{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture1 - This is a scanned copy of written versions of...

Info iconThis preview shows pages 1–18. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 14
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 16
Background image of page 17

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 18
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: This is a scanned copy of written versions of 4.5 of 9 lectures delivered at the Mathematics Research Center, University of Wisconsin-Madison in the autumn of 1 9 79. © 1980 Martin F einberg Lee/tuna on Chenicaz Reaction Netwaalvs Martin Feinberg Department of Chemical Engineering University of Rochester Rochester,“ 1462? Present address: Departments of Chemical Engineering & Mathematics The Ohio State Univerisity 140 W. 19th Avenue Columbus, OH 43210 USA email .‘ feinberg. [email protected] Content» Phefiace Lectuhe l} Inthoduetton 1.A. Motivatton 1.3. Notation Lectuhe 2; Reaetton Netwohhb, Ktnettcb and the Induced Dtfiéehenttat Eguattonb 2.A. Reaction Netwohhé 2.8. Ktnettcb 2.C. The Dtfifiehehttat Equattonb 60k a Reaetton SyAtem 2.0. An Etementahy Connectton betveen Reactton Netvohh Sthuetuhe and the Natute 05 CompoAttion Thajeetohteb 2.E. Open Sybtemb: why Study "Funny" Reaction Netwohhb? Exampfle 2.E.1. Cohttnuoub Ftow Stthhed Tank Reactohb Exampte 2.E.2. Homogeneoub Reactohb wtth Centatn Specteb Coneenthattonb Regahded Conatant Exampte 2.5.3. Intehconnected CettA Lectuhe §5 Igg_Theohemé 3.A. Some Qoeéttoné. Phobtem 3.A.l. The extétence 06 pobtttue equtttbhta Phobtem 3.A.2. The untqueneéb 05 pOAtttve equtttbhta Phobtem 3.A.3. The Atabtttty 06 poottxve equtttbhta Phobtem 3.A.4. The extAtenee 06 pehtodtc compobttton eyeteb 3.8. A Ltttte Vocabutahg 3.C. The Defitctency Zeho Theohem 3.0. The Defitctency One Theohem Lectuhe if Some Deétntttonb Eflg'Paopobtiionb 4.A. Some Mottoatton 4.8. Some Ghaphtcat Aépectb 06 Reaetton Netwohhb 1—1 1-1 ‘1-10 2-1 2—1 2-8 2-12 2-20 2-22 2-24 2-27 3-1 3-2 3-4 3-5 3-10 3-12 3-13 3-19 3-25 4-1 4-2 4-5 4.C. Some In/te/Lplag 06 Stoichiommy and Gwyn/dad same/tune 4-20 4.17. A P/LopaAi/téon Concunéng the Na/tu/Le 06 Equ/Lubua 4-32 Lee/tune 3: Wood gfi ihe Defiioéencg Zeno Thea/Lem 5-1 , 5.A. P1006 . 5-3 5.A.1. Pnoofi 06 pan/u (i) and (LU 5-3 5.A.2. Pnoofi 06 loam (iii), given the exA'A/tence 06 a WAX/ave equiLéb/uium 5-5 5.A.3. Moots 06 the exutence 06 a poultéve equ/Lbéblu'um 5-16 LECTURE 7: INTRODUCTION This lecture is divided into two parts. In the first part I shall try to provide some motivation for everything that follows. In particular, I shall try to explain, at least in an informal way, how chemists and chemical engineers arrive at the differential equations they work with and how these differential equations are tied to reaction network structure. Once this is done we can begin to understand why a reasonably general theory of chemical reaction networks is necessary. Moreover, we can begin to understand why, despite the great complexity of the differential equations involved, such a theory should even be possible in principle. In the second part of this lecture I shall discuss the important but more mundane subject of notation. 7 . A. Motivation Ihese lectures will be about a special but rather large class of ordinary differential equations -— those that derive from chemical reaction networkS. In order that I might provide some sense of how these equations come about it will be useful if I write down an example of a reaction net— work and indicate informally how it induces a system of ordinary differential equations. Then I can discuss the kinds of problems we will want to consider. Suppose that A, B, C, D and E are chemical species, and suppose I believe that the chemical reactions occurringanwng these species are reasonably well reflected in the following diagram: A 222:: 2B A+c :—__-'> D (1.1) B-+E What I have written down is a diagram of a chemical reaction network. It indicates that a molecule of A can decompose into two molecules of B, that two molecules of B can react to form one molecule of A, that a molecule of A can react with a molecule of C to form a molecule of D, and so on. Now suppose that I throw various amounts of my species into a pot. I am going to presume that the pot is stirred constantly So that its contents remain spatially homogeneous for all time, and I shall also suppose that the contents of the pot arev forever maintained at fixed temperature and total volume; This, of.course, is not to say that the chemical composition of the mixture within the pot will remain constant in time, for the occurrence of chemical reactions will serve to consume certain species and generate others. In fact it is the temporal evolution of the composition that we wish to investigate. With this in mind we denote the (instantaneous) values of the molar concentrations of the species by cA(t), CB(t)’.CC(t)’ cD(t) and cg(t), and we abbreviate this list of numbers by the “composition vector" c(t).*- Thus the picture we are thinking about, at least for the moment, looks something like that shown in Figure 1.1.1‘ A molar concentration, say cA, specifies the number of A molecules per unit volume of mixture. More precisely, CA is the number of A molecules per unit volume divided by Avogardro's number, 6.0231(1023. We shall be somewhat more precise about what we mean by the "composition vector" in .Section 1.3. , +The reactor depicted in Figure 1.1 is closed with respect to the exchange of matter with its environment. Our focus on such reactors is temporary and is merely intended to illustrate in a simple context how chemists and engineers formulate differential equations based upon a set of reactions believed to approximate the true chemistry. We will begin to consider "open" reactors in the next lecture (Section 2.E). There we shall indicate how open reactors can be modelled in terms of reaction networks and how the appropriate differential equations, like those for closed reactors, bear a definite relationship to reaction network structure. memosTA-r , l - CJtLCefiLCJtLCJtLCe“) Figure 1.1 We would like to write down differential equations that describe the evolution of the five molar concentrations. Since chemical reactions are the source of composition changes, the key to understanding how to write down differential eQuations lies in knowing how rapidly each of the several reactions occurs._ What is generally assumed is that the instantaneous occurrence rate of each reaction depends in its own way on the instantaneous mixture composition vector, c. Thus; we presume, for example, the existence (e) of a non-negative real-valued rate function fifiA-*2B(.) such that 7% A-*ZB is the instantaneous occurrence rate Of reaction A +23 (per unit volume of mixture) when the instantaneous mixture composition is given by the 1—4 ink . vector c. Similarly, we preSume the existence of a rate function £28+A %A+C->D(.) for the reaction A+C->D, and so on. A kinetics for a reaction network is (') for the reaction 23+A, a rate function an assignment of a rate function to each reaction in the network. Once we presume that network (1.1) is endowed with a kinetics we are in a position to write down the system of differential equations that govern our reactor. Suppose that, at some instant, the reactor is in some composi- tion state c. Let us begin by thinking about the instantaneous rate of change of c Every time the reaction A->2B occurs we lose a molecule of A, and thgt reaction has an occurrence rate %A->2B(C)' 0n the other hand, every time the reaction 2B+A occurs we _g_a_in a molecule of A, and that reaction occurs at rate %2B+A(C)' Similarly, the reactions B+E +A+C and D+A+C produce amolecule of A with each occurrence, while each occurrence of the reaction A+C->D results in the loss of a molecule of A. Thus we write (C) + K ' (C). (1.2) (C) + B+E+A+C E (c) .+ 1CZB+A(C) - 1C A=T7( A+213 A+c'—>D 1CD+A+C If we turn our attention to species B we notice that whenever the reaction A+2B occurs we gain two molecules of B, and whenever ZB+A occurs we lose two molecules of B. When D+B+E occurs we gain one B, and when B+E+A+C occurs we lose one B. Thus, we write E=27€ B A+2B (C) - it (e) (1.3) (c) +1€ (c) -— 27C ZB+A D+B+E B+E+A+C Continuing in this way we can write down equations for ac, ED, era to generate the full system of differential equations that govern our reactor: *9: More precisely, K (c) is the number of occurrences of A+2B A+2B per unit time per unit volume divided by Avogadro's number- 1-5 &A = “kmgm +16 (c) - 1c (c) +f< (c) + 76. (c) ZB+A A+C+D D+A+C B+E+A+C a = 2KA+2B(C‘) ' 212nm“) + KD+B+E(C) ’ KB+E+A+C(C) (C) + £B+E+A+C(C) (1. 4) (e) +71! 0 II c ' k A+C+D D+A+C 5 =16 («9—76 (c) A+C+D(C) ' {Comm D+B+E c K: (c) - X: E ‘D+B+E B+E+A+c ( C.) . Thus far we haven't said anything about the nature of rate functions, and that is what we shall do now. More often than not chemists and engineers presume the kinetics to be of mass action type. With mass action kinetics one can merely look at a reaction and write down its rate function up to a multiplicative positive constant. Here is the way things work: For the reaction A-+ZB we presume that the more A there is in the reactor the more occurrences of the reaction there will be. In fact, we presume that the instantaneous occurrence rate of 'A-*ZB is proportional to the instantaneous value of cA . Thus, we write where a is a positive constant. For the reaction A+C~+D the situation is a little different. An occurrence requires that a molecule of A meet a molecule of C in the reactor, and we take the probability of such an encounter to be propor- tional to the product c c Although we do not presume that every such A C' encounter yields a molecule of D, we nevertheless take the occurrence rate of A+C-+D to be given by %AH+D(C) = YCAce ’ where Y is a positive constant; Similarly, an occurrence of the reaction 2B-+A requires that two molecules of B have an encounter, and we take 2 2B_>A(c) - 8(CB) , where B is a positive constant. Thus, with mass action kinetics the rate functions for network (1.1) take the form KA+2B(C) = occA 2 123 +A(C) = BCCB) KA+C+D(C) = Y CACC ' (1.5) 1CD+B+E(C) = 6°13 zD+A+C(C) = 6CD KB+E+A+C(C) = “13“}: The positive numbers a, B, y, E, 6 and g, called the rate constants for the corresponding reactions, are sometimes estimated on the basis of chemical principles or else one makes an attempt to deduce them from experiments. When a reaction network is presumed to be endowed with mass action kinetics it is the custom to indicate the rate constants (or symbols for them) alongside the corresponding reaction arrows in the net— work diagram. Thus for our example we might have a display like that shown in (1.6). (1.6) 1-7 If we assume mass action kinetics for the network we have been studying, the appropriate differential equations are obtained by inserting (1.5) into (1.4): . 2 cA—-0LcA+ 8(cB) -v‘YcACC+ 6CD+ EchE E= 20Lc -28(c)2+ 8c - EC B A B D BCE CC = - YcAcC +- 6CD + chcE (1.7) CD = 'YcAcC — (5+€)cD rd” II EcD-Ech'E . We have arrived at a fairly concrete system of ordinary differential equations, and we can begin to pose questions about them. Here are some of the questions we might like to ask: (a) Does the system (1.7) admit a positive equilibrium -— that is, an equilibrium at which all species concentrations are positive? (b) Does the system (1.7) admit multiple positive equilibria (in a sense to be made precise in the next lecture*)? (c) Does the system (1.7) admit an unstable positive equilibrium? (d) Does the system (1.7) admit a periodic (positive) composition. trajectory? These are not easy questions, and the answers to them might of course depend on the particular (positive) values taken by the rate constants a, 8, y, 5, E, and g . Even if we could answer these questions for all positive values of the rate constants what would we have accomplished? * y We shall want to know whether there can exist multiple positive equilibria within a stoichiometric compatibility class. In rough terms a stoichio- metric compatibility class is a certain set of compositions which remains invariant under the flow given by (1.7). 1-8 We would have understood one model chemical system fairly well, at least with respect to certain qualitative issues. There are, however, thousands of distinct reaction networks that might, on one occasion or another, command our attention. Each has its own system of differential equations, perhaps more compliCated by far than the system we have been considering. How, then, are we to proceed? It is clear that we cannot rely forever on purely §d_hgg studies of what— ever systems might present themselves for examination. Even if we cast aside the long—term enormity of such an undertaking, there are still two problems that must be faced in the short run. First, questions of the kind we have posed will, for the most part, be confronted by engineers and chemists, not mathematicians. Second, it is by no means clear that ‘mathematicians, even the most expert, are currently in a position to provide much help. The fact is that even moderately large systems of nonlinear differential equations —— in particular polynomial systems like those displayed in (1.7) —— remain poorly understood* in general. It seems to me that what is required is a rather broad-based theory of those systems of differential equations that derive from reaction net- works, a theory which would in some sense cuts across the fine details of individual problems to provide qualitative information about large classes of systems all at once. Moreover, we would like the results of such a theory to be of the kind that engineers and chemists can use easily in addressing questions like those we have posed. This seems like a lot to ask, and I should try to explain why a theory of the kind I have in mind should even be possible in principle. Although we shall also be interested in the more general situation, let me temporarily restrict my attention to reaction networks endowed with mass action kinetics. Thinking back to the source of the system (1.7), we recall that it derived in a rather orderly way from the network (1.6). In fact, we knew how to write down the appropriate * Consider, for example, the remarkable complexity of the seemingly innocuOus Lorenz system [L1], which is composed of three polynomial ordinary differential equations in three variables. differential equations (up to values of the rate constants) merely from inspection of the reaction diagram. Had we begun with a different net— work we would have arrived at a different system of differential equations, but again the essential shape of those equations (up to values of the rate constants) would have derived from the reaction network in a precise way. Indeed, it is the close connection between reaction network structure and the shape of the induced differential equations that lends the-subject of chemical reactor theory its coherency. This is our source of hope. If reactor behavior is determined by‘a system of differential equations which, in turn, is determined by the underlying reaction network in a precise way, then perhaps one can prove theorems which tie qualitative aspects of reactor behavior directly to reaction network structure. Can this in fact be done? I hope that these lectures will help to demonstrate that one can proceed surprisingly far in this direction. In Lecture 3 I will state two theorems, one of which will immediately answer all four questions that we posed about the system (1.7). The answers are ygp, pp, pp, and pp; these answers hold for all positive values of the rate constants a, B, y, 6, 6, Hand 5. Moreover, these answers can be obtained merely from inspection of the reaction diagram (1.6); one need not even write out the differential equations. The fact is that one can delineate a large class of networks —— some extremely complicated -— for which the corresponding differential equations admit solutions of a very limited variety, regardless of the (positive) values the rate constants take. Our objectives will be rather broad, and I should try to make clear what these are. Results of the kind I have just described are typical of those we are after. We seek to classify reaction networks according to their capacity to induce differential equations which admit behavior of a specified type. .When we restrict our attention to networks endowed with mass action kinetics we will ppp_ask, for example, whether the differential equations for a particular network taken with specified rate constants admit periodic orbits. Rather, we will ask if the network is such that the induced differential equations admit periodic orbits for §£_least one set of rate constants -— that is, if the network has the capacity to 1—10 admit periodic orbits. The network itself will be our object of study, * not the network endowed with a particular set of rate constants. 1 . 8. Notation It is not difficult to see that the differential equations induced by a reaction network can be rather cumberSome. With this in mind I want to spend a little time talking about notation. Although the notation I shall use is quite natural to the problems we shall address, it is not entirely traditional. With each reaction network we can associate three sets. The first is the set A; of chemical species -— {A, B, C, D, E} in network (1.1). The second is the set of objects that appear before and after the reaction arrows —— {A, 2B, A+C, D, B+E}' in (1.1). These objects are called the complexes of the network, and the set of complexes will be denoted by the symbol g'. The third is the set 52 of reactions -— {A-*ZB, ZB-+A, A+C+D, D+A+c, D+B+E, B+E+A+c} in network (1.1). With each of these sets we shall want to associate a (finite-dimensional) vector space so that we can, for example, speak of a "vector of species concentrations" or a "vector of reaction rate constants." If m is the number of species, if n is the number of complexes and if r is the number of reactions in a network we can, of course, work in the vector in n Spaces 1R , P. , and 2er . In this way we can speak of the "composition vector c a B9“, c being the molar concentration of the 1th species, 1 i = 1, 2,...,m. And we can speak of the "rate constant vector k 6 Eg,' kj being the rate constant of the "jth reaction," j = 1,2,...,r. This is what tradition would seem to require. * It is perhaps worth mentioning here that, in practice, complete sets of rate constants for intricate networks are hardly ever known with great precision. It is often the case that chemists have a very good sense of what reactions are occurring but can estimate or measure rate constants only to within a considerable margin of uncertainty. For a discussion of the relationship between reaction network structure and the extent to which rate constants can be determined uniquely from certain classes of experiments see [F3] and, for more detail, [F1] and [K]. In [Fl] there is also a discussion of how information about the reaction network itself can, in principle, be inferred from near-equilibrium experiments. 1—11 It turns out, hOwever, that Hf“, IR“, and IRr are somewhat awkward media in which to work; At the very least these spaces require that we number everything in sight so that we can speak of the "ith species," "kth ( I the "jth reaction,‘ or the complex.’ Thus, we must impose an artificial ordering on each of the three sets of objects even before we begin to work, and we must carry that order around thereafter, suppressing or rearranging it whenever it becomes intrusive. There is a much better and far more natural way to do things, and that is what we shall discuss ROW. We denote the real numbers by Hi, the positive real numbers by E and the non—negative real numbers by i; . If I is a set we denote by IRI the vector space of real~va1ued functions with domain I . (Addition of functions and multiplication of a function by a real number are defined in the usual way.) By H’I [resp., lfiz ] we mean the subset of IRI consisting of those functions which take exclusively positive [resp., non~negative] values. Henceforth in this.section we shall suppose that I is a finite set. In this case if x is a vector i...
View Full Document

{[ snackBarMessage ]}