 of a range of experimental studies of hydrocarbon oxidation at temperatures up t o 1300 K Year 1986-1990 (cont)

Ref.

Hydrocarbon

Conditions

Results

[262]

i-butane and n-butane

CFR Motored engine; T = 440 K; P 1.0 MPa; CR = 6.6-15.5; Q, = 1.0.

~631

i-butane

Molecular beam source reactor, MS analysis. 560-820 K; 90-130 kPa.

12 product yields as f( T )

i-butene

JSR; 900-1200K; 0.1-1.OMPa.

Product yields as f(tr,,).

n-pentane

JSFR; 1060-1300 K; 100 kPa; t,,, = 2 ms; 11 mol% n-CSHl2in Hz + diluents.

7 products as f( T ) .

Cyclohexane

5 dm3 closed vessel; 490-510 K; nonisothermal; 100 kPa; [RH] : [O,]: [N,] 2.5 : 1:4.

Cyclohexane consumption as f( T ) .

=

End gas concentrations for 21 stable species as f(engine speed).

*

w

=

4,4-dimethylpentene

Closed vessel; 750 K; 8 kPa; KCI coated vessel; [RH]:[02]:[N2]= 1:6.5:7.5.

Partial pressures of CH4, fuel, acrolein, CH3CH0, CzH4, CO, CH20 and hexadiene as f ( t ) .

n-heptaneli-octane

JSFR; 570-840 K; 202-1212 kPa; 100 cm3.

Ignition boundary. Reaction intermediates and implications to relative knock resistance.

n-heptane, i-octane and 2,3-dimethylbutane

Closed vessel and stirred flow system; 750-1000 K; isothermal.

Relative yields of HZ and CH, as f (extent of reaction).

n-octane and i-octane

Adiabatic turbulent flow reactor; Q, = 1.0 with diluting gases; 1100-1300 K; t,,, = 0125 ms; 100 kPa.

17 product concentrations as f(fres).

i-butene and n-octane mixtures

Adiabatic turbulent flow reactor; @ = 1.0 with diluting gases; 1100-1300 K; t,,, = 0125 ms; 100 kPa.

17 product concentrations as f ( t r e s ) .

3B

% 4


Ref.

Hydrocarbon

Conditions

Results

Toluene

Closed vessel; 773 K; 66 kPa; [ C ~ H S C H: [Hz] ~ ] : I 0 2 1 [Nz]= 2.5 : 140: 70: 287.5.

C6H6, C6HsCH0 and CH4 as f(C6HsCH3 consumption).

~541

n-butylbenzene

Adiabatic turbulent flow reactor; 1069 K; 100 kPa; Q, = 0.98; t,,, = 0-140 rns.

11 aromatics and 5 other product species identified as f(fres).

[661

o-xylene

Adiabatic turbulent flow reactor; 1150 K; 100 kPa; Q, = 0.69, 1.1, 1.7.

10 aromatics plus 5 other aliphatic hydrocarbon products identified as f(tres).

Methane

206 cm3 closed vessel; 793 K; 95 kPa; 2CH4 + Oz.

Ethane

JSR; 800-1200K; 0.1-1.0MPa; 1.5; t,,, = 0-0.3 S.

Q, = 0.1-

7 product yields as f ( t r e s ) .

Ethene

JSR; 880-1250 K; 1-10 atm; t,,,

= 0-0.3.

5 product yields as f(tres).

Propane

1395 cm3 closed vessel; 850-900 K; nonisothermal; 0.6-1.0 MPa.


Propane

JSR; 800-1200 K; 0.1-1.0MPa; 0.15 < Q, < 1.4 in diluting gas; isothermal.

8 R H and oxygenated product concentrations as f ( t r e s ) .

Propane

Pressurized flow reactor; 600-900 K; 1.0 and 1.5 MPa.

Oxidation through the NTC region.

JSR; 1100-1140 K; 0.8 MPa; isothermal;

8 R H and oxygenated product concentrations as f(tr..).

[266]

Propene

t,,, = 0-1.5

Propene

S.

Closed vessel; 753 K; 8 kPa; [RH] [Oz]: [Nz] = 1.2: 1 :3.8.


APPENDIX cont. Chronological review since 1976 of a range of experimental studies of hydrocarbon oxidation at temperatures up t o 1300 K Year

Ref.

Hydrocarbon

Conditions

Results

~

1991-1995 (cont)

[237]

n-butane

Flat flame burner; stabilized two-stage ignition; 670 K; 100 kPa.

Product yields as f (position).

[I731

n-butane

Flow reactor; 563 K; 100 kPa.

Product yields as f ( t ) .

[221

n-butane

RCM; 0.9-1.1 MPa; stoichiometric, lean and rich mixtures; 700-900 K.

7 product profiles and 2-stage ignition; 0heterocycles.

i-butene

JSFR; 800-1200K; 0.1-1.0MPa; isothermal; 0.15 < @ < 4.0 in diluting gas; t,,, = 0-2.5 s.

7 hydrocarbon products as f ( t r e s ) .

n-pentane

Motored engine with fast action, incylinder sampling; 400-735 K.

23 products as f(intake manifold T ) .

n-pentane

JSR; 950-1050 K; 0.1-1.0 MPa; 0.2 < @ < 2.0 in diluting gas; isothermal; I,,, = 0-0.3 S .

7 hydrocarbons and COz, CO and H2 as

f (Ires).

Dimethylpropane

Closed vessel, 753 K; [RH]: [02]: [Nz] = 4 :30: 26; KCI coated vessel and Boric acid coated vessel.


n-heptane

Flow reactor (345 cm3; 485 K; 100 kPa) and CFR engine.

Hydroperoxides formed by-isomerization reactions during oxidation.

n-heptane

JSFR; 35 cm3; 550-1150 K; isothermal; 0.1-1.OMPa.

20 hydrocarbon and some oxygenated product as f( T ).

n-heptane

RCM studies

Products of reaction as f ( t )

;P

5


Ref.

Hydrocarbon

Conditions

m m 0

Results

n-heptaneli-octane

High Pressure JSFR; 550-1150 K; 0.11.0 MPa; 35 cm3.

26 hydrocarbon products as f( To),

[I471

n-heptanel n-tetradecane

Autoignition in an engine. JSFR(100 cm3); 1100 K; 0.2-2.0 MPa. Flow reactor; 500950 K; 3 MPa.

Product concentration as f ( t ) . Oxygenates.

~041

n-heptaneli-octane and n- 100 cm3 JSFR; 550-750 K; non-isothermal; heptaneltoluene 0.2-0.7 MPa.

11 product yields as and T rise as f(fres).

~031

i-octane

7 alkenes and oxygenated products as f(T").

EB ",

[30]

JSFR; To = 600-800 K; non-isothermal; @ = 1.0; 0.1-1.0MPa

[2021

n-decane

35 cm3 JSFR; 550-1150 K; 1.0 MPa.

32 hydrocarbon products as f( T ).

[211,212]

n-decanelgerosene

35 cm3 JSFR: 550-1150 K; 1.0 MPa; isothermal.

32 hydrocarbon products as f ( T ) .

Adiabatic turbulent flow reactor; 1170 K; 0.1 MPa; High dilution by inert gas; t,,, = 0-125 ms.

Concentration vs. f(f,,,) for 6 aromatics, CH,, C2Hs, CzH4, CH2, CO and C 0 2 .

~ 7 1

1-methylnapthalene

R

x3. !

E Ba (D

3.

0

zB %

F& 5. 0

Bg 0

r

Chapter 7

Autoignition in Spark-Ignition Engines D. BRADLEY and C. MORLEY

7.1 INTRODUCTION

Autoignition is of practical importance in a variety of contexts: these include the unwanted ignition of hydrocarbon-air mixtures following an accidental release and the avoidance of autoignition in the premixing duct of some advanced gas turbines. It can arise in the unburned compressed gas ahead of a flame and result in its transition to a detonation. But the desired autoignition in diesel engines and the undesired autoignition in spark ignition (gasoline) engines, manifest as knock, have provided the principal impetus for the development of the subject in recent years. Autoignition occurs in a gasoline engine because the increases in temperature and pressure of the unburned "end gas" cause it to ignite before it has been consumed by the propagating flame. From the outset of the engine era an audible knocking was noticed in spark-ignition engines. This malfunction, which affected petrol but not gas engines, was a serious limitation to the power developed and could lead to engine damage. Hopkinson, in his pioneering researches at Cambridge between 1904 and 1907, assisted by a young engineer named Ricardo, attributed this to detonation. Strictly, the term now implies supersonic combustion in which a reaction wave and a shock wave are coupled. However, knock is now identified, not primarily with such flame acceleration, but with the pressure pulse caused by autoignition in the unburned end gas ahead of the propagating flame. It has become clear that detonation is not an appropriate synonym for knock, despite its widespread use. Phenomenologically, knock, as the word suggests, may be defined as no more than "objectionable noise outside the engine" [1]. During the engine compression stroke the gaseous mixture is compressed by the motion of the piston and is spark ignited before the piston has

662

Autoignition in spark-ignition engines

Ch. 7

reached top dead centre (tdc). Thereafter, as the flame front area increases, the end gas is further compressed, now not only by the motion of the piston, but also by that of the flame front. Even after top centre, when the piston moves downwards in the power stroke, the end gas usually continues to be compressed for about 10° of crank angle by the advancing flame. The flame acts as a secondary piston within the combustion volume. In normal operation the flame eventually reaches the end of its travel, the hot gases expand, the exhaust valve opens, and the gases leave the cylinder. In knocking combustion, autoignition occurs at one or more sites in the end gas and the resulting rapid combustion there leads to a sudden increase in pressure. This excites acoustic resonances of the gas in the cylinder and the engine block. The autoignition is runaway exothermic oxidation chemistry, with temperature rises of more than 1000 K. In addition, particularly with alkane-containing fuels, there is a possibility that "low temperature" chemistry can cause reaction during earlier stages of the end-gas compression. The resulting temperature rise and possible accumulation of reactive intermediates can influence the subsequent explosive autoignition. Sufficiently strong knock can incapacitate the engine by erosion damage to the piston crown, top land, cylinder head and gasket, and might also overload the bearings. Figure 7.1 shows an extreme case of piston damage. Usually, erosion damage is attributed to the associated high heat fluxes and the strong pressure and temperature pulses [2]. Not all autoignitions, however, generate pressure pulses sufficiently strong to lead to damaging knock, and these might be regarded as benign. In this chapter the term autoignition refers to the runaway chemical reaction and knock refers to the acoustic excitation that may, or may not, accompany it. The potential thermal efficiency of the engine [3] increases with the compression ratio. This is a measure of only the piston-induced compression and does not fully characterize the condition of the end gas. It is the ratio of the sum of the cylinder clearance volume at piston top dead centre plus the volume swept by the stroke of the piston, divided by the clearance volume. But an upper limit to the compression ratios of spark ignition engines is set by the onset of knock in the end gas. In the early days of the gasoline engine, compression ratios were no higher than four, but, as a result primarily of improvements to the fuel, they increased to about six in the 1930s and to just above nine in the 1990s. A finer control over the compression of the end gas in each cylinder is provided by either advancing or retarding the timing of the igniting spark.

Introduction

663

Fig. 7.1. Damage to a piston caused by knocking combustion.

This controls the further compression of the end gas by the flame front. Knock can be prevented, or made less severe, by retarding the spark, which delays the main combustion. The critical conditions for autoignition then occur later in the power stroke, when the piston is moving downwards and the pressure is tending to fall. The lower pressure and temperatures make autoignition less likely and, even when it occurs, less damaging. But this also has the effect of reducing the engine efficiency. Engine management systems, triggered by acoustic, ionization, or crankshaft speed sensors, can be designed actively to adjust the timing while the engine is running, so that it operates with trace, rather than damaging, knock. This is usually the optimal point for efficiency. Consequently, the fuel economy

664


Ch. 7

of this type of engine is dependent on the propensity of the fuel to autoignite. The design and characterization of practical gasolines is described in the next section. The knock problem is attractive to chemists for two reasons: first, because subtle differences in chemical structure of the fuel can have a large effect on the behaviour, and second, because it is one of the few practical situations in which gas phase chemical kinetics dominate the behaviour of a homogeneous system. (Atmospheric chemistry, notably, is another.) Useful insights into the occurrence of knock can be gained by assuming that spatial temperature and concentration gradients, diffusional and other transport processes are negligible. In almost all other combustion phenomena these play a crucial role. Of course, the approximation is not completely true and, indeed, the development of knock is dependent on mixture inhomogeneities and, in particular, temperature gradients, as is shown in Section 7.6.2. This chapter aims to show how the chemistry described in Chapters 1 and 6 is applied, and to indicate the progress that has been made in understanding and predicting knock in engines. The progress has been greatest in predicting the occurrence of autoignition. Less certainty surrounds the associated strength of the accompanying pressure pulses, and whether or not they will cause damage.

7.2 FUELS FOR SPARK-IGNITION ENGINES

7.2.1 Composition and manufacture Compared with other possible fuels for spark ignition engines, such as liquified petroleum, natural gas and alcohols, gasoline has the great advantage of a high volumetric energy content in the liquid phase under atmospheric conditions. Furthermore, for stoichiometric mixtures, more energy is inhaled into the cylinder in a working cycle with gasoline [4]. In addition, it readily forms a gas phase flammable mixture with air and is easily handled and stored. Practical fuels are manufactured by blending the products of a number of refinery processes and are, consequently, a mixture of organic species (mainly hydrocarbons) of several different types. This complexity, both in the description and quantification of the autoignition chemistry, creates problems which have only comparatively re-

Fuels for spark-ignition engines

665

cently begun to be addressed, and most current fundamental approaches are not readily extended to practical fuels. A composition of a typical gasoline from a refinery is given in Table 7.1. The main constituents are branched alkanes, aromatic hydrocarbons with one ring and alkenes. Toluene and isopentane (methyl butane) often occur in the largest concentrations. In a refinery the various fractions of the crude oil are broken down, isomerized, or re-assembled to produce useful products. This processing is necessary for two main reasons. First, to convert the heavier fractions which are of low economic value into lighter fractions like gasoline which are not present in the crude oil in sufficient quantities to satisfy the demand balance. Second, to ensure that the gasoline is of a sufficiently high quality. The gasoline has to meet a number of specifications including volatility, storage stability and many others. But it is the attainment of the appropriate resistance to knock that motivates much of the expensive processing. An overview of oil processing can be found in Refs. [5] and [6], but the main processes are summarized briefly here, with emphasis on the types of molecule produced. In catalytic cracking a heavy oil is pyrolyzed on an aluminosilicate catalyst at 700-1000 K, to produce a range of lighter products. The gasoline so produced has a high proportion of aromatics and alkenes. Other cracking processes produce similar, but lower quality products from heavier feedstocks. Hydrocraching combines both pyrolysis and hydrogenation (of cycloalkanes and alkenes) at about 700 K and 150 bar. It is especially useful for upgrading heavy or highly aromatic fractions with low H/C ratios. Reforming is a process which, while not greatly altering the size of the molecules, increases their knock resistance. Among these processes are isomerization of straight chain alkanes to branched hydrocarbons, cyclization and dehydrogenation to aromatics and removal of the side chains of aromatics. The process produces much of the H 2 used elsewhere in the refinery. Alkylation is a synthetic process in which lower alkenes (from catalytic cracking) are reacted in an acid medium (sulphuric or hydrofluoric acid) with small branched alkanes to produce C6 to C 9 branched alkanes. These are probably the most desirable constituents of a gasoline, with good knock resistance and with fewer undesirable properties, such as tendencies to

666


Ch. 7

TABLE 7.1 Composition (weight %) by gas chromatography of a super unleaded gasoline, with RON 98.2, MON 86.2. Shown in order of elution from a methyl silicone gum capillary column. propene propane isobutane isobutene + but-1-ene n-butane trans-but-2-ene 2,2-dimethylpropane cis but-2-en 3-methylbut-l-ene isopentane pent-1-ene 2-methylbut-1 -ene rc-pentane 2-methyl-l ,3-butadiene trans-pent-2-enc 3,3-dimethylbut-l-ene ds-pent-2-ene 2-methylbut-2-ene trans-1,3-pentadiene ds-l,3-pentadiene 2,2-dimethylbutane cyclopentene 4-methylpent-l-ene 3-methylpent-1 -ene cyclopentane 2,3-dimethylbutane 2,3-dimethylbut-l-ene 4-methyl-ds-pent-2-ene 2-methylpentane 4-methy\-trans -pent-2-ene 3-methylpentane 2-methylpent-1 -ene hex-1-ene n-hexane cis + trans-hex-3-ene trans-hex-2-enc 2-methylpent-2-ene 3-methyl-ds-pent-2-ene

0.01 0.06 1.87 0.55 5.41 0.93 0.02 0.25 0.26 11.43 0.74 1.39 2.02 0.09 1.70 0.02 0.93 2.30 0.08 0.03 0.21 0.30 0.08 0.13 0.16 0.77 0.13 0.08 2.07 0.23 1.31 0.37 0.24 0.82 0.62 0.55 0.71 0.42

ds-hex-2-ene 3-methyl-£rajzs-pent-2-ene methylcyclopentane 2,4-dimethylpentane 2,3,3-trimethylbut-l-ene 2,2,3-trimethylbutane benzene + 1-methylcyclopentene 3-methylhex-l-ene cyclohexane 2-methyRrans-hex-3-ene unknown peak 4-methyl-ds/rra«s-hex-2-ene 2-methylhexane + 2,3-dimepentane cyclohexene 3-methylhexane 3,4-dimethyl-ds-pent-2-ene cis-1,3-dimethylcyclopentane trans-1,3-dimethylcyclopentane trans-1,2-dimethylcyclopentane isooctane hept-1-ene 3-methyl-ds-hex-3-ene CI olefin trans-hept-3-ene ^-heptane ds-hept-3-ene 2-methylhex-2-ene 3-methy\-trans-hex-3-ene trans-hept-2-ene 3-ethylpent-2-ene 3-methyl-ds-hex-2-ene d5-hept-2-ene 2,3-dimethylpent-2-ene methylcyclohexane ethylcyclopentane 2,5-dimethylhexane 2.4-dimethylhexane l,trans2 ,c/54-trimethylcyclopentane

0.30 0.53 0.89 0.58 0.09 0.03 1.16 0.04 0.15 0.09 0.03 0.13 1.24 0.07 0.74 0.04 0.22 0.19 0.26 4.34 0.14 0.02 0.12 0.30 0.26 0.16 0.17 0.13 0.16 0.06 0.10 0.14 0.14 0.37 0.12 0.49 0.57 0.07


667

TABLE 7.1 (Contd.) 2,3,4-trimethylpentane unknown peak toluene + 2,3,3-trimethylpentane 2-methyl-£raAzs-hept-3-ene 2,3-dimethylhexane 2,5-dimethylhex-2-ene 2-methylheptane 4-methylheptane 3-methylheptane 2,2,5-trimethylhexane cisl ,3-ethylmethylcyclopentane 1 ,cis2 ,ds3-trimethylcyclopentane n-octane trans-oct-2-ene 2,4,4-trimethylhexane 2,3,5 - trimethylhexane /t-propylcyclopentane 1,1,4-trimethylcyclohexane ethylbenzene ra-xylene p-xylene 4-methyloctane 2-methyloctane 3-methyloctane oxylene n-nonane isopropylbenzene n-propylbenzene m-ethyltoluene p-ethyltoluene 1,3,5-trimethylbenzene oethyltoluene 2-methylnonane 1,2,4-trimethylbenzene n-decane 1,2,3-trimethylbenzene indan C l l alkane 1,3-diethylbenzene m-n-propyltoluene

1.30 0.14 11.70 0.05 0.48 0.11 0.27 0.15 0.37 0.34 0.05 0.14 0.28 0.06 0.07 0.03 0.04 0.08 3.04 6.93 2.58 0.08 0.13 0.13 4.04 0.11 0.16 0.47 1.65 0.70 0.82 0.70 0.07 2.69 0.10 0.61 0.26 0.08 0.07 0.27

p-rc-propyltoluene + 1,4-dietbz «-butylbenzene orc-propyltoluene 4-methyldecane 2-methyldecane 1,4-dimethyl-?-ethylbenzene 1,2-dimethyl-4-ethylbenzene 1,2-dimethyl-3-ethylbenzene /i-undecane 1,2,4,5-tetramethylbenzene 1,2,3,5-tetramethylbenzene 5-methylindan l-ethyl-2-«-propylbenzene 1,3-di-isopropylbenzene naphthalene dimethylindan isomer n-dodecane C12 aromatic C12 aromatic unknown peak unknown peak unknown peak unknown peak 2-methylnaphthalene unknown peak unknown peak unknown peak unknown peak

0.16 0.22 0.04 0.19 0.16 0.27 0.39 0.06 0.05 0.15 0.21 0.13 0.16 0.09 0.28 0.12 0.05 0.08 0.02 0.04 0.02 0.05 0.06 0.04 0.12 0.04 0.02 0.03

Paraffins Isoparaffins Olefins Dienes Naphthenes Aromatics Unknowns

9.2 30.1 16.8 0.2 2.7 40.6 0.6

Total peaks MON. These include both large highly branched alkanes like 2,2,3,3-tetra-methyl hexane (RON = 113 and MON = 92) and the small homologues like propane (RON = 115, MON = 97). For these molecules the peroxy radical chemistry which gives the negative temperature coefficient plays a smaller part. In both the small and the highly-branched alkanes, the peroxy isomerization reactions are not favoured because only short lengths of carbon chain are available. Consequently, the rates are low. For the highly-branched molecules, the reaction is further prevented from going through peroxy radicals by the increased rate of alkyl decomposition reactions. This effect is particularly effective for alkanes


675

with quaternary carbons because of the ease of decompositions to tertiary alkyl radicals [19]. 7.2.5 Blending A difficult practical problem is the prediction of the octane number of a blend of fuels. As yet, fundamental chemical kinetic modelling has made little contribution to its solution. In general, the octane number of a mixture of fuels is not a linear function of the composition. For example, if two fuels with octane numbers N\ and N2 are mixed in the ratio by volume of/x and (1 — fi) the octane number of the mixture of two fuels, Nmix? can be higher or lower than the value given by simple volume weighting, and, in general, AU*/iNi + (l-/i)Ak

(7.1)

In some cases the octane number has a maximum, or a minimum, value at a particular composition. Figures 7.5 and 7.6 show some examples in which the effect is particularly apparent, because here the component gasolines have the same octane number, although the compositions are different. Alkene fuels blended with alkanes tend to produce positive deviations from linear blending for RON (represented by equality in equation (7.1)), and possibly maxima, as in Fig. 7.5. Figure 7.6 shows strong negative effects in MON when alkanes and aromatics are blended. An approximate way to handle the less extreme cases of this non-linear behaviour is to use the concept of blending octane number. This parameter describes the octane quality of a component when it is mixed in fairly small quantities with a base fuel. The definition of blending octane number is illustrated in Fig. 7.7. Here the plot of ON against composition is curved, but at a particular composition the octane number can be described by a linear blending rule, with equality in equation (7.1), but using the octane number of the base fuel and an artificial value, the blending octane number for the component. The latter value is obtained from an extrapolation, shown in Fig. 7.7. This approximation for the mixture octane number can be used in closely related fuel compositions, but the blending octane number is of limited generality, since it depends on the base gasoline and the proportion of the component. The published values of blending octane numbers from the API project

676


Ch. 7

108

0

20

40

60

80

100

Vol % isooctane in mixture with diisobutylene Fig. 7.5. RON of mixtures of iso-octane with di-isobutylene (commercial mixture of 4,4trimethyl pent-1-ene, and 4,4-trimethyl pen-2-ene). This shows synergistic blending: the octane number of the mixture is higher than either component. Data from [12].

45 [10,13] are for pure hydrocarbons at 20% by volume in a RON 60 primary reference fuel (60% iso-octane, 40% rc-heptane mixture). The positive deviations from non-linearity when alkenes are blended with such paraffinic fuels lead to blending octane numbers for alkenes that are larger than the natural octane number of the pure component, as illustrated in Fig. 7.7. The values, especially for di-alkenes, can be very large. For example, the blending RON for isoprene, 202, is much larger than that of the pure compound, 99, and for dicyclopentadiene the equivalent values are 218 and 104. But a primary reference fuel with a RON of 60 is not representative of current gasolines, in terms of either octane number or composition. It was a better match when the API Research program 45 [13], was started in the 1940s (see Fig. 7.3). In normal gasolines the

677

Fuels for spark-ignition engines 102

0

20

40

60

80

100

Vol % xylene in mixture with triptane/heptane Fig. 7.6. MON of mixtures of o-xylene with an alkane mixture (heptane + triptane, 2,2,3trimethyl butane) of the same MON. This shows strong antagonistic blending. Data from [12].

difference between blending and natural octane number is usually much less. Clearly, the non-linear blending behaviour is of great importance in the formulation of gasoline. There is a strong incentive to produce, by blending the various refinery streams, a finished gasoline of exactly the required octane number. If it is too low, it will fail to meet the specifications and, if it is too high, it will increase costs because of the expense of the extra unnecessary processing. To aid this process and allow optimization of refinery operations various methodologies have been published [20]. These may be more complicated than the blending octane number concept, but they are usually empirically based. The complicated and somewhat arbitrary nature of the CFR engine test means that there is no inherent reason to expect a strict compositional

678


Ch. 7

Blending ON of component

a 3

o ON of component

ON of blend

ON of base fuel

20

100

%v of component in blend Fig. 7.7. Non-linear blending of octane number and the definition of blending octane number as used in the API-45 project, illustrated schematically. Mixtures of fuels may show either positive or negative deviations from linearity (solid curves). The blending octane number of a hydrocarbon tabulated in Refs. [10] and [13] is obtained by extrapolation from the ON of the base fuel through the measured octane number of a mixture containing 20% of the hydrocarbon to the righthand axis, as shown.

linearity of octane number. But the existence of positive and negative deviations of octane number from the linear blending law (equation (7.1)), depending on the chemical constitution of the components, indicates that the major cause lies in the chemical kinetics of the autoignition and knock development process.


679

One general approach is to consider the balance between radical production and loss by the different components of a mixture. Alkanes, because of their "low-temperature" chemistry have active chain-branching reactions, while alkenes and aromatics have efficient termination reactions through the production of stabilized radicals, such as allyl and benzyl radicals. While the rates of the branching and termination processes arise from contributions by each of the constituents in the mixture in a way that depends linearly on their composition, the overall rate of the autoignition reactions depends on branching and termination in a non-linear fashion. Croudace and Jessup [21] suggested that the non-linear octane effects they observed between n-hexane and a number of other alkanes were due to the reaction between one fuel constituent and the peroxy radicals derived from another. This seems an unlikely explanation, as this kind of reaction has been shown [22, 23] to be too slow under end gas conditions. Even quite simple modelling shows that the interaction between hydrocarbons in an oxidizing mixture occurs essentially through the small radicals [24]. Leppard [24] suggested that the ease of autoignition, and hence the octane number, of a fuel with a single pure component is governed by the reactivity of the products - while in a mixture it is the initial rate of radical reaction with the fuel that is significant. Compared with alkanes, alkenes produce few radicals in their subsequent chemistry, but are more reactive in their initial reaction with radicals (OH and H0 2 ). This leads to alkenes contributing more inhibition, and hence, higher octane number than expected, when they are present in mixtures. Leppard also suggested that synergistic effects in alkene/alkane mixtures are due to the alkene acting as a radical scavenger in the low-temperature alkane chemistry, while the alkane retards the more active alkene high-temperature chemistry. While some of these qualitative explanations probably have elements of truth, there are undoubtedly a number of effects which contribute to nonlinear octane blending behaviour, with complex interactions between them. Clarification must await a better understanding of the chemistry of fuels other than alkanes which, as pointed out in Chapter 1, is rather limited at present. 7.2.6 Anti-knocks Ever since it was realized that the onset of knock limited the compression ratio, and hence, the efficiency of an engine, chemical remedies have been

680


Ch. 7

TABLE 7.3 Measured changes in RON, ARON in a 96 RON gasoline produced by addition of 0.025 mol/1 of methyl substituted diphenyl oxalate additives [26] Methyl substitution

ARON

none 2 3 4 2,3 2,4 2,5 2,6 3,4 3,5 2,3,5 2,4,6

0 +0.7 + 1.0 0 0 -0.1 +0.2 +0.7 0 + 1.1 0 +0.8

sought by the addition of relatively small amounts of "anti-knock". The first widely successful such additives were lead tetra-alkyls, proposed by Midgley and Boyd in 1919 [25]. These substances have played an important role in the development of fuels and engines since the 1930s, and allowed increases of octane number that would not have been either possible or economic by processing of the base fuel alone. They are still employed extensively in many parts of the world, but their use is in decline because lead both introduces toxic material to the environment and, probably more influentially, poisons the catalysts used to reduce exhaust pollutants. "Ashless" anti-knocks, which do not contain metals and so would avoid these difficulties, could be attractive in gasoline formulation. A range of materials with anti-knock properties are known, but they are much less effective than lead tetra-alkyls. For example, 0.6 mmol d m - 3 of tetra-ethyl lead (which is equal to 0.15 g d m - 3 , currently used in commercial leaded gasolines in the UK) can raise the RON of a paraffinic fuel by 3 numbers. But it requires 90mmol/l of N-methyl aniline (NMA), C 6 H 5 NHCH 3 , a fairly active ashless anti-knock which has been used as a yardstick of activity [26], to produce the same increase in RON. For comparison, 1400 mmol/1 (12% w) of a high octane blending component, methyl tertiary butyl ether (MBTE), would be needed.


681

Many of the ashless anti-knocks are amines or phenols [26] and are related to liquid-phase oxidation inhibitors. They probably work by reacting with active radicals (particularly OH) to produce radicals which are inert. For instance, N-methyl aniline (NMA) C 6 H 5 NHCH 3 probably produces stabilized C 6 H 5 NCH 3 radicals which, because of their resonance stabilization, are unable to react to regenerate active radicals again and may undergo only radical recombination reactions. The rate of radical removal by this process is likely to be limited in the most favourable case by how fast the additive can react with OH to produce stabilized radicals. Although exact rates are not known, this is probably already a fast process for NMA, and unlikely to be very much faster for any other substance. Indeed, the most effective ashless anti-knock found by MacKinven [26] in an extensive study of 970 substances was a tetra-aryl hydrazine, with a molar effectiveness 2.9 times that of NMA. As confirmation of an inert radical production mechanism, iodine compounds are particularly effective because of the production of I atoms. However, there are big deficiencies in our understanding of the details of anti-knock chemistry. This is illustrated by the large differences in antiknock effectiveness shown in MacKinven's measurements between substances with apparently very similar composition [27]. As shown in Table 7.3, some of the methyl substituted diphenyl oxalates are quite good antiknocks, with up to 1.1 times the molar effectiveness of NMA. But another is pro-knock. The mechanism responsible for this structure/property dependence is not known. More recently, high effectiveness has been reported for ashless materials related to dialkyl amino fulvenes [28-31], but no credible mechanisms have been published. No ashless anti-knocks have proved sufficiently cost-effective to be used commercially. Lead tetra-alkyls, on the other hand, provide a relatively cheap way of increasing the octane rating of a fuel. Their much greater effectiveness suggests their mode of action must be different from that of ashless antiknocks. Hypotheses on knock mechanisms (usually incorrect) played a part in their discovery [32]. It was known that kerosene was more susceptible to knock than gasoline. We know now that this is because, in this heavier petroleum fraction, the alkane chains are longer and kinetic branching is easier. In 1916, Midgley and Boyd surmised that it was the volatility that was important, so that increasing the rate of vaporization of a fuel should reduce the knocking tendency. They proposed to do this by colouring the fuel with a dye which, they supposed, would absorb radiation from the

682


Ch. 7

flame. A dye soluble in gasoline was not immediately available and they tried iodine instead. It worked. But compounds of the lighter halogens did not. This can be explained in terms of the reactivity of the halogen atoms: the almost inert I atom acts as a sink for radicals and iodine compounds are reasonably good anti-knocks, whereas CI can easily propagate the kinetic chains. But Midgley and Boyd thought the effect was connected with atomic mass. They systematically searched through volatile compounds of heavy elements, eventually arriving at the very effective lead alky Is. Research on the mechanism by which they work started shortly after their discovery, but in 1971 Ballinger and Ryason [33] perceptively remarked that lead alkyls might be removed from engine fuels before their mode of action was properly understood. Research on the subject has been at a fairly low level in recent years, and there is still no consensus of understanding. Although mechanisms based on homogeneous chemistry have been advocated by such distinguished gas phase kineticists as Norrish [34, 35] and, more recently, Benson [36], the most widely supported mechanism (advocated by Walsh [37]) involves the production of lead oxide smoke in the end gas prior to autoignition. Downs et al. [38] have observed this fog in an engine by a light attenuation technique. Bulk lead oxide is known to be an effective inhibitor, by acting as a catalyst for the decomposition of peroxides [39,40]. The aerosol in the end gas probably removes H 2 0 2 and H 0 2 , reducing their positive contribution to the later stages of the autoignition [41]. Measurement of ignition autoignition delay-times in a rapid compression machine by Kirsch and Pye [42] showed, Fig. 7.8, that in two stage autoignition tetra-methyl lead affects only the second stage (the hot ignition), while the first stage (the cool-flame) is unaffected. The radical scavenger anti-knock NMA affects both. Ballinger and Ryason [33] similarly showed the second stage in a burner stabilized, two stage, heptane flame to be more sensitive to tetra-ethyl lead and that it could be completely quenched with 0.05 mol% in the fuel. The inactivity in the first stage was probably because the temperature was too low for the lead alkyls to decompose fast enough. Measurements of decomposition rates in a shock tube [43] support this, and also demonstrate that, under practical conditions, the condensation of the initially produced elemental lead is sufficiently fast and that it very quickly oxidizes to lead oxide. Differences observed between tetra-

Fuels for spark-ignition engines (a) Inhibition by NMA

100 [NMA] /[fuel]

683

(b) Inhibition by TEL

10 4 [TELj/[fuel]

Fig. 7.8. The effect of two types of anti-knock on the characteristics of two-stage ignition measured in a rapid compression machine. T1 is the time from the end of compression to the cool-flame; r 2 is the subsequent time to the true ignition; AT is the temperature rise over the cool-flame. Conditions: 0.5 stoichiometric mixture of 2-methyl pentane and air at 720 K and total concentration 3.2 x 10~4 mol cm - 3 . TEL (tetra-ethyl lead) affects the second stage, but not TX\ NMA (N-methyl aniline) affects both T\ and r 2 . From [42].

ethyl and tetra-methyl lead [6] possibly arise from their slightly different decomposition rates. The lead oxide aerosol is finely divided, with a particle size R + R R —> R + heat R -» R + B R —> R + Q R —» R + R -» B -> R + R R + Q -> R + B

Thus, there are three intermediate species, as well as fuel, 0 2 , and product. Although formulated in terms of the reactions of these generalized species, the actual values of rate constants used were very largely empirically fitted from ignition delay-times measured in a rapid compression machine [71]. Some of the rate parameters incorporated fuel and 0 2 dependencies, to reproduce those observed experimentally. Figure 7.10 shows the good fit achieved for the total autoignition delay-times of various fuels as a function of temperature and charge density. Simplified schemes share a number of other problems. The course of hydrocarbon oxidation chemistry is governed by two main parameters: the radical concentration and the temperature. Although such schemes are designed to handle the radical generation and removal, the heat generation, the importance of which is explained in Section 7.6.2, is more difficult when all the products are not explicitly modelled. In the Shell scheme, the heat release arises from the main propagation reaction, which is made exothermic to the extent of a complete oxidation to CO, C 0 2 and H 2 0 . Since this is the only source of heat, this implies that the kinetic chains are long which, in fact, is not always the case [72]. Another problem is the lack of mass balance, of importance in CFD modelling, but relatively simple corrections can be made [73,74].

Chemical modelling of autoignition

a—-a O—€1

PRF (RON 90) TRF (RON 89.5) 2-Me-hex2-ene (RON 90.4)

100

2 l/temperature. 10 J K

3 4 5 6 7 Charge density, mole/cm3 x 10 4

Fig. 7.10. Comparison of ignition delay-times measured in a rapid compression machine (points) with Shell model predictions (lines) [71]. Fuels are all RON 90 with different sensitivities: PRF, primary reference fuel, 10% n-heptane, 90% isooctane, MON = 90; TRF toluene reference fuel, 30% heptane, 70% toluene; MON = 77.9; 2-methyl-2-hexene, MON = 78.9. Compression ratio 9.6, 0.9 stoichiometric mixtures, wall temperatures 373 K. (a) Effect of temperature at end of compression; charge density 3.20 x 10~4 mol cm - 3 , (b) Effect of charge density; end of compression temperature 690 K. (Note all end of compression temperatures are averages over whole charge.) From [71].

Because of its large number of empirical parameters, this model can be fitted to a variety of fuels and can reproduce the important autoignition phenomena. This is accomplished with only a modest computational requirement and, consequently, it has been used in a number of engine modelling studies over the past 20 years. Its main deficiencies are the difficulty of extending it to new fuels (which has been done only by the original authors) and its still limited representation of the real chemistry. Because of the generic formulation, the species are rather ill-defined. It would be difficult to use, for instance, in a study of chemical octane requirement increase, discussed in Section 7.5.2.

692


Ch. 7

(Hi) Cox and Cole model In 1985 Cox and Cole [23] showed that the Shell model could be reformulated in terms of elementary reactions of generalized species. The mechanism is a good representation of the known fundamental chemistry and the rate constants are fairly close to those measured or intelligently estimated, without the necessity of empirical fuel and oxidant dependencies. The chemistry is similar to that described in Chapter 1. There are 18 reactions and 9 intermediate species - OH, R, R 0 2 , QOOH, OOQOOH, H 0 2 , ROOH, H 2 0 2 and RCHO. The mechanism includes the significant reversibility of the R + 0 2 = R 0 2 equilibrium, making explicit the cause of the negative temperature coefficient. It retains the labile intermediate of the Shell model (identified as an aldehyde) but one which reacts in the highertemperature part of the mechanism, mainly with H 0 2 , rather than with the more active radicals involved in the low-temperature chemistry: RCHO + H 0 2 -> R + H 2 0 2

H202 + M ^ O H + O H + M H 0 2 + H 0 2 -> H 2 0 2 + 0 2 Together with the decomposition of hydrogen peroxide, this provides a branching route which initiates the true ignition rather than the cool-flame. Hydrogen peroxide is also formed from the bimolecular reaction of H 0 2 , but even when followed by decomposition it is less effective at accelerating the reaction, since the number of radicals is not increased. This outline formulation of the intermediate chemistry was an important addition to the mechanism, but the modelling of the hot ignition was disappointing, giving ignitions which developed rather more slowly than was observed experimentally. The cause is not clear. The model can reproduce much of the autoignition behaviour observed in rapid compression machines, but it has not been applied directly in engine modelling. (iv) Hu and Keck model In 1987 Hu and Keck [75] expanded the Cox and Cole scheme slightly, to include the intermediate chemistry reaction in which H 0 2 reacts with the fuel. They also generated the heat from the enthalpy change in each reaction explicitly. The model was originally validated against experiments in spherical vessels, ahead of spherically expanding flames. There are now


693

12 intermediate species, the concentrations of which need to be calculated. The model has been used in a number of engine modelling studies by the MIT group [58,59,76], giving similar results to those from Westbrook's comprehensive model [59]. This, and all the preceding simplified models, have some difficulty in handling fuels of different chemical structure. For different alkane hydrocarbons, the basic chemistry is likely to be similar but the rate constants will differ considerably, and it is not clear how the generalized parameters in the model should be varied. Hu and Keck handled fuel structure variations by varying a single equilibrium constant for R0 2 QOOH. Blin-Simiand et al. [77] extended the model by including a parallel set of generalized species, the reactivity of which was different (and fitted empirically). The basic structure can be augmented to model selected aspects better. Thus, Li et al. [78] improved predictions of preignition heating and CO production by using 20 species and 29 reactions. (v) Five-step parrot model As an alternative approach to expanding the chemistry, Bradley, Yeo and co-workers [79-81] have attempted to find the minimum number of differential equations that could reproduce experimentally observed engine autoignition delay-times, including the regime of negative temperature coefficient. Five reaction steps and an energy equation proved to be necessary, with only one intermediate species. This became known as the "parrot" model: a consequence of its total absence of profundity, combined with primitive mimicry. It incorporates a single high-temperature reaction and a set of low-temperature reactions, essentially representing a branched chain reaction. The rate constants have been empirically fitted [82] to autoignition delay-times, measured for a range of iso-octane - rc-heptane mixtures in rapid compression machines by Minetti et al. [83, 84] and guided by chemically derived correlations of rate constants with fuel structure [19]. It is used as a demonstration of some basic aspects of modelling in Section 7.5.3, which gives the chemical structure of the model. (W) Four-step model of Miiller, Peters and Lin Also aiming for brevity, this four-step scheme for the high-pressure autoignition of heptane was developed for application to diesel ignition [66]. It is related to a systematically reduced sixteen-reaction scheme, but appears to be less rigorously derived, having, for example, adjustable rate

694


Ch. 7

coefficients. It uses five lumped species: fuel, F = n-C7H16; intermediates, X = 3C2H4 + CH 3 + H and I = H0 2 RO + H 2 0 ; and products, P = 7C0 2 + 8H 2 0. There are essentially five reactions (one step is reversible):

X + 1102^P F + 202OI I + 902^P The first two represent the high-temperature chemistry and the second two the branch chain of the low-temperature chemistry. Griffiths [85] has criticized the model for losing the essential feature of alkane autoignition chemistry, a switch from radical branching to non-branching reactions as the temperature increases, which is responsible for the negative temperature coefficient. The model relies on thermal feedback mechanisms only. (vii) Five-step model of Schrieber et al. Griffiths [85] and Schrieber et al. [86] have shown how the Muller scheme can be modified to incorporate these features by slightly expanding the low temperature part of the mechanisms and adding a further intermediate. While mass and energy balances were intrinsic in the formulation, an empirical approach was adopted both for the form of the additional reaction and for all the rate constants. The latter, for instance, included pressure dependent terms. Good fits were obtained to rapid compression machine and shock-tube autoignition delay-times for heptane, iso-octane, and their mixtures. (viii) Generalizing the chemical structure of the fuel The "unified" model of Griffiths et al. [87] takes account of the number of primary, secondary and tertiary alkyl radicals produced, and allows them to react in ways which are to some extent representative of their structure, such as with different isomerization rates. The larger generalized species are made to decompose to explicit species with three carbon atoms and less, for which a reasonably comprehensive scheme is written. The model has the advantages of providing a systematic method for handling structure, of being applicable (in principle) to mixtures, and of maintaining


695

the elemental balance with no need for an arbitrary heat generation term. Disadvantages are that it is more complex, with more intermediate species than in previous simplified models. A related approach by Morley and Hughes [88] for alkanes of any structure has been to generate systematically a low-temperature mechanism with structurally dependent rate constants for each elementary reaction. A quasi-steady-state can be assumed under almost all conditions and so the system is easily solved for significant parameters, like the generation or removal rates for OH and H 0 2 . These quantities are represented algebraically in the main kinetic integration. The high-temperature chemistry of the fuel and the intermediate products is again treated explicitly. For mixed fuels this approach has the potential advantage of being able to solve most of the low-temperature chemistry independently for each alkane structure, and then to sum the relatively few outputs in proportion to the fuel composition. Most of the complicated structural chemistry is not involved in the main kinetic integration, with considerable saving in computational effort. These, as yet barely tested, methods are intuitive approaches to the problem of handling complex mechanisms in complex mixtures. More systematic methods are described in Chapter 4, although they have not yet been extensively applied to autoignition. It is clear that brute force methods for integration of chemical reaction schemes would be severely challenged when confronted with mixtures like gasoline in which there are several hundred components, each with an oxidation mechanism of several thousand elementary reactions. Of course, there will be considerable overlap of these mechanisms, but even so the computational effort that would be involved seems barely justified by the uncertainties in the rate constants of the tens of thousands reactions that would be necessary. More elegant methods are needed. (he) Significance of Intermediate Products As pointed out in Chapter 1, delocalized radicals, such as allyl and isobutenyl, are particularly unreactive. Their formation, particularly from propene or isobutene, constitutes a radical termination, and fuels which form these alkenes as intermediate products would tend to autoignite with more difficulty than those which do not [89,90]. Such effects are possibly significant for highly branched alkanes, and especially for methyl tertiary butyl ether [18]. Comprehensive models cope with this effect as a matter

696


Ch. 7

of course, but simplified models have not as yet incorporated it, although the structure of the unified model mentioned in the previous section makes this possible. It is notable that there are no simplified schemes specifically written for hydrocarbon classes other than alkanes, reflecting the uncertainties in their chemistry.

7.4 COMBUSTION IN ENGINES

7.4.1 Laminar and turbulent burning velocities Early engine builders recognized that good performance, which avoided knock, was not solely a matter of fuel quality, but also of a successful synergy between that quality and the engine design. Turbulence of the mixture was necessary to obtain a sufficient burn rate, while a short flame travel reduced the propensity to knock. Whether autoignition occurs depends upon the race between the reactions in the end gas and the consumption of that gas by the propagating flame. For this reason, some necessary details of flame propagation are considered in this section. The situation is complicated by the fact that flame propagation and autoignition are indirectly coupled. A development of autoignition that led to knock is shown in Fig. 7.11, by the luminescence from both the propagating flame and the reacting end gases. The six frames are taken from a cinefilm of weak knock in an experimental engine at Leeds University [82]. The flame propagates from left to right and the times are measured from ignition. Three autoignition centres begin to appear at about 3.38 ms, although these cannot be observed on the present plate. There is extensive end gas reaction by 3.50 ms and a rapid rise in pressure, characteristic of knock, occurs at 3.63 ms, as a result of the very rapid reaction of the end gas. The white spots arise from seeding particles, introduced to measure gas velocities. The longer the streak, the higher the velocity. Autoignition drives the burned gas backwards towards the spark plug, with velocities that can exceed 200 m s " 1 . Although the onset of end gas autoignition is influenced primarily by the autoignition chemistry in the end gas, it is also affected by the speed with which the flame propagates across the combustion volume. This is because the pressure and temperature of the end gas depend, not only

5’

Fig. 7.11. Cinefilm sequence leading to weak knock in an engine. Time measured from spark ignition. The flame propagates irom left to right and extensive end gas reaction has occurred at 3.50ms. White spots are seeding particles introduced to measure gas velocities. From [82].

4

698


Ch. 7

upon the motion of the piston, but also upon that of the "flame piston" and the relative magnitudes of the piston and flame speeds, which change from cycle to cycle. Experiments show that, generally, the faster the flame speed relative to the piston speed, the greater is the end gas compression and the tendency to knock [80, 91, 92]. This is because, in the race between autoignition and flame propagation, the autoignition delay-time has been reduced more than the flame transit time. Less typically, under conditions of low octane number and rich mixture, a higher relative flame speed decreases the tendency to knock [93,94]. Under these conditions, the autoignition delay-time is reduced less than is the flame transit time. Because of the importance to autoignition of the flame speed, which is the speed of the flame front that would be observed by a stationary camera, it is necessary to discuss briefly the factors that influence it. The flame speed is the vector sum of the burning velocity and the gas velocity just ahead of the flame. The burning velocity of the mixture is the velocity of the cold reactants, normal and into the plane that comprises the cold front of the flame. With a laminar, one-dimensional, flow this is the laminar burning velocity, uh a physicochemical parameter that depends solely on the mixture composition, temperature and pressure. Numerical values of ut over the full range of engine operating conditions have been uncertain [95,96] due to the experimental difficulties of measurement and also of allowing accurately for the flame stretch rate [97]. More accurate values are becoming available, some of which are computed [98]. In general, they increase with temperature and decrease with pressure. The former causes the larger change during the near-isentropic compression that occurs in an engine. Maximum values tend to occur with slightly rich mixtures. For conventional engine fuels, under atmospheric conditions this value is in the range of 0.37 to 0.43 m s" 1 . There is no correlation of laminar burning velocity with octane number. This is because "low" and "intermediate" temperature chemical kinetics are important for autoignition, whereas "high" temperature kinetics and transport processes control flame propagation. Autoignition is limited by the kinetics of the generation of new radicals. In contrast, radicals are abundant in the hot part of flames and the controlling kinetics for flame propagation involve those radicals that diffuse into the lower temperature regions. Velocity components and curvature that stretch the flame surface can change the burning velocity. This is because the flame structure is changed

Combustion in engines

699

by the additional convective term and the effect of curvature on the fluxes of species and energy. The change in burning velocity is related to the flame stretch rate, (l/A)(dA/dt), of a flame of surface area, A. The stretch rate can be of either sign, but usually is positive. If ut is the unstretched, and un is the stretched, laminar burning velocity, then: ut-un

= L(l/A)(dA/dO

(7.2)

The proportionality constant, L, is called the Markstein length [97]. Equation (7.2) can be written, in dimensionless form, as ({ut - un}/ud =

(VAXdA/dtXSt/uML/S,) Ki Ma

in which Sz is the laminar flame thickness. The first three bracketed terms on the right comprise the dimensionless laminar Karlovitz stretch factor, Kh while the dimensionless group, (L/S/), is the Markstein number, Ma. Usually Ma is positive for hydrocarbon fuel-air mixtures and a sufficiently high value of Kt can quench the flame, when un = 0. With a negative value of Ma, positive stretch can increase un. The mixture motion in the engine cylinder generates sufficient turbulence to affect strongly the burning rate of the charge. The rms turbulent velocity, u', is the root mean square velocity of the turbulent fluctuations about a mean. There is much evidence to suggest that a turbulent flame can be regarded as an array of laminar flames, often called flamelets [48,99], and this simplifies analysis of turbulent burning. Wrinkling of a laminar flame surface by turbulence increases the surface area, even when the cross-section area of the mean flow of the reactants is unchanged. This surface wrinkling increases with u' and, on this account, the burning rate is increased. It is measured by the turbulent burning velocity, ut, which is the mean velocity of the cold reactants normal to the mean flame surface. In the very early stages of flame propagation, just after spark ignition, the flame surface is only wrinkled by the very highest frequencies of the turbulent spectrum, while the lower ones bodily convect the small kernel. As the flame propagates and the kernel size increases, lower frequencies also begin to affect the wrinkling. An effective rms turbulent velocity, u'k, acts on the flame surface and is initially less than u'. As the flame propa-

700

a


o

5

10

Ch. 7

is

20

u \ lu\ Wrinkling Factor Fig. 7.12. Correlation of turbulent burning velocities. Broken curves show RilLe2, with Rt (present notation) evaluated for the fully developed rms turbulent velocity, u' equal to u'k. Different combustion regimes are indicated, conditioned by the value of KLe. Redrawn from [101].

gates, the ratio, u'Ju', approaches unity and its value is a function of a dimensionless elapsed time [100]. The greater the magnitude of uf relative to uh the greater will be the wrinkling and u'klui expresses the wrinkling factor. Shown in Fig. 7.12, taken from [101], is the way it influences the turbulent burning velocity normalized by the laminar burning velocity, utluh In an engine, the wrinkling factor depends upon the engine speed (with which u' increases linearly), the throttling (which reduces the intake pressure) and the mixture composition (which changes ut). For a stoichiometric mixture the factor ranges between about 2 and 5. On the other hand, the turbulence also causes the flamelet surface to be stretched. In addition it also creates a distribution of stretch rates,


701

about an rms value, in which positive stretch tends to dominate. This value is related to the eulerian rms strain rate of the reactants [48] and the turbulent Karlovitz stretch factor, K, is based upon it. A commonly used approximation for it is [101]: £ = 0.157(w7w,)2/?r0-5

(7.4)

in which /?/ is the turbulent Reynolds number (=w7/v), v the kinematic viscosity and / the integral length scale of the turbulence. Physically, the last can be regarded as a measure of the size of the larger eddies, while K is a measure of chemical to eddy lifetimes. When Ma is positive, as it is for most hydrocarbon fuels, an increase in K tends to decrease utlut. The product KMa expresses the influence of the flame stretch rate. For most fuels an increase in the flame stretch rate decreases the turbulent burning velocity. However, in the past there have been few accurate measurements or computations of values of Ma, although this is now being remedied. As a result, the correlation in Ref. [100] of the many experimental measurements of ut that have been made, and which is shown in Fig. 7.12, has instead been in terms of the Lewis number, Le = klpCpD. This is the ratio of the thermal conductivity of the mixture to the product of the density, specific heat and the diffusion coefficient of the deficient reactant [48]. The value of Ma increases with that of Le, but it also depends on the activation energy for the overall rate of combustion. The product KLe comprises a dimensionless stretch rate factor and values of it are plotted by the full line curves in Fig. 7.12. As can be seen, an increase in this factor at constant wrinkling factor reduces utluh Furthermore, its magnitude determines the nature of the flame. Up to a value of about 0.15, the flame comprises a continuous laminar flame sheet, while above it the sheet begins to break up. For values in excess of about 0.3, localized flamelet quenching creates a fragmented reaction zone [102], and eventually the flame begins to extinguish at values of KLe above about 1.5. With stoichiometric mixtures KLe is usually less than 0.6, but it can be larger in lean burn engines. Also shown on the figure, by the broken curves, are values of RtILe2. These values tend to be low because of the small length scales near top centre. An approximate correlation of the burning velocity data presented in Fig. 7.12 is [101]: uju'k = 0.88/(KLe)03.

(7.5)

702


Ch. 7

7.4.2 Engine diagnostics and computational fluid dynamics Successful engine design tailors the cylinder aerodynamics to achieve the desired burn rate. In recent times, this has been aided by laser diagnostics and computational fluid dynamics. In-cylinder diagnostic techniques for production engines include the use of rapid response pressure transducers, ion gauges as markers of flame progress, laser doppler velocimetry and emission spectroscopy. These have been reviewed historically by Witze [103]. The two zone analysis of Chun and Heywood [104] enables the net heat release rate to be derived from the pressure-volume relationship [105]. By subtracting the heat release rate in a non-knocking cycle from that in a knocking cycle it is possible to estimate the heat released by autoignition as a function of crank angle [106]. The variations in heat release rates for three different knock intensities are shown in Fig. 7.13. The onset of knock is indicated by an asterisk and occurs at the maximum heat release rate, which is appreciable at the highest knock intensity. The unburned mass fraction is shown by the axis at the left and it shows that 1.00

K2.50

0.83

+ 2.00

I 067

+ 1.50

is a constant of proportionality and f(6) is the ratio of the value of u' measured by Hall and Bracco [134] at a given crank angle, 0, divided by the value they measured at the ignition crank angle. Shown in Fig. 7.15 are typical computed changes in pressure and fractional mass burned during the cycle, compared with the measured values. The way in which the values of ut/uh that give rise to these, change during combustion is shown in Fig. 7.16, for a different set of conditions for a four-stroke single-cylinder engine with a disc-shaped combustion chamber, central ignition, and a compression ratio of 8. The plots of utlui against u'klui on this figure are drawn from Fig. 7.12 and the curves are loci of the flame condition during the engine combustion. Loci are shown for four different engine speeds, with a stoichiometric mixture of iso-octane-air, an initial pressure of 1 atm and a value of ku> of 0.6. The computations give


—T

-50

1

1

.

0 Crank angle (degree)

,

50

u.u

707

|

-20

i

1

1

1

0 20 Crank angle (degree)

.

[-

40

Fig. 7.15. Changes in pressure and mass fraction burned during compression and expansion. Modelled and experimental results compared. From [132].

the temperature and pressure of the unburned end gas at all points on the loci. The loci commence at the lowest point on the graph with ignition. As the flame kernel propagates, initially with very little pressure increase due to the combustion, the value of u'k develops, as previously described in Section 7.4.1. Combustion occurs at an increasing value of the wrinkling factor, u'kluh but at an almost constant value of the stretch factor, KLe. As combustion proceeds and the unburned charge is isentropically compressed, the value of ut increases, both the wrinkling and stretch factors decrease, and utluh and hence un increase. The value of KLe is highest in the initial stages of flame propagation. Changes in the rate of combustion here have a crucial effect on the overall duration of combustion because the combustion rate is highly non-linear with time. Combustion of the first 1% of the mass of charge occupies as much as one-third of the total combustion time. Variations in the turbulent burning velocity are inevitable, because of the stochastic nature of turbulence and the fuel-air mixing, and these affect the timing of the combustion of the bulk of the charge. This causes appreciable variation in the mean flame speed in different cycles (cyclic dispersion). This, in turn, affects the pressure development and, through this, many aspects of engine performance, including power and pollutant emissions. The end gas is subjected to different temperature and pressure histories for each cycle, which can give rise to cyclic dispersion of autoignition. Sometimes knock may not

708


J.

20-

••+•• --A--*15J -Q-

Ch. 7

_L

750 rpm 1500 rpm 3000 rpm 6000 rpm *s>

*sf>

>£••"

»£..-•

«£

-" *V?'

10-

Q"

0 0

2

T

3

4

5

6

7

0 P. ku I ' I ' I ' 8 9 10

= 1.0 = 1.0 = 0. I ' 11 12

u' k /u, Fig. 7.16. Computed loci of changes in turbulent burning velocity and related parameters during engine combustion at four different engine speeds. Ignition occurs at the lowest point on each curve. As engine speed increases, combustion moves away from the continuous laminar flame sheet regime. From [132].

occur, at other times it may be severe, the latter occurrences usually correlating with the higher end gas temperatures and pressures, consequent upon higher flame speeds [80,91,92]. Whether end gas autoignition occurs, depends upon the mixture and the temporal history of the temperature and pressure. Modelling of this is discussed in detail in Sections 7.5.4 and 7.5.5, and the factors governing knock intensity in Section 7.6. As the engine speed increases, the locus of the flame condition during combustion moves to the right on Fig. 7.16, a consequence of the associated increase in rms turbulent velocity. This is accompanied by an increase in KLe and decline in combustion quality, as the combustion moves from the regime of the continuous laminar flame sheet to that of localized

Autoignition in engines: modelling and experiments

709

flamelet quenching. This, in turn, is associated with increased cyclic dispersion, ultimate loss of power and driveability, incomplete combustion, and increased noxious emissions. A leaner mixture, largely because of the reduced proportion of C 0 2 and H 2 0 in the burned gases and associated increase in the ratio of specific heats, can give a higher cycle efficiency, in addition to a reduced propensity to knock and a reduction of CO, hydrocarbon and NO* emissions. When the mixture is made leaner, the locus of the flame condition during combustion also moves to the right, a consequence this time of the associated decrease in laminar burning velocity. If the mixture is made too lean, this is associated with the well known decline in combustion quality, that is most acute in the early stages of combustion when the value of KLe is highest. This results in intolerable cyclic dispersion. Similar effects are observed with recirculated exhaust gases, introduced to reduce NO* emissions. In stratified charge engines combustion occurs first with a near-stoichiometric mixture, then with a lean mixture. This reduces cyclic dispersion, while giving the benefits of predominantly lean combustion in the form of improved efficiency and reduced emissions of NO*, hydrocarbons and CO. Many changes to traditional engine design might be anticipated as a result of improved control, involving sensors and electronic engine management systems. Charge turbulence, fuel injection and mixing will be better controlled and throttling of the charge reduced. Knock sensors will enable autoignition to be controlled more closely.

7.5 AUTOIGNITION IN ENGINES: MODELLING AND EXPERIMENTS

This section discusses the problems of combining models of engine combustion and of end gas autoignition. But prior to this, some effects observed in engines which complicate this endeavour, involving the carryover of reactive species from one cycle to another, are discussed. 7.5.1 Carry-over of combustion products between cycles (i) Active intermediates Run-on occurs when the engine continues running even after the spark has been switched off. It is caused by autoignition induced by piston compression only. This occurs when active partial oxidation products from

710


Ch. 7

previous autoigniting cycles are present in the residual gas, that remains in the cylinder after the exhaust valve has closed, and which is mixed with the new charge. If the combustion is incomplete, as is likely under the marginal conditions present during run-on, then these products will include such cool-flame products as hydrogen peroxide and organic peroxides. These species, through their thermal decomposition to radicals, will act as potent initiators for the next cycle. Affleck et al. [135] have shown that the influence of the fuel on run-on is described fairly closely by its octane number, and modelling has suggested that quite small amounts of active intermediates can have an important effect [76,79]. Modern vehicles avoid run-on by cutting off the fuel supply along with the spark, but the effect has to be taken into account in motored engine experiments and necessitates a number of cycles being skipped to dilute the residual gases. The possibility of reactive product carry-over through their absorption in deposits is potentially of great practical significance and is considered in the next section. Oppenheim [136,137] has pointed to the advantages of controlled autoignition in engines, with a heterogeneous charge and separate reaction centres. Related to this, attempts have been made to turn the "carry-over" effect to advantage in the Sonex combustion system [138]. Here, a microchamber in the piston and/or cylinder head retains critical intermediate species to seed the entering fresh intake charge and cause autoignition, in the absence of any spark. With hydrogen as fuel, H 2 0 2 has been identified as the dominant intermediate. This suggests a general class of engine in which autoignition occurs by compression of a charge of reactants, hot exhaust gas and active species [139]. (ii) Oxides of nitrogen The combustion products in an engine can contain up to 4000 ppm of NO. The charge contains approximately 10% of residual gas and the nitrogen oxide that is present in this can have a significant effect on autoignition. During the exhaust stroke it is probable that some of the NO will be oxidized to N 0 2 (by the reactions described below), so that its influence also has to be considered. Both NO and N 0 2 can have accelerating and retarding effects on hydrocarbon oxidation. Figure 7.17 shows that ignition delay-times in a rapid compression machine are shortened by small additions of N 0 2 and lengthened by larger amounts [140], and that this behaviour is temperature-dependent. The behaviour of NO is qualitatively

711


0.4 -

c o c Q) O)

v

0.2 -

y

/

/

0.0

c

-0.2

c

-0.4 -

TO J= O

.2 •3 O N 0 2 + OH

(2)

NO + R 0 2 -» N 0 2 + RO

Ch. 7

In the "low-temperature" oxidation chemistry, formation of H 0 2 is effectively a terminating step. Reaction (1) regenerates an active radical, OH, and reduces the termination rate. Reactions (2) of the organic peroxy reactions are not chain branching since both R 0 2 and RO are active radicals, but at low temperatures they probably speed up the chain propagation by providing a faster alternative to the main route, through peroxy isomerization (Section 1.10). Another accelerating type of reaction is the hydrogen abstraction reactions of N 0 2 , which are initiation reactions, (3)

XH + N 0 2 -* X + HONO

where the X—H bond is weak, as for example in an aldehyde. Although rates are not well known, these reactions could be much faster than the equivalent 0 2 reaction (by a factor of 3 x 104 for HCHO at 800 K [62,144]). This indicates that they could be significant in an engine when a few hundred ppm of nitrogen oxides are present. The nitrogen oxides become inhibiting at high concentrations. Termination reactions, such as, (4)

OH + NO + M ^ HONO + M

(5)

OH + N 0 2 + M -> H O N 0 2 + M

are probably responsible. Formation of organic nitrates and nitrites is also likely to occur: (6)

R02 + NO-^RON02

(7)

R + N02->RONO

but, except at the very lowest temperatures, is likely to be reversed because of the weak bond energies. However, nitro compounds also could be


713

formed, which are more resistant to decomposition and whose formation constitutes a termination, (8)

R + N02^RN02

Recognition of the possible importance of nitrogen oxide chemistry in engine autoignition is fairly recent. Before its significance can be properly assessed, further experimental investigation is needed of both the phenomenology in engines and the reactions of NO and N 0 2 with organic substances. 7.5.2 Cylinder wall deposits During normal engine operation carbonaceous deposits, derived from the fuel and lubricant, build up and reach quasi-stationary levels after a driving distance of about 10,000 km, and affect a number of aspects of engine operation. Kalghatgi [145,146] has extensively reviewed the subject and only a brief account of the effect on autoignition is given here. The thermal conductivity of the deposits is lower than that of the surfaces to which they are bonded and, consequently, deposits reduce the heat loss and tend to be appreciably hotter than the metal cylinder wall. Any hot spots that develop are conducive to autoignition. The end gas close to a hot part of the deposit autoignites more easily and the resulting knock increases the heat transfer to the deposit, increasing its temperature still further. This runaway process will be most significant when the engine load and speed are high and it is thought to play a part in "high-speed" knock which causes engine damage (Fig. 7.1). It is much less prevalent than the more usual non-damaging knock at low speed. In leaded fuels the lead oxide in the deposit can catalyze reaction of the fuel-air mixture on its surface and exacerbate the effect. Phosphorus-containing additives have been used to poison such catalysis. The most important effect of combustion chamber deposits is that they make the engine more knock-prone as they build up. This is quantified as an octane requirement increase (ORI), and it typically reaches 5-10 octane numbers. Several mechanisms probably play a part in the phenomenon. Principally these are volumetric, thermal and chemical. The volumetric effect is the increase in the compression ratio caused by the volume of the deposits. This has been estimated [147] to be responsible for only about

714


Ch. 7

20% of the observed ORI effect. The thermal effects arise from the deposits causing the end gas to be warmer when deposits are present. This is due to both the insulating effect of the deposit preventing heat loss to the cooler wall during the compression stroke, and, probably more importantly, the warming of the charge by heat retained in the deposits from the previous cycle. The ORI mechanism of most interest here is chemical ORI: the release of initiating species like peroxides from the deposits during the compression stroke. Such species may arise by adsorption of gas-phase species from the boundary layer near the wall. Combustion in this region is heavily influenced by the cooling effect of the wall and some of the combustion will take place under the "low-temperature chemistry" conditions identified in Chapter 1, which is known to produce peroxides. Alternatively, these species could arise from oxidation of the deposit itself or material, like oil or heavy fuel components, adsorbed on it. It has also been suggested [76] that hydrogen peroxide from the bulk combustion is involved, although this seems less likely than the other possible sources. The release of very small quantities of initiators, which produce radicals by thermally decomposing, can have a significant effect on engine autoignition. During the time when the temperature is low, radical production from the initiator is much easier than from alternative (chain-branching) processes. Ignitionimproving additives for diesel fuels work similarly and autoignition delaytimes are known to depend logarithmically on their concentration [148]. This means that the response falls off with increasing concentration but, conversely, small amounts of initiator can have large effects. The effects of an initiator in a knocking engine have been demonstrated in an engine modelling study by Brussovansky et al. [76]. Their analysis showed that in an engine with substantial deposits, the presence of quite small radical concentrations in the end gas may play a key role in promoting knock. Between 1 and lOppm of H 2 0 2 in the end gas could explain the occurrence of knock (see Fig. 7.18). Although this helps to give credence to a chemical mechanism, the contribution of the various ORI mechanisms is unclear at present, and understanding is hindered by the poor reproducibility of experiments involving deposits in engines. 7.5.3 End gas reactions: basic modelling considerations This section outlines the general way in which the chemical reactions in the end gas can be modelled, when its temperature and pressure are


715

Measured Knock Occurrence (degATC)

£

0

5

10

15

20

25

30



Fig. 7.18. Comparison of measured knock occurrence with that predicted using the Hu and Keck model for various coolant temperatures. 90 primary reference fuel; 1900 rpm. (a) clean engine, showing good agreement, (b) engine with deposits; knock occurs earlier than predicted, (c) engine with deposits, as (b) but modelled with fitted amounts of hydrogen peroxide (1-5 ppm) in the end gas; agreement restored, providing evidence for chemical octane requirement increase. From [76].

716


Ch. 7

changing during the engine cycle as a consequence of piston motion and flame propagation. From the start of the compression of the gasoline-air charge, there is the potential for chemical reaction which, even in a motored engine, can be significant [149]. Whether or not autoignition occurs in the end gas, depends on the cumulative history of these reactions. The end gas is subjected first to compression, then possibly to expansion: consequences of the combined effects of piston and flame motion. Allowance must be made for the changes in the mole density of species that arise from both these effects and also chemical reaction. Let [A] be the molar concentration of chemical species A, in reacting end gas, one mole of which occupies volume v. If the net volumetric chemical source term for moles of A is SA, then the conservation equation for A in one mole of end gas is,

d([A]v) dt

~SAV-

The mole fraction of A, ([A]v)9 is unaffected by volume changes, only by the chemistry. It follows that

d[A] _

[A\6v

— & A ~~

This equation expresses the influence of both chemical and volumetric changes. Such an equation holds for each species. As a demonstration of how these changes might be modelled mathematically, the simplified five-reaction parrot scheme, described in Section 7.3.4(v), is, for the sake of brevity, taken as an example. However, the general approach is readily applicable to more complex schemes. The five reactions of this scheme are:


717

d

F + 12.350 2

>P

direct reaction

F + 02

>C

chain reaction

» 2C

chain branching

C+C

>P

quadratic termination

C

>P

linear termination

6

C q

in which F represents the fuel, C chain carriers, 0 2 oxygen, and P the products of combustion. Each type of reaction is indicated by the symbol above the arrow and these are used as suffixes for the various parameters. As an example, for a 90% iso-octane-10% heptane fuel, stoichiometry is taken to be C7.9H17.8 + 12.350 2 -* 7.9C0 2 + 8.9H 2 0.

(7.9)

In this scheme the "high-temperature" chemistry is represented by a direct reaction with an empirical form for the volumetric source term, following that recommended as a one step reaction by Westbrook and Dryer [150]. This gives a source term for F of -kd[F]a[02]b- The constants a and b have values of 0.25 and 1.5. Similarly, the source term for oxygen in this one equation is -12.35/c^[F] 0 ' 25 [O 2 ] 1 5 . It is interesting, but not surprising, to note in passing that when the values of the rate constants are adjusted to give the experimental autoignition delay-times, theses values are different from the values adjusted to yield the experimental burning velocities. The remainder of the mechanism describes the low-temperature, or "coolflame", chemistry. The three volumetric chemical source terms for F, C and 0 2 consequently are given by, SF = -kd[F]0-25[O2]15

- kt[F][02],

(7.10)

5 C = k,[F][02] + (kb - k,)[C] - 2kq[Cf,

(7.11)

S02 = - 1 2 . 3 5 U * , f 2 5 [ 0 2 ] 1 - 5 - */[F][0 2 ].

(7.12)

718


Ch. 7

These, in turn, are substituted as the source terms in the species conservation equation (7.8), for each species. The chemical energy volumetric source term arising from the five reactions depends on the associated heats of reaction, hA, in the energy equation and is given by, in equation (7.6) was 0.7, the initial conditions 298 K and 1 atm pressure, the angle of advance of the spark 25° before tdc, and the engine speed 1500 rpm. The broken line curves show the variations of end gas temperature, pressure and mass fraction burned predicted by the engine model, without autoignition. The full lines indicate the temperatures and pressures with autoignition, as well as the mole fraction, C, of the chain carrier, as predicted by post processing using the five-step parrot model (Section 7.5.3). Although some slight reaction occurs during compression, the principal, cool-flame, reactions occur at the maximum compression. Advancing the spark ignition from 10

722

Autoignition in spark-ignition engines 1200

120n

7 —"

/

(a)

Autoignition Model Engine Cycle Model

k-

1000

100 H

—

Ch. 7

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800

L

< 600

r

a.

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Low-temperature Combustion and Autoignition, Volume 35 (Comprehensive Chemical Kinetics)

Comprehensive Chemical Kinetics, Volume 33: Catastrophe Theory

Kinetics of Multistep Reactions (Comprehensive Chemical Kinetics)