Relevance of stochastic modeling in chemically reacting systems

Relevance of stochastic modeling in chemically reacting systems. L. K. Doraiswamy, and B. D. Kulkarni. Ind. Eng. Chem. Fundamen. , 1986, 25 (4), pp 51...
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Ind. Eng. Chem. Fundam. 1986, 25, 511-517

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Relevance of Stochastic Modeling in Chemically Reacting Systems L. K. Doraiswamy” and 6. D. Kulkarnl National Chemical Laboratory, Pune 4 1 1 008, India

This paper is concerned with the relevance of stochastic modeling in the study of chemically reacting systems. The need for such an attempt arises from the fact that the importance of stochastic modeling is often not realized. The relevance of this approach is brought out in this paper through appropriate examples.

General Introduction The conventional engineering analysis of a chemically reacting system utilizes macroscopic laws, which in turn have been derived by averaging the more fundamental and detailed microscopic description of the process. The averaging procedure leads to a reduced or projected description of the actual process in which the slowly evolving variables, also called the macroscopic variables, are retained while the fast variables are eliminated. It is these fast variables that constitute the “noise” or fluctuations and, in view of their origin within the system, are referred to as the internal fluctuations. Usually these fluctuations are neglected. The reduced description, also known as the macroscopic description, has been widely employed to study the behavior of reacting systems. It provides a useful insight into the system properties. Note, however, that such a description is approximate a t best; while it has served as a useful tool for the analysis of many situations, it can lead to erroneous conclusions in certain cases. We notice that the macroscopic or deterministic methods are adequate only as long as deviations (fluctuations) from the macroscopic average values remain negligible. The validity of this assertion is established via statistical mechanics at thermodynamic equilibrium where, for a system of order N , the fluctuations around the average values are of the order of N1/2. This result holds a t points far from the points of macroscopic instability (Landau and Lifshitz, 1973). Situations which demand more rigorous formulations than the average macroscopic description can now be identified, and cases subscribing to such treatment come from the classes of systems having a finite number of entities (where the fluctuations, Nl2,are no longer negligible in comparison with N) or those operating near the borders of different stability regions. In these situations, it is anticipated that the fluctuations will assume a dominant role and their incorporation can totally change the system dynamics when compared to cases where they are ignored. In addition to the type of fluctuations described above, another type of “noise” may also appear in the description of macroscopic systems. This happens, for example, when the system is coupled to a fluctuating environment. Typical examples are variations around the mean value of an input parameter such as flowrate, the input temperature of feed stream, etc. Notice that these fluctuations occur a t a macroscopic level and can be included in the macroscopic description once their statistical properties are known. These fluctuations have their origin in sources outside the system and for this reason are also referred to as external fluctuations to distinguish them from the internal noise which has its origin within the system. It is obvious now that, in contrast to internal noise, external noise shows no dependence on the size or the volume of 0196-4313/86/1025-0511$01.50/0

the system. In other words, this type of noise can change the dynamic and steady-state behavior of even large sized, globally stable systems, and the results can markedly differ from those obtained by using the macroscopic model where these fluctuations are ignored. Different approaches have been developed which logically take into account fluctuations in reacting systems. The more popular among these are the master equation and the Langevin equation formalisms. The master equation approach is more convenient when dealing with internal fluctuations, while the latter suits the needs when fluctuations enter the system description from sources outside it (van Kampen, 1981; Gardiner, 1983). Studies on a stochastic approach to chemically reacting systems have been in progress in this laboratory for the past 2 or 3 years. Our analysis has shown that the role of internal fluctuations in chemically reacting systems is usually, but not necessarily, negligible. On the other hand, external fluctuations can exert a significant influence in some situations and bring about a drastic change in the behavior of systems in relation to the predictions of the normally used deterministic methods. In this paper, for the sake of completeness, both internal and external fluctuations are considered. This is followed by a few specific examples to illustrate the dangers of neglecting the role of external fluctuations in determining the behavior of systems. The main purpose is to bring out the relevance of-indeed the need for-a stochastic approach to chemically reacting system analysis, particularly since this does not appear to have been generally recognized.

Internal Fluctuations: Approach Based on the Master Equation Consider a simple chemical reaction where the molecules of the reactant species interact among themselves to produce the final product. Schematically, this is conveniently represented as The macroscopic rate laws model the rates of change of the reactant species or those of production of the product species, in terms of a set of differential equations with appropriate initial conditions. The solution of these equations, obtained analytically or numerically, gives the evolution of the system’s behavior in terms of the concentrations of various species as functions of time. The macroscopic model clearly visualizes the underlying physical phenomenon as a continuous state-continuous time process. A microscopic examination of the underlying physical phenomena, however, indicates that the molecular population can change only discretely and the true reaction therefore represents a discrete-state process, contrary to its description as a continuous process in the macroscopic 1986 American Chemical Society

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model. Also note that the discrete molecular entities in this population interact among themselves in an essentially random way. A stochastic model for this population can be derived based on the concepts of probability theory. The chemical reaction can be fitted into the general framework of the concepts of probability theory. The discrete molecular population characterizing a chemical system can be expressed in terms of the joint probability of the random variables representing the groups of entities in the total population. A significant assumption will now be made concerning the interactions between the various entities: that the system is Markovian. In other words, knowing the present state of the system population, the next state can be predicted without a knowledge of the past history of the system. We define the conditional probability p(x,,tllxo,to)which is the probability that the random variable X takes a value x1 at time t,, provided that it has a value xo at time to. In the event of a vanishing time difference between t , and t,, (tl - tu = T O ) , this conditional probability can be expanded in Taylor’s series as +

P(Xl,t+T;Xg,t) = 6’(X1

- Xo)

-

TCW(X0,X)6 k ( X 1

- Xo)

x

= [I -

+

+ T w ( x o , X ~+) o(7)

T C W ( X O , X ) ] ~-~ Xg) ( X ~ TW(Xo,X1)

+ O(T*)

(1)

I

where W(x,,xl)represents the transition probability per unit time to effect a change in the population from state X, to state X, within the time interval A t . In view of the Markov property, we can simplify the probability p ( x l , t + 7 ) as P(Xi,t+T) =

Cp(xi,t+T;xo,t) p(xo,t)

(2)

IT/(

which, when substituted in ea 4 and in the limit of 0, yields dp(x,,t)/dt = C W ( x , x l )p(x.t)

T

-

The approximated master equation, such as eq 5 , has computational advantages besides its obvious similarity to the convective-diffusion form. Even when the equation cannot be solved exactly, the numerical techniques for such equations are well established. More importantly, the derivation of this equation gives a clue to the identification of the terms ai and a*,which can be found independently without a knowledge of the details of the transition probabilities required in the master equation. This is a great advantage. All one needs to do to set up a Fokker-Planck equation of the form of eq 5 is to take a time interval, I t , so small that X does not change significantly but still the Markovian assumption is valid. The coefficients cyI and 0 1 ~are then obtained respectively as the average change, ((AX)),and its mean square, ((AX)*), over the interval At. The Fokker-Planck approximation of the master equation is based on the assumption that all terms after v = 2 vanish. This is rarely true in practice, however, and a more rational way of approximating the master equation is to systematically expand it in powers of a small parameter which can be approximately chosen. Usually this parameter is chosen to be the size of the system, V. The systematic expansion procedure of van Kampen (1976,1981) is based on the concept that the change in the random variable during a transition is small in comparison with the value of the random variable. The expansion envisages a separation of the macroscopic and random or fluctuating components by an expression such as X = VQ + V‘ 2y,where V is the system’s volume. Substitution of this in the master equation and collection of terms of like orders in V result in the macroscopic equation for 4. The fluctuations, y , are expressed in the form of a FokkerPlanck equation. The details of the mathematical steps, omitted here for the sake of brevity, eventually lead to

(3)

r

Equation 3 is referred to as the “master equation” and can be expressed in alternative forms, depending on the nature of the state variable, x. In reality, eq 3 represents the differential version of the Chapman-Kolmogorov equality which merely expresses the Markovian character of the process. Its conversion into the form of eq 3 makes it applicable to specific situations for which the transition probabilities can be evaluated. Even so, the equation is somewhat difficult to handle, especially when continuous variables are involved and the equation takes the integro-differential form. Alternatively, the master equation can be approximated by a differential equation of the convective-diffusion type. The transition probabilities per unit time can be expressed as functions of the size of the jump, X - X’, and the starting point, X’. For the jump size to be small, one can assume the transition probability to be a sharply peaked function. Also, the solution p ( x , t )is assumed to vary slowly with x. These assumptions permit Taylor’s expansion of terms in the master equation and eventually lead to the equation

This equation represents the Kramers-Moyal expansion of the master equation. In a special situation where u is restricted to two, the equation yields the conventional Fokker-Planck equation

Equations 6 and 7 are the result of the systematic expansion and represent the evolution of the macroscopic and fluctuating components, respectively. As such, these equations can be solved in most cases, especially since eq 7 is a linear Fokker-Planck equation. In many situations, however, the expansion coefficients depend on time, making it difficult to obtain an easy solution. The best way in such instances is to multiply eq 7 by ( y ) or the higher moments, ( y a j ) ,and integrate over all the variables to yield

and

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The covariance can then be obtained as

The equations can also be expressed in terms of the original variable, X. The mean and the variances, while useful, do not give any dynamic infamation regarding the fluctuations. This can be obtained from the correlation functions, which provide a measure of the influence of the value of a random variable at any time t on the values of the random variables a t time t + T . For a Markov process the auto- and cross-correlation functions can be easily derived. Defining the correlation matrix Yl](‘) = (Y,(O) YJ(’))

(13)

the following relation between y L J ( and ~ ) covariances can be established:

In the master equation approach we notice that, in addition to the macroscopic description, we obtain information regarding the fluctuations in terms of their means, variances, and covariances and the corresponding correlation functions. We now present some case examples. The master equation formalism has been applied to a variety of situations, such as a simple monomolecular reaction having first-order kinetics and a bimolecular reaction. The early work in this area has been summarized by McQuarrie (1967). In more recent years Doraiswamy and Kulkarni (1986) have analyzed a variety of more complex situations that frequently arise in practice. Their analysis includes general reaction schemes that can be represented by a polynomial rate expression or by Langmuir-Hinshelwood rate forms involving single-variable and multivariable situations (Tambe et al., 1985a), systems exhibiting multistationarity (Tambe and Kulkarni, 1985) and nonelementary reactions (Tambe et al., 1985b), systems with critical slowing-down features (Tambe et al., 1985c), and systems with diffusional effects (Tambe et al., 1985d). In each of these cases, besides the macroscopic equation, equations for the mean, variance, covariance, and correlation function have been obtained. It suffices to note here that the solutions to these equations are proportional to ( VAT1, where N represents the total population. Since N is of the order of per mole, the fluctuations remain inconsequential unless V assumes correspondingly small values. It is possible, therefore, to say that internal fluctuations are generally inconsequential and the master equation should yield results similar to those obtained by using conventional macroscopic models. It should be remembered, however, that the master equation gives important additional information which may prove useful in certain situations.

External Fluctuations: The Langevin Approach We now consider the effect of external disturbances. The chief problem associated with incorporation of such fluctuations in the system description is that the governing macroscopic equations now become stochastic differential equations. Thus, a typical macroscopic equation for a system coupled to an external noise variable can be written as dXi/dt = F(Xi) + G(XJ l ( t )

(15)

where Xi, generally for an n-dimensional reactor, repre-

sents global macroscopic variables. F ( X J is an n-dimensional vectorial function, and G(X,) represents that part of the function, F ( X , ) ,which is coupled to the noise, [ ( t ) . Note that, in the absence of noise (5 = 0), eq 15 represents the deterministic evolution of the system. The extra term in eq 15 represents the coupling of the macroscopic system with its environment. The term can take different forms depending on the function G(XJ and the noise .$(t). In a typical situation when G(X,) assumes a constant value we have the simple form of coupling called additive coupling and the noise is referred to as “additive noise”. In this form of coupling, the state variable X does not affect the contributions of the last term in eq 15. However, when the function G(X) shows a dependence on the state variable X , we get to a situation that is referred to as “multiplicative coupling”. The noise is then appropriately referred to as “multiplicative noise”. The two types of couplings have quite different effects on the behavior of the system. Indeed, one can expect that the stability properties of the deterministic equation associated with eq 15 will not be modified in the presence of additive noise. The multiplicative noise, on the other hand, may totally change the stability pattern of the system. It is customary to treat t ( t )in the last term in eq 15 as a simple Gaussian white noise. The noise is characterized by a single parameter, D , which measures its strength and of course has no correlation effects. The process X,(t) defined by eq 15, for white noise, has a Markovian character which renders the subsequent treatment easier. As is known, the Gaussian white noise, [ ( t ) ,can be defined in terms of the Wiener process as [ ( t ) = dW,/dt. The unbounded nature of this, however, causes some difficulties because the integration of eq 15 cannot be defined any longer in terms of the Riemann integral. The problem. however, can be overcome by defining a stochastic integral over the Wiener process to assign and complete the meaning of the governing eq 15. To this end one can begin with the usual procedure of taking a limit over a sequence of step functions, approximating the function to integrate. In using this procedure, one quickly realizes that the final result is dependent on the points where the values of the function are taken-a fact that is in contrast to the Riemann integral. We note, therefore, that there are an infinite number of unequivalent stochastic integrals over the Wiener process, d W. Among these, there are a t least two, the Ito and Stratonovich forms, which have been widely employed. The mathematical techniques have been developed largely for the Ito form, while the Stratonovich interpretation of the stochastic process is generally considered to be more physically meaningful. Therefore, the general practice is to treat the stochastic equation formulated from the phenomenological equation in the Stratonovich sense and convert it into the Ito form for subsequent mathematical treatment. The stipulation that the noise in the system must be white makes the process defined by eq 15 Markovian. This is advantageous from a mathematical viewpoint since it is feasible now to convert the stochastic differential equation into an equivalent Fokker-Planck equation describing the probability density, which in this case also defines the transition probability of the process. This property is lost, however, when we consider nonwhite noise that is normally encountered in realistic situations. Nonwhite noise is a broad-band noise having finite timecorrelation effects; white noise is only a mathematical idealization of this in the limit of vanishing correlation time. The simple Gaussian noise in systems is not uniquely

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determined by its intensity, D , alone but requires a knowledge of the correlation time as well. The consideration of { ( t ) in eq 15 as nonwhite noise renders the process X , ( t ) non-Markovian. It is no more feasible then to write a simple Fokker-Planck equation as in the case of white noise. One can, however, still work within the framework of the Markovian theory, provided the noise itself is Markovian. Such a situation arises, for example, when the noise, in addition to being Gaussian, is Markovian and stationary. This noise is now of an Ornstein-Uhlenbeck form, and its correlation function is given by

-

where 7 monitors the correlation time of the noise. In the limiting case of T 0, we of course recover the white noise. The two-variable process XI,{ ( t ) is now Markovian and has a well-defined Fokker-Planck equation associated with it. From a practical viewpoint this is still not very useful since the stationary solution of the multivariable Fokker-Planck equation is generally difficult to obtain. It becomes necessary in many situations to take recourse to direct numerical integration. The highlights of our discussion can now be summarized as follows. We notice that in reality the systems are subjected to fluctuations, both internal and external. The internal fluctuations are unimportant for large-volume systems (proportional to V-1/2)except near the points of instability. The external fluctuations, on the other hand, are important, even for large sized, globally stable systems. Also, in view of the frequent occurrence of such disturbances in practical systems, their due accounting becomes a necessity for a rational engineering analysis. We further notice that the external fluctuations could be of different forms, varying from a noninteractive white noise (the ad-

ditive type) to an interacting noise (multiplicative noise). In addition, the noise should usually have some finite time-correlation effects. A. Examples. Analysis of a Nonisothermal CSTR: Evolution of a Stochastic Diagram. We now consider the simple case of a nonisothermal, continuous, stirredtank reactor in which a first-order, exothermic reaction occurs, to exemplify the effects of external noise in a striking manner. This particular problem is of considerable interest and has been extensively studied earlier from a macroscopic point of view. Further, it provides a simple example of situations wherein a rich variety of features, such as monostable, bistable, and oscillatory behavior, are known to exist over realistic ranges of parameter values. In view of the associated stability characteristics, the example provides a good case for study from a stochastic viewpoint. The governing equations for constant input and output flow rates can be written in dimensionless form as

(18) We have retained the same nomenclature as that followed by Uppal et al. (1974). The dynamic behavior of this set of equations has been classified into six different regions in the parameter space @,a), as shown qualitatively in Figure 1. The lines M, S, and SM in this figure have been obtained by using analytical criteria. Thus, for parameter values of (B,P) lying above the line M, one can expect multiple states for certain Da values. Likewise, the curve S indicates the region in the parameter space where the dynamic condition of stability is just satisfied. Below this

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Table I. Regionwise Solutions and Their Stability Characteristics natureltype no. of stability region of state type solutions characteristics A 1 1 stable node I B 2 1 unstable focus IIa,b 1 stable limit cycle 1 stable node IIIa C 3 1 unstable focus IIIb 1 saddle point D 3 1 stable node IVa 1 stable limit cycle 1 unstable limit cycle IVb E 3 2 stable nodes Va 1 saddle point F 4 1 stable node Vb 1 unstable focus 1 saddle point 1 stable limit cycle G 4 2 unstable focuses VI A, E, F, B 1 saddle point 1 stable limit cycle H 4 2 stable nodes 1 saddle point 1 unstable limit cycle J 5 1 stable node 1 unstable focus 1 saddle point 1 stable limit cycle 1 unstable limit cycle

line the condition will be satisfied; for the region above it, the condition is violated. The line SM represents the case where the equations, det = 0 and tr = 0, have identical roots. The lines bl and a2, signifying the direction of bifurcation, are also indicated in the figure. As the system moves from one region to another, it will satisfy some conditions and violate certain others. This implies that the stability of the system undergoes a change as it moves from one region to another. This has implications on the nature of the final steady state(s) attained, which could be a stable or unstable node or a focus, a saddle point, or a stable or unstable limit cycle. Further, in a given region (i.e., for chosen values of B and p), the nature of the steady state could change with other parameter values such as Da. In a given region, therefore, one may encounter some or all of these states as changes in the parameter value occur. A systematic analysis of this problem reveals that, in the six different regions marked in Figure 1, the number and nature of the steady states realized may vary. The possibilities, regionwise, are summarized in Table I, and the key to the nomenclature of Figure 1appears in the table. This table reveals the variety of behavior patterns exhibited by a nonisothermal CSTR. We now reconsider the behavior of the nonisothermal CSTR in the presence of additive noise. This type of noise, as stated earlier, does not involve any dependence on the state of the system; an example is the random variation in the input temperature of the cooling stream. Indeed, in practice, we can maintain a constant feed temperature only on the average. The actual temperature at any instant can then be written as

x,,= X Z c

+ E(t)

(19)

where the overbar signifies the time-average value and [ ( t ) , as before, represents the random variations around this mean. We define the random variations ((t) as an Ornstein-Uhlenbeck process. The equation describing this can be written in terms of the white noise as d E = - 6- ' f dt where

t-'

+ t-'D

dW

(= 7) refers to the time correlation and

(20)

D is the

intensity of noise. Substitution of eq 19 in the dimensional form of eq 18 and conversion of the resultant equation into dimensionless form lead to dX2 - -x2 + BDa(1 - xl) _

exp

dt

The factor t-l in the last term in eq 21 is introduced to ensure proper reduction to the white-noise case. The mean value of the noise ( E ) is of course zero with a correlation function as defined in eq 16. The statement of the problem is complete with eq 17, 20, and 21. The numerical procedure consists of generating the colored noise for fixed intensities and time correlations of noise using eq 20. The generated data are used in eq 21, which is solved simultaneously with eq 17. The procedure is repeated a number of times for different sets of random numbers, dW, and the results are averaged to obtain a representative sample function. Using this procedure, we have generated the profiles of concentration, xl, and temperature, x 2 , for varying intensity and timecorrelation effects for each of the regions of Figure 1. For the sake of completeness, numerical computations in the presence of noise have been carried out on all the possible types of behavior in each one of the regions listed in Table I. The enormous amount of information obtained this way has been classified, and the results have been expressed in the form of a consolidated stochastic diagram, as shown in Figure 2. This diagram may be regarded as the stochastic counterpart of the consolidated macroscopic diagram of Uppal et al. (1974). The regionwise details of the calculations will be presented elsewhere (Gaikwad et al., 1986), but for the present purpose the main findings are summarized below (see Figure 2). The regionwise solutions permissible from the stochastic model, when compared with those known for the macroscopic model, clearly reveal that the number of permissible solutions in the presence of noise is considerably smaller. In all the regions, for lower values of Da, the stochastic solution approaches the corresponding macroscopic solution and keeps deviating around it without stabilizing. In fact, the extent of deviation depends upon the extent of noise and increases with increase in D or T . In region IIa,b the bistability feature of the macroscopic model realizable for a certain range of Da values (E-type behavior) is seen to be completely destroyed. An interesting situation arises in region IIIa, where the bistability feature persists for E-type solutions. The lower stable solution is realizable, however, only for low levels of noise. With an increase in the level of noise, the only permissible solution is now the upper steady state. Notice that, for the H-type solutions in this region, the upper steady state solution is destroyed in the presence of noise. In region IIIb, for F-type solutions, the system again shows a dependence on the extent of noise. For smaller values of the noise intensity, one may reach the lower stable solution. With an increase in the time correlation of noise, the system trips to another stable solution that corresponds to the stable limit cycle, however. Notice that the noise does not help to create any new states; it only helps the transition from one state to another. In region IVa, the stochastic system shows another interesting feature. For the J-type solutions that are permissible in this region, the incorporation of noise indicates that all other solutions vanish, except the stable limit cycle solution. In fact, even the stable node in this region is transformed into a limit cycle solution in the presence of noise, In regions IVb and Va, likewise, for systems with

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li-” [veDa “‘tI

Do

w

XXXXXX

S T A B L E SOLUTION OSCILLATING AROUND S T A B L E SOLUTION S T A B L E LIMIT C Y C L E

-

_.

Figure 2. Permissible stationary states obtained by using a stochastic model (additive noise).

D-type solutions, the most probable stationary solution is always a limit cycle. In regions Vb and VI, the incorporation of noise indicates great deviations and the solutions never stabilize. In fact, one wonders whether any stochastic stationary solutions exist a t all in these regions. To summarize, the stochastic model indicates that only a small number of solutions are permissible for a nonisothermal CSTR, in comparison with the large numbers known for a macroscopic model. In view of the additive nature of the noise considered, the model does not show any change in the stability properties of the system. The results, however, indicate that the presence of noise can shift a possible steady state. B. Other Examples. Among other examples that highlight the role of external fluctuations, we consider cases of the deterministic equations which show multistationarity and oscillatory behavior. ( 1 ) Bistability on a Catalyst Surface. Tambe and Kulkarni (1985) considered the simple reaction scheme A + X =+AX, AX + 2X + 3X, X Q, which shows multiple solutions for certain sets of parameter values. Incorporation of noise, arising, say, from variations in the partial pressure of component X, destroys the multiplicity and the stochastic system shows only a single stationary solution. In a similar situation where Michaelis-Menten kinetics holds, the analysis indicates that the macroscopic model possesses only one stationary solution. The induction of noise in the system parameters, however, alters the situation drastically and the system now starts showing multistationary behavior.

-

These examples highlight the fact that external noise can induce or destroy multiple solution behavior. (2) Noise-Induced Transitions. Dabke et al. (1985b) have considered a CSTR in which a reaction occurs, typical of that represented by the Brusselator scheme. The macroscopic model for this situation shows that the system passes through regions showing limit cycle, monostable, and bistable behavior on increasing the flow rate. Accounting for random disturbances in the flow, however, changes the situation drastically, and the system now shows only oscillatory behavior in the entire flow region. This example highlights the existence of noise-induced transitions, which obliterate the transitions predicted from macroscopic models. (3) Origin of Oscillations. A large number of skeleton models, such as of Schlogl, Brusselator, or Oregaontor, which have a built-in autocatalytic step, have been used to explain the oscillatory behavior of reacting systems. It is known that all these models contain at least one step in the sequence that is irreversible. In other words, the models do not satisfy the mandatory condition of detailed balancing. The imposition of this additional condition, as well as the use of rigorous mass conservation laws, brings about a drastic change in the behavior of these model schemes: the systems no longer show any oscillatory behavior. However, it is a fact that oscillations have been observed in experimental systems. This leads to a paradoxical situation: the generally accepted reason for oscillations, namely, the existence of autocatalysis, would not be valid in light of the failure of the Brusselator model to explain oscillations in the presence of the mandatory conditions.

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Yet there is the incontrovertible experimental fact that oscillations do exist in such systems. Therefore, one has to find an acceptable reason for these oscillations, either internal or external to the system. Dabke et al. (1985) reexamined the Brusselator scheme, satisfying the detailed balance condition as well as the rigorous mass conservation laws. We now know that the macroscopic model does not show any oscillations under these conditions. Incorporation of fluctuations in the parameter values, on the other hand, shows very interesting features. The oscillations reappear and show some dependence on the level of noise. Thus, they are a derived property of the system induced by noise. In view of the general presence of noise in almost all practical systems, every system should be considered to be prone to oscillations. Explanations based on the mechanistic features of a reaction scheme (such as a built-in autocatalytic step) can be erroneous and misleading.

Summary Practically all analyses of chemically reacting systems that have been attempted so far have used the conventional macroscopic approach. More recently, efforts are being made (mainly by the authors) to apply stochastic methods of analysis. Since in many situations the conventional approach gives identical results, the need for a stochastic approach has often not been felt. Here an attempt has been made to bring out the relevance of stochastic modeling, particularly where external fluctuations are concerned. (Internal fluctuations are often inconsequential.) This has been attempted through examples which clearly indicate the relevance of-indeed the need for-a stochastic approach to chemically reacting system. In certain situations, the conventional (macroscopic) predictions can be drastically in error as, for example, in the case of a nonisothermal reactor with a first-order reaction. The stochastic diagram prepared for this situation (as a result of innumerable computations) is quite distinctly different from the nearly classical macroscopic diagram of Uppal et al. (1974). Nomenclature Ai, Bi = reactant species B = heat of reaction parameter in nonisothermal CSTR case D = intensity of noise Da = Damkohler number F ( X J = n-dimensional vector function G ( X J = part of function F ( x i ) that is coupled to noise

1986 517

N = particle population p ( X , t ) = probability density function V = volume of reactor W = transition probability per unit time Wi = Wiener process X = random variable x = the value of the random variable x l , x 2 = dimensionless concentration and temperature x z , = dimensionless coolant temperature y = fluctuating variable ( ) = mean value of a function Greek Symbols a, = coefficients in Kramers-Moyal expansion Lii, Liij, Sij = expansion coefficients fl = dimensionless heat removal parameter in the nonisothermal CSTR case y = dimensionless activation energy parameters in the nonisothermal CSTR case ~ (- t’) t = correlation function y i j = correlation matrix 4 = macroscopic part of the random variable x 7 = time correlation; also used to denote time difference E(t) = random noise u = stoichiometric coefficient Literature Cited Dabke, N. S.; Kulkarni, B. D.; Doraiswamy, L. K. Paper presented at the 9th International Symposium on Chemical Reaction Engineering, 198% Dabke, N. S.;Kulkarni, B. D.; Doraiswamy. L. K. Chem. Eng. Sci. 1985b, 4 0 , 2007. Doraiswamy, L. K.; Kulkarni, B. D. Analysis of Chemically Reacting Systems: A Stochastic Approach ; Gordon-Breach Science: London, in press. Gaikwad, M. S.; Tambe, S.S.;Kulkarni, B. D.; Doraiswamy, L. K. Paper to be submitted to the International Chemical Reaction Engineering Conference (ICREC-2), National Chemical Laboratory, Pune. India, 1986. Gardiner, C. W. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences ; Springer-Verlag: West Berlin, 1983. Landau, L. D.; Lifshitz, E. M. Statistical Physics ; Addison-Wesley: Reading, MA, 1973; Vol. 5. McQuarrie, D. A. Stochastic Approach to Chemical Kinetics; Methuen and Co. Ltd.: London, 1967. Tambe. S.S.;Kulkarni, 8. D. Paper presented at the National Symposium on Modeling and Simulation, Indian Institute of Science, Bangalore, 1985. Tambe, S. S.;Kulkarni, B. D.; Doraiswamy, L. K. Chem. Eng. Sci. 1985a, 4 0 , 1943. Tambe, S.S.;Ravikumar, V.; Kulkarni, B. D.; Doraiswamy, L. K. Chem. Eng. Sci. I965b, 4 0 , 2303. Tambe, S. S.;Kulkarni, B. D.; Doraiswamy, L. K. Chem. Eng. Sci. I965c, 4 0 , 2298. Tambe, S. S.;Kulkarni, 8. D.; Doraiswamy, L. K. Chem. Eng. Sci. 1985d, 4 0 , 2297. Uppal, A.; Ray, W. H.; Poore, A. B. Chem. Eng. Sci. 1974, 2 9 , 967. van Kampen, N. G. Adv. Chem. Phys. 1976, 3 4 , 245. van Kampen, N. G. Stochastic Processes in Physics and Chemistry; NorthHolland: New York, 1981.

Received for review July 29, 1986 Accepted August 5 , 1986