Guy-Marie Came
2560
Radical Reaction Mechanisms. Mathematical Theory Guy-Marle Came Institut National Polyfechnique de Lorraine, E.R.A. No. 136 du CNRS, E.N.S.I.C., 1, rue Grandville, 54042 Nancy Cedex, France (Received April 26, 1977) Publication costs assisted by Laboratoire de Cinetique Appiiquee
A mathematical theory of a general straight chain radical reaction mechanism has been devised. It is first proved that a radical rate law is always stiff, which is important for the numerical solution of kinetic models. Then, using mainly the singular perturbation theory, quasi-stationary state and long chain approximations (QSSA and LCA) have been examined, in the case of a continuous flow stirred tank reactor. Four adimensional numbers are introduced: p , as a criterion of reactant stationarity; x , as a criterion of pseudo-stationarity state (PSS); 0, as a criterion of QSS; and A, for chain length. The expressions of p , x , 0, and h are function of the rate constants of elementary processes and of concentrations of reactants, excluding radicals. It is therefore possible to give precise conditions for QSSA or LCA to be valid. An efficient PSSA is defined, which differs from QSSA, and which can be applied, even during the induction period. We conclude that the use of those kinetic approximations often is valid. Furthermore, individual criteria T and p allow the partitioning of reactants into slow and fast components, providing a help to the numerical solution of stiff kinetic problems.
I. Introduction In the 1920’s, Bodenstein formulated the quasi-stationary state approximation (QSSA), which has been, since then, the mathematical technique most commonly used in chemical kinetics. QSSA has been questioned lately, particularly by Edelson,l who is asking whether QSSA is fact or fiction, and who recommends that recently developed numerical methods be adopted in future work. QSSA, together with long chain approximation (LCA), allows the simplification of kinetic models so that useful equations can be obtained, more general than particular numerical solutions. Moreover, all preceeding kinetic studies have had recourse, to a large extent, to QSSA, and Arrhenius’s parameter tables now available are mainly based on these works. In both respects, determining conditions of use of QSSA or LCA seems to be interesting. We have previously studied QSSA through direct algebraic methodsa2 In the case of a batch reactor, a general singular perturbation theory of QSSA has been proposed by Aiken, Lapidus, and Liu3 (below referred to as AL). With its bibliography, this work will be considered as the starting point of the present one. Let us now briefly summarize it. AL consider the following ordinary differential equations:
dx/dt = f(x, y) dy/dt= w(x, Y )
x(0)
~(0) (1.1) where the dependent variables divide into nonstiff x and stiff y vectors, AL introduce an artificial small parameter E, which need not actually exist or be identified, such that g(x, y) = ew(x, y) is of the same order as f(x, y), The solution of (1.1)is given in terms of zero-order inner (Xo, Yo)and outer (xo, yo) and first-order inner and outer ( X I , Y l , xl,y1) terms. Let us quote AL: (a) “The zeroth-order outer solution corresponds to the QSSA.” (b) “The region of applicability (of the QSSA), or boundary layer length (or inductions period) is defined as a fractional decay of the zeroth-order stiff inner variable.” (c) “The accuracy of using only the zeroth-order approximation is indicated by the magnitude of the firstorder outer terms.” The Journal of Physical Chemistty, Vol. 8 1, No. 25, 1977
An effective procedure, which does not require the identification of the perturbing parameter t, is proposed by AL to solve points (a) to (c), but numerous questions still remain unanswered (we quote AL): (d) “The preceding analysis is useful only (if) wy has eigenvalues with negative real parts .,, this seems to be true in stiff chemical kinetics.” (e) ‘‘e is an artificial bookkeeping indication of the degree of stiffness of (1.1)”but t is not identified and its physical meaning is not given. Further “the main difficulty would be obtaining a criterion for finding the stiff variables.” The two points are of course connected. (f) “If wi is linear in y, ... a very good estimate of the boundary layer can be made”, but this is not the case for radical chain reactions, which involve combination processes of two radicals. Moreover, threeother questions seem to be of interest: (g) Long chain approximation (LCA) is not considered by AL, though it is commonly used in radical reaction studies. (h) In connection with (e), the partitioning of stiff variables during the induction period (boundary layer) into two groups is examined; QSSA can be applied to nondetermining radicals; we refer to this special QSSA as PSSA. (i) Stationarity of nonstiff variables during the induction period is not treated, though it is often an implicit hypothesis in practice. The main purpose of this work is to answer the previous questions from (d) to (i).
11. Reaction Mechanism a n d Rate Law Let us consider the following general straight chain radical reaction mechanism: A + ( A ) + X, t Xj Xi + ( A ) - Xj t (B) Xi + X j + B + (B)
aij pij Cij
with i, j = 1, 2, ..., q. The As are reactants, Bs products, and X,s radicals. Notation (A) or (B) indicates that the corresponding species may (or not) be part of the elementary step. Let us notice that some As and Bs may be the s min
Pj i
i=ltoq j=n+ 1toq
Let be E = l i p , where p is the smallest rate constant of nondetermining radicals. It is now quite sensible to seek an asymptotic expansion of X1 of singular perturbation type:
xi=XI0 + EX11 + 2 X 1 2+ . . .
(IV.4)
Thus, equations for each of the perturbation functions Xl0, Xll, etc, can be easily derived, by substititing the series into the material balance (IV.3), and then equating terms of equal powers in E . The result is given as
of an isothermal continuous flow stirred tank reactor only. The material balance is thus written x i = tr,
i = 1, 2,
.... q
(IV.1)
We shall consider two main types of radicals: the nondetermining ones, which do not appear in termination processes, the subscript of which varies from 1 to n, and the determining radicals, which do appear in termination processes and which have a subscript from n 1 to q. There are two subclasses of nondetermining radicals: transfer radicals (subscript 1 to m), which are not chain
+
where I is the identity matrix. It must be noticed that X 2 is not expanded into a series of e. Let us make now the hypothesis
7r=pt>>1
(IV.6)
It follows that perturbations P c l / t , etc, are negligible with relevance to I and then
PIXI
-(A1 + PZXZ)
(IV.7)
The Journal of Physical Chemistry, Vol. 81, No. 25, 1977
Guy-Marie C6me
2562
For transfer radicals, which are not chain carriers, Pz = 0, and eq IV.7 is in fact the classical QSSA. For nondetermining chain carriers, for which P2 is not equal to zero, eq IV.7, formally the same as those of QSSA, allows us to calculate X I as function of X 2 , even during the induction period. Since, during the induction period, X 2 determining chain carriers are not in their QSS, it results that nondetermining chain carriers are not in QSS either, though we can use QSSA. That is why we suggest for this case the term pseudo-stationary state approximation which, from now on, will be referred to as PSSA. So, radicals may apparently exhibit typical nonstationary manners and, for them, PSSA may be applied however. This fact has certainly played a part in the criticizing of QSSA. From a practical point of view, PSSA provides a powerful tool to deal with induction period problems.
V. Determining Radicals Now, we are going to examine the case of determining chain carriers. We are assuming that PSSA for nondetermining radicals is granted (i.e., n >> l),so their concentrations are no longer found in the material balance
cpj, = 0 i
but, at least, one minor of order q - n - 1 is different from zero. Then (V.6) has an unique solution, within a multiplicative constant:
tixo
Xi0 =
(V.9)
where tis are only function of propagation rate constants and
xo = cxjo
(V.10)
i
so that
(V.11)
Xti =1 i
Let us sum up eq V.7 on i; we obtain, from eq V.7 to V.ll (V.12) with (V.13)
a = Cui i
i= n
+ 1,n + 2 , . . ,, q
W.1) with pii = Sipu,ais are initiation rates, deduced from ais by use of PSSA and such that Coli
i
=
L:ai = a
W.2)
i
c = (zciil’2tj)*
We supposed the classical relation (only for concise writing): Cij 2
Let us define three adimensional numbers h = p/(ac)1’2 71 = h6 = p t
“=
The symbol a denotes a global initiation rate, c a mean termination rate constant, and p the smallest apparent propagation rate constant associated with determining radicals; t is the mean residence time. By introducing the new variables X and 0 in eq V.l, we obtain the transformed material balance
We shall suppose that chains are long, so that X >> 1. We shall seek an expansion of x i in inverse powers of X: (V.5) Following the same procedure as above, we identify the terms of equal powers in 1 / X . We first obtain 0 = zpjixjo j
S j C i j X i ~ j+0 2ciixi: (ac)‘12 i , j = n + 1 , n + 2 ,..., 4 (V.7) (V.6) is an homogeneous system of q - n linear equations in the q - n xio unknowns. This system has a determinant which vanishes because 2ai
1
+ -cpjjxj1 (ac)’I2 P j
-
The Journal of Physical Chemistry, Vol. 81, No. 25, 1977
2(C..C..)’1* 11 Jl
(V.15)
From (V.12), it derives
6 = t(ac)1’2
xio= 6
(V.14)
i
(:)
112
48 1 + (1+ 1682)1”
From (V.9) and (V.16), we can see that, it 0 QSSA is valid.
(V.16)
>> 1, the
VI, Long Chain Approximation The expression of the first “normalized” perturbation term, with 0 being small, can be written
Consequently, we can see that, if n >> 1 and, of course, This conclusion can be reached more directly, giving a more intuitive and physical meaning to LCA. Let us rewrite equation (V.1) as X
>> 1, LCA can be used.
xi [ 1 + t( lpii I + sjqjxj t ( 2 q + SjPjiXj)
+ 2CiiXi)l = (VI.2)
Let us define A, and X, as chain lengths with reference to initiation and termination rates, respectively A, = SjPjiXj/2Cui (VI.3)
+
h, = lpiil/(sjcijxj 2CiiXi)
(VI.4)
We are supposing, as above, that h>>1
(VI.5)
For small n, A, is much greater than 1,whereas A, is small.
2563
Radical Reaction Mechanisms
However, for T >> 1,X, becomes much greater than 1, so that eq VI.2 reduces to the LCA.
VII. Reactant Stationarity Criterion In the beginning of our theory, we were supposing that reactant concentrations were constant during the induction period. It is now necessary to discuss this point. Let us introduce real propagation rate constants k,, such that
The nondetermining transfer radicals are R-and H.; the nondetermining chain carrier is p.; so the T criterion is written 71
=
min h, t, h, ( f i H ) t , k 2 t
If P >> 1,the PSSA may then be applied to R., He, and pa radicals, even during the induction p e r i ~ d From . ~ now on, we shall assume that this condition is granted. The only determining radical is p., so the 8 criterion is written
where (A) is the reactant concentration participating to the elementary step. Let
If 8 >> 1, the QSSA may be applied, Le., the induction period is over. LCA criterion is written
and
x = zxi
= k3/(Wpp(PH))”2
i
If X >> 1, LCA is valid. Finally, the reactant stationary criterion is
From the material balance, it results that
1
(A) (A),
-2-
Let us define the reactant stationarity criterion At QSS
x
p =
1+ k t x p
= kxt.
= (a/c)”2
So, if p =
k(a/c)”Zt > k 3 ( ~ H ) Let us note that the mechanism is not linear in the stiff variables because of the termination step.
k3(hl/hpp)“2(fiH)”Lt
It is then necessary that p
