Chemistry and combustion

Twenty-Third Symposium (International) on Combustion/The Combustion Institute, 1990/pp. 1-19

CHEMISTRY

AND

COMBUSTION

PETER GRAY Gonville and Caius College Cambridge CB2 1TA

1. Introduction The purpose of this introductory lecture is to speak to an audience much less specialised than the groups who will split up for our many particular sessions. Our size is both a source of satisfaction and a disadvantage for we are now unable to be a single audience so much have we grown since our early beginnings. The success of the Combustion Institute is undoubted, and our sequence of Symposium volumes represents a formidable achievement. But we must recognize that we are beginning to lose something by the very size to which we have grown when we attempt to meet in our traditional style and in our present-day numbers. I hope the Board of Management will think deeply about this. I have chosen as my broad title "Chemistry and Combustion" because of my own training in chemistry. If I expanded upon it, I would say I wish to concentrate on what is called discontinuous behaviour and on its chemical origins. The type of discontinuous behaviour that I have in mind is the abrupt onset of change--an explosive event, or the sudden extinction of a reaction in response to a continuous change in circumstances. Ignition and extinction are instabilities that mirror one another. A rather different form of instability, of interest since early research days, is oscillatory behaviour or repetitive ignition and we shall also consider this. All these exotic phenomena are old friends in the combustion field, though in other circumstances they are seen as strange and uncommon events. The questions I should like to air are: broadly what it is about combustion that encourages them, and how is chemistry so naturally endowed as to make them readily apparent? One sort of answer is engagingly given by the Russian polymath, Lomonossov, 250 years ago. He said he was convinced by what. he read and what he did, that "chemical experiments combined with physical showed peculiar effects." This is what I wish to explore today. Such general words can be made to bear many loads of interpretation. However, the balance between the loss and the production of heat during chemical change, and the influence of temperature upon chemical change are just such conjunctions of physics and

chemistry. Reactions go faster when they are hotter and slower when they are cooler. When a chemical reaction is accompanied by self-cooling, we may see a self-repressing influence. That is a "negative feedback"--such as is exploited in electrical circuits to enhance stability--the kind of internal connexion that le Chatelier may have made when attempting to frame his 20th century synthesizing generalisations of J. H. van't Hoff's earlier, more quantitative studies. When the reverse holds, and chemical change is accompanied by self-heating we meet a self-enhancing influence, a "'positive feedback." Destabilization has been built in. A system apparently "in equilibrium" may now respond very dramatically to a small perturbation. In the language of le Chatelier, the changes within the system may tend to reinforce, not to nullify, the "constraint." We have thus already recognized and in w we examine the features that distinguish combustion from mere oxidation and that give rise to its discontinuities: feedback and non-linear responses. Feedback means the effect of consequences upon their causes. The non-linearity of thermokinetic response is described in the famous empirical rule that a chemical reaction doubles its rate when the temperature is made ten degrees higher. So a span of temperature of one hundred degrees celsius might see a thousand-fold change (2t~ = 1024) in speed; a span of two hundred degrees produces a millionfold enhancement. Chemistry has already entered twice over. Firstly, the fact that rate-processes are so sensitive to temperature reflects the fact that energy needs of breaking bonds takes priority over energy release as we climb the energy contours; and secondly, because the chemical energy so required is returned, with more besides, as stabler final products are formed at the end of the reaction path. Chemistry can also enter combustion in a still more disect manner than that above. This is by the intervention of processes in which the chemical product of an elementary step can catalyze its own production without the need for energy release. The name for this family of self-replicating processes is autocatalysis, and where the overall reaction is accomplished by a chain of elementary processes, it is called chain-branching. These chains of elemen-

HOTrEL PLENARY LECTURE tary steps are of supreme importance in the combustion of hydrogen and they also confer on hydrocarbon oxidation many of its striking features. They are not inevitable in all oxidations, and such features seem not to matter so much, for example, in the combustion of hydrazine. We shall have more to say about them below. It is of some interest to note that whilst the ideas of simultaneous chainbranching and self-heating were very firmly grasped more than forty years ago by Semenov and by Norrish, they were not successfully joined together and coherently expressed until the late 1960s. Prototype model schemes required simultaneous chemical confidence and an adequate mathematical training: mathematics has often seemed a stony pasture to western chemists, and that perception was the reason for the delay. This lecture will not be much concerned with chain-branching and detailed chemistry, because merely to sketch the field would make the treatment too shallow. Even the model schemes presented will have to be mere indications of what is a lively field today. This decision to omit detailed chemical reaction schemes and to place the emphasis on thermal feedback does not mean that the importance of 'real chemistry' is not appreciated. (This point will be taken up again in the concluding section.) Rather the development is to be taken as illustrating how much can be obtained when real chemistry can be fairly represented by certain characteristic laws. The arrangement adopted is as follows. First, an outline is given in w167 of the foundation of what we now call thermal explosion theory. These are the elementary treatments of spontaneous ignition in homogeneous systems, in heterogeneous circumstances and in continuous-flow, stirred-tank reactors (cstr). These are called elementary for several reasons. First, they assume the possibility of stationary states, and they identify critical ignition with their disappearance. This is satisfactory for open systems, but for the first two subjects it implies neglect of reactant consumption. Secondly, they assume uniformity of temperature in the reactant. This is reasonable for small and well-stirred systems but is otherwise either an approximation or restricted to systems where reactants are liquid or gaseous and are mechanically mixed. Nevertheless, these are the investigations that establish, as it were, the eredentials of the subject and allow us to move on to many applications: Amongst these are the treatment of parallel reactions, of autocatalytie exothermie reactions, and the investigatibn of closed systems (w167 in which reactant consumption is not ignored. Other major topics, which could also be accorded the status of foundations, so basic are they, include the treatment (in w of systems in which

temperatures are not uniform but where the intensity of self-heating varies from point to point. Again, unless one has immediate recourse to numerical computation, reactant consumption must be set on one side and convenient approximations made. Here the family of applications subsequently presented have to be chosen selectively from many. Among those illustrated are the use of theory to cope with systematic errors in kinetic parameters (evaluations of reaction order, activation energy, etc.) in the presence of self-heating, the question of safety limits on the 'hot assembly' of potentially dangerous materials (w and the response of exothermic materials to steadily ramped ambient temperatures, such as are often employed in thermal analyses. Inevitably the very diversity of apphcations which deserve to be sketched collectively means that individual explanations must sometimes be overeompressed. Some relief can be found by consulting monographs such as references 2, 18, 25, 32 and 33.

2. Foundations: Representing Non-linearity and Measuring the Intensity of Feedback The systematic study of thermal instabilities begins with N. N. Semenov's (1928) treatment I of "'critical conditions" for the self-ignition of a homogeneous mass. Ten years later (1938) his pupil D. A. Frank-Kamenetskii extended the same ideas to exothermic reactions occurring at a surface such as burning carbon or a platinum catalyst. In 1941, Ya B, Zerdovich offered the same service to the simplest of open systems--the continuous-flow, stirred-tank reactor (cstr). These three great men are unsurpassed in combustion, and these contributions1-3 are fundamental. It must suffice to sketch them briefly. We indicate their broad significance and use a common language to emphasize their inter-connexions and extensions to more complex situations. Amongst the earliest of complexities are (1) looking at temporal evolution of temperaure, (2) considering systems in which temperatures are not uniform but vary in space, (3) broadening the enquiries to chemical reaction schemes that are themselves capable of becoming unstable isothermally. Finally, (4), yet overlapping with all these, comes unstable behaviour of a different kind: oscillatory behaviour. In all of this, we are concerned with essentially static media and with negligible pressure-differences. In his original work in this field neither Semenov nor his colleague O. M. Todes chose to work with dimensionless groups: both Frank-Kamenetskii and Zel'dovich, coming not long after, did. We shall exploit their choices and express some of the early

CHEMISTRY AND COMBUSTION resuhs in these terms. At the present day there are some tendencies to return to older forms: the Appendix gathers some comments together on this.

Non-Linearity of Response: The principal source of non-linearity is the responsiveness of reaction-rate to temperature. The Arrhenius form k ~ exp ( - E / R T ) if not exact, is a splendid approximation over a substantial temperature range, not only for many elementary reactions but for a number of complex ones too. It loses its relevance if T is too low or too high (but 'too high' here means temperatures of order of E/R, say 104 K, that are quite out of reach of selfheating, chemical energy-release). If k oc exp ( - E / R T ) , the responsiveness of k to T is given by: k

dT

(RTZ/E)

This marks the entry of the group of terms RT2/E as the natural yardstick of temperature. In ignition problems, dimensionless temperature-excvsses | are defined by the relationship: T - Try

(aT~//EI '

and relative reaction rates by:

k(T,f)

exp

~-~, ~ exp O

The simple exponential function e ~ is to be regarded as an approximation to the term exp [O/(1 + ~O)] with t = RT,-f/E and eO "~ 1.

Thermal Feedback and the Zel'dovich Number: If an exothermic reaction can heat itself up by an amount AT,d where AT,~, = Q~co/crc~or Qvco/~rcv then the quotient

Oad(or B)

3. Unifying Treatments of Homogeneous and Heterogeneous Systems It is handy to set out Semenov's treatment of the homogeneous exothermic reaction and its stability. When reactant consumption is neglected, the heatbalance equation ~ = .~, can be written:

QkcoV = hS(T - T~), with Q denoting the exothermicity (--AU~ for a system of constant volume, V) and k the reactiunrate constant. S is the surface area of the system, and h the heat-transfer coefficient per unit area. It is assumed that: k = A exp ( - E / R T ) = ka exp{(E/RTa) - (E/RT)}

dk

O

gaining general currency, and in flame theory the quantity B = (EATad/RT2aa) is called a Zel'dovich number.

The use of co expresses one of the major assumptions: that reactant-consumption may be neglected. This allows the problem to be reduced to one of discovering the existence, multiplicity and stability of stationary states. The stationary-state condition can in turn be set out as:

(T-Ta, lEE} RT

hS(RTa2/E) ~ (Se) = ~ - ~ ) - ~ exp

In dimensionless terms and for small temperature excesses: (Se) ~ Oe -~ The group of terms equated to (Se), the Semenov number, may be regarded as a dimensionless measure of exothermicity or a comparison of characteristic rates (or relaxation times) for temperature change by chemical heating or Newtonian cooling. According to the size of (Se), the heat balance equation has either two solutions (O+ and (9_) or none. The point of change (bifurcation) is to be identified with criticality. There, stable (O_) and unstable (O+) stationary states eualesce, and at this point: (Se)~ --- e-l;

aTaa

(nT~/E)

rather than ATad itself is the logical measure of the intensity of thermal feedback. The symbol B is

0 ~ = 1.

A treatment on these lines is generalized extremely readily to more complex conditions with e.g., temperature-dependent heat-losses such as oc~,ur when h is not constant, or non-Arrhenius rates such as T m exp ( - E / R T ) . If we do not simplify the

4

HOTrEL PLENARY LECTURE

treatment by neglecting eO compared with unity (or T - Ta compared with T~) we find: e(Se)cr = 1 + ~; O c t = 1 + 2~.

Heterogeneous oxidation in practice is concerned with two main circumstances characterized by two different examples: the combustion of carbon (coal or coke) and the catalytic oxidation o[ ammonia on e.g., a platinum surface. In one, the surface is eroded; in the other, it is not. Originally, coke combustion was recognized as occurring either very slowly at low or moderate temperatures or very rapidly at elevated temperatures. Emphasis was formerly placed on a mysteriously different chemistry in the two r6gimes. It was an early triumph of thermal theory to unify the two phenomena, to predict conditions for extinctions as well as ignitions and to reveal the connexions between the two examples above and their relationship to the homogeneous system. The treatment of heterogeneous systems2 begins in the same way except that two separate ingredients of stationary states are now needed--the arrival of matter at the reacting surface, and the transport of heat to the (unreacted) gas. The form of ~ is altered to a driving force divided by the sum of two resistances, one chemical and the other diffusive: ~t = QVco/(k -~ + 13-~).

The separate mass and heat-balance equations: k c V = ~S(co - c), QkcV = hs(T

- To),

allow us to connect c and T in the stationary state: c

h S ( T - To)

co

13Qco

and to reach the dimensionless equation: (Vk) .

Oako . . 13

.

Oe - ~

1 - o/o~

The quotient O / O d measures how hot the surface is in comparison with how hot it could become ff reaction were so fast that chemical resistance was effectively nil. Amongst the results that emerge are the existence of a range of conditions under which multiple solutions are possible, bounded by points of ignition and extinction. The requirements of ignition are almost the same as before: Oig ~ 1 + O21 ~ 1, frequently.

The new feature, extinction, occurs when or O ~ t ~ O d -

Td-T-~R~a/E

1

but the basis for dimensionless temperature for extinction is Ta and Oa relates to how far below Ta is the extinction. This marvellously simple analysis was given in 1938 by Frank-Kamenetskii. Not long afterwards, his pupil N. Ya. Buben extended it to reactions of order different from unity (with rate = kcm). Their treatment (which was less than elegant) was improved by Thomas & Bowes (1961) who provide an economical exposition of the elementary case. Real systems often involve distributed temperatures and it is to chemical engineers that we look today for these more general enquiries: Aris's monumental monograph33 is the natural starting point.

4. Extensions to Closed and Open Systems When homogeneous and heterogeneous systems were considered in the early days, reactant consumption was ignored and the problems of real systems were approximated by idealized ones in which stationary states were possible. In open systems, however, truly stationary states are possible. The simplest of open systems is afforded by a well-stirred reactor into which fresh reactants flow continuously and out of which issues a partially reacted mixture of products and reactants. Exothermic reaction leads to self-heating and under certain circumstances to multistability. In the context of combustion, Longwell's unforgettable ~ontribution (1952 paper) is the western landmark. (There was an earlier pioneer, Liljenroth, who seems never to have been noticed much even during his own lifetime.) In the USSR, Zel'dovich and Zysin (1941) were responsible for opening up the field: they too worked with a view to combustion problems. After 1955, chemical engineers ploughed these fields. The treatment is so instructive that it deserves to be sketched in outline. As with heterogeneous systems, conservation equations for matter and energy are the starting point. For matter: V(dc/dt) = uco - uc - Vkc m

or equivalently, for a first-order reaction: dh dt

l-h tres

h exp{ E tch

~o

ET} "

Most of the symbols have obvious meanings; h is the fractional reactant-concentration, c/co; T and To

CHEMISTRY AND COMBUSTION are the exit and entry temperatures. For energy:

5

or Oe -0

(Ze)-~ -

V~cp(dT/dt) = QVkc m - hS(T - Tw) -

selfheating

exothermic reaction

uacpff

-

dt

QC~ (k~176 cFcp

~o

= g(O)

To).

losses heated v/a wall outflow

or equivalently: a'r

-

(1 - O / B )

exp

T - Tw

T - To

(Vcr%/hS)

(V /u)

These expressions have the same structure as those for heterogeneous reaction. They can yield either one or three solutions (not more) and thus predict either monotonic responses to changes in conditions (this is all that happens when AT~a < 4RTOTad/E) or the possibility of 'ignitions' and 'extinctions' and hysteresis when ATaU exceeds this limit. In dimensionless terms, the approximate condition for multiplicity is B > 4. For larger values of B, the approximate locations of ignitions and extinctions are at O_ = 1 and O+ = B - 1; T,g - To = RT~o/E; Tad -- Text = RT~aa/E.

If we use dimensionless temperatures and characteristic times (V/u) = t,~ and 1/k o = tch and (Vacp/ hS) = tN then: 1 dO

km

-~ =--exp n dt tch

-

(Ze)---~

+

This last expression has been simplified in two ways. First it assumes that inflow temperatures and wall temperatures are equal: To = Tw. This restriction can readily be lifted without significant complication when non-adiabatic operation is considered. It also ignores any dependence of the various thermophysical properties (such as specific heat, density and heat-transfer coefficient) on either the extent of reaction or on the temperature. These matters may need consideration in a computational study but they are subordinate to the main thrust.

Adiabatic Operation and the Zel'dovich Numbers: Adiabatic operation is best achieved in practice by externally controlling the wall temperature so as always to equal that of the contents so that Tw = T. (This has the same effect in the equations as letting h = O or tN = ~.) In these circumstances, concentrations and temperatures are linked, and the stationary-state relationship:

AT 0 . . . . ~T~a B

l-k,

is established. Our system is described by a single variable, so oscillations are impossible. The governing equation for stationary states is: (Ze) = (Taa

Qco

~

= Z(T)

[A

E

Non-adiabatic Systems: The reason for commemorating Zel'dovich's name in this context is nowhere more obvious than in the clear way in which he, with Zysin, set up and analyzed the governing equations 3 and evaluated their consequences for non-adiabatic systems. They investigated only the patterns of stationary states (and their 1941 work was rather strikingly rediscovered from 1955 onwards), but what they uncovered was very novel. Their work is especially relevant to what is called (by real engineers) 'glass bulb chemistry' where work in fragile apparatus on a laboratory scale permits only slow flow-rates and guarantees a major role to heat-losses. In terms of conservation equations, that for matter is identical and that for energy preserves the heat-loss term. (The quotient (ucr%/hS) or (tN/tres) or (kftN) is the sole extra term: we call this quotient K and its reciprocal [3). This similarity of structure means that results can be written down at sight for the occurrence and location of ignition, extinction, hysteresis and multiplicity and for the link between stationary-state temperatures and concentrations; First, Oss= B * ( 1 - k , 8 )

KB

where B* = - -

l+K

Next,

B*tres

To)(T - To)

(RT~o/E)(Taa- T) exp

In the foregoing, the term written (Ze) denotes the group (BtreJtch), or more explicitly:

tch

Oe -~

1 -- O/B* '

B

I+~'

6

HOTTEL PLENARY LECTURE

For multiplicity, B* > 4. At points of ignition and extinction (so long as B* is not too close to 4): 1

|

Oext=B*-l.

Although the similarity of structure means that there is once again only a choice between one and three steady states at any particular flowrate, the dependence of B* on flowrate means that for any given excess temperature there are either two flowrates or none. This in turn generates a greater variety of responses to changing flow speeds. In particular as flow-rates are increased from zero, stationary-state temperatures grow and ignitions can be caused by making the flow faster. The temperature-jumps that set in increase quickly to values near the maximum possible: if that maximum is low, the jumps can be quite small. The notable work4 of Griffiths & Hasko on ethane oxidation at low flow rates is a case in point. In rich mixtures, oxygen consumption jumps from tiny values to completion, yet the excess temperature after ignition reaches only 60 K.

5. Temperature-time Histories (for Systems with Negligible Reactant Consumption or Strong Exothermieity)

precedent of using the same symbol for the critical Semenov parameter (which we have called 6) as the one (~) which he had introduced for distributedtemperature cases. This was copied by several authors and has generated confusion to newcomers for many years.) In fact the same functional form does apply to distributed systems, but this was not established until long afterwards in notable studies 19 by Boddington and Kordylewski. The Strongly Exothermic Limit:

To study this, matched asymptotic expansions offer the systematic route. The techniques were introduced5'7 by Kassoy and Lifian and by Lacey. They systematize the earlier work2'6 by Thomas and Frank-Kamenetskii, and open the way to discovering higher-order terms. (Note the misleading use of 8, however.) The conservation equations under examination are: (~rc~V)dT/dt = VQcmA exp(-E/RT) - hS(T - Ta);

T = Ta at t = 0 and -dc/dt

= cmA exp(-E/RT) c = Co at

The earliest treatment of homogeneous exothermic reactions ignored reactant consumption and concentrated on stationary states. It is readily possible under these assumptions to deduce the evolution of temperature in time. When approximations to the Arrhenius law are adequate, sub-critical and supercritical evolutions are very sharply distinct. The former lead to finite temperature excesses (after infinite times); the latter to infinite temperatures after finite times (induction periods). The 'critical trajectory' occurs as the limit of the subcritical family, and dimensionless temperature evolves according to laws like: 13 =

(t/tad)

-

1 + (t/tad)

when

t~

~-qlcr

~'-

1 e

For marginally supercritical circumstances when only slightly exceeds ~Jcr, there is a characteristic dependence of induction period on the degree of supercriticality, (t~/t~cr) - 1. It is described by an equation of the form: t ~ (supercriticality)-1/2 so it lengthens indefinitely as the degree of supercriticality diminishes. This form of result was discovered by Frank-Kamenetskii for uniform-temperature systems. (Rather distractingly, he set a

t= 0

or in dimensionless terms: r

= Cg(w)e ~ - O

B(dw / dr) = g(w)e ~

where the dimensionless time-scale satisfies r = t~ tad. When reactant consumption is neglected and the concept of criticality is clear, the appropriate "small quantity" for asymptotic expansions is k, defined by ~ = qJo(1 + h z) and the appropriate timescale z = kr is used. When reactant consumption is taken into account, z = Ixr is used, where 3 = 1/B for a first-order reaction. This scaling is the most economical and allows an efficient solution to varied circumstances (generalized temperature-dependences of k; generalized isothermal reaction kinetics; ramped ambient temperatures, etc.). The work of Boddington (Leeds), Kordylewski (Wroclaw) and Feng (Beijing) has made innovative contributions in this field. All results require an increase in qJc~from its 'primitive' or zero-order value qJo = e -1 by amounts proportional to (m/B) 2/3. Roughly speaking, ~cr - ~o = 3(m/B) ~/3. This requirement that (m/B) 1/3 be a small quantity is not a trivial restriction. For a first-order reaction, we would expect perceptible deviations from leadingorder correcting terms even if B was as large as 27 = 33; B is unlikely to exceed 100, and 100 1/3 ~ 4.6;

CHEMISTRY AND COMBUSTION for a higher-order reaction, requirements on B are more

severe.

Temperature programming (see also w is commonly used to study exothermic substances from the point of view of safety. As ambient temperature is raised, so the balance between acceleratory effects of self-heating and deceleratory effects of reactant consumption is tilted, and a particular heating-rate may be found at which the latter pair cancel one another out and produce behaviour identical (to within first-order effects) with the primitive case of zero reactant-consumption. These effects were systematized by Kordylewski (1983) and extended by Kay. Values for ~ c r below or in excess of the classical value are possible and the mathematical techniques used are applicable to safety procedures such as heat-removal by surface cooling of a reactive mass.

6. Thermal Runway in Closed Systems: Criticality Versus Sensitivity The ideas explored in the previous section clearly link "criticality" to a jump from one stationary state to another. Stationary states just cease to exist at the point of such a jump and we deal with bifurcations. The treatment is appropriate for the truly stationary states of an open system, like a continuously-fed, well-stirred reactor (cstr). For a closed system, it matches only the simplified model that ignores reactant consumption and not the system itself, though it can be used as a point of departure for asymptotic analysis. In a closed system, temperature evolution and reactant consumption both proceed with differences only in degree, not in kind. All temperature histories (for a single exothermic reaction) have the same gross form with a single maximum value. All histories of extent-of-reaction begin at zero and rise to unity with only varying degrees of steepness. How are we to hope to replace the clear-cut contrasts of criticality by some useful operational criterion? This question turns out to receive a great deal of help from numerical computation, but an analytical approach together with an appropriate choice of reduced variables help us to pose the questions effectively. It is clear that a different concept from "criticality' needs to be used to express ideas more satisfactorily. The idea of sensitiveness (or insensitiveness) to initial conditions is what is needed, and the search is for circumstances where small changes in initial conditions make the most substantial differences in the unfolding of events. We shall see that the principal parameters in a general treatment are (i) the Semenov number (which measures initial rate of heat-release) (Se) = VQkcom/hS(RTa2/E),

7

(ii) the isothermal reaction order m (which can be generalised to more complicated kinetics), and (iii) the Zerdovich number B which measures the system's capacity for self-heating B = Oad ~- (EQco/Cxc~RTaZ).

Analytical solutions are out of reach and results have to be expressed as dependences of events on the values prescribed for (Se), m and B; when B is large, the last two operate together as the quotient (B/m). In what follows we may write O~r simply to aid comparison with the literature, but the preceding remarks about the idea of sensitivity as superior here to the concept of criticality must be borne in mind. In the past the following considerations have been influential: (1) For the primitive case with c = Co, RTa2/E is the upper bound on safe temperaturerises, so in a deceleratory system temperature rises greater than this should exist that belong to the quiescent family. (2) Temperatures that evolve without any surge in gradient must also belong to quiescent systems: points of inflection in temperature-time curves are essential symptoms of runaway. In the 1960s Adler and Enig s suggested that inflexions in the temperature-reactedness curves were sufficient indicators of runaway and on this basis produced a lively analysis with some appealing analytical results for extreme cases (zero order; infinite B). For a reaction of order m, inflexion points set in at Oinfl = 1 + X/m; for an adiabatic system, a minimal value for B is (1 + ~/m) 2. Their results were won for the exponential approximation but extended by P. R. Lee to the Arrhenius form. Despite apparent inconsistencies in their development, these neat formulae will continue to keep this work in mind. A useful present-day study9 of deceleratory reactions in closed systems is that due to Kordylewski and Scott. They begin by adopting the equations: d c / d t = - k c '~ with

k ~ exp ( - E / R T )

cvcrV(dT/dt) = VQkc "n - hS~F - Ta)

but generalize them to arbitrary temperature-dependence of k and arbitrary concentration-dependence of isothermal rate. Then treatment is centred on the maximum value reached by the excess temperature, which they call O*, and how sensitively O* responds to initial conditions (i.e. as expressed by the Semenov number, qJ) for different values of exothermicity (as measured by B). For any large value of B there is a region of great sensitiveness so that d| takes a sharp maximum value (where d20*/dt~ 2 = 0). When B = 20, and m = 1, this occurs at d~ = 0.545. The same features persist for

HOTrEL PLENARY LECTURE any large value of B, but as B gets smaller, so the responsiveness of | to 0 becomes less and less pronounced and the curve O versus 0 loses its inflexion between B = 5 and B = 3. Asymptotic analyses by Lacey7 help to link various approaches together and it is fair to say that the Scott-Kordylewski approach is as attractive as any and promises more rewards from further study-especially of times taken to reach maximum temperature. This is an obviously underdeveloped area. There is broad agreement between their results and Tyler and Wesley's classical numerical computations 1~ at the one extreme, and the Thomas-Kassoy-Lifian analyses 5"~ at the large B range. The other approach also deserves attention. B. F. Gray and Sherrington, who were amongst the earliest (1975) to stress the need n for replacing 'criticality' by 'sensitivity,' concentrated on how whole or start-to-finish temperature-time trajectories were affected by small changes e.g. in ambient temperature. So far their analysis has afforded rather conservative estimates of the range of insensitive ('suberitical') behaviour, with seemingly no stabilising effect being conferred by reactant consumption. We are here discussing sensitivity to initial conditions and we have chosen to combine the various initial conditions (V, S, k, Co etc.) into a single parameter. The rather uninformative description 'parametric sensitivity" (Bilous & Amundson, 1955)1~ could be said to spring from this, though its actual origin in a study of tubular reactors was different. We consider tubular reactors next. 7. Tubular Reactors Continuous-flow processes offer certain economies in chemical industry and tubular reactors exploit them. The reactive fluid (gas or liquid) flows down a tube and emerges partially or completely converted. Though laminar flow down a round tube produces a parabolic distribution of velocities, the simplest model imagines plug flow. In this model, radial dispersion is 100% effective and concentrations and temperatures are beth uniform across the tube diameter. Heat is lost only to the walls. Axial mixing is neglected. In such circumstances, the predicted growth and decay of concentrations or temperatures along the tube's length exactly parallel their evolutions in time in a dosed, stirred vessel (stirred to preserve uniform temperature and concentration fields). Although apparently extremely oversimplified, the model is the natural prototype for real systems, much of whose behaviour is often quite well represented by it. Accordingly the chemical reactor engineering literature throws light on batch reactors as well as tubular ones and parallel studies have been made. For exothermie systems, Bilous & Amundson 12 open

modern studies, although K. B. Wilson (1946) came much earlier and recognized the importance of the group (RTa2/E) as a watershed between moderate and stable spatial temperature profiles and the onset of sensitive, spiky ones. Because of this different background, Barkelew's reactor engineering studies were unknown to Adler & Enig, who rediscovereds his results, and early thermal-explosion work was not much exploited by Amundson though he refers to it. When axial dispersion is taken into account, the strong positive feedback so made possible generates multistability (in space). Jumps from one mode (low conversion, low temperature) to another (high conversion, higl~ temperature) can be expected. Finding these jumps of ignition and extinction is equally a clear indication of feedback. They are expected to be subject to the same features of long passagetimes as were met before: 9

.

.

13

time ~ (degree of supercriticality) -1/2. When radial mixing is not instantaneous, the jumps that occur are now from one stationary-state, radial profile to another. Work in this field seems to be sparse but unifying principles are clearly available. We have in fact made a link between thermal explosion theory and the 1-dimensional laminar flame, which long seemed unconnected. 8. Ramping Ambient Temperature

Differential Scanning Calorimetry: Early studies of potentially hazardous materials proceeded by measuring their response to being placed in surroundings at a sequence of discrete, elevated temperatures. Some very humble materials show very reproducible behaviour, and classic work was done by Walkerzr on the oxidation of fibres like wool and powders like wood flour. Such sequential methods take a long time, and alternative procedures shortening the search for the sensitive region and for recognizable ignition temperatures are therefore desirable. Dynamic methods offer suitable alternatives. In them a sample of known size and shape is subjected to a steadily increasing or 'ramped' ambient temperature. Many automated and computer-controlled instruments are available to provide this. One family of procedures operates by measuring rates of supply of heat required to keep the temperature of an active sample increasing linearly. This is a form of scanning calorimetry; when exothermic chemical reaction sets in, less heat is needed, and so on. Many systematic errors can be avoided by comparing the energy requirements of the active sample with those of an inert standard of the same size and mass; this is the basis of dif-

CHEMISTRY AND COMBUSTION ferential scanning calorimetry. By such techniques, values for thermophysical properties of unreactive materials can be obtained automatically, and commercial devices can be so programmed as to provide values for exothermicities of chemical as well as physical changes. Reaction rates and Arrhenius parameters can also be extracted when reactions are not too complex. The traditional methods of analysis concern themselves with locating temperatures Tm at which the instrumental signal is a maximum. This signal measures reaction rate (or heat-evolution rate) and its location (T,,) depends on the value of the heating rate (W). There are several methods of treatment, and the Kissinger relation (1956) is typical in showing the route to activation energies from a graph of In W or In(W/Tin 2) plotted against 1/RTm: In (W/T,, 2) = const. - (E/RTm) Thermal explosion theory actually offers a very reliable route to analyses of this sort--superior to most of the traditional analyses, which are often actually based on static treatments. It must be clearly recognized that we are dealing here with systems that are very far from constant reactant composition: a slowly increasing ambient temperature is inevitably accompanied by very extensive decomposition long before the end of each 'run. The successful procedure leads to 'critical' or watershed values for Semenov's parameter t~ distinguishing quiescence from runaway. Below Oct, temperature-rate trajectories pass through maxima less than unity and then decay (temperature excesses are expressed on a reduced scale like O). Above Ocr, these trajectories are vertical when the reduced temperature-excesses reach unity and temperatures then run away. The value of ~c~ depends on the kinetic laws for isothermal reaction and must be found by computation. However, only a single computation is needed for a particular rate-law: for a first order reaction, ~Jcr 1.799. The analysis leading to this requires a reference temperature although ambient and sample temperatures are changing: =

da/dt=-ka

with k = A e x p ( - E / R T )

Ramping Ambient Temperature in Systems on the Verge of Runaway (Including Systems With Distributed Temperatures): Theoretical studies of temperature-evolution were first made in systems with uniform temperature in which reactant consumption was ignored. A systematic approach5 including consumption was provided by asymptotic analysis though it is only fair to say that the form of the leading-order corrections had been anticipated by the early and perhaps less 9 ..2 mechanmal treatments of Frank-Kamenetskil and Thomas. 6 Attacks on systems with distributed temperatures began later with the work of Boddington (1983, 1984) and that work was extended to varying ambient temperatures in association with Kordylewsky. The end result was a powerful treatment (1986) that could cope with very general dependences of reaction rate upon temperature, on extent of reaction and on whether material diffuses or not. The asymptotic analysis requires a suitable "small quantity" and the neatest analysis uses a parameter I~ related to the Zel'dovich number: it is ~ B-l/3. The development is fairly straightforward though it involves a whole alphabet of symbols. The timedependent amplitude (I)(z) of the temperature obeys an equation like:

dOP/dz = a~ ~ + bz + c dp--->oo as z--->O+ The case when the stabilizing tendencies of reactant consumption are just balanced by self-heating is represented by b = 0. This is the primitive "zeroorder" case. When external heating is insufficient to compensate for reactant depletion, subcritical and supercritical behaviour of the first-order perturbation are also clearly separated. When external heating more than compensates for consumption, high temperatures are inevitable and all initial conditions lead to runaway. It should be noted that these results are most accurate for vey exothermic reactions near to the point of 'unaided' ignition. The asymptotic treatments spotlight a narrower band of conditions than that of the general realm of scanning calorimetry dealt with in the previous section.

T=Ts+Wt The relationship that T,-s has to satisfy is: (RT2rf/EA) exp (E/RTrf) = W Clearly this must be solved by iteration. An up-todate account of this for small samples of uniform temperature is given by Boddington and Kay (1989) and they also make remarks on the connexions with more limited treatments.

9. Parallel Reactions Although at the outset we instanced the oxidations of carbon monoxide and hydrogen as substantial reaction networks, they are far surpassed by hydrocarbon combustion, of which they also form important elements. All these however are clearly "complex systems" and for thermal explosion theory and chain-branching to be applied to them in a

10

HO'I'TEL PLENARY LECTURE

simple fashion, great simplifications must be made. Before them come the properties of more elementary prototypes--concurrent and consecutive reactions. When materials like cellulose or wood are oxidized, or even merely heated in the absence of air, and logical interpretations demanded, there is a natural tendency to abandon simple principles and to expect irreducible complexity. This fear is not always justified and rather broad chemical principles based on the behaviour of simpler compounds can help a great deal. Scientists from several countries including New Zealand and the Soviet Union have considered14"-16 the possibilities of dealing with chemically independent, concurrent reactions that are linked only by their several responses to heat evolved communally and hence by their exposure to a common temperature. When all the reactions are exothermic, a communal activation energy based upon the responsiveness of overall power output (Watts kg-1) E = RT2 d{ln (power output)}/dT permits the assignment of a common dimensionless temperature excess. When the kinetic and thermal features of the individual reactions are known, it is now easily possible to go forwards with predictions. Unfortunately, the route backwards is not easily mapped. More interesting than these elementary directions are those to be followed15"16 when exothermic and endothermic reactions occur in parallel. If heatabsorbing reactions are at first slower but more sensitive to temperature, thermal runaway is curbed and may even be prevented. Although attention has been mainly directed at systems without any reactant depletion, the results are full of interest, as they may provide primitive models for exothermic oxidations which simultaneously drive off moisture (endothermic). They may turn out to be primitive because the manner of the transport of moisture may play more of a key role in real systems. Some dry materials absorb moisture with heat evolution, and another complication is now possible because ff the dry material is already on the verge of thermal instability, additional heat-release due to its being wetted may take it over the brink and into instability. Once again simple circumstances can generate very complex responses and these models need to be evaluated. Rewarding beginnings16 have been made by B. F. Gray, so far in terms of a uniform composition model; this is another of those fields in which uniformity is a rather strong assumption, and both the transport of water vapour and the reversibility of the exothermic wetting are likely to have significant influences.

10. Distributed Temperatures Theory: The most influential development in thermal explosion theory was its early extension2 to systems in which reactant-temperature varied from point to point. This assessment is based partly upon the importance of the problem itselfis and partly because it was in this context that the dimensionless temperature 0 and the exponential approximation to the Arrhenius equation were first introduced: exp [(E/RTo) - (E/RT)] = exp [O/(1 + r

~ exp O.

These steps were taken in order that analytical solutions to describe how temperature depended on position in systems of simple geometry when heatflow was purely conductive might be derived. Such solutions were found analytically for the infinite slab and later for the infinite cylinder, whilst tables existed for an allied function that subsequently enabled an accurate account to be given for the sphere. All these important results relate to circumstances in which reaction-rates depended only on temperature: reactant-consumption is neglected. The problems are described by Fourier equations of the type: (r%(dT/dt) = V(KVT) + Qkcmo For the 'class A' geometries (infinite slab, infinite cylinder, sphere) when K is constant, stationary states satisfy d20/dp 2 + (j/p)dO/dp + 8e ~ = O, where j = 0, 1, 2 for slab, cylinder and spherical geometries. Temperature is a function of one vari-, able (p = r/a) the reduced radial distance, and Frank-Kamenetskii's parameter 8 is given by ~3 -= [a2Qkacom/K(RTa2//E)] The boundary conditions chosen require zero temperature-excess at the surface. This is an equation which, broadly speaking, has no solution when 8 is too large (8 > 80, say), and has pairs of solutions | stable and unstable, for 8 < 8o. The merging of these solution types at 8 = 80 corresponds to a "critical profile," and 8 = 8o defines criticality: Slab Cylinder Sphere

8o 0.878 2 3.322

Oo 1.199 1.369 1.607

There follows a period of study of different geometries, initiated2 by Frank Kamenetskii himself.

11

CHEMISTRY AND COMBUSTION The 'infinite slab' of thickness 2a is equivalent to a finite slab also of thickness 2a with its four edges perfectly insulated; the infinite cylinder is equivalent to a finite cylinder with insulated ends. Taking away the insulation affects the condition for criticality. Various attacks have been made on the problem--ab initio numerical computation being common from very early days. More satisfactory are intuitive or reasoned analogies such as the idea of the equivalent sphere. New Zealand workers were prominent in these developments, and a survey of their inter-relationships and the existence of rigorous (and sometimes very useful) bounds was later given 17 by Boddinigton in 1971. The next elaborations were provided around 1958 by Thomas and Merzhanov who generalized the solutions to systems with a temperature-step at the surface, corresponding to heat flow affected by surface-resistance as well as by internal responses to thermal conductivity. Many of these treatments is are not exact, but serviceable general approximations are always enlightening and satisfy most needs so far as practical applications are concerned. After this there is a period of study addressed to the problem of abnormally small activation energies. It had been noted by Semenov in 1928 that since critical temperatures and ambient temperatures were related by the quadratic RTZcr/E = Tcr - Ta any activation energy E < 4RTa offered no solution. This result (expressible as err = R T J E < 0.25) may be expected to have analogies in the distributed-temperature case. The problem has intrigued many mathematicians and many papers have concerned themselves with determining ~-tr. Some of these studies were incorrectly made and most were undertaken as intellectual exercises rather than for practical utility. But a watershed in the subject is marked by Kordylewskii's 1978 study 19 which offered a generalized attack on the loss of criticality in distributed-temperature systems (usually when the quotient RT~/E approaches values near 1/4). It must be emphasized that many of these are studies of the pathology of thermal runaway: at such temperatures the Arrhenius expression has lost much or all of its distinctive curvature, and there is no point in persisting with the exponential approximation. Moreover, as these temperatures are often around 104K and chemically inaccessible, their importance for serious practical study is slight.

Distributed Temperature Systems: Unifying Treatments: The manner in which critical central temperatures, etc. vary from one geometry to another often

seems perplexing to those with a purely chemical background. An attempt to show the underlying similarities was made by Boddington and Scott using a treatment (a second-order reversion) which gives analytical results that are not exact but are remarkably precise in terms of a single parameter of the system. This treatment is closely linked to questions of the efficacy of catalyst pellets used in exothermic reactions and it affords instant extension to endothermic systems. The type of result can best be seen in tabular form. The results are 'exact' for the infinite cylinder-i.e., as exact as is the Frank-Kamenetskii approximations (exp O) which usually implies errors only of order e = RTa/E "-~ 1/4o; they are close to 'exactness' for the sphere and the slab.

Distributed Temperatures: Experimental Verification: Despite the universal nature of the acceleratory mechanism, effective searches for excess temperatures that vary from point to point were long delayed. Some early attempts using mercury-in-glass thermometers were unlikely to succeed but when fine-wire resistance thermometers and thermocoupies were used the difficulty of a too-sluggish response to temperature was overcome. The bench. 21 are still those of Gray and Fine on mark studies the thermal explosion of gaseous diethyl peroxide, but pioneering workzz by Vanp6e (Thesis 1956) was done considerably earlier and richly deserves to be celebrated. Briefly, the peroxide studies established that ignition was always preceded by self-heating, that temperature excesses were greatest at the vessel centre and there was little or no detectable excess at the walls. Stable decomposition ceased to be possible at a temperature-excess around 20~ C (close to 1.61 RT2a/E). The level of precision attainable is more severely tested by comparing 'experimentar and 'theoretical" values for ~cr: these are 4.0 and 3.32. The shape of the temperature-position curve was roughly parabolic but thought to be more spiky. Seen in the perspective of twenty years elapsed,

TABLE I Unifying relationships for slab, cylinder and sphere (j = 0, 1, 2 respectively) At criticality Dimensionless rate Central temperature | Boundary gradient Effectiveness factor

2(j + l)(j + 3)/(j + 7) 2 In [(j + 7)/4] 2 (j + 7)/(j + 3)

HOTrEL PLENARY LECTURE

12

TABLE II Unifying treatments for different geometries: parametric results for the infinite cylinder (N.B. x > O, exothermic; x < 0, endothermic) 8x(1 + x)2 ln(1 + x)z 4x/(1 + x)

Oo F

(1 + x)

AE/E 8/fi~ O(p)

x/(1 - x) 4e-~176 - e -~176 2 In[(1 + x)/(1 + xp2)]

some of our earlier certainties sl seem now at the edge of confidence limits but the totality of the evidence is still convincing. Two or three other systems may be mentioned here. Ditertiary butyl peroxide has been studied by Griffiths and his co-workers. It is a very satisfactory illustration of a 'dark' thermal explosion, for there is none of the visible light emission associated with the oxidation of organic compounds. By working in stirred and unstirred systems, Grifliths has been able to show experimentally the relationship between O~t = 1 and Oo~t = 1.61 (uniform or averaged critical temperature excess and central temperature excess) for an unstirred system. This system can also be thermo-chemically tuned. Dilution or partial reaction reduces the value of B (= O~d), obviously leaving E the same; added oxygen increases B (rather less obviously also leaving E the same). This is a matter of great usefulness and importance in testing consistency of prediction and observation, e.g. in flow-through systems. An exothermic oxidation that self-ignites thermally is that of hydrazine. This, too, seems notto involve chain-branching reactions but because its effective activation energy is rather low, very large sub-critical temperature-excesses occur--up to 70 K is common. Finally, if self-heating shows up in exothermic changes, self-cooling should be seen in endothermic ones. Here there will be negative feedback and no discontinuous loss of stability is to be expected. The system chosen to test this is the endothermic pyrolysis of nitrosyl chloride, studied in a cylindrical vessel

where C is predicted to be about -0.18, and found to lie between -0.08 and -0.22.

The Corrections of Reaction-kinetic Measurements for Unexpected Self-heating: Interest has always been strong in certain families of 'unimolecular' decompositions because of the manner in which structure and thermochemistry are reflected in activation energies and preexponential factors. For example there are the differing stabilities towards ring-opening displayed by rings of different sizes or by rings of the same size containing different hetero-atoms. Convenient studies require the elimination of surface reactions--easier in large systems than in small, and the desire for a wide temperature range may be more conveniently met by measuring short half-lives at elevated temperatures than by accepting very prolonged times of study at low temperatures. In this case self-heating is present, and may be sufficient to matter when high precision is the goal. The problem was recognized by reliable experimentalists who stressed the need to measure true reactant temperatures and not merely those of a vessel's surroundings. An early analysis offered by Benson illustrates the difficulties felt even by the ablest of kineticists for it relies on the fortuitous cancellation of errors. The problem yields immediately to thermal-explosion theory, producing results of simplicity and expressiveness. Under uniform temperature conditions, errors in apparent activation energy are very strongly related to temperature excess: AE E

T ( p ) - Ta f 1+ C ] (RTa2/E----' ~ -- - 2 In /L1 + Cp2J~

l-O;

Ta O=(R,I~a/E) T -

An iterative procedure is necessary in principle, but one iteration may suffice. When thermal conductivity governs heat-transport, and temperatures depend upon position, more complex but related expressions can be used. These are approximations but of great utility. For the simple geometries if the centre temperature Oo is known then a single parameter x links it to many practical expressions

NOC! --* NO + i/2C12 AH/kJmo1-1 = 37 Self-cooling is always observed; temperatures are lowest along the axis of the vessel, typically falling 5 K below that (Ta) of the walls. The quasi-stationary temperature position profiles are in reasonable (but not perfect) accord with expectations:

O

Oo = 2In(1 + x) When ~ is known x is the solution of the expression = 8x/(1 + x)2. This formula refers to the infinite cylinder. For the infinite slab and the sphere respectively =

4x(1 - x/f) (1 + x) ~

and

12x(1 + x/10) (1 + x) 2

13

CHEMISTRY AND COMBUSTION The theory can also be made to disclose where to place a thermometer so as to record a representative "average" temperature. For infinite slabs, infinite cylinders and the sphere, these are at distances r = 2ao/3, ao/2 or 2ao/5 from a mid-plane, an axis or a centre respectively. The problem is related to that of evaluating the non-isothermal performance of catalyst beds or particles.

Hot assembly

:

i

o

11. Hot Assembly; Hot Spots; Thermal Explosion of the Second Kind Many real problems of thermal runaway arise when materials capable of exothermic reaction are assembled at a temperature T well above ambient. The classical, stationary-state equation for a uniform-temperature system

r~r

Ambient temperature FIG.

9

Bt._._~N= (Se)= (RT?TE)exp tch

o~ ee

R-T a

admits one or three solutions. When there are three, the lowest is stable, the middle one is unstable and the uppermost solution is at an inaccessibly high value. The equivalent formulation (Se) = O exp [ - O / ( 1 + eO)] -~ Oe -~ with zero or two solutions to its approximate form, highlightS the relevance of the two, physically accessible, stationary branches, the lower being stable (T~) and the upper branch unstable (Tu). If the temperature T lies between these two, the system moves toward the stable state: if T exceeds Tu, thermal runaway sets in immediately. Bowes names these circumstances, "thermal explosions of the second kind." The branch Tu separates quiescent from explosive states and is a watershed. In this approximation, the unstable branch runs from the classical critical point C at Tcr to indefinitely large values. The significance of thermal explosions of the second kind has never been lost on practical technologists working, for example, in the chemical industry but does not seem to have been the centre of attention that the critical point C has been. Of course the question "how far does a system need to selfheat before it runs away" with the dramatic answer "often only by RTa~/E--whieh may be no more than ten or twenty degrees" deserves to catch the attention, but so does the parallel question about how safe it is to stack reactive material that is already preheated. Examples of such circumstances are the storage of hospital sheets after hot laundering and pressing, the manufacture and stacking of materials like fibre-board and the piling of spent, sugar-cane residues (bagasse) in huge amounts.

1.

Ambient temperatures being known, the misleadingly large safe stacking-sizes for piles at those temperatures can distract attention from the dangers of stacking materials that are initially hotter than their surroundings. Several subordinate features arise in any practical discussion of large assemblies. First, the materials presenting these problems are often poor conductors of heat so that uniform-temperature theory is not enough. Secondly, when the materials are close to the watershed stationary state, their dynamic behaviour is not the same as at the classical point of criticality (with the drawn-out time scale obeying the inverse square-root law). Thirdly, whilst for very large amounts of material, characteristic times for temperature-decay are very long, at these more elevated temperatures the characteristic chemical times are abbreviated and there can be much less time to make sytems safe. (The mixture of fact and fancy about the effects of moisture, coupled with the vast amounts of water required simply to cool down such huge systems encourages premature and ill-considered opening of self-heating stacks and may exchange a conflagration for mere smouldering.) The problem of timescales has recently been investigated by B. F. Gray & Merkin, 26 who call the 'hot assembly' problem the critical initial value problem. It aims first to calculate the highest, safe temperature of assembly, safe meaning that the system remains quiescent and does not run away. The second aim is to discover a rule for time-to-ignition for marginally unsafe systems for which the initial temperature T exceeds its stationary value by a tiny amount A. The parallel circumstance near the point of criticality has an exponential growth with a (slow) start obeying ,'foc ~k-1/2

HOTI'EL PLENARY LECTURE

i4

For the new problem: x ~ In(l/A) So escape is brisker partly for this reason but also because characteristic times for chemical reaction to accelerate become shorter as temperatures rise. Bowes' bookz5 describes some of the investigations of these systems when a large pile of material, all at the same elevated temperature, is assembled as quickly as possible from a very large number of individual small 'packets.' There are some of the liveliest practical investigations of the thermal explosion theory ever performed, ranking with Walker's extensive examinations of fibrous, porous and powdered substrates. Broadly speaking, a uniform temperature distribution tends fairly rapidly to the appropriate stationary-state distribution for the same total heat-release rate and it then either accelerates or decays away. 12. Oscillatory Behaviour in Combustion and in Simpler Exothermic Systems In the combustion field, studies of oscillatory reactions begin with the observations by Newitt & Thornes in 1937 of repetitive cool flames in the closed-vessel oxidation of gaseous propane. They are normally to be found (see ref. 28 for a brisk survey) as oxidations proceeding with a feeble, blue chemiluminescence, visible only in a darkened room. Newitt and Thornes reported sequences of up to five successive luminous pulses in propane oxidation. Thus they provided the experimental basis for Frank-Kamenetskii to point to the existence of homogeneous oscillatory oxidation phenomena that did not depend, e.g., on diffusion processes. What seems obvious today seemed strange then and the inescapably oscillatory character was still not stressed elsewhere even when Pease (1940) produced eight such pulses in propane oxidation or even, many years later, when Yokely detected eleven pulses by temperature measurements in the same system. The paths of study divided. To produce and study indefinitely sustained oscillations, open systems with a constant supply of reactant are required. To simplify interpretation, temperatures and species concentrations need to be made uniform: mechanical stirring is generally needed for this. Finally, for a really secure experimental base, more than one variable should be monitored continuously. All this28 had to wait a long time. In early days, only light pulses were measured and only the end-products of oxidation were analyzed. The eventual use of wellstirred, flow-through systems had constituted one of the major advances. By the 1960s, interpretations of real systems could draw on increasingly extensive mathematical theory of the dynamical be-

haviour of simple, exothermic reactions in a cstr as provided by work of chemical engineers (especially of the Minnesota schooll though this was a river with many tributaries). The other streams of work have their origins in early Russian studies. One dealt with closed systems, or at least furnished a basis for modern closedsystem studies. The other sought to make the relationship between single explosive events in closed systems and repetitive explosive events in open systems (in a rather dismissive way). The serious study of thermokinetic oscillations in closed systems goes back to Sal'nikov, who examined the behaviour of the system: P --> A

kop

A~B+heat

ka

i.e., precursor ~ intermediate --+ inert product + energy release. In the simplest case, (a) the precursor was assumed to generate the intermediate at a constant rate (precursor itself being maintained at constant concentration), (b) the second step was both exothermic (Q # O) and responsive to temperature with k cc exp(-E/RT). The scheme was broadened3~ by B. F. Gray and C, H. Yang and has recently received rather intensive investigation in Leeds and in Sydney. It is a remarkable example of how a simple model can generate rewardingly rich behaviour. (It does not adequately explain hydrocarbon oxidation but it contains many lessons: it was the distinction of Sarnikov's original work that it insisted on a rigorous proof of oscillatory behaviour.) It is instructive to set out the conservation equations governing mass and heat2balance in this elementary system. For a concentration of P maintained constant at Po with a constant ambient temperature Ta, the two conservation equations are: dt

kopo - kaa exp

kopo - ka

; a

dT

Qka

-dt- = -eCv -

hS

Vcrc~(T - Ta).

The energy-conservation expression can be written in dimensionless form: dO - -

~

ore O

--

O.

d, We choose O = (T - Ta)/(RTa2/E) as usual and eL = a/aft, where arf = (hS/QVka)(RTa2/E) = earn. Here aign is the concentration of the intermediate

CHEMISTRY AND COMBUSTION A that, if present alone, would just become explosively unstable. In these terms the mass-balance expression becomes

d~ -

Ki~

-

~e~

d~ The two parameters K and r satisfy K = kat N and o" = (kotNPo/a~f), so o" = (kopo/koaq). (In their study, Scott & Kay work in terms of K and ~, tr, so ,~ = IJ,/K). This system gives a unique stationary state with elegant parametrie expressions: O~s=,X

and

ass=,re-L

"lq~e stability of this stationary state is conditional. When loss of heat is slower than consumption of A so that -> (~r - 1)e-" -> e -z ~- O. 13, there is stability everywhere. When K < e -2 there is a middle range of unstable behaviour between the two values of ,~ satisfying (o" -

1)e -r

=

K.

At these values, oscillatory behaviour sets in or ends. It should be noted that although values of the stationary-state concentrations do not exceed those predicted to be explosively unstable by simple thermal explosion theory, values of the (stationary,) excess temperature 0 certainly do exceed the normal stable limit (Oig, = 1). It is also worthy of note that the amplitude of the oscillations is very substantial and that oscillatory temperatures may range from near zero to more than 19 = 10 when their (unstable) focus is at | ~ 4. Precursor Decays: p = Po exp (-kot):

All this development accepts the idea that a stationary state is achievable and that in turn depends on assuming a constant concentration for the precursor. If the precursor decays slowly from an initially large value to zero, then an evolution in time can be described as oecuring in four phases. In terms of temperature excess we may find: I First, a fast jump of temperature 0 to the stationary state value appropriate to initial conditions. The approach may be monotonic, or by overshoot or by overshoot plus damped oscillations. II Secondly, an exponential downward drift of 0 in time, hugging the stationary-state locus more closely as time passes. III Thirdly, as the 'upper' Hopf bifurcation point is

15

passed and this locus ceases to attract, the 3dimensional trajectory winds out on to a surFace and travels around it in a helical t~shion. This is the oscillatory phase. IV Next, after the second (lower) llopf bifurcation point, the stationary, state locus again becomes attractive. Oscillations die out well before all the reactant is consumed. V Finally, eventual exponential precursor-decay is mirrored by the exponential decays of the concentration of 'intermediate' and excess temperature towards zero. It is quite remarkable how so simple a model so graphically represents the major features of many real systems in closed vessels. Moreover, it links many properties to the parameters of the system and so to one another. Amongst these are such properties as the duration of the oscillatory and preoscillatory modes, the frequencies and amplitudes of oscillations at birth and death, and the values of the various concentrations and chemical reaction-rate coefficients, activation energies etc.

13. Chaotic Behaviour of Thermokinetic Origin Even systems of only one variable (such as the adiabatic cstr) can show multiplicity of stationary states and hence ignitions and extinctions. These are called static instabilities. Systems of two variables--a single reaction in a non-adiabatic cstr-can show not only nmltiplc stationary states hut also sustained oscillations. These are dynamic instabilities. In recent years much attention has been fucussed on oscillations that show period-doubling sequences and chaotic behaviour. For this kind of versatility, a mininmm of three variables is required. Certain prototype systems deserve to be listed. They all relate to theory rather than experiment. In practice a key question is, what are the effects of the small but inescapable variations in conditions that always occur in "'real life"? Ultimately, of course, there is an irreducible minimum of fuctuations that is related to the moleeular nature of matter, and most theoretical eflbrt has gone into studying that. However, real exothermie systems in thermal contact with their surroundings will show, e.g., temperature fluetuations far in excess of this minimum. Useful analyses of these important questions seem scarce: perhaps this admission of ignorance will generate a lively response. The tahle lists a few early examinations of "unforced' systems. Period-doubling leading to chaos in response to forcing had been established earlier, but the unforced examples are, even ff only at first sight, more surprising. One of the most suggestive of recent investigations is a numerieal-plus-ana[ytieal study of virtually isothermal autocatalysis

HOTrEL PLENARY LECTURE

16

TABLE III Minimal systems showing chaotic responses to thermal feedback System

Variables

CSTR Concurrent reactions A---> C, B---> C

CA, Ca, T

CSTR Consecutive reactions A ---> B --* C

CA, Ca, T

CSTR Single reaction Two heat capacities

CA, TI & T2

Semi-batch reactor Autocatalysis plus selfheating

CA, Ca, T

P~ A

ko = koo exp[(E/RTo) - (E/RT)]

A---~ B

ka

A + 2B---~ 3B B --~ C

k~ ka AH = -Q;

Q.~ RTo

The strictly isothermal system (with Q = 0) is one of only two variables when p is kept constant. It shows oscillations conditionally (so long as k, < (1/8)ka) over a range of values of Ix satisfying (ixz) +_ = 1 + 2K -+ (1

-

8K) 1/2

where K = k,,/kd and IX = kopo(kdk 3d)1/2

These oscillations are of simple form. As soon as the reaction becomes non-isothermal we have a 3dimensional system and if any of the steps in the scheme is made sensitive to temperature, we also have a feedback mechanism. The intensity of feedback is measured by the group ~ = ATad/(RTo~/E). As this quantity increases, so does the possible variety of response. At first oscillations remain simple but a value of ~ is then reached at which complexity sets in. For such values of ~j, period-doubling sequences exist in the oscillatory region as IX is varied. These sequences can lead into chaos and out again. What is perhaps unexpected is the very small value of exothermicity or responsiveness to temperature that is needed to cause these complications, because all real chemical reactions are accompanied by heat effects and the displays calculated here are probably waiting to be found in many real systems.

Investigators Lynch D. T., Rogers T. D. and Wanke S . E . Math. modelling 3, 3, 103.

Date

1982

Aris R. A. and Jorgenson D. V. Chem. Eng. Sci. 38, 45

1983

Planeaux J. B. & Jensen K. F. Chem. Eng. Sci. 41, 1491

1986

Scott S. K., Kay, S. R. & Gray P. J. Phys. Chem. 94, 3394

1989

14. Unifying Themes This Review Lecture began with the intention to emphasize broad lessons in the relationships between chemistry and combustion and to offer a varied set of illustrations. In fact, "real chemistry" has taken rather a back seat as the value of simplified representations and model schemes has been exploited. It has been commonplace in the technical exploitation of combustion to hear it stated that the chemistry does not matter, and that physical processes (like mixing) hold the key to understanding and to advances in applications. To the extent that chemical processes are not only multiple, full of alternative routes and above all rapidly increasing in rate with temperature, neglect of chemical details in engines and furnaces is not a bad first simplification, especially for a chemist, to make. However, as limits to performance are approached, so deeper understanding is called for. Problems of knock, of unwanted emissions and of substitute fuels all demand this depth. One of the most interesting challenges at present is the need for some convenient but not oversimplified "global kinetics", intermediate between schemes involving hundreds of elementary steps (these are already with us), and optimistic mass-action laws involving only expressions like rate ~ [fuel] ~ [oxygen] t3. Most of the discourse has concentrated on the interplay between heat-release, self-heating, reactant consumption and reaction rate, and thus on topics that are lumped together under the umbrella of thermal explosion theory. The first unifying theme is the concept of feedback: consequences of the change having effects upon the change. We have not said much about the chain-branching reactions that offer isothermal acceleration. They take the form of free radical multiplication, and different steps are

17

CHEMISTRY AND COMBUSTION important over different ranges. In heptane combustion, for example, all of the following branching steps can be important, each under different conditions:

symbols and ideas of thermodynamics, constant use casteth out fear and a sympathetic notation certainly eases the process.

HO2CTH1402H ~ HO + OC7H1402H

Nomenclature

H202+ M~2HO+

M

H + O2--~ HO + O Instead we have concentrated upon thermal feedback, with energy release serving to heat the system, so causing the energy-releasing reactions to go faster. The combination of heat-release and responsiveness to temperature has been expressed by a dimensionless group B, the Zel'dovich number. The constituents of B are ATad which involves the exothermicity, and RTrfZ/E which involves the activation energy. To each reaction corresponds its own B and it is their collective outcome that we must establish in complex systems. The second is less a unifying principle than an observation of the near-universality of non-linear responses in chemistry, so that feedback is non-linear, our rate-equations are non-linear and the texture rich. But this pervasive quality also means that quite simple models can enjoy a rich pattern of responses. The Sal'nikov2~ scheme as the prototype redeveloped here for oscillatory behaviour, and extended to the circumstances of closed systems is just such an illustration. More fundamental in its implications than this is the choice of starting point that we should take. We begin with the conservation laws of matter and energy. If we choose to suspend them it should be briefly. One example is the primitive foundation for studying thermal explosion, where reactant-consumption is at first ignored and ignition treated simply as a sudden jump away from a stationary state. This jumping between stationary states can, of course, be realized for an open system like the cstr. Otherwise it is an invaluable first approximation but not the whole story. Chemical prejudices in the combustion field encourage us to disregard the reversibility of chemical reactions because combustions very obviously tend very near indeed to completion. So in combustion, any approach which required overall reversibility as a vital ingredient of understanding unstable behaviour would deserve little encouragement. The opposite attitude has been fostered in the isothermal oscillatory field without perceptible gain: the approach from oscillatory combustion is not a bad preparation. On a lower level of generality but with a bonus of convenience comes the use of dimensionless groups. There is something in a chemist's education that impedes their adoption. Like the

Symbol Meaning a species concentration ao radius, half-width B, B* dimensionless excess temperatures c, Co species concentration cv, cp specific heat capacity E activation energy k reaction velocity coefficient rate of heat loss m reaction order p concn of species P Qv, Qp exothermicities R universal gas constant rate of heat production S surface area t time T temperature u flow-rate V volume W steepness of temperature ramping parameter X Zeldovich number Ze

SI unit mol m -3 m

mol m -3 J K -1 kg -1 J mo1-1 s -1 (conc) 1-m W mol m -a J mol -I J K-lmo1-1 W m2 s K mas- 1 m3 K s -1

Greek symbols

ot 13 F e ~l A O 8 kO K tr I~

p k

dimensionless concentration diffusive transfer coettqcient boundary t e m p e r a t u r e gradient (=RT~/E) effectiveness factor increment d i m e n s i o n l e s s excess temperature Frank~Kamenetskii parameter Semenov parameter (Se) dimensionless time thermal conductivity parameter dimensionless reactionrate intensity of thermal feedback dimensionless distance fractional concentration

-m s -1 --

-----W m -x K -1 -----

18

H O T r E L PLENARY LECTURE

Subscripts and superscripts pertaining to different conditions rf ad cr ign ext a o

res ch N

reference, as in adiabatic, as in criticality ignition extinction ambient, as in initial value, as in residence, as in chemical newtonian cooling

Tff

18.

ATad 19. Ta

Co tres

20.

21.

Acknowledgments

22.

I am grateful to Dr. Richard Gann for his unfailing courtesy as an editor and to the Combustion Institute and its British Section for their support.

23. 24.

25. REFERENCES 1. SEMENOV, N. N.: Z. Physik. 48, 571, 1928. 2. FRANK-KAMENETSKII,D. A. : Diffusion and Heat Transfer in Chemical Kinetics, 2rid ed., Plenum Press, 1969. 3. ZEL'DOVlCH, YA. B.: Zh. Tekh. Fiz. 11, 501 (1941); ZEL'DOVICH,YA. B. AND ZYSIN,: Zh. Tekh. Fiz. 11, 510 (1941). 4. GRAY, P., GRIFFITHS, J. F. AND HASKO, S. M.: Proc. Roy. Soc. A396, 227, 198ft. 5. KASSOY, D. AND LINAN, A.: Q. J. Mech. Appl. Math. 31, 99 (1978). 6. THOMAS, P. H.: Proc. Roy. Soc. A262, 192 (1961). 7. LACEYA. A.: Int. J. Eng. Sci., 21, 501 (1983). 8. ADLER, J. AND ENIC, J. W.: Comb. Flame 8, 97 (1964). 9. KORDYLEWSKI,W. AND SCOTT, S. K.: Proc. Roy. Soc. A390, 13 (1983). 10. TYLER, B. AND WESLEY, T. A. B.: Eleventh Symposium (International) on Combustion, p. 1115, The Combustion Institute (1967). 11. GRAY, B. E. AND SHERRINGTON, M. E.: Comb. Flame 19, 435, 445 (1972). 12. BILOUS, O. AND AMUNDSON,N. R.: A. I. Chem, Eng. J. 1, 513 (1955). 13. GRAY, P., KORDYLEWSKIW. AND KRAJEWSKI,Z.: Proc. Roy. Soc. A412, 45 (1987). 14. BODOINGTON, T., GRAY, P. AND WAKE, G. C.: Proc. Roy. Soc. Lond. A393, 85 (1984); A401, 195 (1985). 15. GRAHAM-EAGLE, J. G. AND WAKE G. C.: Proc. Roy. Soc. A407, 183 (1986). 16. GRAY, B. F. AND WAKE, G. C.: Comb. Flame, 79, 2 (1990). 17. See e.g. references in BODDINGTON, T., GRAY,

26. 27. 28.

29. 30. 31.

32.

33.

P. AND HARVEY, D. I.: Phil. Trans. Roy. Soc. A270, 467 (1971). GRAY, P. AND LEE, P. R.: Thermal Explosion Theory in Oxidation and Combustion Reviews, Vol. 2, p. 1, Elsevier 1967. KORDYLEWSKY,W.: Kommunikat 1-20/K-020/78 Tech. Univ. Wroclaw (1978); BODDINGTON, T., FENG, C. G. AND GRAY, P.: Proc. Roy. Soc. A390, 247 (1983). BODDINGTON,T., GRAY, P. AND SCOTT, S. K.: J. Chem. Soc. Faraday Trans. Vol. 78, p. 1721 & 1731 (1982). FINE, D. H., GRAY, P. AND MACKINVEN, R.: Proc. Roy. Soc. A316, 241, 255 (1970). VANP~E, M.: Compt. rend. Acad. Sci. Paris 241, 951 (1955). GOODMAN,O. AND GRAY, P. : Trans. Faraday Soc. 66, 2772 (1970). BODDINGTON,T. AND GRAY, P.: Proc. Roy. Soc. A320, 71 (1970); BODDINGTON,T., GRAY, P. AND SCOTT, S. K. : Proc. Roy. Soc. A378, 27 (1981). BOWES, P. C.: Self-heating: Evaluating and Controlling the Hazards, London H.M.S.O., 1984. GRAY, B. F. AND MERKIN, J.: J. Chem. Soc. Faraday Trans. 86, 597 (1990). WALKER,I. K.: New Zealand J. Sci. 4, 775 (1961) and many subsequent papers. GRAY, P., GRIFFITHS, J. F., HASKO, S. M. AND LIGNOLA, P. G.: Proc. Roy. Soc. A374, 313 (1981); A396, 227 (1984). SAL'NIKOV,I. E.: Zh. Fiz. Khim. 23, 258 (1949). GRAY, B. F. AND YANC, C. H.: J. Phys. Chem. 69, 2747 (1965); 73, 3395 (1969). GRAY, B. F. AND ROBERTS, M. J.: Proc. Roy. Soc. A416, 391 (1988); GRAY, P., KAY, S. R. AND Scoa'r, S. K.: Proc. Roy. Soc. A416, 321 (1988). GRAY, P. AND SCOTT, S. K.: Chemical Oscillations and Instabilities, Clarendon Press, Oxford 1990. Ares, R. : The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Clarendon Press, Oxford, 1975.

Appendix: Limitations on the Use of O and the Form exp O Recently B. F. Gray, J. Merkin and G. C. Wake (GMW) have urged the avoidance of O and exp O and urged the use of the Arrhenius expression in the form exp-(1/u), where the variable u stands for RT/E. This is the choice made around 1956 by Vulis (following Todes, 1936) and later by Hardee et al. The point of view of GMW is that of the student of differential equations rather than that of combustion but they have interesting comments to make that should not be ignored. For example they correctly suggest errors of order RTa/E in many so-

CHEMISTRY AND COMBUSTION lutions. Vulis' book is the best starting point for a host of examples for uniform-temperature systems. For distributed-temperatures, the variable u is less user-friendly, leading immediately to numerical computation--analytical solutions are completely out of reach. An interesting recent example of features highlighted by the new variable u and not by FrankKamenetskifs O is the discussion of "disjoint solutions" in the simplest of open systems--a firstorder exothermic reaction in a cstr. The exact expression for turning points on a stationary state locus (see section 2) is:

1+

T,~a - To~ T2 E?R J - (Tad'4- W~ + W~

=

O.

The physically realistic ordering is To ~ Taa E / R since To ~ 500 K, Tad ~ 3000 K and E / R

19

30,000 K. Under these circumstances, which are the norm, ignition and extinction s~tisfy: T_ - To ~- RToZ/E; Taa - T+ ~- R T 2 a d E

If, with GMW, we go outside the normal realm of combustion conditions (abandoning strongly T-dependent reaction-rates) and allow Tad - To to be as big as E/R, an extinction point can be located at T+ = E/2R, when To is absolute zero. Values of T+ on the "wrong side of the To axis" are also solutions. These lively and unexpected results (or forgotten results--see Vulis) remind the combustion scientist not only to avoid absolute zero as a reference temperature for O but also to be sure that his chemical system is indeed strongly responsive to temperature, and not up in the region of Te =

E/R.