Lumped kinetics of many irreversible bimolecular reactions

Pergamon

Chemical Engineering

LUMPED

Corporate

Scimce. Vol. 49, No. 5, pp. 781-795, 1994 Copyrighr 0 1994 Elssvier Scisncc Ltd Print& in Great Britain. All rights mscrvcd ow%2509/94 S6.M + 0.m

KINETICS OF MANY IRREVERSIBLE BIMOLECULAR REACTIONS

B. S. WHITE, T. C. HO’ and H. Y. LI* Research Labs, Exxon Research and Engineering Co., Annandale:NJ (Received for publication

18 August

08801, U.S.A.

1993)

Abstract-We generalize previous results that the lumped kinetics of many symmetric irreversible bimolecular reactions at large times is usually of second order. However, this behavior can be violated in special situations, for which we give conditions. We show how the asymptotic second-order rate constant depends on the feed composition in a way that allows only a finite number of possibilities, which can be expressed in terms of the underlying rate constants. Each of these possibilities usually corresponds to a unique asymptotic mixture composition, but a continuum of asymptotic compositions can also exist. The system exhibits a long-term memory when the asymptotic behavior depends on the feed composition. Stability theory is used to develop an algorithm for determining observable asymptotic compositions and rate constants. Numerical examples are provided for binary and ternary mixtures to illustrate some structural features of general bimolecular systems. Implications of the present results for experimental kinetic&s and process modelers are discussed.

INTRODUCMON This paper is a continuation of our previous work cm kinetic lumping, an area which has fortunately at-

tracted Professor G. Astarita’s attention in recent years. Professor Astarita has had a powerful influence on this field. In particular, it is he who gave the first treatment of the aggregate behavior of many interacting reactions, which is the subject of this paper. We dedicate this paper to him on his 60th birthday. Our objective in this study was to understand the aggregate behavior of a finite number of bimolecular reactions. Reactions of this type are commonplace (e.g. coke formation, pyrolysis, polymerization/ oligomerization, disproportionation, alkylation, etc.). In many practical situations, it is the aggregate behavior, not the individual behavior, that matters. For example, a central concern in catalytic cracking is the total production of coke whose primary source comes from the bimolecular reactions between olefins and aromatics. Another example is that in kinetics studies, what one measures is the total reaction rate over an ensemble of energetically heterogeneous catalytic sites. Recently, several workers have determined the lumped kinetics of many symmetric irreversible bimolecular reactions based on some simplifying assumptions. In one group of studies (Astarita and Ocone, 1988, 1992; Astarita, 1989; Chou and Ho, 1989; Astarita and Nigam, 1989; Ho, 1991), the assumptions were that the coupling between the reactions is uniform and that the number of reactants is so large that the system can practically be treated as a continuum (Aris, 1968). This approach generally requires a priori specification of the distributions of feed composition and reactivity. In another group of ?Authorto whom correspondence

should be addressed. *Current address: China Technical Consultants, Inc.. P.O. Box 88, Toufen, Miaoli, Taiwan 35102, Republic of China. 781

studies (Li and Ho, 1991a; Scaramella et al., 1991), the assumption was that the coupling is weakly nonuniform. For an overview of recent developments in kinetic lumping, the reader is referred to the books edited by Astarita and Sandier (1991) and Sapre and Krambeck (1991). It is surely desirable to be able to treat the general case without specification as to the mode of coupling and the number of reactants. Li and Ho (1991b) have made an initial attempt to address the general problem. They showed that when all reaction rates are nonzero, the lumped kinetics at large times is second order. In special cases they expressed asymptotic second-order rate constants as functions of the underlying rate constants. Their analysis revealed some interesting behavior even for binary (obviously, here the word “binary” refers to the number of reactants) mixtures and raised some unsettled questions. This prompted us to undertake the present study to understand better the structural features of general bimolecular reaction systems. Our interest primarily lies in the high-conversion regime (or long reaction time) so some rather general statements regarding the asymptotic behavior of the system can be made, as was done for continuous mixtures (Ho et al., 1990). Practical applications of the asymptotic theory have been discussed by Ho (1991). Specifically, the main results we obtain here are: (1) In most cases the asymptotic, lumped kinetics is second order, but exceptions do exist. Conditions under which such exceptional cases arise are determined. (2) When the asymptotic kinetics is second order, the corresponding lumped rate constant may or may not be unique. It may depend on the feed composition. (3) In most cases, each asymptotic rate constant corresponds to a unique steady-state composition.

782

B. S. WHITE et al.

However, under special conditions, a continuum of steady-state compositions can exist, but the number of lumped rate constants is always finite. (4) The lumped rate constant can be c@ixted by the fast reactions of the system. (5) When the lumped rate constant is a complex function of the underlying rate constants, its temperature dependence can still be well approximated by the Arrhenius equation under most practical conditions.

convenient to work with reactant mole fractions defined as

The remainder of the paper is organized as follows. We first derive evolution equations for the overall lump and the individual mole fractions. We then discuss the structural features of the system. Next, we present in some detail results for binary mixtures which serve as a prototype for more complex mixtures. Implications for experimental kinetic&s are discussed where appropriate. Following this, the linear stability of possible steady-state compositions is examined. Finally, we demonstrate that a system with as few as three species can exhibit a rich variety of behaviors, consistent with the conditions given above. The appendices contain some supplementary information.

with

LUMPED

KINETICS

dci - = - ,ti k+ic,, dt

c,(O) = Cl/2 0.

(I)

Here k,, are the bimolecular rate constants. The subscript f refers to the feed mixture. Note that s(t) 2 0 is preserved by eq. (1). The reactions are irreversible and symmetric so that

k, 2 0

(21

k, = kii.

(3)

When k, = k,k,, the coupling is called uniform (Astarita and Ocone, 19881,which is a special case of the pseudomonomolecular system discussed by Wei and Prater (1962). When k, = L + sij with Isill Q E, the coupling is then weakly nonuniform (Li and Ho, 1991a). The total reactant concentration, or the lump, is defined as N cj.

i=,

Hence dC -dt = - ? 5 kvcicj, *=, ,=i

C(t)

C(0) = ,$i

c~I

=

~0.

> 0.

The obvious advantage is that since c xi = 1, at least one of the xrs is positive even as t 4 m . Equation (5) then becomes (7)

a(t) w xrKx 2 0

(8)

where K is the symmetric matrix composed of k, and x the vector composed of xi_ Equation (7) can be thought of as a pseudo second-order kinetics with an instantaneous rate coefficient a(t) which in fact is a quadratic mole fraction average of the k,p at any time. This simple “mixing rule” (a term borrowed from thermodynamics) is hardly surprising because the individual kinetics are second order. Equation (7) implies that C is strictly decreasing unless cz= 0. Upon integration, C(t) can be cast in a form frequently used for second-order kinetics 1 ---= C(r)

AT ALL TIMES

Consider the following class of bimolecular kinetics in an isothermal system (batch- or plug-flow reactors) having N different types of reactants. In dimensionless form, the governing equations in terms of molar concentrations are

c= 2

x,(t) = *

J

1

r

0

Co

cr(t’)dt’.

(9)

As will be shown later, a(t) decreases as t increases and eventually converges to 8 nonnegative constant 8. So the plot of (l/C) - l/C, vs t is concave downward and approaches a straight line with slope equal to d at large times. Numerical examples of this will be given later. We mention in passing that for very early times, C decays at an asymptotic zero-order rate with rate constant a,, i.e. C(t) -

co -

Qo'

=

CO-

x C ki,xi,xu i i

>

as t 4 0. EVOLUTION

t (10)

OF MOLE FRACTIONS

Differentiation of eq. (6) and use of eqs (1) and (3) yields .

(11)

Under the action of the two competing terms on the right-hand side, Xi can increase or decrease while the corresponding ci always decreases. Note that xr 2 0 is preserved by eq. (11). Also, c x1 = 1 is preserved at all times because

(5)

As stated earlier, our main objective is to determine the long-time behavior of C. To put the problem into a form that reveals what goes on at large t, it is

To eliminate C, we introduce the warped time r

t=Jf

C(f) dt’.

0

Lumped kineticsof many irreversiblebimolecularreactions We now show that r(t) is strictly monotone increasing and maps [O, co] onto [0, 031 so the behavior of x at large t can be deduced from its behavior at large 7. That 0 d k, c co and 0 < x1 d 1 implies that there must exist a positive number o such that a(t) < u x a,. From eq. (7) d

1

zc 0

That c xi = 1 reduces the above equation to da -= d7

- 2PH

where N P(X) = c Xl(Yi - =)2. *=1

(23)

Thus p B 0, and eq. (22) implies that a[~(@] increasing i.e.

Therefore,

Thus, C > 0 for finite t. From eq. (13), dr/dt = C z=-0, so the mapping from t to 7 is strictly monotone and is therefore one-to-one. Putting inequality (15) into eq. (13) yields

= i In (1 + &at). d

(16)

So as 2+ co, T 4 a3 at least logarithmically (more on this later). Also, z(O) = 0, so the r-to-t mapping is onto. From eqs (11) and (13), we obtain the following autonomous system:

=

(17)

gitx)

7 =

0,

xi

=

xip

A possible steady state or critical point j2 of eq. (17) is given by setting gi = 0, which yields For each i, either fi = 0 or (KzX)~= frKf

= 8. (18)

In most cases there should be a discrete set of at most (2N - 1) isolated critical points, or possible steady states, corresponding to whether Xi is or is not zero, for each i. The permissible number of steady states may be less than (2N - 1) due to the constraints 0 d xi d 1 and c xi = 1. Some of the permissible steady states may not be stable and therefore are not physically observable. A continuum of steady states may also exist, as will be. discussed later.

TIME EVOLUTION

y=KElx

-

xi (Yi

The equality in eq. (24) holds if and only if p = 0, which from eq. (23) occurs if and only if x is a steady state, eq. (18). Another important consequence is that

IdXV = F xftn- aI2G F

xdy, -

a)l

*

p(x).

Because a(7) 3 0 and decreases along the trajectory x(t), a converges to a nonnegative value B as T + 00. Since then p -+ 0, eq. (25) implies that the limit set of any trajectory consists only of steady states 1. Thus, if there are only a finite number of steady states, x(t) converges to one of them. In the general case we must have Z = Y?Kf for any E in the limit set. Two additional consequences of eq. (24) are worth mentioning here. First, the decreasing a indicates a dissipative process. There are no limit cycles because x cannot return to the same point as time proceeds. Also, x will not spiral around j2, as will be shown in the section on linear stability. The function V(x) = b(x) - B can be regarded as an “excess free energy” which drives the decay process. Second, the instantaneous order of the lumped kinetics n, defined as

n_dH14>1 d In C

(26)

is greater or equal to two because n=2+i($)($)-’

ASYMPTOTIC

so that CL= xry and eq. (17) takes the form dr

(24)

3 2.

(27)

OF 01

We introduce the variable

-=

is non-

da - < 0. dr

(15)

-

al*

Note that a is a scalar function of time through the trajectory x. It can be shown after some algebra that the time derivative of a along x is g=

783

-2[~x,,?-(~xiY#~].

(21)

SECOND-ORDER

KINETICS

From the foregoing development we see that the decay of C obeys an asymptotic second-order kinetics with a rate constant & if %#O. A necessary and s&lcient condition that E w 0 for all feed compositions is that k, > 0 for every i, i.e. all self-reactions must be present. This can be proved as follows. First, suppose that B = 0. Then k,.-Z,.T, = 0 for every i and j. In particular, k& = 0 for every i. Since kli # 0, St must be zero for every i, a contradiction to the requirement c x1 = 1. Conversely, suppose that kii = 0 for some i. Then the constant trajectory consisting of the steady state % with fi = 1, %, = 0 for j# i, has

784

B. S. WHITE

B = %‘K% = kii = 0. A more complete characterization of solutions with E = 0 will be given later. To summarize, regardless of the feed composition, as long as kii r 0 for every i, the decay of all reactants as a whole obeys the following asymptotic second-order kinetics: dC dt-

- 6C2

and 8>0

as t+oO.

(28)

If kii = 0 for some values of i, eq. (28) may still be satisfied but E > 0 depends on the feed composition. The question now is how to find 8. STEADY-STATE

COMPOSITIONS

AND ASYMPTOTIC

RATE

CONSTANT

Here we derive general expressions for Z and Cr.At each steady state, some fis may be zero, with the remaining Xis being positive. There must be at least one Xi > 0. So essentially we are dealing with two submixtures. One contains relatively reactive species which become exhausted as t + 00 (Xi = 0). Another contains relatively refractory species which survive the reaction in the sense that their Xi > 0. To distinguish these two subsystems, we introduce the following notation: IV,: S:

&: K,:

e:

the number of surviving species whose Xi > 0, N, d N; a set of N, indices chosen from the set I = {1,2,. . . , N} and corresponding to the surviving species; an N,-dimensional composition vector whose components are positive; an N, x N s submatrix of K obtained by deleting all rows and columns whose indices are not included in S; an N.-dimensional vector of all ones, e = (1, 1, . , lp-.

Obviously, K, is symmetric. The set S spans all (2N - 1) possible nonempty subsets of I; each may give a possible steady state or a continuum of steady states. For a ternary mixture, I = { 1,2,3} and S can be {1,2,3), {1,2}, {1,3}, (231, 111, 121, and (3). In what follows our interest is in the surviving species (signified by the subscript s). After “factoring out” all the exhausted species, a reordering of the surviving species and their associated rate constants is needed for accounting purposes. To avoid complexity in indicial notation, a simple way is to use new indices. Accordingly, we use the Greek indices p and v to enumerate only the surviving species. Thus, p and v both range over { 1,2, , . . , N,}. (The indexes i and j are used to enumerate all species, whether surviving or exhausted.) Take N = 4 and N, = 2 as an example. Suppose that S = {2,4} (i.e. species 2 and 4 survive), then the first and third rows and columns are deleted from K. And K, and X, take the forms

et al.

That is, (K,h, = kzz, (K,h2 = kz4, (%,), = Ez, etc. Similarly, the exhausted species need to be reordered as well, which will be done in the section on steadystate stability. Since Zi = 0 for i 6 S, for a steady state t 2 = x=Kx = x;K 52 I.

(30)

The equation satisfied by Es is K,%, = (%fK.%,)e = 5e

(31)

with the constraint eTf, = 1. To solve 2, and Or,there are two subcases, depending on the invertibility of K,. Invertible K, For nonsingular K,, 9, can be directly solved from eq. (31) i.e. ji, = hK,‘e

(32)

where K; 1 is the inverse of K,. For FL.to be a valid steady state, we require (K,‘e),,

> 0

for all p.

(33)

Using er%, = 1 and eq. (32) gives 1 or a=1

5

?(I(;‘),.

i[ Id=, v=l

1

, p,v=l,2

,...,

N,

(35)

which is a generalization of the result of Li and Ho (1991b) for binary mixtures. To sum up, when K, is inuertible, the procedure for determining E and P, isfirst to check if (K; ’ e), 2- Ofor 011p and then compute ii and ?ts according to eqs (34) (or 35) and (32), respectively. There is at most one possible 2. for a given choice of S. Note that E can be used to calculate the asymptotic rate constant for the individual surviving species. Since cV (KS)PY(AS)Y = E, we obtain from eqs (1) and (6) that

dc,

dt-

ast+oo

~=1,2,...,N,.

(36)

This says that each surviving species p also decays asymptotically at a second-order rate with rate constunt L%/(&),. Thus, after the reactive species become exhausted, the system is described by simpler, uncoupled equations since we are dealing with a relatively homogeneous group of species. Li and Ho (1991b) have shown that each of the reactive species (Zi = 0, i# S) gets eliminated asymptotically at a power-law rate with an order less than two. We remark here that after a long time C and ci decay to zero in such a “balanced” manner that the mixture’s composition remains time invariant. In this sense, the system appears to reach an apparent “equilibrium”. Noninvertible K, The matrix K, may not be of full rank. Physically, one could think of a mixture containing a discriminat-

Lumped kinetics of many irreversiblebimolecular reactions ing species which can tell the difference between, say, two isomers. Once this species becomes exhausted, the mixture’s behavior is governed by the submatrix K. which is nearly singular due to the presence of the two isomers of nearly equal reactivities. Systems of this type must have constraints (or conservation laws), which are discussed in Appendix A. Here solutions of eq. (31) depend on the Fredholm alternative [e.g. see Griffel(198l)l. That is, a solution %, exists if and only if e is orthogonal to any solution of the corresponding homogeneous equation K,w = 0 (w is therefore in the null space of K,). In other words, solutions exist if and only if N. eTw = C wP = 0 L=,

(37)

where w,, are the components of the vector w. If this condition is met, the general solution f, takes the form x, = It, + /3w

(K, ‘e), + j?w,, > 0

w2

for all p

(39)

which imposes a constraint on the permissible values of p = a/?. Although in this case there are possibly infinitely many steady states, all give the same asymptotic rate is because d = (erK; ‘e)-l constant 8. This = [e’(K; le + pw)] - ’ by virtue of eTw = 0. Thus, irrespective of invertibility, there are ut most (2N - 1) possible values of 8, corresponding to the choice of S. Some numerical examples of the noninvertible case follow.

=

1 -

2w,,

WI

=

(%)a.

(42)

Thus, there exists a line segment of steady states. As will be shown later (Fig. lo), this steady-state line is asymptotically unstable. We can use the above example to provide a simple geometric interpretation of eqs (37) and (38). Shown in Fig. 1 is the triangle plane defining the composition space. The vector e is normal to this plane. Since both 2, and II, are composition vectors, they terminate at some points on the reaction triangle. So w must lie entirely on the reaction plane, implying that w is orthogonal to e [eq. (37)]. Example 2 This example illustrates that f, may not exist because inequality (39) cannot be satisfied. Consider the following K, matrix of rank two:

(38)

where B is an arbitrary scalar, %. any particular solution, and w any solution of the homogeneous equation K,w = 0. If K;’ is interpreted as any inverse of K, (restricted to the orthogonal complement of its null space), then the formulas are the same as in the invertible case, i.e. *, = SK; le and B = (e’K;‘e)-‘. Again, before using these formulas, one must make sure that the following inequality is satisfied:

785

then

K, =

1 312 [ 2

312 2 5/2

2 5/2 3

1 .

(43)

Here w = [1, - 2, l]r and a particular solution is = [ - 2,2,0]‘. No B can be found to satisfy inequality (39). Since erK;‘e = 0, E cannot be wmputed from eq. (34). Solution of the form eq. (38) does not exist. Using the stability criteria developed later, one can determine that this system has only one stable steady state fl = 1.

K;‘e

Special cases of noninvertible K, 6 = 0. A special case of noninvertible K, occurs when & = 0. Obviously, the long-time asymptotic kinetics will not be second order. A simple example of this

Example 1 This example gives rise to a continuum of steady states. Let K, be 1

K,=

2 3 1

2 2 2

1

3 2 1

(40)

which has a rank of two. It can be found that w = [l, - 2, l]‘, a particular solution is K; le = [O, l/2,0]=, and B = (eTK; ‘e)- ’ = 2. These results lead to

+]+++[l+]

where 0 < /!l < l/2. The steady-state

(41)

composition

is

Fig. 1. A geometric interpretation of eqs (37) and (38) for ternary mixtures. The ends of the composition vectors 1, and kz lie on the reaction triangle. w has no components along the normal to the reaction plane.

B. S.

186

WHITE et al.

is given later. A necessary and st@cient condition for E = 0 with limit set containing % having Xi > 0 for i E S is that K, = 0. To prove this, first let & = 0. Then if fi > 0 for all i E S, N.

& =

N.

c c

#I=1*=1

(K,),,,f,f,

= 0.

(44)

Since all terms in the sum are nonnegative, we have (K PI1” ) XcX Y = 0

for all JJ, Y.

(45)

But then Z,, > 0 for all p implies (K& = 0 for all p and v. So K, must be a zero matrix. Conversely, assume K, = 0, then K& = Ee = 0. This implies that d = 0. Note also that if K, = 0 then any Zs with (E.),, > 0 and e*Z, = 1 is a steady state.

reactivity differences defined as 61 = kll - kiz,

&, = k22 - klz.

(50)

The relative magnitudes of kl 1 vs kz2 are irrelevant. For the discussions to follow, we call a system robust if it has only one realizable steady-state composition and hence a unique &, whatever the feed composition. A nonrobust system will then have multiple a values, depending on the feed composition. That is, the system has a long memory. By putting x2 = 1 - xi, we obtain the governing equation dxi -=dT

x,(1 -

XI)C62

-

(4

+ S2hl.

(51)

It is convenient to introduce the new parameter Separable kinetics. In the case of uniform kinetics, K and hence KS is of rank one. The behaviors of this system and its continuum analog have been discussed elsewhere (Astarita and Ocone, 1988, 1992, Astarita, 1989; Aris, 1989; Chou and Ho, 1989; Li and Ho, 1991b). Here we examine the long-time behavior of the system based on the present theory. K, is of rank one and therefore can be written as K, = kk’,

or

k,, = k,k.

for some vector k with components K.m, = tie, we have kk%, = Ore.

(46) k, 3 0. Since (47)

When kT%, # 0,

k=Le.

k=%,

Hence, k, is constant generality, let k, = 6, Then K,%, = %e gives

for every cc. Without loss of so k,, = k,, for all p and v. k,,

=a

(49)

because eTx, = 1. Thus, when all kip are distinct, there can be only one surviving species at sufficiently large times. Li and Ho (1991b) reached the same conclusion by using a different approach. The behaviors of the individual species, whether surviving or exhausted, are discussed in Appendix B. Thus far our interest has mainly been in the determination of possible 5 and steady states for general bimolecular mixtures. In what follows the foregoing development is applied to binary mixtures, with a dual purpose. First, the binary system serves as a prototype and building block for larger mixtures. Practical implications of the results can be readily inferred. Second, a straightforward analysis can be made to illuminate the role of stability in determining physically realistic steady states under any given circumstances. BINARY MIXTURES

This section represents an extension of the result of Li and Ho (1991b). The system is governed by two

6* i a2/(6, + 6,).

(52)

There are three possible steady states: two terminal points, Z1 = 0, 1; and one interior point, f, = S*, which is permissible if 0 < 6* -Z 1. A stability analysis can be made for this one-dimensional problem by merely checking the sign of dxl/dr near the steady state in question. Table 1 summarizes the results. As can he seen, when a1 and fi2 are both positive (case I), the interior steady state Xl = b* exists and is stable, with the two terminal steady states being unstable. The final composition satisfies X16, = X,6, (somewhat reminiscent of the “principle of detailed balance” in reversible reaction systems). Both species coexist at ail times. From eq. (34), the overall asymptotic rate constant a: is calculated to be [see also Li and Ho (1991b)l (53) The slowest reaction is the “pairing up” of species 1 and 2. The supply of such pairs depends on the rates of the self-reactions. This is why B depends on kl 1 and kzz_ Based on a brief literature survey, we list in Table 2 examples of such systems for radical-radical reactions. Note that there are large uncertainties associated with the klz values reported here. When k,, is smaller than k,, and kz2, its measurement is very difficult. Apparently, more experimental work in this area is needed. If k,, and kzz can be determined from pure-component experiments. Then ktZ can be calculated from kI 1, kz2, and B according to eq. (53). In the limit of k12 40, eq. (53) reduces to ti = kIL klJ(kIt + kzz), which is the result of Dole et al. (1975) for analyzing the decay of total free radicals from an irradiated polymer. Here one deals with two independent second-order reactions. Neither of the two species dominates the long-time behavior of the system. (Note that this is not the case for two independent first-order reactions.) We shall come back to this point in the section on linear stability. Case I represents a robust system. This is not so for case II. When both species are inherently unreactive (6, < 0 and 6, < 0), the one with a sufficiently higher

Lumped kinetics of many irreversible bimolecular reactions

787

Table 2. Examples of bimolecular radical-radical reactions k,, x lo- I3 1.8’ CZ] 1.8’ [Z] 1.8’ [2] 0.03’ [Z] OS+ [X2]

Reactant

k‘i x lo-‘3

HCO

2.7 [I]

CHz

3.2 [27

CIH

1.99 [2]

CH,CO

3.5 [3]

HOa

0.12 [S]

CH,O

2.9 [S]

t-&H,

3.4 [6] 1.0 [7] 0.68 [S]

0.5 [14] 2.0 Cl51

CH&

= CH

3.4 [9]

CHs

2.5 [lo]

CH,C(CH,)=CH,

2.6 [11-j

Units of rate constants: cm3/mol/s, T = 298K. +Estimated value. *Recommended value. Cl1 Temps, F. and Wagner, H. G., 1984, Ber. Bunsenges Phys. Chem. 88,410. PI Tsang, W. and Hampson, R. F., 1986, J. Phys. Chem. Re/: Data 15, 1087. c31 Adachi, H., Basco, N. and James, D. G. L., 1981, Int. J. Chem. Kin&. 13, 1251. c41 Kurylo, M. J., Ouellette, P. A. and Lauder, A. H., 1986, J. Phys. Chem. 90,437. Heicklen, J., 1988, Adu. Photo&em. 14, 177. :z; Arthur, N., 1986, J. Chem. Sot. Faraday Trans. 82, 1057. c71 Adachi, H. and Basco, N., 1981, In&.J. Chem. Kinet. 13, 367. PI Anastasi, C. and Arthur, N. L., 1987, J. Chem. Sot. Faraday Trans. 83, 277. PI Alkemade, U. and Homann. K. H.. 1989, Z. Phys. Chem. ldl,t9. Cl01 Hippler, H., Luther,K.. Ravishankara,A. R. and Tree, J., 1984, Z. Phys. Chem. 141.1; and Arthur, N. L., 1986, J. Chem. Sac. Faraday Trans. 82, 331. Cl11 Bayakceken, F., Brophy, J. H., Fink, R. D. and Nicholas, J. E., 1973, J. Chem. Sot. Faraday Trans. 49, 228. Batt, L. and McCulloch, R. D., 1976. Inr. J. Chem. WI Kin& 8, 491. Chemistry (Edited by Cl31 Warnatz, J., 1984, in Combustion W. C. Gardiner, Jr.), p. 197. Springer, Berlin. Cl41 Wu, C. H. and Kern, K. D.. 1987, J. Phys. Chem. 91, 6291. 1151 Tsang. W., 1973, Int. J. Chem. Kinet. 5, 929.

will survive the reaction. As depicted in Fig. 2, the interior steady state is unstable: as t --c GQ, xl(t) -P 1 for any xl1 > 6*, and x1(t) -P 0 whenever x1/ -z S*. A practical implication of this result is that one can get a desired final composition by a judicious choice of the feed composition. Thus, there are two realizable terminal steady states and a, depending on the feed composition. Cases III and IV represent the “mixed bag” situation, i.e. one species is inherently reactive and the concentration

788

B. S. WHITE

t+-

et al.

t+-

.

s*

0

:

x,(t) Fig. 2. Instabilityof interior steadystate for case II.

other is not. Obviously, the latter will dominate the system’s long-time behavior and the system is robust. The limiting case where all reactions have the same rates is shown as case V in Table 1. There is no motion at all (dx/dr = 0 for all 7), and any feed composition is a neutrally stable steady state, i.e. %, = xl1 and f2 = x2/. Temperature

1.75

1.65

1.55

10-3/T Fig. 3. Arrhenius plots for k, i, k,z, k,, and B. A,, = ICI-~, A,, = 10’. A,, = 1.0; E,, = 10 kcal/mol, El2 E 30 kcal/ mol, E,, = 20kcal/mol.

dependence

For case I, E.is a complex function of the k,,s [eq. (53)]. Let k,, = A,, exp (- E*,/RT), where A, and Eij are the preexponential factor and activation energy, respectively. It is easy to show that d has the desired property dE/d(l/T) < 0. However, there is no fundamental reason why B in general should follow the Arrhenius behavior unless by chance all the individual rate constants have the same activation energy. It turns out that in most practical situations, the temperature dependence of d is well approximated by the Arrhenius equation. This is becuase RT/E is typically very small (e.g. RT/E = 0.045 for E = 25 kcal/mol and T = 300°C). It then follows that El1 i Et2 and E22 -z El2 as long as 6, >O and 6* > 0. With no loss of generality, let El, -z Ez2 < EIZ. so that klz Q kz2 < kll. Then it can be seen from eq. (53) that d N kzz, so the activation energy for B is approximately Ez2. An example of this is shown in Fig. 3. Some numerical examples Figure 4 shows (l/C - l/C,) vs t plots for a binary mixture (k, 1 = 3, k12 = 1, and kz2 = 2; case I) with three different feed compositions. Here d = 5/3 = 1.67. Curve A is essentially a straight line since the feed composition is not far away from the steady-state composition (X1 = l/3, X2 = 2/3).Curves B and C show slight curvature at small times. If one fits the C(t) curves in Fig. 4 by an empirical nth-order kinetic model, dC/dt = - ICC”, then n 3 2 [eq. (27)]. Table 3 lists the results of such a fit for a reaction time up to t = 5, corresponding to total conversions of 89.3, 90.0, and 91.7% for curves A, B, and C, respectively. As can be seen, the overall behavior of the mixture, for practical purposes, may very well be modeled by second-order kinetics, Had we used a longer reaction time, n would be closer to two. Note also that here the values of k, span a factor of three.

40 rl I

s 0

4

12 t

Fig. 4. Second-orderkinetic plots for three different feed compositions.Co = 1, k,, = 3, k,, = 1, k,, = 2. Curve A: X1, = 0.4,xzJ = 0.6;curve B: x - 0.7,x2, = 0.3;curve C: XII = 0.9,Xi; = 0.1.

Table 3. Fitting of curves A, B, and C in Fig. 4 by power-lawmodel

n

K

A

B

C

2.00 1.678

2.08 2.04

2.11 2.66

provide some insight. In the limit of vanishing k, I and k 22, we have for k12 = 1

dc,

dt=

-

c,c2.

dc, -&

=

-

ClC2.

(54)

Since kll = k22 = 0, h = 0, whatever the feed composition. Without loss of generality, let c2f = cl1 + y and y b 0. Then we have the conservation law cl(t) = c,(t) + y for all t. The y = 0 case is trivial because dc, /dt = - c: and c1 = cz. For y z 0, it is easy to show that

-

Non-second-order kinetics Before ending the discussions on binary mixtures, we show an example of E = 0 which admits an analytical treatment. While the example seems trivial, it does

20

cl(t) =

7,

IYe 1 - qesY*

where q = c~~/c~~.As t + co, cl(t) drops to zero exponentially but cl(t) + y, i.e..the species which is initially

Lumped kinetics of many irreversible bimolecular reactions

present in excess of the stoichiometric amount remains unreacted even as c 4 co. The overall behavior at 011 times becomes for all t.

(56)

Since C + y as t + co, the above kinetics asymptotically reduces to dC dt m - yC + 7’

for large t.

In other words, the asymptotic kinetics is not second order; rather, it is first order with a constant generation term. Obviously, the presence of y in eq. (57) signifies a long-term memory effect. Summarizing, the above results for binary mixtures are illustrative of the structural features of more complex mixtures. The interactions between the reacting species are such that the system can exhibit a iongterm memory. It is clear that stability plays a central role in determining the system behavior. Accordingly, in what follows we establish stability criteria for the general case. But before doing so, we describe another property of R. which is actually related to system stability. The property is this: t, represents an extremal point of the quadratic function jirK,t, where % is an arbitrary vector constrained by e% = 1.

EXTREMA

OF frK If

Using the method of Lagrange multipliers, we seek the extrema of the function I defined as I(f) = fTK$

+ cer%

(case I), then E,,,= (kllkzz )

789 - kT2)/(kll

+ kz2 - 2k12)

k,z.

It follows from the above that C for any feed composition is bounded by two second-order kinetics with rate constants tiM and a,,,, i.e. 1 l/Co + i&t

=s c(t) <

1 l/C, f a,t’

(62)

The lower bound fohows from eq. (14) with u = &,. The upper bound follows by a similar argument, using a,,,2 8. The upper bound may be used to construct a simple one-parameter model for C at any time (Li and Ho, 1991b). Note also that inequality (62) implies that, for E, # 0, r(t) goes to infinity logarithmically for t + a. LINEAR

STABILITY

OF THE STEADY STATES

Since the limit set of the trajectory x(z) consists only of steady states, x(7) will eventually pass close to some steady state % satisfying eq. (18). The linear stability of % can then be analyzed by studying the Jacobian matrix asi/dxj evaluated at Z. From eq. (17), we have 67i -=

-

S,(j$

-

b!)

-

5Zi(kij -

2jj)

axj

(63)

where 6, is the Kronecker delta and 9 = Kf. Recall that the indices i and j enumerate all species, whether surviving or exhausted. Let zi = Xi - xi be a small perturbation from Xi. The set of equations governing linear stability of the steady state is then i=l,2

(58)

,...,

N.

W)

with 5 being the multiplier. Setting the derivative a//&% As a result of mass balance, the perturbation z must equal to zero gives satisfy K$=

-;e

which reproduces eq. (31) by letting c = - 2~. Thus, all X,s are extremal points of iTK.ii within the whole composition space, whether or not W satisfies eq. (17). Further, it can be shown that if K, is positive definite, then j2, minimizes iirK%. Since x’Kx decreases along any trajectory x in the composition space, the %. giving a local (global) minimum of grK.?Z must be locally (globally) stable since at least the neighboring trajectories move toward this steady state. By the same reasoning, a local maximum occurs at a globally unstable %., since all trajectories move away from it. To distinguish these two extrema, let tlM = max %JK,%, s,P,

(60)

E& = min ffK,f.. s,%

(61)

Take binary mixtures as an example. When and 8z < 0 (case II in Table l), the interior state is unstable and CM = tkrt&z - k:z)/(ki, - 2k12). On the other hand, if 6, > 0 and

eTx = eTx - eTx = 1 - 1 = 0.

(59)

6, < 0 steady + kzz 6, > 0

(65)

That is, z must be perpendicular to e at all times. Figure 1 can be used to provide a geometric interpretation of this situation for N = 3 if Bw, P,, and 2. are replaced by z, x, and f, respectively. The invariance property given by eq. (65) will be used later. Recall from eq. (20) that yi = & = X’y for i E S and Xi = 0 for i $ S. Substituting eq. (63) into eq. (64) yields dz; -=-xi dr

~ (k, - 2yj)Zj

[ j=i

dzf -= ds

- (ji - B)zi

1

for i E S

for i$S.

(66) (67)

Note that the summation in eq. (66) is over all species. It is useful to break the summation into two parts: one contributed by the surviving species, the other by the exhausted species. To do so, summing eq. (66) over all ieS gives $&zI=j.Tz.

(68)

Similarly, it follows from eq. (67) (upon summation)

B. S. WHITE et al.

790

(73) can be written in a matrix form as

that

&[:I] =u[:I]

(69) Addition of the above two equations gives

(7%

where U is a block matrix of the form

& (eTz) = ti(eTz)

(70)

(80)

which confirms that eTz = 0 [eq. (65) J for all r if eTz = 0 at t = 0. This leads to the following relation:

The stability of the above N-dimensional system is determined by the eigenvalues of U. In this connection, it turns out that U has a nice property: rZis an eigenoalue of U if and only if ;I is either an eigenvalue of A or an eigenuaIue of D (or both). The proof is as follows. Suppose Izis an eigenvalue of U. Then there exist z+ and z- such that

(71) Therefore,

c

IES

jjZj

c azj = -

=

c

iiT

jrS

(72)

zj.

AZ+ + Bz- = lz+

j+S

With eq. (72), eq. (66) can be rewritten as

dzi

iF=

-

1 fik;jZj + g [%i(2yi - 2a jsS

Dzkij)lzj

for i E s.

The stability of jz is thus governed To put them in a more compact notation is introduced. For ~1,Y = be a vector with components rz + =zr, ZP

(73)

by eqs (67) and (73). form, the following 1, 2, . . . , N,, let z+ defined as (74)

icS

and A be. an N, x N, matrix with components a,,” defined as 4,

(75)

= - (%),(Ks),v.

Evidently, z+ and A are associated with the surviving species, and the indexes p and v are one-to-one functions of i or j. For the exhausted species, we deal with complement of S, denoted by SC,i.e. SC= {1,2, . . . , N}\S. Thus, the total number of elements in S is N - N,. Again for bookkeeping purposes, we use the Greek indexes 0 and 4 to enumerate these elements. We then define a vector z- with components z; defined as z;

= zr

for 0 = 1,2, . . . , (N - NJ and i E S’

(76)

and an (N - N,) x (N - N,) diagonal matrix D with elements de, d,

= - (j$ - EC)&,,

0, q5 = 1,2, . . . , (N - NJ and i E SC.

(77)

Again, 0 and 4 are one-to-one functions of i or j. The coupling between the surviving and exhausted species is characterized by an N, x (N - N,) matrix B whose components are bpB = Xi(2yj - 2a( - ki,) P =1,2

,...,

N,, i E S,

e-i,2

,...,

(78) (N-NJ

j E SC.

With the above notation and partition, eqs (67) and

(81)

= lz-.

W-4

If z- # 0, then Dz- = dz-, implying that I is an eigenvalue of D. If z- = 0, then AZ+ = AZ+, implying that I is an eigenvalue of A. Conversely, consider the case where rl is either an eigenvalue of D or of A. First, let us suppose A is an eigenvalue of A, then there exists a vector v such that Av = Iv. But then [v’, OIT is an eigenvector of W with eigenvalue 13. Next, suppose 1 is an eigenvalue of D, but not of A. So DV = .it and (A - 2) is invertible. But then

A [ 0

B D’

? I.

I[

-(A--I)-‘Bf

(83)

Thus, 1 is an eigenvalue [( - A - lI)-‘BP)T, CT-jr.

of U with eigenvector

Stability criteria Since D is a diagonal matrix, the stability of the exhausted species can be readily determined. The stability of the surviving species depends on the sign of the eigenvalues of A. This is discussed as follows. One can show that A has an eigenvalue - & with eigenvector jz,. Obviously, this special eigenmode is stable. To find the sign of other eigenvalues, it is often useful to “derive” a symmetric matrix from A or AT (A and AT have the same eigenvalues). To this end, we define a symmetric matrix Q with components then Q has the same 4rv = - m(K,),,,m; eigenvalue A as AT with &v’ as the eigenvector. Here v’ is the eigenvector of AT, with its components u\ satisfying V$r (K&(&L

u: = 20;.

(84)

The symmetry of Q implies that all its eigenvalues, and hence those of A, are real. These eigenvalues are negative if and only if Q is negative definite. This is true if and only if K, is positive definite. The preceding results lead to the following stability criteria based on the eigenvalues of D and A.

Lumped kinetics of many irreversible bimolecular reactions

An isolated steady state R is asymptotically (Coddington and Levinson, 1955) if (i) j$ > c

fir every i $ S, and

(ii) K, is positive

stable

(85)

definite.

(86)

Condition (ii) guarantees that all eigenvalues of A are negative. Of course, the eigenvalues of K, are not the same as those of Q,. The positive definiteness of K, does not imply that of K. As mentioned before, x will not spiral around t because the eigenvalues of A and D are real. In the case of independent reactions (i.e. k,, = 0 for i$i), it can be shown that condition (i) is always violated if S is nonempty. So the physically observable case is where 2, > 0 for all i. Instability is implied if either (1) yr < & for some i$S, or (2) a’%~ < 0 for some vector a. The above stability criterion for the surviving species [eq. (86)] can also be defined in terms of the transition probability matrix of an N&ate Markov chain. The ma&x is defined as P = - (l/@A’, with its elements P,,y having the following properties: (87) Thus, stability is implied if all eigenvalues of P are positive. The properties of the eigenvalues of P can be found in standard textbooks on stochastic processes (e.g. Karlin, 1969). For instance, the largest eigenvalue of P is unity with eigenvector e. The absolute values of all other eigenvalues are less than unity. Note that the disturbances, which decay exponentially in 7, usually (for B # 0) decay in powers oft based on the logarithmic estimate of r(t). However, as discussed earlier, z must be confined to the hyperplane c zi = 0, where zi are the components of z. It remains to be shown that, if A or D have a positive eigenvalue, such an unstable eigenmode can indeed be excited to cause an instability within the hyperplane by some initial z” satisfying C zp = 0. Instability with invariant eTz = 0 We first suppose that I is an eigenvalue of A and 1 > 0, i.e. Av = Iv. Since A = - &P’ and Pe = e, it follows that (A + B)eTv = 0.

(88)

But I # - L < 0, so erv = 0 or C ui = 0.

(89)

Thus, we may take anlnitial disturbance z” = [v, O]=, and the disturbance will grow exponentially as _ _ V z= eAv (90)

11 0

*

791

is an eigenvector of U. So the disturbance z = z0 exp (17) grows exponentially. Hence, we need only to show that z” satisfies c z$’ = 0. Since D is a diagonal matrix and D1= ,G, we can take ? to be any coordinate vector, say e,, so ‘& Ei = 1. Then 1= - (j, - &) with 2E S’. Thus, as long as we can show er[ - (A - II)-‘BS] = - 1, then Cizp = 0 will be automatically satisfied. Let q = (A - II)-‘BT, then our problem is to show erq = 1. Using A = - BP’ and e’A = - tier yields eTBq = - (1 + h)eTq. Note that, if tYojer = C& follows:

we can compute e’B+

(92) as

eTM = c (Bi?), = c b,, Ir Ir = 1 Xi(2jj - 2& - k,)

i

i e S, j E S

(93)

= 2(Y, - a) - y, = - (1 + E). This, coupled with eq. (92), implies that eTq = 1 because R > 0 and Or2 0. Stability of continuous steady state In the general case the continuum of steady states lies on a hyperplane. Here we show how a stable steady state iz,, when perturbed off the plane, will eventually return to the plane, either at the original point 2, or at a point f:, provided that Ji > B for every i$S. From eqs (79) and (80), we have dz+ p= dr

AZ+ + Bz-

dz~ = Dz-. dr

(94) (95)

Upon integration, eq. (95) gives z- (7) = eD’z- (0).

(96)

Thus, z- + 0 as t + ao. That is, the perturbation returns to the hyperplane of the surviving species. The behavior of z* can be analyzed by projecting it onto the eigenspaces of A. We do so only when the eigenvectors span N. dimensions, and no eigenvalue of A is also an eigenvalue of D. Suppose A is an eigenvalue of A but not of D and 1 d 0. Let v be a left eigenvector of A i.e. vTA = Avr. Then we need to show that vTz+ converges for all such v euen zY~ = 0. Multiplying eq. (94) by v’.yields .$“TZ’)

= lVTZf + v=Bz-.

(97)

The solution of the above equation is

We next suppose that Iz > 0 is an eigenvalue of D, but not of A. Then

zo =

-

(A - iLI)-‘W G

1

(98) The integral in the parentheses can be evaluated as

B.

792

s. WEmE

et 01.

follows: r eAr

I

e(D- i’)s&

= [e”’

-

e”“] (D -

AI)-’

(99)

J. < 0

and

.o

0

which as T + cc approaches -(D-_I)-‘ifR=O.Thus,asr+a~ vrz+(z)40 ~~z+(~)+v=z+(O)

0

if

ifRc0 - v’B(D

-

x2

(1W AI)-‘z-(O)

if 1 = 0.

(101)

As can be seen from the above development, the exhausted species have a considerable stabilizing influence on the surviving species. NUMERICAL

EXAMPLES:

TERNARY

MIXTURES

Some numerical examples are given here to illustrate possible behaviors of ternary mixtures. The maximum number of isolated steady states is seven: three terminal ones at the vertices of the reaction triangle (cf. Fig. 1); three edge ones on the sides of the triangle; and an interior one inside the triangle. The behavior of the system should necessarily be more complex than that of binary systems. In the figures to follow the reaction trajectories are projected onto the x1-xz plane. Shown in Fig. 5 is a system with seven steady states; of them the only stable one is the interior steady state, with % = CO.364, 0.333, 0.3033r. The system is therefore robust. This system gives the appearance of a reversible reaction mixture in the sense that there exists an “equilibrium” composition for any feed composition. Figure 6 shows a situation where all three terminal steady states are observable. There are three subregions in the reaction triangle, within each of which an arbitrary feed composition converges to a terminal steady state. Hence three possible & can be realized. The mixture represented by Fig. 7 has a robust edge steady state lying on R1 = 0. This is consistent with the fact that here kz3 is smaller than any other k@

Xl Fig. 6. Phase-plane trajectories for k,, = k,, = k,, = 1, k,, = 3, k,3 = 4, k,, = 5. Seven steady states:(0) unstable;

(0)

stable. Here the stable steady states are [l, 0, OIT, [0, I, O]=, and [O,O, 13=.

0.0‘

x1

Fig. 7. Phase-plane trajectories for k,, = 4. k,, = 5, k,, = k,, = 3, k,, = 1, k,, = 2. Five steady states: (0) unstable; (0) stable. Here the stable steady state is co, 1/3,2/31T.

I.

x2 x2

0.0

Fig. 5. Phase-plane trajectories for k,, = 3, k,, = 1.5,

k,, =

4,

k,, = 1, k,, = 0.05, k,, = 5. Seven (0) unstable; (0) stable.

steady

states:

I .o

Fig. 8. Phase-plane trajectories for k,, = k,, = 2. k,, = 3, k,, = 4, k,, = 1, k,, = 5. Five steady states: (0) unstable; (0) stable. Here the stable steadystatesare [O, 0.8,0.2] Tand

c~.o,w.

Lumped kineticsof many irreversiblebimolecularreactions

However, the converse is not necessarily true: k13 can be the smallest rate constant, but this does not guarantee the robustness of an edge steady state lying on x1 = 0. An example of this is shown in Fig. 8. A related situation is shown in Fig. 9: here kli is not smaller than k,,, yet % = [l, 0, O]r is the only stable steady state. This conveys another message: an f = [l, O,O]r does not necessarily imply /cl1 is the smallest rate constant. Finally, we show examples of continuous steady state. Shown in Fig. 10 is an unstable continuum of steady states lying on fz = 1 - 2P1. The stable steady states are the two isolated terminal points: [l. 0, O]r and [0, 0, 11’. By contrast, Fig. 11 demonstrates a stable continuum of steady states. The system also has a stable isolated steady state [0, 1, OIT. As seen, different feed compositions will lead to different points on the line segment X, = 1 - 22, or the isolated steady state. The above examples clearly point to the importance of varying the feed composition in kinetics studies, including deleting some species in the feed. The

I.0

Fig.

11. Phase-ptane trajectories for k,, = k,, = 1.5, k,, = k,, = k,, = 1, k,, = OS. Infinite number of steady

states, denoted by the line segment which is stable: (0) stable;(0) unstablestates are [0, 0, llT and [l, 0, OIT.

long-time data can be used either to estimate kinetic constants or to sharpen the estimation. CONCLUDING REMARKS

have developed an asymptotic theory to describe the aggregate as well as individual behaviors of a finite number of irreversible bimolecular reactions. The theory is valid for all possible feed compositions and no restrictions are imposed on the number of reactions and on how the individual reactions are coupled. In some cases the behavior of the system as a whole has a long memory and in other cases it has no memory at all. Extension of the present treatment to more complex systems, such as those including unimolecular and/or reversible reactions, should be of both fundamental and practical value. We

I.

x2

0.0

x1

I .o

Fig. 9. Phase-plane trajectories for k,, = k,, = 1, k,, = k,, = 2, k,, = 3, k,, = 4. Four steady states:(0) unstable;(0) stable.

Fig. 10. Phase-planetrajectoriesfor k,, = k,, = 1, k,, = k,, = k,, = 2, k,, = 3. Infinitenumberof steady states,de-

noted by the line segmentwhich is unstable:(0) unstable; (0) stable steady states are [0, 0, 11’ and [l, 0. OJT. CES

49:5-o

Acknowledgment-H. Y. Li acknowledges Exxon summer internship.

the support

of an

NOTATION an arbitrary vector components of A, defined in eq. (75)

an N, x N, matrix preexponential factor of k, components of B, defined in eq. (78) an N, x (N - N,) matrix vector with elements ci concentration of ith species feed concentration of ith species total reactant concentration components of D, defined in eq. (77) an (N - NJ x (N - N,) diagonal matrix an N,-dimensional vector of all ones activation energy of k, function defined in eq. (17) [ = - X&G - @I function defined in eq. (Bl) index for enumerating all reacting species set of N integers (1,2,3, . . . , N) index for enumerating all reacting species

794 ii I k kiJ

K KS 1

n N N. PW :

B. S. WHITE et al.

mean rate constant minimum of k, for separable kinetics rate constant vector for separable kinetics bimolecular rate constant for species i and j bimolecular rate constant matrix rate constant matrix for surviving species function defined in eq. (58) power-law index total number of species number of surviving species for t -+ 0D components of matrix P transition probability matrix [ = - (lp)AT] matrix with comuonents

Lagrange multiplier, eq. (58) a ratio defined as c1,/c2/ index for enumerating the exhausted species rate constant for power-law kinetic model eigenvalues index for enumerating the surviving species in&x for enumerating the surviving species function defined in eq. (23) a positive constant such that a < a, eq. (14) warped time, eq. (13) eigenvalues of K, eq. (Al) index for enumerating the exhausted species

components of ma&x Q rank of K a set of N, indices chosen from I for surviving species complement of S time temperature eigenvector of K, eq. (Al) matrix defined in eq. (SO) eigenvectors eigenvectors eigenvectors excess Vree energy” ( = a - h) vector satisfying K,w = 0 reactant mole fraction of species i reactant mole fraction of species i in feed mole fraction vector mole fraction vector formed from surviving species steady-state reactant mole fraction vector (t+m) particular solution defined in eq. (38) vector defined as Kx y at steady state (t + 00) components of vector y ( = xj kijxj) vector composed of small perturbations (=f-x) perturbation vector associated with surviving species perturbation vector associated with exhausted species

Aris, R., 1968,holegomena to the rational analysis of systems of chemical reactions. II. Some addenda. Arch. Rational Mech. Anal. 27, 356. Aris, R., 1989, Reactions in continuous mixtures. A.1.Ch.E. J. 35, 539. Astarita, G., 1989, Lumping nonlinear kinetics: apparent overall order of reaction. A.1.Ch.E. J. 35, 529. Astarita, G. and Nigam, A., 1989, Lumping nonlinear kinetics in a CSTR. A.I.Ch.E. J. 35, 1927. Astarita, G. and Ckone, R., 1988, Lumping nonlinear kinetics. A.I.Ch.E. J. 34, 1299. Astarita, G. and Ocone, R., 1992, Chemical reaction engineering of complex mixtures. Chem. Engng Sci. 47, 2135. Astarita, G. and Sandler. S. I., 1991, Kinetic and Themwdynamic Lumping of Multicomponent Mixtures. Elsevier, Amsterdam. Chou, M. Y. and Ho, T. C., 1989, Lumping coupled nonlinear reactions in continuous mixtures. A.1.Ch.E. J. 35,533. Coddington, E. A. and Levinson, N., 1955, T?zeory of Ordinary Differential Equation. McGraw-Hill, New York. Dole, M., Hsu, C. S., Pate& V. M. and Patel, G. N., 1975. Kinetics of two simultaneous second-order reactions occurring in different zones. J. phys. Chem. 79, 2473. Griffel, D. H., 1981, Applied FunctioMI Analysis. Wiley, New York. Ho, T. C., 1991, A general expression for the collective behavior of a large number of reactions. Chem. Engng Sci. 46,281. Ho, T. C., White, B. S. and Hu, R., 1990, Lumped kinetics of many parallel nth-order reactions. A.1.Ch.E. J. 36, 685. Karlin, S.. 1969, A First Course in Stochastic Process. Academic Press, New York. Li, B. Z. and Ho, T. C., 199la, Lumping weakly nonuniform bimolecular reactions. Chem. Engng Sci. 46, 273. Li, B. Z. and Ho, T. C., 199lb, An analysis of lumping of bimolecular reactions, in Kinetic and Thermodynamic Lumping of Mulricomponent Mixtures (Edited by G. Astarita and S. I. Sandler). Elsevier, Amsterdam. Sapre, A. V. and Krambeck, F. J., 1991, Chemical Reactions in Complex Mixtures. Van Nostrand Reinhold, New York. Scaramella, R., Cicarelli, P. and Astarita, G., 1991, Continuous kinetics of bimolecular systems, in Kinetic and Thermodynamic Lumping of Mulricomponent Mixtures (Edited by G. Astarita and S. I. Sandier). Elsevier, Amsterdam. Wei, J. and Prater, C. D., 1962, The structure and analysis of complex reaction systems. Adv. Catal. 13, 203.

REFERENCES

4rv R s s t T ui U V i V’

V W Xi Xif X X* %

It

Y 7 Yi 2 2+

Z-

Greek letters a defined by a = xTKx, eq. (8) minimum EiK.% a, maximum of f I‘K .lf aM & asymptotic second-order rate constant asymptotic zero-order rate constant, eq. (10) a0 coefficient defined in eq. (38) coefficient ( = B/a) ; constant defined by clf = cl/ + y reactivity difference, eq. (50) :I reactivity difference, eq. (50) 62 6* defined in eq. (52) Kronecker delta (= 1 for i = j and 0 for sij i #i) small quantities, lsi,l + Z QJ

APPENDIX

A: CONSERVATION

LAWS

Suppose that K is of rank R; then it can be represented its eigenve-ctor q and eigenvalue ui as K-;

u,u,u:.

by

(Al)

i-1

ui = 0 for

i = R + I,. . . , N so that u:K = 0, i= R + 1,

.

, N.

WI

Lumped kinetics of many irreversible bimolecular reactions Equation (1) can be written in matrix form d In c -= dt

-Kc

h(t)=T;k,c,,exp(

(A3)

where e is the concentration vector composed of ci. Multiplying eq. (A3) by UT for i = R + 1, . , N yields $(tt:lne)=-(tt~K)c=O,

i=R+l....,N

(A4)

i=R+l,...,N

(A5)

i = R + 1,. . . , N

tW

or alternatively

h(t)-z,kc,,exp

where S’ is the

fi cp’* = constant, ,=a

-k,jIkdt’).

(B3)

Now consider the general case where there are a finite number of refractory species, all of which have a common rate constant k. At large times, the system is governed by these species, that is to say,

which in turn gives the following (N - R) constraints at all times: tt: In c = constant,

795

and

(-kj:hdr’)

set {i: kl = inf, k,}. Thus,

where (n& are the N components of ul.

APPENDIX

B: LARGE TIME 3EHAVIOR KINETICS

OF SEPARABLE

While this system ean be analyzed by the general theory discussed before, it is instructive to compare with the more detailed calculations, which are possible for this case. Specifically, we derive expressions for the long-time asymptotic behaviors of both the surviving and exhausted species: dcr _= dt

- kici C k,c, = - k‘e,h,

h = kTc.

(Bl)

Then c,=c,;exp(-k,J;hdt’)

(B2)

Upon integration, and after some algebra, it can be shown that

VW and WE

for i#S’.

(337)

Thus as t - co, the refractory species decays as l/t (an apparent order of two), while the reactive ones decay as (l/tp’r (an order less than two).