J. Phys. Chem. 1988, 92, 1110-1 119
1110
them would produce a small enhancement at low energy in the total cross section.
Acknowledgment. We are grateful to Drs. C. D. Eley and S. S. Parmar for their helpful discussions. This work was supported
by grants from the U S . Army Research Office and the National Science Foundation. Registry No. Rb, 7440-17-7; CH,I, 74-88-4; BH,, 13283-31-3; CO, 630-08-0; NO, 10102-43-9; C2H,, 74-85-1; O,, 7782-44-7; H,O, 773218-5.
Scaling Relations and Self-Similarity Conditions in Strongly Coupled Dynamical Systems Herschel Rabitz* Department of Chemistry, Princeton University, Princeton, New Jersey 08544
and Mitchell D. Smooke Department of Mechanical Engineering, Yale University, New Haven, Connecticut 06520 (Received: September 23, 1986)
Dynamical equations arising in a number of physical areas typically involve many dependent variables as well as numerous parameters. In some cases these equations may contain a dominant dependent variable (e.g., the temperature in flame systems, etc.), and the consequencesof such an identification are examined in this paper. Particular emphasis is placed on the behavior of dynamical Green’s functions and system sensitivity coefficients with respect to the parameters residing in the particular model. In the case of a single dominant dependent variable for a system of ordinary differential equations, it is possible to reduce the Green’s function matrix to knowledge of one of its columns and often to one independent element. Furthermore, when there are N dependent variables and M parameters, the N X M matrix of sensitivity coefficients reduces to knowledge of only two characteristic vectors of lengths Nand M , respectively. These various reductions are referred to as scaling relations and self-similarity conditions. The consequences of a dominant dependent variable are illustrated with examples drawn from various areas of combustion and kinetics modeling. A brief discussion in the Appendix is also presented on similar organizing principles in multidominant dependent variable systems and cases described by partial differential equations.
I. Introduction Mathematical modeling in a host of physical areas is becoming an increasingly powerful tool for analysis and design purposes. A natural consequence of these developments has been the increased interest in employing sensitivity analysis tools for probing the role of physical parameters residing in a particular model under study.’ A growing literature exists in the latter area with extensive numerical calculations becoming available. In many respects the numerical sensitivity analysis calculations are providing a data base from which to seek out any generic or general trends transcending particular physical systems or problems. The existence of such rules of thumb could be extremely important for providing a basis for simplifying complex systems and perhaps even interrelating different models of the same process. We will refer in this paper to connections of this type as scaling and self-similarity relations among the system’s dependent and independent variables. By definition, the dependent variables are those whose solution is sought in the modeling process and the independent variables are those parameters residing in the equations, boundary conditions, or initial conditions. For purposes of physical clarity the independent coordinates or time will be distinguished from the parameters themselves. The basic thesis of this paper is that scaling relations and self-similarity conditions may occur in dynamical systems where one or, at most, a few dependent variables dominate the physical behavior of the remainder. Situations like this may arise in a number of circumstances with the case of combustion receiving particular attention in this paper. In combustion problems the temperature is a natural controlling variable if it varies appreciably (1) (a) Tomovic, R.; Vukobratovic, M. General SensifiuifyTheory; American Elsevier: New York, 1972. (b) Tilden, J ; Costanza, V.; McCrae, J.; Seinfeld, J. In Modelling of Chemical Reaction Systems; Ebert, K., Deuflhard, P., Jager, W., Eds.; Springer-Verlag: Berlin, 1981. (c) Rabitz, H.; Kramer, M.; Dacol, D. Annu. Rev. Phys. Chem. 1983,34, 419. Rabitz, H. Chem Reo. 1987, 87, 101
0022-3654/88/2092-1110$01.50/0
over the spatial and/or temporal range of interest. The dominant role of the temperature has been long recognized in combustion. Although combustion phenomena motivated the developments described below, the conclusions may also apply to other problems where similar controlling dependencies occur. We will show that a dominant dependence has fundamental consequences for the relationship between all the dependent variables and parameters of a system. To provide some necessary background on the development of the dominant variable scaling and self-similarity conditions, a brief introduction to the relevant‘aspects of sensitivity analysis will be presented in section 11. This will be followed by the derivation of the scaling relations and self-similarity conditions in section 111 for problems described in terms of ordinary differential equations with a single dominant dependent variable. The approach taken is based on the presentation of a critical conjecture upon the form of the kinetic solutions (cf. eq loa), and the sensitivity consequences are then developed. The justification of the conjecture resides in its successful prediction of sensitivity behavior under appropriate conditions. An accompanying Appendix will give an introduction to an extension of the theory into the realm of multidominant dependent variables and cases described by partial differential equations. Some illustrations will be given in section IV and more extensive examples will be presented elsewhere covering the detailed physical aspects of the problems. Finally, in section V we present some concluding comments.
11. Basic Operating Equations Sensitivity analysis concepts are central to the issues of this paper. As a result, a summary of germane asp+ of the theory will be presented here for completeness. A specific physical problem of interest concerns the study of steady premixed laminar flames2 Realistic laboratory situations of this type may be (2) Buckmaster, J. D.; Ludford, G. S. S. Theory of Laminar Flames; Cambridge University Press: London, 1982.
0 1988 American Chemical Societv
The Journal of Physical Chemistry, Vol. 92, No. 5, 1988 1111
Strongly Coupled Dynamical Systems reasonably modeled as one-dimensional two-point boundary value problems described by the following chemical species and energy equations3 dYi dx
M-
d + -(pYiK) dx
where M is a constant and P = pRT/ p.In the above equations the index i labels properties of the ith species, p is the mass density, Y, are the species mass fractions, V, is the diffusion velocity, W is the mean molecular weight of the mixture, P is the pressure, C, is the constant pressure heat capacity of the mixture, T i s the temperature, and X is the thermal conductivity of the mixture. In addition to the differential equations, appropriate boundary conditions would be specified and these latter conditions might also contain parameters of physical interest. The vector a is defined to contain all relevant physical parameters including any arising in the boundary conditions. Although eq l b has the same inherent form as eq l a , it is naturally separated off due to its distinct physical meaning. In addition, the dependence of both 1; and g upon components of the species vector Y is typically quite distinct from that of T. In particular,J and g are often algebraically nonlinear in the components of Y, while the temperature usually enters in a highly nonlinear exponential (Arrhenius) fashion. This strong driving nonlinear dependence through the chemical rate coefficients is the origin of the strong dependence of the overall reaction rate upon temperature. Models more complex than those arising in eq 1 occur under a variety of generalized conditions. For our purposes, however, all of these problems may be described abstractly in terms of a vector set of differential equations
L(0,a) = 0 where O(x,a) is the vector of dependent variables sought after by solution of eq 2 and L, is an appropriate differential operator for the ith equation. The mathematical modeling process alone would consist of specification of all the necessary system parameters and conditions followed by solution of eq 2 by suitable numerical means. At this juncture the role of sensitivity analysis enters as a means for assessing the importance or contribution of the various system parameters a with respect to objectives of interest. In some problems these objectives might be a function or functional of the raw output vector O(x,a)while in other cases components of this latter vector would themselves constitute the observable objectives of interest. The present paper will take the latter perspective, and this is often the case arising in combustion. Furthermore, the choice of other reduced function or functional objectives obscures the basic scaling relations and self-similarity conditions sought out in section 111. Therefore, we naturally consider the sensitivity coefficients of interest to be (3) which provide a direct measure of how the jth parameter controls the behavior of the ith dependent variable at point x . Higher order gradients of various types may also be calculated (4)
but the present paper will only examine first-order derivatives. The gradients are understood to be evaluated at the nominal parametric operating conditions for the model. The subject of this paper is not concerned with how to compute the sensitivity gradients in eq 3, but a formal solution for these gradients will be important to the self-similarity analysis. For this purpose we may first identify the equation that the sensitivity coefficients satisfy by simple differentiation of eq 2l (3) See, for example, Smooke, M D.; Miller, J. A.; Kee, R. J. Combust. Sei. Technol. 1983, 34, 79.
s)(2) ( 2 )
=O
+
(5)
The solution to these equations may be conveniently expressed in terms of the system Green’s function defined to satisfy the following equation:
+f,(Y,T) = 0
a*oi(x,a) /aa++
f(
where it may be shown that the elements of G have the following interpretation:]
(7) The elements G,,,,,are, in principle, measurable in the laboratory and Correspond to the response of the nth observable dependent variable O,(x) at point x with regard to a disturbance of the flux J,(x’) of the dependent variable O,(x’) at point x’. The solution to eq 5 may now be written in terms of Green’s function
where, for convenience, we have assumed that Green’s function satisfies the same boundary or initial conditions as the sensitivity coefficients in eq 3. Regardless of how the sensitivities are calculated, the fundamental role of Cis evident in eq 8. In particular, all the system sensitivities are expressed in terms of a convolution of the Green’s function with the explicit parametric derivatives of the differential equations. The actual calculation and utilization of sensitivity coefficients can involve many other operations beyond the basic input-output interpretation implied by eq 3. Details of these issues may be found in the literature,’ but the above synopsis suffices for the purpose here. 111. Strong Dependent Variable Coupling: Scaling Relations and Self-similarity Conditions The existence of a single dominant dependent variable is certainly problem dependent but the situation can readily arise as already indicated in eq 1. Without any loss of generality we may consider eq 2 as separated into two parts
LI(0,C.l) = 0 L,(O,a) = 0, i = 2 ,
(9a)
..., N
(9b)
where it is understood that the equation L,(O,a) = 0 is a differential equation for O,(x,a) which typically depends in some algebraic fashion on the remaining dependent variables O,, n’ # n to produce the coupled set in eq 9. For convenience we have separated off the differential equation for O,(x,a) which is now assumed to be the dominant strongly coupled dependent variable. The fundamental conjecture of this paper is that this circumstance will imply the relation On(x,a)
Fn(O,(x,a))
(loa)
where Fnis an appropriate function. Equation 10a suggests the generalization On(x,a) = Fn(x,a,OI(x,a))
(lob)
where eq 10b is, in fact, not an approximation and it is just written here to indicate that the nth dependent variable may be thought of as having an explicit dependence on x and a as Well as perhaps an implicit dependence through Ol(x,a). On the other hand, eq 10a is the essential conjecture that the x and a dependence of O,(x,a) arises solely as a function of the dependence arising in the dominant controlling dependent variable Ol(x,a). Before we proceed to the mathematical consequences of eq loa, the conditions for the validity of this statement need to be addressed. The answer to this question is not fully known but a number of comments can be made. First, the coordinate de-
1112 The Journal of Physical Chemistry, Vol. 92, No. 5, 1988
pendence alone implied by eq loa could be exact in some problems. In particular, if O l ( x , a ) is a monotonic function of x , then this function may be inverted such that a one to one correspondence exists O1 x . However, the parameter dependence implied by eq loa makes this an entirely different matter. Equation 9 generally implies a functional relationship between the components of O ( x , a ) (or equivalently a relation between O ( x , a ) and its spatial derivatives). The form arising in eq 10a may occur in the singular perturbation limit of eq 9 and this circumstance is valid (e.g., the steady-state limit of chemical kinetics) at least in some systems. For example, strong coupling among a pool of chemical radicals can result in the identification of a single member O I ( x , a ) as tightly coupled to the remainder. Equation 10a likely may be derived under a number of assumptions and it is suggestive from the behavior of eq 1 that the degree of nonlinearity of the dependent variables may be a reasonable indicator for establishing a special dominant dependent variable. At the present time the validity of the conjecture resides in the apparent broad numerical verification of the conclusions that follow from it (a sample of these results will be given in section 111). Furthermore, there may also exist milder or more encompassing conjectures which will yield the same results below. The remainder of this section is divided into two subtopics on scaling relations and self similarity conditions that follow from eq loa. A . Scaling Relations. The function structure of eq 10a has some immediate and simple consequences of basic importance to sensitivity analysis. We could utilize the general structure in eq 10b to decompose O,(x,a) into explicit and implicit gradient behavior, but the results are not of practical utility. The working eq 10a is our starting point with the goal being a set of sensitivity scaling relations. These relations may be derived by first functionally differentiating eq 10a with respect to J,,,(x?
Rabitz and Smooke
-
Exactly analogous equations also apply to Green’s function elements through eq 12. The simplification implied by these results is quite dramatic and the validity of the relations in eq 12 and 14 are easily tested once sensitivity information has been calculated. For example, a simple consequence is that the sensitivity coefficients and Green’s function elements of the nth dependent variable will change sign as a function of x whenever an extrema aO,,/ax = 0 exists. Illustrations of these formulas will be given in section 111. The Appendix gives an introduction to strong variable scaling when there is more than one dominant dependent variable or when the equations are of a partial differential nature. B. Self-similarity Conditions. The basic operating equation (eq loa) immediately led to the scaling relation between elements of the Green’s function matrix in eq 12. Equations 8 and 12 also implied the scaling of all system sensitivities in eq 14, and we now show that eq 12 leads to a further level of simplification under mild physical assumptions. By definition, the elements of Green’s function matrix satisfy eq 6 and this should still be true within the degree of approximation leading to eq 12. Substitution of eq 12 into eq 6 leads to the following differential equation:
where Li(x) =
and similarly, differentiation of eq 10a with respect to x will produce
(z) (z)(z) N
(Ilb)
This equation simply states that all the elements of the fundamental Green’s function matrix may be reduced to knowledge of the first column of that matrix along with coordinate gradients of the dependent variables. The latter gradients are assumed to be naturally available from solving eq 2 and the scaling implied by eq 12 corresponds to a reduction of the N X N dimensional Green’s function matrix down to knowledge of a single vector of dimension N . The scaling relation in eq 12 in itself is a considerable simplification but it also has further implications for the sensitivity coefficients. Substitution of eq 12 into eq 8 produces the result
Application again of eq 8 with n = 1 leads to identification of the sum in eq 13 to yield the scaling relation for the sensitivity coefficients N
z)(
% ) I
-
z)( ?)I
(16b)
Equation 16a is clearly a Green’s function type equation but with a somewhat unusual structure. In particular, if we define the column vector g(x,x’) to have components gd(x,x? = 6 0 1 ( x ) / 6Jnt(x3,the eq 16a is equivalent to L(x).gT(x,x?
By eliminating the derivative d F n / a O l from eq 1l a and 1 lb, we obtain the first scaling relation
(y)(?)(
”)(
E( n don
= -h(x - x?
(17)
A ”normal” Green’s function type equation (e.g., see eq 6) would have a matrix differential operator acting on a matrix Green’s function. The unusual structue of 16 or 17 is evident from the observation that for i # n’we have
This equation suggests that the fundamental Green’s function vector is approximately separable in terms of a universal x-dependent function 60,(x)/6Jd(x’) N X(x)fd(x9 where, as yet, fd(x’) is to be determined and X(x) satisfies the equation L i ( x ) X ( x ) N 0, for all i (19) This form for the fundamental Green’s function is not entirely adequate since the singularity at x = x’must still be taken into account by the equation for i = n’
This equation is of the classic Green’s function variety and will have a solution of the following form:
(14)
The same comments above about a matrix vector reduction of Green’s functions are also applicable to eq 14. Equation 14 also immediately implies the following relationships:
where the functions Ad* and fd* satisfy the appropriate boundary conditions of 6 0 1 ( x ) / S J d ( x ? in accord with the coordinate inequalities of eq 21. Equation 21 is the most general circumstance satisfying eq 20 but the argument above leading to eq 19 suggests
Strongly Coupled Dynamical Systems that a milder ansatz may be valid while still satisfying the inhomogeneous eq 20
where X(x) satisfies eq 19. The form in eq 22 is most appropriate for the case that L(x) is a first-order differential operator and likely would become increasingly accurate in the limit when the advective term dominates the diffusion term in a second-order differential equation. Although we may proceed with eq 21 as the fully general case, a number of numerical results, including some which are cited in section IV below, suggest that eq 22 is a reasonable statement. Since X(x) is determined from eq 19, it remains to establishfdi(x’) by proper satisfaction of boundary conditions and the jump condition of eq 20. In order to establish the jump conditions, we first write L,(x) as
where the functions p:(x) will generally depend on the solution to eq 2. There are two circumstances arising from eq 20 depending on whether the highest derivative term in Ld(x) is of first or second order first-order equations
(pZn’ =
0):
second-order equation:
In the case of first-order differential equations causality would demand thatfd-(x? = 0 in eq 24a. The second-order case results in a discontinuityfd+(x? # fflt-(x’)in order for eq 24b to be satisfied. However, the magnitude of this discontinuity could be quite small and numerically insignificant if )pz“(x’) dX(x’)/dxl >> 1. In practice some of the numerical results on second-order differential equations of the type in eq 1 appear to show a small (smoothed) numerical jump across x = x’ (cv. Figure 3 below). In this regard, it is worth noting that eq 21 for second-order differential equations exhibits continuity of the Green’s function (although not its slope) across the point x = x’, and the situation in eq 22 could then be argued if X,,*(x) only has a weak dependence on its or - indexes and n’. In any case we will proceed with eq 22 due to its confirmed sensitivity consequences argued below and illustrated in section IV. Finally, we point out that the above procedure could, in principle, be used to calculate X(x) andfdi(x?. However, caution is called for since the necessary differential equations are approximately valid. The most significant utility of the analysis resides in the selfsimilarity conditions that follow from it. The consequences of eq 22 are important with regard to sensitivity behaviar. This comment is evident first upon substitution of eq 22 into eq 12 to produce the result
+
where A,(x) = h(x)(
2)(2) ’
Equation 25 is the first of the self-similarity conditions, which in this case is applicable to Green’s functions. Note that if the more general eq 21 was applied, the only modification to eq 25 would be a or - index added to the function A,(x). The physical significance of eq 25 is evident when it is recalled that the Green’s functions are, in principle, measurable in the laboratory. In particular, in the case n = 1, the function X(x) could be established by a disturbance in the flux of any dependent variable. In turn,
+
The Journal of Physical Chemistry, Vol. 92, No. 5, 1988 1113 by successively disturbing each of the dependent variables, the functions fdi(x’) could be determined. Even without this experimental effort in establishing the fundamental Green’s functions, the self-similarity structure of eq 25 is theoretically important. Since the statement implies that the multitude of IV Green’s function surfaces reduces to knowledge of a vector of surfaces SO,(x)/lJd(x’) which themselves are simple product functions indicated in eq 22. A further consequence of eq 22 arises by substitution of eq 25 into eq 8
where uj
=
The self-similarity condition in eq 27 has a surprisingly simple structure. In general, uj may depend on x due to the discontinuity betweenfd+(x) andf,;(x). However, as argued above in eq 24, this discontinuity seems to be weak in some practical calculations illustrated in section IV, and in this situation uj will only be weakly dependent on x. We will assume here that the limiting case of constant uj is valid and the breakdown of the assumption can be assessed from numerical calculations. Therefore, eq 27 states that under its conditions of validity, all system sensitivities reduce to knowledge of a vector of functions A(x) and a vector of characteristic constants u . The vector u has the same length as the parameter vector and is accordingly labeled in the same fashion; however, its components are a complicated function of all the system parameters through eq 28. Insertion of eq 27 into eq 1Sa immediately leads to the result
This equation states that the sensitivity of a given species or dependent variable, with respect to a sequence of parameters, will be described by a self-similar set of curves in (coordinate) space all related by the constants in the vector u. The factorized structure of eq 27 also has consequences for the sensitivity of objectives’ which are functions or functionals of the dependent variable vector O(x). Various objectives can arise when considering bulk properties (e.g., thermodynamic functions) of a physical system which are superpositions of all the dependent variables. In the case where the objective is a function Q = Q(O(x)) of the dependent variables we have
where
In a situation where the objective is a functional Q = J dx Fo(x,O(x)) of the dependent variable vector with Fo being an appropriate function we have the parameter sensitivity
where
Again, the self-similarity conditions in eq 30 and 31 have a very
1114 The Journal of Physical Chemistry, Vol. 92, No. 5, 1988
simple structure. Equations analogous to eq 15 may be readily identified by taking ratios of the particular cases arising in eq 30 and 31. Although applications of this type have not been considered thus far, the validity of eq 30 and 3 1 totally relies on eq 27 which has been numerically examined and will be illustrated in section IV. It is significant to observe that the functionsfn,*(x? (as well as Xd*(x), if eq 21 is employed) may likely only depend on n’in a rather weak fashion. To see this point, consider eq 16 again and sum it over the index i 1(X)(6O,(X)/6Jfll(X~) -6(x - x ?
(32a)
where This equation is valid for all values of n’although the index does not appear on the right-hand side of eq 32a. In practice, we have found cases wheref,,*(x? appears to be nearly independent of n‘ (cf. Figure 1 below), but this circumstance does not seem to arise generally. The main conclusion of this section is the scaling and selfsimilarity behavior embodied in eq 14 and 27. The essential assumption underlying these results is eq 10a and its validity resides in the degree to which the consequences in eq 14 and 27 are verified in actual calculations. In many instances the approximation in eq 10a may only be valid for a subset of the systemdependent variables or parameters. Nevertheless, within the confines of these cautionary comments, a growing body of numerical results has justified the premises and conclusions, at least qualitatively and even quantitatively, in some cases. Particular numerical illustrations will be presented in the next section.
IV. Numerical Examples In this section several examples will be considered illustrating the scaling and self-similarity relations presented above. Quantitative assessments of these relations will be made, although the ultimate conceptual utility of this work likely resides in its qualitative applicability. Therefore, emphasis will be placed on the validity of these relations as exemplified by the qualitative structures evident in sequences of sensitivity profiles. Examples A and B will be particularly concerned with the scaling and self-similarity behavior of system Green’s functions. Examination of the two-dimensional Green’s function surfaces in these cases will be used to verify the qualitative predictions of eq 12 and 22. In example C parametric sensitivity coefficients will be examined and a quantitative assessment of eq 14 and 27 will be given. Each of the illustrations will be drawn out of context from detailed, individual studies, and no attempt will be made here to elaborately discuss or analyze each problem. These systems are treated in thorough detail elsewhere in separate papers covering other physical and chemical aspects besides strong coupling behavior. A . A Model Premixed Steady Flame.4 The basic eq 1 corresponding to a one-dimensional steady premixed flame may be illustrated by the simple linear chemical reactions C,
- ci -
c,
(33)
where the subscripts r, i, and p, respectively, refer to reactant, intermediate, and product. Although the chemistry in eq 33 is linear, the overall flame is strongly nonlinear through the temperature entering into the rate constants in the differential eq la,b. The modeling and sensitivity analysis of this system was carefully examined with one primary conclusion being the strongly dominant role played by the temperature. The temperature also played the dominant role in examples B and C below. Scaling and selfsimilarity were clearly exhibited in the calculations which are exemplified by the Green’s function matrices in Figure 1. Since the product C , in eq 33 is entirely determined by a knowledge of the reactant and intermediate, there are only three dependent variables: temperature, reactant, and intermediate. In the notation (4) Reuven, Y.; Smooke, M. D.; Rabitz, H. J . Comput. Phys. 1986, 64, -7
L!
Rabitz and Smooke of the previous section we identify O,(x) = T, 0 2 ( x ) = C, and 03(x) = C,. The nine Green’s function surfaces corresponding to these variables are depicted in Figure 1. A simple glance at the figure clearly shows similar structure from one surface to another especially along the rows. To understand this structure, we refer to eq 12 with the dominant dependent variable O,(x) identified with the temperature T ( x ) . Consider first the reactant Green’s function surfaces G,,,(x,x’) given by
Since the temperature is a monotonically increasing function of position x > 0 (the flame was modeled on the domain 0 I x I 10 with reactant flux entering at x = 0), the temperature slope in eq 34 will always be positive. In contrast, the reactant concentration monotonically falls with increasing distance x and its spatial gradient appearing in eq 34 will always be negative. Therefore we can understand that Green’s function surfaces of the reactant G2*,in the second row of Figure 1 are basically the negative of Green’s function surfaces of the temperature G , , in the first row. Furthermore, the structure of the intermediate Green’s function surfaces G 3 d ( ~ , ~ may ’ ) be similarly understood. The intermediate concentration, which is initially zero at the left boundary, rises to a maximum in the flame zone and then diminishes. The spatial gradient of the intermediate will then change sign while passing ,through the flame with the result that the intermediate Green’s function surfaces G3,,,(x,x’) prior to the flame location should look similar to the temperature Green’s functions G,,,(x,x? and change sign at aC,/ax = a03/ax = 0 upon passage through the flame. This behavior is again evident in Figure 1. All the analysis in this case thus far has been based on the Green’s function scaling relation in eq 12. The self-similarity conditions in eq 22 and 25 are also evident in Figure 1. In particular, an examination of the temperature Green’s functions surfaces GId(x,x? will show that a cut through any of the surfaces as a function of x for a given value of x’will be approximately the same function X(x) implied by eq 22. Furthermore, all three temperature Green’s function surfaces are themselves similar corresponding to the comment below eq 32 that the characteristic functions f,,*(x’) may be weakly dependent upon the index n’. Therefore, the scaling and self-similarity conditions combined ultimately reduce the nine surfaces for the reaction in eq 33 down to knowledge of a single independent temperature Green’s function surface. The consequences of this behavior naturally show up as prescribed by eq 14 and 27 in all the sensitivity coefficients which are evident in the graphs of ref 4. Finally, it is important to understand that the relations derived in section I1 and discussed here are only approximations. This point is evident in Figure 1 and also the remaining illustrations by the apparent deviations from the scaling and self-similarity “rules”. The particular disparities evident in Figure 1 are generally found at the boundaries but, in general, other differences might also arise depending on the problem and the degree to which a single dominant variable may be identified. B . Steady Premixed Carbon Monoxide-Hydrogen-Oxygen Flame.5 This system is similar to that of example A except now a realistic model of the chemistry was included involving 52 chemical reactions. The model contained 11 chemical species as well as the temperature thereby producing 144 Green’s function surfaces and numerous sensitivity profiles. Reference 5 presents an elaborate physical analysis of this system and only three Green’s functions surfaces will be presented here to illustrate qualitatively the validity of the self-similarity condition of eq 22. Figure 2 presents the Green’s function surfaces for the dominant temperature variable O,(x) = T ( x ) with respect to itself as well as the chemical intermediate hydrogen peroxide H202. Considering first Figure 2a with the surface 6T(x)/6JT(x’), it is evident from the graph that the factorization of eq 22 is operative. In particular, a cut through the surface at any fixed value of x’should produce (5) Mishra, M.; Yetter, R.; Reuven, Y.; Rabitz, H.; Smooke, M. D., to be submitted for publication.
The Journal of Physical Chemistry, Vol. 92, No. 5, 1988 1115
Strongly Coupled Dynamical Systems G, ,tx.x’)
G, .JX X )
G,&X,X’)
10’
0 -10-
0
0
a*dX,X7
Q2JX,X’)
1
10-’ 0
lo-’
-10
-10
0
0
-10
0 0
0
0
Figure 1. Green’s function surfaces Gnd(x,x’)relating the temperature T = O,(x), the reactant C,= Oz(x) and the intermediate Ci = 0 3 ( x ) . The system corresponds to a steady premixed flame in one dimension with linear chemistry described by eq 33. Scaling and self-similarity are evident in the related structure of the surfaces. In particular, all of the surfaces may be reduced to knowledge of Gll(x,x’)and the spatial gradients of 0,and 03-
a curve proportional to the characteristic function XT(x)and this latter function may be identified by following along any one of the corresponding surface lines of the figure. The function XT(x) is seen to sharply rise and then exhibit a slow decay at increasing values of x. The function fT*(x’)behaves similarly except without a long range tail. Equation 22 states that the Green’s function ST(x)/6JH2O2(x’) in Figure 2b should have the same x dependence as Figure 2a described by the function XT(x). An examination of parts a and b of Figure 2 with regard to their x dependence at fixed values of x’ indicates the same type of behavior. Cuts through these surfaces at fixed values of x show that the function JYH2O2(x’) is distinct from its temperature analogfT*(x’) in that it shows an immediate positive value near the cold boundary followed by a peak and an inversion in sign before asymptotically going to zero. A careful examination of cuts through Figure 2b illustrates the approximate factorization relation ST(X)/~JH,O,(X’) XT(x)f+H202(x?An interesting point concerning Figures 1 and 2 regards the discontinuity fd*(x’) # fnt-(x’) at x = x’implied by eq 22 and 24b. Neither of these figures shows any apparent evidence for this discontinuity, although other Green’s function surfaces in the CO-H2-02 system exhibited such evidence. For example, Figure which clearly 3 illustrates the Green’s function 6O2(x)/6JO2(x’) contains a mild jump or smoothed discontinuity along the diagonal x = x’. This behavior is particularly apparent both before and after the main response structure of the figure. All of the diagonal Green’s functions G,,(x,x’) for this system contain mild jumps, and selective off-diagonal Green’s functions also contain this structure. Nevertheless, in general, the results thus far appear
more like those of Figures 1 and 2 implying only a mild discontinuity or that the more general result in eq 21 is actually valid with the circumstance X’(x) N X-(x) being true. The near continuity of frit+ andf,; implies that aj in eq 28 is essentially constant, and the detailed numerical parameter sensitivitis~~ verified this through l h e satisfaction of eq 27. Example C will present another quantitative illustration of this behavior. C. A Premixed Hydrogen-Air Flame.6 This example again is similar to those in illustrations A and B except we will now focus on the parametric sensitivities rather than on the Green’s function surfaces. Both qualitative as well as quantitative tests of scaling and self-similarity relations will be presented. In addition, the conclusions drawn from these illustrations also carry over to the system Green’s functions (not shown here) due to their intimate connection apparent in eq 8, 12, and 13. The physical model for the hydrogen-air flame is like that of examples A and B with the temperature again identified O1(x) = T ( x ) as the dominant variable.6 Figure 4a-presentsprofiles of the temperature and species mass fractions as a function of position. The cold boundary is at x = 0.0 and the rapid rise in the temperature near x 0.055 indicates the location of the flame front. The constant curve labeled by N2 indicates the presence of the nitrogen diluent from air. The scaling relation in eq 14 involves the slope of the particular species being probed and in this regard particularly observe the H atom
-
(6) Smooke, M. D.; Reuven, Y.; Rabitz, H.; Dryer, F. “Application of Sensitivity Analysis to Premixed Hydrogen-Air Flames”, Combust. Flame, in press.
Rabitz and Smooke
1116 The Journal of Physical Chemistry, Vol. 92, No. 5, 1988
A
TABLE I: Percentage Error bBnHfor the Diffusion Coefficient Scaling Relation" X
DH,
0.05 0.06 0.07 0.09
6.0 3.5 13.0 42.1
~~~
~
DO, ~
6.1 1.o 2.4 26.4
6,H DH 6.5 1.8 3.8 19.9
Do
Don
4.1 2.3 5.5 23.3
6.4 2.3 7.7 29.1
OResults here and in the following tables are from a hydrogen-air stoichiometric steady flame. The percentage error is defined in eq 36. The scaled sensitivities span over an order of magnitude at the x values sampled.
TABLE II: Percentage Error bDnoHfor the Diffusion Coefficient Scaling Relation 6D? X
0.05 0.06 0.07 0.09
Dnz 5.6 0.6 0.3 0.6
Doz
Dn
Do
6.3 10.0 3.1 0.3
6.4 2.9 0.3 0.9
6.1 6.0 0.2 1.o
Don 15.0 9.0 0.0 0.6
later.) Equation 14, written in terms of the current circumstances has the form
Figure 2. Two Green's function surfaces arising in a steady premixed CO-H2-02 flame. The behavior of the surfaces provides an illustration of the self-similarity condition in eq .22. In particular, a cut through either surface for a fixed value of x' has the same characteristic shape X(x). In addition, a cut through parts a and b of Figure 2 at a fixed value of x will reveal the functions fT*(x) and SHZo2, respectively.
Figure 3. Green's function surface 602(x)/6Joz(x') arising in a steady premixed CO-H2-02 flame. The figure clearly exhibits evidence for a mild discontinuity along the diagonal x = x'as implied by eq 22 and 24b.
curve in Figure 4a. The curve exhibits an extremum, and hence a zero derivative dH/dx = 0, at x N 0.078. With this observation consider now Figure 4b containing sensitivities dH(x)/dD, of the H atom concentration H ( x ) with respect to the system diffusion coefficients 0,.The species labeling each curve in the figure indicates the correspondingdiffusion coefficient sensitivity being evaluated. In a similar fashion Figure 4c presents the sensitivity of the temperature dT(x)/dD, with respect to the same diffusion coefficients. In both figures a logarithmic scale is used for the sensitivities with numbers of absolute value less than set to zero. We will now show that the behavior found in parts b and c of Figure 4 quantitatively corroborates the scaling and self-similarity relations in eq 14 and 27. (The one especially notable exception to this conclusion involves the hydroxyl diffusion coefficient case in Figure 4b. However, a possible reason for this will be mentioned
First, the equation states that the hydrogen atom sensitivities should be proportional to the slope d H / d x of the hydrogen mass fraction profile in Figure 4a. In partichlar, as commented above, this derivative goes through zero at x N 0.078and this corresponds quite accurately with the multiple traversals through zero of the sensitivity curves in Figure 4b at the same point. Second, the proportionality between d H ( x ) / d D , and d T ( x ) / d D , appears in two ways: (a) the ordered sequence of the labeled curves in Figure 4b exactly follows that of Figure 4c with the expected reversal at the aforementioned extremum of the H atom mass fraction profile, (b) the extrema in the temperatue sensitivities in Figure 4c at x E 0.05 is reflected in the prominent bump at the same location in the plots of the H atom sensitivities of Figure 4b. Thus far the analysis of parts b and c of Figure 4 based on eq 14 has been qualitative and it is instructive to make a quantitative evaluation of the validity of the latter equation. As a simple assessment of this issue we may define
as the percent deviation of the left- and right-hand sides of the scaling relation in eq 35. Table I presents 6,H at four representative values of x for five diffusion coefficients. For further comparison, Table I1 presents the percentage deviation SD?" for the scaled sensitivity dOH/dD, of OH with respect to the same diffusion coefficients. The percent deviations of Tables I and I1 are actually quite small especially when it is realized that the sensitivities themselves span over an order of magnitude in values at the sampled points and in the case of H a sign change of the sensitivities is also involved. This behavior is typical of the diffusion sensitivities to all of the chemical species and good results were found except at points where the slopes on the right-hand side of eq 14 approach zero. Nevertheless, even near the latter points systematic qualitatively correct scaling behavior was found (cf. the tight multiple set of crossings of all the sensitivity curves in 0.078where d H / d x = 0). Figure 4b near x As an illustration of the predictive quality of eq 14 for typical system rate constants, Table I11 presents the analogous percentage deviation &.HZ for several rate constants in the mechanism. The results in Table I11 are even more dramatic when it is realized that the sensitivity coefficients being scaled span 4 orders of magnitude over the range of x values being sampled. Similar good
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The Journal of Physical Chemistry, Vol. 92, No. 5, 1988
Strongly Coupled Dynamical Systems TABLE 111: Percentage Error bknHz for the Rate Constant Scaling Relation” &.H2
0.01 0.04 0.06 0.08
2.0 6.0 0.3 0.8
1.2 5.8 0.4 0.8
1.9 5.9 0.4 0.9
3.4 7.4 0.3 28.3
2.6 6.6 0.5 14.3
”The percentage error is defined analogously to eq 36. On the range of x values chosen the sensitivities being scaled span approximately 4 orders of magnitude. The rate constant sensitivities refer to those of the nth reaction with its equilibrium constant held fixed. The reactions are as follows: (1) H2 OH = H 2 0 + H; (2) H O2 = OH 0;(3) 0 H2 = OH + H; (4) H + 0 2 M = HO2 + M (M = H 2 0 H2); (5) H 0 2 + M = HO2 + M (M = 0 2 ) .
+
+
+
+
+
+
+
behavior was also found for the scaling results of the other chemical species with respect to all of the system rate constants. Tables 1-111 provide a quantitative measure of the accuracy of the basic scaling formula in eq 14. An even more encompassing result is embodied in eq 27 implying the constant (or nearly constant) self-similarity ratio in eq 29. Equation 29 is obviously qualitatively satisfied by practically all of the sensitivities in Figure 4b,c. This conclusion follows from the parallel nature of the curves in each case. As a quantitative comparison, Table IV presents 02, the ratio (a In O,/a In D , ) / ( a In U,/a In Dj)for 0, = T,HZ, and H. According to eq 29 this ratio should be DHuHJD~u,independent of 0,. The table clearly shows that the self-similarity condition is satisfied to a high degree of accuracy. As a final point of comparison, Table V shows the analogous data for the ratio (a In 0,la In k , ) / ( a In O,/a In kj) which should be klulJkjaj independent of 0, according to eq 29. Again the high accuracy of the self-similarity condition is quite evident. A final matter concerns the fact that the numerical results in Tables IV and V are changed only by a few percent upon evaluation at other values of x (except near points where the sensitivities are zero since the ratio is then undefined). This behavior confirms the suggested spatial independence of uj in eq 28. The link between the parametric self-similarity-scaling behavior and the underlying Green’s function is provided by eq 22, 25, and 27 where it is evident that the same common characteristic function X(x) drives the spatial dependence of Green’s functions and sensitivity coefficients. This point is illustrated in Figure 4c where the heavy solid curve is a cut through the temperature Green’s function surface 6T(x)/6J02(x’)evaluated at x’= 0. It is apparent that the Green’s function and the temperature sensitivities are quantitatively governed by the same characteristic function X(x) = TT(x). By virtue of the quantitative validity of TABLE I V
(a In O,/a DJ
DH1 DO2 DO DOH DH02 DHzO DH201
the scaling and self-similarity conditions presented in the tables, it immediately follows that this function also drives the spatial sensitivity behavior of the other dependent variables. The quantitative comparisons presented here for the hydrogen-air system are representative of the remaining variables and parameters in this system. In addition, similarly good quantitative results were found in cases A and B above. Nevertheless, it is important to emphasize that significant exceptions to the scaling and self-similarity behavior can arise and a case in point is that of the OH profile in Figure 4b. The precise reason for this latter behavior is not clear but one possibility may be associated with the fact that OH itself is a strongly controlling variable in the oxidation process. Thus, prior to the rapid rise in the flame temperature the OH may break away from the dominant control of the temperature and behave in an independent fashion. Exceptions to the scaling and self-similarity rules may occur for various reasons, and further analysis and computations are needed to explore the matter.
V. Conclusions This paper presented a set of generic scaling and self-similarity conditions that followed from an ansatz of there being a single dominant dependent variable in the dynamical system. The consequences of the scaling and self-similarity conditions go beyond the issue of mathematical relations existing between the sensitivities in such systems. In particular, the implied interdependence of the variables or parameters is very suggestive of system lumping being a reasonable approximation under these conditions. Other consequences may also follow and it remains for further numerical calculations, as well as analytical work, to exploit the relations developed in this paper. The theoretical development in section I1 was itself guided by observations of numerical calculations, and the number of such sensitivity and modeling calculations is growing rapidly. In the context of this paper the remaining fundamental question concerns the validity of the basic dominant variable ansatz in eq loa. It is evident that this ansatz must be an approximation, since the model differential equations provide relations between the dependent variables and their derivatives rather than the dependent variables alone. Nevertheless, the results presented here and elsewhere show it to be a reasonable approximation in several systems. Although a milder assumption than eq 10a may possibly lead to the same conclusions, no such choice has been found thus far. Establishing the conditions in the original differential equation under which eq 10a or its analogues are valid remains an important task. Another related point is that eq 10a with 0, as the dominant variable leads to eq 15 which shows, upon comparison with eq 14, that any of the strongly coupled dependent variables could
In D d / @ In O./d In D J as a Test of the Self-similarity Relation in Equation 29’ (a In T/a In D H ) / (a In H2/a In D H ) / (a In 02/a In DH)/ (a In T/a In 0,) (a In H2/a In 0,) (a In 02/a In 0,) -2.57 -9.68 13.88 18.23 -3.41 x 104 -24.46 -80.18
-2.30 -10.25 14.10 18.76 -1.66 x 104 -23.89 -80.06
1117
-2.59 -8.31 13.66 18.37 -1.47 x 104 -21.56 -75.65
(a In H/a In DH)/ (a In H/a In 0,) -2.62 -9.61 13.95 18.48 -1.77 -26.39 -80.65
X
lo4
“The self-similarity relation in eq 29 would be exact if the numbers in each row were constant. The gradients were evaluated at x = 0.6, and the numbers were nearly invariant to x on the significant flame range 0 5 x 5 0.125 (except near the null value H atom sensitivity at x E 0.078 where the tabulated ratio is undefined). TABLE V (8 In O./a In k , ) / ( a In O./dkh as a Test of the Self-Similarity Relation in Equation 29“ (a In H2/a In k,)/ (a In T/a In k l ) / (a In O,/a In kl)/ (a In T l a In k,) (a In H,la In k , ) (a In o,ia In k J k, k2 1.21 1.22 1.21 k, 1.72 1.72 1.71 k; 2.42 2.43 2.43 13.30 13.47 kS 13.43 “See the footnote to Table IV. The rate constants refer to the mechanistic steps listed in Table 111.
(a In H/a In k , ) / (a In H / a In k,) 1.21 1.72 2.30 12.95
Rabitz and Smooke
1118 The Journal of Physical Chemistry, Vol. 92, No. 5, 1988 100
just as well be taken as the 2500 NZ----NZ------NZA 1-N2 2000 c
x
u
15002 1 -.)
m
1000~
$
e 500
0
IO2
0
dominant member. Only through constraints on the dependent variables could the key member be truly identified. Combustion phenomena of the type discussed in section I11 may continue to provide a fruitful testing ground to explore these questions. The interesting body of literature on central manifold theory reduction' has an apparent relation to the work in this paper. The reduction of variables inherent in central manifold theory may be argued in terms of singular perturbation theory. It is intriguing that although the physical arguments seem to be distinct from those presented here for the reduction of variables, both theories argue an entrainment of dependent variables such as found in eq loa. The additional implications for multiparameter dependencies discussed in this paper do not seem to have been explored with central manifold theory. A deeper comparison and analysis of these two approaches to system reduction would be valuable. The basic implication behind the existence of dominant variable dependence is that strongly coupled systems may, in fact, behave in a simpler fashion than first believed. It is curious that this behavior appears likely to be more valid in problems which are inherently nonlinear and normally thought of as having more complex behavior than arising in linear problems. In a sense, the strong mixing often found in nonlinear problems can lead to a surprising level of parametric simplicity under appropriate conditions.
Acknowledgment. The authors acknowledge support for this research from the Office of Naval Research and the Air Force Office of Scientific Research. We also thank Dr. Y . Reuven for special help in assembling some of the numerical results.
Appendix: Multivariable Dominant Dependence The basic ansatz in eq 10a was restricted to a system having one dominant variable 0, and one independent coordinate x. Many real problems could be dominated by large numbers of dependent variables as well as have additional spatial coordinates or time. There is little known about scaling or self-similarity conditions in these cases, but some comments can be made. First, note that the case of many strongly coupled dependent variables may still reduce to the situation of eq loa. The strong coupling among the dependent variables implies that one of them may be chosen as a representative member with the others related in a fashion like that of eq The other cases listed below have certain distinguishing aspects not treated in the body of the paper. ( 1 ) A Single Dominant Variable with Two Independent Coordinates. This case is similar to that of e4 10a with the assumed relationship being QI(x,y,a) = Fil(0,(X,YP) (A. 1) The coordinates x, y in a partial differential equation model may both be spatial variables or one could be time as the situation dictates. Two scaling relations exactly analogous to eq 14 may be derived
Figure 4. Results from modeling and sensitivity analysis of a hydrogen-air stoichiometric steady premixed flame. (a) The temperature and species mass fraction profiles as a function of position in the flow with the combustible mixture entering at the origin x = 0. Note the hydrogen atom curve with its extremum near x N 0.078. (b) Hydrogen atom sensitivities with respect to the diffusion coefficients of the particular species in the flame. Each curve is labeled by the species whose diffusion coefficient is being probed. (c) Temperature sensitivities with respect to the same diffusion coefficients. The heavy bold curve in Figure 4c corresponds to 6T(x)/6Jo,(x? at x ' = 0. The same shape exhibited by this curve, as well as the sensitivities, verifies the existence of a single x-dependent function A,(x) governing Green's function and parameter sensitivities as claimed in eq 25 and 27. In addition, the curves in Figure 4c are quantitatively related (except for the OH diffusion sensitivity) to those in Figure 4b through the scaling relation in eq 35.
The existence of these two conditions in turn implies that
)(:
2)-' (z)( 2)"
(A.3)
N
and this equation is equivalent to the statement
(n),: (n), =
(A.4)
(7) Ruelle, D.; Takens, F. Commun. Math. Phys. 1971, 20, 167. Fernandez, A. Phys. Reu. A 1985, 32, 3070. (8) Yetter, R.; Dryer, F.; Rabitz, H. Combust. Flame 1985, 59, 107.
1119
J . Phys. Chem. 1988, 92, 11 19-1 126
Equations A.6.b and A.6.c may be used to express the derivatives dF,/dO, and dF,/dO, in terms of spatial gradient information. Therefore, solving for these latter two quantities and substitution into eq A.6.a will finally lead to an expression for the sensitivity (dOn/daJ)in terms of the sensitivities of 0, and 0, and their various spatial gradients. This result is an exact analogue of eq 14. A similar statement for Green's functions would also follow in this case. ( 3 ) Two Dominant Dependent Variables with One Independent Coordinate. Under these conditions eq 10a takes on the form
The latter relation would imply, for example, that the velocity measured with respect to Ol is the same as that determined by 0, when x is the time t and y is a spatial coordinate. These various relations have been verified in a steady propagating combustion wave simulating a flame front moving through a combustible mediume9 Analogous relations may be derived for Green's functions. ( 2 ) Two Dominant Dependent Variables and Two Coordinates. This case is a direct extension of that discussed in item 1 above except now an additional dominant dependent variable 02(x,y,a) is assumed to exist. Recalling the discussion at the beginning of the appendix the two variables 0, and 0, are assumed to act independently of each other. Therefore, eq 10a is now replaced by the form
On(X,Y,a) E Fn(Ol(X,Y,a),O ~ ( X , Y P ) )
On(x,a) E F ~ ( O I ( X P ) , O ~ ( ~ P ) ) ('4.7) Differentiation of this equation with respect to ai and the coordinate x will lead to equations of exactly the same form as (A.6.a) and (A.6.b). However, in this case we no longer have t h e y coordinate to produce the remaining equation (A.6.c). A third independent equation of this type is necessary to again eliminate the gradients dF,/dOl and dFn/dO,. Such an equation can be found by differentiating eq A.7 with respect to a different parameter ak to produce the analogue of eq A.6.c. Following the same operations discussed above, a scaling relation will be found between the spatial gradients and sensitivities involving both parameters a, and &. The above brief treatment of multivariable systems only touches on a number of possibilities and even more complex relations might exist. Furthermore, we only considered scaling relation arguments and the development of self-similarity conditions has not been explored. As discussed in the conclusion, the basic remaining question concerns the derivation, or at least understanding, of the general conditions under which conjectues of the type in eq 10a are valid.
('4.5)
Simple differentiation of this equation will produce the following relationships:
(z)( (z)($) (z)($)
(2) 2)+ (7) ($( $)+ N
N
(A.6.b)
(A.6.c)
(9) Reuven, Y.; Smooke, M. D.; Rabitz, H., to be submitted for publica-
tion.
Rate Constants for the Reactions OH Products
+ HOC1
-
H,O
+ CIO and H + HOC1
-
C. A. Ennist and J. W. Birks* Department of Chemistry and Biochemistry and Cooperative Institute for Research in Environmental Sciences (CIRES). University of Colorado, Boulder, Colorado 80309 (Received: December 29, 1986; In Final Form: July 23, 1987)
A new laboratory source of gaseous hypochlorous acid (HOCl) has been used in two kinetics investigations in a mass spectrometry-resonance fluorescence discharge flow system. Two potential removal reactions of stratospheric HOCl were studied. The rate constant for the reaction OH + HOCl H 2 0 C10 (1) at 298 K was found to be lower than the NASA estimate by a factor of about 2-12; a value in the range (1.7-9.5) X lo-" cm3 molecule-' s-l for k l is reported here. The reaction of CI,O + OH interfered in the study of k , and was the subject of a preliminary investigation. Its rate constant was determined to be (9.4 f 1.0) X cm3 molecule-l s-l at 298 K. The rate constant for the reaction H + HOCl products (2) was determined to be (5.0 f 1.4) X cm3 molecule-' s-I at 298 K. Although branching ratios for three possible product channels could not be determined, OH was identified as a product. The results of this work imply that reactions 1 and 2 are not competitive with direct photolysis in the removal of HOCl from the stratosphere.
-
Introduction Hypochlorous acid (HOC1) began to attract attention in 1976 when an atmospheric modeling study suggested that HOCl may serve as a temporary reservoir for chlorine atoms in the stratosphere.' Evaluation of this possibility requires values of the rate constants for all reactions that form and destroy HOCl to a significant extent. In earlier work our group has reported on several such reactions.2-5 Here, we present rate constant measurements for two previously unstudied reactions of HOCI. The primary focus of our study is the reaction OH + HOC1 H,O + CIO (1)
-
Present address: National Center for Atmospheric Research, Boulder, CO 80307. *
0022-3654/88/2092-1119$01.50/0
+
-
Using a low-pressure, mass spectrometry-resonance fluorescence discharge flow system, we have measured the rate constant of reaction 1 ( k , ) at 298 K. Additionally, we have found that the reaction H + HOCl products (2)
-
(1) Prasad, S. S. Planet. Space Sci. 1976, 24, 1187. (2) Leck, T. J.; Cook, J. L.; Birks, J. W. J. Chem. Phys. 1980, 72, 2364. (3) (a) Cook, J. L.; Ennis, C. A.; Leck, T. J.; Birks, J. W. J. Chem. Phys. 1981, 74, 545. (b) Cook,J. L.; Ennis, C. A,; Leck, T. J.; Birks, J. W. J. Chem. Phys. 1981, 75, 497. (4) (a) Ennis, C. A,; Birks, J. W. J. Phys. Chem. 1985.89, 186. (b) Ennis, C. A. Ph.D. Thesis, University of Colorado, Boulder, CO, 1985. (5) Mishalanie, E. A,; Rutkowski, C. J.; Hutte, R. S.; Birks, J. W. J. Phys. Chem. 1986, 90, 5578.
0 1988 American Chemical Society
