Capítulo 2. Marco contextual
2.3 El uso de las tecnologías de la información, la comunicación, el conocimiento y el
V.A.VOLPERT
Laboratoire d’analyse numérique, Université Lyon, France A.I.VOLPERT
Department of Mathematics, Technion, Haifa, Israel Abstract
This paper is devoted to a new approach to study combustion waves and branching chain flames with complex kinetics. We develop a mathematical theory of travelling wave solutions for some classes of parabolic systems, and show that reaction-diffusion systems desribing chemical waves under some conditions can be reduced to these classes of systems. We obtain conditions of uniqueness and stability of combustion waves and cold flames, and find their velocity.
Key words: combustion, cold flames, complex kinetics.
1. Introduction
Development of the modern combustion theory is determined to the large extent by the works of Zeldovich and Frank-Kamenetskii on the thermal propagation of flames and by the works of Semenov on branching chain flames. In the first case the chemical reaction is strongly activated, and the propagation of the flame occurs basically due to the heat diffusion. The simplest example can be given by the one-step chemical reaction of the first order,
A→B.
The reaction rate under the mass action law in this case has the form
where A and B denotes also the concentration of the corresponding species, T is the temperature, E the activation energy, R the gas constant, k is the pre-
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exponential factor. If the activation energy E is sufficiently large, then the reaction takes place in a narrow temperature interval near the adiabatic temperature, and the reaction zone is localized in space. It allows to apply the infinitely narrow reaction zone method
developed by Zeldovich and Frank-Kamenetskii [28],[29] and to find the wave velocity and the condition of its stability.
The simplest reaction describing branching chain flames is the following one-step autocatalytical reaction:
A+B→2B.
The propagation of the flame in this case occurs due to the diffusion of the active center B. If we suppose that the diffusion coefficients of the species A and B equal to each other, then the model describing the propagation of the cold flame in this case is the same as considered by Kolomogorov, Petrovskii, and Piskunov in [9]. We know that the waves exist for all values of velocities greater or equal to the minimal velocity, they are stable in a certain sense, and the value of the minimal velocity is found explicitly.
It appears that even these very simple examples describe the basic properties of flames. However the aim to describe a more realistic complex chemistry seems very attractive, and there are many works devoted to this problem.
The method of infinitely narrow reaction zone can be generalized for more complicated chemical reactions [6]–[8],[13],[25]. The best studied examples are independent reactions:
A→B, C→D,
consecutive reactions:
A→B→C,
and parallel or competitive reactions:
A→B, A→B.
If each reaction occurs in its own narrow reaction zone and the interaction between them is only due to the temperature field, then the infinitely narrow reaction zone method is applicable. There are some other examples and generalizations [1],[26], [27]. Sometimes physical arguments help to simplify a problem and to reduce it usually to one of the examples above [14],[15].
There are also some other approaches to study more mathematical questions of existence of solutions [2]–[5],[10],[19] and to study some properties of attractors of the corresponding dynamical systems [11].
In our works [18]–[24] we develop another approach. It is based on the following two ideas:
• We develop a mathematical theory of travelling wave solutions for the monotone parabolic systems:
(1)
where u=(u1,…, un), F=(F1,…, Fn), a is a diagonal matrix, and the function F satisfies the following condition
Fire engineering and emergency planning 52
(2)
We find, in particular, the wave velocity and show its stability,
• We show that under some conditions the models describing combustion waves or branching chain flames with complex kinetics can be reduced to the systems of equations from this class.
Thus we can apply the developed mathematical theory to study chemical waves.
2 Reaction-diffusion systems
We consider a chemical reaction of the general form
(3) Here A1,…, Am are concentrations of the reactants, αij, βij the stoichometric coefficients.
Under the usual approximation of constant density the distribution of the temperature and the concentrations can be described by the reaction-diffusion system
(4) (5) where T is the temperature, κ, and d are the coefficients of the heat and mass diffusion, respectively, qi is the adiabatic heat release of the i-th reaction, γij= βij−αij, αij, Wi is the rate of the i-th reaction,
The functions Ki(T) determine the temperature dependence of the reaction rate and usually have the form of the Arrhenius exponent,
where is a constant, Ei is the activation energy of the i-th reaction, R is the gas constant.
We consider the case where the diffusion coefficients of all species are equal to each other and to the coefficient of thermal diffusivity κ. It means physically that we consider a reaction in the gaseous phase, and the molecular weights of gases are close to each other. Some of the results, namely those on the wave existence, can be obtained without this condition. However in this work we discuss basically the questions of wave stability, unicity and velocity. It is well known for the simple one-step kinetics that conditions of
Modelization of combustion with complex kinetics 53
stability depend on the relation between κ and d. If and the dimensionless parameter called Zeldovich number, is sufficiently large, then the flame is unstable [28]. (Here Tb is the adiabatic temperature.) If κ=d, then it is always stable.
We can put the question whether the flame with complex kinetics is stable in the case where κ=d. The answer is known. For the parallel reactions (see the examples in Section 1) the wave can be nonunique [7],[8]. Under some conditions on parameters there are three waves propagating with different velocities. Two of them are stable and one is unstable.
So we can say that there are two types of instabilities of chemical waves: the thermo-diffusional instability which can appear if even for a simple kinetics, and the kinetic instability which can appear for a complex kinetics even if κ=d. We discuss here conditions of the kinetic stability. Using the results for monotone systems [19]–[23], we can say that the wave is unique and stable if the reaction-diffusion system can be reduced to the monotone system. Thus the conditions of reducibility to the monotone systems give the conditions of stability and uniqueness of the wave.
3 Conditions of reducibility
We consider the kinetic system of equations
(6) where A is the vector of concentrations, W is the vector of reaction rates, Γ the matrix of stoichometric coefficients,
Denote by r the rank of the matrix Γ. Without loss of generality we can suppose that the first r columns of this matrix are linearly independent. We introduce the new functions ui, i=1,…, n by the equalities
(7)
where are positive constants which determine the balance polyhedron. Since the columns r+1,…, n of the matrix Γ are linearly dependent of the first r columns, then we have
(8) where are some numbers. Substituting these expressions into the system of equations (6), we obtain after some transformations
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(9) If the reactions are not isothermic, we need also the expression of the temperature through the new variables:
(10) where T+ determines the value of the adiabatic temperature, qi is the heat release of the i-th reaction.
After making the indicated substitutions, we denote the right hand-side of (9) by Fi(u):
(11) We find now the conditions of monotonicity, i.e. the conditions of validity of the inequality (2). We assume that all functions Ki(T) are nondecreasing. This condition is satisfied, in particuluar, for the Arrhenius temperature dependence of the reaction rate.
The direct computations give:
Sufficient conditions of monotonicity take the form
for all i, j=1,…, r, , l=r+1,…, n, k=1,…, m.
These conditions are not necessary. For example we can consider the heat releases qj
of alternating signs. Moreover these conditions are not convenient for application. They can be given in a different form and simplified [19]. However here we restrict ourselves to some more simple particular cases and examples.
An important particular case is that of linearly independent reactions. It means that the rank of the matrix Γ is n and its columns are linearly independent. In this case the conditions of monotonicity become very simple:
If each species is consumed in no more than one reaction and all reactions are exothermic, then the condition of monotonicity is satisfied.
Thus we obtain the following result. If the reaction does not contain parallel stages and all elementary reaction are exothermic, then the flame in kinetically stable and unique. Its velocity admits the minimax representation.
We recall the result of Khaikin and Khudyaev which we mentioned in the previous section. For an example of two parallel reactions they show that the combustion wave can be nonunique [7],[8]. Thus if the conditions of unicity and stability are not satisfied, the solution can really be nonunique and unstable.
Modelization of combustion with complex kinetics 55
Here are some examples of reaction without parallel stages.
Independent reaction:
A1+A2→…, A3+A4→…, A5+A6→…,
sequantial reactions:
A1→A2→A3→…
or
A1+A2→A3, A3+A4→A5, A5+A6→A7, nonbranching chain reactions
A2+B→AB+A, B2+A→AB+B,
We note finally that we obtain the conditions of monotonicity for reactions with parallel stages, with heat releases of alternating signs, for linearly dependent reaction, for reversible reaction, and for other classes of reactions [19]–[23].
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