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
School of Chemistry, University of Leeds, Leeds LS2 9JT, United Kingdom
C. MOHAMED
School of Chemistry, University of Leeds, Leeds LS2 9JT, United Kingdom
C. MORLEY
Shell Research Ltd., Thornton Research Centre, P.O. Box 1, Chester CHI 3SH, United Kingdom
M.J. PILLING
School of Chemistry, University of Leeds, Leeds LS2 9JT, United Kingdom
S.H. ROBERTSON Molecular Simulations Inc., 240/250 The Quorum, Barnwell Road, Cambridge CB5 8RE, United Kingdom S.K. SCOTT
School of Chemistry, University of Leeds, Leeds LS2 9JT, United Kingdom
P.W. SEAKINS
School of Chemistry, University of Leeds, Leeds LS2 9JT, United Kingdom
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 + 02^OH + 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
Basic chemistry of combustion
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
Basic chemistry of combustion
-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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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 ^OkJmol" 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
Basic chemistry of combustion
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 + 02^OH + 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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Review of experimental methods
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
Basic chemistry of combustion
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
Review of experimental methods
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
Basic chemistry of combustion
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
Review of experimental methods
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 (<5% consumption). (v) Method V. Direct studies of hydrocarbon oxidation Although, as already indicated, direct studies of hydrocarbon oxidation have not been conducive to the determination of kinetic data for elementary reactions, in specific cases, under carefully selected conditions, particular key processes can be isolated, mechanisms elucidated and kinetic data obtained. The approach will be illustrated by studies on the oxidation of C 3 H 6 over the range 650-800 K at pressures ^60Torr. The complexity of the system is limited under these conditions by three factors: (a) The homolysis of H 2 0 2 , which corresponds to secondary initiation, is dramatically reduced by the low temperature, low pressure used and by focusing on initial products where [H 2 0 2 ] is very low. (b) As shown in Table 1.3, the equilibrium CH 2 =CHCH 2 + 0 2 ^ CH 2 =CHCH 2 0 2 is well to the left so that radical branching via a sequence such as the following is negligible. CH 2 =CHCH 2 0 2 + R H ^ CH 2 =CHCH 2 OOH + R
1
CH 2 =CH 2 CH 2 0 + OH
24
Basic chemistry of combustion
Ch. 1
(c) The "stabilized" allyl radical undergoes mutual recombination to give hexa-l,5-diene and, from the rate of its formation, [CH2CHCH2] can be calculated accurately. Furthermore, as propene oxide is formed uniquely in reaction (17) and k17 is known reliably [37], [H0 2 ] can be calculated with precision giving a very well-defined system. (17)
H 0 2 + C 3 H 6 -* C 3 H 6 0 + OH
Extensive kinetic information on allyl reactions has been obtained, together with Arrhenius parameters for the important initiation reaction (lp) [38]. (lp)
C 3 H 6 + 0 2 -> 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
26
Basic chemistry of combustion
Ch. 1
extremely early in reaction. For a mixture containing [CH4] = [0 2 ] = 100 Torr over the range 700-1200 K, the rates of secondary initiation (R s ), through reactions (If) and (13), and primary initiation (R p ) become equal when 10" 4 % of CH 4 has been converted into H 2 0 2 and 10" 2 % into HCHO (If)
HCHO + 0 2 -» HCO + H 0 2
(13)
H 2 0 2 + M -> 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
Primary initiation reactions
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
Basic chemistry of combustion
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
Primary initiation reactions
(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
Basic chemistry of combustion
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
^IP[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.
Primary initiation reactions
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&
<2 7.1 16.1 <1
<2 8.3 18.5 <1
i-C4H8* tC4H8-2 butene-1 pentene-1
5.5
T/K
log(A/cm3molecule-' s-')
EM mol-'
693 693 693 693
-11.10 -11.2 -11.7 -11.7
2.25 x 4.7 x 2.25 X 5.5 x
693 693 693 743
-11.7 -11.0 -10.9 -12.7
145 151 143 122
<6 X 2.85 x 6.3 x
713 713 713 713
-11.5 -11.7 -12.0
163 149 140
lo-=
rf: 0.44
161.2 158 146 145
6.4
f i;'
B
&. * v1
-
4
%
8
3
E. a
*Arrlienius parameters from direct studies (Table 1.5). **Arrhenius parameters recommended are those for G H 6 + 02.
0
P
33
Primary initiation reactions TABLE 1.8 Summary of kinetic data for RH + 0 2 —> 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. ^Obtained 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
Basic chemistry of combustion
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
Primary initiation reactions
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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 + 02^OH + 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
Basic chemistry of combustion
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 + 02^OH + 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
Basic chemistry of combustion
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
Basic chemistry of combustion
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 (<600K) relative to A:(CH3 + CH 3 ) 1/2 and extrapolation to high temperatures requires care due to potential pressure and non-Arrhenius effects.
1.8 HOMOLYSIS OF ALKYL RADICALS
Over the whole temperature range (500-1000 K), three overall reactions, (3), (5A) and (5C), summarize the competition for alkyl radicals, where AB', AB, and OR, are lower alkenes, conjugate alkenes, and Oheterocyclic compounds, respectively. The overall activation energies of (5A) and (5C) are small (typically <20kJmol" 1 ), but E 3 « 1 3 0 150kJmol _1 . Increase in temperature thus favours (3), and homolysis reactions become dominant for most radicals above about 850 K unless [0 2 ] is very high. In competition with (5A) and (5C), successive homolysis may occur until one of the 4 "basic" radicals CH 3 , C 2 H 5 , 2-C3H7 and tC 4 H 9 , is formed, as illustrated by the sequences below (3)
R (+ M) -* R' + AB' (+ M)
(5 A)
R + 02-»AB + H02
(5C)
R + 0 2 ^ O R + OH
(a)
CH 3 C(CH 3 ) 2 CH 2 CH 2 CH 2 CH(CH 3 )CH 2 CHCH 3
I
CH 3 C(CH 3 ) 2 CH 2 CH 2 CHCH 3 + CH 2 =CHCH 3
I
CH 3 CH(CH 3 ) 2 CH 2 CH 2 + CH 2 =CHCH 3
1
(CH3)3C + C 2 H 4
Homolysis of alkyl radicals
(b)
45
CH 3 CH 2 CH 2 CH 2 -> CH 3 CH 2 + C 2 H 4
In many situations, these homolyses represent termination reactions because the "basic" radicals are unable to undergo reactions to give reactive species such as OH radicals and instead (CH 3 apart) react rapidly with 0 2 to give the conjugate alkene and the unreactive H 0 2 radical in yields of virtually 100% (discussed later) C2H5 + 0 2 - * C 2 H 4 + H 0 2 CH3CHCH3 + 0 2 -> C 3 H 6 + H 0 2 (CH3)3C + 0 2 -> *-C4H8 + H 0 2 Relatively few rate constants are available for the alkyl homolysis reactions mainly because clean sources of the alkyl radical have proved difficult to find. Consequently, the data are not always reliable, but some check is available [64, 65] from thermochemical and kinetic data for the reverse reaction. Direct photolysis of azo-compounds and mercury-photosensitized decomposition of alkanes have so far provided the most reliable (although old) data [64]. For good results, the method depended on precise product analysis in the early stages of reaction, with equation (1.9) used to determine k3/kl/2, where R A B ? and R R _ R are the initial rates of formation of AB' and recombination product R—R, respectively. RAB' pl/2
k\'2
?
(1.9)
R - * R' + AB' R + R-^ R - R Baldwin, Walker and co-workers successfully used the competition between the homolysis of R and its oxidation to obtain kinetic data for R + 0 2 , as illustrated for 2-C4H9 radicals. When butane was added to slowly reacting mixtures of H 2 + 0 2 (Method I), C 3 H 6 and butene-2 were observed [23] in high yield in the initial stages. As the two compounds cannot be formed from 1-C4H9 radicals (in any plausible way), then they arise uniquely from reactions (3b) and (5Ab) of the 2-C4H9 radical
46 (5Ab) (3b)
Basic chemistry of combustion
Ch. 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=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 (A^IO13-8^0-^"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
Basic chemistry of combustion
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
The reaction of alkyl radicals with 0 2
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
Basic chemistry of combustion
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
The reaction of alkyl radicals with 0 2
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
The reaction of alkyl radicals with 0 2
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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(^ipa/^i4)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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Oxidation of cyclic alkanes
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
Basic chemistry of combustion
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
Oxidation of cyclic alkanes
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
^o^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
Oxidation of cyclic alkanes
formation of the 1,3 oxide, and rupture occurs to produce the energetically more favourable hex-5-en-l-al.
A H'A^A
^
H'V^A
^ 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
Basic chemistry of combustion
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
Oxidation of cyclic alkanes
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
Basic chemistry of combustion
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
YH2
O
W
5-oxabicyclo[2.2.2] octane °CH3
^
O
HO. CH2 O
__ ^d __
+ OH
CH2—O
^
6-oxabicyclo[3.2.1] octane
Both have considerable stability and should be observable at 750 K.
Oxidation of alkenes
83
QOOH chemistry will then remain of considerable importance for substituted cyclohexanes and may be influential in determining research octane numbers.
1.16 OXIDATION OF ALKENES
1.16.1 Introduction As emphasized earlier, alkenes are major primary products with 5099% yields of conjugate alkene in the oxidation of C2—C6 alkanes between 600 and 800 K. Increasing amounts of lower alkenes are formed above 800 K, as homolysis competes favourably with oxidation. 1-C3H7 (+ M) -> 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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Oxidation of alkenes
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
Basic chemistry of combustion
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
Oxidation of alkenes
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
Basic chemistry of combustion
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
Oxidation of alkenes
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
Basic chemistry of combustion
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-
Oxidation of alkenes
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)
Basic chemistry of combustion
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
Basic chemistry of combustion
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.
Atom and radical addition to alkenes
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
Basic chemistry of combustion
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
Atom and radical addition to alkenes
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
Basic chemistry of combustion
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
Atom and radical addition to alkenes
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
Basic chemistry of combustion
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
Atom and radical addition to alkenes
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
Basic chemistry of combustion
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.
Atom and radical addition to alkenes
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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
Basic chemistry of combustion
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)
Basic chemistry of combustion
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 (<20%) of abstraction will occur from the ring below 1000 K. In this region, the benzyl radical is thermally stable and is unreactive towards 0 2 due to the unfavourable position of the equilibrium C 6 H 5 CH 2 + 0 2 ^ C 6 H 5 CH 2 0 2 (Table 1.3), so that radical-radical reactions are important as found in allyl and methylallyl oxidation chemistry.
115
Oxidation of aromatic compounds
At about 750 K, benzaldehyde is formed in high yield from toluene oxidation, almost certainly through the sequence [115]. CH2OOH
CHO
CH20
O^OUO^D o2
C 6 H 5 CHCH 3 radicals are able to react with 0 2 to give styrene quite rapidly, with a rate constant a factor of 102 lower than for alkyl + 0 2 . Radical-radical reactions are hence considerably reduced in importance. CHCH3
/£XTH(OOH)CH3
0
CH(O)CH3
^ v X H O + OH+CH3 COCH3 s £ ^
Table 1.34 summarizes the rather limited kinetic data for O, OH, H 0 2 and H attack on benzene and the simple alkyl aromatics. Little other kinetic information is available. The Princeton group [116,117] have used their flow reactor to investigate the oxidation of a number of alkylbenzenes in the region 800-1200 K. Although relatively good measurements of product-time profiles were made, mechanisms were not fully elucidated and no kinetic data were obtained for the elementary reactions involved. There is, however, a consensus view that the ring in alkylbenzenes is largely retained until the later stages of reaction. Where relatively high concentrations of H atoms exist, addition of H followed by loss of an alkyl group is an efficient way of reducing alkylbenzenes and poly aromatics to benzene which, of course, is of considerable significance in aromatic cracking processes.
116
Basic chemistry of combustion
Ch. 1
TABLE 1.34 Kinetic parameters for X + alkylbenzenes based on k = AT" t~B,T Reaction
i4/(cm
O + C6H6 - ^ OH + C6H5
* molecule
(*i + k2) -=
1
n
5.9 x 10~23
11 k2- = 3.7 x 10"
-^>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
Basic chemistry of combustion
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
Basic chemistry of combustion
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.
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[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.
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[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.
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[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.
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[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
Elementary reactions
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
Elementary reactions
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
Elementary reactions
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
<^1 obs
/Torr s - 1 1.02 6.0 1.20 4.5 1.41 3.00 9.1 1.09
x x x x x x x x
/cm3 molecule -1 s - 1
10 - 5 10 - 6 10 - 5 10" 5 10 - 5 10 - 5 10" 5 10" 4
average = a
1.3 2.3 2.3 2.9 1.8 1.9 2.0 1.8
x x x x x x x x
(2.03
10" 23 10" 23 10" 23 10 - 2 3 10 - 2 3 10" 23 10" 23 10 - 2 3 0.47) x 10 - 2 3
^ 1 corr
/cm3 molecule -1 s - 1 3.3 x 10 - 2 3 2.7 x 10 - 2 3 2.7 x 10 - 2 3 3.4 x 10 - 2 3 1.9 x 10 - 2 3 2.1 x 10 - 2 3 2.3 x 10 - 2 3 2.0 x 10 - 2 3 (2.55
0.58) x 10 - 2 3
Balance to 60 Torr total pressure with N 2 .
and the final corrected values of kx are given in column 5 of Table 2.1. The Arrhenius parameters for initiation for formaldehyde and propene are given in Table 2.2. The activation energies are, within experimental errors, equal to the enthalpy of reaction, suggesting, as might be expected, that the reverse reaction has zero activation energy. In the light of these and other observations Stothard and Walker [2] suggest the following parameters for primary abstractions: (a) For alkanes, alkenes and probably oxygenated compounds where no electron derealization exists on the radical formed k = 1.66 x 10" 11 exp(-&H/RT)
cm3 molecule -1 s" 1 .
(2.2)
(b) For alkenes, dienes, alkyl benzenes and related compounds where an TABLE 2.2 Kinetic data for the initiation reaction RH + 0 2 Compound
A /cm3 molecule * s *
£/kJmol-1
Exptl temperature range/K
HCHO C3H6
3.4 x 10 - 1 1
163 164
673-813 673-793
-12
6 5
134
Elementary reactions
Ch. 2
electron-delocalized radical is formed k = 1.66 x 10~12 exp(-A///7?r) cm3 molecule"1 s" 1 .
(2.3)
The difference of a factor of 10 in the pre-exponential factor is due to the increased loss of entropy in the formation of the electron-delocalized radical.
2.3 ABSTRACTION REACTIONS
2.3.1 Introduction The initial propagation step in the oxidation of fuel is it's conversion from unreactive hydrocarbon to reactive carbon centred radical by hydrogen abstraction. The most important abstracting species are OH, H and O radicals (7)
OH + R H - * H 2 0 + R
(8)
H + RH^H2 + R
(9)
O + RH-^OH + R
with lesser contributions from H 0 2 and other hydrocarbon radicals. The other important abstraction reaction by 0 2 has already been briefly mentioned in the previous section and is discussed in greater detail in Chapter 1. The rate at which the abstraction occurs is crucial for calculating overall properties such as flame speeds but in addition, the relative efficiency of the abstraction at each of the different sites (primary, secondary or tertiary) of the hydrocarbon will shape the chemistry of the subsequent oxidation steps. As abstractions involve the reaction of one radical species with a stable substrate (as opposed to radical-radical reactions) it is relatively easy to isolate these reactions for detailed study. Section 2.3.2 examines an elegant method for measuring hydroxyl abstraction rate coefficients using laserflash photolysis coupled with laser-induced fluorescence detection of the OH radical. Absolute values of abstraction rate coefficients and their temperature
Abstraction reactions
135
dependence are generally interpreted via transition state theory (TST). The application of TST to abstraction reactions is outlined in Section 2.3.3. TST is especially useful in interpreting isotope effects (Sections 2.3.4, 2.5.3). Branching ratios for specific reactions are much harder to determine in real-time experiments and investigations have tended to concentrate on measuring the overall rates of reactions of prototypical compounds. Measurements of the rates of abstraction from ethane or neopentane determine rate coefficients per primary bond. Related measurements can then be used to calculate the rates of secondary or tertiary abstraction assuming that the overall rate coefficient is of the form k = npkp + nsks + ntkt
(2.4)
where np, ns and nt are the number of primary, secondary and tertiary C—H bonds and kp, ks and kt are the characteristic rate coefficients for primary, secondary or tertiary abstraction. The validity of this assumption is discussed in Section 2.3.5. Theoretical efforts in this field have included the use of TST to investigate non-Arrhenius behaviour, the validity of additive empirical rate expressions and attempts have been made to calculate rate coefficients for these relatively simple reactions from first principles. We conclude Section 2.3.5 with a brief review of abstractions by other radical species and a comparison of time-resolved measurements with alternative techniques. 2.3.2 Time resolved studies of OH abstraction reactions A series of studies by Tully and coworkers [5-8] illustrate the technique of laser-flash photolysis and time-resolved spectroscopic detection used for the study of a number of classes of elementary reactions including those of H-atom abstraction by OH or other radicals (Fig. 2.3). The example we shall consider is the reaction of OH with a number of ethane isotopomers, e.g., (10)
OH + C 2 H 6 - • H 2 0 + C 2 H 5
In general the kinetics of the system and the experimental complexity are greatly reduced by having the stable ethane reagent in great excess, making
136
Elementary reactions
MUITI CHANNEL
/
AN U Y Z E R
Ch. 2
PDF 11 7 3 MINI COMPUTER
Fig. 2.3. Schematic figure of laser flash photolysis, cw laser induced fluorescence apparatus.
the reaction kinetics pseudo-first-order. Under these conditions we do not need to know the absolute concentration of the radical reagent but rather a relative measurement of concentration as a function of time, a measurement that can usually be obtained by a number of spectroscopic techniques, in this case by laser induced fluorescence. The time dependence of the OH radical concentration ([OH]t) is given by [OH] t = [OH]0e i-k't)
(2.5)
where [OH] 0 is the initial concentration of hydroxyl radicals produced by a pulsed photolysis laser, t is the time after the photolysis flash and k! is the pseudo-first-order rate coefficient given by
^ = UC 2 H 6 ].
(2.6)
Abstraction reactions
137
c >» *-*
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
Elementary reactions
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<248 nm) -> 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<—X), is generated in a cw ring dye laser and directed into the reaction cell. A fraction of the ground state OH radicals are raised into this upper excited state from which a portion of the molecules will relax back to the ground state by the emission of a photon (fluorescence). Unlike the directed light from the exciting laser, this fluorescence is isotropic and can, therefore, be detected by a photomultiplier tube placed perpendicularly to both the photolysis and excitation lasers to minimize the observation of scattered light. Under the conditions used, the observed fluorescence intensity is proportional to the concentration of OH radicals. Each time the photolysis laser fires a complete profile of the OH decay is obtained and recorded. A number of such decays are stored and averaged to reduce the signal to noise and bimolecular rate coefficients are obtained at a number of different temperatures by the procedures outlined above. The availability of suitable cw dye lasers is somewhat limited and for many systems pulsed dye lasers have to be used. The obvious difference is that the relative concentration of the radical is obtained for only one time-delay each time the photolysis laser is fired. Thus, the production of a full decay curve, of say 100 points, can take considerably longer than with a cw probe laser. Figure 2.5 shows how a decay trace is built up in these experiments. The reaction cell could be heated and bimolecular rate coefficients were
139
Abstraction reactions i
i
/—V
»
LIFSi portion*il to [0
X
•
Data points
• #
id)
o
• • • Mi \ t
\
t
m
• + + «
^
Time
Excimer laser Dye laser fires, generating fires, probing initial [OH] for [0H] t
/. the time d elav ^ Q i asers> 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
Elementary reactions
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
Abstraction reactions
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
Elementary reactions
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
Elementary reactions
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
Elementary reactions
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
Abstraction reactions
•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
Elementary reactions
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.
Abstraction reactions
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 (<540K) the [H0 2 ] decay was determined by the self reaction leading to H 2 0 2 , and once the residual absorption of the hydrogen peroxide had been accounted for the decay could be fitted with simple second-order kinetics. However, above 540 K the residual absorption due to H 2 0 2 increased, an observation which JemiAlade et al. [19] attributed to the chain reactions (22)
H 0 2 + H 2 CO -> 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
Elementary reactions
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
Elementary reactions
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-
Radical decomposition reactions
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
Elementary reactions
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]
(2.20)
and the associated rate coefficient in this low pressure region is denoted as k° (=ka). Figure 2.13 is a sketch of the pressure dependence of a unimolecular reaction showing the two limiting conditions. The region joining the two extremes is known as the 'fall off' region. Theories of unimolecular reactions have advanced considerably since Lindemann's initial proposal but they are still based on the same physical ideas so clearly highlighted in the Lindemann mechanism. The decompositions of larger radicals (>C 6 ) will be in their high pressure limit, i.e., independent of pressure, under typical ignition conditions but for modelling the reactions of smaller radicals this pressure dependence needs to be accounted for. Kineticists have studied the pressure dependence of unimolecular reactions in order to understand the underlying physics of the energy transfer processes that control such phenomena and to parameterize the dependence so that unimolecular (and related termolecular reactions) can be realistically incorporated into combustion models. Section 2.4.3. describes some of the models which have been used to model pressure dependence. Experimental data on radical decompositions will remain limited for the forseeable future, therefore modellers will be reliant on extrapolated experimental or theoretical data. These studies are of relevance because our confidence in data extrapolated out of the experimental region will be greatly enhanced if it is based on
157
Radical decomposition reactions
-2.5
-2.7
^
h
-2.9
-3.1
h
-3.3
-4
-3
-2
-1
lg{[A]/mol dm3} Fig. 2.13. Typical pressure dependence of a unimolecular reaction.
an experimentally validated physical model. Historically there has always been a trade off between accuracy of the model and the computer time required to generate the required data. However, this is an interesting period for the reaction modeller as developments in computing power mean that realistic physical models of unimolecular reactions can be incorporated into the overall combustion model. (ii) Parameterized fits and representations Considerable progress in developing techniques to analyse fall-off data has been made in recent years. By far the most popular technique is that developed by Troe and co-workers [25-27]. The basis of this technique is the Lindemann-Hinshelwood model for unimolecular fall-off:
k(T,[M])
kxku[M] k°° + k°[M]
n
LH>
(2.21)
158
Elementary reactions
Ch. 2
wherefc°°is the high pressure unimolecular rate coefficient, k is the low pressure bimolecular rate coefficient and F L H is the Lindemann-Hinshelwood factor. The rate coefficient calculated from this model gives a pressure dependent curve which, when compared to experiment, is observed to fall-off more rapidly than the data would suggest. Troe and co-workers corrected for this deficiency by introducing factors to the right hand side of equation (2.21) which have become known as the broadening factors. There are two main components: the first, F s c , accounts for the effects of the energy dependence of the microcanonical (energy dependent) rate coefficients, and the second, F w c , accounts for weak collision effects (i.e., that complete deactivation of an energized molecule may require more than one collision). (2.22)
k(T, [M]) = ^ F L M F S C F W C .
F S c is the more important of the two factors accounting for most of the fall-off curve broadening. It is dependent on the ratio k°\M\lk™ (sometimes referred to as the reduced pressure), SK the effective number of oscillators in A, and Z?K, the ratio of the threshold energy to kBT where kB is the Boltzmann constant. Troe has suggested methods of determining these parameters from experimental data and has developed formulae to determine F s c . Troe and co-workers have also developed formulae for F w c . Again these depend on BK and 5 K as well as a third variable, the collision efficiency, j8c, which is a multiplicative factor applied to the collision frequency to account for weak collisions, i.e., the effective collison frequency is fica) where co is the actual collision frequency. Troe has also suggested a simpler form of the broadening factor product (F) which is adequate for most purposes log F =
^ ^
,
(2.23)
where N - 0 . 7 5 - 1 . 2 7 log F c .
(2.24)
159
Radical decomposition reactions
As can be seen this representation depends on the three parameters k°, k™ and Fc. Fc is temperature dependent and is usually represented by the approximate equation F c = (1 - a) exp(-T/T***)
+ aexp(-T/T*)
+
exp(-T**/T). (2.25)
Equations (2.23-2.25) can be used to fit the experimental data using A;00, k°, and Fc as temperature dependent variable parameters. In addition, they are widely used to parameterize evaluated or recommended rate coefficients for combustion modelling (Chapter 3) [28]. For the reaction (25)
C 2 H 5 ^ H + C2H4
for example, an evaluation of the available rate data led to the following recommended parameters, for helium as the diluent gas k°° = 6.6 x 1(T15 T 1 ' 28 exp(-650/T) s" 1 ,
(2.26)
k° = 1.3 x 10" 29 exp(-380/r) cm3 molecule" 1 s" 1 ,
(2.27)
Fc = 0.24 exp(-7740) + 0.76 exp(-771025).
(2.28)
2.4.2 Real time experimental techniques for radical decompositions The laser flash photolysis/photoionization mass spectrometry (PIMS) technique developed by Gutman [29] and Bayes [30] has recently been applied to the direct determination of radical decomposition rates. The experiment combines the technique of laser photolysis for generating the radical species, with a flow tube at pressures of 0.5-30 Torr, and real time mass spectrometric radical detection. The low energies used in the photoionization technique ensure that little radical ion fragmentation occurs. A schematic diagram of the apparatus is shown in Fig. 2.14. The apparatus has been used by Gutman and co-workers to study a number of radical decompositions which together cover the complete range of unimolecular behaviour. The studies of the formyl [31] and ethyl [32] radical decompositions are in the low pressure region under the conditions of the experiment, isopropyl [33] is in the fall-off region and neopentyl [34] close to the high pressure limit. After brief comments on the ethyl and neopentyl radical decompositions, we shall use the isopropyl radical
160
Ch. 2
Elementary reactions Quadrupole Mass Filter Ion-focusing Lenses lyZuy^
Gas Inlets \
Conical Skimmer -"* \
Vacuum UV Ionizing Light
/ Thermocouple
Heatable Tubular Quartz Reactor
/
Quartz Window
Vacuum Chamber 1
\
Vacuum Chamber 2
To Pumps
y
Fig. 2.14. Laser flash photolysis/photoionization mass spectrometry apparatus for studying reaction (24).
as a more detailed example to illustrate the experimental method and modelling techniques used to study radical decomposition reactions. The data for the neopentyl radical showed that the rate coefficient is close to the high pressure limit under the experimental conditions employed, allowing an accurate extrapolation to be made. Analysis gives: £°° = 10
5
exp(-129
4 kJ moV1 IRT) s~
(2.29)
This expression is central to the analysis of the isomerization of the neopentylperoxy radical discussed in Section 2.7. The ethyl radical decomposition is, effectively, at the low pressure limit under the PIMS conditions and the rate data were used, in conjunction with a master equation analysis, to obtain energy transfer parameters. This analysis is discussed further in Section 2.4.3. In the laser flash photolysis/PIMS technique the radicals are generated uniformly along the entire length of the heated flowtube in two ways, either by the direct photolysis of a suitable precursor (26)
(/-C3H7)2CO + hi<193 nm) -> 2/-C3H7 + CO
Radical decomposition reactions
161
or indirectly via the generation of chlorine atoms (27) (28)
CC14 + hz<193 nm) -> CC13 + CI CI + neo-C5H12 -> neo-C 5 H n + HC1
Conditions of laser power and precursor concentration are chosen such that very low initial radical concentrations (~3 x 1010 molecules cm3) are generated so that radical recombination reactions are minimized. The absence of such reactions is verified by the invariance of the observed radical disappearance rate coefficient with radical concentration. A small portion of the mixture leaks out of a pinhole and is formed into a beam (Fig. 2.14). The radicals are ionized by light from a halogen, hydrogen or rare gas resonance lamp, and then mass selected. Temporal ion signals are recorded on a multichannel scaler including a short portion of background signal before the laser is fired. The radical decays are exponential in nature confirming that only first order processes are responsible for the removal of the radicals. However, there are two removal processes that need to be considered. Firstly, a small loss (—20-50 s _1 ) of radicals is observed even at low temperatures, due to removal of the radicals on the wall of the reactor. This heterogeneous loss rate is approximately constant with temperature in the range 300500 K. At higher temperatures the rate of radical removal begins to rise rapidly due to the gas-phase decomposition of the radical (Fig. 2.15). The experiment measures the loss of radicals due to both processes but the gas phase rate can be determined by subtracting the heterogenous component, assuming that this remains constant over the temperature region of interest. To allow for the possibility of slight changes in the heterogeneous rate Gutman and coworkers have only considered data where the radical decomposition is at least three times greater than the extrapolated heterogeneous rate, determining a minimum value for the radical decompositions which can be measured with this technique. Maximum rates of approximately 600 s - 1 are controlled by the flight time of the radicals to the observation region. Pressures can be varied from approximately 3-30 Torr of helium and the estimated uncertainties in the radical determinations are % in the middle of the temperature range to % at the extremes. Figures 2.16a,b show typical data sets for isopropyl and neopentyl decomposition.
162
Elementary reactions
Ch. 2
500 •
400
# 300
• 200
• •
100
#
L
500
• •• • • • L_
•
••
L_
600 700 Temperature /K
1
800
Fig. 2.15. Variation of k24 with temperature showing the clear onset of the unimolecular decomposition against a background of slow heterogeneous loss at the reactor wall.
The major advantages of the PIMS technique is its flexibility (it is able to monitor a large number of different hydrocarbon radicals and the products of the reactions such as methyl radicals or propene), and sensitivity. The low concentrations of radicals ensure the decomposition reaction is isolated for study. Unfortunately there are two significant drawbacks to the technique. The use of mass spectrometric detection and a flow system limits the pressure of the experimental technique. Neither of the two reactions shown in Fig. 2.16 is in the high pressure limit and therefore modelling and extrapolation are required to obtain this parameter. Secondly, the restrictions of the maximum first-order rate coefficient which can be studied with the technique limit the temperature range of the studies to relatively narrow regions. 2.4.3 Modelling radical decompositions In this section we shall concentrate on modelling the isopropyl data. Located as it is between the low and high pressure limits, data in this fall
163
Radical decomposition reactions
CchL
C3H;
CH 3 +
C3H;
10 Time / ms
20
0
10
20
Time / ms
Fig. 2.16. Typical experimental traces for the decomposition of the neo pentyl (C 5 Hn) and isopropyl (C3H7) radicals and production of the corresponding products, CH 3 and propene (C 3 H 6 ). For neopentyl decay, [He] = 2.40 x 1017 molecules cm - 3 , [C5H12] = 7.84 x 1013 molecules cm - 3 , [CC14] = 4.70 x 1012 molecules cm - 3 , T=600K. For isopropyl decay 12 T= [He] = 3.0 x 1017 molecules cm" [isopropylketone] = 9.4 x 10 molecules cm 780 K.
off region provide the greatest challenge to the modeller. Figure 2.17 shows that the unimolecular rate coefficient for iso-propyl decomposition is a complex function of temperature and bath gas concentration. The analysis of such data and the extraction of physically meaningful parameters is a crucial part of any kinetic investigation. The great advantage of the Troe approach is that one can obtain good representations of the data using relatively simple analytic formulae, an important consideration in the modelling of combustion or atmospheric processes where rate coefficient determination has to be rapid. A disadvantage of this method is that it is not always possible to extract meaningful parameters from rate data or to assess the validity of theoretical models.
164
Elementary reactions
0
5
Ch. 2
10
15 16 20 25 30 35 [He]/10 molec/cm 3 Fig. 2.17. Pressure and temperature dependence of reaction (24). The solid lines are a calculated fit using an inverse Laplace transform/Master Equation technique described in Section 2.4.3.
A better understanding of unimolecular processes has been obtained by examining the assumptions of the Lindemann model. As a result advances have been made in two aspects of the theory: Firstly, the rate at which an activated molecule dissociates is assumed to be constant in Lindemann theory. More realistic models express the rate of dissociation as a function of energy, the greater the energy the more likely it is that sufficient energy will be located in the dissociating mode and the faster the reaction. The other important assumption is that activation and deactivation are single step processes, an event which is extremely unlikely even for the strongest colliders. A more realistic model treats activation and deactivation as multi-step processes, a given molecule executing a random walk in energy until eventually attaining enough energy to dissociate. Such motion in energy space is described by a probability distribution over the various states in which the molecule can exist. Both of these corrections to the Lindemann model are embraced by the
165
Radical decomposition reactions
set of techniques based on the so-called master equation (ME). For a unimolecular decomposition reaction the ME is written as,
-p(E,
i) = (o\
dt
P(E/E')p(E',
t)dEf -
Jo (2.30)
where p(£, t) is the time dependent probability density that a given molecule will have rovibrational energy E at time t, co is the collision frequency, P(E/E') is the transition probability density of the molecule going from energy Ef to energy E on collision and k(E) is the microcanonical rate coefficient for decomposition. There are a number of restrictions on P(E/Er): the amount of reactant present is not altered by collisional excitation/de-excitation processes, hence it must be normalized, J P(E/Ef)dE=l. Jo
(2.31)
Furthermore, in the absence of reaction the system must approach the equilibrium Boltzmann distribution, f(E), that is the system must obey detailed balance, P{EIE')f{E')
= P(E'/E)f(E)
= P(E'/E)N(E)
exp(-/3£)/<2(/3), (2.32)
where (3 = l/kBT, N(E) is the rovibrational density of states function, and <2(/3) is the corresponding rovibrational partition function of the molecule. From equation (2.30) it can be seen that p(E, i) is dependent on co, the form of P{EIE') and k{E). OJ is most often taken to be the LennardJones collision frequency i.e., the hard sphere collision frequency which is rectified for the effects of intermolecular forces by the inclusion of a collision integral factor. The most common form for P{EIE') is the exponential down model, P{EIE') = A(E') exp(-a(E'
- E))
E' ^ E.
(2.33)
166
Elementary reactions
Ch. 2
where a =
d \
(2.34)
(A£) d being the mean energy transferred in a deactivating collision. A(E') is an energy-dependent normalization coefficient. This model is very widely used and there is some theoretical justification for its use from scattering theory [35]. Currently there is considerable debate as to its suitability particularly in view of the increasing evidence that so-called supercollisions have significant effects on rate coefficients [36, 37]. However, it should be emphasized that the ME technique is not tied to any one form of P(E/Ef) and, so long as the above normalization and detailed balance conditions are met, any model for energy transfer can be used. The determination of the microcanonical rate coefficient k(E) is the subject of active research. A number of techniques have been proposed, and include RRKM theory (discussed in more detail in Section 2.4.4) and the derivatives of this such as Flexible Transition State theory, Phase Space Theory and the Statistical Adiabatic Channel Model. All of these techniques require a detailed knowledge of the potential energy surface (PES) on which the reaction takes place, which for most reactions is not known. As a consequence much effort has been devoted to more approximate techniques which depend only on specific PES features such as reaction threshold energies. These techniques often have a number of parameters whose values are determined by calibration with experimental data. Thus the analysis of the experimental data then becomes an exercise in the optimization of these parameters so as to reproduce the experimental data as closely as possible. One such technique is based on Inverse Laplace Transforms (ILT). The ILT technique may be used to calculate microcanonical rate coefficients from experimental Arrhenius expressions of the high pressure canonical rate coefficient. The idea was first proposed by Slater [38] who observed that the standard expression that relates the microcanonical and canonical rate coefficients can be written in terms of a Laplace transform
k
*W
=
7 ^ f k(E)N(E)
&
M-PE) d£,
(2.35)
167
Radical decomposition reactions
k^)
(2.36)
= -^j^[k{E)N{E%
where Ki(fi) is the high pressure limiting rate coefficient. Equation (2.36) shows directly how the microcanonical rate coefficient may be obtained if we have &J(/3). Moreover, the uniqueness theorem of Laplace transforms guarantees that there is a one-to-one correspondence between k(E) and k^(p). The difficulty with the equation (2.36) is that k^(fi) needs to be determined over a wide range of temperatures (strictly an infinite range), which presents experimental difficulties because of the strong dependence of k^(fi) on j8. Errors in /c^(/3) can have dramatic effects on the functional form of k(E). Such difficulties are less pronounced for the association rate coefficient, ka(p), which in general is a much weaker function of p. k^(fi) and ka((Z) are connected via the equilibrium constant K(p) and so equation (2.36) can be rewritten as, k(E) = 2-\Q(p)K(p)k:(p)]/N(E)
(£>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
Elementary reactions
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
Radical decomposition reactions
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
Elementary reactions
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
Radical decomposition reactions
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
Elementary reactions
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-
Radical decomposition reactions
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
Elementary reactions
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
Radical decomposition reactions
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
Elementary reactions
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
Elementary reactions
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
Elementary reactions
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
Radical recombination and association reactions
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.
Radical recombination and association reactions
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
Elementary reactions
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-
Radical recombination and association reactions
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
186
Elementary reactions
g=-R'(t)M-1cl>.
Ch. 2
(2.53)
The observed rate of association, Rob, can be calculated in one of two ways. Viewed from the product, the overall rate of association is equivalent to the rate into the absorbing barrier, hence the rate coefficient is given by, ka = a,lPajgi = a>dPa,,(M-V),.
(2.54) (2.55)
where Pai is the transition probability from state i to the absorbing barrier. An advantage of inversion methods is that they can be very much faster than straightforward eigenvalue methods, an important consideration as the evaluation of each \ 2 point may require the calculation of hundreds of rate coefficients. As with the dissociation methods of Section 2.4.3. the calculated rate depends on co, P(E/Ef) and k(E). In the results discussed shortly co and P{EIE') were calculated in the same way as above. k(E) was calculated using the ILT method the only difference being that this time it is the parameters A00, n° and £T° that are to be determined. In the iso-propyl case the parameters A//$ and (A£) d were floated but for the methyl association scheme these are known from previous experiments and ab initio calculations. The fitting of all three parameters simultaneously would yield a x2 hypersurface which in principle can be explored and minimized just as any other surface. For reasons related to the fitting of the isopropyl data it was found convienent to reduce the number of parameters fitted. This was achieved by prescribing that the high pressure rate coefficient at 300 K which was fixed at 6.0 x 10" 11 cm3 molecule -1 s" 1 , a value that has been accurately measured by a number of groups (see Baulch et al. [28]). A°° was written as a function of ET and n° so that the fixed value of /c00 at 300K was reproduced for any E00 and n°. The \ 2 surface for f00 and n° was then calculated and is shown in Fig. 2.25. As with the isopropyl system the x2 surface has a flat region rather than a well defined minimum and as a consequence it is not possible to assign unequivocal best fit values to the parameters.
187
Radical recombination and association reactions
200 ~
^Oo
\1*~ A50"
'»«
Fig. 2.25. x2 surface for the methyl + methyl reaction as a function of ET and n°.
188
Elementary reactions
Ch. 2
There are a number of reasons for the wide range of acceptable parameter values among which are: • The modified Arrhenius model is incorrect. • The data are drawn from different experiments and may contain inconsistencies. • There are not enough data to be able to extract parameter values. An acceptable fit is obtained using the parameters A°° = 5.4 x 10~ 8 cm 3 molecule" 1 s - 1 , n°° = -1.10 and E°°/R = 160 K and is shown in Fig. 2.22. Over the whole range of acceptable parameter sets, a negative temperature dependence is obtained for &32 • 2.53 CH3 + H->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:
Radical recombination and association reactions
• • • •
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
Elementary reactions
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. <
CO
CD
Q:
0.0
2.5
5.0
7.5
Time / ms Fig. 2.26. Experimental decay traces of (A) H atoms and (B) CH 3 radicals in the study of reaction (31). Total pressure (He) = 100.2 Torr, [CH3COCH3] = 1.06 x 1015 molecule cm - 3 , percentage photolysis = 0.53, Temperature = 504 K. The residuals refer to best fits according to equations (2.57) and (2.48) respectively.
Radical recombination and association reactions
400 600 800 Helium pressure/Torr Fig. 2.27. Fall-off curves for reaction (31) as a function of helium pressure. A, 300 K, O, 400 K, O, 500 K, A,600K.
of N 2 0 at 193 nm in the presence of exeess deuterium: (11) (12a) (42)
N 2 0 + hi/(193 nm) -> 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
Elementary reactions
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
Radical recombination and association reactions
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
Elementary reactions
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-
Radical recombination and association reactions
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
Elementary reactions
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
Radical recombination and association reactions
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
Elementary reactions
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
Elementary reactions
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
Elementary reactions
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 (<2 atm). Keiffer et al. studied the methyl/oxygen system over an extensive range of conditions and developed a global fitting procedure using the Troe formalism [106] to obtain temperature dependent values for the limiting high and low pressure rate coefficient. The values obtained at room temperature are approximately a factor of 2 lower than the directly determined value of Cobos et al. [97]. Pilling and Smith [107] attributed this discrepancy to neglect of radical radical reactions such as (48)
CH 3 0 2 + CH 3 -> 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
Elementary reactions
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
Elementary reactions
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
Elementary reactions
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
Elementary reactions
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
Elementary reactions
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
Elementary reactions
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
<<
0.6
0.4
0.2
^44-4J
0.0 15
16
17
18
log([0 2 ]/molecules c m 3 ) Fig. 2.39. Variation of the A+ value returned from the analysis of OH temporal profiles according to equation (2.67). Inset shows the initially unexpected rapid rise in A+ as the oxygen concentration is lowered.
rate determining step in the route to OH and A+ tends to k62> As the concentration of [0 2 ] is decreased A:6i[02] becomes comparable with both k60 and k-61. This means that a greater fraction of the neopentyl radicals decompose, reducing the yield of OH, but also that the formation of the peroxy radical becomes reversible leading to the observed increase in A+. 2.7.3 Discussion Table 2.9 presents a comparison of the direct and indirect studies of the neopentyl isomerization. The experimental parameter, R, measured by Baldwin et al. [120, 121] and listed in column 2a is K61k62/k60. The direct comparison, shown in column 2b is calculated from the measured Arrhenius parameters for k62, and values of k60 and k61 taken from the studies
19
220
Elementary reactions
Ch. 2
TABLE 2.9 Comparison of direct and indirect determinations of k62 Temperature/K
R/MT cm molecule -1 s - 1
W10~17 molecule cm - 3
wio 4 s-1
A:62/KT2 s" 1
660
2.92
670
4.04
690
(a) (b)
3.87 4.29
5.11 0.407
1.08 1.01
7.67 98.7
700
(a) (b)
3.48 3.23
4.66 0.313
1.49 1.39
10.42 133
715
(a) (b)
2.22 2.15
3.05 0.215
2.37 2.20
16.23 205
730
(a) (b)
1.22 1.46
1.71 0.149
3.69 3.41
24.84 311
750
42.65
of Slagle et al. [34] and Xi et al. [105]. Within the experimental errors the agreement is excellent. Columns 3a and b contrast the values of the equilibrium constants either measured in the direct studies (3a) or estimated from group additivity by Baldwin et al. and used in their calculation of k62 (3b). It can readily be seen that it is this discrepancy (discussed in Section 2.6), rather than the values used for k60 (columns 4a and b) that leads to the differences in k62 (column 5a and b). Baldwin et al. combined their data from neopentylperoxy, ethylperoxy and several other peroxy radicals to give a set of Arrhenius parameters for a variety of intramolecular hydrogen transfers from values of rate coefficients measured at 753 K. According to transition state theory the A factors for hydrogen transfer will decrease with increasing ring size due to the increasing loss of entropy on formation of the ring intermediate. The formation of the ring restricts the free internal rotations that can occur in the parent peroxy radical. Baldwin et al. estimated that the loss of entropy is about 1 7 J m o l _ 1 K _ 1 per internal rotation restricted in the transition state. Carrying this estimate through the transition state calculation they
221
Peroxy radical isomerization TABLE 2.10 Arrhenius parameters for H atom transfer reactions Reaction
Type
A (per C—H E bond) /kJ mol
CH 3 CH(0 2 )CH 2 CH 3 -> CH 2 CH(0 2 H)CH 2 CH 3 CH 3 CH 2 0 2 -» CH 2 CH 2 0 2 H (CH 3 ) 3 CCH 2 0 2 -* CH 2 C(CH 3 ) 2 CH 2 0 2 H (CH 3 ) 2 CHCH 2 0 2 -> CH 2 CH(CH 3 )CH 2 0 2 H CH 3 (CH 2 ) 3 0 2 -* CH 3 (CH 2 ) 3 0 2 H CH 3 (CH 2 ) 4 0 2 -> CH 3 (CH 2 ) 4 0 2 H CH 3 CH(0 2 )CH 2 CH 3 -» CH 3 CH(0 2 H)CHCH 3 CH 3 CH(0 2 )(CH 2 ) 2 CH 3 -* CH 3 CH(0 2 H)CHCH 2 CH 3 CH 3 CH(0 2 )(CH 2 ) 2 CH 3 -> CH 3 CH(0 2 H)CH 2 CHCH 3 CH 3 (CH 2 ) 4 0 2 -> CH 3 CH(CH 2 ) 3 0 2 H (CH 3 ) 2 CHCH 2 0 2 -^ (CH 3 ) 2 CCH 2 0 2 H
1,4 1,4 1,5 1,5 1,6 1,7 1,4 1,4 1,5 1,6 1,4
1.41 1.41 1.76 1.76 2.20 2.75 1.41 1.41 1.76 2.20 1.41
p p p p p p s s s s t
x x x x x x x x x x x
1012 1012 1011 1011 1010 109 1012 1012 1011 1010 1012
155.1 154.1 122.7 124.9 104.9 92.5 137.5 133.2 108.7 90.7 117.9
estimate that the A factor reduces by a factor of 8 per C—H bond as the transition state increases by one carbon atom. Baldwin et al. qualified the absolute accuracy of their results by noting that the calculated values, as we have seen, are dependent on values of alkyl radical decomposition rates and the alkyl/alkylperoxy radical equilibrium constant. The work of Hughes et al. [122] now means that the data of Baldwin et al. can be appropriately scaled to give the results shown in Table 2.10. The activation energies determined by Hughes et al. are approximately 10 kJ mol - 1 greater than those estimated by Baldwin et al. and this affects estimates of the ring strain occurring during the formation of the cyclic transition state. The overall activation energy for an H atom transfer consists of two components: the activation energy for the abstraction of an H atom (primary, secondary or tertiary) by the peroxy group, which can be estimated from the corresponding H 0 2 activation energies (85.6 kJ mol - 1 for primary abstractions), and the ring strain. Using the activation energies in Table 2.10 it is possible to estimate values for the ring transition state strain energy of reactions involving primary H atoms as follows: eight membered (1,7 p) 6.9 kJ mol - 2 1 , seven membered (1,6 p) 19.3 kJ mol - 1 , six membered (1,5 p) 38.2 kJ mol - 1 , five membered (1,4 p) 69 kJ mol - 1 . Using the earlier lower estimates Baldwin et al. postulated that the residual strain energy in 1,7 transfers was negligible and hence that the measured activation energies for 1,7 p,s and t transfers
222
Elementary reactions
Ch. 2
corresponded only to the H atom abstraction activation energy. However, the more recent determinations predict that even the 7 and 8 membered rings have some ring strain energy. Further evidence for the validity of the approach adopted by Baldwin et al. and Hughes et al. comes from a comparison of activation energies for the formation of the same sized rings but formed by either primary, secondary or tertiary hydrogen abstractions. According to the 'two component' hypothesis advanced above the ring strain component should be identical in both cases and this is borne out by the following observations: £(l,4s) - £(l,5s) = 26.7 kJ mol - 1 £(l,4p) - £(l,5p) = 30.8 kJ mol - 1 £(l,5s) - £(l,6s) - 18-OkJmor 1 £(l,5p) - £(l,6p) = 18.9 kJ mol" 1 The data obtained by Hughes et al. gives an indication of the level of detailed knowledge that can be gained from direct studies of appropriate chemical systems and also the complementary nature of direct and indirect studies. However, even in these ideal reactions the chemistry can rapidly become increasingly complex and this may represent the limits of direct studies.
2.8 THEORETICAL AND DYNAMICAL STUDIES OF THE HYDROGEN/OXYGEN SYSTEM
2.8.1 Introduction Figure 2.36 shows the rate of oxidation vs temperature for propene. We have seen that the minimum in this figure is related to the shifting of the R + 0 2 equilibrium back towards reactants and an alternative mechanism for alkyl oxidation. As the temperature is increased further, the break down of large hydrocarbon radicals becomes very rapid and the main propagating species are H, Q and C 2 radicals. Hydrogen atoms play an increasingly important role in the overall combustion process and therefore
223
Theoretical and dynamical studies TABLE 2.11 Basic reaction mechanism for the H 2 / 0 2 reaction Reaction
Reaction type
A/cm3 molecule s _ l a
El kJmoFla
(66)H 2 + 0 2 ^ 2 0 H (67) OH + H 2 - * H + H 2 0 (68)H + 0 2 - + O H + 0
Initiation Propagation
2.8 x 10" 11 4.2 x 10" 11 3.8 x 1(T10
200 22 70
Branching (69)0 + H 2 ^ O H + H (70)H + O 2 + M - + H O 2 + M
4.8 x 10" 11 5.8 x 10" 3 O r _ 1
42 b
c 1.0 x 10" 12
c 77
3.0 x 10" 11 1.0 x 10" 7 d
4 111
Termination (71-73) H, O, OH, -> wall (74)H0 2 + H 2 - * H + H 2 0 2 Restore the chain at high temperature (36)2H02->H202 + 0 2 (37)H202^20H a
Several of the reactions, and especially those with low activation energies, have rate coefficients with a complex temperature dependence. The data [28] given above are approximate and should only be used at temperatures reasonably close to 800 K. b Third order reaction, with units cm6 molecule -2 s _ 1 . Note the negative temperature dependence. Very dependent on the nature of M. c The rates of these reactions depend on diffusion coefficients (and hence on pressure) and on the vessel dimensions and surface characteristics. d Unimolecular reaction in the second-order regime, hence the large A factor.
much can be learnt about high temperature hydrocarbon combustion from the study of the hydrogen/oxygen system. Table 2.11 lists the most important reactions in the hydrogen oxygen system. Reactions (68) and (69) are chain branching and so the system is potentially explosive. However, the pressure dependent reaction, (70), is in direct competition with the H + 0 2 chain branching reaction leading to a complex pressure and temperature dependence of the explosion limits as shown in Fig. 2.40. Considerable experimental attention has been devoted to the hydrogen/oxygen system, not only because of its practical importance but also because the simplicity of the reactions involved has attracted the attention of both the dynamics and theoretical communities who are looking for simple (i.e., small numbers of relatively light atoms) yet relevant reactions
Ch. 2
Elementary reactions
third limit °: 400 V-<
ignition
00
5 700 800 ambient temperature, T/K Fig. 2.40. Pressure temperature diagram of the hydrogen/oxygen reaction.
as test beds for models of bimolecular reactions. In the following section we shall examine some of the more recent theoretical and dynamical studies relevant for combustion chemistry. 2.8.2 State specific rate coefficients So far our discussion has focused on the measurement of thermal rate coefficients for a number of different classes of reactions involved with low temperature combustion. A significant proportion of the kinetics community is interested in looking beyond thermally averaged rate coefficients (k(T)) towards quantum state specific data kvi, the rate coefficient for reagents in the quantum state v,/. The two quantities are related via equation (2.71)
k(T) = HPvjkvjJ v
(2.71)
j
where PVJ is the fraction of molecules in quantum state v,/ in a Boltzmann distribution. The aims of such studies are diverse and do not appear to significantly overlap with, say, the aims of the low temperature combustion modeller. It is, however, worth briefly reviewing the aims of such studies,
Theoretical and dynamical studies
225
how they contribute to our present understanding of reaction kinetics and what information may be available in the near future. The main goals of dynamical studies are to develop theories for chemical reactions and gain an insight into the molecular mechanisms of elementary reactions. The value of a validated and workable theory for reaction kinetics is obvious, rate coefficients would no longer have to be measured but could be calculated for any set of conditions. More specifically, good models of a specific elementary reaction, validated against dynamical data, allows rate coefficients to be calculated at any temperature. 2.8.3 Theoretical treatment of bimolecular reactions Abstraction reactions are typical bimolecular gas phase processes and as such can be treated using standard theoretical tools for such systems. Broadly speaking the theoretical treatment of bimolecular reactions can be divided into two categories: dynamical and statistical. These categories can be further sub-divided into techniques that are based on quantum or classical mechanics. Of the dynamical techniques available the most rigorous and informative are the quantum mechanical dynamics methods. These methods are, however, the most sophisticated and computationally intensive to employ. Two of the most widely used quantum dynamics techniques are quantum scattering (QS) [35] and wavepacket (WP) [125] analysis. In the quantum scattering approach the collision is modelled as a plane wave scattering off a force field which will in general not be isotropic. Incident and scattered waves interfere to give an overall steady state wavefunction from which bimolecular reaction cross-sections, cr, can be obtained. The characteristics of the incident wave are determined from the conditions of the collision and in general the reaction cross-section will be a function of the centre of mass collision velocity, w, and such internal quantum numbers that define the states of the colliding fragments, represented here as v and j . Once the reactive cross-sections are known the state specific rate coefficient, kvj, can be determined from,
k vj =
ucr(u:vj)f(u)du
(2.72)
226
Elementary reactions
Ch. 2
where/(w) is the Maxwell-Boltzmann velocity distribution. The canonical rate coefficient is obtained from kvj by the canonical average given by equation (2.71). The complexity and computational demand of crosssection calculations escalates rapidly with the number of degrees of freedom of the system and much effort is presently being expended to find accurate but rapid approximations to full scattering calculations. In WP analysis the time evolution of an initial wavefunction (or wavepacket) is obtained by the solution of the appropriate time-dependent Schrodinger equation. The initial wavefunction is determined by the conditions of the collision. The Schrodinger equation is then integrated, which given the complexity of the potentials usually has to be performed numerically. Information about the crossections can be obtained from this technique and again canonical rate coefficients obtained by the above averaging procedure. Apart from the complex nature of these techniques and their intense computational demands a detailed knowledge of the potential energy surface is required, which is in itself a demanding problem for ab initio quantum chemistry. As a consequence these techniques are limited in the systems to which they can be applied - rigorous calculations involving four atoms have only recently started to appear - and so direct application to reactions of importance in combustion is not possible at present. One important exception is the reaction, (67)
OH + H 2 - > H 2 0 + H
which has recently been tackled by Clary et al. [126] and Bowman et al. [127] For larger systems relevant to combustion the rigorous quantum dynamical approach must be abandoned. If the quantum mechanical effects such as tunnelling can be ignored then classical trajectory techniques can be employed. The basis of this technique is the modelling of a reactive collision as a classical trajectory moving across the PES that describes the reaction. The trajectory is obtained by integrating Hamilton's equations of motion for the system, which requires the specification of initial conditions for the coordinates and conjugate momenta that describe the system. As with the quantum mechanical treatment the quantity of principle interest is the reactive cross-section and to obtain this a large number of trajectories have to be run with initial conditions drawn from a statistical
Theoretical and dynamical studies
227
distribution. The initial conditions are subject to certain constraints: energy, angular momentum and a variable called the impact parameter are fixed. The impact parameter, 6, is the smallest distance that two fragments would have during a collision if there were no interaction potential. The probability that reaction will occur, Pr9 is a function of the impact parameter. In classical trajectory studies the probability of reaction is defined as the ratio of reactive trajectories to the total number of trajectories run. The reaction cross-section in terms of Pr is given by [124], f b max
a{u\ v,y) =
Pr(u, b, vj)b db.
(2.73)
Jo Having obtained the cross-section the canonical rate coefficient can then be determined by using equation (2.71). The approach is valuable for systems that contain heavy atoms whose dynamics are well approximated by classical mechanics. Classical trajectories are relatively easy to calculate on modern machines and so this approach has the advantage of being much faster than the quantum dynamics methods. However, as with the quantum dynamics methods, it still requires a detailed PES. Furthermore, as abstraction reactions in combustion systems exclusively involve H-atoms, corrections for quantum mechanical tunnelling are required. Though dynamical techniques yield a great deal of information about state specific processes, a lot of this detail is eliminated when individual micro-canonical rate coefficients are averaged to give thermal rate coefficients. It is this averaging which forms the basis of the so-called statistical theories, the most important being transition state theory (TST) which was discussed in Section 2.3.3. The application of TST theory falls into two categories, scattering reactions and association reactions. The latter have been discussed in previous sections. The former are usually characterized by a well defined barrier along the reaction coordinate, the transition state being located at the top of this barrier. Current developments, however, show even that in these cases, this may not be the best location of the transition state. For reactants whose total energy is close to the reaction threshold this is indeed a good location. However, for reactants whose energy is rather more than the threshold energy the best location of the TS is at the point where the reactive flux is at a minimum (cf. association reactions). These flux minima
228
Elementary reactions
Ch. 2
tend to occur in the vicinity of the the PES saddle point. The location of the TS is thus a function of the total energy of the reactants and has led to a number of theories that can be broadly grouped under the title of variational transition state theory (VTST). Truhlar and co-workers [128] have examined the VTST theory at the microcanonical level. At the microcanonical level the rate coefficient is given by,
««*>-S
(274)
-
where G{E\ R) is the sum of states at the transitions state (also referred to as the accumulated reaction probability for energy E), N(E) is the density of states of the reactants and h is Planck's constant. G, and consequently k, are functions of the reaction coordinate value and it is G that has to be minimized with respect to the reaction coordinate. Note the similarity with the equivalent association/unimolecular formulation. The canonical rate coefficient can be obtained by taking the canonical average
oih(E\R). Transition state theory can also give us some insight into the nonArrhenius behaviour of rate coefficients as epitomized by Fig. 2.6 for the OH + ethane reaction. Curvature of the Arrhenius plot can arise from a number of factors. (a) The partition functions in the pre-exponential term of equation (2.11) are temperature dependent. As the temperature dependences of the various degrees of freedom (translation T 3/2 , rotation, T (linear), T3/2 non differ and there are linear, vibrational Qv = [1 - exp^hv/k^T)]'1) changes in the degrees of freedom moving from the reactants to the transition state, the ratio of partition functions {QTSIQAQB) will be temperature dependent. (b) Equation (2.71) shows that the overall rate coefficient is the weighted sum of the state specific rate coefficients for reaction from various quantum states. It is easy to envisage that these state specific rate coefficients could vary dramatically, even between various quantum states which have the same overall total energy. For example we would intuitively expect an abstraction reaction to Iproceed significantly faster if vibrational energy is located in the bond to be broken (this is indeed the case and we discuss some specific examples in Section 2.8.4). As the temperature is
Theoretical and dynamical studies
229
increased the relative population of vibrationally excited reagents, which react faster than their vibrationally cold counterparts, increases and hence the overall rate coefficient curves upwards with increasing temperature. (c) For hydrogen abstraction reactions, where a light particle is being transferred between two molecules, tunnelling of the hydrogen atom wavepacket through the activation barrier can be a significant contribution to the overall reaction, especially at lower temperatures. The relative tunnelling contribution decreases as the temperature is raised once again leading to deviations from Arrhenius behaviour. 2.8.4 Reagent excitation A common goal in reaction dynamics is to investigate the effects of specific reagent excitation on the rate or dynamics of a reaction. Translational excitation, especially of hydrogen atoms, can be produced by laser photolysis, e.g., (75)
HI + hv(l93 nm) -> H + I + 320 kJ mol" l
Conservation of momentum dictates that the H atom must carry over 99% of the energy released resulting in an initial velocity of 22 km s _ 1 (In fact the H atom velocity distribution is bimodal as the iodine atoms can also be formed in the first excited state). Wolfrum and coworkers [129, 130] have carried out a number of hot H atom studies for example the endothermic reactions (76)
H + H 2 0 - » O H + H2
(77)
H + C 0 2 - + O H + CO
The reverse reactions are crucial steps in combustion liberating a significant proportion of the overall reaction exothermicity. The effects of vibrational excitation have been studied by a number of groups. As might be expected reagent excitation is most effective when the bond to be broken is excited. For example a thousand fold increase in k7S is observed when the HC1 bond is vibrationally excited by a pulse
230
Elementary reactions
Ch. 2
from an IR HC1 laser [131]: (78)
O + H C l - ^ O H + Cl
Work is currently in progress to extend these experiments to polyatomic reagents. These are significantly harder experiments to perform and interpret as intramolecular energy transfer can remove the energy from the target bond. Conversely vibrational excitation of the OH reagent in reaction (10) has virtually no effect on the rate coefficient, the energy is located in a 'spectator' bond, not in one of the C—H bonds that is to be broken. In a series of elegant experiments Crim and coworkers [132, 133] have managed to effect bond selective chemistry for reaction (79) (79a) (79b)
H + HOD -* H 2 + OD HD + OH
By depositing 4 quanta of vibrational energy in either the OD or OH local mode they were able to control to a high degree whether the OD or OH bond is broken.
References [1] S.W. Benson, Thermochemical Kinetics (Wiley-Interscience, New York, 1976). [2] N.D. Stothard and R.W. Walker, J. Chem. Soc, Faraday Trans. 87 (1991) 241. [3] R.R. Baldwin, A.R. Fuller, D. Longthorn and R.W. Walker, J. Chem. Soc, Faraday Trans. I, 70 (1974) 1257. [4] C.A. Morgan, M.J. Pilling, J.M. Tulloch, R.P. Ruiz and K.D. Bayes, J. Chem. Soc, Faraday Trans. II, 78 (1982) 1323. [5] F.P. Tully, M.L. Koszykowski and J.S. Binkley, Symp. (Int.) Comb. 20 (1984) 715. [6] F.P. Tully and J.E.M. Goldsmith, Chem. Phys. Lett. 116 (1985) 345. [7] F.P. Tully, A.T. Droege, M.L. Koszykowski and C.F. Melius, J. Phys. Chem. 90 (1986) 691. [8] W.P. Hess and F.P. Tully, J. Phys. Chem. 93 (1989) 1944. [9] G. Hancock, Measurements of Thermal Rate data, in: Modern Gas Kinetics, eds M.J. Pilling and I.W.M. Smith (Blackwells, Oxford, 1987). [10] R. Atkinson, W.P.L. Carter, S.M. Aschermann, A.M. Winer and J.N. Pitts Jr, Int. J. Chem. Kinet. 16 (1984) 469.
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Chapter 3
Kinetics Databases D.L. BAULCH
3.1 DATA FOR COMBUSTION MODELLING
3.1.1 Modelling combustion chemistry Computer modelling of large-scale chemical processes has long been a major goal for scientists and technologists since it provides rapid and cost effective access to data essential for the design, development and control of a wide range of chemical systems. Great progress has been made as the power of modern computers has increased and more efficient software has developed to the point where the use of computer simulation of processes such as pyrolysis, ignition and combustion is now widespread. It is possible to predict properties of combustion systems such as induction times [1], chemical composition [1] and ignition properties [2] and to model pollutant abatement processes such as thermal De-NO x and RAPRENO x [3]. This information is wide ranging and can be obtained relatively cheaply and rapidly. Thus, it can offer data for the design and testing of new combustion systems where experiment is difficult or impossible, and in favourable cases the response time of the simulations may be sufficiently rapid for it to be used "on-line", providing feedback to control the process being monitored. Nevertheless, a number of problems remain in modelling both the physical processes (fluid dynamics, heat transfer), the complex chemistry and the coupling between them. There are major limitations on the applicability of computer models and in the accuracy which simulations can achieve. It is of the utmost importance for model users to be aware of the main sources of error; blind belief in the output from models can be dangerous and expensive. In this chapter we consider the major source of uncertainty in chemical simulations, whether full or reduced mechanisms are used, namely the quality and quantity of the available kinetics data.
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The present volume deals with the low-temperature oxidation of hydrocarbons but in this chapter it is necessary to consider a much wider range of combustion conditions and to draw upon information drawn from the chemistry of a broad range of compounds. The two main reasons for this are, first, the experimental methods used to obtain data required to simulate low-temperature oxidation often operate at temperatures and pressures well removed from those of the oxidation processes. Second, in trying to assess the quality of data for the relatively narrow range of low-temperature oxidation conditions, it is necessary to consider all of the experimental results available over the whole range of temperatures and pressures accessible. It also follows from these points that it is necessary to extrapolate and interpolate experimental results to conditions relating to low-temperature oxidation. Such techniques are an important aspect of evaluation and are also considered in this chapter. 3.1.2 Data needs for modelling The basic data required for simulation of the complex chemistry of oxidation of the simplest hydrocarbon potentially involves many hundreds of possible elementary reactions. Many can be eliminated as unimportant by relatively straight-forward chemical considerations but, in the case of a hydrocarbon of even modest size, several hundred are still likely to remain. All techniques for further reducing this number depend in some way on a knowledge (measured or estimated) of the rate parameters of the reactions (Chapter 4). However, of the many hundreds of reactions possible in the wide range of practical combustion systems, probably no more than about 300 have been studied in sufficient depth to make their measured rate parameters reliable input into a model. Taking this at its face value the task of obtaining the required data appears to make modelling combustion hardly worthwhile, but the situation is mitigated to a degree by a number of factors. First, the mechanisms of combustion, in air or oxygen, of compounds containing C, H, N, O (most fuels), have many reactions in common. Thus, when mechanisms used for modelling methanol [4] and ethylene [5] combustion are compared, we find that all of the 83 methanol reactions are included in both and only an extra 10 reactions are added to model the ethylene system. Second, for larger compounds the number of reactions in the model
Data for combustion modelling
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may become very large e.g., 760 have been used to model iso-octane combustion [6]. The experimental kinetic data available for reactions of these larger species, and the radicals derived from them, is extremely meagre but it becomes possible to use estimation techniques, based on relating rate parameters to structural features of the reacting molecules, to calculate rate constants for reactions of the larger species from those of their smaller homologues (Section 3.4). Third, the chemical and physical properties of combustion systems are more sensitive to the rate parameters of a relatively small number of reactions in the mechanism than the many others participating. There are well established computational techniques available to identify these core reactions and to characterize the degree of sensitivity ('sensitivity analysis', Chapter 4). The combustion conditions, particularly the ratio of fuel to oxidant, will affect the sensitivity so that different reactions may predominate under different circumstances, and even as the reaction progresses, with changing temperature and pressure, we can expect the importance of different reactions to wax and wane. Nevertheless, identification of such key reactions remains feasible, enabling the experimentalist to concentrate on the determination of their rate parameters. Even with these redeeming limitations on the number of key reactions in the mechanism the amount of data required is formidable and in excess of what is currently available. It should also be remembered that to establish that a reaction plays only a minor role in a mechanism some knowledge, albeit imprecise, of its rate parameters is required. However, as our knowledge of combustion chemistry grows, mechanisms are becoming better defined, key reactions are being identified and becoming clearer targets for the efforts of experimentalists and theoreticians. 3.1.3 Other uses of kinetic data in combustion This chapter focuses on data required for direct use in computer simulations but a wider range of kinetic data also has an indirect but important impact on combustion modelling. The most obvious of these other uses lies in the methods for obtaining kinetic data experimentally (Section 3.2). For example, in shock tube measurements, which are the most reliable source of high-temperature data, there are often several interfering or competing reactions. It is necessary to model the overall behaviour, using existing kinetic data, to obtain
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information on the target reaction. Even more direct use is made in relative rate constant measurements where rate constants are determined relative to those of a reference reaction, which may not be relevant in combustion, but the kinetics of which must be known as accurately as possible. In Section 3.4 we discuss the use of estimation methods for obtaining rate data for reactions which have not been studied experimentally. One method is to make use of data on reactions chemically analogous to the combustion reaction of interest. The greater the quantity and the better the quality of the available ancillary data the easier it is to make this kind of extrapolation. Similarly, in critically assessing the quality of the data to be used in modelling it is often necessary to draw on data from reactions outside combustion. The extreme form of the estimation procedures is ab initio calculation from theory. There is every reason to hope that improvements in theory and in computational methods will eventually reach the stage where such data can be used confidently in models, and even at its present level of accuracy such data are useful for checking experimental results. However, further progress in theory will depend on validation by experimental kinetics data, drawn from a wide range of sources. Finally, mention must be made of the contribution made by kinetics measurements to the provision of thermodynamic data. Thermodynamic data are vital ingredients in all combustion models. They originate from a variety of sources but the data on heats of formation of many of the radical species come most reliably from measurements of their reaction rates.
3.2 PRIMARY SOURCE OF KINETIC DATA: THE NEED FOR EVALUATION
3.2.1 Experimentally measured rate constants Experimentally measured values reported in the literature provide the primary source of kinetic data for the modeller. However, this information may be widely scattered and of variable quality. This has generated a need for bibliographies, reviews and critical assessments of the reported data to aid scientists and technologists who are not expert in chemical kinetics. In this section we review the main limitations of the primary data indicating how the need for compilation and critical evaluation has arisen. Combustion processes span a wide range of temperatures (700-3000 K)
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and pressures (0-30 atm). Furthermore, as will be seen later, rate-constant values obtained at much lower temperatures are often required to interpret the high-temperature measurements. No single experimental technique is capable of measuring rate constants over this wide range of conditions. The major techniques used are shock tubes, pulsed photolysis combined with kinetic spectroscopy and fast-flow discharge methods. These three techniques are broadly complementary, operating over different ranges of pressure and temperature and together covering a wide range of both. Their operation has been described elsewhere in this volume; their advantages and limitations, insofar as providing rate data for elementary reactions, is considered in the following sections, together with a number of other less widely used methods. (/) The chemical shock tube Shock tubes (Chapter 6) provide rate constant values at temperatures above 1000 K and over a substantial pressure range. Thus, they provide rate data determined at temperatures and pressures directly applicable to combustion. The limitations associated with their use arise largely from the problem of 'isolating' the particular elementary reaction that it is desired to study. There are two main factors at work affecting this isolation. First, it is necessary to have a 'clean' source of the radical (or radicals) under study, i.e., a compound present which will undergo rapid thermal decomposition under the conditions of the experiment. Second, it must be possible to choose conditions where the reaction in question dominates the rate of removal, or production, of the radical being monitored. These two conditions can rarely be achieved without introducing a degree of chemical complexity and hence potential sources of error. At the high temperatures being used there are usually a number of competing chemical processes which affect, to a greater or lesser extent, the radical production and the progress of the reaction under study. Therefore, it is important to know enough about the chemistry of the system to be able to correct for the effects of this complex chemistry, which, as indicated (Section 3.1.3), may call on data for reactions which have no direct relevance to combustion. The desired rate data are usually extracted from the measurements by fitting the observed temporal changes in concentrations to an assumed chemical model, varying the rate constant of the reaction under study to achieve the best fit. If it is possible to
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choose conditions where the desired reaction dominates the change in concentration of the species being studied i.e., the reaction is 'isolated' from the rest of the chemistry in the system, then the corrections stemming from the other concurrent reactions may be quite small and this, in essence, amounts to an absolute measurement of the rate constant. However, this is not always possible; competing processes may be significant and under these circumstances the analysis of results will be sensitive to the chemical mechanism employed and the quality of the data used to model it. The final result derived is then virtually a relative rate constant measurement, critically dependent on the other data. In a few instances reactions of a radical with a mixture of two compounds has been used deliberately to yield relative rate data [7]. The need to achieve conditions in which the desired reaction is isolated usually limits the range of temperatures and pressures that can be covered in a particular study but, with modern signal capture and processing methods and advances in modelling the complex chemistry involved, the quality of data from shock tube studies is comparable with that from the lower temperature techniques. (ii) Pulsed photolysis/kinetic spectroscopy This technique has been described fully in Chapter 2. It was developed for operation at temperatures close to ambient but in recent years determined attempts have been made to extend this range. It is now routinely operated from sub-ambient temperatures up to —1000 K and a number of studies at even higher temperatures have been reported [8]. It has the great advantage of being able to cover a very wide pressure range from a few Torr up to many atmospheres but most studies have been carried out in the range 10-760 Torr. Insofar as combustion is concerned, all of the data that have been obtained relate to temperatures well below those achieved in flames. The ignition region is covered by some of the studies, but most of the data refer to temperatures below 500 K. Despite that, as will be seen later, to be able to assess the quality of data for a particular elementary reaction often requires values of the rate constant over as wide a temperature range as possible, making the results from pulsed photolysis studies invaluable. The same problems of interfering chemistry, as encountered in shock tube studies, may also occur in pulsed photolysis systems but to a lesser degree. The temperatures involved are lower and the radical sources are
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photolytic, and hence, largely independent of the temperature used. Photolysis may produce other unwanted species, either from the source compound or other compounds in the system, but such complications can often be minimized by appropriate choice of photolysis wavelength or by using a variety of source molecules. In general, it is much easier to isolate the reaction studied than is the case in shock tube work. The optical techniques used to monitor the progress of the reaction may also encounter problems of sensitivity, collisional perturbation of the detection signal and products absorbing or emitting. Despite these potential difficulties, modern studies typically produce results with a precision of 10% (lor) in most instances. This precision, combined with the ability to isolate the reaction of interest, makes it potentially an extremely accurate technique. It is the most flexible and prolific technique for kinetics measurements, especially for the determination of pressure-dependent rate constants. (Hi) Discharge-flow methods This technique is described in Chapter 2. Like pulsed photolysis methods it was developed to operate at temperatures close to ambient. Similarly, in recent years its temperature range has been extended to cover much the same regime (sub-ambient up to —1000 K) as pulsed photolysis, but most studies are still confined to <500 K. Efforts are being made to extend this range and studies are beginning to appear at temperatures in the range 1000-2000 K [9]. Unlike pulsed photolysis the technique is most convenient to use at relatively low pressures (0.1-10 Torr). In a very few instances, studies have been made at higher pressures (up to 380 Torr) but, in this region, the fluid dynamics are more complex and operation of fast-flow systems at high pressures consumes large amounts of carrier gas making the measurements very expensive [10]. The difficulties which arise in interpreting discharge-flow measurements are rather similar to those encountered in pulsed photolysis studies. These are mainly associated with the chemical complications of interfering reactions but, as well, wall reactions may be important. Because the generation of the reactive radicals is spatially separated from the reaction zone, source complications associated with radical sources are rather less than is the case in pulsed photolysis. The discharge-flow technique also has some advantages in detection. It is a steady state system, making extended periods of observation and information gathering possible, and making feasible detection methods alternative to the spectroscopic methods usually
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employed with this and other techniques. In particular, mass-spectrometry [11] can be used to identify and monitor the concentration of a variety of species in the same system, some of them e.g., large hydrocarbon fragments, that would be difficult to handle by spectroscopic means. This has advantages in mechanistic studies quite apart from the measurement of particular rate constants. Similar to pulsed optical techniques, the precision of rate constant determinations is within 10% (lcr) in favourable cases, but systematic errors usually reduce the accuracy slightly. Overall the technique has not been as prolific as pulsed photolysis methods in producing rate constant data for combustion reactions but, in some instances, it is applicable where pulsed photolysis is not, and the two provide a valuable check on the results from each other. (iv) Other techniques There are a number of other techniques used to generate data for modelling combustion processes but none have the general applicability of the three techniques discussed previously. At high temperatures, flame studies have been a useful source of data. Values of rate constants of participating reactions are obtained by monitoring the concentrations of as many species as possible and then fitting the results to an assumed mechanism by computer modelling. In this respect, the data analysis is rather similar to some shock tube studies. However, in general, the difference is that there is much less control possible over the flame chemistry, which is usually very complex and not understood in detail. This introduces considerable uncertainty into the accuracy of the modelling and leaves residual doubts as to the reliability of the rate constants derived. Furthermore, there is much less control over the range of temperatures and pressures that can be covered. Although flames are still sometimes used to derive kinetic data, in general they have largely been superseded by shock-tube studies. The technique based on studies of H 2 /0 2 mixtures close to the second explosion limit described in Chapter 1 has also proved to be an extremely useful source of rate data. Measurements of the relative rate of consumption of hydrogen and an organic compound added to the mixture in trace amounts gives rate constants for the reactions of H and OH with the additive and provides information on the reactions of the radicals derived from such additives. Although the technique is limited to operating over
Primary source of kinetic data
243
a limited pressure range and over a very small temperature range close to 770 K, this happens to be a temperature region inaccessible to shock tube studies and rather higher than that usually used in fast-flow discharge and pulsed photolysis techniques. In a similar manner, product analysis studies on the thermal reactions of mixtures of organic compounds and oxygen have yielded data on the oxidation of a radicals in the temperature range 400-900 K. These techniques are described in Chapter 1. At temperatures close to ambient, steady-state photolysis experiments have produced a substantial amount of relative rate data. Most of this work has been aimed at reactions of importance in atmospheric chemistry but much of it is also relevant to low-temperature oxidation of hydrocarbons. 3.2.2 The quality of kinetic data In the previous sections we have reviewed very briefly the main features of the techniques used to obtain kinetics data for modelling combustion processes, emphasizing the factors limiting their range and the quality of data derived from them. The same factors are among those that must be considered when the data are being assessed. The experimental techniques reviewed are potentially capable of producing good precision rate data from sub-ambient temperatures to 3000 K and beyond, and of covering a wide pressure range. Despite that, the quality of the data available for modelling leaves a lot to be desired. The main problems encountered are: (/) Absence of data There are many reactions which, on good chemical grounds, are postulated to occur in combustion processes but for which there are no measured rate data. In some cases this may be due to the lack of a convenient source of the species involved but often this is not so. Techniques for studying the reactions exist but the financial resources have not been made available to do so. This same resource limitation often applies also to extending existing measurements into regions of temperature and pressure pertinent to combustion processes. When experimental data are lacking recourse must be made to theory and estimation methods (Section 3.4).
244
Kinetics databases
Ch. 3
(ii) Systematic errors in the measurements: scatter Although a standard deviation of about 10% is now quite usual for rate constant measurements from fast-flow discharge and pulsed photolysis studies, it is still found that the same reaction studied in different laboratories by the same technique may give results of similar precision, but which differ by far more than would be expected on the basis of that precision. Potential sources of such discrepancies are many but, with experience, likely errors can sometimes be identified in particular cases. For example, reactions between short lived radical intermediates are common in combustion processes. The measurement of the rate constants of such reactions, where the reaction is second order with respect to the transient species, pose particular but well recognized difficulties stemming from the need to determine the absolute concentration of the reacting radical (Chapter 1). This is difficult to achieve and has been a common source of error in this type of determination as exemplified in the series of studies on the rate of the reaction CHO + CHO -* CH 2 0 + CO. In the recent measurements of the rate constant of this reaction at 295298 K the values obtained for 1011 A:/cm3 molecule -1 s" 1 are (3.0 1.2) [12], (2.3 0.5) [13], (3.4 0.9) [14], (4.5 1.7) [15], (7.5 2.9) [16]. In some cases the error limits quoted are based solely on the reproducibility (precision) of the measurements and, in others, an attempt has been made to allow for other sources of error. However, it is clear from the differences in the values of k that in estimating the limits not all errors have been accounted for. The differences do not correlate with the various pressures used, which is sometimes a source of such discrepancies, but appear to be related to errors in the values of the optical absorption coefficient used in measuring the concentration of the CHO radical. In most of the studies optical detection methods were used and, in these cases, very similar experimentally measured values of k/s were obtained, where e is the optical absorption coefficient of CHO at the wavelength used. Differences in the values used for e account for the differences in the quoted values of k [16]. In this case the likely source of error can be identified fairly readily; other cases are far less transparent and the reasons for scatter outside the quoted error limits can often not be identified. It should be noted that this kind of discrepancy used to be extremely common but is now far less so. The quality of rate data has improved enormously in recent years, both in terms of precision and in concordance among studies from different laboratories. This is largely due to improve-
Primary source of kinetic data
245
ments in the equipment for making kinetics measurements among the most important of which are the use of lasers for both the production and detection of species, better light detection and signal processing systems, and the use of computer modelling to check the chemical assumptions used in setting up the experiments and analysing the data. (Hi) Limited coverage of the relevant pressure and temperature range Fast-flow discharge and pulsed photolysis systems usually operate at temperatures below 600 K while shock tube studies are carried out at temperatures above 1000 K. Thus for many reactions there is a preponderance of data at high and/or low temperatures with much less in the range 500-1500 K. Only a limited number of reactions have been thoroughly studied over a wide temperature range. One such is the reaction OH + C 2 H 6 -» H 2 0 + C 2 H 5 for which there are rate constant values over the range 226-1610 K. However, even in this case it can be seen from the Arrhenius plot (Fig. 3.1) that there are rather fewer measurements between 800 and 1200 K. A more typical example, with far fewer data, is the reaction O + HCHO-* OH + CHO, where the Arrhenius plot (Fig. 3.2) shows no values of k between 750 and 1400 K. This situation is being addressed slowly, mainly by pushing the low-temperature techniques to higher temperatures, but for many reactions much more data are required in this intermediate temperature region. For some reactions the trends observed in the rate constant with change of temperature in the high- and low-temperature regions are readily compatible, the one extrapolating reasonably smoothly to the other as illustrated with the data for CH 3 + HCHO reaction (Fig. 3.3). This is not always so, as shown in Fig. 3.4 where, for the CHO + 0 2 reaction, it can be seen that the exact relationship between the high- and low-temperature data is far from clear. For some reactions this may not be a concern; if the high-temperature data are well defined for combustion conditions this may be all that is required. Nevertheless, if the relationship with the lowtemperature data is not understood there remains the nagging doubt that there may be features of the reaction mechanism persisting at high temperature which are also not understood. The pressure and temperature dependence of rate constants of combination reactions and their reverse decomposition reactions are complex. Measurements over a wide range of both temperature and pressure may be required to fully characterize the rate parameters. For example, atom
246
Kinetics databases
Ch. 3
IO°T~7K~ Fig. 3.1. Arrhenius plot of experimental data for the reaction OH + C2H6—> H 2 0 + C2H5: ( • ) [73]; (V) [74]; (B) [75]; (x) [76]; ( • ) [77]; ( • ) [78]; ( • ) [79]; (A) [80]; (O) [81]; (A) [82]; ( • ) [83]; ( J ) [84]; (O) [85]; (+) [86]; ( • ) [87]; (O) [88]; the rectangle centred on 298 K envelops the range of measurements in this region too numerous to plot clearly.
247
Primary source of kinetic data T/K
-10.0
-
2000
1000
500
I
1
1
1
400
300
1
1
c -io.5 •-
i
r~" -n.o jn
3 u
o
1 -".»
n
0
O
o • 0
J3 -12.0
• o
a•
B0 ~
-12.5
• D
-13.0
•
-13.5
«
1
0.5
1.0
1 1.5
! 2.0
1
L_
1
2.5 1
3.0
3.5
»
1
IOVVK" Fig. 3.2. Arrhenius plot of experimental data for the reaction O + H C H O ^ OH + CHO: (V) [89]; ( • ) [90]; (H) [91]; ( • ) [92]; ( • ) [93]; (O) [94]; ( • ) [95].
248
Kinetics databases
Ch. 3
T/K 2000 T
1000
0.5
1.0
1.5
2.0
loV/K" 1
2.5
3.0
3.5
Fig. 3.3. Arrhenius plot of experimental data for the reaction CH 3 + HCHO-> CH 4 + HCO: ( - - ) [96]; ( • ) [97]; ( — ) [98]; (•••) [99]; ( ) [100]; ( - ) evaluation [36].
249
Primary source of kinetic data T/K 1000 i
2000 1
1
r
r—i
o
S G
500
400
300
1
1
1
1
1
i-i
o
o4
o
©
o -a
Fig. 3.4. Arrhenius plot of experimental data for the reaction CHO + 0 2 -> products: (—) [101]; (H) [95]; (A) [102]; (O) [103]; ( • ) [104]; (O) [105], [106]; (V) [107]; ( • ) [108]; (O) [109]; ( • ) [110]; ( • ) [111]; (•••) [112]; (T) [113].
combination reactions such as H + H + M and O + N + M, are in their third order region throughout the pressure and temperature ranges normally encountered in combustion and the effects of pressure can be described simply in terms of the third body concentration in the rate law i.e., the reactions are in their 'low-pressure region' throughout. The same will be true of some atomic + diatomic systems but for reactions of larger species the effects of pressure leading to 'fall-off in the measured rate constant, and ultimately to a change in the reaction order, must be taken
250
Kinetics databases
Ch. 3
into account. The tendency in combustion processes to move to ever higher operating pressures exacerbates this need. Some idea of the effort required to fully characterize pressure-dependent rate constants can be gained from the data on the reaction CH 3 + CH 3 + M—> C 2 H 6 + M. This reaction has been extensively studied for a number of reasons; it is important in combustion and in pyrolysis processes, it is a widely used reference reaction for methyl radical reactions, it has been the subject of a number of theoretical studies and, together with its reverse, it provides information on the C—C bond strength in ethane. Since 1976 there have been some 23 studies (and about 25 before then) over a temperature range of 200-1800 K and covering a range of gas densities of a factor of 104 for several bath gases [17]. Some of the data relating to the pressure dependence of the rate constant are plotted in Fig. 3.5 for Ar as the 'bath gas'. There are a reasonable amount of data at temperatures relevant to low-temperature oxidation of alkanes but, even there, scarcely enough to satisfy the needs of modellers over the whole range of pressures and temperatures, and the sparsity of data at higher combustion temperatures ( r > 1 5 0 0 K ) is particularly noticeable. Therefore, it is necessary to analyse these data and parameterize the rate constants within the framework of an appropriate model. As explained in Chapter 1, and discussed later, the model usually used is that due to Troe et al. [18]. The combination of methyl radicals is a relatively favourable case. The rate parameters of most pressure-dependent rate constants are based on much less extensive data. It is now realized that the need for studies of the pressure dependence of rates of combustion reactions is much greater than had been appreciated. This arises from the fact that many reactions formally written as bimolecular processes occur, unexpectedly, by an addition mechanism with formation of a short-lived adduct. Such reactions will have rate constants that are pressure dependent although not necessarily in a regime relevant to combustion. For example, the reaction (1)
OH + C 2 H 2 -* H 2 0 + C2H
could proceed by direct H abstraction but, because the energy barrier to addition is so much lower than that for abstraction, the mechanism is much more likely to be addition, isomerization, and decomposition of the
-10.0
r
I
In
r
-3 -u E Q
Q
“s.
-11.0
Y 0
0 -1
-12.0
15
16
17
18
19
20
21
Log([Ar]lmolecule cm-3)
Fig. 3.5. Plot of In k vs bath gas concentration for the reaction CH3 + CH3 + M)zH6+ M experimental data at selected temperatures are plotted. The lines are calculated from the expression recommended by Baulch et al. [17], derived from a fitting to all of the available data: (a) [114]; (W) [115]; ( 0 )[116], [117]; (0)[117]; ( 0 )[117]; (A) [116], [117]; ( 0 )[117]; (v)[116], [117]; (0) [118]; (*) [1181; (A)[1191; ( 0 )[1191.
252
Kinetics databases
Ch. 3
energized intermediate. Such a mechanism will lead to a pressure-dependent rate constant and would present no surprise. In other cases the occurrence of an addition mechanism is not so obvious. Thus it would be reasonable to expect that the reaction (2)
0 2 + CH 2 OH -> H 0 2 + HCHO
might be bimolecular, occurring by direct H abstraction, since this is a well known reaction of 0 2 with other molecules. Such a channel may be operative in some conditions but the experimental results show that the rate constant is pressure dependent and, here too, the mechanism must involve addition. Before leaving this section, one other aspect of the data on pressuredependent rate constants should be mentioned. The rate constants of these reactions are dependent on the nature of the 'third body' in the reaction. For example, the reaction H + 0 2 + M —> H 0 2 + M is of major importance in the H 2 /0 2 systems particularly at the second explosion limit. Water molecules have an extremely high efficiency as 'third bodies' in this reaction being some six times as efficient as N 2 [19]. Thus as the reaction proceeds the product formed increases the rate of this reaction, all other factors being equal. This must be taken into account when modelling the system. Indeed, it is impossible to explain fully the oscillations observed in the H 2 /0 2 reaction near the second explosion limit without acknowledging the exceptional efficiency of H 2 0 in this case. Most pressure-dependent rate constants are measured in kinetics studies using inert gases as 'third bodies'. But air is the usual oxidizing agent in combustion systems, and therefore, the main 'third body' is N 2 , for which few data exist. Furthermore, there is little empirical or theoretical guidance in extrapolating data from inert gases to more complex molecules. (iv) Product identification and multichannel reactions Many chemical reactions occur by more than one reaction channel giving rise to corresponding different sets of products. For example, the reaction between H atoms and H 0 2 radicals can proceed by three paths (3) (4)
H + H02-+20H -*H2 + 0 2
Primary source of kinetic data (5)
253
-> H 2 0 + O
Some of the products here are reactive atoms or radicals which are likely to undergo further reaction in any system in which they are produced, whereas the channel yielding the molecular products H 2 + 0 2 is less likely to propagate further reaction. Therefore, it is important to know the fraction of the reaction occurring by each route over the relevant temperature range i.e., the rate parameters of each of the channels. The problem is particularly acute at high temperatures. At low temperatures relatively small differences in the energy barriers for the possible channels may ensure that one path dominates. At high temperatures not only are data lacking but these energy barriers provide less discrimination because of the greater thermal energy of the reactants and, as a result, the branching ratios may become more comparable. To determine the branching ratios in multi-channel reactions requires measurements to be made of the rates of appearance of the various products. Unfortunately this is usually a far more difficult task than measuring the rate parameters for removal of the reactants. The reasons for this are discussed in Chapter 1; briefly, using pseudo first-order conditions for reactant removal, the overall rate constant can be determined without a knowledge of the absolute concentration of the reactant being monitored, whereas to determine the rate constant for appearance of a product, requires a method for the detection of what may be a small product yield and a means of calibrating the signal monitored, either in terms of the absolute concentration of the product, or its concentration relative to that of the minority reactant. Consequently, few studies have been made of rates of product appearance. The rate constant data for the various channels of the H + H 0 2 reaction are shown in Figs 3.6 to 3.8. The branching ratios have been extensively studied at ambient temperatures because of the importance of the reaction in atmospheric chemistry and are believed to be well known (the results of Keyser [20], which agree with those of Sridharan et al. [21], are usually taken as definitive). However, there are very few studies at higher temperatures and no reliable values above 1000 K. This is not unusual. In most cases there is no information at all for combustion conditions. Current ignorance of reaction pathways in multichannel reactions is possibly the major uncertainty in modelling high-temperature processes.
254
Kinetics databases
Ch. 3
T/K
IO 3 T~VK _ 1
Fig. 3.6. Arrhenius plot of experimental data for the reaction H + H02—> OH + OH: (A) [120]; (H) [121]; ( • ) [122], [123]; ( • ) [21]; ( • ) [20]; ( - ) evaluation [17].
3.2.3 Critical evaluation As indicated in the previous sections, published kinetic data are often in conflict, demonstrating differences in rate constant values well outside the precision to be expected from the technique used. Modellers are usually not practising kineticists and are not in a position to assess the quality of the data but, nevertheless, must select from them however inexpertly. This problem was first fully appreciated in the 1960s, prompted, first, by the rapid expansion of the kinetics literature as the result of the development of pulsed photolysis, fast-flow discharge and shock tube techniques; second, by the growing significance of combustion problems particularly exhaust emissions; and third, by the developments in digital computing which enabled complex chemical systems to be modelled. In response to the need for reliable data a number of groups of kineticists in the UK, USA and USSR began to compile and critically assess the rate data for
255
Primary source of kinetic data
-10.5
00
o
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
loV/K'1 Fig. 3.7. Arrhenius plot of experimental data for the reaction H + H 0 2 : (A) [120]; (H) [121]; ( • ) [122], [123]; ( • ) [21]; ( • ) [20]; ( - ) evaluation [17].
homogeneous gas phase reactions. This activity is still required and is now accepted as an essential part of the modelling process. It is important to distinguish between compilation and critical evaluation. The former lists the published experimental data, presenting it uncritically, while in the latter the published information is assessed in a variety of ways described in Section 3.3. There is a continuous range of possible types of review between these two extremes. All are valuable, and have their uses, but the modeller requires a set of recommended rate parameters for each reaction together with some indication of reliability of these recommendations in the form of error limits assigned to the rate parameters. Only critical evaluation can supply this. Collections of evaluated data save the modeller time spent searching the
256
Kinetics databases
Ch. 3
T/K
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Fig. 3.8. Arrhenius plot of experimental data for the reaction H + H 2 0 + O: (A) [120]; (H) [121]; ( • ) [122], [123]; ( • ) [21]; ( • ) [20]; ( - ) evaluation [17].
Evaluation of kinetic data
257
literature and provide a data source in which some confidence can be placed. If the collection commands the respect of a wide range of modellers it will provide a common data-set the use of which should allow a more meaningful comparison of the output from different models. This in turn leads to an important element of iteration in the provision of experimental data. Modelling can identify the reactions to which particular systems are most sensitive and, together with the critical evaluation of existing kinetic data, can indicate to the experimentalist the direction most profitable for further kinetics studies.
3.3 EVALUATION OF KINETIC DATA
The critical assessment of the experimentally measured rate parameters of an elementary reaction aims to establish which data are reliable and from them derive an expression for the rate constant of the reaction as a function of temperature and pressure. The methods and criteria used are outlined in this section. It is difficult to set them out in extensive detail since much depends on the experience and judgement of the evaluators. To that extent there is a degree of subjectivity involved. Because data are often very limited, or even non-existent, for a particular reaction the evaluator is also called on to use these sparse data to provide rate constant values for the conditions of interest to the modeller, which may be well outside the range of experimental studies. The interpolation, extrapolation, and estimation procedures involved are touched on in this section but treated in more detail in Section 3.4. 3.3.1 Evaluation groups The earliest evaluations were carried out by small research groups working in relative isolation from each other. It was soon appreciated that there are some advantages in working in larger groups offering, among its members, access to a wider range of expertise and self-criticism. Output from such a Panel also carries with it the authority of all of the experts in the group and, therefore, might be expected to gain wider acceptance. This is an important factor when comparisons must be made between the output from different models. The use of Panels has grown but, nevertheless, individuals and small groups also continue to make distinguished
258
Kinetics databases
Ch. 3
contributions to the evaluation of data on combustion and atmospheric chemistry. There are two large groups (IUPAC, NASA) which meet regularly to evaluate data on photochemical and chemical kinetics data for atmospheric modelling and one (CEC) which evaluates kinetic data for modelling combustion. Smaller groups and individuals still producing evaluations periodically are Tsang (NIST), Baulch (Leeds), Cohen (Aerospace Corporation) and Atkinson (Riverside, CA). 3.3.2 Collecting the information The basis for a successful evaluation is an adequate bibliography. The most complete collection of articles on gas phase chemical kinetics is held by the National Institute for Science and Technology (NIST), Gaithersburg, USA [22]. This collection is now accessible through a commercially available diskette to run on a PC. The cost is modest considering the expense of collecting, indexing and updating the collection over many years. A similar collection has been built up at the University of Leeds, UK. The NIST diskette contains bibliography on some 5000 reactions which covers the literature up to within about one year of the publication of the diskette. It is updated regularly. However, although a range of information from each paper held is presented, there is no evaluation of the material. There is no corresponding up-to-date hard-copy compilation. The bibliographies prepared by Kerr and Moss, which cover much of the literature through to the late 1970s are extremely useful for older material [23]. Kinetic data can age quite rapidly as advances in techniques produce more accurate data, but such older bibliographies can be of occasional value. Modern review articles can be useful in providing not only up-to-date bibliography on the rate measurements but also references to related topics which the evaluator often needs to consult. Published evaluations themselves also provide some of the background references as well as references to the source of the measured data. Westley has published a list of all of the evaluations for combustion processes up to the late 1970s [24], and more recent evaluations are discussed in Section 3.5. The information which an evaluator requires from a published account of kinetics measurements depends on the depth of the evaluation but, in general, it is desirable that: (i) there should be sufficient experimental
Evaluation of kinetic data
259
detail to allow the quality of the work to be assessed; (ii) the results should be reported in sufficient detail for them to be reanalysed and reinterpreted by others; and (hi) there should be estimates of the precision and, if possible, the accuracy of the results. These general principles have been elaborated on in a COD ATA document setting out standards for reporting kinetics data [25]. Unfortunately, even modern accounts do not always meet these criteria. 3.3.3 Evaluation procedures Some of the factors to be considered in evaluating published material in kinetics have been outlined by Hampson and Garvin [26], Baulch and Montague [27] and Cohen and Westberg [28]. The factors fall into two categories, first the evaluation of the details of the technique and data analysis and, second, comparison of the results with material external to the study, e.g., other measurements of the rate parameters, theoretical predictions etc. (/) Assessment of the technique and data analysis Potential sources of experimental error in rate constant determinations are many and varied; some have been mentioned in Section 3.2. Some are general, e.g., are the criteria for pseudo first-order conditions truly met?, while others are peculiar to the particular technique, e.g., in a fast-flow discharge study has there been adequate investigation of wall reactions? The experience of the evaluator in kinetics work in general and, if possible, with specific techniques plays a key role in the evaluation at this stage. Discussions with colleagues with the desired expertise and the collaboration of groups of evaluators has advantages in providing a pool of experience. Another valuable source of information on the deficiencies of techniques comes from the publications of other authors working in the same field, who provide some of the best critical evaluations of the works of others. Some rate constant determinations depend on a knowledge of the reaction mechanism and/or data on the rates of competing reactions. When new information on these features becomes available it is possible to reanalyse old data if it is presented in sufficient detail. For example, rate constants derived from a measurement of a rate constant ratio can easily be recalculated when new values of the reference rate constant are pub-
260
Kinetics databases
Ch. 3
lished, but it is much more difficult to reassess systems where the chemistry is more complex and many competing reactions are involved. The incorporation of new data on reference reactions is done in most evaluations but the reanalysis of more complex systems is rarely embarked on. Sometimes there are not enough published details to do so but this mainly occurs because it is extremely time consuming, even though there are now computer programs which make it feasible in many cases. There are exceptions to this. In their evaluations, Cohen and Westberg [29] have carefully reanalysed a number of high-temperature rate measurements and Cohen [30] has also reanalysed a number of results from fast-flow discharge measurements of atomic oxygen reaction rates to show that they are likely to be in error because of the over-simplified assumptions made in the original analysis. Although such reanalysis is rare in evaluations, fortunately, individual authors are now undertaking it more frequently in making comparisons between their own results and those of others. (ii) Comparison with other measurements As indicated in Section 3.2, even modern techniques, with much improved precision compared with older equipment, still produce a wider scatter of results than expected, due to small systematic errors in the measurements. Often no specific reason can be found by the evaluator for these differences and, in these circumstances, all that can be done is to use some form of averaging and assign error limits wide enough to encompass the variation. At the other extreme, occasionally very large differences are found when basically the same technique is used. This usually indicates different chemistry being involved in the two studies, e.g., different sources of the radical being studied leading to complicating side reactions in one or both cases. Unless the source of the discrepancy can be identified the evaluator will be reluctant to recommend one in preference to the other unless there are other indications, e.g., theory, estimated values, chemical reasons etc. why one is to be preferred. If one of the two is preferred then large error limits are likely to be assigned until confirmatory experimental evidence is forthcoming. Between the two extremes already mentioned there is a continuous variation. On occasions the evaluator is faced with a number of measurements of a rate constant in good agreement but with perhaps one measurement very different from the others. This situation can be particularly
Evaluation of kinetic data
261
difficult to evaluate if the consensus measurements are rather older and the "odd one out" is by a reliable modern technique. Usually the author of the most recent result will have discussed likely sources of error in the older measurements and, if the arguments are convincing, these may be accepted by the evaluator. But if no good reasons can be found for the differences, evaluators tend to react conservatively recommending a value based on the concordant results, perhaps with wide error limits, until the new result can be confirmed by an independent measurement. The comparison of older and newer measurements often presents problems. Measurements carried out in earlier days by the major techniques, involved detection methods requiring relatively high concentrations of atoms and radicals, thus, enhancing the probability of interfering chemistry and significant departures from the pseudo first-order conditions assumed in the data analysis. Modern detection techniques are usually sensitive enough to avoid such complications. However, this progress over the last 30 years has been continuous and at a particular time in the past it is often difficult to know whether differences between older and newer results are due to improvements in technique or whether there are genuine differences in the chemistry, if different chemical methods are used. So far, we have discussed procedures for handling discrepancies in rate constant measurements obtained under the same conditions of temperature and pressure. Sometimes the situation occurs in which there is agreement between values of k determined at a particular temperature but differing values of the temperature coefficient of k as the temperature is changed. The high-temperature data for the reaction CH + N 2 —• HCN + N, shown in Fig. 3.9 provide an example. There are four sets of data, two from measurements on flames [31, 32] and two from shock tube studies [33, 34]. In the evaluation by the CEC group [17] the shock tube studies were taken as the basis for the recommendation because they were believed to be freer of chemical complications. The two expressions from the shock tube work were k = 1.0 x 10~12 exp(-7000/T) cm3 molecule" 1 s" 1 and k = 7.3 x 10" 12 exp(-11060/7) cm3 molecule -1 s _ 1 . These have values of E/R differing by amounts greater than the scatter on each set of results and cannot be explained by differences in the conditions used, but the two expressions when extrapolated give virtually identical values of k at 2000 K. The recommended expression for k, based on these results, was obtained by averaging the values of E/R and, using the Arrhenius equation, deriving a value for the pre-exponential factor based on this value of E/R and the
262
Kinetics databases
Ch. 3
T/K
loV/K"1 Fig. 3.9. Arrhenius plot of high-temperature experimental data for the reaction CH + N2 -^products: ( ) [31]; (•••) [32]; ( ) [33]; ( ) [34]; ( - ) evaluation [17].
value of k common to both studies at 2000 K leading to k = 2.6 x 10" 12 exp(-9030/T) cm3 molecule -1 s - 1 . This is shown as the solid line in the diagram. Where there are reliable data over a wide temperature range the evaluator is able to fit an expression to the data with confidence, but the way the fit is performed may differ. In some cases a least squares fitting is carried out giving equal weight to all data points that are considered reliable. In other cases the fitting is performed 'by eye' by drawing in a line on the Arrhenius plot and the numerical expression to the fit derived subsequently. The outcome is usually much the same in both cases. Superficially, the statistical fitting is more objective but some of that is lost in
Evaluation of kinetic data
263
the lack of weighting of points, which must vary considerably in their reliability, and also in the fact that some derivations fit log k while others fit k which leads to different weightings of the points in the two cases. The statistically derived standard deviations to the fitting parameters are of some value but, as discussed later, the final error limits recommended are invariably derived subjectively. There are many more situations that can arise but these are among the commonest and provide some 'feel' for the ways in which evaluators come to decisions. In any evaluation the reasoning behind the derivation of the expression for k should be made clear in the discussion of the data and its analysis. (Hi) Comparisons with thermodynamic data Thermodynamic data may be used in several ways in evaluating kinetics data. Its most obvious application is through the relationship Kc = kf/kr where Kc is the equilibrium constant for the reaction written in terms of concentrations and k{ and kr are the rate constants for the forward and reverse steps in the reaction. Where there are values available for both kf and kr this provides a valuable check on their consistency. The relationship also provides a route to kf knowing Kc and kr (or kr from Kc and kf). An example of the value of the relationship in checking for consistency is provided by work on the reaction (6)
H C 1 ^ H 2 + C1
where, for some time, measured values of k{ and kr from a number of studies gave values of Kc inconsistent by a factor of 2 or 3 with the accurately known values of Kc from thermodynamics. The discrepancy led to several new studies of the rate constants of the forward and reverse reactions. Rate constants measured for the H + HC1—>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 <2#, QA and QB are, respectively, the partition functions of the transition state and those for the reactants, A, B and E0 is the reaction threshold energy. The partition function ratio can also be expressed in terms of the thermodynamic quantities AS^ and AH^, the entropy and enthalpy change in going from reactants to the transition state. In turn AS9* can be related to the pre-exponential factor in the expression for the rate constant, and AH # related to the activation energy. The partition functions or thermodynamic properties of A and B are usually well known and to estimate a value of k all that is required is a knowledge, or estimate, of Q*, or of the thermodynamic properties of the
266
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Ch. 3
transition state. These cannot be measured but from values of established rate constants it has been possible to build up a large body of semiempirical knowledge and understanding of the properties of transition states which allow useful estimates of rate constants to be made by those well practised in the art [38]. This information can be used to, (a) check that values of k derived from experimental measurements are reasonable, (b) transfer information gained on well understood reactions to the evaluation of data on less well characterized reactions, and (c) estimate the effects of temperature on rate constants. These are discussed in more detail in Section 3.4. An example of the value of this approach is provided by the earlier evaluation of the data on the reaction (10)
CO + O H - > 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
Evaluation of kinetic data
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
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(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
Evaluation of kinetic data
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)
[*=*> (?-£)]
<33)
where A log k{T) is the error in log k at temperature T, A log A:(298K) is the error in log k at 298 K, and A(E/R) is the error in E/R. This expression assumes that the rate constant is known with minimum error at 298 K. Often this is so, but the expression can be readily modified to deal with the cases when the minimum uncertainty lies at another value of T within the range covered by the experimental studies. Evaluations for combustion modelling are usually concerned with a much wider range of temperatures, much greater variability in the extent of experimental measurements at the various temperatures in the range, and considerable change in the value of E/R with change in temperature. These evaluations usually give values of A log k at the extremes of the temperature range without trying to specify exactly how the error limits will change from one extreme to the other. However, the implication is that they will lie between the two values of A log k quoted. In some cases there may be a large amount of good quality data at an intermediate temperature, at which point the rate constant is better defined than at the temperature extremes. In that case three values of A log k may be quoted, that at the best defined value of k and those at the extremes of the
272
Kinetics databases
Ch. 3
temperature range. Thus for the reaction H 0 2 + C 3 H 5 —> 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].
Interpolation, extrapolation and estimation procedures
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
Interpolation, extrapolation and estimation procedures
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.
Interpolation, extrapolation and estimation procedures
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
Interpolation, extrapolation and estimation procedures
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
Kinetics databases
Ch. 3
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
Kinetics databases
Ch. 3
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].
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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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[53] [54] [55] [56] [57] [58]
[59] [60]
Kinetics databases
Ch. 3
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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Kinetics databases
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292 [148] [149] [156] [151] [152] [153] [154]
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T. Ohta, J. Chem. Kinet. 16 (1984) 879. R. Atkinson and S.M. Aschmann, Int. J. Chem. Kinet. 16 (1984) 1175. G.P. Smith, Int. J. Chem. Kinet. 19 (1987) 269. F.P. Tully, Chem. Phys. Lett. 96 (1983) 148. E.D. Morris, Jr. and H. Niki, J. Phys. Chem. 75 (1971) 3640. A.V. Pastrana and R.W. Carr, Jr., J. Phys. Chem. 79 (1975) 765. H.W. Biermann, G.W. Harris and J.N. Pitts, Jr., J. Phys. Chem. 86 (1982) 2958.
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
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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
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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
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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-
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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-
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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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315
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).
Sensitivity and uncertainty analysis
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
Sensitivity and uncertainty analysis
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.
Sensitivity and uncertainty analysis
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
Sensitivity and uncertainty analysis
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.
Mechanism reduction without time-scale analysis
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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:
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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.
Mechanism reduction without time-scale analysis
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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
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Mathematical tools 2000-4 1800
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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
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Mechanism reduction without time-scale analysis
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,
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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
Mechanism reduction without time-scale analysis
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
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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
Mechanism reduction without time-scale analysis
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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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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
O
o Full model
15.0 H
14.5 0.0
Reduced model i
1 0.1
1
F
r0.2
0.3
1
0.4
1
1
0.5
t/S
Fig. 4.5. Comparison of log hydrogen concentrations for full and reduced schemes: (a) T = 790 K; (b) r = 1 0 0 0 K .
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Mechanism reduction without time-scale analysis
2000
i
Full model and first reduction
1800 -j 1600 - I
T/K
14
/
o°i
2nd / reduction
1200
1000 -A 800-5 10
15
20
25
30
t/S
Fig. 4.6. Comparison of temperature traces from the 22- and 16-step reduced models with the full 47-step non-isothermal model at Ta = 790 K. The removal of reactions 14, 27, 29, 35, 37 and 45 causes only a small decrease in the oscillatory period.
and a sample trace showing results from both the 16-step scheme and the larger 22-step scheme compared with the results from 47-step full mechanism are given in Fig. 4.6. Information contained in the overall rate and temperature sensitivities can also be used to follow the switching between branching and termination mechanism as the reaction proceeds through ignition cycles. In the weak ignition cycle where hydrogen is being replenished into the reactor only 6 reactions are of importance (reactions 3-7 and reaction 10). In the "strong" ignition-stage reactions 13, 14, 16 and 24 become significant and as the radical concentrations increase and the concentration of H 2 falls to a minimum, reaction 23 becomes an important propagation step. As the system moves in and out of ignition the sensitivities clearly show the switch between the branching reactions 13 and 4, 5. In particular the reactions of OH with itself and other radicals are important at higher temperatures, and their omission from the scheme results in a significant qualitative change in the shape of the oscillatory waveform.
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4.6.5 Discussion Finding a subset of a reaction mechanism with identical applicability to the full mechanism, should be the final step of every mechanism generation, and the first step of any mechanism reduction work. However, most published mechanisms contain plenty of species and reactions which are redundant over the range of experimental conditions they are intended to cover. A systematic search for redundant species is almost never carried out, and redundant species are usually identified either accidentally or on the basis of detailed chemical knowledge of the mechanism studied. Two techniques are described here which allow the identification of redundant species in a systematic way. Comparison of reaction rates, called rate-of-production analysis, is a frequently applied technique and is the basis of limiting the size of a newly created mechanism. However, this technique requires a lot of manual effort. Algebraic rate sensitivities are the partial derivatives of production rates with respect to rate parameters. These measures are equal to normed reaction rate contributions. Inspection of algebraic rate sensitivities, based on either the sum of squares of the coefficients (overall sensitivities) or principal component analysis, is a simpler and more automatic way for the identification of redundant reactions than that based on rate-of-production analysis.
4.7 FORMAL LUMPING PROCEDURES
Using the techniques outlined in Section 4.6 it is possible to reduce a full mechanism to a reduced scheme containing no redundant species. For 3-D calculations, however, the number of species contained in typical reduced schemes will still be larger than even present supercomputers can handle. Even the 16-step reduced hydrogen scheme would still be too large to be included in most 3-D turbulent combustion calculations. Rate sensitivities have shown that each term in the remaining equations is important, so the question remains as to how the model can be further reduced without losing information about these terms. To achieve any further reduction it is necessary to condense the information contained in the resulting set of equations into a smaller set, or to replace the equations by some other mathematical representation of a lower dimension than the
Formal lumping procedures
343
full equations. This stage of the reduction process is often named species lumping, since it is based on representing information about groups of species using a single variable. The new lumped variables are related to the original variables by a function called the lumping function which can be either linear or non-linear, and will depend not only on the original species concentrations, but also on the important parameters. In mathematical terms lumping can be described as the reduction of the ^-dimensional system described in equation (4.1) to an /z-dimensional lumped set, ^ = f(c),
(4.20)
at where h ^ n and c = h(c), (4.21) where h is some linear or non-linear function of the original variables c. If we think in terms of the phase space of the species concentrations, we can see a geometric picture of lumping as corresponding to a mapping from the composition space of original variables to a lower dimensional space. The important questions to answer are: 1. What degree of lumping can be achieved while maintaining the accuracy of the reduced scheme, i.e., how small can the lower dimensional space be? 2. What is the optimal way of choosing lumping functions which give the smallest reduced model for any general system of equations and still satisfactorily represents the behaviour of the full scheme? So far, the main aims of research into lumping have been to try and find mathematical procedures which apply to any general reaction system, and provide an automatic algorithm for reduction. It is not clear yet whether or not such a technique will ever exist which can be applied for any mechanism. Certainly if a method is to be developed it will have to be founded on rigorous mathematical principles. Quite a large amount of
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work has been carried out in this area and by describing the results obtained we can begin to answer the questions posed above. 4.7.1 Linear lumping The simplest form of lumping is linear lumping, where the new variables are linear combinations of the original ones, Ci = raMC! + miac2 + • • • + mifncn. Therefore, equation (4.21) simplifies to c = Mc,
(4.22)
where M is a n x n real constant matrix called the lumping matrix. The new h set of odes for the lumped system is given by dc — = Mf(c). dt
(4.23)
For exact lumping Mf (c) must be a function of c so that the reduced system can be expressed in terms of the new variables. Therefore, we need to know the inverse of M since c = M _ 1 c.
(4.24)
The inverse mapping from the c space to the c space is equally important as the forward mapping, not only because it provides a link between the lumped species and the original species, but because its existence is a necessary and sufficient condition for exact lumping. For a reduced system n
AA1" A = A
(ii)
A 1 AA 1 = A 1
Formal lumping procedures
(iii)
(AA f )* = AA f
(iv)
(A f A)* = A f A
345
where A* is the conjugate transpose of A and A 1 is an n x m matrix [106]. Wei and Kuo [107] and more recently Li and Rabitz [108,109] have set out conditions for the exact and approximate linear lumping of ordinary differential equations, and give examples of techniques which can be used to find the lumped schemes. In the linear case the method involves finding suitable lumping matrices of a chosen dimension and their inverses, i.e., an invertible mapping. The equivalent problem is finding invariant subspaces of original equations, i.e., invariant subspaces of the transpose of the Jacobian J r (c) so that the eigenvalues of J r (c) and J r (M _ 1 Mc) are identical. The eigenvalues of the reduced system at a fixed point form a subset of the eigenvalues for the full equations. By choosing which eigenvalues are retained it is possible to ensure the equivalence of the full and reduced systems. For example, if the full system exhibits oscillatory behaviour, for the qualitative dynamics to be expressed by the lumped system, all the complex eigenvalues cannot be lumped. Finding the eigenvalues involves the conversion of J r (c) into some special or canonical form. For non-linear systems, such as those found in combustion, J r (c) will not be a constant matrix and, hence, converting it directly into a canonical form will be a difficult task. It is possible to express J r (c) in the form n
r
J (c) = A0 + 1 A*
(4.25)
k=\
where ak(c) are functions of c which for non-isothermal systems will be simple polynomials. Li and Rabitz have proved that the invariant subspaces of A0 and Ak or A = XZ=oAk a r e equivalent to the invariant subspaces of J r (c). The simplest technique for finding the invariant subspaces is to determine eigenvalues and eigenvectors of A, If the eigenvector matrix is given by X=(xl9x2,..
. ,x„),
(4.26)
where x1? x 2 ,. . . , xn represent the columns of X, then the subspaces are
346
Mathematical tools
Ch. 4
given by the span of these columns. For example, the one-dimensional invariant subspaces are given by Span{0}, Spanjxx}, Span{x2}, Span{x3} etc. and the two-dimensional subspaces by: Span{x1? x 2 }, Span{x 2 , x3} etc. The lumping matrices M of different dimensions can then be formed by taking the columns of X or any linear combinations of them. (/)
Example To illustrate the above approach we include here a short example from Li and Rabitz's work [108]. We consider a first-order reaction system with 3 species described as follows: 2,
C2
fc/= 3,kr = 2
3,kf=W,kr
=6
Ci^C3,A/=10,Jfcr = 4 The corresponding kinetic equations are dc — = - 1 3 c ! + 2c 2 + 4c 3 dt
(4.27)
den — = 3 d - 12c2 + 6c 3 d£
(4.28)
— = 10ci + 10c2 - 10c 3 , dt
(4.29)
where rate constants have been chosen arbitrarily. In vector notation this corresponds to
Formal lumping procedures
dc - = Kc, at
347 (4.30)
where K is the rate constant matrix, and '-13 2 r J (c) = [ 3 - 1 2 10 10
4\r 6 • -10/
(4.31)
The eigenvector matrix X is given by
(4.32)
Any subspace spanned by a subset of the eigenvectors is J (c) invariant. For example, Span{xl9 x2} can give a two-dimensional lumping matrix,
-G
; -a-
The new form of equations is given by
— = MKc, At
(4.34)
and we need to find the generalized inverse M _ 1 in order to express the righthand sides in terms of c. The inverse will not be unique but the form of the inverse does not affect the form of the lumped equations. If we arbitrarily choose
348
Mathematical tools
M"x=
0.5 0 ,
Ch. 4
(4.35)
the lumped kinetic equations become — = MKM dt
1
c=( c. V 10 10/
(4.36)
The resulting lumped scheme is
where Ci = ci + c2 + c3 C2
=
Ci + c 2 "~ C3.
For this simple reaction scheme we can see that it is possible to relate the reduced set of differential equations to a kinetic reaction scheme. 4.7.2 Nonlinear lumping The above example is a simple first-order scheme so that the Jacobian is constant in time. For such systems linear lumping can potentially provide a high degree of reduction and can do so in an automatic way. However, combustion systems are highly non-linear and often linear lumping cannot achieve the degree of reduction required to give an efficient model. There are two options available. The first is to consider the system as being locally linear so that the above analysis can be applied over short-time periods. In this case, different lumped schemes would be necessary for different time periods. The problem with this approach is that for highly non-linear situations, such as ignition, a large number of lumping schemes may be required to cover the desired reaction period. The switching between different reduced schemes could potentially slow down the calculation to a point where using lumped schemes would no longer provide a
Formal lumping procedures
349
time saving. The second option is to develop non-linear lumping methods which will give more flexibility in how the lumped species can be represented as a function of the original species. This means that the reduction to a lower dimension may be possible with non-linear lumping rather than with linear lumping, and the possibility of producing a reduced scheme which will apply over the whole reaction zone will be greater. The disadvantage of this second approach is that the generalization to a nonlinear theory is quite complicated. For linear lumping the analysis was based on simple matrix calculations which could be applied automatically on a computer. A non-linear analysis will involve complicated analytical theory, but steps have been taken in this direction by Li et al. [110] and we will outline here the developments which have been made. First, we define the necessary and sufficient conditions for exact nonlinear lumping. If we start with the system of equations (4.1) then the new lumped variables are defined by an n-dimensional non-linear transformation c = h(c) and the new n-dimensional equation system becomes ^ = f(c(0). at
(4.37)
The conditions for exact lumping can be expressed in different ways. If we define the Jacobian of the transformation h(c) as Dhc(c) = dh/dc, then Dh,c(c)f (c) = Dh,ch(h(c))f (h(h(c)))
(4.38)
is a necessary and sufficient condition for exact lumping. As in the linear case, exact lumping depends on the existence of the generalized inverse transformation h. Since h is now a non-linear function, the calculation of h becomes difficult for high-dimensional or highly-coupled systems. Comparison with the linear case becomes more apparent if we redefine the system using a linear partial differential operator A, which, using indices notation, is given by n
A=2fi(c(t))-rThe original differential system now becomes
(4-39)
350
Mathematical tools
T
= A(c).
Ch. 4
(4.40)
By comparison with previous work on linear lumping, it is clear that finding the non-linear lumping function h depends on finding canonical forms for the operator A and/or on finding the invariant manifolds of the original system. This parallels the linearized case where we searched for canonical forms for the Jacobian J and its invariant subspaces. As for the linear case, one way of finding canonical forms for the operator A would be to search for its eigenfunctions relating to eigenvalues which are no longer constant but are functions of c(t). Li et al. [110] have already shown in their preliminary work that finding a full space of eigenfunctions for a non-linear system is not an easy task. We shall see in a Section 4.9 that finding approximate canonical forms for A is easier, and can be guided by using the physical properties of chemical systems, such as the separation in reaction time-scales between species. 4.7.3 Chemical lumping The methods described above have been deliberately developed for general reaction systems. Another approach to defining a lumping algorithm is to take into account the structure of the reaction network and to develop procedures accordingly. One example of this approach is chemical lumping. For example, Frenklach [9] describes a polymer system where the chemical reactions describing polymer growth are of the same type, and the rate parameters and thermodynamic data vary only sightly between different polymer sizes. The lumping procedure is then guided by chemical structure, and similar species are treated as part of a lumped group. For soot formation the reaction is described by a distribution function for the degree of polymerization, and a repeating reaction cycle for particle growth [4, 111]. The structure and rate constants for each repeating cycle are treated as being the same. We now discuss an example from Frenklach's review to illustrate how chemical lumping is applied. The mechanism Frenklach describes is the production of polycyclic aromatic hydrocarbons (PAH) in flame environments. Here, the PAH growth proceeds by a replication process, and so the lumping can be guided by the similarity in structure of the hydrocarbon
Formal lumping procedures
351
species in the replicating sequence. Using Frenklach's review we start with the following reaction sequence:
(/,i)
A/ + H - A ; + H2
(7.2)
AJ + C2H2 ^ A,- CHCH*
(7.3)
A.CHCH* + C2H2 -> A /+1 + H
O'+U)
A /+1 + H - A - + 1 + H2
(i +1,2)
A;+1
(i+1,3)
A /+1 CHCir + C2H2-+ Ai+2 + H
(7 + 2,/ + 3, . . . , n)
+ C 2 H 2 - A f . + 1 cHor
. . . etc.
where A, denotes an aromatic molecule containing i fused aromatic rings. A* is an aromatic radical formed by the abstraction of an H atom from A, and A.CHCH* is a radical formed by adding C 2 H 2 to A*. The reaction groups (7,1) to (7,3) and (7+1,1) to (7+1,3) are chemically similar, each completing the building cycle of one ring. It is assumed that the replicating process continues to infinity. This reaction system can be described by the following set of rate equations: ^
at
= r0- fcx[A,][H] + ^ [ A ^ H d
^ 1 = fci[A,][H] - ^ [ A ^ H d - A:2[A'][C2H2] + fc_2[A,CHCH*] d[A,CHCH-] = ^ [ A ; ] [ C 2 H 2 ] _ £_ 2[A( . C HCH-] at - A:3[A,CHCH*][C2H2] d[A i+1 ] = ^3[A,CHCH*][C2H2] - A:![A,+1][H] at
+ k-^K+iW
352
Mathematical tools
at
l
= kl[Ai+1][U]
Ch. 4
- /^[A^JtHj
- fc2[A*+1][C2H2] + fc_2[A/+1CHCH*] d[A +1
^ H C H l = A:2[A-+1][C2H2] - /c_2[A,+1CHCH*] at
'
- /r3[Ar+1CHCH-][C2H2] etc.
(4.41)
where r0 is the rate of formation of At by initiation reactions, kj is the rate constant of reaction j which is assumed to have the same value for each cycle since the species are chemically similar. It is this assumption which enables lumping to be achieved on a chemical basis. If we sum equations (4.41) we get d[A,] dt
|
d[Al] dt
|
d[A,CHCH-] dt
{
d[A, +1 ] dt
|
\ /
°'
or ^
= ,o,
(4.43)
d£
where M 0 = [A,-] + [A]] + [A,CHCH#] + [Af + J + • • •. This is a one-dimensional lumped system which describes the evolution of the total PAH concentration M0. However, it does not describe the detailed dynamic behaviour of the system of equations, and to do this we need to move to a higher-dimensional lumped system. If we multiply each of equations (4.41) by an integer roughly corresponding to the molecular mass, i.e., the first equation by ra0, where ra0 is the number of carbon atoms present in A,-, the second equation by m0 and the third equation by (m0 + 2) etc., and again sum the equations, we get
iMiJmo^l dt
\
dt
+ mi>m+ dt
(mo + 2)f!lA£Han dt
353
Formal lumping procedures
dt = m0r0 + 2k2[C2H2] X [A*] - 2/fc_2 2 [A, CHCH*] /
i
+ 2A:3[C2H2] I [A,CHCH*],
(4.44)
where Mx = m0[A,] + m0[A*] + (m0 + 2)[A,CHCH#] + (m0 + 4)[A /+1 ] + (m0 + 4)[A*+i] + • • •, is the total number of carbon atoms accumulated in the PAH's, or the first moment of the PAH distribution function. We can see now that there are lumped concentrations of the reaction system and can define a new set of variables c where ci = 2 [A,] i
h = l [A*]
(4.45)
i
c3 = 2) [A,CHCH*]. i
The corresponding lumped equation system is as follows: ^ = r0 - ^ [ H f o + £-i[H 2 ]c 2 + £3[C2H2]c3 at ^
= ^ [ H f o - fc-![H2]c2 -
fc_2c3
(4.46)
dc3 —- = &2[C2H2]c2 - k-2c3 - k3[C2U2]c3. at This example illustrates that chemical lumping is nothing other than a special case of linear lumping, and that this system is a good example of the application of exact linear lumping. In fact, we can express the relationship between the original and lumped species in the same formal way as we described in the previous section:
354
Mathematical tools
c = Mc,
Ch. 4
(4.47)
or
fcA /l 0 0 1 0 0 c2 I = 10 1 0 0 1 0 \c3) \0 0 1 0 0 1
(4.48)
Here we can see that M is a 3 x n lumping matrix which will be an invariant subspace of the full system. This type of chemical lumping obeys all the rules of the formal methods of exact lumping. The only point of caution is that the simple relationship between the full and lumped systems has been achieved as a result of the assumption that the rate parameters for the different species are the same. In reality there may be some errors induced by making this assumption. However, to couple this approach with sensitivity analysis, which would give some quantification of these errors, could be a promising way forward. Similar techniques have also been applied for atmospheric models such as the Carbon Bond Mechanism [112], in hydrocarbon cracking schemes [113] and in low-temperature alkane combustion [114]. Such methods seem to work well for schemes containing large numbers of hydrocarbon species. 4.7.4 Lumping of continuous mixtures Another important application of lumping is the lumping of continuous mixtures where an indefinitely large number of species are involved in a reaction [115-125]. Such techniques will be useful especially in processes which involve complex hydrocarbon mixtures such as petroleum feedstocks etc. The discrete system is then represented by a continuum and the lumping by an integration rather than a summation over the original variables. As an example we consider an isothermal system containing N different
Formal lumping procedures
355
reactant types [119] which react irreversibly with an nth-order rate. Such a system can be described by the following mass balance equations: ~=-kiCl. at
(4.49)
If the reaction order is assumed to be constant for all species then the only species dependent parameter is kt. Therefore, a species can be defined by its concentration ct and its reactivity kt. Then, for N discrete species, the reactant lump can be expressed as TV
C(t)= I c ^ - , 0 .
(4.50)
If N becomes sufficiently large k may be treated as a continuous function and used to designate individual species. The lump C(t) can then be expressed in the integral form C ( 0 - f g(k,t)dk,
(4.51)
where A: is a continuous variable and k and k are the lower and upper limits for the mixture in question. The product g(k, t)dk is the total concentration of a species with rate constants between k and k + dk and needs to be interpreted as a concentration distribution function. As the number of species approaches infinity the separation between k and k becomes larger and for mathematical convenience it is assumed that k—>0 and k-^ oo. This gives C(0«
f
g(k,t)dk
(4.52)
Jo
as the conventional lumping equation for continuous mixtures. To determine g, the conventional approach has been to place restrictions on the lump based on the premise that the continuous mixture must be kinetically consistent with the underlying discrete mixture. This leads to the following conservation equation:
356
Mathematical tools
d[g(M)d£] At
=
_%(Mdfc]»
Ch. 4 (453)
This equation is only kinetically consistent if n = 1 which leads to a restrictive form for the kinetics. The assumption that the lumped equation should follow the form of traditional kinetics seems to be ill-founded and, therefore, the above equation is meaningless. Chou and Ho [119] have expressed this problem and have developed a new lumping procedure which also takes into account the fact that the number of reactant types per unit range of k can vary along the k axis. The basis of their approach is to approximate equation (4.50), which is a sum over reactant type /, by an integral over reactivity k. We may regard this as a one-to-one mapping between "/-space" and "A:-space". The Jacobian of this map D{kt) is given by 0(*i)=™'
(4-54)
and a large value of D(kt) indicates that the reaction mixture has a large number of reactants with rate constants near kt. Corresponding to D{kt), there is a continuum of species concentrations C(k, t) defined at a point k. The properties of the continuous mixture are thus characterized by k, c(k, t) and D{ki). The lump concentration is given by C ( 0 ~ I c(k,t)D(k)dk.
(4.55)
Jo The equations describing the time-dependence are given by c\C
f°°
— ~D(k)kc(k,t)ndk. (4.56) dt Jo The D(k) acts as a weighting factor which takes into account the fact that in the discrete system thefc,-'smay not be equally spaced along the k axis. It depends on the feed type and, therefore, has to be determined for each
Formal lumping procedures
357
feed or each group of similar feeds. This technique is useful for evaluating the effective overall reaction order. 4.7.5 Discussion Formal lumping techniques have been sought which will provide methods of reducing any general reaction system to a lower dimensional system of equations. This is achieved by representing information about groups of species using fewer variables. General methods for both linear and non-linear lumping have been investigated. Linear methods, where the new species are represented as linear combinations of the original ones, work well for linear systems and also provide some degree of reduction for non-linear schemes. Non-linear methods are much more general, but can involve complicated algebraic theory which might limit their use. The main advantage of using lumping approaches for mechanism reduction is that searching for the lumping functions can be done systematically based only on the form of the equations rather than any intuitive knowledge of the chemical processes. However, there are problems associated with some lumping methods. For example, in linear lumping there are large numbers of possible lumping matrices for each dimension of reduced scheme and the number of possible lumped schemes increases with the number of original variables. A combinatorial explosion may occur, and there has to be some way of choosing the optimum lumped scheme for each problem. The calculation of the inverse matrix M _ 1 provides further problems and is not unique, although the choice of inverse matrices does not affect the form of the lumped equations. Rather than testing each reduced scheme to check whether or not its dynamics matches that of the original scheme, there is a need to combine lumping with physical information about the system in order to select appropriate lumping matrices. This has been carried out to some extent in chemical lumping, but what is needed are general rules which can be applied to any reaction mechanism. One way forward is to incorporate knowledge about the time-scales involved in the reaction into the lumping process. The criterion on which we base the success of the lumped scheme is usually its agreement with the full scheme for the important variables. This agreement may be within certain error limits and so the existence of exact lumping schemes may not be necessary. Therefore, the development
358
Mathematical tools
Ch. 4
of approaches for approximate lumping is important, and as we shall see, when coupled with a time-scale analysis, provides a practical way forward for non-linear lumping.
4.8 REDUCTION BASED ON THE INVESTIGATION OF TIME-SCALES
Chemical systems are computationally expensive to solve not just because of the large number of species involved, but because of the large range of time-scales present in any mechanism. In combustion systems, intermediate radical species may have relaxation times up to ten orders of magnitude less than stable species, leading to stiff systems of equations requiring special numerical techniques. The very fast time-scales are usually associated with local equilibrium processes. It is often possible to decouple these processes, and assume a local equilibrium with respect to the fastest time-scales leading to a reduced set of equations. The stiffness of a system is related to the ratio of the fastest-to-slowest time-scales so that the removal of the fast species will automatically lead to a less stiff system where perhaps a cheaper explicit integration scheme could be used. Therefore, it is natural to base any method for the reduction of species on some kind of time-scale analysis. Again we should seek to base this analysis on formal methods rather than on an intuitive understanding of the competing reaction steps. In any case, for large, complex mechanisms such an intuitive approach would be virtually impossible. The reduction techniques which take advantage of this separation in scale are described below. They include the quasi-steady-state approximation (QSSA), the computational singular perturbation method (CSP), the slow manifold approach (intrinsic low-dimensional manifold, ILDM), repro-modelling and lumping in systems with time-scale separation. They are different in their approach but are all based on the assumption that there are certain modes in the equations which work on a much faster scale than others and, therefore, may be decoupled. We first describe the methods used to identify the range of time-scales present in a system of odes.
Reduction based on the investigation of time-scales
359
4.8.1 Low-dimensional systems For low-dimensional equation systems, the investigation of such timescales can be carried out through a non-dimensionalization process. For example, consider the 2-variable scheme ^ = f(x,y,k), dt e^ = g(x,y,k), at
(4.57) (4.58)
where x and y are dimensionless quantities. Here e is a small positive quantity and we can immediately see that the time-scales for x and y are different. If g(x, y, k) is zero then dyldt will be zero (steady-state). If g is non-zero and negative then the prescence of the small parameter e means that dy/dt will be initially large and quickly approach zero. Therefore, the long-time behaviour of the system can be represented by the equation set dx - = f(x9y,k), at
(4.59)
g(x9y,k)
(4.60)
= 0,
reducing the description of the fast variable j to a simple algebraic equation. 4.8.2 Jacobian analysis An important quantity in studying the time-scales and stability of a set of differential equations is the Jacobian, and this is calculated as the integration proceeds, although not generally output by most packages. This type of analysis is useful in large systems where a non-dimensionalization procedure would prove difficult. In order to understand the information contained in the Jacobian, we must first introduce the concept of perturbation analysis. By studying the evolution of a small disturbance or perturbation to the non-linear system it is possible to reduce the problem to a locally linear one. The resulting set of linear equations is easier to
360
Mathematical tools
Ch. 4
solve, and information can be obtained about the local time-scales and stability of the non-linear system. If we disturb the system of equations (4.1) at a particular fixed point fx with a small perturbation Ac, so that ct = cf* + AQ
(4.61)
and expand as a Taylor series, we get dCj
dt
\dcjfx
\dc2/fx
+ .-. + ( ^ ) Acn+-.-. \BcJfx
(4.62)
Substituting
ac L = a£f + a A £ L = a*
a/
dt
{
SAC, dt
then to leading order, we are left with a set of linear equations describing the motion of the perturbation,
The solutions to such a set of linear equations can easily be found and are of the form: A Q ( 0 = CxeK* + C2eK* + • • • + CneK\
(4.65)
where Ct depend on the size and direction of the perturbation. Alternatively, we can write the above equation in matrix form:
Reduction based on the investigation of time-scales
Ac(0 = AeJ(,)'
361 (4.66)
where J (t) is called the Jacobian matrix which is given by dc{\ \ dc„ dt
_d_
dcx \ dt
(4.67)
dc d
(dCn
dcx \ dt
d
(dcn
dc„ \ dt
The exponents A, are the eigenvalues of the Jacobian matrix calculated at the fixed point. Therefore, each eigenvalue of the Jacobian matrix is associated with a different time-scale of the locally linear solution to the full equations. The largest negative eigenvalue corresponds to a perturbation which decays very quickly and is associated, therefore, with the fastest time-scale. This approach is essentially local and if we are studying non-linear trajectories then the Jacobian will be time-dependent. Thus, the ordering of time-scales may change as the reaction proceeds. We see in later sections that any separation in time-scales provides an ideal basis for the reduction in both the size and stiffness of a reaction mechanism. The Jacobian provides a simple way of investigating time-scales since it is obtained at many time-points during the numerical integration as a matter of course, allowing information to be extracted at very little extra cost. Programs based on Computational Singular Perturbation (CSP) theory have also been developed and give an automatic way of ordering timescales. 4.8.3 Computational Singular Perturbation (CSP) theory The CSP technique [126-130] has been developed by Lam and Goussis as a formal method for analyzing reaction mechanisms. It is specifically designed to enable the user to establish partial-equilibrium and quasisteady-state relationships without using any chemical intuition or expertise. The basis of the technique is to rewrite equations (4.1) and (4.2) in terms of a new set of basis vectors, rather than the physical representation using
362
Mathematical tools
Ch. 4
stoichiometric and reaction rate vectors. By choosing this new basis set in an "optimum" way, information about the ordering of time-scales can be seen more easily. Again we consider the basic rate equation (4.1). As described in equation (4.2), the reaction rate column vector f is usually written in the form R
f=y2\jRj
(4.68)
j
where R is the number of elementary reactions included in the mechanism and \j is the stoichiometric coefficient for reaction /. This physical representation is not unique, however, and the reaction rate equations can be written in terms of another basis. Let a;(f), i = l , 2 , . . . , N, be a set of N linearly independent column basis vectors with inverse bl(t). The reaction rate vector f can be rewritten as TV
f=2aI-di,
(4.69)
where TV
a, = 2 ykA?,i = l,,..,N,
(4.70)
k=l
and R
l
d = b\f = I (b\vy)R;-, i = 1,. . . , N,j = 1,. . . , R.
(4.71)
7=1
This new representation is mathematically identical to the equations (4.1) and (4.2) regardless of the choice of the matrix of new basis vectors Af. Each of the N terms on the righthand side of the "rate" equations (4.69) no longer has any physical meaning in terms of elementary reactions, but represents some linear combination of the R reactions and is called a reaction group or mode. As equations (4.70) and (4.70) show, each a, is
Reduction based on the investigation of time-scales
363
a linear combination of the elementary stoichiometric vectors, while each d1 is a linear combination of the elementary rates. What the CSP method does is to choose A? in some optimal way so that the rate of each of the new reaction groups is associated with a single time-scale. The reaction groups are ordered in ascending time-scales so that reaction group 1 is the fastest and reaction group N the slowest. As the reaction runs its course then one by one the fastest modes become exhausted and the rate of that particular reaction group approaches zero. These exhausted reaction groups are called dead modes and can be discarded from the righthand side of equations (4.69). If the mechanism has E elements, then automatically d' = 0 for / = N — E + 1, N. The simplified righthand side of equation (4.69) becomes »M+id
+ aM+2d
+ • • • + a^_^d
and represents a reduced mechanism of N - (M + E) steps, where M is the number of dead modes. The M algebraic relations, Dl = (bVi)/?1 + (b'v2)/?2 + • • • + (VyR)RR = 0, represent conservation equations which are generalizations of quasi-steadystate and partial equilibrium relations. They can also be interpreted as N - (M + £)-dimensional manifolds within an TV-dimensional space. We will return to this point in the next section. Discarding dead modes leads to a less stiff system of equations, so that computational savings could be made by using a cheaper solver. If the vectors a, are determined computationally, the savings provided by CSP are outweighed by the effort required to recompute the new basis at each time-step. In that case, CSP has been mainly used as a tool for investigating mechanisms rather than a computationally efficient technique in its own right. Using CSP it is possible to identify both important species and reactions so that it can be used as a formal method for reducing reaction mechanisms. Reduced mechanisms for methane ignition [131] and methanol oxidation [127] have been generated using the CSP method. It is also possible to use the information about the fast modes to identify quasisteady-state species and partial equilibrium assumptions. Recent papers by Goussis [132] and Katsabans and Goussis [133] present an algorithm for constructing reduced mechanisms using algebraic relations derived from
364
Mathematical tools
Ch. 4
the CSP method. The advantage of the CSP method over simple eigenvalue/eigenvector analysis is that it can reveal information about which species and reactions are associated with the fastest modes and, therefore, eliminated using the approximation methods we are about to describe. 4.8.4 Slowlinertial manifolds The main problem in the solution of non-linear ordinary and partial differential equations in combustion is the calculation of their trajectories at long times. By long times we mean reaction times greater than the timescales of intermediate species. This problem is especially difficult for partial differential equations (pdes) since they involve solving many dimensional sets of equations. However, for dissipative systems, which include most applications in combustion, the long-time behaviour can be described by a finite dimensional attractor of lower dimension than the full composition space. All trajectories eventually tend to such an attractor which could be a simple equilibrium point, a limit cycle for oscillatory systems or even a chaotic attractor. The attractor need not be smooth (e.g., a fractal attractor in a chaotic system) and is in some cases difficult to compute. However, the attractor is contained in a low-dimensional, invariant, smooth manifold called the inertial manifold M which locally attracts all trajectories exponentially and is easier to find [134,135]. It is this manifold that we wish to investigate since the dynamics of the original system, when restricted to the manifold, reduce to a lower dimensional set of equations (the inertial form). The inertial manifold is, therefore, a useful notion in the field of mechanism reduction. The idea of long-time solutions and attractors may seem strange when we are thinking of ignition systems, since clearly the interesting region of reaction is not close to the steady-state region. However, if we consider the whole range of time-scales present, the ignition time is actually fairly long. Fast radical processes can occur on a scale as small as 10 - 1 0 s, and it is these processes that we wish to decouple. It will help to consider a more geometrical picture of how a trajectory looks as the reaction proceeds, and the dimension of the space in which it moves. The geometrical interpretation of the time dependence of a reaction proceeding can be described using the idea of trajectories relaxing onto a series of increasingly lower-dimensional manifolds and eventually to equilibrium. Roussel and Fraser present an interesting picture of the relax-
Reduction based on the investigation of time-scales
365
ation of a chemical system as a "cascade through a nested hierarchy of smooth hypersurfaces (inertial manifolds)" [136-141]. As the reaction proceeds, the dimension of the surfaces containing the reaction trajectory will become smaller and smaller as more and more of the fast processes equilibrate. The aim of mechanism reduction is to find the lowest dimension hypersurface which will still represent the important features of the reaction. Clearly, in the case of autoignition, we must not exclude timescales which are longer than the time to ignition. In homogeneous systems it is the concept of the "slow manifold" which is particularly useful since the slow manifold describes the long-term behaviour of the system once the fast modes have died away [142,143]. The existence of a slow manifold depends on the natural decomposition of time-scales which is common in chemical systems and is indicated by a large gap in the eigenvalues of the linearized system, i.e., a large spectral gap. The concept of the slow manifold or intrinsic low-dimensional manifold (ILDM) has been successfully applied to a number of both homogeneous and reactive flow systems by Maas, Pope and co-workers [6, 7, 144148]. The required outcome of this geometrically-based technique is to identify the points in composition space which are in equilibrium with respect to the fastest time-scales. Mathematically this corresponds to a subspace in equilibrium with respect to its most negative m eigenvalues. A typical slow manifold and the evolution of fast trajectories onto the manifold for the simple system , B—>C is shown in Fig. 4.7. We can see that all trajectories tend exponentially to this manifold before relaxing to the steady-state. We can also see that points on the manifold can be defined by assuming zero speed of motion perpendicular to the manifold. We must now define equations describing this motion. Following the approach of Maas and Pope [6] we represent the thermochemical state of a fluid by the variables 6=(61, 02> • • • 0N). These variables will generally correspond to mass fractions. In fluid flow the composition field 0(x, t) evolves as
^ ^ - = S[0(x,t)] + r(x,t). at
(4.72)
Here S[0(x, t)] is the rate of change of 0 due to chemical reaction and T(x, t) due to transport effects such as convection, diffusion etc. The
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Fig. 4.7. A phase plot showing the approach of trajectories to the slow manifold for the reaction system A ^ B, B —> C.
simplified scheme, based on the ILDM approach, will be used to determine the chemical component S[0(x, t)]. Since, in the simulation of combustion problems, the calculation of the reaction terms must be carried out many times, it is important that this is done efficiently. For this reason look-up tables are often used in preference to an explicit calculation of each function such as the rates of change of species concentration. Evaluations can then be performed using interpolations from the table rather than by integrating large systems of non-linear equations. The main purpose of the ILDM technique is to produce tables of species and system properties relating to points on the slow manifold for use in the CFD code. The reduction of the dimension of phase space by its restriction to a manifold reduces the size of the tables and, therefore, the burden on computer storage and look-up times. (/) Application to homogeneous systems Let us first illustrate the method for homogeneous systems. Using vector notation the evolution of the mass fractions is given by the following equation:
s-'<*
(4.73)
Reduction based on the investigation of time-scales
367
where if/= (h,p, cf)1, c/>2, . . . , (f>ns)T and h and p are the specific enthalpy and pressure of the reaction. fa is the specific mole number of species / defined by fa = wJMt, where Mt is the molar mass of species i and wt the species mass fraction. ns is the dimension of the full system. Assuming constant pressure and enthalpy, and making use of conservation laws, we can say that the chemical reaction corresponds to motion along a trajectory in (ns - ne) dimensional space where ne is the number of chemical elements. The elements of the Jacobian J are given by dFJdWj and the eigenvalues A, of the Jacobian identify the ns different time-scales at each point in the state space. The corresponding eigenvectors describe the directions associated with these eigenvalues. The motion corresponding to the slow manifold, i.e., the long-time behaviour, is perpendicular to the eigenvectors associated with the fast eigenvalues. For a system with a well-conditioned Jacobian it is possible simply to work with the eigenvectors as a basis. However, for an ill-conditioned matrix, where degenerate or almost degenerate eigenvalues exist, it is better to work with a modified basis. Maas and Pope chose the Schur vectors as their basis vectors. The Schur-decomposition of a matrix is a method of triangularization which leaves the eigenvalues as the diagonals (or as 2 by 2 blocks in the case of complex pairs) of the resulting matrix, in order of descending real parts. Therefore, it is ideal for the ordering of time-scales. The Schur-decomposition is defined by QT3Q = T where Q is the Schur matrix and T is the resulting triangular matrix. The required dimension of the reduced system nc must first be chosen. If Q is defined as
where q, are the Schur basis vectors, then the low-dimensional manifold is defined as the set of points in state space for which
368
Ch. 4
Mathematical tools T
\
T H2 + n c +«c+2
0 = GlOA)FOA) =
FW-
T
-
V
-
q«
-
(4.75)
-
J T
£)£ is the submatrix of Q corresponding to the ns- ne- nc fastest variables, and so the slow manifold is defined by assuming that motion in the direction of the Schur vectors associated with the fast variables is zero. This means that the system is in local equilibrium with respect to its fastest time-scales. The ILDM technique provides a method for identifying the optimum slow manifold for a given dimension nc and mapping out its points in phase space. Any value of nc can be chosen although the accuracy of solution will decrease with decreasing dimension of the reduced manifold. In practice, each system will have an intrinsic dimension and it is sensible to use the information contained in the eigenvalues to guide the choice of manifold dimension. For example, in a system which may contain complex oscillations it would not be sensible to choose nc = 2. By analyzing the time-scales first and then choosing nc accordingly, it is possible to ensure the differentiable equivalence of the slow manifold and the full phase space. We have already discussed how the existence of a slow manifold is indicated by a large gap in the spectrum of the Jacobian. Therefore, the dimension nc should be chosen as the number of eigenvalues which lie above this gap, i.e., those which do not have large negative values, minus the number of conservation laws ne. In this way the reduced manifold is determined by the physical properties of the system rather than by an arbitrary choice by the modeller, and the errors minimized. An important point is that for non-linear systems, time-scales change over the course of a trajectory. The CSP method has indicated that in some cases there are modes which remain fast for some parts of the computational domain, and return to being slow modes at a later time. To use a constant value for nc, therefore, requires a time-scale investigation over the whole domain so that only those modes which are always fast should be eliminated. For manifolds which are highly non-linear, the choice of reference values at which to create the tabulation has to be made very carefully. In regions where steep solution gradients occur, errors can easily arise when interpol-
Reduction based on the investigation of time-scales
369
ating between values at neighbouring points in the table. Maas et al.y have proposed a technique using adaptive grids to solve this problem. Regions of composition space with steeper solution gradients are then tabulated with a higher resolution and errors due to interpolation from tabulated values can be avoided. (if) Example To illustrate the nature of the equations solved in order to describe the manifold, we consider a mechanism describing CO oxidation [6]. There are thirteen species in the mechanism composed of three elements, H, C and O. Along with equation (4.75), an additional 2 + ne + nc parameter equations define the manifold. ne + 2 equations describe the conservation of pressure, enthalpy and elements. nc equations are used to define the concentrations of the chosen reference species. In this example we have chosen nc = 2 and the specific mole numbers for H 2 0 and C 0 2 as the nc reference parameters. The full set of equations defining the ILDM is, therefore, h - hIet
(4.76)
p-pIef
(4.77)
*H(«A) - r H
(4.78)
* C ( 0 - rC
(4.79)
xO(iff) - r O
(4.80)
TH2O
(4.81)
0CO 2 - T C 0 2
(4.82)
QlF{$)
(4.83)
4>H2O -
where T indicates a reference value for a particular parameter. We fix enthalpy, pressure, choose a set of constant values for H20 and <£C02, and then solve the remaining equations to define points on the manifold. The result is a graph of points on the two-dimensional manifold as illus-
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trated in Fig. 4.8a. In Fig. 4.8b only one parameter, H 2 0 , has been chosen and so the bold curve represents a one-dimensional manifold. (Hi) Application to reactive flow problems In the application of the ILDM method to reactive flow problems, physical processes (e.g., transport) are considered as a disturbance to the chemical reaction system. These disturbances may perturb the system within, or out of the manifold. Only the time-scale of the perturbation is important rather than its size, which may be very large. If the time-scale of the disturbance is of the same order as that along the slow manifold, then the components of the perturbation in the fast subspace will have a negligible effect since the chemistry will relax them back to the manifold. The components in the slow subspace will directly couple with the slow chemistry and, therefore, move the state within the manifold. Therefore, the basic idea for flow systems is to create a projection operator P, which projects any physical perturbation onto a perturbation within the manifold. This means that interaction between the physical processes and the slow timescale chemistry can be represented, but other processes can be neglected since they are equilibrated by the chemistry in a short time. The nature of the projection operator P will depend on the local characteristics of the manifold. We now define the form of the projection matrix P. Extending equation (4.73) to include perturbations arising from physical processes we get
^ = F(<0 + E ( 0 ,
(4.84)
at where again F(i//) represents the chemical rates of change and E(i/0 now represents the physical perturbation. The state ^ is a function of the N parameters 6 which determine the manifold. If X denotes the matrix of eigenvectors, the components of the perturbation E(t/>) in the local eigenvector basis are given by TT = Ori, 772, . . . , 7Tn) =
X'^ljj).
Separating the eigenvector basis into its fast and slow subspaces 7rf and
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371
•H20 (molo/k9) 0.5
r
0.4 h
0.3 h
0.2
0.1 0.0 2.0
4.0
6.0 8.0 4>C02 (mole/kg)
10.0
Fig. 4.8. Intrinsic low-dimensional manifolds for the CO-H 2 -air system: (a) two-dimensional manifold, 0O = w 0 /M 0 , Qx = (f)C02 = w C o 2 /M C o 2 , 02 = H2O = H>H2O/MH2O, units of mol/kg; (b) one-dimensional manifold projected into C 0 2 - H 2 0 plane. The bold dotted line represents a one-dimensional manifold obtained from stead-state and partial equilibrium assumptions (reproduced with permission from U. Maas).
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77s, and remembering that perturbations in the fast subspace relax to zero, the approximation to the perturbation is given by
T I if=(iTs,0) =( o
^X-'EOA),
(4.85)
where / is the N by N identity matrix and all the other elements of the matrix are zero. Transformation back into the original basis for the "effective" perturbation gives,
~(t) = X=(Io
^X^EOA).
(4.86)
If the manifold is parametrized by 6 = Cij/, where C is the parametrization matrix determined by the standard ILDM technique, then the equation describing the flow problem restricted to the slow manifold is
- = S(6) + Pu(H0)), at
(4-87)
where P denotes the projection matrix
P=Cx(Io°ojX-\
(4.88)
The slowest chemical time-scale which is decoupled must be faster than that of the physical processes. This is the only restriction which applies, however, there are no restrictions on the nature of the perturbation which may be caused by diffusion, convection, mixing processes etc. The technique has been successfully applied to laminar reacting flows and diffusion flames and could potentially be applied to autoignition systems, although it is unlikely that the degree of reduction will be as great as that found for diffusion flames [144].
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373
(iv) Discussion The main advantage of this technique is that it requires no information concerning which reactions are to be assumed in equilibrium or which species in quasi-steady-state. The only inputs to the system are the detailed chemical mechanism and the number of degrees of freedom required for the simplified scheme. This means that the dimension of the slow manifold is chosen by the user. The intrinsic dimension of the manifold may be greater than that chosen by the user, so care must be taken to determine the accuracy of each reduced scheme as compared with the full calculation. A further consequence of the ILDM technique, is that the set of initial conditions or boundary values from the full equation system may not lie within the manifold and, therefore, may not be contained in the reduced system. Preliminary applications of the technique have led to speed-ups of over a factor of ten and, therefore, it is applicable for large-scale numerical calculations. ILDM is necessarily better suited to numerical problems since it does not result in sets of rate equations which are directly related to the inputs of the model. Therefore, the reduced model cannot be related to global reaction mechanisms which give an indication of the underlying kinetics of the system. For this purpose the slow manifold is mainly useful as a conceptual idea which can be coupled with other methods to produce an inertial form for the equations giving a greater insight into the chemistry. There are two ways to find such sets of reduced equations describing the motion of slow modes. The first is via the application of the quasisteady-state approximation and the second is via the lumping of species. 4.8.5 The quasi-steady-state approximation (QSSA) The application of the quasi-steady-state approximation is a well established technique introduced at the start of this century. The importance of the early applications led to the analytical solution of non-linear reaction systems which, without the aid of computer technology, could not otherwise be solved at that time [149-154]. Since the advent of computers and advanced software for the solution of stiff systems of equations there have been suggestions that the QSSA is an obsolete technique. Even if such an argument was valid, an understanding of the basis and applicability of the QSSA would still be needed, as emphasized by Come [155]: "since the QSSA has been used to elucidate most reaction mechanisms and to
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determine many rate coefficients of elementary processes, a fundamental answer to the question of the validity of the approximation seems desirable." Furthermore, the use of chemical mechanisms in reactive flow calculations led to significant demands on even present computer power. For this reason the QSSA is still of importance for the conversion of systems from stiff to non-stiff forms, and for a reduction in the number of variables to be solved. The QSSA has close connections with slow manifold techniques in that it depends on the existence of time-scale separation in the variables. Specifically, it involves finding those species which react on a very short time-scale so that the system can be assumed to be in equilibrium with respect to their motion. The application of the QSSA to mechanism reduction implies that the concentrations of fast intermediate species can be expressed algebraically in terms of other species, since it is assumed that their rates of change can be decoupled from the differential equations and the righthand sides set to zero. The application of the QSSA to a subset of the original species converts equation (4.1) into the following system of differential-algebraic equations: dc (1) ~ = f (1) (c,k), dt
(4.89)
0 = f (2) (c,k),
(4.90)
where c (1) and c(2) define the concentrations of the non-QSSA and the QSSA species, respectively. The rates of change of the non-QSSA and QSSA species are given by f(1) and f(2), so that by setting f(2) to zero, the number of differential equations is reduced since some have been replaced by algebraic expressions. (i) Criteria for the application of the QSSA An important aspect of applying the QSSA is the choice of QSSA species. The ultimate criterion must be the accuracy of the resulting model, but retaining this accuracy is aided by employing objective procedures for identifying QSSA species. Frank-Kamenetskii [156] was the first to investigate general criteria for the applicability of the QSSA which could be used for any mechanism and any number of intermediate species. For
Reduction based on the investigation of time-scales
375
an English translation see [157]. The formulae given below stem from Frank-Kamenetskii's early ideas. The difference in the concentrations of the QSSA species calculated from differential equation (4.1), and algebraic equation (4.90), while the concentrations of the non-steady-state species are the true concentrations calculated from equation (4.1), is the instantaneous error of the quasisteady-state approximation. The instantaneous error induced by the application of the QSSA to a single species Ac* [158] can be used to identify the possible steady-state species and is calculated by the following expression:
W^TA
1 Ac
(4.91)
Ju at
where d/;(c,k)l hk ~
.
dck _T
Since the local lifetime of a species is equal to the reciprocal of the diagonal Jacobian element for that species, Lifetime = — ,
(4.92)
J«
small errors stem from small lifetimes and/or slow rates of change for a species. Therefore, it is useful to study species lifetimes since this can also indicate possible QSSA species. Furthermore, as discussed previously, a large value of Ji7 indicates the fast decay of perturbations, and often quite large errors can be tolerated if the lifetimes are very short. An important feature of QSSA species is that their concentrations are completely determined by the concentrations of other species through equation (4.90). If the concentrations of the QSSA species calculated in equation (4.1) are perturbed slightly, this perturbation must vanish within a short time. This property of QSSA species has been noted previously, (see, for example, Klonowski [159], p. 83) who stated that "the fast
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components 'forget' their initial values". Thus, a perturbation analysis can also be used as a probe of the OSS A behaviour. Assuming a QSSA species is perturbed from a fixed point by an amount Ac,-, and using the relationship
<\t
dt
dt
where ct is the true concentration of the steady-state species and Q its concentration as calculated from equation (4.90), we can define the speed of relaxation of the perturbation as
= JAQ
^
(4.94)
dt since dCJdt = 0. We can now see that the rate of relaxation of a perturbation depends on the distance of the perturbed solution from the true solution (or fixed point), and on the magnitude of the Jacobian. If the fixed point is stationary, then the perturbed solution tends exactly to the fixed point. This case is illustrated via the reaction Y with kf= kb = 10 s" 1 (Fig. 4.9) which has a simple steady-state corresponding to an equilibrium with X = Y = 1. Solutions with initial conditions away from the steady-state eventually relax to the steady-state. If we consider a quasi-stationary system, then the fixed point moves with a speed dc(2)/d£. The Jacobian elements for the QSSA species will be large and negative as discussed in Section 4.8.2, and so the initial approach will be fast. However, since the fixed point is moving in time, the perturbed solution will never exactly reach it, but will remain at a distance Ac(2). The k
\
k
i
_ -,
scheme X >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,
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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,; ky^i
(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.
Reduction based on the investigation of time-scales
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
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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] =
K^A][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.
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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
Reduction based on the investigation of time-scales
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
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-
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 .
Reduction based on the investigation of time-scales
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
Reduction based on the investigation of time-scales
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.
Reduction based on the investigation of time-scales
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
QTAQ=T=D
+ N,
(4.111)
Approximate lumping in systems with time-scale separation
393
where D = diag(A1?. . . , An) and NC ( # " X " is strictly upper triangular. Q can be chosen such that the eigenvalues A, appear in any order along the diagonal. If Q = [ql9. . . , qn] is a column partition of the unitary matrix Q, then the q, are referred to as Schur vectors satisfying k-\
Mk = A*q* + 2 nikqi\
k = 1,. . . , n.
(4.112)
i=l
This implies that the subspaces S* = span[ql9 . . . , q„]
(4.113)
are invariant for A. At least one A:-dimensional invariant subspace is associated with each subset of k eigenvalues. Therefore, the Schur vectors can be used to form lumping matrices. Q is an orthogonal transformation so that QQT= QTQ = I making the calculation of M _ 1 simple because it is the transpose of the original matrix
GThe linear approximation to equation (4.1) is given by ^ = Jc,
(4.114)
d£
where J is the Jacobian matrix. We define new variables Z by choosing an n x n-dimensional lumping matrix QT such that Z = erc.
(4.115)
For systems with large time-scale separation there will be large gaps in the spectrum of J, i.e., a large gap in the absolute value of the eigenvalues. If we choose QT so that the eigenvalues appear in descending order on the diagonal then rj,(ll) T
Q JQ = l
0
J'(12)-|
j,(22)}
(4-116)
394
Mathematical tools
Ch. 4
where J' (11) correspond to the n - n most negative eigenvalues (i.e., the fastest relaxing time-scales) and J' (22) to the h positive or small negative eigenvalues. Therefore, the linear system for Z about the reference point is given by dZ dt
j,(11)
J'(12)-j
. 0
J' (22)
(4.117)
(ii) Partitioning Because of the gap in time-scales, Z can be partitioned as (4.118) where F is an (n - n)-dimensional vector and S is of dimension n. The equations describing the linear system can then be partitioned as follows: dF = JF + J (12) S, dt
(4.119)
dS = J (22) S. dt
(4.120)
Therefore, the system can be divided into two parts and S is essentially decoupled from F. It can be used to describe the dynamics of the whole system if the fast variables F are to be neglected. The matrix QT can also be partitioned as QJ =
(4.121)
where Qf is^.of dimension (n - h) x n and Qj of dimension h x n. The matrix Qj is, therefore, the ft x n lumping matrix spanning the slow subspaces which determines the lumped variables. The connections with the slow manifold theory of Maas and Pope now become clear. The calculation of the points on the slow manifold involves the solution of the equation
Approximate lumping in systems with time-scale separation G/f(c,k) = 0,
395 (4.122)
which is directly comparable to the Maas and Pope equation (4.75). This equation is of the same form as the steady-state equations but we no longer associate a particular species with each time-scale. The description of the motion along the slow manifold in time involves the solution of the lumped set of equations:
— =QT(c,k) = QTQJS).
(4.123)
(Hi) Problems associated with the linear approach The above approach will work well for linear systems where the Jacobian is a constant matrix. Its drawback is that for non-linear systems, where the sections of the Jacobian associated with the slow modes change significantly with time, the lumping matrix will also have to change since it depends on the form of the Jacobian. For combustion mechanisms, the Jacobian will change quickly during the ignition period and a large number of different lumped schemes may be necessary to describe any trajectory. Approaches such as CSP which are based on a linear analysis suffer from this drawback, and although they are useful for analyzing the structure of reaction mechanisms and finding locally linear reduced models, the continuous updating of local models is time consuming. Although simulation of the reduced models is faster, the whole procedure does not necessarily lead to significant savings in computational time. Linear techniques may not be applicable for highly non-linear problems such as autoignition and it is necessary to extend the ideas described above. The ILDM technique can overcome this problem by using a linear formulation but by tabulating the rate functions at selected points in the manifold. This gives the necessary information required by a numerical model but does not give the kinetic form required to provide a deeper understanding of the mechanism or for analytical investigations. For this purpose it will be necessary to investigate some non-linear extension of the above ideas.
396
Mathematical tools
Ch. 4
4.9.2 Approximate non-linear lumping in systems with time-scale separation For a system of equations such as (4.57) and (4.58), the operator A described in equation (4.39) is of the form A = A0 + eAx + eA2 + • • •,
(4.124)
where e is a small parameter. In order to lump the system, or to find equations describing motion in its slow manifolds, we need to find a canonical form for the above operator. This will be some special form enabling the separation of groups of variables just as the diagonal form did in the case of linear lumping. For example, the diagonal form of a non-linear operator with a basis of eigenfunctions
AlKico)^)-^— <-=i
(4.125)
d
where the eigenvalues A,-(o>) are the equivalent of the diagonal elements in the linear case discussed. Finding canonical forms for each of the A^'s is an easier task than for any general form A. A technique has been developed at Princeton [189191] for finding such canonical forms using an algebraic method in nonlinear perturbation theory. For the application of this technique the leading operator A0 must be chosen to be in canonical form. The basic approach is then to find a non-linear transformation C such that the resultant operator, Y=C1AC
(4.126)
is in a canonical form similar to A0. The resulting system of equations, which can be written down automatically as dc/dt = Y(c) once the form of Y has been found, can then be separated into slow and fast components. The slow components can then be selected as the lumped variables and will be given as
Approximate lumping in systems with time-scale separation c=C~1c.
397 (4.127)
Direct comparisons can be made with the Schur transformation in the linear case. Other techniques based on the use of algebraic sets for the approximation of non-linear inertial manifolds have also been developed [192] but have not yet been applied in combustion systems. We shall see, when we discuss an example in the next section, that nonlinear lumping provides a higher-order correction to the QSSA for a system of equations of any form. This higher-order correction can enable further variables to be lumped so that the low-order lumped system will be significantly less stiff than the original. The increased accuracy provided by using extra higher-order terms, however, has to be weighed up against the consequently increasing complexity of the lumped equations. Furthermore, the determination of such non-linear canonical forms may contain some complicated algebra, although manipulation packages such as REDUCE, MAPLE or MATHEMATICA could be used. The main issue is not how many variables are present in the model, but the degree of coupling between the variables. For large systems which are highly coupled it is not clear that using an algebraic technique such as non-linear lumping will be feasible. Even if computer packages can be utilized, the resulting expressions may be too complicated to lead to any significant understanding of the underlying kinetics. However, if the technique is applicable, then it can provide advantages over the tabulated methods of ILDM or the repromodelling approach, since the lumped form provides a direct link with rate parameters from the full mechanism. Furthermore, the original rate data could be modified within reasonable limits without having to recalculate the reduced scheme.
4.9.3 Application of approximate non-linear lumping to the hydrogen oxidation example In this section we will again use the hydrogen oxidation scheme to show how approximate lumping methods can lead to improved solutions over the QSSA, but to a more complex form for the reduced scheme. For the illustration of this point we will use the core set of reactions used in the QSSA section which represent the most basic ignition system.
398
Mathematical tools
Ch. 4
For the convenience of calculation, we set ?i = [H 2 ], y2 = [02],
y 3 = [H 2 0],
>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
- ye-
Ch. 4
(4.131)
The original equations can now be expressed in terms of a new basis, y{ (i = 1, 2, 3, 4) and
k4yxcp2 0(Pi
.
(4.132)
0
The algebraic method in non-linear perturbation theory can now be used to convert the operator Al into a similar canonical form. The purpose is to decouple the new purely fast variables from the remaining species so that we can assume equilibrium with respect to their motion. We seek to do this using non-linear analysis so that the resulting lumped scheme will be valid over a wide range of conditions. The reader is referred to the original texts for the details of calculation [189,190]. Once a canonical form has been achieved, leaving the new fast variables decoupled, we then assume that their motion has become exhausted. This is essentially the same idea used in both the CSP and ILDM methods except that here we have used non-linear transformations. Assuming that the variables
-kiyiy2
— = 2k3y2y4 dt
- k3y2y4 - k5y2y4{y1 + 0Ay2 + 6y3), — a, k2k4
(4.133)
Approximate lumping in systems with time-scale separation
- r = ktftf2 + 2k3y2y4 - k5y2y4(y1
dt
- k6y4
401
+ 0Ay2 + 6y 3 )
2(k2 + k4) — a. k2k4
where k3y2 a = —7- [k1y1(y1y2 - y y4 + ^ 4 ) + k3y4{2yxy2 ~ yiy* + 3y2y4) - *syiy4(y2 + ydiyi + o.4y2 + 6y3) - k&iy4].
(4.134)
Notice that the lumped scheme (equations (4.133) and (4.134)) is simply the QSSA scheme but with some extra terms added. Figure 4.15 shows the improvement in accuracy over the QSSA method gained by using only the first-order correction. Using the QSSA the maximum temperature rise is significantly overpredicted. The increased accuracy of the first-order lumped scheme is gained at the expense of an increasing complexity of the righthand sides of the rate equations. The simplified scheme can no longer be expressed in terms of a global kinetic mechanism, although it does contain a small number of variables. Again, we can see that if we wish to produce a global kinetic scheme we must discard the first-order corrections with a resulting loss of accuracy. 4.9.4 Discussion The method of lumping using time-scale separation provides a general method of reducing non-linear systems. New fast variables are defined which can be non-linear combinations of the original variables and these fast variables can be eliminated from the scheme using singular perturbation methods. The resulting system of equations essentially describes the motion along the slow manifold of the system and is perfectly good for long-time behaviour. These general methods improve on the QSSA in two ways. First, the new fast variables are combinations of the original species, so that more fast variables than QSSA species may be found. Second, a higher-order accuracy can be achieved so that the methods may apply in regions of composition space where the QSSA does not. The
402
Ch. 4
Mathematical tools
— exact solution zeroth order approx. — first order approx.
0.08
2e+17 1.5e+17
,
,
J
, — exact solution
'•A
ilcs
|
'
0.1
zeroth order approx.
le+17
first order approx.
j
Mil
J
5e-fl6
cone
c
c
=
-5e-fl6 -1o_L1 7
-j
j i
i
0.02
0.04
1
0.06
1
0.08
1
0.1
t, s Fig. 4.15. A comparison of the zeroth- and first-order lumped schemes for an initial mixture H 2 : 0 2 = 2 : 1 , pressure = 20 Torr, Ta = 800, 1000 K: (a) Temperature profiles; (b) Concentration of H 2 ; and (c) Concentration of H.
main problem with the general approximate non-linear lumping technique is that it can lead to complicated algebraic expressions which might be difficult to manipulate automatically. The resulting equation system, although being more accurate than that generated by lower-order methods such as the QSSA, may contain complex righthand sides. Therefore, in-
1.6e+17
I
1.4e+17
h
conc(intra tion, mole cules cm"
Fitted kinetic models
1.2e+17
f-
B.
0
T
403
i
l
l
— e x a c t solution zeroth order approx. — f i r s t order approx.
'.'••.. :'"••• : "*'••...
le+17 8e+16 6e+16 4e+16 2e+16
1
.U t
-2e+16 0
0.02
i
0.04
l
0.06
l
0.08
0.1
t, s Fig. 4.15c.
creasing generality and accuracy is gained at the expense of ease of application and efficiency.
4.10 FITTED KINETIC MODELS
The systematic reduction of large detailed reaction mechanisms using the application of a series of approximations leads to a smaller number of kinetic equations. As shown in the previous sections, this smaller model can be formulated as a set of a few global reactions in many cases. The rates of these global reactions can be related to the rates of the elementary reactions in the original scheme. Another way of generating reaction schemes consisting of global reactions is to fit the parameters of one or a few global reactions to some kind of kinetic data. The data may come from experiments or detailed chemical kinetic simulations. Fitting a small kinetic scheme requires the optimization of the parameters of a system of ordinary differential equations. If the reaction scheme consists of elementary reactions, and the parameters are optimized only within the physically feasible boundaries, the approach of the GRI mechanism (Section 4.3.7) is reproduced. The fitting of parameters in small schemes is described in the following sections.
404
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4.10.1 Fitting a system of odes to experimental data The creation and parameterization of small global reaction schemes has a long history. Such empirical kinetic models were used before detailed kinetic models. Even now, when large detailed mechanisms are available for many fuels, and there are techniques for the systematic reduction of these mechanisms to global reactions, empirical mechanisms have some merits. Elementary reaction mechanisms may not accurately reproduce all global features, and the kinetic assumptions applied at the deduction of the global mechanism may allow calculated results to differ from bulk experimental results. Empirical mechanisms fitted to experimental data may accurately reproduce bulk experimental observations. Some reaction mechanisms, such as the combustion of large hydrocarbon molecules, are so complex that global reaction schemes obtained by systematic reduction have not been published. An even more important point is that for the combustion of some practically important fuels and fuel mixtures such as petrol, detailed reaction mechanisms have not been created, yet it is possible to produce empirical schemes by fitting to experimental data. Global kinetic models for high-temperature gas-phase reactions were reviewed by Abdalla et al. [193] and Babushok and Dakdancha [194]. Empirical global kinetic schemes for the low-temperature oxidation of hydrocarbons have been recently reviewed by Griffiths [11]. Here, only some characteristic models are mentioned and for a full account the reader is referred to the review by Griffiths. The first empirical model for hydrocarbon ignition was created by Gray and Yang [195] in 1965. This simple model, given below, has two variables: a single concentration variable x and temperature T. —» X X —» X —» 2 X X —>
initiation termination 1 branching termination 2.
Temperature-dependent rate parameters and heat-release rates were assigned to the reactions, including different values for the two termination reactions. The concentration of the starting fuel, and therefore, the rate of initiation, was held constant. This model was found to be in close agreement with the main thermokinetic features of the experimental/? - Ta
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diagram obtained from propane oxidation in a closed vessel. The GrayYang model is not able, however, to reproduce multiple-stage ignition and, therefore, was extended by Wang and Mou [196]. They added a further branching reaction and allowed the concentration of the fuel to vary. A very good match to multiple-stage ignition data was obtained by the careful selection of rate parameters. Other derivatives of the YangGray model have been used in models for the simulation of autoignition in engines [5]. Unlike the Gray-Yang model, the Shell model and its derivatives use both real and lumped chemical species. The Shell model was developed over a number of years, and in its most general representation [8] takes an empirical form. It has been developed further by Cox and Cole [3], Keck and Hu [197] and by Griffiths et al. [114]. In all these models the basic structure of the oxidation scheme is realistic, although some of the species are lumped and include a general alkane RH and general alkyl radical R, etc. The starting rate parameters for each reaction are taken from the literature, but some of them are subsequently tuned to help the model reproduce bulk experimental data such as ignition delays. A two-step global mechanism has been derived by de Soete [198] for the description of NO and N 2 generation from fuel nitrogen. Parameters of the model were fitted to concentration profiles in flames. This model is widely used in post-processor packages of commercial CFD codes, although it works poorly under fuel-rich conditions. The first global empirical model for moist CO combustion was created in 1956 [199]. In a more recent article, Yetter et al. [200] compared previous global models for CO oxidation and suggested a procedure for a more systematic generation of global models based on initial concentration sensitivity calculations. Probably the first global scheme for the high-temperature oxidation of hydrocarbons was presented by Edelman and Fortune [201]. Their mechanism combined overall and elementary kinetics. The oxidation of hydrocarbons to CO and H 2 was represented by an overall expression, while the subsequent oxidation of CO and H 2 was described by elementary steps. A drawback of this model was the absence of coupling between the oxidation of fuel, CO and H 2 . In a real oxidative environment there is a competition of the radicals for all the above molecules, which results in a reduction of the CO and H 2 oxidation rates. Experimental results from the Princeton turbulent flow-reactor have
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indicated that the oxidation of hydrocarbons takes place in several steps. As a first step, hydrocarbons decompose to olefins. In the olefin-producing stage, even after the consumption of half of the original fuel, the temperature remains almost constant. The oxidative pyrolysis of the original paraffin to olefins is endothermic, but this endothermicity is balanced by the oxidation of H 2 . The oxidation of olefins to CO is also endothermic, and most heat is generated during the oxidation of CO to C 0 2 . The global mechanism of Hautman et al. [202] includes all these major stages: n C„H2„+2 —> - C 2 H 4 + H 2 C 2 H 4 + 0 2 ^ 2 C O + 2H 2 CO + - 0 2 ^ C 0 2 2 H2 + - 0 2 - > H 2 0 2 The rate of each reaction step was given as the multiple of rate parameters in Arrhenius form, and various powers of concentrations of species, which were present or not present in the given global reaction. The Arrhenius parameters and the powers of concentrations of the four global reactions were fitted to experimental data, and the model accurately described the flow-reactor and shock-tube results over a wide range of equivalence ratios, initial temperatures and pressures. By fitting the pre-exponential factor of the first global reaction, the scheme also reproduces experimental data for a series of other hydrocarbons, excluding methane and ethane. Westbrook and Dryer [203, 204] have studied the application of several global mechanisms for the oxidation of hydrocarbons in flames. Their onestep model was of the following form: Fuel + nx02 -> n2C02 + n 3 H 2 0, kov = ATn expf—\ [Fuel]fl[02]6.
(4.135)
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Parameters nu n2 and n3 were dictated by the type of fuel, while parameters A, n, Ea, a and b were fitted to experimental flame velocity data. These parameters were determined for 17 different fuels and, in all cases, the predicted flame speeds agreed well with the experimental data over a wide range of conditions. However, these one-step models overpredict the burnt-gas temperature, mainly due to the lack of accurate calculation of C 0 2 and H 2 0 concentrations at high temperatures. The applicability of one-step mechanisms for the description of flames was debated by Coffee [205]. The following two-equation model [202], provides a better adiabatic flame temperature and a reasonable estimate of the CO concentration at equilibrium: Fuel + n102-^
n2CO + n 3 H 2 0
co + §o2->co2 As a third step, the global model for hydrocarbon combustion was extended with a detailed model of 21 elementary steps for moist CO combustion, resulting in a quasi-global model similar to that of Edelman and Fortune. This model [202] predicted the species profiles as well. A similar model was almost simultaneously proposed by Duterque et al. [206] for five fuels. More recent work on the generation of global reaction schemes for the combustion of alkane hydrocarbons up to isobutane in flames, has been carried out by Jones and Lindstedt [207], Their global model included two fuel consumption steps: Alkane + n102 —» n2CO + n3H2 Alkane + n 4 0 2 —> n5CO + n 6 H 2 H2 + | 0 2
20
CO + H 2 0 ^ C 0 2 + H 2 The rates of the last two reactions were obtained by partial equilibrium assumptions, while the rates of the first two reactions were tuned to reproduce both the measured flame speed and the observed CO to H 2
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ratio in methane and propane flames. This model provided an accurate reproduction of experimental data such as flame speed and species profiles. Examples of high-temperature hydrocarbon oxidation are interesting because they highlight the advantages and limitations of empirical kinetic model development. In general, these kinetic models are less successful than the equivalent systematically reduced global models. However, the one-step models are able to reproduce a single experimental feature over a wide range of experimental conditions. Multistep models are able to provide a description of several experimental features over a limited range of conditions. Although five to seven parameters are generally fitted for the rate expression of each global reaction, these expressions are rather simple compared to the complex analytical expressions which describe the dependence of global reaction rates on the concentration of all species present in the model. These relatively simple expressions provide, in several cases, an insufficient approximation, but on the other hand allow the fast evaluation of rate expressions. In all the above cases the global reactions were formulated on the basis of chemical knowledge. However, mathematical methods could be used for planning the structure of global reactions mechanisms. Target factor analysis [208] can be applied to aid decisions about whether or not an assumed global model is compatible with experimental data. The target stoichiometric vector is projected onto the stoichiometric space of the experimental data. If the target and the projected vector are close, based on an F-test, then the target is taken as an acceptable reaction stoichiometry. This approach is useful for either deriving simple, approximate stoichiometric models for complex systems, or for testing possible stoichiometrics and thereby investigating reaction pathways. In another approach, the reduction of the originally proposed reaction mechanism and the estimation of parameters are combined, using parameter adaptive random search and ridge selection analysis. This method is called the concomitant parameter estimation-model reduction (CPEMR) method [209-214]. Here, some of the rate parameters from the original mechanism are assigned to zero, and all the others are refitted to the experimental data, resulting in a smaller and better fitting kinetic model. This method, together with other numerical procedures, was built into an expert system [215] for chemical kinetic model identification.
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4.10.2 Fitting a system of odes to detailed kinetic simulations While fitting parameters of a small system of odes to experimental data lies in the realm of empirical modelling, fitting parameters in a similar way to results of calculations from a detailed kinetic model can be considered as a branch of mechanism reduction. The following two-variable empirical model was used by Korobeinikov et al. [216] to give a simple approximation of ignition delay times in shock waves: da dt ~
dl dt
=
1
~k2p2
(4.136)
[M-I
(4.137)
where r, is the induction time of the reaction, a is the fraction of induction time remaining, /3 is the fraction of reactant remaining, Q is the chemical energy of the mixture, k2 is the reaction rate and E2 is the activation energy. During the induction period, no energy is released so long as a > 0. The chemical composition of the fluid is not assumed to change during the induction period. This model was used later by Taki and Fujiwara [217] in two-dimensional shock-wave calculations. Neither Korobeinikov et al. [216] nor Taki and Fujiwara [217] fitted the parameters to detailed mechanisms, unlike Oran et al. [218, 219] and Clifford et al. [220] who derived similar shock-tube models. Oran et al. [218, 219] developed a global parameterized model which describes the chemical induction time as a function of temperature and pressure. Parameters of the induction time function were determined for stoichiometric hydrogen and methane in air mixtures. The parameters were fitted to numerical results obtained from the simulations based on detailed reaction mechanisms. This technique allowed a 22-times faster calculation of the induction time and reduced the simulation time in a onedimensional model by a factor of 7.5. The fitted model was used in twodimensional shock-wave simulations. The work of Oran et al. was continued by Clifford et al. [220], who used
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the model of Korobeinikov et al. and fitted its parameters to detailed kinetic simulations for the description of shock-induced ethylene combustion. They used the fitted model in a two-dimensional shock-tube calculation, where the variables were a, /3, temperature and pressure. In this case, calculations with the fitted model were 20 times faster than with a reduced model of elementary reactions, and the internal energy, the j8 profile and the propagation of the shock were modelled fairly well. The first attempt to derive overall reaction rates systematically using information available from a detailed laminar-flame model was made by Coffee et al. [221, 222]. The overall reaction rate parameters were found from a least squares fit of the heat release profile. The single reaction model gave accurate results for flame speed and temperature profiles, but the optimal kinetic parameters varied with equivalence ratio, initial temperatures and pressures. Kiehne et al. [223] have described the hightemperature oxidation of propane by an eight-step mechanism including seven reactive species. The rate parameters of the eight global reactions were varied until a good agreement was achieved with detailed kinetics predictions. The model reproduced well gross features such as flame speed, flame temperature, postflame composition and ignition delays. In a recent article, Revel et al. [224] demonstrated a systematic method for the production of a global mechanism fitted to a detailed mechanism for NO generation in methane flames. The global reactions were deduced on the basis of element flux calculations and pathway analyses. The global mechanism of 10 species and 6 reactions is able to reproduce accurately the ignition delays, temperature profiles, and concentration profiles of major species and NO over a wide range of fuel-to-air ratios and initial temperatures. Jiang et al. [225] fitted a five-step global mechanism to a detailed methane oxidation mechanism. They identified the dominant reactions in the detailed mechanism and determined the rate expressions for the steps of the global mechanisms on the basis of the rate of the corresponding reaction rates of the detailed model. The reduced model was also tested in reactive shock calculations. 4.10.3 Fitting algebraic equations to detailed kinetic simulations Chemical kinetic models usually take the form of ordinary differential equations or differential-algebraic equations. Even when reaction phe-
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nomena are constant in time, such as in the case of stationary reactors, the solution is calculated via the stationary point or points of odes. It is clear that in the case of stationary reactors, a model could be represented by algebraic equations, calculating the stationary concentrations as a function of input-flow concentrations, temperature, pressure and residence time. Similarly, as in the previous sections, these algebraic equations can either be deduced from detailed kinetic models, or fitted to experimental data. At a first glance, it is not so obvious that algebraic models can be used for the description of non-stationary kinetic processes where a description of changes in species concentration is required over time. The key lies in the replacement of the kinetic odes by difference equations: Y(f + Af) = F(Y(0,p),
(4.138)
where Y is the vector of system variables, p is the vector of parameters, and F is an algebraic expression that produces the value of system variables at a time M later on the basis of the previous values of system variables and parameters. Consecutive calculations of the expression F of the above difference equation provides the time history of the dynamical system. The first application of this approach was by Dunker [226] who used it to give a reduced representation of a smog reaction mechanism. Dunker assessed the typical ranges of system variables (species concentrations) using a full mechanism, and formed a grid in the space of variables which included all realistic concentrations. First- and second-order initial concentration sensitivity coefficients were calculated by the decoupled direct method (see Section 4.5.2) using a detailed reaction mechanism. Based on this sensitivity information, the algebraic function F was given as secondorder Taylor polynomials. A drawback of this approach is that a different set of polynomials were related to the neighbourhood of each node in the grid. Coefficients of more than 20,000 polynomials had to be stored in a huge look-up table, and finding the right polynomial slowed down the calculations. In spite of this, the calculation of concentration curves using the polynomials was 300 times faster than the simulation of the original kinetic model. Almost simultaneously, Marsden et al. \221], as discussed in the review by Frenklach [24], developed a different approach for the determination of function F. In their method, the original kinetic model was used in a
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series of calculations, and the concentrations at each time t were arranged according to a factorial design. Coefficients of a single second-order polynomial were fitted to the calculation results. Unfortunately, they did not demonstrate that this single second-order polynomial is capable of describing chemical changes for a wide range of concentrations. The method of fitting of parameters of algebraic equations to detailed kinetic simulations was developed further [228,229] on the basis of the repro-modelling approach [230]. Repro-modelling is a generally usable method for the efficient simulation of large computational models by identifying the computationally-intensive submodels and replacing them by simple fitted models. Many kinetic models are used as the chemical component of a CFD code. Frequently, the method of operator splitting is used so that the calculation of the chemical changes, and concentration changes due to advection and diffusion, are separated. In terms of the CFD code, it is unimportant whether the chemical changes during each time-step are calculated by an ode solver or by approximating algebraic equations. Therefore, the repro-modelling approach seems a feasible option for generating reduced chemical representations for use in CFD models. The first stage in the formulation of a repro-model is the generation of suitable data. Usually, several thousand typical data points are considered and a detailed model is used for the calculation of the concentrations as a function of time. The evaluation of a large number of data points requires quite intensive computation, since, large sets of odes must be solved for all conditions. However, this effort has to be expended only once, and when the data have been fitted to polynomials, the repeated kinetic calculations, which are generally carried out during every CFD calculation, are avoided. The second stage in generating the repro-model is to fit suitable algebraic relations to the input data. Application of high-order polynomials has many advantages, since such polynomials can approximate very non-linear relations in a simple way. General high-order polynomials have a large number of coefficients and the application of such polynomials is feasible only if most of the coefficients are set at zero. At most there should be a few dozen non-zero parameters if the greatest time savings are to be achieved. The procedure has to be suitable for fitting a polynomial to tens of thousands of data points, determining the non-effective parameters and fitting the effective parameters using the method of least squares. This task
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has been solved in [231], and another, possibly a more elegant solution, was proposed in [228] based on orthonormal polynomials. Let xl = (JCI, xl2,. . . , xlm), i = 1,. . . , n, denote a set of data, and let 0 ; , j = 1,. . . , / be a set of scalar-valued functions of an m-vector. A scalar product of functions 7 and (f>k is given by n
(
H^.(X')<MX').
(4.139)
i=l
This scalar product belongs to the set of data x' and non-negative weights wi9 i = 1,. . . ,n. Functions j and k are said to be orthonormal with respect to this scalar product if, and only if, (7, k) is equal to zero for j y£ k and 1 for j = k. Note that this is a discrete orthonormal property unlike the usual orthonormality of functions based on integrals. Any function G can be approximated using an orthonormal set of functions (7, ; = 1 , . . . , / ) by an orthonormal (Fourier) expansion, F = I (G, ,) ,,
(4.140)
7=1
When this method is applied to the polynomial approximation of mechanisms, the function G is the response of the kinetic model calculated using the original detailed reaction mechanism, and is a series of orthonormal polynomials constructed by a Gram-Schmidt orthonormalization process using the data set. The function F, defining the final algebraic model, is constructed in such a way that only the significant members of the summation are considered. The result of the fitting procedure is a set of multivariate high-order polynomials. The efficiency of evaluating such polynomials can be improved [231] significantly by using appropriate factorization. This can be achieved by converting the polynomials to the multivariate equivalent of Horner equations and an algorithm is given in [228]. The above procedure has been applied for the description of the homogeneous ignition of wet CO [229] and also for the spread of H 2 /0 2 shock waves [232]. In both cases the agreement between the algebraic model and the solution of the kinetic odes was very good and the simulation of the algebraic model required several orders of magnitude less computer time.
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The algebraic model has been successfully used in Monte-Carlo simulations of turbulent jet-diffusion flames [233]. Variables in the algebraic model can be identical to those in the approximated system of differential equations [228]. Another possibility is to approximate only a subset of the original variables, including the important variables and those which have the greatest influence on the important variables of the model. For example, a repro-model can be constructed which describes only the evolution of the slow variables [229]. This means an indirect utilization of the very different time-scales characteristic of combustion systems. In most applications of time-scale separation the basic problem is to derive expressions which describe the time derivative of slow variables as a function of the same slow variables. The algebraic equations, resulting from the repro-modelling, contain these dependencies and therefore constitute an implicit application of slow manifolds. There are possible alternative ways for the construction of algebraic models. If the look-up tables of the intrinsic low-dimensional manifold method are fitted by polynomials [234], the result is an algebraic model similar to a repro-model describing only slow variables. Polynomials can be fitted to the integrated solutions of few-step global mechanisms [233]. Such integrated solutions are found in the look-up tables used in the Monte-Carlo method for the simulation of turbulent flames. Another fitting procedure, based on the application of neural networks has also been developed [235-238]. The artificial neural-network approach has been shown to produce algebraic models that are good approximations of kinetic mechanisms. Simulations based on this model are much faster than the solution of differential equations but slower than the use of an algebraic model obtained from the polynomial approximation. 4.10.4 A fitted algebraic model for wet CO ignition The first application of the polynomial approximation to combustion systems describes the temperature and CO and 0 2 concentration profiles during the ignition of a wet carbon monoxide-air mixture. The detailed mechanism was identical to the CO mechanism that has been used as an example in ILDM calculations [6]. All in all, 300 initial conditions, uniformly distributed between ambient temperatures of 990 and 1010 K and fuel-to-oxygen ratios 0.5 to 1.5, were used in spatially homogeneous simulations. Temperature and mole fractions of CO and 0 2 were recorded every
Fitted kinetic models
0.180
415
r
0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 X(CO)
0.40
Fig. 4.16. Carbon monoxide-oxygen phase plane. The solid lines give the trajectories corresponding to a starting temperature of 1000 K and fuel-to-air ratios 0.5, 1.0 and 1.5, respectively, up to a simulation time of 0.01 s. The upper ends of the lines belong to the initial unburned mixture. For the preparation of the repro-model the initial composition of the mixture was uniformly distributed between 0 = 0.5 and 1.5. Dots represent the values of CO and 0 2 concentrations used for the generation of the repro-model. Only a part of the 30,000 data points are plotted on the figure for clarity.
10~4 s up to the final simulation time of 0.01 s. The distribution of the data is illustrated in Fig. 4.16. The values of temperature, [CO] and [0 2 ] at the time (t + 10~4) s were fitted as a function of the three variables at time t. The fitted functions were fourth-order polynomials with between 22 and 25 non-zero parameters. The average deviations between the results of detailed simulations and values calculated from the polynomial were below 0.26% over the range of conditions covered. A FORTRAN program was used to convert the polynomial coefficients into FORTRAN functions optimized for multiplications. An example of such a computer-generated function is given in Fig. 4.17. An alternative version of the FORTRAN function-generator program has also been elaborated. In this version, the powers of variables are calculated sequentially and then applied to the independent calculation of
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function h31(y) Implicit real'8(a-h t o-z) dimension y(*), a(22) data a/ 1.25887E+04,-9.64505E+00,-4.77828E+04, 6.51249E+01, •-1.27202E+01,-1.73475E+04,-2.09878E+04,-8.07198E+04. 1. 70698E-06, •-2.62401E-02, 1.73978E-02, 2.16879E+01, 1.51500E+04,-1.13259E+06, •-3.28281E-10, 3.26475E-06.-4.65053E-06, 1.36031E-03, 3.39191E+02, • 7.03503E+04, 1.98323E+06, 3.01339E+06/ h31 =a(l)fy(3) , y(3)"a(8)*y(2) B (a(3)*y(3)*(a(7)+y(3) # (a(14)+y(3)« •a(22)))*y(2)»(a(6)+y(3)'y(3)*a(21)+y(2)Ma(13)+y(3)'a(20))))+y(DMa(2)+y(3)»(a(5)+y(3)"a(12))+y(2) B (a(4)+y(3)-y(3)»a(19))+y(l)«(y(3 •)'a(ll)+y(2)-(a(10)+y(2)-a(18))+y(l)»(a(9)+y(3)'a(17)+y(2)*a(16)+y •(l) B a(15)))) return end Fig. 4.17. An example of a fully optimized computer-generated multivariate Horner equation. The repro-model obtained consists of a set of such FORTRAN functions. This function calculates the temperature 1 0 - 4 s later, given the temperature and mole fractions of CO and 0 2 .
monomials. The evaluation of FORTRAN functions in this way proved to be very much less efficient on a single-processor workstation than the fully optimized functions, but these types of function (see Fig. 4.18) may have advantages on multiprocessor computers. Concentration-time and temperature-time curves calculated from the polynomial approximation are in close agreement with the simulation results from the detailed kinetic model (see Fig. 4.19). Calculation of the curves with the polynomial approximation was 11,700 times faster. This gain in computer-time is unexpectedly high, although a 104-times increase in calculation speed has been reported in another application of repromodelling [230]. 4.10.5 Discussion Fitting small mathematical models to kinetic data is an alternative to the reduction of large kinetic models. The mathematical models can be systems of ordinary differential equations (odes) or algebraic equations (including systems of difference equations) and the kinetic data may be either simulation results of a detailed model or experimental data. When simulation results are fitted, the original experimental information might be distorted by the inaccurate large model. However, experimental data
Fitted kinetic models
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function s31(y) dimension y ( a ) , a (22) data a/ 11.25887E+04, -9.64505E+00, 4.77828E+04, 6.51249E+01, •-1.27202E+01 -1.73475E+04, - 2 . 09878E+04, - 8.07198E+04, 1.70698E-06, •-2.62401E-02 1.73978E-02, 2 . 16879E+01, 1.51500E+04, - 1 . 13259E+06, •-3.28281E-10 3.26475E-06.-4 .65053E-06, 1.36031E-03, 3.39191E+02, • 7.03503E+04 1.98323E+06, 3 .01339E+06/ yll = y(l) y l l • y(l) yi2 yl2 • yd) yi3 yl3 • yd) yl4 y(2) y2i y21 • y(2) y22 y22 * y(2) y23 y(3) y3i y31 • y(3) y32 y32 y(3) y33 s31=a(l)+a(2)»yll+a(3),y21+a(4),yll,y21+ a(5)»yll*y31+a(6)*y22+a(7) ••y21»y31+a(8)*y32+a(9),yl3+a(10)»yl2"y21 +a(ll)-yl2*y31+a(12)*yll-y *32+a(13),y23+a(14),y21,y32+a(15)-yl4+a(l 6)*yl3»y21+a(17)*yl3»y31+a •d8)«yl2»y22+ad9)*yll*y21»y32+a(20)'y23 •y31+a(21),y22*y32+a(22)«y •21»y33 return end Fig. 4.18. This equation is identical to the equation in Fig. 4.17, but here the powers of variables are calculated sequentially and then applied to the independent calculation of monomials.
are usually too sparse and not appropriate as a basis for fitting a small model that is intended for use as an extrapolation and interpolation tool. The key problem in making a small fitted ode model is not the determination of the values of the parameters, but finding a small set of odes with optimal structure. So far, the main approach has been to set up a skeleton mechanism that corresponds to chemical kinetic knowledge about the system. Arrhenius-type expressions are used for the description of the temperature dependence of the reaction rates, and the powers of concentrations in the rate expressions are parameters to be fitted. This way of setting up the small systems of odes is heuristic, but the fitting of parameters has been an automatic process based on the least-squares method. So far, fitted kinetic odes have been less successful than reduced mechanisms. One reason for this limited success may be a result of the hybrid chemical-mathematical approach for finding the ode models. It is expected that in the future, fully mathematical methods will be used for finding
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0.00 0.000
0.001
0.003
0.004 0.006 time/ s
0.007
0.009
900 0 .010
0.00 0.000
0.001
0.003
0.004 0.006 time / s
0.007
0.009
0.010
Fig. 4.19. (a) Concentration-time and temperature-time curves for a stoichiometric mixture. Solid lines represent the solution obtained from the original detailed mechanism. The dotted lines were calculated using the three-variable repro-model. Calculation of the curves with the repro-model was 11,700 times faster; (b) Concentration-time and temperature-time curves for the same initial conditions as in (a). Here the dots were calculated using the two variable repro-model, and the calculation of the curves with the repro-model was 24,000 times faster.
Conclusions and future directions
419
optimal small systems of odes which can accurately describe the kinetic data fitted, and can be used as an extrapolation tool. Fitted algebraic models offer very high simulation speed and also good accuracy over the parameter space for which they are fitted. In spite of some successful applications, it is not yet clear if the high simulation speed is accompanied by robustness and flexibility. If algebraic models prove to be robust enough, they may have wide applications in combustion simulations. In combustion models, thermodynamic data such as the heat capacities of each species, are not calculated from first principles, but stored in the form of high-order polynomials as a function of temperature. Maybe in the foreseeable future complex kinetic models for many fuels and conditions will also be available in the form of polynomials. It will then be possible for combustion simulations to concentrate on reproducing complex fluid behaviour, rather than spending a large amount of computational effort on chemical kinetic calculations.
4.11 CONCLUSIONS AND FUTURE DIRECTIONS
In this chapter we have tried to give an introduction to most of the mathematical methods that have been used for the construction, investigation and reduction of combustion mechanisms. In terms of application, these methods differ very much in their present state and future prospects. Some have become well established techniques, and consequently are likely to be widely applied in the future. Others have only been introduced over the last few years, and although they have proven successful on a few test systems, they have yet to prove themselves on a wide range of applications. Although not presently available, we expect that a generally applicable program for mechanism generation will be made available in the future. Such a program will produce a complete mechanism, and will be linked to ever-expanding kinetic databases (as discussed in Chapter 3) to provide Arrhenius parameters and the corresponding thermodynamical data. For those reactions which are not available from existing databases, an essential module of this package will be for the estimation of such rate parameters. Once generated, there is a need to validate the mechanism both against experimental data, and in terms of the uncertainty of the parameters it contains. Of all the techniques we have discussed, the most well established
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are the simulation packages available for the numerical investigation of homogeneous combustion systems and one-dimensional boundary value problems. All of these packages incorporate well-tested integration methods which can easily cope with stiff systems. Most packages are also able to calculate local concentration sensitivities and therefore provide, not only raw output data for comparison with experiment, but also additional tools for the investigation of the mechanism. These local sensitivities can be used for a rough estimation of the effect of parameter uncertainties. We hope, in the future, to see the use of global sensitivity methods for uncertainty analysis of combustion systems. Such calculations have not been done previously because of the high computer-time requirements and the limited availability of uncertainty data for the reaction coefficients. Generally, local concentration sensitivities are used for finding the parameters that have to be known with high precision, and for the identification of rate-limiting steps. An important achievement in the field of sensitivity analysis has been the introduction of principal component analysis as a method for the interpretation of the large amount of information contained in the sensitivity matrix. In the future, a more wide-spread application of principal component analysis is expected for the interpretation of concentration sensitivity results. This would help the extraction of further mechanistic details, or could be used for the detection of redundant reactions. The calculation of initial concentration sensitivities is very useful in some cases, but most simulation packages cannot calculate these sensitivities and there is no sign that such a feature will be incorporated. For the reduction of chemical mechanisms, reaction-rate analysis has probably the largest record of success. A novel way for the inspection of rates is based on the study of algebraic-rate sensitivities and the Jacobian matrix. These methods can be used for the automatic identification of redundant species and reactions, to produce a reduced mechanism consisting of a subset of the original mechanism. The use of algebraic manipulation in techniques such as the QSSA and lumping, make the production of a reduced mechanism essential and make subsequent calculations as simple as possible. CSP is an alternative to rate-of-production and sensitivity methods for mechanism reduction. Again it provides an automatic selection of the important reactions, and also provides a time-scale analysis which can be used in the later stages of the reduction process. Generally, the removal of redundant terms from the equations does
Conclusions and future directions
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not result in the size of mechanism being suitable for large-scale CFD calculations. Therefore, it is necessary to condense the remaining set into some lumped form. This may lead to the use of new variables, or even in the representation of the reduced set of variables by a look-up table or a polynomial function. We have shown that the application of the optimum lumping scheme generally relies on some physical insight into the problem. The lumping could be based on the structure of the reaction scheme as in chemical lumping, or on the range of time-scales involved in the reaction. A number of different techniques exist which are based on a time-scale analysis and these are likely to remain at the centre of mechanism reduction. The simplest and most widely used technique involving the separation of time-scales is the quasi-steady-state approximation or QSSA. It is likely to remain the most generally applied method both as a thinking aid, and for the construction of systematically generated global mechanisms. However, it is expected that the intuitive usage of QSSA will be replaced by more formal mathematical techniques for the selection of QSSA species. The use of empirical global reaction schemes, fitted to either experimental data or a detailed chemical model, will continue to have applications in the future. A breakthrough in the automatic generation of such models is expected using more sophisticated mathematical tools. A prediction of the errors induced by such low-dimensional schemes might be developed using similar ideas to those applied to the application of the QSSA. A possible limitation of the QSSA is that it may not provide the minimum low-order system. Frequently, the number of fast modes, as determined by an eigenvalue analysis of the linearized system, is greater than the number of QSSA variables. The slow manifold in this case is of a lower dimension than the remaining set of differential equations after the application of the QSSA, and a further reduction could be made. This problem is avoided by the slow-manifold technique of Maas and Pope which is based on the generation of look-up tables describing the properties of selected species on the slow manifold. This technique seems entirely appropriate for those combustion systems where the eventual aim is to carry out CFD calculations. If the emphasis is on quick evaluation of chemical changes, another promising technique is the generation of polynomials based on the repro-modelling approach. Initial results have shown speed-ups of several orders of magnitude beyond those capable using lookup tables. Such polynomials could also be obtained from fitting look-up
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tables. It is expected that these polynomials will provide an efficient way of representing chemical change in transport models. Both the polynomials and the look-up tables lead to efficient representations of the chemistry, but if representation by equations is required, then we must turn to methods such as approximate non-linear lumping, where the species are transformed into purely fast and slow variables and equations for the motion along the slow manifold defined. So far, this technique has been applied only to simple systems, and it is possible that for a large and highly coupled mechanism the resulting equations would be of a very complex form. The automatic application of this technique could be developed with the use of tools based on algebraic manipulation packages such as MATHEMATICA, but it is likely that it will be applicable to only special cases, at least in the near future. Chemical lumping (a special form of linear lumping) could prove very useful in areas such as the combustion of hydrocarbon mixtures or soot formation. The automation of chemical lumping could benefit from the formal procedures developed for linear lumping. A World Wide Web site [239] has been set up to provide software for the investigation and reduction of combustion mechanisms. The programs available include MECHMOD, a code for the automatic modification of CHEMKIN format combustion mechanisms, and KINALC, which is an almost automatic program for the investigation and reduction of gas-phase reaction mechanisms. KINALC is a postprocessor to CHEMKIN-based simulation packages SENKIN, PREMIX, OPPDIF, RUN1DL, PSR, SHOCK and EQLIB. MECHMOD can transform all reversible reactions to pairs of irreversible reactions, eliminate selected species from the mechanism, convert the pre-exponential factors and the activation energies to different units. KINALC processes the concentration sensitivity information, calculates the sensitivity of objective functions and carries out principal component analysis of the sensitivity matrix. The program can suggest a list of ratelimiting steps or carry out a rate-of-production analysis giving a summary of important reactions. Alternatively, it includes a method for finding redundant species and also redundant reactions on the basis of algebraic rate sensitivities. Uncertainties of simulation results can also be calculated from reaction uncertainties and local concentration sensitivities. QSSA analysis can be carried out from the calculation of species lifetimes and the estimation of instantaneous QSSA errors, thus enabling the proper
Conclusions and future directions
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selection of QSSA species. Other methods based on the utilization of different time-scales are also included such as CSP analysis, the calculation of ILDM dimensions and the effect of perturbation on the movement on the manifold. All the programs described here, including KINALC, although providing a certain amount of automation to the reduction process, are based on the programming style of the seventies. The input of these programs is based on keywords and output is a text file. It is expected that before long, program packages will appear which can combine both the simulation stage, and the mechanism investigation stage, the output being in a visual form such as a picture or a movie. The development of a visual based interactive package for the investigation of mechanisms would be ideal. Any physical insight required by the modeller could then become partially automated, aided by the visualization of the reaction trajectory. Since many of the techniques we have described are dynamical systems based, a moving image of the reaction dynamics would be very useful in providing extra insight. Virtual reality is another unexplored area for combustion models, and the present graphics hardware available on even the cheapest workstations provides an excellent background for such programs. One of the first steps in this direction is the program XSenkPlot, a graphical postprocessing program for CHEMKIN. XSenkplot utilizes X Windows and OSF/Motif Graphics routines that are supported on UNIX workstations. The inputs to this analysis package are the chemical mechanism and the time/temperature-dependent species concentrations describing the chemically reacting system. The later input is the solution to the actual initial and/or boundary value problem provided by CHEMKIN programs Senkin, Premix or PSR. The screen for each chemical kinetic calculation simultaneously displays a variety of characterizations in multiple windows and allows analysis of time/temperature-dependent species and reaction information including species concentrations, species steady-state analysis, individual reaction rates, net production/destruction rates, reaction equilibrium analysis and the temperature/time history of the system. The interactive user-sorting of the species and reaction information from the numerical simulations is mouse/cursor driven. An additional feature also allows interactive analysis and identification of dependent and independent species and reaction pathways, on-line reaction network analysis and pathway/flowchart construe-
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tion. The program is downloadable from the World Wide Web site of the NIST [240]. Throughout this chapter the emphasis has been on the formalization of procedures. The eventual aim, in a perfect world, would be to produce a completely general and automatic package for the construction, investigation, reduction and visualization of a chemical mechanism, with no input from the user except in the area of application and the limits of accuracy. It is doubtful whether or not it is possible to achieve such a completely automatic and general method which could be applied in any situation. However, an existing set of tools can be selected according to the needs of the application. We have shown throughout the chapter, that the methods for reduction are based firmly on the dynamics of the chemical system, and these, in turn, are based on the nature of the mechanism and physical situation. Since models in combustion are expected to cover a wide range of conditions, it is natural to expect that a different approach might be used for different cases. As the application base of these tools becomes wider, it will become more apparent which will be the most successful in a given situation.
ACKNOWLEDGEMENTS
One of the authors (TT) thanks the financial support of the Hungarian National Science Foundation, OTKA under project No F014481. We are also grateful to EPSRC for a Visiting Research Fellowship for TT.
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Chapter 5
Global Behaviour in the Oxidation of Hydrogen, Carbon Monoxide and Simple Hydrocarbons S.K. SCOTT
5.1 INTRODUCTION
This chapter describes in some detail the information and approaches to data analysis that can be drawn from the relatively new conceptual structures of non-linear mathematics in the context of combustion chemistry. Chemical reactions that evolve with time are part of the class of dynamical systems: other examples arise throughout physical, biological and engineering science, but chemical kinetics is an area almost tailormade for the new mathematics developed in the past 15 years or so. The two simplest oxidation/combustion reactions, H 2 and CO, have been the most intimately studied via the non-linear mathematics approach, but some features of the combustion of hydrocarbons, the so-called cool-flame phenomena, can also be usefully discussed in this context. The main interest here will not be in the details of the individual elementary steps by which a chemical reaction occurs but in the global behaviour of the reaction under particular experimental operating conditions such as the ambient temperature, mixture composition and total pressure. This "behaviour" may be for the reaction to respond to the operating conditions by establishing a steady-state reaction rate in which only a small fraction of the initial fuel and oxidant reacts or, for slightly different conditions, to exhibit an explosive reaction. The changes in the qualitative nature of the global behaviour, e.g., from slow reaction to ignition, in response to a small change in the experimental operating conditions (known technically as the system parameters) are known as bifurcations of the system. This chapter will aim to show how the concepts and general understanding of non-linear dynamical systems can be used to explain how such bifurcations from one type of reaction behaviour to another arise, and to predict
440
Global behaviour in simple oxidations
Ch. 5
the existence of additional responses not yet observed as the operating conditions are changed. Additionally, some relatively new types of response, such as chaos, will be discussed. If a chemical reaction is evolving chaotically, then traditional methods for predicting the future evolution of the system in time are no longer appropriate and new approaches and statistics are required. Before discussing particular reactions, some basic concepts will be introduced in the context of combustion chemistry. The first is non-linearity, which we will see is actually an "old friend" amongst chemists and is the rule for chemical reactions rather than anything special. The second feature is that of feedback. There are two main mechanisms by which feedback occurs in combustion processes: chemical feedback and thermal feedback and often these occur together in the systems of interest here to produce thermokinetic feedback. Without these feedback processes, none of the phenomena we associate with the science of combustion would exist (and presumably there would not be such an active field as combustion chemistry). Because the coupling of non-linear feedback, and the various other processes such as fluid flow and spatial inhomogeneity that arise in combustion, simplified representations of the chemical feedback processes can be very useful in seeking generalized theories. Some of the more widely exploited models or caricatures of the global chemistry will also be discussed. The previous chapter in this book describes more formal approaches to the representation of complex mechanistic sets of elementary reactions by reduced but accurate and "lumped kinetics".
5.2 NOTATION
symbol a, b a{ a0, b0 A cp E F J
quantity concentrations of species A, B relative third body efficiency of species i inflow concentrations of species A, B pre-exponential factor specific heat capacity activation energy steady-state condition Jacobian matrix
units mol dm 3 mol dm 3 cm 3 molecule - 1 s" J K_1 kg-1 J mol-1
Notation
k P P
s tch *N
'res
T T 1
a
^ad TATT
To V Xi
a
P A>
#ad
© K
A A; V
p T
reaction rate coefficient ("rate constant") pressure vector of parameters = - A H , exothermicity of reaction i rate of reaction / surface area = l/k(T0), chemical timescale = CppV/xS, Newtonian cooling timescale mean residence time thermodynamic (absolute) temperature ambient temperature adiabatic (flame) temperature =E/R, Arrhenius temperature inflow temperature volume mole fraction of species /
cm3 molecule Pa Jmol-1 molecule cm - 3
= a/a0, dimensionless concentration of A = b/a0, dimensionless concentration of B = b0/a0, dimensionless inflow concentration of B = (T - T0)E/RTQ9 dimensionless temperature rise = (Tad ~ T0)E/RTQ9 dimensionless adiabatic temperature rise function in Gray-Yang model parameter in Salnikov model primary bifurcation parameter eigenvalue parameter in Salnikov model stoichiometric coefficient extent of reaction density = t/tch, dimensionless time = *isr/*ch> dimensionless Newtonian cooling time = tTes/tch> dimensionless residence time net branching factor
_ -
™2
m s s s K K K K K ™3
m -
kgm"
442 X
Ch. 5
Global behaviour in simple oxidations
surface heat transfer coefficient = 0adTN> Semenov number
Wm
2
K
subscripts and superscripts a ambient ad adiabatic cr critical i relating to species / 0 inflow values ss steady-state
5.3 NON-LINEARITY AND FEEDBACK IN CHEMICAL KINETICS: STOICHIOMETRY AND ELEMENTARY STEPS
Chemical reactions can be represented by an overall stoichiometric equation which indicates the relative molecular proportions with which the reactants combine to form the products of the particular reaction. Some simple examples from combustion chemistry are (i)
2H 2 + 0 2 ^ 2 H 2 0
(ii)
2CO + 0 2 - ^ 2 C 0 2
(hi)
C 4 H 10 + 6§0 2 -> 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 + 02^OH + 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-
Stoichiometry and elementary steps
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,
^o2=-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
Global behaviour in simple oxidations
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 (^u/^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 + 02^OH + 0
rate = fc2[H][02]
Stoichiometry and elementary steps
(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
= ^o[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
Global behaviour in simple oxidations
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
Stoichiometry and elementary steps
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
Global behaviour in simple oxidations
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
= kv^Ar[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)
Global behaviour in simple oxidations
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
Chemical feedback: Branched-chain ignition
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
Global behaviour in simple oxidations
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] <* £ so that we obtain quadratic autocatalysis
R = kq[A]l^(l - £)
giving rise to the parabola in Fig. 5.1(c). In the initial phases of the reaction the deceleratory effect of the decreasing concentration of A is more than compensated for by the increase in the rate as [B] increases. These effects balance as the two species concentrations become approximately equal and, in the later stages, the consumption of the initial reactant is more significant in determining the sign of the slope of R. In a similar way, the feedback mechanism giving rise to the cubic curve can be given the stoichiometric representation cubic autocatalysis
A + 2B —>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
Chemical feedback: Branched-chain ignition
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
Global behaviour in simple oxidations
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
Chemical feedback: Branched-chain ignition
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
Global behaviour in simple oxidations
If the same model is examined in a CSTR, then the governing equations are
d[A]_([A]0-[A]) Qf
d\X]
(5.17a)
k^A] - 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
Chemical feedback: Branched-chain ignition
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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
Ch. 5
version determined earlier we have for the adiabatic case (i.e., with TN^oo)
^ - = 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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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
Thermokinetic feedback
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
Global behaviour in simple oxidations
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
Thermokinetic feedback
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
Global behaviour in simple oxidations
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-
Thermokinetic feedback
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
Global behaviour in simple oxidations
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.
Thermokinetic feedback
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
Global behaviour in simple oxidations
(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
Thermokinetic feedback
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
Global behaviour in simple oxidations
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.
Thermokinetic feedback
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
Global behaviour in simple oxidations
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
Thermokinetic feedback
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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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
\
N^AT = 3 K
^
4T = 2 K ^ ^
AL
N A T = 2K
AT =
AT = 3K k s
^
iks
WIKV
AT = 0 K V > - ^ ^ignition
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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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
Flow reactor studies of the The H2 + 0 2 reaction
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 < Ta^op{, 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
Global behaviour in simple oxidations
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
Flow reactor studies of the The H2 + 0 2 reaction
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
Global behaviour in simple oxidations
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 T^Uopf 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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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-
Mechanistic modelling of complexity in H2 + 0 2 reaction
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
Global behaviour in simple oxidations
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 ^a,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
Mechanistic modelling of complexity in H2 + 0 2 reaction
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 + 02^OH + 0 0 + H2^OH + 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
Global behaviour in simple oxidations
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
Global behaviour in simple oxidations
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" (<0.1% H 2 ), and there is then no detectable accompanying temperature excursion, i.e., the oscillatory glow can be an isothermal process. Even when the oscillatory response dies in a given experiment, there is typically 50% of the original fuel remaining in an initially stoichiometric mixture. The reaction may exhibit a period of steady glow (generally of diminishing intensity) following the end of the oscillatory phase, but this also ceases well before complete reactant consumption. With higher concentration of H 2 , the oscillations are more intense (sufficiently so to be confused with an ignition if only monitored by eye) and accompanied by significant reactant consumption and self-heating. This means that only short trains, perhaps only 2 or 3 pulses, are possible with the oscillatory phase again ceasing before complete consumption. Some exmples of the oscillatory glow behaviour observed in these systems are shown in Fig. 5.30. An important feature of these experiments was that oscillatory and steady glow were observed for the same ranges of pressure and ambient temperature, lying in a p-Ta peninsula with "first" and "second limits", as indicated in Fig 5.31. It appears that a conditioning of the surface arising from repeated experiments in the same vessel is responsible for determining whether a given experiment will exhibit oscillatory or steady luminescence. A newly-cleaned vessel will generally exhibit a steady glow for early experiments at a given pressure and ambient temperature, but eventually the system will support the oscillatory response for, perhaps, 100 consecutive experiments at the same/? and Ta. Eventually, the surface will age too much and the response will return to the steady glow until the vessel is cleaned again by acid washing. At high H 2 concentration (wet systems) the genuine ignition response invades the p-Ta region for glow. However, the inevitable effects of reactant consumption again indicate that these various phenomena are better studied quantitatively and characterized under continuous flow conditions. 5.11.2 Open systems In well-stirred flow reactors, the CO + 0 2 reaction supports five different modes of response [63,64]: steady slow reaction, steady glow, oscil-
517
The CO + 0 2 reaction
(a) 82
17 c
O)
JUu uuu LLULiUUuLJu 1000
i94 J_JL
Time/s
9000
(b)
(c) c £
O
o o o
2:
0
200 400 Time/s
Fig. 5.30. Representative trains of oscillatory glow for CO + 0 2 mixtures in a closed system; (a) chloropicrin-treated reactor with p = 33 Torr, Ta = 841 K giving a total of 94 oscillations over 3 hr; (b) and (c) light emission and reactant consumption in a clean reactor with p = 22 Torr and Ta = 900 K. (Reprinted with permission from reference [60], © Royal Society of London.)
Global behaviour in simple oxidations
518
Ch. 5
-(b)
duration decreases 800
900
1000
1000
ambient temperature, T,/K Fig. 5.31. The co-existence of (a) steady glow and (b) oscillatory glow for the CO + 0 2 reaction (dry mixtures) in an ageing reactor. (Reprinted with permission from reference [60], © Royal Society of London.)
latory glow, oscillatory ignition or a steady high reaction state ("steady flame") - the first and last two of these being the counterparts of the same phenomena in the H 2 + 0 2 system. We will also see that the oscillatory ignition response can show complex periodicities although perhaps of a different nature from those described in Section 5.7 for hydrogen. Figure 5.32 shows the p-Ta diagrams appropriate to five different mixtures of increasing H 2 content, indicating the different regions of the parameter plane for the basic responses. Steady slow reaction is always found at the lowest pressure and ambient temperature conditions. In this mode, the reaction may sustain a steady temperature excess of the reacting gases above the ambient, vessel temperature with corresponding extents of reaction of up to 10% of the inflow concentration of the reactants. If the system is disturbed, perhaps by injecting a cold inert gas or by perturbing the gas flow, the system returns to the same steady-state after the perturbation is removed and does so in a simple, exponentially decaying manner, indicating stable nodal character of this steady-state. Steady and oscillatory glow responses occupy separate regions of the parameter plane under continuous flow conditions. It is possible that the flow of reactants in the period before the reaction is monitored (and this period must be rigorously reproduced in any given set of experiments if reproducible results are to be obtained) provides a consistent degree of surface conditioning in such experiments [65]. Oscillatory glow, which is
519
The CO + 0 2 reaction
750
775
800
TJK
Fig. 5,32. The p-Ta regions for steady dark reaction, steady glow, oscillatory glow and oscillatory ignition, and the influence of added H 2 for the CO + 0 2 reaction in a flow reactor with fres = 8s: (a) no added H 2 ; (b) 150 ppm added H 2 ; (c) 1500 ppm added H 2 ; (d) 7500 ppm added H 2 ; (e) 10% H 2 in final mixture. (Reprinted with permission from reference [64], © Royal Society of London.)
now sustained for as long as the flows are maintained, typically occupies a closed region at lower pressure within a steady glow peninsula. There is a different response to perturbation of the steady glow from that for steady dark reaction, with a damped oscillatory return in the case of glow indicating stable focal character. As the oscillatory glow is also clearly a stable limit cycle response, we can assume that the boundary between steady and oscillatory glow corresponds to the locus of Hopf bifurcation
520
Global behaviour in simple oxidations
Ch. 5
points, Section 5.4, with these generally being of a supercritical character as the oscillations have vanishingly small amplitude at the boundary itself. For systems with low concentrations of H 2 (i.e., if no H 2 is added and for which the natural impurities are probably less than 250 ppm H-containing species, or with 150 ppm H 2 added to these impurity concentrations), the oscillatory glow is effectively isothermal as no temperature excursion can be measured (the level of detection is approximately 4 K) and no reactant consumption can be detected. As the concentration of added H 2 increases, so temperature excursions and reactant consumption accompanying the oscillatory glow events can be recorded. Ignition is also a feature for such systems. Ignitions are readily distinguished from glow by having temperature excursion well in excess of 200 K, effectively complete consumption of the fuel (in equimolar or stoichiometric mixtures) and significant emission at wavelengths other than that corresponding to CO*. With increasing H 2 concentrations, the p-Ta peninsula for ignition invades the accessible p-Ta region and begins to "cover" the regions of oscillatory and steady glow. The boundaries for the latter do not appear to be very sensitive to the H 2 concentration and even with 10% H 2 added, there is a remnant of the steady glow region with the limit separating it from steady dark reaction in much the same position as for the mixture with no added H 2 . In a sequence of experiments, Johnson [66, 67] has investigated the development of the oscillatory ignition waveform for mixtures containing a "significant" addition of H 2 (typically 2% of the fuel and so 1% of the total mixture). The p-Ta diagram for a system with an average mean residence time of 16 s is shown in Fig 5.33. The slow reaction at low Ta gives way to oscillatory ignition as the (second) ignition limit is traversed. As r a is increased further, so the waveform changes quantitatively (in amplitude and period). More interestingly, perhaps, is that for a range of operating pressure from approximately 15 to 45 Torr, there is a qualitative change in the waveform as the system traverses a region of "complex oscillation". A flavour of the evolution of the waveform can be obtained by allowing the ambient temperature to drift upwards through this region at a fixed total pressure. Such a sweep through this region at a relatively high pressure, p = 40 Torr, where the region of complex response is relatively narrow, is shown in Fig 5.34(a). Complexity here evolves through a perioddoubling bifurcation, with the period-1 type oscillation giving way to a
521
The CO + 0 2 reaction
700
750
800
ambient temperature, Ta/K Fig. 5.33. The p-Ta regions of slow reaction, simple and complex oscillatory ignition and steady ignited state for CO + 0 2 reaction with 0.5% H 2 in a flow reactor with fres = 16 s. (Reprinted with permission from reference [66], © Royal Society of Chemistry.)
waveform that has one large + one small peak separated by approximately the same period per repeating unit. Thus, the repeat period is approximately twice that for the original period-1 response. Notice that as this change is encountered, the difference in amplitude between "large" and "small" grows smoothly: it is possible to sustain the system just inside this region such that the two peaks have amplitudes that differ by as small an amount as desired (at least within the experimental control allowed in this experiment). Thus, this is a supercritical period-doubling bifurcation. The experimental record in Fig. 5.34(a) shows that the system has virtually settled into a sustained period-2 responses before the ambient temperature is increased further. The response to this further increase in Ta is that the system undergoes a supercritical period-halving bifurcation and settles onto a small amplitude period-1 oscillation. A further increase in Ta causes the system to traverse a supercritical Hopf bifurcation and the oscillatory nature of the ignition event gives way to a stable steady-ignited state. At a lower total pressure, e.g., p = 20Torr, the region of complex oscillatory behaviour is of wider extent in Ta. The traverse through this region uncovers more exotic responses after the initial period doubling, as indicated in Fig. 5.34(b). To characterize the various responses in more detail, the system must be allowed to settle to a steady ambient temperature (and all other experimental parameters) at various points within this range. Selected time series obtained in this way at different Ta are shown
522
Global behaviour in simple oxidations i
0
1
Ch. 5 1
1 2 time/min
120 H
P 80H < I
806.1K
808.3K
809.6K •
Ta slowly increasing
i
i
i
120 ^80
< 0 777.3K
- - -•-•• 778.0K 781.8K
782.1K
782.6K
782.8K
• T, slowly increasing
Fig. 5.34. Traverse through region of complex ignition in Fig 5.33 for p = 40Torr (upper trace) and p = 20Torr (lower trace) showing evolution of complex oscillatory waveform. (Reprinted with permission from reference [67], © American Institute of Physics.)
in Fig. 5.35(a)-(f). The original period-1, (a), undergoes period doubling to give period-2 (b). The period-2 response also period-doubles as Ta is increased to give a period-4 response, (c). Evidence for period-8, following another period doubling, and even for period-16 has also been reported, but the control over the experimental conditions in the apparatus is not sufficiently high that such high periodicities can be sustained for data capture. Figure 5.35(d) corresponds to a complex response of no apparent periodicity, but for certain very narrow ranges of ambient temperature within this basic chaotic evolution, the system manages to find periodic states which repeat every 5- or 3-ignition events, (e) and (f), respectively. As the region of chaos is exited at high Ta a reverse period-doubling cascade is observed down through period-8, period-4 and period-2 to a small amplitude period-1 oscillation that eventually undergoes a Hopf
The CO + 0 2 reaction
.(d)
— i
1—
Time / mins
Fig. 5.35. Transient-free records of different periodic states from region of complex oscillation in Fig. 5.33: (a) period-1; (b) period-2; (c) period-4; (d) aperiodic trace; (e) period5, (f) period-3. (Reprinted with permission from reference [66], © Royal Society of Chemistry.)
bifurcation as the steady-ignited state about which all this temporal complexity is evolving gains stability. The reconstructed limit cycles for the five periodic states from Fig. 5.35(a-c, e, f) are shown in Fig. 5.36. The aperiodic time series in Fig. 5.35(d) has no repeating unit, but is not a random sequence of ignition. That there is an underlying order can be revealed by plotting the so-called next-maximum map. In this construction, the amplitude of one ignition (as measured by any appropriate and convenient device, so here we use the signal from the internal thermocouple junction which is an approximate measure of the self-heating
524
Global behaviour in simple oxidations
dtgtialfeed ihermocouple record
Ch. 5
digiiatized thermocouple record
Fig. 5.36. Reconstructed attractors corresponding to time series in Fig. 5.35. (Reprinted with permission from references [66 and 67], © Royal Society of Chemistry and American Institute of Physics.)
of the gas) is plotted against the amplitude of the next peak. This process of finding x-y pairs is repeated for every ignition in the sequence. If the response were random, then the points on the graph obtained in this way would be randomly scattered. As shown in Fig. 5.37, however, the CO ignition lie so as to construct a relatively simple, single-humped maximum
The CO + 0 2 reaction
525
+ o
is
amplitude of n-th maximum Fig. 5.37. Next maximum map obtained from chaotic sequence in Fig. 5.35(d). (Reprinted with permission from reference [66], © Royal Society of Chemistry.)
map. This is a characteristic feature of deterministic chaos and helps distinguish complex responses from noise. We can also reconstruct the attractor corresponding to the time series using the method of delays described above. The strange attractor arising for this chaotic sequence is also shown in Fig. 5.37 and its relationship to the simple limit cycles from the periodic behaviour can be recognized. Complex behaviour is favoured in the CO + 0 2 system by long residence times and by low concentrations of H 2 . With residence times of 10 s or less, the region of complex responses includes only period-2: no further period doublings are evidenced. At tres = 16 s, the region of complex ignition moves to higher ambient temperatures as the H2-content decreases. The full period-doubling cascade to chaos described above for 0.5% H 2 is lost as the concentration of H 2 increases. With 1.33% H 2 , period-2 and period-4 are found, while for 2% H 2 we find only period-2 responses. The region of complex responses covers a decreasing range of pressure as [H2] increases. The behaviour of "pure" CO, i.e., the cylinder gas (CP grade) without added H 2 is of interest in that no period-doubling is observed, but a remnant of the mixed-mode type of oscillation seen for the H 2 + 0 2 system arises.
526
Global behaviour in simple oxidations
Ch. 5
5.11.3 Mechanistic interpretation and modelling The reaction steps involving CO and 0 2 are rather limited: (12)
CO + 0 2 - * C 0 2 + 0
(10)
CO + O + M ^ C O f + M
the first being an initiation step and the second a termination. Thus, a "pure" CO + 0 2 system thus has no branched chain processes and could only support a thermal ignition at best. The fact that all systems include some hydrogen-containing species, and the experiments described in detail above actually have H 2 added, indicates that we should also consider the H 2 + 0 2 reactions and any additional reactions involving CO with intermediate species in that mechanism. The most important step is (13)
CO + O H ^ C 0 2 + H
which is the primary source of C 0 2 in such systems. It is a chain propagation step, converting one chain carrier OH into another H and so neither enhances nor diminishes the propensity to chain-branching. Another step (14)
CO + H 0 2 -> 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 + 02^OH + 0
(3)
0 + H2^OH + 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
Global behaviour in simple oxidations
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
Hydrocarbon oxidation
529
mode is reset by the direct inflow of fresh H 2 and, for ignition, of CO [68]. In a closed system, much the same chemistry must operate (there may be some differences arising from the slightly different p-Ta range for the oscillatory glow in these systems compared with flow reactors and this, in turn, may be related to different states of the reactor surface). The major problem in interpreting oscillatory glow in the above terms, i.e., as an H 2 + 0 2 ignition in a system heavily diluted with CO which is acting primarily as an inert gas, is to understand the resetting mechanism by which H 2 is regenerated. Suggestions due to Babushok et al. [69] and to Gray et al. [64] invoke the "water gas shift" process (15)
CO + H 2 0 ^ C 0 2 + H 2
perhaps occurring on the surface, hence, the sensitivity of closed vessels studies to the pre-treatment processes mentioned above. Chinnick and Griffiths [70] have successfully simulated some of the closed vessel behaviour in these terms. The complex periodicities in the CO + 0 2 reaction are only now yielding to simulation, but are again arising from a mechanism based on an H 2 + 0 2 scheme with the few steps involving CO described above.
5.12 HYDROCARBON OXIDATION
"Exotic" oscillatory and other types of non-linear behaviour are also features of most hydrocarbon oxidations [71-74]. The next chapter will provide a detailed mechanistic description of the basis for cool-flames etc., and their relevance in various situations. It is interesting, however, to apply the classification system developed in the previous sections to the global behaviour in these systems. We start with a description of the oxidation of acetaldehyde (ethanal) and again concentrate on modern studies in flow reactors where the effects of reactant consumption (which are much more significant in closed systems for these cases than for CO) are not a feature.
530
Global behaviour in simple oxidations
Ch. 5
5.12.1 Oxidation of acetaldehyde (a) Experimental features The oxidation chemistry of small, partially-oxygenated fuels is of great interest in combustion chemistry as these are important intermediates in the combustion of virtually all commercial hydrocarbon fuels. Fuels with long carbon backbones react in their early stages mainly through a sequence of reactions that cause chain rupture, yielding smaller hydrocarbon fragments such as radicals. These then typically react with 0 2 to produce precursors of aldehydes, ketones etc. Not all of the features of acetaldehyde chemistry are completely representative of hydrocarbon oxidation, but this point is developed in the next chapter. The basic global behaviour of a mixture of acetaldehyde vapour in 0 2 is illustrated by reference to the p-Ta "ignition" diagram, Fig. 5.39. Up to five regions of qualitatively different responses are characteristic [75]. At low ambient temperature and pressure, the system exhibits a steady dark reaction, Region I. This may support a measurable steady-state tem150
100
50
450
500
550 Ta/K
600
650
Fig. 5.39. Experimental p-Ta diagram for CH 3 CHO + 0 2 in a flow reactor with tTes = 3 s showing five regions of different qualitative behaviour. (Reprinted with permission from reference [75], © Royal Society of London.)
531
Hydrocarbon oxidation
perature excess, as the reacting gases self-heat through the exothermic reaction beyond the oven temperature, of up to 40 K. The temperature excess ATSS increases with increasing Ta through this region, indicating that the rate of chemical heat release di has a positive temperature coefficient. The oxidation process is incomplete under these conditions, with the major product species detectable including per ace tic acid (CH3CO3H), methanol (CH 3 OH), formaldehyde (CH 2 0) and methane (CH4) in addition to C 0 2 and H 2 0 . If the system is perturbed in this region, the steady-state reveals a stable nodal character. Region V corresponds to the other steady-state form, that of steady glow, and lies at the highest ambient temperatures. As the name indicates, reaction is accompanied by a steady chemiluminescence. In contrast to the CO oxidation, however, the molecular source is not excited C 0 2 but excited formaldehyde HCHO*. Reaction is also accompanied by a steadystate temperature excess. In this region, however, ATSS typically decreases as the reacting gas temperature increases, as illustrated in Fig. 5.40. This phenomenon, termed the negative temperature coefficient (n.t.c), indicates that the rate of chemical heat release decreases as Tgas increases. (It is important to be careful on two counts here: first it is the gas temperature rather than the ambient temperature that is referred to even though it is the ambient temperature that determines whether the system is in region V or not: second, the rate of heat release is not the same as the "rate of
2"
40
20
v
c o
80
X
c o U
3 cj
-Uo
©
cu
£ S c
3 8
500 600 700 © reactant temperature, T/K Fig. 5.40. The steady-state heat release rate from CH 3 CHO + 0 2 in regions I and V as a function of reacting gas temperature showing negative temperature dependence in region V. (Reprinted with permission from reference [75], © Royal Society of London.) 400
532
Global behaviour in simple oxidations
Ch. 5
reaction" - the latter being a somewhat ambiguous quantity anyway.) The major products in this region are CH 3 OH and other partially oxidized, C 0 2 , CO and H 2 0 . The steady-state is a stable focus as perturbations decay in a damped oscillatory manner. The degree of damping decreases as we approach the boundary with region IV. If the ambient temperature is decreased, the system enters region IV in which the reaction exhibits oscillatory cool-flames. Each oscillatory event is accompanied by emission from HCHO* and by a temperature excursion. Cool-flames were first described by Davy [76] who noted that they were not able to ignite paper held in the flame (he stabilized a cool-flame above a bed of sand in which the liquid reactant was absorbed). The temperature rise may be as much as 200 K, but is typically less than this: cool-flames in which the temperature excursion is only 10 K or so are regularly observed. Indeed the oscillatory amplitude seems to tend to zero as we approach the boundary with region V. This latter point indicates that the cool-flame corresponds to a simple limit cycle oscillation with the species concentrations and gas temperature oscillating around an unstable steady-state. The steady-state regains stability as the ambient temperature Ta is increased emerging as the stable focal, steady glow state in region V. Thus, the boundary between regions IV and V is that of a locus of (supercritical) Hopf bifurcation points in the terminology developed in Section 5.5. The products detected from reaction in region IV include CO, H 2 0 , CH 2 0, CH 3 OH and CH 4 and lower quantities of peracetic acid, ethane (C2H6) and hydrogen peroxide (H 2 0 2 ) all of which oscillate with the coolflame period as indicated in Fig. 5.41. On decreasing the ambient temperature further, we enter the regions II and III (the convention of splitting this region into two subregions has some convenience, but there is in fact a coherence to the behaviour across this whole range). At the lowest Ta in this region (i.e., in region II) the behaviour corresponds to a two-stage ignition event (see Chapter 6). Basically, the behaviour is that of an ignition in which a high transient temperature excess develops accompanying consumption of the primary reactants and the production of CO and H 2 0 and other "final" products. There is, however, some "fine structure" to the ignition waveform, with a distinct shoulder preceding the ignition proper. During this initial development phase, the products of the cool-flame response are observed to build up. Thus, we have a cool-flame as a precursor stage, building up intermediate species and the gas temperature before the transition to a
Hydrocarbon oxidation
533
-IJLLLL «
*
0
«
20 Time/s
Fig. 5.41. Typical experimental records in cool-flame region IV. (Reprinted with permission from reference [75], © Royal Society of London.)
fully-fledged ignition. The transition from regions I to II is accompanied by a "hard excitation" similar to that observed at the transition from slow reaction to oscillatory ignition in the H 2 + 0 2 system in the sense that it corresponds to a saddle-node bifurcation of the low reaction steady-state with the system then moving to a large amplitude limit cycle solution existing around the unstable high reaction steady-state (in this case corresponding to a complex oscillation). Region III corresponds to other types of multistage ignition events [77]. In this type of response, a two-stage ignition event is preceded by a number of distinct cool-flame oscillations, with some examples shown in Fig. 5.42. Thus, just across the border from region II, we observe a single cool-flame followed by a two-stage ignition event, giving rise to an overall 3-stage ignition. The number of cool-flames interspersing the two-stage ignition peak increases as we move across region III to higher Ta, giving 4-stage, 5-stage ignitions etc., with the ignition peak then disappearing from the sequence as we enter region IV. A schematic bifurcation diagram summarizing these changes in behaviour as a function of the ambient temperature is sketched in Fig. 5.43. We may also note that at yet higher Ta, beyond region V, a simple ignition limit may be encountered.
534
Global behaviour in simple oxidations
Ch. 5
CH3CHO
nn n 1
AC
HoO
t
1
1 •
1 !
i
CH20 CH3OH
\ \
20 Time/s
;
•
1
•
j
1
^ •M
mm
AT
1
i ;
AT
40
Fig. 5.42. Typical experimental records in two-stage ignition region III. (Reprinted with permission from reference [75], © Royal Society of London.)
(b) Basis of mechanistic interpretation A full mechanistic account follows in the next chapter. Here we simply indicate the important features of the currently-accepted interpretation of the above facts. The key feature in these thermokinetic phenomena is that there are both thermal and chemical feedback processes combining to produce the various "exotic" responses, including the ntc. At the heart of the clockwork is the equilibrium involving the methyl radical CH 3 and molecular oxygen [78] (16)
CH 3 + 0 2 ^ CH 3 0 2
It is evident that A5?6 < 0, indicating that the equilibrium will shift from
Hydrocarbon oxidation
535
Fig. 5.43. Schematic bifurcation diagram for CH 3 CHO + 0 2 reaction showing separate branches corresponding to dark reaction and steady glow, with upper branch losing stability at a Hopf bifurcation as Ta is reduced to give limit cycle (cool-flame) oscillations. The simple limit cycle also loses stability as Ta is reduced further and a complex oscillation corresponding to the multi-stage ignition will emerge but cannot be adequately represented in this 1-D diagram.
the right (CH 3 0 2 ) to the left (CH 3 + 0 2 ) as the temperature of the reacting mixture increases. Thus, at low gas temperatures we can expect the chemistry to be that characteristic of the peroxy species CH 3 0 2 but at high Tgas it will be that of the methyl radical. The reaction steps at the pressures and temperatures of interest that arise from the formation of CH 3 0 2 are the following: (17)
CH 3 0 2 + CH 3 CHO -> CH 3 0 2 H + CH 3 CO
a propagation step in the sense that one radical is consumed and one (CH3CO) is produced. However the "molecular" product CH 3 0 2 H is unstable to decomposition into two more radical species (18)
CH 3 0 2 H -» CH 3 0 + OH
so there is a degenerate branching in which three chain carriers eventually emerge from the CH 3 0 2 species. The increasing radical/chain carrier pool leads to an increasing reaction rate and, hence, to an increasing rate at which heat is released by this exothermic reaction channel.
536
Global behaviour in simple oxidations
Ch. 5
As r g a s increases in response to the increased rate of heat release, so the equilibrium (16) swings over to the lefthand side in favour of the methyl radical. Under the conditions typical of this reaction, the major fate for CH 3 is recombination to form ethane (19)
CH 3 + C H 3 ^ C 2 H 6
This termination step reduces the radical pool and alters the overall reaction exothermicity, both having the effect of reducing the rate of chemical heat release. In turn, the drop in 9t causes the gas temperature to fall, allowing the equilibrium to shift back to the right. For some conditions, this cycling in Tgas and the CH 3 or CH 3 0 2 chemistry continues to give sustained, limit cycle oscillations (cool-flame); for other operating conditions, a balance can be attained so that a steady-state (steady glow) is established. The negative temperature coefficient corresponding to steady glow also emerges naturally as the equilibrium lies further in the direction of the unbranched CH 3 chemistry leading to a lower 9t, the higher Tgas. Full scale numerical computation on relatively large mechanisms [78] have reproduced the major features (see also refs [79-84]). Simplified model interpretations have also been built on the Gray-Yang scheme discussed earlier [18-21]. This is a two-variable model (chain carrier x and gas temperature T) and can account qualitatively for the dark reaction, cool-flame, steady glow and n.t.c. phenomena and provide hints of the ignition-type events. For the complex oscillations, a three-variable model is needed and an elaboration on the Gray-Yang scheme has been provided by Wang and Mou [22] (see also refs [85-87]). Interestingly, although the computations and extended models can produce "mixed mode" multistage ignitions, one feature of the experimental responses is not matched. In the theoretical approaches, the increasing number of cool-flames as we move across region III is achieved by adding two extra cool-flames at each stage, so we see sequences of 2-, 4-, 6-stage ignitions etc., or of 3-, 5-, 7-stage etc., but not the 2-, 3-, 4-stage etc., reported in the experimental studies. 5.12.2 Non-isothermal oxidation of alkanes Cool-flames arise for some ranges of experimental conditions for many other hydrocarbon oxidation reactions. Chapter 6 presents many p-Ta ignition diagrams for different hydrocarbon fuels, all of which have a
Hydrocarbon oxidation
537
richness matched by the acetaldehyde system. An earlier discussion of the detailed chemistry relevant to such oxidations has already been presented in Chapter 1, but it is worth also stressing that the contributions of the heat generation and heat loss (thermal feedback) processes are as important as the kinetic complexity. If we denote a general hydrocarbon as RH, then initiation processes involving 0 2 may give rise to the corresponding alkyl radical R. We may anticipate an analogous equilibrium to that involving CH 3 produced in acetaldehyde oxidation of the form (20)
R + 02 ^ R02
with the same general dependence of the equilibrium on reactant temperature. This is indeed the case, but the subsequent reactions are not simple generalizations of the acetaldehyde schemes [74]. The major fate of R if the equilibrium lies to the left (the high Tgas regime) involves abstraction of an H atom from the C atom adjacent to the radical site (if this has a C—H bond), giving rise to the conjugate alkene and H 0 2 . More significant, however, is the difference in the R 0 2 chemistry in the low Tgas regime. The feasibility of the reaction (21)
R 0 2 + RH -> R 0 2 H + R
which would then be followed by the degenerate branching (22)
R 0 2 H ^ R O + OH
is greatly reduced for a typical hydrocarbon. For an aldehyde, the —CHO group provides a relatively labile H atom for abstraction by R 0 2 , but in the absence of an activating functionality, the basic C—H bond is too strong to allow this to be a sufficiently fast process at the cool-flame temperatures. This point is developed in the next chapter where the route involving intramolecular hydrogen abstraction is described. This mechanism allows us to account for the different cool-flame characteristics of different fuels and, in particular, for the different propensities of different isomers, such as n- and /-butane to show cool-flame behaviour (see Fig. 6.14) on the basis of their molecular structure.
538
Global behaviour in simple oxidations
Ch. 5
5.13 CONCLUSIONS AND FUTURE DIRECTIONS
The application of the methods of non-linear dynamics to chemical reactions is still in its relative infancy, but it's advocates should not rely solely on its novelty. It is time to ask the question "what can this approach really do for combustion chemists"? At the lowest level, it provides a new system with which to label and categorize different forms of global response (stable node, limit cycle etc.). Beyond this, however, is the power of then being able to bring to bear the body of knowledge pertaining to generic features of how, say, a stable node can give way to a limit cycle as some (experimental) parameter is varied. Important in this is the basic recognition of the role played by steady-state solutions of the governing reaction rate equations. First, if we use flow reactors, such steady-state solutions genuinely exist! (If we operate in a batch process, quasi-steadystates may arise in which the gradual background change in reactant concentrations plays the role of a slowly varying parameter rather than a significant reactant: we can then construct steady-states by ignoring reactant consumption as a first approach to modelling, see Chapter 4.) Steadystate solutions are, in principle at least, obtainable from solutions of algebraic equations (in spatially homogeneous systems) rather than requiring solutions of differential equations (in spatially-distributed systems, we may hope to reduce to problem to the solution of ordinary rather than partial differential equation if there are symmetries or restrictions on the geometry). Thus, we can exploit more efficient computational approaches and obtain the steady-state solutions directly. Second, it is possible (and not particularly unusual) to have more than one steady-state solution for the same set of experimental conditions (parameters). If steady-state multiplicity occurs, there are usually 3 (or less frequently 5 or 7) steady-states for the system to choose from. Multiplicity is the underlying cause of the phenomena of ignition and extinction. In the simplest case, with three co-existing steady-states, two may be stable (and, hence, physically attainable for the reaction system) and the other "middle" solution is unstable. The unstable solution is not attainable but does play an important role in separating the basins of attraction of the competing stable states, i.e., in separating initial states (concentrations and gas temperature) that evolve to the slow reaction state from those that evolve to the ignited steady-state. Thus, this separatrix is of potential real interest: methods for constructing the separatrix in the vicinity of the
Conclusions and future directions
539
unstable state and following it away across the concentration "state space" are already well developed in simple situations. As an experimental parameter is varied, so the different steady-states change smoothly in response. However, it is possible that two of the steady-states can approach each other and merge at some "critical" parameter value. Beyond this, the two states vanish and the system must move to the only remaining state: a discontinuous response to a small, smooth change in the parameter characteristic of an ignition. Thus, we can thus compute ignition points in terms of a condition in the steady-state behaviour: a saddle-node bifurcation. Such points can be located directly by adding the condition for a zero eigenvalue or, equivalently, a zero determinant for the corresponding Jacobian matrix as an additional algebraic condition to the steady-state solution. Furthermore, we can then follow this condition as some other parameter is varied, allowing our numerical package (or even our algebraic analysis in the simplest cases) to construct an ignition limit directly. In addition to merging with other steady-states, a stable steady-state can also give rise to a qualitative change in the behaviour of the chemical system by losing its stability. An important example of this is the Hopf bifurcation which often gives rise to the onset of oscillatory behaviour. By adding the condition for this bifurcation (a pair of purely imaginary eigenvalues) to the steady-state condition, such points can again be solved for directly and followed as a second parameter is varied. Additional characterization of a Hopf point will suggest whether the system shows soft or hard excitation to oscillations. At present, computational techniques really only allow us to work in this way with stiff systems of small numbers of intermediate species (<6). As model reduction techniques become further developed and more sophisticated, and as computing power increases, there is the prospect of more combustion systems being tackled in this way. We also now know that complex oscillations evolve as simple limit cycles become unstable, bifurcating to more complex limit cycles. Only a small number of bifurcation sequences account for all known scenarios. We have seen examples of mixed-mode sequences (H 2 + 0 2 ) and period-doubling cascades (CO + 0 2 ). A third route involving quasi-periodic responses is known and arises in some chemical system [88], but has not yet been observed in combustion systems (except in some special studies in which the ambient temperature or some other parameter is "forced" to vary in some sinusoidal or other periodic manner [89]). The important lesson then
540
Global behaviour in simple oxidations
Ch. 5
is simply that systems do not (except in very, very rare special cases) change straight from a simple response to a very complex response in one step. Except in pathological cases there will always be one of the above sequences that effect the transition from simplicity to complexity and a small number of additional observations (made with some guidance form the above theory) should suffice to decide which is actually responsible in a particular instance. Once the bifurcation sequence has been identified, it is usually possible to predict which other types of response will be found "close by" in terms of the experimental conditions. For instance, a concatenated mixed-mode state consisting of alternate l 1 and l 2 waveforms has been observed recently for a narrow, but finite range of ambient temperature between the l 1 and l 2 states in the H 2 + 0 2 system. Another important lesson arises for modelling/mechanism development. In the past, the observations of oscillations could be regarded as unusual and modellers were perhaps tempted to conclude that any hand-waving argument that supported such behaviour must be correct. It is now realized that the requirements for oscillations are not particularly stringent (some moderately strong feedback) nor unusual (chain-branching, self-heating) and that typically several different oscillatory clockwork mechanisms can be produced for chemically-complex systems. It is now imperative that such suggestions are subjected to quantitative tests. Simply reproducing some oscillating concentration records is not enough. However, we do now know in what form we should obtain data for such quantitative tests: we can observe how oscillatory amplitudes grow as we move away from the onset conditions for oscillations; we can see whether the period lengthens logarithmically or as the inverse square-root of the ambient temperature as we approach an extinction limit. Similarly, although not discussed at any great length here, we now have answers to the question "what should I do with a very complex (perhaps chaotic) record"? First, there are well-specified tests to determine whether the signal is actually chaotic or just random noise (or chaos with noise superimposed). One important, but frequently overlooked approach in this regard, is to vary the experimental conditions slightly to see if period4, period-2 or mixed-mode states etc. arise - a good indication that the signal does indeed have a chaotic basis. Next, we can gain from the data some coefficients known as Lyapounov exponents that give as a measure of how quickly the system is "losing information". This is relevant in chaotic systems for making forward predictions of the subsequent evol-
References
541
ution. System with parameter values such that the response is chaotic are extremely sensitive to their initial conditions. (Note that this sensitivity is not about whether the system will be chaotic or not: that depends only on the parameter values, not on the initial conditions.) This sensitivity, means that two virtually identical systems will typically begin to evolve in a similar way, but will sooner or later begin to behave differently, i.e., they will become uncorrelated. Thus, systems in their chaotic mode are essentially unrepeatable and also consequently unpredictable in the long term. The Lyapounov exponent tells us whether these differences develop quickly or more slowly. This is important because it allows us to calculate a timescale on which the behaviour is predictable within a certain tolerance. For instance, we might find that the value of the exponent is such that we will be able to predict the evolution to % accuracy for lOmin, but if we require % accuracy, we will only be able to predict for 10 s. It is perhaps too early to expect major modifications to rate data to be made on the basis of trying to match oscillatory or other non-linear behaviour (although this has happened in other areas of chemistry where some "guestimated" rate constants were revised by a factor of 105 on the basis of an analysis to match the observed speed of chemical waves in solution). Nevertheless, the semi-quantitative matching of experiment and numerical predictions for a whole p-Ta ignition diagram is a stringent test both of a proposed mechanism and of the associated rate data (and any assumptions about the heat and mass transfer processes): and the more "interesting" the p-Ta diagram in terms of regions of oscillations, complex oscillations etc., the more stringent the test. We can hope to see chemical kineticists and combustion scientists becoming more familiar with the concepts of nonlinear dynamics to the extent that they become recognized as additional extra tools in our "detective kit" for teasing out our understanding of the behaviour of complex chemical systems.
References [1] H. Richtering, in: Low Temperature Combustion, ed W. Jost, Chapter 2 (Gordon and Breach, 1965) pp. 37-82. [2] B. Lewis and G. von Elbe, Combustion, Flames and Explosions of Gases, 3rd edn. (Academic Press, 1985). [3] N.N. Semenov, Chain Reactions (Clarendon Press, Oxford, 1935).
542
Global behaviour in simple oxidations
Ch. 5
G. Dixon-Lewis and D J . Williams, Comprehensive Chemical Kinetics, eds C H . Bamford and Tipper, Vol. 17, (1977) pp. 1-239. P. Gray and S.K. Scott, Chemical Oscillations and Instabilities (Oxford University Press, 1994) S.K. Scott, Oscillations, Waves, and Chaos in Chemical Kinetics (Oxford University Press, 1994). N. Chirkov, Acta. Phys. Chim. URSS 6 (1937) 915. V.G. Vorokonov and N.N. Semenov, Zh. Fiz. Khim. 13 (1939) 1695. R.R. Baldwin and R.W. Walker, in: Essays in Chemistry, eds J.A. Barnard, R.D. Gillard and R.F. Hudson, Vol. 3, (1972) pp. 1-37. T. Boddington, P. Gray and G.C. Wake, Proc. Roy. Soc. Lond. A 357 (1977) 403. S.R. Kay, S.K. Scott and A.S. Tomlin, Chem. Eng. Sci. 44 (1989) 1129. J.L. Liu and S.K. Scott, Dynam. Stab. Sys. 8 (1993) 273. D.A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics (Plenum Press, 1969). I.Ye. Salnikov, Dokl. Akad. Nauk SSSR 60 (1948) 405. I.Ye. Salnikov, Zh. Fiz. Khim. 23 (1949) 258. (a) P. Gray, S.R. Kay and S.K. Scott, Proc. Roy. Soc. Lond. A 416 (1988) 321. (b) S.R. Kay and S.K. Scott, Proc. Roy. Soc. Lond. A 416 (1988) 343. (a) J.F. Griffiths, S.R. Kay and S.K. Scott, 22nd Symp. (Int.) Comb. (1988) 1597. (b) P. Gray and J.F. Griffiths, Comb, and Flame 78 (1989) 87. C H . Yang and B.F. Gray, J. Phys. Chem. 73 (1969) 3395. C H . Yang and B.F. Gray, Trans. Faraday Soc. 65 (1969) 1614. B.F. Gray, Trans. Faraday Soc. 65 (1969) 1603. C H . Yang, J. Phys. Chem. 73 (1969) 3407. X.-J. Wang and C.Y. Mou, J. Chem. Phys. 83 (1985) 4554. P. Gray and S.K. Scott, in: Oscillations and Traveling Waves in Chemical Systems, eds R.J. Field and M. Burger, Chapter 14, (John Wiley, 1985) pp. 493-528. J.F. Griffiths, S.K. Scott and R. Vandamme, J. Chem. Soc. Faraday Trans. 1, 77 (1981) 2265. R.R. Baldwin, D. Jackson, R.W. Walker and S.J. Webster, Trans. Faraday Soc. 63 (1967) 1665. R.R. Baldwin, D. Jackson, R.W. Walker and S.J. Webster, Trans. Faraday Soc. 63 (1967) 1676. R.R. Baldwin, D. Jackson, R.W. Walker and S.J. Webster, 10th Symp. (Int.) Comb. (1965) 423. R.R. Baldwin, M.E. Fuller, J.S. Hillman, D. Jackson and R.W. Walker, J. Chem. Soc. Faraday Trans. 1, 70 (1974) 635. P. Gray, J.F. Griffiths and S.K. Scott, Proc. Roy. Soc. Lond. A 394 (1984) 243-58. P. Gray, J.F. Griffiths and S.K. Scott, 20th Symp. (Int.) Comb. (1984) 1809. P. Gray, J.F. Griffiths, A.J. Pappin and S.K. Scott, in: Complex Chemical Reaction Systems, eds J. Warnatz and W. Jager (1987) pp. 150-159. [32] J.F. Griffiths, W. Kordylewski, A.J. Pappin and A.F. Sykes, in: Spatial Inhomogeneities and Transient Behaviour in Chemical Kinetics, eds P. Gray, G. Nicolis, F. Baras, P. Borckmans and S.K. Scott (Manchester University Press, 1987), pp. 237-253.
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[33] J.F. Griffiths, B.R. Johnson and S.K. Scott, J. Chem. Soc. Faraday Trans. 87 (1991) 523. [34] D.L. Baulch, J.F. Griffiths, B.R. Johnson and R. Richter, Proc. Roy. Soc. Lond. A 430 (1990) 151. [35] D.L. Baulch, J.F. Griffiths and R. Richter, Chem. Eng. Sci. 46 (1991) 2315. [36] D.L. Baulch, J.F. Griffiths and R. Richter, Proc. Roy. Soc. Lond. A 434 (1991) 399. [37] D.L. Baulch, J.F. Griffiths and R. Richter, Phil. Trans. Roy. Soc. Lond. A 337 (1991) 199. [38] D.L. Baulch, J.F. Griffiths and R. Richter, Comb, and Flame 85 (1991) 271. [39] B.R. Johnson, PhD thesis (University of Leeds, 1991). [40] D.L. Baulch, J.F. Griffiths, A.J. Pappin and A.F. Sykes, Comb, and Flame 73 (1988) 163. [41] W. Kordylewski and S.K. Scott, Comb, and Flame 57 (1984) 127. [42] D.L. Baulch, J.F. Griffiths, A.J. Pappin and A.F. Sykes, J. Chem. Soc. Faraday Trans. 1, 84 (1988) 1575. [43] K. Chinnick, C. Gibson, J.F. Griffiths, and W. Kordylewski, Proc. Roy. Soc. Lond. A 405 (1986) 117. [44] K. Chinnick, C. Gibson and J.F. Griffiths, Proc. Roy. Soc. Lond. A 405 (1986) 129. [45] B.R. Johnson, S.K. Scott and A.S. Tomlin, J. Chem. Soc. Faraday Trans. 87 (1991) 2539. [46] H. Kuhl, Z. Phys. Chem. 44 (1903) 385. [47] M. Bodenstein and F. Ohlmer, Z. Phys. Chem. 53 (1905) 166. [48] B. Topley, Nature 125 (1930) 560. [49] V.E. Cosslett and W.E. Garner, Trans. Faraday Soc. 26 (1930) 190. [50] E.J. Buckler and R.G.W. Norrish, Proc. Roy. Soc. Lond. A 167 (1938) 292. [51] G. Hadman, H.W. Thompson and C.N. Hinshelwood, Proc. Roy. Soc. Lond. A 138 (1932) 297. [52] G. Dixon-Lewis and J.W. Linnett, Trans. Faraday Soc. 49 (1953) 756. [53] D.E. Hoare and A.D. Walsh, Trans. Faraday Soc. 50 (1954) 37. [54] A.S. Gordon and R.J. Knipe, J. Phys. Chem. 59 (1955) 1160. [55] M. Prettre and P. Laffitte, C.R. Hebd. Seances Acad. Sci. 189 (1929) 177. [56] P.G. Ashmore and R.G.W. Norrish, Nature 167 (1951) 390. [57] P.G. Dickens, J.E. Dove and J.W. Linnett, Trans. Faraday Soc. 60 (1964) 539. [58] J.W. Linnett, B.G. Reuben and T.F. Wheatley, Comb, and Flame 12 (1968) 325. [59] J.R. Bond, P. Gray and J.F. Griffiths, Proc. Roy. Soc. Lond. A 375 (1981) 43. [60] J.R. Bond, P. Gray, J.F. Griffiths and S.K. Scott, Proc. Roy. Soc. Lond. A 381 (1982) 293. [61] B.J. McCaffery and A.L. Berlad, Comb, and Flame 26 (1976) 77. [62] E.N. Aleksandrov and V. Azatyan, Combust. Explos. Shock Waves 12 (1977) 407. [63] P. Gray, J.F. Griffiths and S.K. Scott, Proc. Roy. Soc. Lond. A 397 (1985) 21. [64] P. Gray, J.F. Griffiths and S.K. Scott, Proc. Roy. Soc. Lond. A 402 (1985) 187. [65] P. Gray, J.F. Griffiths and S.K. Scott, J. Chim. Phys. 84 (1987) 49. [66] B.R. Johnson and S.K. Scott, J. Chem. Soc. Faraday Trans. 86 (1990) 3701. [67] B.R. Johnson, J.F. Griffiths, and S.K. Scott, Chaos 1 (1991) 387.
544
Global behaviour in simple oxidations
Ch. 5
[68] J.F. Griffiths and A.F. Sykes, J. Chem. Soc. Faraday Trans. 85 (1989) 3059. [69] V.I. Babushok, E.A. Novikov and V.S. Babkin, Dokl. Akad. Nauk SSSR 271 (1982) 878. [70] K. Chinnick and J.F. Griffiths, J. Chem. Soc. Faraday Trans. 2, 82 (1986) 881. [71] J.F. Griffiths, in: Oscillations and Traveling Waves in Chemical Systems, eds R.J. Field and M. Burger, Chapter 15 (John Wiley, 1985) pp. 529-564. [72] J.F. Griffiths, Ann. Rev. Phys. Chem. 37 (1985) 77. [73] J.F. Griffiths, Adv. Chem. Phys. 64 (1986) 203. [74] J.F. Griffiths and S.K. Scott, Prog. Energy Comb. Sci. 13 (1987) 161. [75] P. Gray, J.F. Griffiths, S.M. Hasko and P.G. Lignola, Proc. Roy. Soc. Lond. A 374 (1981) 313. [76] H. Davy, Phil. Trans. Roy. Soc. Lond. A 107 (1817) 77. [77] P. Gray, J.F. Griffiths, S.M. Hasko and P.G. Lignola, Comb, and Flame, 43 (1981) 175. [78] J.F. Griffiths and A.F. Sykes, Proc. Roy. Soc. Lond. A 422 (1989) 289. [79] M.P. Halstead, A. Prothero, and C.P. Quinn, Proc. Roy. Soc. Lond. A 322 (1971) 377. [80] M.P. Halstead, A. Prothero and C.P. Quinn, Comb, and Flame, 20 (1971) 211. [81] M.P. Halstead, L.J. Kirsch, A. Prothero and C.P. Quinn, Proc. Roy. Soc. Lond. A 346 (1975) 515. [82] L.J. Kirsch and C.P. Quinn, 16th Symp. (Int.) Comb. (1976) 233. [83] A.H. Harrison and L.R. Cairnie, Comb, and Flame 71 (1988) 1. [84] A.J. Harrison, R.M. Furzeland, R. Summers and L.R. Cairnie, Comb, and Flame 72 (1988) 119. [85] X.-J. Wang, Comb, and Flame 75 (1989) 107. [86] X.-J. Wang and P. Gaspard, in: Spatial Inhomogeneities and Transient Behaviour in Chemical Kinetics, eds P. Gray, G. Nicolis, F. Baras, P. Borckmans and S.K. Scott (Manchester University Press, 1987) pp. 687-690. [87] P. Gaspard and X.-J. Wang, J. Stat. Phys. 48 (1987) 151. [88] S.K. Scott, Chemical Chaos (Oxford University Press, 1993). [89] S.A. Pugh, M.A. Schell and J. Ross, J. Chem. Phys. 85 (1986) 868.
Chapter 6
Experimental and Numerical Studies of Oxidation Chemistry and Spontaneous Ignition Phenomena J.F. GRIFFITHS and C. MOHAMED
6.1 INTRODUCTION
This chapter is concerned mainly with experimental observations and measurements, one purpose of which is to show how the mechanistic structures for hydrocarbon oxidation (Chapter 1) lead to the observed combustion characteristics, and to describe more recent chemical evidence from combustion studies in support of those interpretations (Section 6.5). It also paves the way to the discussion of spontaneous ignition, or autoignition, in spark-ignition engines, in Chapter 7. Historically, the underlying theme in the interpretation of the lowtemperature oxidation of hydrocarbons has been the formation of a molecular intermediate of limited stability which acts as a chain-branching agent. As the temperature is increased the rate of decomposition of the chain-branching agent is enhanced, but also there are competing nonbranching processes which are increasingly favoured at the higher temperatures. These competing processes lead to a reduction in the overall reaction rate as the temperature is raised (Chapter 1). Since the early 1960s, the key to this thermokinetic switch has been recognized to involve the alkyl/alkylperoxy radical equilibrium, (1)
R + 02
2
in which the alkyl moiety, R, takes whatever forms are appropriate as a result of H atom abstraction from the fuel molecule. Consequently, differences in behaviour emerge from different isomeric
546
Experimental and numerical studies of oxidation chemistry
Ch. 6
structures within a hydrocarbon series [1], but it is not easy to establish quantitative relationships between the measurements made in different experimental systems because the dependences on experimental parameters and mechanistic features are complex. One way forward to a unified interpretation is through numerical modelling. These aspects are addressed in Section 6.6. Chain-branching is known to arise from further reactions of R 0 2 , but a fundamental distinction of current interpretations of the chain-thermal interactions leading to cool-flames and spontaneous ignition, from those invoked formerly, is that the simplistic concept, that "a critical concentration of a molecular intermediate" must be reached before ignition can occur, is no longer a requirement. The subtle balance achieved between branching and non-branching reactions in a changing temperature environment is sufficient to give rise to the complex, non-isothermal combustion phenomena that result from the exothermic oxidation of hydrocarbons, the principles of which are discussed in Chapter 5.
6.2 MEASUREMENTS OF REACTION RATE AND ITS DEPENDENCE ON EXPERIMENTAL CONDITIONS
6.2.1 Introduction The majority of chemical systems exhibit a monotonic increase in reaction rate as the temperature is increased. Perhaps the most important feature of alkane oxidation, and certainly a distinguishing characteristic from most other chemical systems [2, 3], is the existence of a temperature range, typically 700-800 K, within which the overall reaction rate decreases as the temperature is increased. This is often referred to as the "region of the negative temperature coefficient (ntc) of reaction rate". It was first revealed by Pease during the decade from 1929 in studies of the oxidation of propane and n-butane using both flow and static systems [4,5]. Not only has this kinetic phenomenon become a widely recognized feature of the low-temperature oxidation of many (but not all) classes of organic compounds (Table 6.1), but its importance in determining some global characteristics of the non-isothermal behaviour has been unequivocally established. The isothermal kinetic origins of the ntc have been discussed in Chapter 1, but much of the complexity of the combustion of hydrocarbons
547
Measurement of reaction rate
TABLE 6.1 Experimental evidence for the existence of a region of ntc in hydrocarbon oxidation (terms and abbreviations are defined below) Results
Ref.
Method
Methane/0 2 Ethane/0 2 Propane/0 2 /N 2 Propane/0 2
[6] [7] [4] [8]
Closed Closed Closed Closed
Propene/0 2
[9]
Closed vessel
[10] [11] [12] [13] [14]
Closed vessel Closed vessel CSTR Closed vessel PFR.
Reactants
Propane/0 2 Propane/0 2 Propane/0 2 /N 2 Propane/air Propane/air
vessel vessel vessel vessel
«-butane/0 2 z-butane/02 rc-butane/02
[9]
Closed vessel
[15]
Closed vessel
z-butane/02
[16]
Ai-butane/02 z-butane/02 «-butane/02/N2 «-butane/02/N2 rc-butane/02/N2/Ar Butane/0 2 /N 2 /Ar «-pentane/0 2 n-pentane/Air Cyclopentane M-pentane/Air neo-pentane/0 2 M-heptane/N2/02
[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
Kinetic and analytical investigation (560-700 K) Closed vessel CSTR Closed vessel Closed vessel Burner/RCM RCM Closed vessel Closed vessel Closed vessel Annular flow reactor Closed vessel Flow reactor
^-heptane/air rc-heptane/N2/02 Ai-octane/air /-octane/air z-octane/N2/02 Alkanes Alkenes/air
[29] [30] [31] [32] [31] [33] [34]
Shock tube JSR Flow reactor JSFR JSR Closed vessel Flow system Low pressure
Aromatics/0 2 Xylene (meta-/para-)
[35] [36]
Closed vessel Closed vessel
Aplt vs. 7 AT and p max vs. 7 Pmax
p-t measurements 0 2 consumption logio ( ) loglo(l/0) vs. 1/7 Ap vs. 7 Pmax VS. 7
AT-t records Q vs. 7 (6) and (d/?/dr)max vs. Ta AT vs. T Fuel consumption vs. T lOglo(Pmax)
log1(,(l/0) vs. 1/7 log 100/0 and logp max vs. 1000/7 0and logp max vs. 1/7 A/? and 6 A7 max vs. 7 and Q vs. 7 0vs. 7 p max and 6 vs. 7 T, VS. Tc Tj VS. Tc
p vs. 7 /vs. 7 Pmax VS. 7
[RH] and [0 2 ] vs. Ta T, and p max vs. 7 Extent of reaction, p and A7 vs. 7 T, VS. 7
[RH] vs. 7 [0 2 ] vs. Ta Q vs. 7 [RH] vs. 7 Formation of CO and / Extent of pre-flame reaction, %CO in product gases vs. 7 lOg f20-60 VS. 1/7
p-t measurements
CSTR = Continuously-stirred tank reactor, PFR = Pressurized flow reactor, RCM = Rapid compression machine, JSR = Jet-stirred reactor, JSFR Jet-stirred flow reactor. Aplt = p = rate of change of pressure, (A/?)max = maximum pressure increase, p max maximum rate of change of pressure, 6 = induction period = time delay from admission of reactants to attainment of p max , A7=Reactant temperature excess, Ta = wall temperature, Tc = Compressed gas temperature, r, = ignition delay, / = light emission intensity, t2o-6o = period of maximum rate of reaction given that ('20-60)-1 = kpn, Q = heat release rate.
548
Experimental and numerical studies of oxidation chemistry
Ch. 6
emerges as a result of the interaction of thermal feedback with the kinetic development. One particularly important consequence of the ntc is a complex dependence of the ignition delay on initial or control temperature. The two-stage development of ignition during hydrocarbon oxidation from temperatures below about 750 K also arises from the kinetic interactions which give rise to the ntc of reaction rate, coupled to thermal feedback. 6.2.2 The dependence of reaction rate on temperature (i) Concentration or mole fraction measurements In the earliest studies in tubes under laminar flow (see Section 6.3), the existence of an overall negative temperature dependence was interpreted from the extent of reactant or oxidant consumption at the outlet when constant inlet concentration and flow rates were maintained as the tube temperature was raised [4]. Complementary to this is the extent of reactant consumption at the outlet of a well-stirred flow reactor (or continuous stirred-tank reactor, CSTR) at constant mean residence time, tres. For reaction in a CSTR, the rate of reaction is given by v = [A]i(l-a)/tm
(6.1)
in which [A]t represents the inlet fuel or oxidant concentration and a represents the fraction of reactant remaining at the outlet. The formal interpretation of behaviour in a CSTR is discussed in Chapter 5, and the experimental systems employed in hydrocarbon oxidation studies are described in Section 6.3. Direct evidence for the existence of a negative temperature coefficient of reaction rate, and the distinctions between alkanes of different structure were demonstrated in recent studies of n-heptane and 2,2,4-trimethylpentane (/-octane) oxidation in a well-stirred flow reactor (Figs 6.1 and 6.2). The experiments were performed by Dagaut et al. [30] at 1 MPa and a mean residence time of 1 s over the temperature range 550-1150 K. The fuel at the reactor inlet was set at a fixed mole fraction, 10~3, and the [RH]: [0 2 ] ratio was chosen in the proportions 1:36,1:22,1:11 and 1:7.5. These correspond to the stoichiometric proportions = 0.3, 0.5, 1.0 and 1.5 respectively. The reactants were diluted with an excess of N 2 in order to maintain isothermal conditions in the reactor. The results shown in Fig. 6.1 for n-heptane and in Fig. 6.2 for 2,2,4-
Measurement of reaction rate
549
1.2X10" 3 -,
T/K Fig. 6.1. Variation in measured mole fraction of /i-heptane as a function of vessel temperature. Jet-stirred reactor at 1 MPa over a range of fuel:oxygen ratios () = 1:3.6 (0.3) ( • ) , 1:22 (0.5) ( • ) , 1:11 (1.0) (A) and 1:7.5 (1.5) ( • ) at a mean residence time of 1.0 s. Reactants diluted with excess N2 to maintain isothermal conditions. (After Dagaut et al. [30].)
trimethylpentane represent the variation in mole fraction of each alkane in the reactor at stationary conditions over an appropriate temperature range for each [RH]:[0 2 ] mixture. The mole fraction is proportional to the fraction of reactant remaining (a) to a good approximation because there was a sufficiently high dilution of the reactants by inert gas for there to be a negligible change in numbers of moles of material as reaction proceeded. If reactant concentrations were to be determined quantitatively from these data a correction proportional to T~l would have to be applied across the temperature range. As expressed in (6.1), the reaction rate is proportional to the variation in mole fraction of reactant consumed (1 - a). Thus, at a constant vessel temperature, not only do the fractional extents of reaction of fuel and oxygen give information about the stoichiometry of the reaction, but also the differences between fractional extents
550
Experimental and numerical studies of oxidation chemistry
Ch. 6
1.2x10°-,
T/K Fig. 6.2. Variation in measured mole fraction of 2,2,4-trimethylpentane as a function of vessel temperature. Jet-stirred reactor at 1 MPa at a mean residence time of 1.0 s. See Fig. 6.1 for other conditions and symbols. (After Dagaut et al. [30].)
of reaction for different reactant compositions contain information about the dependences of the overall rate on the reactant concentrations. Although there are variations in the measured alkane mole fractions, which affect the reaction rate, by far the greatest effect on the overall rate must be that of the differences in oxygen mole fractions, which were varied over a very wide range in these experiments. Whereas extensive oxidation of ^-heptane occurred from about 550 K (Fig. 6.1), the oxidation of 2,2,4-trimethylpentane became prominent only above 750 K (Fig. 6.2) under the conditions of these experiments. The extents of reaction that occurred below 750 K were limited by the residence time in the reactor. The considerable difference of reactivity of these two alkanes, may be expected to arise from the relative difficulty for 2,2,4trimethylpentylperoxy radicals to undergo isomerization reactions compared with the reactivity of n-heptylperoxy radicals (Chapters 1 and 2). Related arguments are presented at the end of this chapter. Although the temperature range within which the ntc of reaction rate was observed in
551
Measurement of reaction rate
ra-heptane oxidation was not noticeably effected by different [RH]: [0 2 ] ratios, the variation in extent of reaction within this temperature range was very marked indeed. Only a limited negative temperature dependence of the extent of reaction was detected in the richest composition, which also exhibited a more complex form than that found in the more lean compositions.
(ii) Pressure change in closed vessels The reaction rate dependence on temperature cannot be evaluated so easily from the rate of change of reactant concentration in a closed vessel, and in many studies the global rate has been measured from the rate of pressure change, the pressure change itself being taken to be proportional to the extent of reaction (Fig. 6.3). The underlying assumptions are that the reaction is isothermal, which is the case only at the lowest temperatures and pressures in hydrocarbon oxidation, and that the oxidation is accompanied not only by a change in numbers of moles of material (reactants—> 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)(£),
<">
where [F] 0 and [F] are the initial fuel concentration and its concentration at time t respectively, n is the total number of moles present. Invoking the ideal gas equation (6.5) < > • yields
The autocatalytic nature of the reaction (see Chapters 1 and 5), which may be a limitation on the validity of the underlying assumption about a constant reaction stoichiometry (i.e., constant y and z), requires there to be a specific reference point at which the rate of pressure change may be compared. The usual reference is taken to be the maximum rate of pressure change in each experiment (p max )- This approach has been exploited extensively over the years, and there are very many other examples to be found in the literature (Table 6.1). For example, Wu et al. [15], investigating the gaseous oxidation of butane in the temperature range 470-810 K, were
553
Measurement of reaction rate 3.0-.
2.0
S
1.0-
\
100/0 d
0A
-1.0 1.40 1.55 1000/T(K)
1.25
1.70
Fig. 6.4. Relationship between the maximum rate of pressure change (p max ) and reciprocal inclination time (100/0) as a function of closed vessel temperature and the time to the attainment of the maximum rate of pressure change, the induction time (f,-). (After Shu and Bardwell [15].)
able to confirm that as the temperature increased from 667-703 K the rate of reaction decreased. It was also shown that the time, T, from admission of reactants into the vessel to attainment of the maximum rate of pressure rise reached a maximum when the maximum rate was itself at a minimum (Fig. 6.4). Knox and Norrish [7] showed that, during the oxidation of ethane in a closed vessel, the vessel temperature at which p m a x exhibited a maximum and a minimum coincided with a maximum and a minimum of an associated temperature change, AT (Fig. 6.5). Whilst these observations are a qualitative confirmation that there is a negative temperature dependence of overall rate for ethane oxidation, the relationship of the rate of pressure change in a closed vessel under non-isothermal conditions to the reaction rate is a complicated one [37]. Differentiating (6.5) with respect to time gives the rate of pressure change,
dt
V\
\dtj
\dt
(6.7)
554
Experimental and numerical studies of oxidation chemistry
Ch. 6
h80
30
60
5 40
10
k20
600
700
650 T/K
Fig. 6.5. Relationship between the maximum rate of pressure change (p max ) as a function of vessel temperature (Ta) and the maximum temperature excess reached (AT) during reaction in a closed vessel. (After Knox and Norrish [7].)
Invoking (6.2) and (6.3) yields dp dt
R{(z[F]0 + (l+y-
z)[F]} +
dT RT(l + y d/ (6.8)
Equation (6.6) is recovered under isothermal conditions (d77dr = 0), and (dp/dt)max is then found to coincide with (d[F]/dt)max. The rate of reactant consumption under non-isothermal conditions is given by d[F] _ 1 dp dt RT{\ + y + z) dt
[F] +
dlnT z[F]o (1 + y + z)) dt
(6.9)
which shows that there is a correction term to be applied to the rate of pressure change for the representation of the reaction rate under nonisothermal conditions. Unfortunately, the additional term requires knowledge of the progress of reaction itself in order to obtain information about
555
Measurement of reaction rate
the temperature of the system and its rate of change, which tends to defeat the initial purpose of exploiting the measurement of pressure. Supplementary points that relate to Fig. 6.5 are (i) whether or not, the maximum rate of pressure change is a useful practical parameter for the interpretation of the reaction rate under non-isothermal conditions, and (ii) whether or not, the range of vessel temperatures within which the ntc is found to exist from pressure measurements is a satisfactory representation even if there is some quantitative discrepancy from the rate of reactant consumption. The correction term in equation (6.9) may not be negligible, and it is certainly not constant, since it is known that the intervention of temperature change during the autocatalytic oxidation of alkanes causes the maximum rate of reactant consumption to occur at increasing extents of reaction [38,39]. A condition for the maximum rate of pressure change during reaction under non-isothermal conditions is obtained by differentiating (6.7) with respect to time and equating it to zero [37].
By substituting (6.3) and its first and second differentials with respect to time in (6.10) the condition for a maximum rate of pressure change during non-isothermal reaction corresponds to
At2
At
At
\
(1 + y + z)J At2
This is not the same as the condition for a maximum rate of reaction defined by the consumption of the reactant,
^ 3 = 0. At2
(6.12)
v
J
Clearly, some caution must be exercised in the interpretation of data which involve measurements of rates of pressure change under non-isothermal conditions. Unfortunately, whether or not self-heating has occurred in
556
Experimental and numerical studies of oxidation chemistry
Ch. 6
much of the early work cannot be readily discerned because reactant temperature measurements were not made. (Hi) Temperature change in well-stirred closed vessels The use of temperature change under non-isothermal conditions as a monitor of reaction rate gives a quite different perspective. With uniform temperature throughout a closed reaction vessel and a Newtonian heat loss rate at the vessel surface, the energy conservation equation with respect to the rate of reactant consumption (see Chapter 5) is,
-^)—(-fHf)^• in which Cv is a specific heat capacity at constant volume and a is a molar density for the system (both assumed constant), AC/ is the internal energy change per mole of fuel reacted, \ *s a heat transfer coefficient and S/V is the surface to volume ratio. (T— Ta) represents the mean temperature excess of the reactants above the vessel wall temperature. The first term on the right-hand side represents the heat release rate per unit volume, and the second term represents the volumetric loss rate. In a stationary (or quasi-stationary) state dT/dt = 0, so that (-AC,)(-fil)-(|)(7--T.).
(6.M)
Thus, the heat release rate is proportional to the measured temperature excess. The heat release rate and its dependence on conditions may be the fundamental information required for the interpretation of many combustion phenomena, but once again the heat transfer coefficient must be evaluated. Heat transfer coefficients for low-pressure gases in closed, stirred, spherical reaction vessels were measured by Gray and co-workers [40-42]. A simple quantitative relationship of the rate of reactant consumption to the temperature excess requires that the overall exothermicity does not vary with reaction conditions. As is implicit in Chapter 1 from the variations in reaction stoichiometry, and discussed further in Section 6.5, this is certainly not the case during alkane oxidation over the temperature range 500-900 K.
Measurement of reaction rate
557
Given ideal, well-stirred conditions, the heat release rate could be interpreted from (6.13) under non-stationary conditions, but accurate measurements of (dT/dt) would also be required. The rate of temperature change is always more important than the heat loss rate during the late stages of the development of ignition, because the chemical time-scale is much shorter than the Newtonian cooling time-scale. Temperatures during hydrocarbon oxidation have been measured by thermocouples in closed vessels, well-stirred flow systems and laminar flow tubes or burner studies. Fine wire thermocouples, suitably placed in the reacting volume, are required in order to measure temperature changes satisfactorily in atmospheric or lower pressure gases in closed and wellstirred flow systems. They must have as low a thermal inertia as possible, but they must be supported in a sufficiently robust, but least obtrusive manner to survive use in numerous experiments. Satisfactory vacuum sealing is also necessary for sub-atmospheric pressure studies. Such devices were pioneered in experimental investigations of thermal ignition in gases in the 1960s [43, 44]. The most responsive yet robust junctions are usually gas-welded from platinum/platinum + rhodium alloy wires (25 /mm diameter) which are then coated in a thin film of silica to prevent catalytic activity of the surface. Response of such junctions is in the range 10100 ms, according to the bead size and the gas pressure, the response being slowest at the lowest pressures. Commonly, the internal junction is coupled to one located on the external surface of the vessel so that the signal gives a measure of the temperature difference. Unfortunately well-stirred conditions have not been maintained in most applications. (iv) Temperature change in well-stirred flow reactors The stationary-state heat release rate may also be interpreted from the measured temperature excess in well-stirred flow systems. The energy conservation equation for a well-stirred flow system is similar to equation (6.13) but an additional term is required to represent heat transport via the outflowing gases (aCp(T- Ta)/tTes) as shown in equation (4.4). The inflowing gases are assumed to be pre-heated to the vessel temperature, Ta. Under constant pressure conditions, normally applicable to flow reactors, Cp replaces Cv and AH replaces AU in equation (6.13). The heat release is obtained from a summation of the product of individual reaction rates and their enthalpy change ((-AH)jRj) in equation (5.4)). The relative rate of wall losses to mass transport losses is governed by
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Experimental and numerical studies of oxidation chemistry
Ch. 6
the ratio tres/tN (= x^reJVaCp from equation (5.4)), where tN is the Newtonian cooling time. In small reaction vessels, which have relatively high surface to volume ratios ( S / F > l e m _ 1 ) , operated at or below 100 kPa, tN is found to be about 0.5 s [45]. Thus, for mean residence times in excess of 5 s, more than 90% of the heat is lost through the vessel walls. These conditions are typical of laboratory studies of alkane combustion in low pressure CSTRs. Heat transfer coefficients have been obtained by Caprio et al. [18] and Lignola et al. [46] from a mechanically-stirred flow reactor. Heat release rate measurements from propane oxidation under stationary state conditions in a mechanically-stirred flow reactor were reported by Gray and Felton [12]. For the reactant composition [C 3 H 8 ]:[0 2 ] = 1:2.2 at a pressure of 62.5 kPa (Fig. 6.6) there was a range of reactant temperatures at which the rate of heat release was virtually independent of temperature. However, a negative temperature dependence of the heat
300 PRF70
250
£ 200
i
3£ 150
2
^ 8
100 50 -| 0 550
—,— 600
650
700
750
T/K Fig. 6.6. Heat release rates measured during alkane combustion in well-stirred flow reactors. Data for propane in oxygen, [RH]:[0 2 ] = 1:2.2, were measured by Gray and Felton [12] at p - 62.5 kPa and tres = 125 s. A region of hysterisis was obtained, as shown by the broken line in the heat release rate curve. Data for /-butane in an oxygen and nitrogen mixture, [RH]: [0 2 ]: [N2] = 1:2:1 were obtained by Caprio et al, [18] at p = 100 kPa and fres = 20 s. Also shown are the heat release rates obtained by Lignola et al. [47] for primary reference fuel mixtures, i-C8H18 and n-C 7 Hi 6 , given as the volume percentage of /~C8Hi8, at p = 100 kPa and tr< lis.
Measurement of reaction rate
559
release rate was established over a similar temperature range in equimolar propane + oxygen mixtures. Caprio et al. [18] measured heat release rates from /-butane oxidation at 100 kPa in the reactant mixture [/-C4H10]: [0 2 ]: [N2] = 1:2:1 (Fig. 6.6). A considerably higher heat release rate than that from propane occurred in this system despite the lower partial pressure of reactant. It seems likely that the differences in residence times in the respective experiments is contributory since variations in the heat release rate from /-butane oxidation were obtained when the mean residence time was changed [18]. However, Lignola et al. [47] showed that, at constant fres, the heat release rate from primary reference fuel mixtures, comprising /-C8H18 and nC 7 H 16 , depended on the proportions of the fuel components (Fig. 6.6). The negative temperature dependences shown in these heat release rate curves are governed both by reaction rate and enthalpy. A relationship to the rate of reactant consumption over a range of temperatures is subject to constant enthalpy change. This requires there to be no change in reaction stoichiometry, which is not the case for the low-temperature oxidation of alkanes. 6.2.3 Ignition delay times The ignition delay is defined as the time interval from the initiation of combustion to the occurrence of flame or some other manifestation of ignition. Suitable quantitative monitors of the ignition event could be the maximum in light output or the maximum rate of temperature or pressure change. Ignition delays may be measured in closed vessels, shock tubes or rapid compression machines. These experimental techniques are discussed in Section 6.3. Provided that premixed gaseous fuels are used, the ignition delay arises solely from the development of the heat release rate in the chemical process in competition with the heat loss rate from the system. Although these influence the ignition delay during droplet combustion also, they are not the only factors [48]. Liquid evaporation and mixing may contribute to the overall elapsed time before ignition occurs when droplets are injected into heated gases, and they may even be the most important factors if the chemical time-scale is short compared with the "physical" time-scale. If kinetic interpretations are being sought, it is important to limit the analysis to a range of conditions in which the reaction is under kinetic control.
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Experimental and numerical studies of oxidation chemistry
Ch. 6
CO
K-tf*—h—H
t/ms Fig. 6.7. Pressure and light output records typical of those observed from the two-stage ignition of hydrocarbons. These results were obtained by the authors during the combustion of n-pentane in a rapid compression machine. The duration of the compression stroke was 22 ms, as shown in the lower pressure record. The first- and second-stage time intervals, measured from the end of compression, are marked as TX and r 2 respectively. Compressed gas temperature derived from (6.16) was 740 K [49].
At initial temperatures above about 850 K, ignition occurs in a single stage. However, at lower initial reactant temperatures a two-stage ignition can be discerned from the pressure record that develops during alkane ignition, and that of other organic compounds. Subsidiary components of the overall ignition delay may be interpreted from these structures (Fig. 6.7). The first stage of the ignition involves the development of a coolflame. This gives a modest pressure increase in a closed vessel, which signifies that some heat release has occurred and the reactant temperature has increased because of it, typically to 800-850 K. The rate of pressure change is deceleratory, such that a temporary "plateau" is often observed in the pressure record (Fig. 6.7) [49, 50]. The time from initiation of reaction (marked as the end of compression in Fig. 6.7) to the maximum rate of pressure rise in the first stage, which corresponds to the maximum in an accompanying light output, is defined as TI. The interval from this point to the maximum rate of pressure rise in the second stage is defined as T 2 . When the variation of these parameters is plotted as a function of the isentropically compressed core gas temperature (see Section 6.3.5) it is found that TX decreases monotonically with increasing temperature, whereas r 2 increases at first, before decreasing
561
Measurement of reaction rate
50-,
40 J
30-J C/)
E
20 J
10-J
650
700
750
800
850
900
950
T c /K Fig. 6.8. Variation of rl and r 2 as a function of compressed gas temperature (Tc at end of compression) in a rapid compression machine, n-pentane/air ( 0 = 1 . 0 ) ; compressed gas density, 130 mol m" 3 ; Pc = 0.65 - 0.75 MPa [49].
(Fig. 6.8). The overall consequence is for the ignition delay rT(= ri + T 2 ) to decrease at both low- and high-compressed gas temperatures, but to exhibit an intermediate temperature range within which rt increases (Fig. 6.8) [49-51]. The origin of this complex dependence of rT on compressed gas temperature, Tc can be traced back to the kinetics that also determine the ntc of reaction rate in isothermal conditions, but the reactant temperature itself changes dramatically throughout the ignition delay. Thus, while the increase of ignition delay throughout an intermediate range of control temperatures is a consequence of the complex dependence of the overall kinetics on temperature, it does not constitute a direct determination of "the ntc of reaction rate", as is tacitly assumed. This distinction becomes important when comparisons are being made between behaviour in different experimental systems. It is possible for the ignition delay at a given reactant density and control temperature to be reproduced quantitatively
562
Experimental and numerical studies of oxidation chemistry
Ch. 6
in different systems only under adiabatic conditions. When heat losses occur, the measured ignition delay at given conditions is a characteristic of the experimental system. This may mean that, for a particular reactant composition and density, even the range of control temperatures in which a negative temperature dependence of ignition delay is measured varies from system to system. Such differences should be taken into account in numerical modelling if differences in the heat transfer rates can be quantified.
6.3 EXPERIMENTAL METHODS
6.3.1 Introduction In this section we describe the main experimental techniques that have been used to study physical and chemical aspects of hydrocarbon oxidation, especially at temperatures below 1000 K. Methods of chemical analysis are also reviewed. 6.3.2 Closed, constant volume reaction vessels Experiments in closed, constant volume vessels involve admission of reactants to an evacuated vessel. The vessel is maintained at constant temperature in an oven so that an experiment is performed at a given initial reactant pressure (p) and vessel temperature (Ta). The reaction vessels used in studies at pressures below one atmosphere are normally made from Pyrex glass. Silica is not a pre-requisite because most of the "interesting" behaviour occurs at temperatures below the softening point of Pyrex glass. Optical access within the visible transmission range also presents no problem with glass vessels. Metal vessels (normally stainless steel) are required for experiments at pressures at, or above, atmospheric. There may be particular difficulties with surface activity in such cases, although glass vessels are not entirely free of such problems. For any interpretation of kinetic parameters from chemical measurements of exothermic elementary reactions it is essential that the reaction be performed under isothermal conditions so that the reactor temperature itself is the control parameter. This is normally the case in experiments which are set up to investigate elementary reactions. However, low-tem-
Experimental methods
563
perature hydrocarbon oxidation is sufficiently exothermic that there must be an accompanying temperature change in closed vessels, which can be very large when cool-flames occur (AT > 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
Experimental and numerical studies of oxidation chemistry
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 <& = 1.0 but with a high dilution by nitrogen. Very high gas flows are maintained so that radial gradients are eliminated by turbulence, leading also to short contact times (t< 200 ms). Operation of the tube is regarded to be virtually adiabatic and axial temperature increases of up to 200 K occur as the reactants progress along the tube. Kinetic studies in a well-stirred flow reactor by Malte and co-workers at the University of Washington [68-70] also apply to temperatures in excess of 1000 K. These studies provide information on the transition from the low-temperature combustion processes to those at normal flame temperatures, in the final stages of spontaneous ignition. These and related studies are summarized in the Appendix. A turbulent, high pressure, flow reactor has been developed by Koert et aL [14] for studies throughout the temperature range from 600 to 1000 K and at pressures up to 1.5 MPa. Results from it pertaining to propane oxidation are discussed in Section 6.4. Vertical, upward flow, laminar flow reactors which have been used to study stabilized cool-flames and two-stage ignitions are a development of the atmospheric pressure, vertical, laminar flow reactors, pioneered at the Naval Research Laboratories in Washington [71]. These types of systems have been used by other research groups [72-74]. The flow tube that was
Experimental methods
565
operated at BP Research Centre in the 1960s [72] was set up with a temperature gradient. The hottest point was at the top (—1000 K) and decayed to —500 K at the base. Under certain conditions, two cool-flames could be stabilized prior to the onset of the second-stage ignition. This appears to be the only reported case of a spatial periodic structure of coolflames in flowing gases. Only one cool-flame is established in a uniformly heated tube. The stabilization of a cool-flame and two-stage ignitions in tubes were reported in very early work also, but usually with a horizontal orientation, which gave rise to considerable distortion of the combustion fronts, as a result of buoyancy effects which disturbed the horizontal, laminar flow [75]. Heated flat-flame burners also have been used for the stabilization of cool-flame and two-stage ignitions. These systems were pioneered by Agnew et al. [76], and a definitive study of the spatially resolved chemistry of diethyl ether combustion has been reported [77]. Ballinger and Ryason [78] have described this type of burner in detail and have reported the relative behaviour of some n-alkanes, PRF fuel mixtures, and the effect of anti-knock additives. An important application of the heated flat-flame burner is the photolytic perturbation of the post cool-flame gases downstream from the stabilized cool-flame of n-heptane [83]. The chemical perturbation was initiated by an incident laser beam which dissociated H 2 0 2 to OH radicals. The relaxation of the excess OH, measured by laser induced fluorescence, was shown to be tied to the principal reaction modes in alkane oxidation, namely propagation, termination and net branching. Global rate parameters were quantified from the decay curves. Carlier et al. [21] stabilized the two-stage spontaneous ignition of nbutane on a heated flat-flame burner at a pressure of 0.18 MPa. The temperature profile was measured by a chromel-alumel coated thermocouple and samples at different points along the axis of the reactor were removed for analysis by gas chromatography, polarography or electron spin resonance (esr) spectroscopy. The esr signals gave information about the spatial variation of H 0 2 and R 0 2 radical concentrations. 6.3.4 Shock tubes The combustion of hydrocarbons covers an exceedingly wide temperature range (600-2500 K), so it is essential that kinetic data of many elementary reactions be obtained over a similar range. Elementary reaction rate
566
Experimental and numerical studies of oxidation chemistry
Ch. 6
data can be derived from flame studies, but another source of data at high temperatures is through study of reactions behind a shock wave under controlled experimental conditions in a shock tube. Methods of measuring rate constants of reactions, or other combustion characteristics such as spontaneous ignition delay times, under carefully controlled conditions behind shock waves are outlined in this section. Kinetic studies in shock tubes are also discussed in Chapter 2. A high temperature and pressure is rapidly attained in a shock heated gas, governed ideally by the pressure ratio across the shock front and the ratio of heat capacities of the gases [80-82]. The discontinuity means that a reactant gas is raised virtually discontinuously to the shocked gas temperature, T0. The time interval available before a rapid cooling occurs, 10-1000 JUS, is kinetically significant at high temperatures. Heat losses are negligible over this reaction time interval. These are ideal circumstances in which quantitative kinetic measurements may be made. A plane shock wave is produced in a long, closed tube by the sudden bursting of a diaphragm (e.g., aluminium foil) which separates one gas at high pressure, the driver gas, from another at low pressure, the test gas. For kinetic measurements, the reactants in the test gas are usually heavily diluted with >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
Experimental and numerical studies of oxidation chemistry
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
Experimental and numerical studies of oxidation chemistry
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].
Experimental methods
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
Experimental and numerical studies of oxidation chemistry
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
Experimental methods
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
Experimental and numerical studies of oxidation chemistry
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
Experimental and numerical studies of oxidation chemistry
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
Experimental and numerical studies of oxidation chemistry
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 <& = 1. In fact, methane is alone amongst the hydrocarbons in exhibiting a minimum ignition temperature in a mixture with air which is slightly leaner than the stoichiometric mixture ([CH 4 ]:[0 2 ] = 1:2) [130]. 6.4.3 Spontaneous ignition in flowing gases Most of the phenomena observed in closed systems have their counterparts also in flow systems. In fact, the first observations of cool-flames were made in moving gases by Sir Humphry Davy [131] in (1815) and investigated further by Perkin [132] in 1888. Cool-flames have been stabilized in heated, laminar flow tubes or above heated burners (Section 6.3.3). Their velocities are relatively low, 0.05-0.1 m s _ 1 compared with the normal burning velocity for alkanes of about 0.4 m s _ 1 . By gradually increasing the proportion of oxygen in the gaseous mixture it is possible to cause the second stage, hot flame also to be stabilized downstream from the coolflame, giving two-stage ignition. The benefit of this mode of generation is that, by moving the probe system relative to the flame structure, thermocouple measurements and detailed chemical analysis can be made by continuous sampling from the regions below, between and above the flames [71-74, 77], thus, building up a spatial profile which relates to the temporal evolution of cool-flames and spontaneous ignition in closed systems. In well-stirred flow reactors, oscillatory cool-flames are sustained indefinitely as long as the reactor temperature and flow control are held constant.
Global combustion phenomena
579
Two-stage or multiple-stage ignitions also become repetitive events. The combustion of acetaldehyde [55, 56] and of a number of alkanes [18, 47, 58, 117, 133] have been studied in this way at atmospheric pressure or below, and experiments at pressures up to 1 MPa have been made in wellstirred flow reactors [32, 133]. The time scales of events are some orders of magnitude shorter than those in sub-atmospheric pressure studies. Investigations at high pressures are related to conditions in the combustion chambers of spark-ignition or diesel engines. 6.4.4 Ignition diagrams for selected hydrocarbons It is important to distinguish the consequences of different molecular structure from the effects of different reaction vessel size or shape, or the proportions of fuel:oxygen in the reactant mixture. Usually, it is not easy to draw kinetic conclusions directly from comparisons between the experimental results obtained by different groups of workers because there are considerable differences in the choice of reaction vessels and reactant compositions. The variation in location of ignition and cool-flame boundaries in reaction vessels of different size and shape may be demonstrated with respect to results published for propane oxidation at similar C 3 H 8 : 0 2 ratios (Fig. 6.11). Newitt and Thornes [134] did not distinguish multiplestage ignitions in their early work (Fig. 6.11B). The effect of variations of the ignition and cool-flame boundaries as a result of changes of the percentage by volume of C 3 H 8 in air is shown in Fig. 6.12. These compositions lie either side of the stoichiometric mixture, 0 = 1, which contains 3.8% by volume of C 3 H 8 . These results show that, throughout the temperature range to 900 K, spontaneous ignition occurs with increasing ease as the fuel and air mixture is enriched. However, there must be a stage of enrichment at which spontaneous ignition starts to become increasingly difficult. There is very little information available about this aspect of spontaneous ignition. Qualitative differences may also arise as a result of changes in reactant compositions. Bardwell [136] obtained the (p-Ta) ignition diagrams for n-butane + oxygen mixtures in a 250 cm3 cylindrical reaction vessel at [nC 4 H 10 ]:[O 2 ] ratios of 1:1 and 1:2 (Fig. 6.13). Although the quantitative differences in location of the cool flame and ignition regions are not very significant, there is a major qualitative difference in the development of a region of multiple-stage ignition, called delayed ignition by Bardwell [136]
580
Experimental and numerical studies of oxidation chemistry
Ch. 6
801
ouIGNITION
60-
rf
|2S
fim
40-
20-
OP
X
60
-'
40
20
* _ * / SLOW REACTION
550
600
20
SLOW REACTION
650
550
600
650
SLOW REACTION
550
600
650
Ta/K
Fig. 6.11. (p-Ta) ignition diagrams for the combustion of propane in oxygen [C 3 H 8 ]: [0 2 ] = 1.0) in unstirred closed vessels of three different sizes. A, 1000 cm3 sphere [56]; B, 400 cm3 sphere [34]; C, 300 cm3 cylinder [124].
1.0
7.5% CO Q.
0.5H
500
600
700
800
900
Ta/K
Fig. 6.12. Ignition and cool flame boundaries (broken lines) in the (p-Ta) ignition diagram for the combustion of propane in air as the proportion of propane (% by volume) in air is varied [135].
Global combustion phenomena (a)
500
400 H
300 A
*
I
IGNITION
8
1
200 H
IOOH
K
7 %
} COOL FLAMES?
\
W° 470
—T" 570
—I 770
670 Ta/K
400i
(b)
300H
TWO-STAGE IGNITION
&
5 200-J COOL FLAMES
100H
«S
SLOW COMBUSTION
540
560
580
600
620
Ta/K Fig. 6.13. A comparison of (p-Ta) ignition diagrams for rc-butane + oxygen in the proportions (a) 1:1 and (b) 1:2 respectively. (After Bardwell [15].)
582
Experimental and numerical studies of oxidation chemistry
Ch. 6
in the 1:2 mixture. Multiple-stage ignitions were not observed by Berry et al. [137] in a very similar reaction mixture but in a smaller cylindrical reaction vessel (200 cm3), which may indicate a moderating effect on the reactivity by enhanced heat losses. If direct comparisons of reactivity of different hydrocarbons, or different isomeric structures are to be made, it is desirable that the measurements be made in the same vessel, and with [RH]: [0 2 ] in equivalent proportions. For isomers of a particular fuel, [RH]: [0 2 ] = 1 or some other fixed ratio would seem satisfactory. For example, the differences of reactivity of nC4Hio and /-C4H10 are demonstrated by their respective (p-Ta) ignition diagrams in a closed vessel at the same reactant compositions, [RH]:[0 2 ] = 1:2, Fig. 6.14. Whereas the ignition boundary of n-C 4 Hi 0 was fully developed in these mixtures at a reactant pressure above 20 kPa, there was no evidence of ignition of i-C4H10 even at pressures exceeding 40kPa [127]. Multiple cool-flames were detected in both systems in a similar temperature range, although the threshold pressure for i-C4H10 was higher than that for n-C 4 H 10 . A B IGNITION
0-}
550
1
1
600
650
IOOH
550
,
,
600
650
Ta/K
Fig. 6.14. A comparison of (p~Ta) ignition diagrams for (A) ra-butane + oxygen and (B) ibutane + oxygen In the molar proportions 1:1 measured in the same reaction vessel. (After Pollard [127].)
Global combustion phenomena
583
0.75
0.5Q.
i 0.25
0
. 550
600
650
700
Ta/K
Fig. 6.15. A comparison of the (p-Ta) ignition diagrams for the oxidation of w-butane and rt-hexane in stoichiometric proportions in air in a closed vessel. Points a-d are discussed in the text. (After Kane et al. [138].)
For fuels of different carbon atom numbers, or different C:H ratios, some relationship to the respective mixture at 4> = 1 would seem preferable. For example, a comparison is made in Fig. 6.15 of the ignition and cool-flame boundaries of n-C6H14 and n-C4H10 in air obtained at = 1 in a nickel steel cylindrical 190 cm3 vessel [138]. The considerably enhanced reactivity of n-hexane over that of n-butane is apparent throughout almost the entire range of conditions. However, that the reactivity of n-hexane approaches or may even be lower than that of n-butane at about 650 K may be attributed to an "inversion" created by the underlying chemistry in the ntc region (see Section 6.5). A complementary ignition diagram to that for «-C4H10 in an unstirred closed vessel was obtained in a jet-stirred flow reactor by Proudler et al. [58]. The reactor was a 500 cm3 stainless steel cylinder, operated at a mean residence time of 9.4 0.4 s with a reactant mixture [n-C 4 H 10 ]: [0 2 ] = 1.13:1 (Fig. 6.16). Oscillatory cool-flames and ignitions were detected within narrow temperature ranges, but comparable with those of the closed
584
Experimental and numerical studies of oxidation chemistry
Ch. 6
450H
CF
350H
250H
J 542
569
579 581 /K
Fig. 6 16. (p-Ta) ignition diagram obtained for the combustion of rc-butane + oxygen in a well-stirred flow reactor at a mean residence time of 9.4 0.4 s. Different phenomena are identified as follows: L, low-temperature stationary states; H, high-temperature stationary states; CF, oscillatory cool-flames; I, oscillatory ignitions. A hysteresis of the boundaries was measured as a result of increasing (solid lines) and reducing (broken lines) the vessel temperature. (After Proudler et al. [58].)
Global combustion phenomena
585
vessel studies. A distinction from closed vessel studies was the existence of a hysteresis in the location of the cool-flame boundaries obtained by an increase of the vessel temperature followed by a decrease through the same range at constant pressure and molar flow rate. Ignitions were not observed under conditions of decreasing temperature. Similar phenomena were detected by Gray and Felton [12] in their studies of propane oxidation in a well-stirred flow reactor. Hysteresis phenomena cannot be investigated in closed vessel experiments. Lignola et al. [32,133] have characterized oscillatory cool-flames and ignition and other non-isothermal reaction modes for n-heptane and /octane in a high pressure jet-stirred flow reactor (100 cm3). The respective ignition diagrams are shown in Fig. 6.17. Stoichiometric compositions were studied in each case at a mean residence time of 0.4 s. A wide range of non-isothermal phenomena were observed in similar temperature ranges for each fuel + air mixture. However, spontaneous ignition of n-heptane occurred in the "ignition peninsula" (Fig. 6.17, region b) at an appreciably lower pressure than that required for /-octane ignition (0.2 vs 0.9 MPa). A marked contrast is observed in the behaviour of the simplest aromatic hydrocarbon-air mixtures at high pressures. No cool-flame phenomena or an ignition peninsula in the (p-Ta) diagram are observed. These are found only when sufficiently reactive aliphatic side-chains are associated with the aromatic ring. Burgoyne et al. [129] showed this to be the case for npropylbenzene in a closed vessel (Fig. 6.18). The ortho- and meta-isomers of the xylenes also showed a similar reactivity. Benzene, toluene and ethylbenzene were found to undergo spontaneous ignition at temperatures only above 700 K. Standard tests (ASTM, BS or EEC) are used to establish the minimum autoignition temperature (or AIT) of flammable gases or vapours in air at one atmosphere in fuel-rich mixtures. In each case liquid fuel is injected into a hot vessel of prescribed size and shape, typically a conical vessel of 200 cm3 capacity, which is open to the atmosphere. Whether or not ignition occurs is determined visually. Experiments are repeated at different vessel temperatures until the minimum temperature for spontaneous ignition is found. Although autoignition temperatures are composition dependent, the AIT is established in these experiments because a considerable range of compositions exist within pockets of gas in the vessel as evaporation of liquid fuel, or mixing of the injected gaseous fuel, occurs. The minimum autoignition temperatures of a number of isomers of the
586
Experimental and numerical studies of oxidation chemistry
Ch. 6
alkanes, determined by the standard procedure, are shown in Fig. 6.19. The behaviour that isolates one group of compounds or isomers in a hightemperature range from another in a lower temperature range may be rationalized from the shape of the (p-Ta) ignition boundary (e.g., Fig. 6.15). In mixtures at O = 1 and/? = 0.3MPa, both n-hexane and n-butane have minimum spontaneous ignition temperatures below 600 K on the "low-temperature branch" of the ignition boundary (Fig. 6.15c and d). However, at 0.2 MPa the minimum spontaneous ignition temperature of n-hexane is at 550 K on the "low-temperature branch", whereas that of butane is beyond 700 K on the "high-temperature branch" of the ignition boundary (Fig. 6.16a and b). The latter distinction also exists at 0.1 MPa
1.2-
1.0H
0.8-H 2 0.6H
0.2—
600
700
800
Tr/K Fig. 6.17a. (p-Ta) ignition diagram for 2.2.4-trimethylpentane. See Fig. 17b for details.
587
Global combustion phenomena
0.8-
>t
i
•
0.6H ft.
2 0.4-
#4
.•.
t t
0.2-
/
600
700
800
Tr/K Fig. 6.17b. (p-Ta) ignition diagram for stoichiometric «-heptane-air measured in a high pressure, jet-stirred reactor, residence time 0.4 s. (a) slow combustion; (b) oscillatory, multiple-stage ignitions; (c) oscillatory cool-flames; (d) jump approach to stationary state; (e) oscillatory single-stage ignitions. (After Lignola etal. [32].) Broken lines show boundaries in a closed vessel.
in richer mixtures, but if the reactant pressure is raised to only 0.15 MPa, the minimum spontaneous ignition temperature of n-butane shifts to the "low-temperature branch" [139]. The minimum autoignition temperature established by standard tests cannot be regarded as a satisfactory diagnostic for conditions of safe operation at pressures above 0.1 MPa. Experimental studies of relationships between fuel structure and the ease with which hydrocarbons undergo spontaneous ignition have received considerable attention from engineers through the characterization of
588
Experimental and numerical studies of oxidation chemistry
Ch. 6
15 -i
n-propy. benzene
benzene
10 H
5H
ethylbenzene
523
623
723
823
923
T/K Fig. 6.18. (p-Ta) ignition diagrams for (a) benzene, (b) toluene, (c) ethyl benzene, and (d) n-propyl benzene. (After Burgoyne et al. [35].)
knock ratings of fuels in research engines. Such studies began in the early 1920s and are outlined in Chapter 7. However, the study reported by Lovell [140] shows with remarkable clarity the way in which the isomeric structures of pure components relate to engine-knock resistance, measured by the critical compression ratio. Some of the results for isomers of alkanes are shown in Fig. 6.20, but Lovell [140] has recorded similar data for alkenes, acetylenes, naphthenes (cyclic alkanes) aromatics and bicyclic hydrocarbons. There is more than a passing resemblance in Fig. 6.20 to the increase of minimum autoignition temperature of the alkane isomers,
589
Global combustion phenomena 800-
C->H„ (CH3)3CH
750
'
•
(CH3)3CCH2CH3
n-C4H10
*
(CH3)3CCH2CH(CH3)2
((CH3)C(CH3)2CH3 •
700
(CH3)3CCH(CH3)2
#
• (CH3)3CCH(CH3)CH2CH3
(CH3)2CHCH(CH3)2 H
650 (CH3)2CHCH(CH3)CH2CH3
600-)
(CH3)2CHCH2CH2CH3
550
n-C5H12
CH3CH2CH(CH3)CH2CH3 n-C H
500
-i
r-
n_C
7Hi6
-i
5
1
8
Carbon Number Fig. 6.19. Minimum autoignition temperatures for various alkane isomers (ASTM-E 65978).
shown in Fig. 6.19. With the exception of two of the isomers of C 8 Hi 8 and the relative positions of the compounds in each of these figures are very similar. Both the autoignition temperature and the onset of engine knock depend upon the ease of oxidation at low-temperatures, as discussed in Section 6.5. In a somewhat similar context Reverchon and Lignola [141] demonstrated that the period of oscillatory cool-flames in a CSTR could be correlated with the octane number during the combustion of primary reference fuel mixtures. 6.4.5 Ignition delays of alkanes and other hydrocarbons The time taken for spontaneous ignition to develop, the ignition delay, is important in many applications, and it is also important to know how the ignition delay is affected by changes of pressure, temperature and composition. These data are relevant to not only industrial combustion hazards but also combustion in reciprocating engines, but many complications are associated with measurements in engines themselves. Thus, the understanding and interpretation of spontaneous ignition (or autoignition) is best approached in more simple devices. Low-pressure closed vessels can be used but engine operating conditions are more closely represented
590
Experimental and numerical studies of oxidation chemistry
Ch. 6
(CH3)3CCH(CH3)2
10 (CH3)3CCH(CH3)CH2CH3
C3Ha
8-J
(CH3)2CHCH(CH3)2 (CH3)3CH
(CH3)3CCH2CH(CH3)2 (CH3)2CHCH(CH3)CH2CH3 •
O
o
n-C4H10
(CH3)3CCH2CH3 CH3C(CH3)2CH3
•
(CH3)2CHCH2CH2CH3
•
CH3CH2CH(CH3)CH2CH3 n-C5H12
• n-CcH^
n-C^H.,
Carbon Number Fig. 6.20. Critical compression ratios (CCR) for selected alkane isomers, measured in a CFR engine as discussed in Chapters 1 and 7. (After Lovell [140].)
by shock tube (Section 6.3.4) or rapid compression machine studies (Section 6.3.5), and more closely still by motored engines (Section 6.3.6). Ignition delays of liquid fuels have also been measured by gravity feed of droplets to the top of a vertical heated tube [48], which is relevant both to hazard evaluation and diesel engine combustion. In a shock tube there is a clearly defined initiation of reaction (t = 0) at passage of the incident or reflected shock through the reactant mixture. Very wide ranges of pressure, temperature and composition are also accessible, but measurements of ignition delay in mixtures at temperatures below 1000 K are not without difficulty, as discussed in Section 6.3.4 [29]. Data obtained from shock tubes generally pertain to shocked gas temperatures in excess of 1500 K [89, 90]. If ignition delays measured in different devices under the same conditions are brought together it becomes clear that there are incompatibilities that cannot be attributed to differences in composition or pressure. In rapid compression machines the discrepancies between different sets of results may arise from different rates of compression [142], which can affect the rate of heat transfer in the early stage of the post-compression interval as a result of the extent of gas motion that is created by the piston motion [50,102]. Ignition delays become longer as the heat loss rates are
Global combustion phenomena
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increased, and there may be a failure of spontaneous ignition to occur in the limit of very high heat loss rates. Differences in behaviour can also arise because, with a slow compression of very reactive fuels in air, quite significant extents of reaction can occur during the final stages of the compression stroke. Much less reaction would occur during the shorter interval available in the final stage of a compression at a higher rate [50]. This is not usually a problem in fuels of low reactivity. The last point is of practical significance because spontaneous oxidation of the fuel + air charge admitted to the cylinders of a spark ignition can occur during the compression stroke, which is a preliminary to the eventual autoignition of the end gas. Thus there are attributes to the application of rapid compression machines to spontaneous ignition studies which relate to combustion engines, especially with regard to an understanding of the development of "end-gas" combustion in association with engine knock. Spontaneous ignition is easily detected in a motored engine and the threshold conditions for its occurrence, which may be the data of primary interest, can be readily obtained [143]. Comparative measurements of the ignition delay of different fuels are also possible, but absolute measurements are difficult because the start of reaction cannot be defined with any accuracy. The interpretation of spontaneous ignition data in engines sometimes relies on a match between numerical simulations of the pressure record to the experimental observations [143]. The duration of the ignition delay in a diesel engine can be measured accurately from the start of fuel injection, but the overall elapsed time also includes droplet evaporation in the hot, compressed air charge [144]. The typical initial temperature and pressure ranges of interest in studies of ignition delay related to engine combustion are 600-1000 K and 1.03.0 MPa (10-30 atm). The ignition delay and how it is affected by the compressed gas (or shocked gas) temperature and pressure is related to the type of oxidation processes that can occur in these ranges of conditions. For reasons related mainly to thermal feedback the measured ignition delays are also strongly dependent on the particular system in which they are obtained, so it is necessary to apply predictive approaches through numerical modelling if data from one system are to be interpreted in the context a different application. The qualitative features associated with some hydrocarbons are described here. The light emission and pressure traces give clear indication of coolflame phenomena, characterized by a relatively small pressure rise and
592
Experimental and numerical studies of oxidation chemistry
160H
Ch. 6
|nC 4 H 1 0
140J nC4H10
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4(H
20H
T ad /K Fig. 6.21. Ignition delays as a function of compressed gas temperature for the normal alkanes, /t-butane to /t-heptane, /-octane (2,2,4-trimethylpentane) and toluene in stoichiometric proportions in air. The results were obtained in a rapid compression machine at a compressed gas density of 128molm - 3 (0.65-0.75 MPa). Ignition in the ntc and lower temperature range was not observed under these experimental conditions [50].
low intensity light output (Fig. 6.7). The overall negative temperature dependence of the ignition delay for the alkanes within a limited range of compressed gas temperature is a consequence of the two-stage ignition (Fig. 6.8). The overall delay time may be divided into two components for the first, TX and second stages, r 2 , shown in Fig. 6.8 for a rapid compression machine. Whereas TX decreases as Tc is raised, the duration of r 2 lengthens (Fig. 6.8). The underlying kinetic reasons for the increase of T 2 as Tc itself is raised are connected in part with a diminishing ability of the degenerate branching reactions in the first stage to raise the reactant temperature sufficiently high at the onset of the second stage [50, 51]. The overall ignition delays of the (C4 - C7) n-alkanes and for toluene,
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3000 ^
2